Yes, so let's get started. Yeah, I'll talk about finite neural networks from the landscape point of view today. And this is, the talk is mostly based on our ICML paper from 2021, but I will also present some material from my thesis, well, and some more numerics that we put up recently. And this is joint work with Francois, Flavio, Francesco Arter, and my supervisors, Lemona and Wolfram, and also Johani. So I want to first start with some motivation for finite with neural networks. So here I am pulling some empirical results from a PEC nomination reverse paper from 2017. So here's a classic shell of neural network that is trained on the Amnesty dataset. And here, like at 32, neural network fits, hits a zero training error. And with further over-parameterization, we see the test error is decreasing, and it saturates around with 128 or so. And the similar trends carry over for other datasets, like CIFAR and SVHN. What I'm seeing in this plot is that the further the over-parameterization is, the generalization error does not gradually decrease, but it first decreases at the onset of over-parameterization, and then it saturates. And in this talk, I'm going to try to understand what is going on at the onset of over-parameterization and also further on, but in particular, this behavior, like what makes over-parameterization good for the generalization error? And this is now a known fact, I think, that over-parameterization helps with training as we saw in the first slide, but I want to show here a toy example. So here's a target function generated by two hyperplanes, and we can pick the hyperplanes in different-like orientations, so that generates another target function, and we can generate more-like tasks of this sort. And then we can train a two-neuron student network. So here, each row represents a different sampling of the hyperplanes, so it's a different dataset, and we are training 20 seeds of students, and here, the dark brown indicates a zero loss solution, so the loss is 10 to the minus 16 in that case, and the yellow dot is a high loss, basically. So we see in some fractions, like the gradient flow converges to the zero loss solution, but in some other seats, it fails to find the solution, even in this very simple problem. So this is the zero over-parameterization case, and if we keep over-parameterizing further, we see that the number of dark blue dots increase, so there is this reliable convergence already at factor two for this simple problem. Sorry, factor four, I want to say, already at factor two, like there's a lot of dark blue dots, but at factor two, it's almost completely dark blue, so it's almost, every seed converges to zero loss. And similarly, if we arrange a similar problem generated by four hyperplanes, then we see in the zero over-parameterization case, for some teachers, the problem turned out to be quite difficult, actually, because the last half of the rows, we see no cases of success, so it's all yellow dots. So it's a very non-convex problem. And in particular, I'm gonna try to answer these questions. I mean, give an answer to these questions, actually, in this talk. So why is it hard to train without over-parameterization, and what is the benefit of over-parameterization? So the go-to answer, I think, is like, yeah, there's a fixed number of data points, and if we over-parameterize further, there are more solutions, so we can find smoother interpolators, so that's a benefit of over-parameterization. But I will give an answer from a landscape point of view to this question. And this is the big question, like, how much over-parameterization do we need to prove convergence to a zero loss solution in finite-width scenarios?  This is very problem-specific, and I think it's a very difficult question, like, even to study in your favorite problem of choice in a shallow neural network setting, but I will give an average case answer to this question that one can't keep in mind when choosing a problem. And this is my final motivation slide, and the analysis gives also a perspective on this recent work. So there is over-parameterized solutions are equivalent up to permutations, and here we are looking at a picture from Mentezari. So there are, like, there's these basins that are identical to each other up to permutations, so each of these solutions are identical, and also recent work on the Geetree Basin, they come up with a similar argument, like, they can't find a good permutation to map solutions to each other that are, in turn, like, linearly mode-connected, and they attribute to this implicit regularization of stochastic gradient descent, but we will see in our model, like, that all solutions are identical to each other, and SGD sometimes, if the network is wide enough, SGD finds them, so they are, like, they are naturally linearly mode-connected in this regime, and so this is, and, yeah, with the characterization of the global minima manifold, we are, like, we are able to see that their solutions are connected, but also we know more about the structure of the solutions, like, we know the exact parameters of the solutions, and not only the functional form, but how, what should each neuron do, basically? And using this formalization, we can prune wide neural networks, and here we are seeing, like, on the, so this is an amnesty experiment, and we are generating a teacher network that has 30 neurons in three hidden layers, and it reaches a high loss level, this is represented by the orange dots on the left, and then we are over-parameterizing a neural network by a factor of three, and then, like, okay, like, with over-parameterization, we are finding better solutions that are shown by the purple dots, and then we can prune them down to the original network size, so that gives a solution of the original non-convex problem. That's one way to solve it. Okay, so, yeah. So I wanna talk about the neural landscapes of neural neural networks, but I wanna talk about it from the symmetry point of view, and so there are already, like, a lot of example loss landscapes with full of symmetries, so, for example, dictionary learning, clustering, tensor decomposition, and also in Risa's talk, like, we see a simplest case of symmetry, a subtle point between the two solutions, and there are, like, broader classes of symmetries for loss landscapes, like, here, in the first picture, there's a rotational symmetry for the global minimum or discrete symmetries, which is the case we are gonna be studying, and, yeah, there's a, there are also, like, more, like, more versions of it, so this is a line of critical points on the third picture, and I'm gonna talk this a little bit more in detail. So there are the symmetries, and there is the complexity for neural networks, and so this complexity analysis was done for spherical spin glasses and more models, so the relevant thing to do in this analysis is to count the number of critical points, because then it gives an idea about how difficult this problem is and how non-com, non-comax it is in some ways, so we are gonna take this approach to study neural networks, and we are gonna count the number of critical manifolds. So neural networks are symmetries, so it's not gonna be the number of critical points in this case because they, because the critical points perform manifolds due to symmetries, so we are supposed to count the number of the manifolds, so I'm gonna do it in six steps. First, yeah, I'll introduce the symmetry approach to landscape complexity, then I'll present the scaling laws, so that's the number of critical manifolds, and then I will look at one such manifold, and actually, even one such manifold is, it shows quite a non-trivial behavior, so there is on the same line, all critical points achieve the same loss, but some part of it is the set of points, full of set of points, and the other part has local minimum, so we will look at this transitions from a static point of view, and I'll introduce the landscape complexity measure, and then the equivalence class of global minimum, and yeah, if time permits, I'll have a look at this pruning application in the last slide. So here I'm taking a shallow neural network that is generated by an neurons, so WJ is the incoming vector, and AJ is the outgoing weight, and theta is the parameter vector, so it's a concatenation of all the vectors of the neurons, and so I'm gonna use these cartoons for my counting arguments, so here's a width and neural network, and each neuron is represented by a color, so when I refer to a neuron, I mean a concatenation of the incoming vector, W1 and the outgoing weight, A1, and I'm picking a loss function, so I'm defining a cost, C, and any target function that is F star, so cost, C measures the difference, and I'm averaging over an input data, that is denoted by P, it could be discrete or it could be continuous, so at this stage, it could be an ERM, or it could be the population loss, and I'm assuming C and sigma are twice differentiable, and that the cost function is zero, if and only if the prediction and the target match, so this is gonna be the all assumptions that I'm using, so the first principle is the neuron spitting, so if we take one of the neurons, the green neuron, let's say, I'm sitting in two pieces, that is, copy the incoming vector, W1, and spit the outgoing weight into two pieces by a vector mu, so mu times A1 is the one outgoing connection, and one minus mu times A1 the other, so the sum of the two neurons is the same as the original network function, this is rather trivial, but it was observed by Fukumusi and Mari in 2000, if the first configuration is the critical point, then the second configuration is also a critical point of the loss landscape, so this comes from a simple gradient calculation, so this is known, and then there is another similar principle, that is the zero neuron addition, so we can add two neurons to this expanded network, so the incoming vector is W prime, that can be anything, and outgoing weight is A prime and minus A prime, so if we sum them together, it is a zero neuron, it doesn't add anything to the function, it doesn't change the network function, okay, so this doesn't preserve the criticality, but it preserves the network function because the neurons cancel out each other, and here I am showing a group of two zero neurons, but it could be a group of five, or one group of two, and another group of three, and this is gonna factor in our computations of the number of manifolds, so there are two symmetric principles that I'm gonna study, and with that, now I can already introduce the scaling law, and here I'm putting an extra assumption, that is the, for all the data set, that the data set is generated by a finite width neural network, that is let's say a width k neural network, so already with a width k network, the neural network, the data set can be interpolated by a neural network, so this is also known as a teacher assumption, but I wanna say this is a structured teacher, so it is a teacher that fits the data set perfectly, and k can be as large as we want, but it is finite, that's the assumption, so in this, now I can introduce also what I mean by overparameterization, underparameterization, so there's a width k neural network, and the smaller neural networks are underparameterized, and bigger ones are overparameterized, okay, so I'm picking an underparameterized network of width and now, and here I'm showing you an irreducible parameter, that is the network function generated by this four neurons cannot be represented with a smaller number of neurons, so that is to say, there's no spitted neurons inside of this configuration, so I couldn't merge them together, it's all neurons are distinct from each other, that's why I'm using different colors in this cartoon, so, and then with neurons spitting, so yeah, here the green neuron is spitted for three times, and orange is spitted two times, and I'm ending up in this configuration that I'm also showing in the bucket, and okay, so yeah, let me just write, get into it, so what do I mean by scaling law, is the number of manifolds that are generated by this width and perimeter, in the width and neural network, so we are looking at the loss landscape of this wider neural network, of this am neural network, and the question, and the thing is we have am neurons to start with, but we wanna fill am positions with this am neurons, so how can we do it? It's actually a partition problem, like let's look at the first second formula, so for each partition that I'm denoting by K1 to KN, so there's a number of permutations inside the wider network, so that is denoted by this multinomial vector, and I have to use each neuron for once, so KIs are all bigger than one, and then for each partition, I have to use the count as permutations, and this can be written in another way, so that is, so basically this is a sterling number of the second kind, and that counts the number of functions from a set of am elements to another set of am elements, a set of surjective functions, right? But once there's the surject, once there's the function that maps, then there's the coloring, because each am neurons were different to start with, so that brings us another and factorial, and there's the second invariance that we looked at, so I want to also get the scaling low for this function, to preserve the function, and that is denoted by this number T, and here, so N was free in the first case, because from any under parameterized network, I can map into a wider neural network, it's gonna give a different number, but I want to consider the zero loss configurations, I want to count the number of ground states, so that's why I'm fixing K, so that was the data generating width, and now in the over parameterized network with am, what is the number of manifolds? So this is the number of manifolds that I get from neuron splitting only, that is denoted by this factor G, and then there's an additional factor, because there is also the zero neuron addition, so that gives me an extra factor, okay? So I want to zoom into one of these manifolds, so this is not gonna affect the complexity calculations, but I just want to show an example of how funky these things actually look like. So I just showed, here I'm from Fikumizu, and they had this preserving of the critical points, but we didn't talk about the second order derivatives, so is it a local minimum or is it a subtle point? So here I'm showing, yeah, maybe it's good to look at the example first, so here we are looking at the case when only one neuron is split into two pieces, and there is a factor of mu that is free, so when I change mu, for every mu there's a critical point, but I can't move the mu, so that gives me a line of critical points. So I want to study the second order derivatives on this line as a function of mu. So basically, with the composition of the Hessian, we can get this theorem on the signs of the eigenvalues of the Hessian. So I want to know the signs of the eigenvalues of the Hessian on the symmetry induced critical points for a given mu, so that's on the left-hand side. So the theorem says you should look at the Hessian spectrum of the irreducible critical points, theta, and then you should look at this new sub-metrics Y that I'm writing the formula here, comes from the Hessian decomposition, and the nice thing is that this Y and the factor of mu splits from each other, so I get a factor of one minus mu times mu, and this fixed matrix Y, that depends on the neuron that I'm splitting, and then there is an extra zero because they are living on a line, so there's a zero eigenvalue due to the symmetry, and I'm gonna show an example of a local minimum here in a four-neuron network, and I'm gonna split this first neuron, W1A1, into two, so I'm gonna plot the minimum five eigenvalues of the Hessian as a function of mu, and here I see the minimum eigenvalue is negative for negative mu, so on the negative side of the line, we get strict settles, and then in between zero and one, this factor of one minus mu times mu is positive, so I get a local minimum in between, and at mu zero, that the second part of the composition vanishes, and in that case I get only zeros, so we get a non-strict settle connecting the settles and the minimum on the line, and here is a cartoon representation of this, so here we see the red part shows the local minimum, and this is what happened in this actual example that I showed here with the teacher-student cartoon, but it could have also been like, if y is a negative definite matrix, then we would get in the line rays, we would get rays of local minimum, and because there's permutation symmetry, the line is symmetric from mu is 0.5, and then in between we get the line segment of strict settles, and they are, but in any case, like independent of what is the spectrum of y, because this middle factor vanishes at mu equals zero and one, it's always connected with non-strict settles, so now I'm gonna introduce the landscape complexity analysis and I'm gonna kind of forget like the second order derivatives, and I'm just gonna treat them all equally, and I'm gonna focus on the number. So we've looked at this first scaling law that came from neuron spitting, so this is the number of critical manifolds, okay, so yeah, I wanna look at this number in some limits to get some idea about how it works, so, and we are studying the loss landscape of the width am neural network, right? So now if I take the large am limit, so I'm in an infinitely wide neural network, so what is the number of manifolds coming from super small neural networks? So for small am, this expansion factor is exponential in am, so it grosses and to the power am, and if we look at the, and this is high-loss manifolds in the sense that the small network can only solve the dataset to some extent, so it's gonna achieve high-loss configurations, and yeah, so after the neuron spitting, the function is exactly the same, so the, I mean, network is identical to a narrow neural network in that sense, so it achieves a high-loss values, and, but if am is close to am, so we get this low-loss manifolds, and in that case the number, the scaling number is factorial in am, so it's faster than the exponential, so there are more low-loss manifolds in that sense, and okay, a better way to study these numbers is actually when we scale both am and am at the same speed with a factor of alpha, and I'm normalizing with am to the power am in the log scale, so it's similar to the low-loss, a scaling of the low-loss manifolds, and here we see a unimodal distribution, so the more common manifolds are the ones that come from this intermediate stage of issuing could factor like alpha, and actually, yeah, so this numbers were, as I was kind of hinting, they were studied in the context of communicatorics as we realized more lately, so it is, this g factor is equal to the number of surgexons from a set of size am to another set of am, and also they were interested, like where is the peak of this unimodal distribution, and so it's in this cool math overflow of discussion, and they're like, oh, it is achieved at one over two log two, and that's also what we see in our cartoon, so the most dominant critical manifolds are those coming from the neural network, that is one over two log two times smaller than the landscape we are analyzing, so it is kind of, yeah. Okay, I wanna, or maybe, yeah, I wanna, yeah, let's do this slide first, then I can ask for questions. I wanna also introduce the landscape complexity measure, so that is, yeah, that I'm denoting by C, so we take a teacher network with K, and we are studying now the over parameterized network with M, so I'm defining this as the number of critical manifolds that come from the network with K minus one, so that's the scaling law G, and then there's a scaling of the number of zero loss manifolds, that's the scaling law of T, and I wanna compare, like, which one scales faster, so if there are too many settled manifolds, then I'm gonna conclude, this is a very non-convexed loss landscape, so it's hard to reach zero loss manifolds, but if there are, like, two many zero loss manifolds, and I, okay, this is still a non-convexed loss landscape, because there are still all the settled manifolds that are exponentially growing, at least exponentially, or maybe factorial, but there is much more zero loss manifolds, so it's a benign non-convexity, so I wanna study how this loss landscape complexity grows as a function of M. So, yeah, similar to the limits that we did before, so when M goes to infinity, in the infinite width limit, it goes down to zero, so this is nice, infinite width loss landscape is also benign from this point of view, so there is incredible amount of zero loss manifolds, and the number of settles are small in comparison, so, and there is a second limit, there is a, I think it's interesting, so it's the infinite data complexity limit, when the teacher width goes to infinity, and we have limited over-parameterization, so M grows in the same speed, and complexity explodes, so in this regime, it should be very hard to train the neural network, so here, also I'm not only in the limits, but we see the complexity grows gradually, so this is, so the over-parameterization gradually decreases the complexity, okay, here I have some toy examples, and I wanna talk about the deep case, actually, so I've been discussing the shallow case only so far, but we only use the permutation symmetry to make these scaling low computations, and in the case of deep networks, we can apply permutation symmetry in every hidden layer, so then it will give an exponential factor on the landscape complexity, so we applied one after the other layer, basically, so that means in the mild over-parameterization case, when the complexity is high, so if the network is also deep, it's gonna be even higher, so and we see it here, in the mild over-parameterization case, the gradient flow doesn't find any solutions, so the permutation symmetry prevents it from converging, like in rough terms, basically, but if we over-parameterize further, again, the number of zero loss manifolds take over, and when the complexity is smaller than one, exponentially, I think it is gonna make it just much smaller, so it's much more easier in the vest over-parameterization, so deep depth just makes it much, difference much more pronounced between mild and vest over-parameterization, and so here I have an amnesty experiment, we created a teacher network from the amnesty with 13 neurons, and this was funny to see, I mean, we trained it with Adam, I think, and we trained over-parameterized networks of 50 seats each, and we see in this onset of over-parameterization, it's a non-comic problem, so it doesn't converge, and then there is this intermediate regime, where the complexity is closer to one, and it flattens out, and it comes from this flattening of the expansion factor G at one over two log two, so there's this intermediate phase, and there's only a fraction of seats converging, and with further over-parameterization, okay, it's converging, more seats are converging towards zero loss, so this, I don't wanna make a lot of claims from this figure, but it was, there is a little bit of parallel between this and the complexity plot, and from the level of analysis, I didn't expect to see any match between these two, maybe, so it was surprising that there are some, there is maybe some indications that the complexity analysis gives us on the training dynamics, so for very wide neural networks, do you mean? Yeah, I guess so. Yeah, so we can look at actually this unimodal curve, sorry, it's actually, the switch comes from really, I'm going in the opposite direction, yeah, yeah, so what happens for, yeah, let's do it like this, what happens for wide neural networks is, so whatever K is, K is gonna be like a small compared to M, let's say, and this is, the T is always larger than G, because there's the, due to the zero neurons, so let's say, let's even forget the zero neurons, let's consider only G, and the, I mean the ratio is G, basically K minus one M, right, so that is the K minus one factor is gonna be lower than the K factor, because it's increasing, the G function is increasing in this case, and why it flips in the rest of our parameterization is because when K is close to M, then the settled manifolds are larger than the zero loss manifolds, exactly, yeah, and it's a, exactly, yeah, that's the right way to read it, for choosing the parameters of the teacher? Yeah, I'm wondering basically, oh, I see, yeah, that's a very good question, so yeah, so for sure, I know some teachers that are not gonna be difficult, in the sense that even with zero over parameterization, you can learn these teachers, like the teacher's infinitely wide, depending on the selection of the parameters, so if you choose a unit orthonormal teacher with all the activation functions, like this is, so if you sample them, like if D is larger than the K, right, they are gonna be, the first layer is gonna be orthogonal to each other, so it's gonna be effectively a unit orthonormal teacher, so this teacher is easy to train, so I mean, this, my argument is gonna fail, actually, but I think this is an extra simple case, and it's not like a teacher fitted to MNIST, like where each neuron has to do something different, like learn a different component of the dataset, so we are trying to construct, like, more like, less random teachers in some sense, like this is by choosing the hyperplanes, for example, and creating, like, this XOR constellations, it's a bit the idea. Okay, yeah, we didn't do this, so I wanna go back to the questions that I was hinting at the beginning of the talk, so I was, yeah, why is it hard to train without over-parameterization? So that is, because they are, yeah, I argue that it's extremely non-convexity, but I wanna say, like, this is not true for all the teachers, but for average case teachers, I think this is a good picture to keep in mind, and here I'm plotting a cartoon, like, that's, so the X, the Y, yeah, the vertical axis is the loss of the critical manifolds, and this first circle is the critical manifolds coming from small neural networks, so the G, the expansion factor G is small, and as we increase the width of the under-parameterized network, there's more number of critical manifolds, so, and we are going lower in the loss levels. After 1 over 2, a factor of 1 over 2 log 2, where there is this, like, most numerous, this is the most numerous critical manifolds at this loss level, and then if we go further over-parameterized or further down in the loss level, there is a smaller number of manifolds, and we eventually hit the zero loss part of the loss landscape. So this is the case of zero over-parameterization, but now if you over-parameterize the neural network, this loss landscape, the global structure of it changes because what happens is that we don't fall onto the second phase of the unimodal curve that we looked at before, so we are on the onset of that curve, and the saddle manifolds, like, grow gradually as we decrease the loss level, and the zero loss manifold is the most numerous, and so it's kind of, it's easier to converge in some sense, so this is a, I don't know if these pictures make sense, but yeah, it's, yeah, maybe we can take it offline. If someone is interested, I would be very happy to talk about them. And the question also that I hinted at the beginning, like, how much over-parameterization do we need in average case problems? So this complex analysis, at least to look to, and in the cartoon's practical scenarios that I showed that first it looked like a factor of four, but yeah, I have no ways to make a strong claim on this, like, I think, yeah, one should choose a problem and just, yeah, focus on, it's very problem-specific, but it's nice that it gives a kind of an average case analysis, like, with this analysis. Yeah. So I wanna talk about the equivalence class of global minimum. So here I'm assuming some more assumptions. The input distribution has full support, and the teacher network is unique up to permutation. So there's only one function that interpolates the dataset. And then I'm assuming that the activation function is analytic, doesn't cross zero, it's zero, and it has non-zero derivatives for infinitely many odd and even values.  So I wanna study the simplest symmetry scenario. And in this case, we get that all zero loss points in an over-parameterized network are identical to the teacher network, up to Neuron Spitting and Neuron Edition. So if you have a zero loss point, it has to be identical to the teacher up to these two symmetries. So this is the ground state of the over-parameterized neural network. And I wanna study the topology of this manifold. So I wanna, yeah, let's look at a toy case with the two teacher's student neurons and three student neurons. So the factor returns 12. And in this case, each dot represents an f i sub space of zero loss points and they are connected to one another. If the two f i sub spaces have an intersection, I put an edge between them. So that gives me the connectivity graph of the global minimum manifold. And then I can put it in another case with three teacher neurons and four student neurons.  So that are only, I mean, green and blue type of f i sub spaces are different because the blue one has a neuron that is split it and the green one has an extra neuron that is a zero neuron. So with the zero neuron, I can move the parameter to the split it neuron. And then from the split it neuron, I can move one of the split it, the other split it neuron to another zero neuron. So that's how we walk towards like within inside this global minimum manifold. So maybe any questions that I'm gonna talk about this pruning application. So here's a cartoon of what I was talking describing before. Like it's a with zero over parameterization. We have a non-convex optimization loss landscape and it's not only non-convex, but it's also hard to train. So it's not like a benign non-convexity in the sense that all settles are strict and we can still converge to the global minimum manifold. So we have non strict settles, we have degenerate settles on the line of critical points. So, and then the way to solve this, one way to solve this optimization problem is to go to an over parameterized neural network with a factor of row, and then train a bunch of seats and then collect the neurons of all these seats together and just cluster them together. And this is in Flavio's paper, we put it in the archive recently. And here we, again, like the examples that I showed similar to the first slide. So we are creating some teacher networks that has input dimension eighth, first layer with four and the second layer with two. And then this is a, oh yeah. If we train this teacher network, we reach like high loss levels, but if you train an over parameterized student networks, as expected, we reach lower loss levels for different seats. But if we looked at the neurons of these networks and combine them together, using this, I mean kicking out the zero neurons and then merging together the spitted neurons together, we go back to the shallower neural network because we are cutting a not shallower neural network because we are cutting off some neurons. And it actually achieves much better loss because we are also removing the noise from the neurons that comes from training. So this works in some settings, in some toy settings that I'm showing here and in an amnesty setting too. And so here I have like, I'm showing more experiments in line with Sebastian's question. So here like we have teacher networks generated by this XOR like hyperplane configurations. And we see like, so this is factor two of over parameterization and so on. So as the larger the over parameterization factor is, the general trend is that it gets easier to converge, but it's also like too, it's hard to like analyze these questions more in finer detail because it doesn't even converge for every seat even at the factor of four. So we see some like green dots. So, there's just somebody, thank you. Oops, sorry, really bad at this. Okay. It's basically observing that it's always convenient to go more over parameterization you're using here. But if I believe this plot, of course I believe this plot. Let's say, why are not, that's the standard you achieve, but no. So that plot is basically telling you that feature learning in fine-tuning method is not too important. I mean it's not, I would like to know your opinion, but it's also something for the next discussion. Yeah, it's a great question actually. I mean, another way to, so I didn't make these plots first, I took them from these papers, but yeah, so in these plots, yeah, infinitely wide networks are working better, but you don't wanna train an infinitely wide network in practice, you don't want to train an infinitely wide neural network. You don't need to train there. Yeah, so I'm not sure if this solution is exactly like identical to the anti-K solution actually, because even an infinitely wide network, there's the like, if you scale your parameters large, then you converge to the anti-K solution, anti-K plus NNGP solution. But if your parameters are slow, then we don't have a closed form expression for the function. So we don't know like, we have to train the infinitely wide neural network to get the small initialization solution in the mean field regime. So that's the first thing. And the, so yeah, it's an interesting question, like how the anti-K solution would fall onto this term, but on this curve, I think for feed the forward neural networks, Francesco Spas and Metivia, and their group has a lot of work on showing like the anti-K solutions are better than the mean field solutions, but for convolutional networks, it's the other way around. So I don't think by any means, it's the end of the story actually, but and also like, so yeah, when you don't have this closed form correspondence through the anti-K, like let's say you want to recover the mean field solution in some ways. And the question is like, what is the network that you want to train? Like should it be a super wide network or like with a factor of 10, you already converge to that solution like effectively. And that's, so I find it interesting that this curve like plateaus.  And I think like the other picture would be like that this curve decreases smoothly. So it would be the variance reduction picture, like what we get from the anti-K analysis. But it's actually, I picked these kind of pictures, like to be honest with you, because there are like other pictures that show exactly the same experiments and this curve like the case lovely. And that's compatible with the anti-K point of view. But I mean, these things also came from that line of work. So I was like, okay. And it fits my story, I guess.  Exactly, yeah. Well, this honestly, like with my symmetry arguments, I like it has zero sensitivity to like the training, the scaling of the training dataset. I mean, in Reza's slides, we saw that he like he hit this nice picture with the ERM landscape. Like it is more funkier than the population loss landscape. And I think it looks like that. So it's, and I mean, to study like how this, how the critical points disappear with like the, as the sample complexity increases and how many new ones that you get, like in the finite sample scenarios, I think that's, I have no ways to do it with this, with this approach. Okay, yeah, this. If I have finite, if I have a finite training dataset, then it should be kind of effectively like, it should have converged to the population loss. This I agree. But to study the scaling of it, like it's, I think it's a very like, I wouldn't know. I mean, yeah, this is a difficult one. The horizontal line. The vertical line, the vertical line, it is represents the first time it hits the zero, zero training area. So it would be my K, the width of the teacher network. Good question. So, I mean, there is a little bit of peak. There's a little bit of peak here. Yeah. No, it's a, maybe there's early stopping that's a, that could be one thing. Early stopping could be any kind of regularization, could be no added. Thanks. Last question. It's, I mean, so one thing for sure is I'm only using the symmetries to make all the arguments and it's a, it's, but I'm also using the network parameterization because this neuron spitting is a special symmetry to neural networks. If I didn't have a second layer, if the second layers were fixed, for example, it would be still permutation symmetric, but I wouldn't have the neuron spitting invariance. So, so it's not even only, only permutation symmetry. I think it's a really neural network thing that I'm analyzing and counting here, but it's also true, like in a lot of physics problems, there's lots of symmetries. So, there's the rotation symmetry actually thinks, okay, what I can say for linear neural networks, in that case, when there's no activation function, like this manifolds merge together with the rotational symmetry. So there's, it's a more like thick, big one or two manifolds and I lose all the numbers. So with the rotation, it's more like, yeah, it's a, I think I would lose these scaling arguments. But it's, we can't talk more like. I would say let's, yeah, let's take again, I'll just pick up some.


6. Paul's Second Deprivation Polly was beset by so many misgivings in the morning that but for the incessant promptings of her black-eyed companion she would have abandoned all thoughts of the expedition and formally petitioned for leave to see number 147 under the awful shadow of Mr. Dombie's roof. But Susan, who was personally disposed in favor of the excursion, and who, like Tony Lumpkin, if she could bear the disappointments of other people with tolerable fortitude, could not abide to disappoint herself, through so many ingenious doubts in the way of this second thought and stimulated the original intention with so many ingenious arguments that almost as soon as Mr. Dombie's stately back was turned and that gentleman was pursuing his daily road toward the city, his unconscious son was on his way to Staggs Gardens. This euphonious locality was situated in a suburb known by the inhabitants of Staggs Gardens by the name of Camberling Down, a designation which the Strangers' Map of London, as printed, with a view to pleasant and comodious reference on pocket handkerchiefs, condenses with some show of reason into Camden Town. Hither the two nurses bent their steps, accompanied by their charges, Richards carrying Paul, of course, and Susan leading little Florence by the hand and giving her such jerks and pokes from time to time as she considered it wholesome to administer. The first shock of a great earthquake had, just at that period, rent the whole neighborhood to its center. Traces of its course were visible on every side. Houses were knocked down, streets broken through and stopped, deep pits and trenches dug in the ground, enormous heaps of earthen clay thrown up, buildings that were undermined and shaking, propped by great beams of wood. Here a chaos of carts overthrown and jumbled together lay topsy-turvy at the bottom of a steep unnatural hill. There confused treasures of iron soaked and rusted in something that had accidentally become a pond. Everywhere were bridges that led nowhere, thoroughfares that were wholly impassable, babble towers of chimneys wanting half their height, temporary wooden houses and enclosures in the most unlikely situations, carcasses of ragged tenements and fragments of unfinished walls and arches and piles of scaffolding and wildernesses of bricks and giant forms of cranes and tripods straddling above nothing. There were a hundred thousand shapes and substances of incompleteness wildly mingled out of their places, upside down, burrowing in the earth, aspiring in the air, moldering in the water, and unintelligible as any dream. Hot springs and fiery eruptions, the usual attendance upon earthquakes, lent their contributions of confusion to the scene, boiling water hissed and heaved within dilapidated walls, whence also the glare and roar of flames came issuing forth, and mounds of ashes blocked up rites of way, and wholly changed the law and custom of the neighborhood. In short, the yet unfinished and unopened railroad was in progress, and from the very core of all this dire disorder trailed smoothly away upon its mighty course of civilization and improvement. But as yet the neighborhood was shy to own the railroad. One or two bold speculators had projected streets, and one had built a little, but had stopped among the mud and ashes to consider farther of it. A brand new tavern, redolent of fresh mortar and size, and fronting nothing at all, had taken for its sign the railway arms, but that might be rash enterprise, and then it hoped to sell drink to the workmen. So the excavator's house of call had sprung up from a beer shop, and the old established ham and beef shop had become the railway eating house, with a roast leg of pork daily, through interested motives of a similar immediate and popular description. Lodging housekeepers were favorable in like number, and for the like reasons were not to be trusted. The general belief was very slow. There were frowsy fields and cowhouses and dung hills and dust heaps and ditches and gardens and summer houses and carpet-beating grounds at the very door of the railway. Little tumuli of oyster shells in the oyster season and of lobster shells in the lobster season and of broken crockery and faded cabbage leaves in all seasons encroached upon its high places. Posts and rails and old cautions to trespassers and backs of mean houses and patches of wretched vegetation stared it out of countenance. Nothing was better for it, or thought of being so. If the miserable waste ground lying near it could have laughed, it would have laughed it to scorn, like many of the miserable neighbors. Stag's gardens was uncommonly incredulous. It was a little row of houses, with little squalid patches of ground before them, fenced off with old doors, barrel staves, scraps of tarpolin and dead bushes, with bottomless tin kettles and exhausted iron fenders thrust into the gaps. Here the stag's gardeners, trained scarlet beans, kept fowls and rabbits, erected rotten summer houses. One was an old boat, dried clothes and smoked pipes. Some were of opinion that stag's gardens derived its name from a deceased capitalist, one Mr. Stag's, who had built it for his delectation. Others, who had a natural taste for the country, held that it dated from those rural times when the anthlert herd under the familiar denomination of stag's, had resorted to its shady precincts. Be this as it may, stag's gardens was regarded by its population as a sacred grove not to be withered by railroads, and so confident were they generally of its long outliving any such ridiculous inventions that the master chimney sweeper at the corner, who was understood to take the lead in the local politics of the gardens, had publicly declared that on the occasion of the railroad opening, if ever it did open, two of his boys should ascend the flues of his dwelling, with instructions to hail the failure with derisive jeers from the chimney pots. To this unhellowed spot, the very name of which had hitherto been carefully concealed from Mr. Dombie by his sister, was little Paul now born by fate and Richards. That's my house, Susan said Polly, pointing it out. Is it indeed Mrs. Richards, said Susan condescendingly? And there's my sister Jemima at the door.  The sight added such an extensive pair of wings to Polly's impatience that she set off down the gardens at a run and bouncing on Jemima, changed babies with her in a twinkling to the utter astonishment of that young damsel on whom the air of the Dombies seemed to have fallen from the clouds. Why Polly, cried Jemima, you, what a turn you have given me. Who thought it come? Come along in, Polly. How will you do look to be sure? The children will go half-while to seal Polly, that they will. That they did, if one might judge from the noise they made, and the way in which they dashed at Polly and dragged her to a low chair in the chimney corner, where her own honest apple face became immediately the center of a bunch of smaller pippins all laying their rosy cheeks close to it, and all evidently the growth of the same tree. As to Polly, she was full as noisy and vehement as the children, and it was not until she was quite out of breath and her hair was hanging all about her flushed face that her new christening attire was very much disheveled, that any pause took place in the confusion. Even then the smallest toodle but one remained in her lap, holding on tight with both arms round her neck, while the smallest toodle but two mounted on the back of the chair and made desperate efforts with one leg in the air to kiss her round the corner. Look, there's a pretty little lady come to see you, said Polly, and see how quiet she is.  This reference to Florence, who had been standing by the door, not unobservant of what past, directed the attention of the younger branches toward her, and had likewise the happy effect of leading to the formal recognition of Miss Nipper, who was not quite free from a misgiving that she had been already slighted.  This is my sister, Jemima. This is. Jemima, I don't know what I should ever do with myself. If it wasn't for Susan Nipper, I shouldn't be here now but for her. Oh, do sit down, Miss Nipper, if you please, quote Jemima. Susan took the extreme corner of a chair with a stately and ceremonious aspect. I never was so glad to see anybody in all my life. Now, really, I never was, Miss Nipper, said Jemima. Susan, relaxing, took a little more of the chair and smiled graciously. Do untie your bonnet strings and make yourself at home, Miss Nipper, please, and treated Jemima. I am afraid it's a poorer place than you're used to, but you'll make allowances, I'm sure. The black-eyed was so softened by this deferential behavior that she caught up little Miss Toodle, who was running past, and took her to Banbury Cross immediately. But where's my pretty boy, said Polly. My poor fellow, I came all this way to see him in his new clothes. Ah, what a pity, cried Jemima. He'll break his heart when he hears his mother has been here. He's at school, Polly. Gone already? Yes, he went for the first time yesterday. For fear he should lose any learning. But it's half-holiday, Polly. If you could only stop till he comes home, you and Miss Nipper least ways, said Jemima, mindful in good time of the dignity of the black-eyed. And how does he look, Jemima? Bless him. Faulted Polly. Well, really, he don't look so bad as you suppose, returned Jemima. Ah, said Polly with emotion. I know his legs must be too short. His legs is short, returned Jemima, especially behind, but they'll get longer Polly every day. It was a slow, prospective kind of consolation, but the cheerfulness and good nature with which it was administered gave it a value it did not intrinsically possess. After a moment's silence, Polly asked, in a more sprightly manner. And where's Father Jemima, dear? For, by that patriarchal appellation, Mr. Toodle was generally known in the family. There again, said Jemima, what a pity. Father took his dinner with him this morning and isn't coming home till night. But he's always talking of you, Polly, and telling the children about you, and is the peace-sublist, patient-ist, best-tempered-ist soul in the world, as he always was and will be. Thank you, Jemima, cried the simple Polly, delighted by the speech and disappointed by the absence.  I say the same of you sometimes, and think it too. In spite of the double disappointment, it was impossible to regard in the light of a failure a visit which was greeted with such a reception. So the sisters talked hopefully about family matters, and about Beiler, and about all his brothers and sisters, while the black-eyed, having performed several journeys to Banbury Cross and back, took sharp note of the furniture, the Dutch clock, the cupboard, the castle on the mantelpiece with red and green windows in it, susceptible of illumination by a candle-end within, and the pair of small black velvet kittens, each with a lady's ridicule in its mouth, regarded by the stag's gardeners as prodigies of imitative art. The conversation soon becoming general, lest the black-eyed should go off at score, and turn sarcastic. That young lady related to Jemima a summary of everything she knew concerning Mr. Dombie, his prospects, family, pursuits, and character. Also an exact inventory of her personal wardrobe, and some account of her principal relations and friends. Having relieved her mind of these disclosures, she partook of shrimps and porter, and evinced a disposition to swear eternal friendship. Little Florence herself was not behind hand in improving the occasion. For, being conducted forth by the young toodles to inspect some toadstools and other curiosities of the gardens, she entered with them heart and soul on the formation of a temporary breakwater across a small green pool that had collected in a corner. She was still busily engaged in that labour when sought and found by Susan, who, such was her sense of duty, even under the humanizing influence of shrimps, delivered a moral address to her, punctuated with thumps on her degenerate nature while washing her face and hands, and predicted that she would bring the grey hairs of her family in general with sorrow to the grave. After some delay, occasioned by a pretty long confidential interview above stairs on pecuniary subjects between Paul and Jemima, an interchange of babies was again effected. For Polly had all this time retained her own child and Jemima little Paul and the visitors took leave. But first the young toodles, victim of a pious fraud, was deluded into repairing in a body to a Chandler shop in the neighborhood for the ostensible purpose of spending a penny. And when the coast was quite clear Polly fled, Jemima calling after her that if they could only go round towards the city road on their way back, they would be sure to meet little Boiler coming from school. Do you think we might make time to go a little round in that direction, Susan, inquired Polly when they halted to take breath? Why not, Mrs. Richards, returned Susan. It's getting on toward our dinnertime, you know, said Polly. But lunch had rendered her companion more than indifferent to this grave consideration, so she allowed no wait to it, and they resolved to go a little round. Now it happened that poor Boiler's life had been since yesterday morning rendered weary by the custom of the charitable grinders. The youth of the streets could not endure it. No young vagabond could be brought to bear its contemplation for a moment without throwing himself upon the unoffending wearer and doing him a mischief. His social existence had been more like that of an early Christian than an innocent child of the 19th century. He had been stoned in the streets. He had been overthrown into gutters, bespattered with mud, violently flattened against post. Entire strangers to his person had lifted his yellow cap off his head and cast it to the winds. His legs had not only undergone verbal criticisms and revilings, but had been handled and pinched.  Thus it fell out that Byler, on his way home, sought unfrequented paths and slunk along by narrow passages and back streets to avoid his tormentors. Being compelled to emerge into the main road, his ill fortune brought him at last, where a small party of boys headed by a ferocious young butcher were lying in wait for any means of pleasurable excitement that might happen. These, finding a charitable grinder in the midst of then, unaccountably delivered over, as it were, into their hands, set up a general yell and rushed upon him. But it so fell out likewise that at the same time Polly, looking hopelessly along the road before her, after a good hour's walk, had said it was no use going any further when suddenly she saw this sight. She no sooner sought than uttering a hasty exclamation and giving Master Dombie to that black-eyed. She started to the rescue of her unhappy little son. Surprises like misfortunes rarely come along. The astonished Susan Nipper and her two young charges were rescued by the bystanders from under the very wheels of a passing carriage before they knew what had happened, and at that moment it was market day. A thundering alarm of mad bull was raised. With a wild confusion before her of people running up and down and shouting and wheels running over them and boys fighting and mad bulls coming up and the nurse in the midst of all these dangers being torn to pieces. Florence screamed and ran. She ran till she was exhausted, urging Susan to do the same, and then stopping and wringing her hands as she remembered they had left the other nurse behind, found with a sensation of terror not to be described that she was quite alone. Susan Susan cried Florence, clapping her hands in the very ecstasy of her alarm. Oh, where are they? Where are they? Where are they? said an old woman coming hobbling across as fast as she could from the opposite side of the way. Why did you run away from him? I was frightened, answered Florence. I didn't know what I did. I thought they were with me. Where are they? The old woman took her by the wrist and said, I'll show you.  She was miserably dressed and carried some skins over her arm. She seemed to have followed Florence some little way at all events, for she had lost her breath, and this made her uglier still as she stood trying to regain it, working her shriveled yellow face and throat into all sorts of contortions. Florence was afraid of her and looked hesitating up the street of which she had almost reached the bottom. It was a solitary place, more a back road than a street, and there was no one in it but herself and the old woman. You needn't be frightened now, said the old woman, still holding her tight. Come along with me. I don't know you. What's your name? asked Florence. Mrs. Brown, said the old woman. Good Mrs. Brown. Are they near here? asked Florence, begging to be led away. Susan ain't far off, said Good Mrs. Brown, and the others are close to her. Is anybody hurt? cried Florence. Not a bit of it, said Good Mrs. Brown. The child shed tears of delight on hearing this and accompanied the old woman willingly, though she could not help glancing at her face as they went along, particularly at that industrious mouth, and wondering whether bad Mrs. Brown, if there was such a person, was at all like her.   Opening the door with a key, she took out of her bonnet. She pushed the child before her into a back room, where there was a great heap of rags of different colors lying on the floor, a heap of bones, and a heap of sifted dust or cinders. But there was no furniture at all. And the walls and ceiling were quite black. The child became so terrified that she was stricken speechless, and looked as though about to swoon.   Florence obeyed her, holding out her folded hands in mute supplication. I'm not going to keep you even above an hour, said Mrs. Brown. Do you understand what I say? The child answered with great difficulty. Yes. Then, said good Mrs. Brown, taking her own seat on the bones. Don't vex me.   I could have you killed at any time, even if you was in your own bed at home. Now let's know who you are and what you are and all about it. The old woman's threats and promises, the dread of giving her offense, and the habit unusual to a child, but almost natural to Florence now of being quiet, and repressing what she felt and feared and hoped, enabled her to do this bidding, and to tell her little history or what she knew of it. Mrs. Brown listened attentively until she had finished. So your name's Domby, eh? said Mrs. Brown. Yes, ma'am. I want that pretty frock, Miss Domby, said good Mrs. Brown, and that little bonnet and a petticoat or two and anything else you can spare. Come, take them off. Florence obeyed as fast as her trembling hands would allow, keeping all the while a frightened eye on Mrs. Brown. When she had divested herself of all the articles of apparel mentioned by that lady, Mrs. B examined them at leisure and seemed tolerably well satisfied with their quality and value. Humpf! she said, running her eyes over the child's slight figure. I don't see anything else, except the shoes. I must have the shoes, Miss Domby. Poor little Florence took them off with equal alacrity, only too glad to have any more means of conciliation about her. The old woman then produced some wretched substitutes from the bottom of the heap of rags, which she turned up for that purpose, together with a girl's cloak, quite worn out and very old, and the crushed remains of a bonnet that had probably been picked up from some ditch or dung hill. In this dainty raiment she instructed Florence to dress herself, and as such preparations seemed apprelude to her release, the child complied with increased readiness, if possible. In hurriedly putting on the bonnet, if that may be called a bonnet, which was more like a pad to carry loads on, she caught it in her hair, which grew luxuriously, and could not immediately disentangle it. Good Mrs. Brown whipped out a large pair of scissors and fell into an unaccountable state of excitement.  I beg your pardon. I don't know what I have done, panted Florence. I couldn't help it. Couldn't help it, cried Mrs. Brown. How do you expect I can help it? Why, Lord, said the old woman, ruffling her curls with a furious pleasure. Anybody but me would have had him off, first of all. Florence was so relieved to find that it was only her hair and not her head, which Mrs. Brown coveted, that she offered no resistance or entreaty, and merely raised her mild eyes toward the face of that good soul. If I hadn't once had a gal of my own, beyond seas now, that was proud of her hair, said Mrs. Brown, I'd have every lock of it. She's far, far away. She's far away. Oh, ho, oh, ho. Mrs. Brown's was not a melodious cry, but accompanied with a wild tossing up of her lean arms. It was full of passionate grief, and thrilled to the heart of Florence, whom it frightened more than ever. It had its part, perhaps, in saving her curls. For Mrs. Brown, after hovering about her with the scissors for some moments, like a new kind of butterfly, bade her hide them under the bonnet, and let no trace of them escape to tempt her. Having accomplished this victory over herself, Mrs. Brown resumed her seat on the bones, and smoked a very short black pipe, mowing and mumbling all the time as if she were eating the stem. When the pipe was smoked out, she gave the child a rabbit skin to carry, that she might appear the more like her ordinary companion, and told her that she was now going to lead her to a public street when she could inquire her way to her friends. But she cautioned her with threats of summary and deadly vengeance in case of disobedience, not to talk to strangers, nor to repair to her own home, which may have been too near for Mrs. Brown's convenience, but to her father's office in the city, also to wait at the street corner where she would be left until the clock struck three. These directions, Mrs. Brown enforced with assurances that there would be potent eyes and ears in her employment cognizant of all she did, and these directions Florence promised faithfully and earnestly to observe. At length, Mrs. Brown, issuing forth, conducted her changed and ragged little friend through a labyrinth of narrow streets and lanes and alleys, which emerged after a long time upon a stable yard with a gateway at the end whence the roar of a great thoroughfare made itself audible, pointing out this gateway and informing Florence that when the clock struck three, she was to go to the left, Mrs. Brown, after making a parting grasp at her hair, which seemed involuntary and quite beyond her own control, told her she knew what to do and bade her go and do it, remembering that she was watched. With a lighter heart, but still sore afraid, Florence felt herself released and tripped off to the corner. When she reached it, she looked back and saw the head of Good Mrs. Brown peeping out of the low wooden passage where she had issued her parting injunctions. Likewise, the fist of Good Mrs. Brown shaking towards her, but though she often looked back afterwards, every minute at least, in her nervous recollection of the old woman, she could not see her again. Florence remained there, looking at the bustle in the street, and more and more bewildered by it, and in the meanwhile the clocks appeared to have made up their minds never to strike three any more. At last the steeples rang out three o'clock. There was one close by, so she couldn't be mistaken, and after often looking over her shoulder, and often going a little away, and as often coming back again, lest the all-powerful spies of Mrs. Brown could take offense, she hurried off as fast as she could in her slip-shod shoes, holding the rabbit-skin tight in her hand. All she knew of her father's offices was that they belonged to Dombien's son, and that that was a great power belonging to the city. So she could only ask the way to Dombien's sons in the city, and as she generally made inquiry of children, being afraid to ask grown people, she got very little satisfaction indeed. But by dint of asking her way to the city, after a while, and dropping the rest of her inquiry for the present, she really did advance by slow degrees toward the heart of the great region which is governed by the terrible Lord Mayor. Tired of walking, repulsed and pushed about, stunned by the noise and confusion, anxious for her brother and the nurses, terrified by what she had undergone, and the prospect of encountering her angry father in such an altered state, perplexed and frightened alike by what had passed and what was passing, and what was yet before her. Florence went upon her weary way with tearful eyes and once or twice could not help stopping to ease her bursting heart by crying bitterly. But few people noticed her at these times in the garb she wore, or if they did, believed that she was tutored to excite compassion and passed on. Florence, too, called her her aid all the firmness and self-reliance of a character that her sad experience had prematurely formed and tried, and keeping the end she had in view steadily before her, steadily pursued it. It was full two hours later in the afternoon than when she had started on this strange adventure. When escaping from the clash and clanger of a narrow street full of carts and wagons, she peeped into a kind of whore for landing-place upon the riverside, where there was a great many packages, casks, and boxes strewn about. A large pair of wooden scales and a little wooden house on wheels, outside of which, looking at the neighbouring masts and boats, a stout man stood whistling, with his pen behind his ear and his hands in his pockets, as if his day's work were nearly done.  If you please, is this the city? asked the trembling daughter of the Dombies. Ah, it's the city, you know that well enough, I daresay. Be off, we haven't got anything for you. I don't want anything thank you, was the timid answer, except to know the way to Dombian sons. The man, who had been strolling carelessly toward her, seemed surprised by this reply, and looking attentively in her face rejoined. Why, what can you want with Dombian sons? To know the way there, if you please.  Joe, he called to another man, a laborer, as he picked it up and put it on again. Joe, it is, said Joe. Where's that young spark of Dombies, who's been watching the shipment of them goods? Just gone by the other gate, said Joe. Call him back a minute. Joe ran up an archway, bawling as he went, and very soon returned with a blithe-looking boy. You're Dombie's jockey, ain't you? Obedient to the indication of Mr. Clark's hand, the boy approached towards Florence, wondering, as well he might, what he had to do with her. But she, who had heard what passed, and who, besides the relief of so suddenly considering herself safe at her journey's end, felt reassured beyond all measure by his lively, youthful face and manner, ran eagerly up to him, leaving one of the slip-shot shoes upon the ground, and caught his hand in both of hers. I am lost, if you please, said Florence. Lost? cried the boy. Yes, I was lost this morning, a long way from here, and I have had my clothes taken away, since. And I am not dressed in my own now. And my name is Florence Dombie, my little brother's only sister. And, oh dear, dear, take care of me, if you please, saw Florence, giving full vent to the childish feelings she had so long suppressed and bursting into tears. At the same time her miserable bonnet falling off, her hair came tumbling down about her face, moving to speechless admiration and commiseration young Walter, nephew of Solomon Gill's ship's instrument maker in general. Mr. Clark stood wrapped in amazement, observing under his breath. I never saw such a start on this wharf before. Walter picked up the shoe and put it on the little foot, as the prints in the story might have fitted Cinderella's slipper on. He hung the rabbit skin over his left arm, gave the right to Florence, and felt not to say, like Richard Whittington, that is a tame companion, but like St. George of England, with the dragon laying dead before him. Don't cry, Miss Domby, said Walter, in a transport of enthusiasm. What a wonderful thing for me that I am here. You are as safe now as if you were guarded by a whole boat's crew of picked men from a man of war. Oh, don't cry. I won't cry any more, said Florence. I am only crying for joy. Crying for joy, thought Walter, and I'm the cause of it. Come along, Miss Domby. There's the other shoe off now. Take mine, Miss Domby. No, no, no, said Florence.  These do better. These do very well. Why, to be sure, said Walter, glancing at her foot.  What am I thinking about? You never could walk in mine. Come along, Miss Domby. Let me see the villain who will dare molest you now. So Walter, looking immensely fierce, led off Florence, looking very happy, and they went arm in arm along the streets, perfectly indifferent to any astonishment that their appearance might or did excite, by the way. It was growing dark and foggy and beginning to rain, too, but they cared nothing for this, being both wholly absorbed in the late adventures of Florence, which she related with the innocent good faith and confidence of her years, while Walter listened as if, far from the mud and grease of Thames Street, they were rambling alone among the broad leaves and tall trees of some desert island in the tropics. He as very likely fancied for the time they were. Have we far to go, asked Florence at last, lifting up her eyes to her companion's face? Ah, by the by, said Walter, stopping. Let me see, where are we? Oh, I know. But the offices are shut up now. Miss Domby, there's nobody there. Mr. Domby has gone home long ago. I suppose we must go home, too, or stay. Suppose I take you to my uncle's, where I live.  Won't that be best? I think so, answered Florence. Don't you? What do you think? As they stood, deliberating in the street, a man passed them, who glanced quickly at Walter as he went by as if he recognized him, but seeming to correct that first impression, he passed on without stopping. Why, I think it's Mr. Karker, said Walter. Karker in our house, not Karker our manager, Miss Domby, the other Karker, the junior. Hello, Mr. Karker. Is that Walter gay, said the other, stopping and returning? I couldn't believe it, with such a strange companion. As he stood near a lamp, listening with surprise to Walter's hurried explanation, he presented a remarkable contrast to the two youthful figures arm in arm before him. He was not old, but his hair was white, his body was bent or bowed as if by the weight of some great trouble, and there were deep lines in his worn and melancholy face. The fire of his eyes, the expression of his features, the very voice in which he spoke, were all subdued and quenched, as if the spirit within him lay in ashes. He was respectively, though, very plainly dressed in black, but his clothes, molded to the general character of his figure, seemed to shrink and abase themselves upon him, and to join in the sorrowful solicitation, which the whole man from head to foot expressed, to be left unnoticed and alone in his humility. And yet his interest in youth and hopefulness was not extinguished with the other embers of his soul, for he watched the boy's earnest countenance, as he spoke with unusual sympathy, though with an inexplicable show of trouble and compassion, which escaped into his looks, however hard he strove to hold it prisoner. When Walter, in conclusion, put to him the question he had put to Florence, he still stood glancing at him with the same expression, as if he had read some fate upon his face, mournfully at variance with its present brightness. What do you advise, Mr. Parker? said Walter, smiling. You always give me good advice, you know, when you do speak to me. That's not often, though. I think your own idea is the best, he answered, looking from Florence to Walter and back again. Mr. Parker, said Walter, brightening with a generous thought. Come, here's a chance for you.  It may do you some good. Sir, I'll remain at home. You shall go. I returned the other? Yes, why not, Mr. Parker, said the boy. He merely shook him by the hand in answer. He seemed, in a manner, ashamed and afraid, even to do that, and bidding him good night, and advising him to make haste turned away. Come, Miss Domby, said Walter, looking after him, as they turned away also. We'll go to my uncles as quick as we can. Did you ever hear Mr. Domby speak of Mr. Parker the junior, Miss Florence? No, returned the child mildly. I don't often hear Papa speak. Ah, true, more shame for him, thought Walter. After a minute's pause, during which he had been looking down upon the gentle, patient little face moving on at his side, he said, The strangest man Mr. Parker the junior is, Miss Florence, that you ever heard of. If you could understand what an extraordinary interest he takes in me, and yet how he shuns and avoids me, and what a low place he holds in our office, and how he is never advanced, and never complains, though year after year he sees young men passed over his head, and though his brugger, younger than he is, is our head manager, you would be as much puzzled about him as I am. As Florence could hardly be expected to understand much about it, Walter bestowed himself with his accustomed boyish animation and restlessness to change the subject, and one of the unfortunate shoes coming off again, opportunely, proposed to carry Florence to his uncles in his arms. Florence, though very tired, laughingly declined the proposal, lest he should let her fall, and as they were already near the wooden midshipmen, and as Walter went on to cite various precedents from shipwrecks and other moving accidents, where younger boys than he had triumphantly rescued and carried off older girls than Florence, they were still in full conversation about it when they arrived at the instrument maker's door. Hello, Uncle Saul, cried Walter, bursting into the shop and speaking incoherently and out of breath from that time forth for the rest of the evening. Here's a wonderful adventure, here's Mr. Dombie's daughters lost in the streets and robbed of her clothes by an old witch of a woman found by me, and brought home to our parlour to rest, look here. Good heavens, said Uncle Saul, staring back against his favourite compass case. It can't be, well I know, nor anybody else, said Walter, anticipating the rest. Nobody would, nobody could, you know. Here, just help me lift the little sofa near the fire, will you, Uncle Saul?  Throw those shoes under the grate. Miss Florence, put your feet on the fender to dry. How damp they are. Here's an adventure, Uncle, eh? God bless my soul, how hot I am. Solomon Gills was quite as hot by sympathy and in excessive bewilderment. He patted Florence's head, pressed her to eat, pressed her to drink, rubbed the soles of her feet with his pocket handkerchief, heated up the fire, followed his locomotive nephew with his eyes and ears, and had no clear perception of anything, except that he was being constantly knocked against and tumbled over by that excited young gentleman, as he darted about the room, attempting to accomplish twenty things at once, and doing nothing at all. Here, wait a minute, Uncle, he continued, catching up a candle. Till I run upstairs and get another jacket on, and then I'll be off, I say, Uncle, isn't this an adventure? My dear boy, said Solomon, who, with his spectacles on his forehead and the great chronometer in his pocket, was incessantly oscillating between Florence on the sofa and his nephew in all parts of the parlor. It's the most extraordinary! No, but do, Uncle, please, do, Miss Florence, dinner, you know, Uncle. Yes, yes, yes, cried Solomon, cutting instantly into a leg of mutton, as if he were catering for a giant. I'll take care of her whiley, I understand. Pretty dear, famished, of course. You go and get ready, Lord bless me, Sir Richard Whittington Thrice, Lord Mayor of London. Walter was not very long in mounting to his lofty garret and descending from it, but in the meantime Florence, overcome by fatigue, had sunk into a dose before the fire. The short interval of quiet, though only a few minutes in duration, enabled Solomon Gill so far to collect his wits as to make some little arrangements for her comfort, and to darken the room and to screen her from the blaze. Thus, when the boy returned, she was sleeping peacefully. That's capital, he whispered, giving Solomon such a hug, that it squeezed a new expression into his face. Now I'm off. I'll just take a crust of bread with me, for I'm very hungry. And don't wake her, Uncle Saul. No, no, said Solomon. Pretty child. Pretty indeed, said Walter. I never saw such a face, Uncle Saul. Now I'm off. That's right, said Solomon, greatly relieved. I say, Uncle Saul, cried Walter, putting his face in at the door. Here he is again, said Solomon. How does she look now? Quite happy, said Solomon. That's famous. Now I'm off. I hope you are, said Solomon, to himself. I say, Uncle Saul, cried Walter, reappearing at the door. Here he is again, said Solomon. We met Mr. Corker the Junior in the street, queerer than ever. He bade me good-bye, but came behind us here. There's an odd thing. For when we reached the shop door, I looked round and saw him going quietly away, like a servant who had seen me home, or a faithful dog. How does she look now, Uncle? Pretty much the same as before, Wally, replied Uncle Saul. That's right, now I am off. And this time he really was. And Solomon Gills, with no appetite for dinner, sat on the opposite side of the fire, watching Florence in her slumber, building a great many airy castles of the most fantastic architecture, and looking in the dim shade, and in the close vicinity of all the instruments, like a magician disguised in a Welsh wig and a suit of coffee colour, who held the child in an enchanted sleep. In the meantime, Walter proceeded toward Mr. Dombie's house at a pace seldom achieved by a hack-horse from a stand, and yet with his head out of window every two or three minutes, in impatient remonstrance with the driver. Arriving at his journey's end, he leaped out and breathlessly announced his errand to the servant, followed him straight into the library, where there was a great confusion of tongues, and where Mr. Dombie, his sister, and Miss Tox, Richards, and Nipper were all congregated together. Oh, I beg your pardon, sir, said Walter, rushing up to him, but I'm happy to say it's all right, sir, Miss Dombie's found. The boy, with his open face and flowing hair and sparkling eyes, panting with pleasure and excitement, was wonderfully opposed to Mr. Dombie as he sat confronting him in his library chair. I told you, Louisa, that she would certainly be found, said Mr. Dombie, looking slightly over his shoulder at that lady, who wept in company with Miss Tox. Let the servants know that no further steps are necessary. This boy, who brings the information, is young gay from the office. How was my daughter found, sir? I know how she was lost. Here he looked majestically at Richards. But how was she found? Who found her? Why, I believe I found Miss Dombie, sir, said Walter modestly. At least I don't know that I can claim the merit of having exactly found her, sir, but I was the fortunate instrument of. What do you mean, sir? interrupted Mr. Dombie, regarding the boy's evident pride and pleasure in his share of the transaction, with an instinctive dislike, by not having exactly found my daughter, and by being a fortunate instrument. Be plain and coherent, if you please. It was quite out of Walter's power to be coherent, but he rendered himself as explanatory as he could in his breathless state, and stated why he had come alone. You hear this, girl? said Mr. Dombie sternly to the black-eyed. Take what is necessary, and return immediately with this young man to fetch Miss Florence home. Gay, you will be rewarded tomorrow. Oh, thank you, sir, said Walter. You are very kind. I'm sure I was not thinking of any reward, sir. You are a boy, said Mr. Dombie, suddenly and almost fiercely, and what you think of, or affect to think of, is of little consequence. You have done well, sir. Don't undo it. Louisa, please to give the lad some wine. Mr. Dombie's glance followed Walter Gay with sharp disfavor, as he left the room under the pilotage of Mrs. Chick, and it may be that his mind's eye followed him with no greater relish as he rode back to his uncles with Miss Susan Nipper. There they found that Florence, much refreshed by sleep, had dined, and greatly improved the acquaintance of Solomon Gill's, with whom she was on terms of perfect confidence and ease. The black-eyed, who had cried so much that she might now be called the red-eyed, and who was very silent and depressed, caught her in her arms without a word of contradiction or reproach, and made very hysterical meeting of it. Then, converting the parlour for the nonce into a private tiring room, she dressed her with great care in proper clothes, and presently led her forth as like a Dombie as her natural disqualifications admitted of her being made. Good night, said Florence, running up to Solomon. You have been very good to me. Old Saul was quite delighted and kissed her like her grandfather. Good night, Walter. Goodbye, said Florence. Goodbye, said Walter, giving both his hands. I'll never forget you, pursued Florence. No, indeed I never will. Goodbye, Walter. In the innocence of her grateful heart, the child lifted up her face to his. Walter, bending down his own, raised it again, all red and burning, and looked at Uncle Saul quite sheepishly. Where's Walter? Good night, Walter. Goodbye, Walter. Shake hands once more, Walter.  And when the coach at length moved off, Walter on the doorstep gaily returned the waving of her handkerchief, while the wooden midshipman behind him seemed, like himself, intent upon that coach alone, excluding all the other passing coaches from his observation. In good time Mr. Dombe's mansion was gained again, and again there was a noise of tongues in the library. Again, too, the coach was ordered to wait for Mrs. Richards, one of Susan's fellow servants ominously whispered as she passed with Florence. The entrance of the lost child made a slight sensation, but not much. Mr. Dombe, who had never found her, kissed her once upon the forehead, and cautioned her not to run away again, or wander anywhere with treacherous attendance. Mrs. Chick stopped in her lamentations on the corruption of human nature, even when beckoned to the paths of virtue by a charitable grinder, and received her with a welcome something short of the reception due to none but perfect Dombe's. Ms. Tox regulated her feelings by the models before her. Richards, the culprit Richards, alone poured out her heart in broken words of welcome, and bowed herself over the little wandering head as if she really loved it. Ah, Richards said Mrs. Chick with a sigh. It would have been much more satisfactory to those who wished to think well of their fellow creatures, and much more becoming in you if you had shown some proper feeling in time for the little child that is now going to be prematurely deprived of its natural nourishment. Cut off, said Ms. Tox in a plaintive whisper, from one common fountain. If it was my ungrateful case, said Mrs. Chick solemnly, and I had your reflections, Richards, I should feel as if the charitable grinder's dress would blight my child and the education choke him. For the matter of that, but Mrs. Chick didn't know it, he had been pretty well blighted by the dress already, and as to the education, even its retributive effect might be produced in time, for it was a storm of sobs and blows. Louisa, said Mr. Dombie, it is not necessary to prolong these observations. The woman is discharged and paid. You leave this house, Richards, for taking my son, my son, said Mr. Dombie, emphatically repeating these two words, into haunts and into society which are not to be thought of without a shudder. As to the accident which befell Ms. Florence this morning, I regard that as, in one great sense, a happy and fortunate circumstance, in as much as, but for that occurrence, I never could have known, and from your own lips, too, of what you had been guilty. I think Louisa, the other nurse, the young person, here, Miss Nipper sobbed aloud. Being so much younger and necessarily influenced by Paul's nurse, may remain. Have the goodness to direct that this woman's coach is paid to. Mr. Dombie stopped and went to Staggs Gardens. Polly moved toward the door, with Florence holding to her dress and crying to her in the most pathetic manner not to go away. It was a dagger to the haughty father's heart, an arrow in his brain, to see how the flesh and blood he could not disown, clung to this obscure stranger, and he sitting by. Not that he cared to whom his daughter turned, or from whom turned away. The swift sharp agony struck through him, as he thought of what his son might do. His son cried lustily that night at all events. Soothe to say, poor Paul had better reason for his tears than sons of that age often have, for he had lost his second mother. His first, so far as he knew, by a stroke as sudden as that natural affliction which had darkened the beginning of his life. At the same blow his sister, too, who cried herself to sleep so mournfully, had lost as good a true a friend. But that is quite besides the question. Let us waste no words about it. All LibriVox recordings are in the public domain. For more information or to volunteer, please visit LibriVox.org. Dombie and Son by Charles Dickens, Chapter 7 A bird's-eye glimpse of Miss Tox's dwelling place. Also of the state of Miss Tox's affections. Miss Tox inhabited a dark little house that had been squeezed at some remote period of English history into a fashionable neighborhood at the west end of the town, where it stood in the shade like a poor relation of the great street round the corner, coldly looked down upon by mighty mansions. It was not exactly in a court, and it was not exactly in a yard, but it was in the dullest of no thoroughfares, rendered anxious and haggard by distant double knocks. The name of this retirement, where grass grew between the chinks in the stone pavement, was Princess Place, and in Princess Place was Princess's Chapel with a tinkling bell, where sometimes as many as five and twenty people attended service on a Sunday. The princess's arms was also there, and much resorted to by splendid footmen. A sedan chair was kept inside the railing before the princess's arms, but it had never come out within the memory of man, and on fine mornings the top of every rail, there were eight and forty, as Miss Tox had often counted, was decorated with a pewter pot. There was another private house besides Miss Tox's in Princess's Place, not to mention an immense pair of gates with an immense pair of lion-headed knockers on them, which were never opened by any chance and were supposed to constitute a disused entrance to somebody's stables. Indeed, there was a smack of stabbling in the air of Princess's Place, and Miss Tox's bedroom, which was at the back, commanded a vista of muse, where hostlers at whatever sort of work engaged were continually accompanying themselves with effervescent noises, and where the domestic and confidential garments of coachmen and their wives and families usually hung like Macbeth's banners on the outward walls. At this other private house in Princess's Place, tenanted by a retired butler who had married a housekeeper, apartments were let, furnished, to a single gentleman, to it a wooden-featured, blue-faced major, with his eyes starting out of his head, in whom Miss Tox recognized as she herself expressed it, something so truly military, and between whom and herself an occasional interchange of newspapers and pamphlets and such platonic dallions was effected through the medium of a dark servant of the majors, who Miss Tox was quite content to classify as a native without connecting him with any geographical idea whatever. Perhaps there never was a smaller entry and staircase than the entry and staircase of Miss Tox's house. Perhaps taken all together, from top to bottom, it was the most inconvenient little house in England, and the crookedest, but then Miss Tox said, what a situation. There was very little daylight to be got there in the winter, no sun at the best of times, air was out of the question, and traffic was walled out. Still Miss Tox said, think of the situation. So said the blue-faced major, whose eyes were starting out of his head, who gloried in Princess's place, and who delighted to turn the conversation at his club, whenever he could, to something connected with some of the great people in the great street round the corner, that he might have the satisfaction of saying they were his neighbors. The dingy tenement inhabited by Miss Tox was her own, having been devised and bequeathed to her by the deceased owner of the fishy eye in the locket, of whom a miniature portrait with a powdered head and a pigtail balanced the kettle holder on opposite sides of the parlour fireplace. The greater part of the furniture was of the powdered head and pigtail period, comprising a plate warmer, always languishing and sprawling its four attenuated bow legs in somebody's way, and an obscure harpsichord illuminated round the maker's name with a painted garland of sweet peas. Although major bagstock had arrived at what is called in polite literature the grand meridian of life, and was proceeding on his journey downhill with hardly any throat and a very rigid pair of jaw bones and long flapped elephantine ears, and his eyes and complexion in the state of artificial excitement already mentioned, he was mightily proud of awakening an interest in Miss Tox, and tickled his vanity with the fiction that she was a splendid woman who had her eye on him. This he had several times hinted at the club in connection with little jocularities of which old joe bagstock, old joey bagstock, old jay bagstock, old josh bagstock and so forth was the perpetual theme it being, as it were, the major's stronghold and dungeon keep of light humor to be on the most familiar terms with his own name. Joey B. Sir, the major would say with a flourish of his walking stick, is worth a dozen of you. If you had a few more of the bagstock breed among you, Sir, you'd be none the worse for it. Old joe, Sir, needn't look far for a wife even now if he was on the lookout, but he's hard-hearted. Sir is joe. He's tough, Sir, tough and devilish sly. After such a declaration, wheezing sounds would be heard and the major's blue would deepen into purple while his eyes strained and started convulsively. Nevertheless, his very liberal laudation of himself, however, the major was selfish. It may be doubted whether there ever was a more entirely selfish person at heart, or at stomach is perhaps a better expression, seeing that he was more decidedly endowed with that latter organ than with the former. He had no idea of being overlooked or slighted by anybody, least of all had he the remotest comprehension of being overlooked and slighted by Miss Tox. And yet Miss Tox, as it appeared, forgot him, gradually forgot him. She began to forget him soon after her discovery of the Toodle family. She continued to forget him up to the time of the christening. She went on forgetting him with compound interest after that. Some thing or somebody had superseded him as a source of interest. Good morning, ma'am, said the major, meeting Miss Tox in Princess's place some weeks after the changes chronicled in the last chapter. Good morning, sir, said Miss Tox very coldly. Joe Bagstock, ma'am, observed the major with his usual gallantry, has not had the happiness of bowing to you at your window for a considerable period. Joe has been hardly used, ma'am. His sum has been behind a cloud. Miss Tox inclined her head, but very coldly indeed. Joe's luminary husband out of town, ma'am, perhaps, inquired the major. I, out of town? Oh, no, I have not been out of town, said Miss Tox. I have been much engaged lately. My time is nearly all devoted to some very intimate friends. I am afraid I have none to spare even now. Good morning, sir. As Miss Tox, with her most fascinating step and carriage, disappeared from Princess's place, the major stood looking after her with a bluer face than ever, muttering and growling some not at all complimentary remarks. Why, dammy, sir, said the major, rolling his lobster eyes round and round Princess's place and apostrophizing its fragrant air. Six months ago the woman loved the ground Joe Backstock walked on. What's the meaning of it? The major decided, after some consideration, that it meant man traps, that it meant plotting and snaring, that Miss Tox was digging pitfalls. But you won't catch Joe, ma'am, said the major. He's tough, ma'am. Tough is JB. Tough and devilish sly, over which reflection he chuckled for the rest of the day. But still, when that day and many other days were gone and past, it seemed that Miss Tox took no heed whatever of the major and thought nothing at all about him.  But now she never gave the major a chance and cared nothing at all whether he looked over the way or not. Other changes had come to pass too. The major, standing in the shade of his own apartments, could make out that an air of greater smartness had recently come over Miss Tox's house, that a new cage with gilded wires had been provided for the ancient little canary bird, that diverse ornaments cut out of colored cardboards and paper seemed to decorate the chimney piece and tables, that a plant or two had suddenly sprung up in the windows, that Miss Tox occasionally practiced on the harpsichord, whose garland of sweet peas was always displayed ostentatiously, crowned with that Copenhagen and bird waltzes in a music book of Miss Tox's own copying. Over and above all this, Miss Tox had long been dressed with uncommon care and elegance in slight mourning. But this helped the major out of his difficulty, and he determined within himself that she had come into a small legacy and grown proud. It was on the very next day after he had eased his mind, by arriving at this decision, that the major, sitting at his breakfast, saw an apparition so tremendous and wonderful in Miss Tox's little drawing-room, that he remained for some time rooted to his chair. Then, rushing into the next room, returned with a double-barreled upper glass, through which he surveyed it intently for some minutes. It's a baby, sir, said the major, shutting up the glass again, for fifty thousand pounds. The major couldn't forget it. He could do nothing but whistle and stare to that extent, that his eyes compared with what they now became, had been in former times quite cavernous and sunken. Day after day, two, three, four times a week, this baby reappeared. The major continued to stare and whistle. To all other intents and purposes, he was alone in Princess's place. Miss Tox had ceased to mind what he did. He might have been black as well as blue, and it would have been of no consequence to her. The perseverance with which she walked out of Princess's place to fetch this baby and its nurse, and walked back with them, and walked home with them again, and continually mounted guard over them, and the perseverance with which she nursed it herself, and fed it, and played with it, and froze its young blood with airs upon the harpsichord, was extraordinary. At about the same period, too, she was seized with a passion for looking at a certain bracelet, also with a passion for looking at the moon, of which she would take long observations from her chamber window. But whatever she looked at, sun, moon, stars, or bracelet, she looked no more at the major, and the major whistled and stared and wondered, and dodged about his room, and could make nothing of it. You'll quite win my brother Paul's heart, and that's the truth, my dear, said Mrs. Chick one day. Miss Tox turned pale. He grows more like Paul every day, said Mrs. Chick. Miss Tox returned no other reply than by taking the little Paul in her arms, and making his cockade perfectly flat and limp with her caresses. His mother, my dear, said Miss Tox, whose acquaintance I was to have made through you. Does he at all resemble her? Not at all, returned Louisa. She was. She was pretty, I believe, faltered Miss Tox. Why, poor dear Fanny, was interesting, said Mrs. Chick after some judicial consideration. Certainly interesting. She had not that air of commanding superiority, which one would somehow expect, almost as a matter of course, to find in my brother's wife, nor had she that strength and vigor of mind which such a man requires. Miss Tox heaved a deep sigh. But she was pleasing, said Mrs. Chick, extremely so. And she meant, oh, dear, how well poor Fanny meant.  If the major could have known how many hopes and ventures, what a multitude of plans and speculations rested on that baby head, and could have seen them hovering in all their heterogeneous confusion and disorder round the puckered cup of the unconscious little Paul, he might have stared indeed. Then would he have recognized, among the crowd, some few ambitious moats and beams belonging to Miss Tox. Then would he perhaps have understood the nature of that lady's faltering investment in the Dombie Firm. If the child himself could have awakened in the night and seen, gathered about his cradle curtains, faint reflections of the dreams that other people had of him, they might have scared him with good reason. But he slumbered on, a like unconscious of the kind intentions of Miss Tox, the wonder of the major, the early sorrows of his sister, and the stern visions of his father, and innocent that any spot of earth contained a Dombie, or a son. End of chapter seven.