Welcome to the third class of the Deep Learning 2022 Fall Edition, Fall semester. So we cover a few things so far, a bit of history on the first class. Then yesterday we cover a little bit of back propagation and gradient descent. And then I actually pointed out in the chat that gradient descent and back propagation are not exclusively used for training. So we're going to be learning about that today a little bit more. But today, so today topic is going to be only about inference. What is inference? Inference is about using neural networks. Just you're given a network and then we're going to be learning today how we use it, how we make predictions. Before that, a few announcements. Why is that?  Usually you do. I mean, most of the time. So one of the things I really like is going to be this book from Edward Taft. He actually sent me all his books. And this is called Beautiful Evidence, which is basically talking, well, it's a book about quantitative visual art. This is the title. You can see. There's some glare. OK, you can see, right? Duck. So it's super nice. And it tells you basically how to properly convey information with a visual type of media. So you should be able to convey information with different type of media. You don't have just to provide text. As you're going to see in my classes, I like a lot to provide mathematics with colors such that I believe at least it's using the way to make it. It makes easier to make connection between the charts and then the diagrams and the mathematics. Anyway, so that's one thing I wanted to show you. Another thing was the I told you last time to have a look about the introduction to linear algebra or whatever linear algebra animated by Glenn Sanderson or three blue, one brown, right? I really encourage you to binge watch his videos on YouTube such that you can start acquiring a visual understanding of how these mathematical operators behave, OK? Such that you can reason in terms of pictures, in terms of images. That's how at least I find it very convenient to think about things visually. About that, there is a blog post, which I really, really, a couple of blog posts from this guy, Gregory Gunderson. He has this amazing blog site, blog blog.  There is like show all posts. He's a New Yorker. So you're going to see also things about New York. But the two things I wanted to show you here is the matrices. Where is it? A geometrical understanding of matrices. This is, I think, really good. And it shows you basically how to think about matrices. So after you watch the videos from Grant, I would really recommend you to read through this blog where you're going to get some additional intuitive understanding of how these operators work. And actually, we're going to be covering some of these things today as well. Another blog, which is super nice, which I also got inspired by, is the singular value decomposition here. So who doesn't know what is singular value decomposition? Tell me in the chat. Is there someone who doesn't know? I know. Of course, you won't say, oh, I don't know. I don't know. The point is that one thing is the mathematical introduction of the singular value decomposition.  But let's say we might have forgotten, which is possible. But again, if you forget something, means you never understood it. If you understand something, it's part of you. So you cannot forget understanding. You can forget knowledge. But knowledge and understanding are different things.  Actually, this is the interpretation I will tell you about today. But if you scroll down, there's a preview from the book. OK, too much stuff. OK, here. So this is a lecture in the mathematical style from Gilbert's strength. So here you have singular value decomposition. And this one is called diagonalization of matrices. So what is another word for diagonalization of matrices? What kind of decomposition is this one? Type, type in the chat. Type, type. Eigenvalue decomposition, right? So do we know what our eigenvalues are, eigenvectors? What could be a quick description of eigenvector in the chat? What is the physical intuition? What is the geometrical intuition behind eigenvectors? Do we know? Do we not know? Direction of maximal variance? Absolutely not. OK? So that's the main point I want to stress, right? So thank you for pointing out the fact that you say something that is completely wrong. It makes me understand that the things I'm trying to teach, perhaps, will give you something, right? So vectors whose directions do not change with the application of a matrix. Yeah, that's correct, right? And so I would actually say that there is no eigenvector. But I would actually say eigendirection. So all the vectors that are on that line, that direction, will not change their direction when after a linear transformation, OK? And so again, I wrote a bit about that. So if you go, again, in this case, on my blog post, on my blog, I mean, why not do an advertisement first time, right?  And then here I talk about a little bit of these eigenthings, eigenstaff. It's the only article I've written. But here I basically introduced this eigendirection rather than eigenvectors, because I don't like the vector part. And then basically here you select two eigendirections. And then given that you select them, you can drag the eigenvalues, right? You can have 1, 2, minus 1, minus 2. So here you change the eigenvalues. And then before we just chosen the two eigendirections, which are these directions, these lines, right? And this is written in JavaScript. You can also check the code if you click on the link here, right? But also I explain to you how this works in the part below here, OK? So again, this is, again, yes, it's a little bit of advertising what I'm doing. But the point is that you should be getting a visual understanding or geometric or whatever you want to call it, understanding a physical understanding of all these mathematical things we are using. Otherwise, it's really hard to really have like an intuition about what the heck these things are, right? Especially there is a lot of confusion arising by using terminology which we don't quite have a inner understanding, right? Another question that I asked, I asked also Jan, and he got it wrong, so I get some pleasure in that. My question is, do eigenvectors need to be orthogonal? Yes or no? Type in the chat. OK, so for those that type yes, I'm asking. So subsets, OK, fine. Stop typing. Asking those who type yes. Are these two lines that are drawn on the screen orthogonal? Just those that type yes before? Yes? Wait, what? Did you see these lines, right? You see my screen. There are two white lines, right? Or gray lines, this one and this other one, right? Orthogonal means, are they at 90 degrees? Are they intersecting at 90 degrees? The answer is, I believe, obviously no, right? And so the answer to my question are eigenvector orthogonal? It's no, of course not. I just show you on the screen, these are not orthogonal eigendirection. So again, please, again, no shame in answering incorrectly because, I mean, we are in class for learning. But the point is that if you find yourself doubting about the understanding of something, just dig further. So again, read through these little plots and you will see that it doesn't really take that much effort to understand these concepts. But again, you need to actually put yourself up to these things, right? One more thing, and then I stop with these other types of reasoning, advertising, otherwise we don't go anywhere today. It's going to be the mathematics for machine learning, OK? This book by a lot of people, but I only know Mark in person. So this book is available for free online, right? So you just go to this link. And then you should really read the vector calculus for next week because you will have to do a homework. And this is really required, right? So I mean, you should, in theory, already have this knowledge from previous classes. Still a refresher wouldn't hurt. OK, just have a look to this one, right? This one, the two videos from Gilbert, right? Wouldn't hurt. The blog post, I mean, this is maybe less necessary. But again, if you find this stuff kind of rusty, you're self-wracking, these topics have a look. I really also recommend checking out this one, right? The geometrical understanding of matrices. And finally, the introduction, the YouTube one, right? Where is it here? Linear algebra, linear algebra, three blue, three blue, one brown, right? So there is like a playlist, a sense of linear algebra. I just watched all of this, right? I mean, recommendation. You're not, don't have to, but this is all I can tell you to. I can try to help you, right?  So also, again, here you can look for here from me, right? And then you want to look for probability data science, OK? I think this is also a very good one. This book from Stanley Chang, it's really highly recommended. And it explains probability for a data scientist, so with codes and interactive way. So sometimes I see, although we won't be using much probability in this course, in other courses, you might need probability. And sometimes it's hard to get your head around these topics. So just, you may want to check out these resources, OK? And now I'm done with these resources, these advertising. I hope this is helping someone, at least, right? If he helps one person, I'm glad. But there were a few of you who got the eigenstaff wrong. So, you know, Jan included can gain something by checking out these resources. Anyway, moving on, let's close this thing. What are we talking about today? We talk about, we said, what do we talk about today? Remember? No one say anything. Inference, yes, they're correct. All right, so this is, it's called draw.io, or now they changed the name into app.diagrams.net. But you can just type drawdraw.io, OK? And press Enter. You're going to get to the same place. What is this thing, right? So finally, we see here, somehow there is a module, right? We said that we use this kind of bullet-shaped, like, figures, shapes, bullet-shaped, you know, a bullet shape for representing a deterministic transformation of something into something else, OK? So in this space, in this specific case, I have a X here, which is just denoting a arbitrary input in this case. It's not necessarily a data point yet. So I have an input here that goes inside this thing, and then it gets a specific output, OK? So in this case, I have this circle here, which is shaded, right? You can see it's different color, the background from this one. So this means it's observed. So I provide an observation to my predictor, which is spitting out a Y tilde, OK? So what equation would we write to represent this thing, right, in mathematics? How would you type in the chart, right? How would you describe this diagram in mathematical terms? So yeah, but yeah, there you are, right? So Jack said what I wanted to point out. So you have Y tilde equal F of X, right? So the order of the items are all flipped, right? So Y tilde, which is the extreme right, is going to be the first to the left. Then you have the, it's OK, you don't have to keep typing. Then you have the predictor, which is going to be still in the middle, right? But then it's operating on the X, which is on the, well, the X is on the extreme right, right? And then finally, on the extreme left, you're going to have the Y tilde, right? So things are flipped, I believe, right? At least in my opinion, right? So this is going to be called the forward pass, right? You put something inside the network and these don't speed something else out. But again, the order of things are inverted, which is somehow counter-intuitive. Question. So if this is, we are trying to do maybe regression, we expect this Y tilde to be how, with respect to my ground truth, Y, far close, close, right? And so, oh, equal, hopefully, yeah. So this tilde here on top, although the font is broken because I don't know whatever issue the website has right now, so the tilde means it's like an approximation for the Y, OK? And somehow there has been a process, which is training, but we don't talk about training today, which has found the, well, has optimized or has given us the parameters, the weights, for this specific model, such that it converts the specific input X in something Y tilde, which is something close to my target, OK? Anyway, so how about I have something else, right? So in this case, over here, what's the major difference here? Despite the font being broken, I don't know what, yeah, what X tilde, I have an X tilde here, and also the circle is no longer shaded, right? You see? And the other thing I have is the blue Y, right? So now the Y is blue and also has the shape. So in this case, I'm given the Y, but I'm not given the X. So how do I get the X out of a model given that I only have the output? This is also called the inference. So we can always go, we can go in both directions. Once you, this one is called the fit forward, OK? And in the fit forward, you feed something to the network, this model, this predictor is the model.  But again, I can't really reason in terms of mathematics here.  In this case, I can reason, right? I feed something inside and this one spits out some sort of prediction. How about this case? I have something that I would like to be this value. How to find this one, right? So someone say, yeah, invert the F, right? Or invert the predictor. Let's say the predictor is not invertible, right? So in machine learning, that's actually what we do very often, actually, the case. So most of the time, what we are actually doing is actually this one is going to be our parameters. And given that we observe both the X and the Y, we somehow find which are the parameters that allows us to connect the X to the Y. Anyway, so what we are going to be doing is going to be the following. We're going to be trying to find, so this X tilde is going to be the arc minimum of all possible X's.  My output is going to be swinging, right? The Y tilde is going to be swinging until I get the Y tilde that is minimizing this difference. So whenever I'm going to get that Y tilde that is the closest in terms of square distance to my target, then I found the X which is producing that specific Y. Is there one single X? No, no one said that. Is there an X which can produce that? Not either, not necessarily. I can only just find the best X, given the specific network configuration, right? It doesn't have to be exactly the same. You don't have exactly to have zero in terms of error. You can get the X that is giving you the smallest error. So we infer the predictor from X and Y. That will be the training. We talk about that next week. This week, instead, we're talking about inference. So the inference, there are two types of inference. There is the inference that you achieve the easy way, like you just perform a fit forward, meaning you put points, you put a specific value at the input of the network. The network speeds out something. Say we're going to be playing with this in a bit with the code. Or the other way to do inference, but we won't be doing that with the code today, is the other way around, right? We basically, given that we are given a predictor, like a network, a model, whatever, and given that we know what is our objective, we can try to minimize the discrepancy between the output of the model and the target in order to get the A possible X that produces that output. But this one that I just show you right now, let me hide this bar on the left. Can I? Yes, awesome. But this is like the wrong notation, okay? So this is abuse of notation. Let me show you now what is going to be the full notation and therefore introducing the first time we introduced the energy term, okay? So far, are there questions? I can also show you afterwards how to draw these diagrams, okay? There are no questions. You can type, I can wait 10, 20 seconds and see whether there are questions. So far it is clear, right? What I've been trying to explain. Okay, all right. Moving on, my question to you, right? So yesterday in the chat, and then I answer a lol was, I put a statement, right? I said, back propagation and gradient descent are not only used for training, okay? Why? Answer, type in the chat. They are used also for inference, right? It's very important this, right? So again, what is back propagation doing? What does back propagation do? What is another name for back propagation? Change rule, compute the gradients, okay? Gradient descent, what is it? I'm reading the chat, right? Update, not necessarily the weights, not the weights, right? That's too specific, right? Not parameters, neither parameters, right? Because if it's operating parameters, it's already learning. It's already specifying something, right? So finding something, right? Finding some variables, I guess, right? By minimization of a cost, let's call it this way. And now we understand, it's sort of understanding why we talk about loss, cost, and now we'll be introducing this energy. Again, all these are not necessarily synonyms, although they are all scalar functions, okay? Anyway, the loss, we don't talk about loss this week with me, right? Loss, we talk about next week. Everything we talk about today is gonna be either costs or energies. What are these things? So let me show you what are these things. So despite the font being broken and which is really upsetting me today, I'm sorry, I'm upset, it's really bugging me, but okay, I can't do anything because I'm using online free resources and that's what you get by doing this. So this is the full notation, okay? So for the moment, let's ignore this shaded box. And so I have here a X, with the check, we talk about the check afterwards. The X goes inside the predictor and the predictor gives me a Y tilde. Y tilde is a approximation for possibly the Y. Then on the other side, I have my blue Y, which is my target, okay? I'll tell you in the future why it's blue, but for now it's just blue, a color of my choice. This blue Y and this violet Y tilde, both goes inside the square, okay? And the square represents a scalar function, meaning it gets inputs, maybe vector, so in this case, it gets into two scalars, so you can think about one vector of two elements. And it returns you a value. In this case, C stands for cost. Cost is the penalty you pay for making a misprediction, okay? So if Y tilde is the same as Y, you pay zero cost, right? You have a very small penalty, zero penalty, right? Otherwise, you're gonna pay equal to the square Euclidean distance from my target, the blue Y, okay? So here it's written underneath that my cost is going to be the cost between Y and Y tilde. And since I'm using this interface, I cannot have color math, but later on we have color math. This is simply equal to the square Euclidean distance. And then finally, which is not such a big thing, but again, it's a different thing, we define this capital F being our energy and our energy is function of the inputs. What are the inputs of the system? Tell me. What goes inside the big X and Y, right? So X and Y are the only input to the system. Later on we will have also Z, X, Y and Z, but today we just have X and Y. So F is going to be a level of incompatibility between X and Y, which is equal to the basically single scalar value, right? So again, no big deal. The C is a give a scalar value. With this scalar value is the same value that this F has, right? There's a change of variable, right? So C is a function of Y, the target and the Y tilde, the prediction, whereas F is a scalar function, function of the inputs of the system, right? So this Y tilde is function, right? We computed before, right? Y tilde is a function of the X, right? And so eventually the C is also a function of X, right? But the inputs are Y and Y tilde, right? So these are the two arrows. You can see also the two arrows going into the dash box is going to be the X, X guy and the Y. Finally, the X check, check means it looks like an arrow pointing down, right? Downward. So that's like the minimizer, a minimizer of the energy. So this capital X should be a calligraphic X. I really, the font got broken today, I'm sorry.  So in this case, the Y is observed, right? So it's shaded. So I have this value, right? So like a conditional almost type of energy. I have this Y and then this function changes as I change the values in this position here. So X check is gonna be the value that is minimizing the, how do we call this F? What is it? What is F? F is the type in the chart. The definition of F, the energy and what is the representation? What does it mean? The energy represents the level of, let's type it down on the chart, level of incompatibility between X and Y, okay? Very good. Definitions we have to keep in mind. Very good. Okay, so in the next part of the lesson, we are gonna be going through the notebooks. So I was thinking to, so you had two options, but I already decided which one to do, but I show you where you can find notebooks that are going to be similar to the one I'm covering today, but I was thinking to show you the notebook. Actually, I work on for my book, right? So I'm writing a textbook about this class, basically. I'm very late. I wanted to actually get it done a major part before starting this semester, it didn't happen, but so I will show you the notebook I made for the book, which is very similar to the one you can find online, right? So where to find the resources online? So you go on GitHub, right? Then there is the 2021 edition, but here I only have a few notebooks, which we are gonna be covering in the future. And most of the notebooks are from the 2020 edition. I haven't changed them too much. Well, I changed them for the book, right? But the repository for the book is not yet available online because it needs to be fixed, right? I will answer the question in a second. So the one we are gonna be going over right now is gonna be the number two, space stretching or some variation of this, right? Is there, before I answer the question, is there anyone in class not familiar with NumPy? I repeat, is there anyone in class out of 90 people not familiar with NumPy or NumPy or NumPy or whatever you want to call it in French or Spanish? NumPy? I'm like, yeah, no, I'm just kidding. Sorry. Okay, so it seems like that either people didn't hear the question that people didn't know or didn't hear or at least maybe everyone knows about this, right? Anyway, so I'm leaving this tensor tutorial for yourself, right? And so just go over this first notebook by yourself. It explained a little bit how to use Jupyter notebooks, how to pull out the documentation and so on, okay? So this is like very kind of basic, basic tutorial. I don't want to go over these in class because it's gonna be mostly a waste of time for most of you since, again, it seems that you already have some sort of exposure to NumPy and PyTorch has a very similar interface for actually design now, right? PyTorch was not, it's not based on NumPy actually, right? Someone posted that on Twitter a few weeks ago, it's incorrect. PyTorch derives from Torch which we developed initially using Lua scripting language which just for the reason because it was super light and there was no Python community at the time and it didn't make any sense. Then my student decided to port basically Torch to Python and came up with PyTorch during like a few months internship at Facebook and now the whole world is using this framework and I feel so proud about that, right? Anyway, that's mostly the reason why we are using that but also because it's like state of the art, right? Anyway, today we're gonna be covering this one here, space stretching, okay? So how to get everything running, right? So here underneath you have instructions about how to get your mini-Konda environment up and working in 15 different languages. If you find bugs and typos in any of the translations and any language language you speak, please send a poor request. This is really highly appreciated, right? I don't speak all these languages so these are being like all contribution from people that gave some time to this course. Anyway, so if you install all these things, you're gonna get the different environments. There's gonna be the PDL, PyTorch, deep learning and then you can either run the JupyterLab or the notebook, the lab is gonna be much nicer, I think, but these have been developed with Jupyter notebook. So some things might not quite work out of the box for the JupyterLab. Anyway, so I was going to tell you that instead of going through this notebook here, you can check this out, right? So my expectation and also someone asked me today, do we follow along with a notebook in class? No, well, you follow what I'm doing in class, right? And then afterwards, you're kind of supposed to spend at least one hour per notebook by your own time, right? Such that you get familiar, you check the documentation, you have some bugs, you check the dependencies, I don't know, you should get familiar with the code such that once, when you have to actually do the homework, you already know most of the things, right? How they work. So instead of doing that, we'll just use the book one. Let me see, there was a question. What is the difference between the check sign and the hat over x? Okay, awesome, that's a good question. So we don't have yet a hat over the thing, right? So far in this today, we saw either the tilde, which means a approximation because it's like when you write, you know, pi equal more or less 3.14, right? Pi is not 3.14, but it can be approximated as 3.14 so that tilde means it's an approximation. It's not exactly the value you get, right? So y tilde tries to be similar to y. How we do that by plugging here, basically a spring. What is a spring, right? Have you taken physics? Yes, what is the force of a spring? Like, like, like, like, f equal minus kx, very good, right? And this is called hook law. What is the energy of a spring? Oh, also, let's call it x naught to be the central equilibrium point, right, of the spring, of relaxation of the spring. One half, yeah, let's use the relaxation point at x naught, right? Right, otherwise you have the spring that flips at zero, right? Like, or let's have a reference system that you plug the spring at zero and you have it at equilibrium at x zero, right? There we go, right? So you have one half kx minus x naught square, right? And that really looks very similar to this thing, right? Well, despite the one half and k, right? But again, those are just constant, right? Whatever, we don't care. So if you look at this thing, this basically represents the energy contained in a spring which is being displaced by y tilde minus y, okay? You see that? And so once, when we will be training the system later on, we'll basically tune the parameter such that all the springs will get relaxed. Or on the other case, when we will try to infer the y check, right? Sorry, the x check, right? How to find this x check? You have a specific y, you connect the spring between the y and the y tilde and then you start moving the x check such that you try to relax the spring, okay? I hope it makes sense. If it doesn't, just think more about what I said. If you don't remember what I said, check the record. It doesn't, okay? It's okay. You have 14 more lessons to get acquainted with all these things. Don't worry. Also remember, I'm sarcastic a lot of time. In the last class, you said you will be telling the difference between y tilde and y hat, okay? We haven't seen y hat yet. And if you see a y hat, if you have seen any y hat so far, it's just a typo, okay? I know, I don't have the typo, but Jan may have it in his slides because I've been trying to update that. We only have y tilde for prediction, y check for the output of minimization, okay? So the check points downwards, which means it's a low energy value, okay? And then you're gonna see later on, we're gonna have a hat. Possibly next week, I will show you, next week or the week afterwards, I will show you an example of y, of x hat or y hat or whatever hat. That is called a contrastive sample, okay? It's supposed to be a sample to which we want to assign a high energy. Again, we doesn't make any sense right now because we haven't really talked about high energies. We talk about low energy, right? So there is this paper from a co-worker of mine, which collected, well, Larry Alpinto, which has collected several examples of grasping some object with a tool, right? With a robot. And so you have the picture of the grasping or like the orientation. You have picture of the object, picture of the orientation of the player I think it's called in English. And then you have either it was successfully grasped or it was unsuccessfully grasped, okay? So you want to learn what is the correct orientation. You could do that by only using the small fraction, let's say 1% of your data.  You just use the successful cases and you train a regressor over the correct orientation of your player given the object, okay? That would be horrible, you know, because you throw away all the data point, right? The alternative is to use the classification. Let's say you discretize all possible orientation into, I don't know, 10 possible angles or whatever 36 possible angles. So you'll have 10 each, 10 degrees each. And then you use Y, the blue Y for the try to increase the probability that the model will select that orientation. Or you're gonna consider a Y hat, which is a negative, a contrastive sample in order to decrease the probability of that specific orientation, okay?  So you can run gradient ascent in the contrasting sample when you had a contrasting sample or just put the minus, right? So if you want to run gradient ascent for the contrasting sample, you can simply take the loss and put a minus in front. Like you're gonna get the flipped gradient, right? So the point is that in that case, once you collect correct and incorrect cases for classification, you want to tell that that specific image should not have that orientation associated because it's not gonna work, okay? So that's a good question, right? So Y hat in this case is used for classification kind of a proxies for regression which wouldn't actually use all the data you collected. This is super cool, I think. You can check out the paper. Anyway, so let's move on, right? Otherwise we don't go anywhere. So let me show you a little bit of the actual book. And then a few, a bit of notebook, okay?  So if you think about a neural net, I'd like you to think about a sandwich, okay? So what happens if you have like a sandwich made only of slices of bread? How do you call it a sandwich made of only slices of bread? It's a loaf, right? It's a loaf of bread, right? So in order not to have a loaf of bread, you need to put what? Something in between the layers, right?  Instead, we're gonna need to have like a sandwich between linear and non-linear layers. We saw that also from the slide with Jan yesterday with that kind of chain of multiple things. So again, we're gonna be using this kind of notation which is helping us understanding the flow of information through the network. So I will call my generic input R and then the output is going to be S. Where can you find this textbook? That's a very good question. On my website, when I finish writing the first draft, so far you just get a preview. It's not being peer review, so it's not yet available for others to read. First, Jan has to fix my errors and then it's gonna be available for everyone to read. Anyway, the things I'm telling you, at least I know they are correct. You're gonna have this linear layer and how do we think about linear layers? So again, talking about this SVD, any linear layer can be decomposed into three items. It's gonna be a rotation, a scaling and another rotation. Moreover, whenever you are in high dimensional space, we just reason and think about this linear layer basically just as one big rotation. And basically we discard this other two. We just, usually we just think about rotation because let's say you are in a hundred dimensional space. In how many ways can you change the orientation of a vector? Let's say it moves around a hypersphere. If you're in a hundred dimensional space, in how many directions can you, let's say if you have two dimensions, right? Two dimensions, how many directions can you rotate the thing, right?  In two dimensions, right? So you have one degree of freedom to change the orientation. And then one degrees of freedom of changing the length. If you are in a three dimensional space, you're gonna have a surface, so I have a two infinite way of changing, two degrees of freedom of changing the orientation. And then one degree of freedom of changing the length, right? If you are in 100 dimensional space, you're gonna have 99, yeah, 99 degrees of freedom for changing the orientation. And then just one degree of freedom of changing the length. And so once, when we want to compare vectors in high dimensional space, we care about the alignment of vectors more than the length of vectors, right? Because the alignment can be much more descriptive, right? It can contain much more information. It has a lot of more capabilities of expressing differences, right? Of encoding of representation, whereas the length is gonna be just one degree of freedom. So again, although a linear transformation can be split up in rotation scale and rotation, usually when you just reason about linear transformation in high dimensional space, we just reason in terms of rotations, okay? So most of the time when you hear me talking about rotation, it means a linear transformation in a high dimensional space. Anyway, one more thing is the fact that whenever you write the math in this way right here, it's annoying again because things are flipped, right? So R is the input. Then a first operator is this V, which is a rotation. Then you have this Sigma, which is having the scaling factor. Then you have this U, another rotation. Then you get the output, right? So everything is flipped. Whereas here you can clearly see that which one is happening first and which one is happening later. Anyway, so a little bit of math, we don't care. And this is what is the big picture. So whenever you have a matrix, it's just a bunch of numbers. And this is, they don't really tell you much. Instead, if you perform a SVD decomposition, it's like making a radiography of your transformation, such that you can actually clearly tell what's happening. Any matrix, any linear transformation rotates a pair of orthogonal axes in the input space into a different orientation here in the output space. And then it changes its magnitude independently. So you have one big rotation from this side here to the right side over here. And then each of them have its own length change. So what happens is that you have a circle that is getting rotated and then squashed in one specific direction. Moreover, the first singular value, since they are actually ordered, is gonna tell you which is gonna be the maximum deformation, maximum length of a transform vector. So let's say you have your whatever input vectors and you have sigma one equal three. You know that this matrix multiplication can provide you up to a zooming of factor equal three, okay? It can scale up things up to three. Why do I say up to? Because let's say sigma two is gonna be zero. It might kill completely all other vectors that goes in this direction, for example. Anyway, so this is two mathematical, two boring. And so instead, what we are gonna be doing is gonna be actually having a little bit of fun now with the notebook, right? I'm gonna be showing you exactly this.  I will have a random matrix, a two by two matrix, which is performing, what is a matrix doing? Matrix is gonna be a linear transformation and we said that it performs, we can be thinking about it as performing what operation. Type, type, type, type. Quick, quick, quick, quick. Rotation and some stretching and then some other rotation, okay? Again, in 2D, we can actually think about combining this into like shearing and other things, but again, it becomes a little bit too messy. We don't care about that in high dimensional space. So we have some skating, some rotation. We're gonna get basically this circumference rotated. You can see these two axes that are telling you what is the orientation of this circumference. And you can see these green points are in correspondence of V1. They get rotated down here, right? On the U1 and then they get scaled. In this case, a little bit less than one here, which I showed now with the notebook. So let's have a look a bit at the notebook and so we can learn how to use PyTorch. Again, my recommendation is after you, we are done with this lesson in 10 minutes, you pull out the Jupyter notebook, either the one on the GitHub or you just try type yourself and just get yourself familiar with all these different functions we talk about today, okay? In order to get some sort of hands dirty, right? Hands on experience on playing with mathematical tools, right? This is a numerical calculus kind of discipline. It's a empirical discipline. You need to actually try out things, not only to make sure your mathematics is correct, but also to acquire a intuition, right? About how these things work, right? Without intuition, it's really hard to make educated guesses about how something could work. If you just think about coding up everything you might come up with in your mind, you will never have enough time to try everything. But instead, if you use your mind as a predictive model, you can come up with good possible ways of performing some, getting a good reward, a low cost in the future. But in order to train a predictive model and what you need to do, how do we train predictive models? Do we know? I mean, we haven't yet talked about predictive models, but do we know, do you know the answer of this question? We're in the same, yes, but on a lot of self, a lot of data, right? You have to play with observations and get your mind to be able to predict the future, right?   The repository we talked about before, but we're gonna be going now inside Book and Python, which is not yet available to you. I know that. I'm gonna have, you're gonna be doing Conda, activate, activate, you're gonna be doing basically activate PDL for PyTorch Deep Learning, but I will be doing activate Book, right? Always use environments. And then I also have alias for JL, which is JupyterLab, such that it's quickly to launch it. Let me know if anything is not clear on the code, right? Because sometimes I go too fast in the code and I don't realize that. I try to go to a decent speed.  And then I also, from Torch, I import Optime and NN, which are going to be useful later on. Then I import other utility functions I define, which are gonna be slightly different in the version on Github. Doesn't really matter. These are just the plopping routines. Here I create a folder for saving the results that just for generating the book. And here what I'm doing is gonna be sampling P, capital P, and we asked yesterday what it was. Someone, P is gonna be the total number of samples. So it's gonna be 100 samples from a random normal distribution, right? With 100 samples of two dimensions. Then I have my labels equal all zeros. It's gonna be one if you are basically going to be in the second quadrant. Two, if you're gonna be in the third quadrant. And three, if you're gonna be in the fourth quadrant. Why is that? Such that you can tell the orientation of things and whether there is a flipping. When do we have a flipping when once we apply a linear transformation? Do you know what I'm talking about? When do we have a reflection? When the determinant is negative, yes. Okay, so here I plot the input space, which we call R, right? So we have R1 and R2 with a different four colors, okay? And then here I'm gonna be creating, I just use torch.manual seed in order to have always the same random numbers whenever I create these linear layer such that things are consistent, especially for writing the textbook. And in this case, it's gonna be a two by two linear transformation without bias. What does it mean? So if there is a bias, there is going to be also an additional shifting after applying the rotation. A rotation, well, rotation like linear, right? Rotation, linear transformation plus a bias is also called a fine transformation. Again, this helps us get in the model more degrees of freedom to play with, okay? But usually, yeah, no, usually, that's it. So here I just generate this linear two by two, which is basically a matrix, a matrix, okay? What I'm doing here below is going to be the following. I say torch, please, please, do not track the computational graph. What does this mean? In PyTorch, every time you use a model that has weights inside, has parameters, PyTorch keeps track of the computational graph such that once I'd like to perform backward, it knows what are the operations you need to get the Jacobian and multiply the output gradient with, okay? We saw yesterday that the forward pass, basically you put some input, you have the X input, whatever, inside the model, you get something at the output. Eventually, you're gonna get a final number one, which is the output derivative. You basically, that's the output gradient. You send it backward, and how does it get affected by those modules that went, where the signal went forward first? Well, you have to multiply the output gradient by the Jacobian of that individual module, given that we fed a specific input before, right? So most of the time we also need to keep the previous input nearby, because that's more often time used in the backward pass. So the output gradient needs to be multiplied by the Jacobian, and then it keeps getting propagated backwards through the model, okay? Again, the model, the PyTorch, therefore creates a computational graph with the sequence of operations that we used in the forward pass. Since today we're gonna be talking about just fit forward, we don't care about going backwards, we're gonna be using these, what's calling in Python I forgot, with, it's called the help me out, context manager, is it correct? Context manager, yeah. So this is gonna be a context manager which disables PyTorch tracking of sequence of operations, okay? And so basically by running these, I send R, what was R? R was the bunch of points, right? The 100 points in two dimensional space. So I send the whole R inside the model, and this guy spits me out, S, right? Which is going to be what? How many points? You're following? How many points are in S? 100, yes. So it's gonna be the same 100 points but then rotated basically, right? Rotated because there is no bias, right? There is no shifting. Here I compute these U, Sigma and V transpose with these torch dot linear algebra dot singular value decomposition, to which I input the weights, so which is the metrics, and I use this dot detach, right? So this is also important. When I do dot detach, basically I disconnect the computational graph that was made, any computational graph that was made before. So this dot detach is giving me the actual metrics without, okay, how does it work? Let me explain again. PyTorch keeps creating, it creates a computational graph every time you use something that require a gradient, okay? So the weights are some parameters in PyTorch-jergon which automatically require gradient. Now, if I call this dot detach, I stop, I prevent a computational graph to be performed, to be created, right? So one way to do that was with this way, right? Here I say, do not create a computational graph and just proceed forward with this thing. And here I didn't have access to that weights, right? So here this is a module, right? This is object in Python. I'm sending something inside the object and then something comes out. So I cannot cut the graph or whatever, right? Or I could have said here at the end, dot detach. So I could have created something which has a computational graph and then I could have done detach here, right? But then this one does create a computational graph. Instead, why didn't I do that here below because it's a different way of performing something similar? Here I'm getting as input, this is a parameter, but then I say, oh, it's no longer a parameter. This is just the values, the tension, right? And then I compute this singular value decomposition of basically the values, not the parameters, okay? Let me see the question here. Does it initialize the weights to some random by default? Yes, so this here and then dot linear, you can check the initialization on the PyTorch documentation, okay? PyTorch initialization, okay? So you can check later on the manual how things are initialized, okay? I would really, really recommend you going through the whole documentation of PyTorch to be sure you know what's going on there, okay? So here I compute this singular value decomposition to find the orientation of things and to find the initial orientation of that pair of axes in the input space, find the orientation on the outer space. Both of them are orthogonal, right? So these are called singular vectors. Left singular vectors, right singular vectors. The property of singular vectors are that they are orthogonal. These are orthogonal, not the eigenvectors, okay?  Here I transpose such that we have column vectors and then this U sigma, I just scale these vectors by those scaling factors. So here I show you again my, what do I do? Here I create a circle, such that I can plot things, okay? Is the touch like a deep copy? Absolutely not. There is no copy going on. I'm just using this as input and nothing will change to this thing. This is just an input of the system. It's not a copy at all. You want to do clone, right? If you want to do a deep copy of the vector, right? You want to do clone. Now that would be a new instance in the memory, okay? But we don't need that here. Okay, so here I create a circle, right? With cosine and sine, tau is two pi, right? But it's better tau. Anyway, here the interesting thing. The circle, what is this circle? How many points are inside the circle? Do you see? 100.  And then I put this inside this linear transformation and I get an ellipse. And then here I show you the input space and the output spaces, okay? This is going to be the input space with the V1 and V2. And then it's going to be the output space which has the U1 and U2 in different orientation and shrunk, right? So you can tell that the largest, again, well, not again, something. This is the largest possible transformation that this matrix is. It's going to be something like 0.9 or something, okay? Finally, quickly, quickly, such that I don't waste, I don't take too much of your time. We said this is going to be, these are the linear transformation which are basically rotations. But we also have something else. And I will take those questions later after the class, right? So these are going to be called activation functions which are acting most of the time on the singular level, single item individually, okay? In this case, I have the positive part or reload or we had the leaky reload, which has like a compression factor which is not compressed to infinitely zero, right? It has a compression factor of, in this case, 120th. Other options are going to be the hyperbolic tangent and the sigmoid. What happens now, again, these are going to be the transfer function. It tells you how the output, the A activation output changes given that you change the S, the summation, the linear output you get from the previous part. And so these are going to be the four options you may get, right? If this is my input on the top left, on the top right, you're going to have this leaky reload.  Here you just compress 20 times the horizontal part. You compress 20 times both sides, coordinate, and you compress the vertical coordinate by 20. If you have hyperbolic tangent, you basically get a square out of the circle. And the last one, you're going to get some smooth kind of square centered in 0.5, 0.5, okay? How did I do that? Simply I initialize here. I just used torch dot hyperbolic tangent, logistic sigmoid, or this leaky reload. And then I just exactly do the same as I showed you on before. Let me show you once here. I just send my point, the 100 points through this hyperbolic tangent, the sigmoid or the leaky reload, okay? Finally, once you can put this together, you're gonna get the fully connected architecture, which is the most generic architecture we encounter. This has no assumption over the input. It has the first part, which is going to be the linear part, right? This kind of rotation. And then each branch, each dimension, you can see here, gets F, this non-linearity, which is spitting out this activation value, okay? And the activation are the value of the non-linear function, okay? Whereas this S is just the linear summation of this initial components, okay? And this is called FC for fully connected model. Again, we use this kind of bullet shape to represent all these operations here. And what happens if you do that? You're gonna get this thing here. I show you a three-layer, three fully connected layers, one two to 20, 20 to 20, and then 20 to two. So I have one, two hidden layers, right? The output is gonna be S and green because this is gonna be a linear output. I don't have a non-linearity on top. I just have a non-linearity here and non-linearity here. So these are again my input points, sample from that normal distribution, with a few more level lines such that we can see what happens. And here on the bottom part, you can see the network which has the hyperbolic tangent, which is kind of smooth kind of transformation, but it's not the best for learning arbitrary transformations. And on the right hand side, you can see this network with the positive part, with the real part, okay? So this is pretty much all I wanted to show and I really went over with the time. So are there, I would say, just go over yourself with like after class, go over these two notebooks, okay? The first two, the tensor tutorial, okay? And the space stretching. And lesson is dismissed. Sorry for going a little bit over, okay? More questions in the kappa-pa-pa-pa? Yes, thanks, bam, Wikipedia done, finished, right? I think I answer everyone. You're very active, I like it. You're more active than last semester.  I hope it's gonna keep this way, right? Especially later on when the concepts get a little bit harder to understand. If you don't ask questions, it's hard for us to address your doubts because again, we can't really read your mind. A bit we can, right? We've been doing this for a while. It's been 10 iterations now, but still you should speak up, speak your mind and no shame in asking any kind of question that is related to the class, right? Okay, enough. Bye-bye, see you next week. Enjoy the weekend, I go dancing. If you want to dance, it's a very nice studio. It's called Empire Mambo. We are starting the new cycle in two weeks. Again, advertisement, I don't get paid, but it's so good. It's like a school number one in the world for dancing. We have it here in New York City, so. Anyway, bye-bye, see you next week.


Good afternoon everyone, and welcome to Entrepreneur India's Resilient Series. During this tough period, while investors have become more observant about the ongoing trends, founders have looked for ways to increase their cash runway, while some truncated their expenses, others pivoted their businesses to keep the cash flow and earn investors confidence during the pandemic. These unprecedented times, despite its cruel nature, has made the bond between investors and founders stronger, or so would we believe. So, today we are going to talk about just that. I'm Saurav Kumar, editor of Special Projects Entrepreneur India, the moderator for this session. I'll quickly lay down the ground rule for our attendees. The discussion will go on for 30 minutes followed by a 15 minute Q&A session. If you have any questions during the course of the discussion, you can post them through the Q&A or comment option. Mention in your question if it's directed towards any specific panelists and we'll take them up post the discussion. Let me now introduce our panelists for the day. We have with us today Mr. T.C. Minakshi Shundaram, founder and managing director of Chirate Ventures, India Advisors. Mr. Rishabh Sanghvi, co-founder and CTO Railroad 5, Mr. Praveen Agarwal, co-founder and CEO of Better Places. Welcome everyone. So, you know to start with T.C.M, I'll come to you first in a nutshell, if you can please tell us that. I think it was calling me a bit earlier, but I was on a call with entrepreneur India because T.C.M. linked for me and what N.D. Winker went with it. No, no, I'm there. Okay, so we'll just continue. So T.C.M. I'll just come to you. So in a nutshell, if you can please tell us that how has been the last six odd months as an investor? There have been studies that more than 50% of the startups will close down, etc. So what was going through your mind as an investor? What were the emotions, the ups and downs, if you can just briefly tell us. You know, interesting question. To look back, I think the first four to eight weeks was a time of focusing on saying that how to extend the runway for the companies which are burning capital, right? And that was the most important thing and entrepreneurs are obviously by DNA optimistic, right? And so to that extent, they, and they were people who drive at 320 kilometers, right? 200 miles per hour. And that's why they are growing as fast compared to when the economy was growing much slower. So it took, I think one or two weeks for them to realize saying that this is actually here for longer impact, right? COVID has changed. Initially, I think they were saying that, okay, I will run faster. I'll continue to run faster. And that is at a time when what I would say is the rest of the world had stopped and going in reverse, right? That means, you know, the effort that is involved to do that and that was not even necessary. So once they realized it, I think saying that this is here for long, I think pretty much everyone moved into, you know, managing the burn. And it was an intense period between, I think, investors and entrepreneurs, it was brutal on them, I should say. And I remember it's easy, right? We can be a lot more objective because we are not in the firing line, right? In terms of going out and telling employees saying that, look, we cannot continue, right? Or saying what, going through those. So it's not easiest of things, but it was like, I can tell you saying that this is like learning by the fire, right? They went through the firewall. But the good thing is once that got over and we are tech VC, so to that extent, one of the important things all entrepreneurs realized is that tech is actually a COVID, COVID is actually a tech tailwind event for the tech companies. So that has actually helped them start thinking about how they can move things more and more online, launch new products, new offerings, new channels, which is phenomenal actually. So after the first six, eight weeks of looking at the burn, reducing it, which was brutal, their optimism came back and then started saying that, okay, now how do we grow? And I can tell you saying that our portfolio, which in March had a $1 billion revenue run rate collectively by September is already back at and above several of them actually are significantly higher than pre-COVID peaks is at and above billion dollar run rate. So which is phenomenal. Yes. And this is at a time when we are talking about economy having contracted 24% in the June quarter and then even in September quarter, it will be closer to 10%. And so that is the resilience of the startup entrepreneurs and management teams. Right, absolutely. And this has been shown by many entrepreneurs in the way they reacted to the opportunity change their businesses. So I'll come to two entrepreneurs that we have here and who were at the receiving end when this all started. So Praveen, I'll come to you first. But TCM very well put it that they're back to where they were pre-COVID almost and things like that. So what was your conversations with the investors when all this started? What was the conversation around? Was it like that the boss do something, change things, or was it okay to come down? Let's take a step back and think about how we are going to move from here. Praveen. First of all, this was perhaps the best experience that one can live with, right? I mean, from a journey perspective, the kind of learnings that we got, we will never get perhaps and hopefully we will never get it. But if you look at in the mid-March, one of our investors will pull his best out of US. He started asking me, what are your plans for next 18 months and COVID is coming? And we were in denial more. Like TCM said that, why should you worry so much? And so many things happened in past like dengue and SARS and other things. Don't worry about it. But every day it was changing. And when the 25th of March came, the world was already upside down with the lockdown in the picture and business going down dramatically. Some parts of the business went down by almost 80% all of a sudden. We had to rethink, right? The good thing happened that we have only one view. The investors have multiple views, right? They talk to 15, 20, 100 of their portfolio companies. They understand not only what's happening in India, but also in US and in China and other things. So we started doing a weekly making almost as a board and with investors and trying to figure out what are they learning from others and what are the things that we should do, right? And the first request was that, okay, let's see that we have a run rate for next 18 months at least assuming that our revenue goes down to zero, right? Even if we have a revenue that only helps us to increase the run rate, but let's assume that the revenue is zero and then what would be the run rate? So we started preparing ourselves for the next 18 months in a way where there is zero revenue. Now the good thing is that the investors said, don't worry much whether whatever we have planned for this financial year, if it's delayed by a year, it's delayed by a year. But if we are there and if we learn from the situation, we will bounce back again. And so that's the situation we were with. So we started cutting down costs and we also got a lot of help from the investors. Again, as I said that they have multiple views which helps us to learn from others. They also started collecting data from different portfolio companies saying what are they doing, what are the different initiatives which are being run and then we learn from each other. For example, we just left one office because anyway, we're not going to work from office for a long time. So we said, let's leave this office and we will save on rent and other things. So there are all those smaller things, bigger things that we started putting together. And then in the month of early June, we started seeing the positive trend for ourselves. And we went back to the investors and said, okay, we do have a run rate for 18 months, but this is the time to grow. This is not time to sit back and wait for things to happen. But this is the time to make things happen. And I have to put my acceleration in place. I have to go from gear two to three to four to five. And this is perhaps the time to raise capital as well for the growth. There were some discussions whether this is the right time to do it or what happens if we fall flat. But I think everybody started backing seeing the results. So the questions were also very helpful so that you don't make a decision like he said, we are optimistic people right but optimism to some extent is good in difficult times, we have to be even more careful. So investors and all of our investors, they started asking those difficult questions, which was actually helping in streamlining the process better. Today, I'm proud to say that we are far better than the pre COVID levels, we are at least 50% higher than that. We have raised capital because new investors also see that there is a potential to grow right because if you can show growth in this time and there is definitely potential to keep growing in the future. So overall if I see I think the investors work more like partners in this game rather than seeing a VC who has given money and looking for a return by hook or by growth. I think that was a very positive thing for us and our experience and I assume that most of the. So no numbers were required to be reject or anything. No, we had to we had done a projection in the January with the growth numbers. And in mid March, I said that I will stay with these numbers, but my investors kept asking me that I should reduce it. Take first quarter off the hook completely and then bring it down to 50% or 40% because them. And then I said that okay COVID is coming and it will go off by this thing there was still views that it's a cycle it will keep coming and going there will be waves. So it should be a little bit more pessimist rather than optimistic. I think we wanted to have more the investors kept pushing us. So which which was the target that you achieved the one that you thought you would do or the one that investor. We are in between right now. Okay, all right, Rishabh I'll come to you Rishabh what was your conversation with TCM during this period. What was TCM telling you. Yes, I mean it really depends on the market that you're in. So for the transportation and travel space COVID led to a high period of uncertainty, both in terms of short term impact on business and long term impact on the micro environment in which we are building right. And we had to take some difficult decisions over the next few quarters how to tactically execute and strategically how to plan for the next one to three years. And luckily we have great investors like TCM on board who guided us through this right. There was no panic at all created because they understood the market and the macro very well. We thought through the tactical steps to increase the runway and conserve cash at this moment which is very important for our market. And second, they TCM helped us think through our short term as well as long term strategies. Again, in space like transportation and travel, it's difficult to predict how long the long term consequences of this will be right. So we also in the board developed a plan B in case there's a prolonged disruption in the market beyond what any of us could imagine right. So it was a great exercise in the boardroom where we got, you know, short term tactical as well as strategic and plan B in place with the guidance of TCM. Just a story on TCM at the disc of sounding corny. It was the investor equivalent of love at first sight for us with TCM right because in our first meeting we got to know that TCM has been a train traveler for 30 years. And we knew it with the right partner because he just got it right like he knew the real consumer problem. So we're lucky to have investors who understand the market and the founders empathize with the founders at this stage.  So did you get any such such comfort from your investors? So for us, as I said, because of the uncertainty, we had to take decisions to extend our runway. We did extend our runway. And of course, investors, in terms of our projections of revenue growth, everybody understood right that although we saw green shoots of recovery, nobody knew it's going to be a V shaped or a U shaped recovery. So because of that understanding, you know, the projections in terms of revenues and other growth metrics, obviously, even the marketing expense. And I think it was clear that until you see a sustained, you know, return to something close to normal, it doesn't make sense to kind of kick on with growth at this stage. So we spend the time doing like other things in terms of building products which were there further in our pipeline to kind of expand our market size and a lot of other productive things instead of like growth for us. And I remember a conversation in the call. So obviously, it was a no brainer saying that don't spend on marketing right that's that was very easy. I think where we and they were sitting on pretty much all the money that we gave them. So they didn't have a huge run, you know, burn at that point in time. So I think what we challenge them is look, you are dependent on travel. So two things we asked was one to say that look you had a progressive rollout plan. Can you use this time to be ready to go national in one shot. And you needed actually a lot of this is AI, which means more and more data is important, but data is not available. So in the, in the limited data that you have, how can you go national whenever the markets open up. So literally now they are ready and I can say they are ready that what they would have taken, let's say 12 months plus to be national. Now they are ready whenever the markets open up on travel side to actually go national on day one. Right. So that is something that they use the time. So that is what we were able to use the time and challenge and then save and find innovative solutions on the product front. The second was more interestingly, saying that look, suppose this is going to continue what is your plan B in terms of which other. So you are basically a driven company. So where can this skill be used. Right, in any other market, do you want to look at that. So was this evident across all companies or like obviously talked about today if I but they could be why I'm asking this to you because you know towards the beginning we were hearing a lot of stories that you know investors were not honoring the term sheets and also was that sector specific or was it a knee jerk reaction at the initial phase and then they realize that okay it can be salvaged so what was it going on. So a couple of things. I think from an industry perspective. Now we did actually have an industry group which was exchanging notes and then as to how to guide the entrepreneurs, the ACT which was done by all the VCs together with some VEs also joining. We put out some of the guidelines in terms of even how to manage cost reduction, you know, different ideas and at that time, if you know NDMA had issued a notification so you can't actually let go anybody. Right, without getting notices. So those are all the challenges right. But one of the, I think there was a. There was a lot of I would say half baked information which got interpreted differently. You know that typically of the startups, only about 1015% or 20% are funded by VCs. Right, there are many which are self funded, many which are angel funded friends and family funded. But they do not have that much where with all and run ability to continue giving money more. Right, so when when a lot of newspaper reports came saying that you know 75% of the companies will die from a quantity perspective, is it true potentially. But are they VC funded ones or are they self family, family, friends funded angel funded ones was not identified so everybody said that started with no companies will die. Right. Definitely, COVID has played a role in in some companies deciding not to pursue not to continue. Right, because they don't have the runway. Most VC funded company because I also spoken to others in the ecosystem. Most VC funded companies found ways to extend runway to 12 months and beyond. There are many which are 18 and 24 months also. I don't think they were at the risk of suddenly falling off the cliff. Obviously, some of them were affected much more where revenues possibly straight went to zero and things like that. In some of those cases possibly they had to manage cost better or sometimes they got merged into others. What is very important in every company we have had to have this discussion. It is see many of the entrepreneurs who are in the 20s and 30s have not seen a downturn really even global financial places our startup ecosystem, which is like 15 years old has not seen many of them. Right. So to that extent they have not gone through this to take the tough decisions. Also for entrepreneurs, it is a very lonely journey. Right, you can't even share this with family you can't share this with your management team in terms of all the rest. And so, we had to give them a lot of like as a sounding board. We are also giving confidence saying that look we have seen multiple downturns we have seen how what is a playbook in terms of how to handle it. And sometimes you need to do things in the larger good where some people get affected. Right. And give them the confidence saying that do it like a karma Yogi and not like saying that look you are you have done something wrong. And I should say the management teams also stepped up. There are a few companies in our portfolio where there was zero attrition, but 100% of the organization went through cost reduction. Everybody anywhere from 20% to 50, 75% cost cut, a salary cut was taken by people voluntarily. So I think everybody stepped up hats off to them. And you know that is why they could remain as a cohesive unit and then you know come back and take the market when markets are opening up.  You know so you have both the site so you know what what did you read out the situation when these kind of news were coming in as TCM explained that you know it was not the VC back really was it was, you know funded by maybe you know angel family friends which were kind of so what was your reading of the situation. So I think I think the CM has already spoken about most part of it, but if you if you look at the first priority was at least the company sales through the situation and sales through basically means that you at least have a visibility of 12 to 18 months right and then you will see what happens. And the second thing is that now that your business in the current form shape might not be running. So what are the other monetization opportunity or other ways of keep doing something which will add value at a later stage right. So, giving you an example of one of the companies where I have made initial investments is confirmed ticket, which is similar to what Richard does. But at confirmed ticket we said okay trends are close like Richard already explained, there's no revenue coming it's completely shut, but there is a lot of data. Now what are the things that we can do with this data and work with companies that might need to acquire these users and see if there is a partnership that we can develop and make some kind of strategy around it right. So I'm not sitting idle and waiting for things to happen. It was all about making things happen right and every day is about making perhaps more trial and errors rather than waiting right so I always tell my team that taking no risk is the biggest risk right I mean, if you take risk you might fail if you don't take risk you will definitely fail. So here we started thinking about different things same with better place. A lot of things we started working on new products, new solutions. We never thought that some of the solutions that we have done in last six months we would have done if not pandemic. And those are actually becoming or being adopted more than some of other solutions that we have. So this is a time to also learn. It's no time to say that I'm going to have to wait. I think that's the wrong attitude and I'm sure that none of the entrepreneurs, none of the VCs go in that path. Secondly, you said that I have seen my some of my friends who had started some businesses, whether this is brick and mortar, or some of the new take side of the investment where they had started. And for them this was a bigger challenge because they don't see clear horizon in terms of when they would be able to make money, how long it is sustainable. The people that they have they cannot pay salaries they have their own commitments. So this becomes a very difficult journey for them so either they pivot or they go back to what they were doing previously right. So there we have seen otherwise in my circle I have not seen any big negative impact where they had to shut down. One of the areas that we have felt there is definitely some impact is the fintech side, especially in the lending business they had to stop the business because you don't know the NPA situation and you have to pivot then figure out the right way of doing it. So we are in discussion with some of them, how can we partner and create a distribution channel for larger audience. But it's something that we have to learn. Yeah, I mean that's basically what I would like to add. So before I go to the next question I'll again request our attendees if there are questions, keep them coming we'll take them another 5-10 minutes maybe. So Risha I'll come to you so Praveen just talked about pivot and utilizing the downtime to think of what data can be done.   So what were the thoughts that you within your organization were you thinking that I should change my strategy or should I pivot or should I look for other avenues? So I think what TCM alluded to earlier is that we are very early in our journey and we were sitting on most of the cash we raised anyway so for us you know the burn rate was anyway very low so that pressure wasn't there. For us it was more about I mean firstly just on a separate note the first thought that came in our mind was a humanitarian crisis right the first few weeks we thought of it that way. It was crucial to focus on the health and safety of our employees that was the first thing that we did before any of these other thoughts came in our head. So in that line like there were no layoffs at Raylofi we started doing drills before the lockdown we were well prepared on the IT process and communications front right so we've also assisted some of our team members medically in the pandemic. But coming back to your point there was really no panic in our minds because for us we were very early on in our journey and we were sitting on most of the capital as such. We just wanted to use the time more productively and with our team we wanted to make sure that one efficiency doesn't go down in work from home. And everybody knows it's bound even it can happen if you don't have the right set of team members or the right culturally the right hires right and second was that the communication the coordination with the team is good. So we just put that in place and we of course came up with a plan being case this is longer or we had you know discussions on extending runway and cash but yeah I don't think we really panic on that stage. He said I'll come to you know how did the investors reacted to pivots. I know a lot of companies did pivot but that was not the original plan for a lot of them but they did to survive the period. So how would investors looking at it. Yeah so in our portfolio nobody did a pivot but they added extra lines. So there's a company which is the B2B you know in the packaging solutions and they looked at the opportunity of all medical supplies that are required BP requirements. They actually started an additional line on BP, which this year is likely to be 150 to 100 crores of business for them. Right. So that was opportunistically they were able to jump into it because their core competency was able to find suppliers qualify them and also find buyers right and then so create a marketplace for them. I think, but lately we are looking at companies which raised seed and then said that they pivoted. And I think the important question there that I try to get into is that obviously you know pivoting is a very pragmatic approach there's no question about it. I also try to understand what went what was going in their mind in terms of making the pivot and why they felt that the previous thought process eventually you know pandemic will go. It's not in six months maybe nine months or 12 months. So what made them to make a pivot which is completely abandoned the previous one and do something completely new. That is more assessment of the team than assessment of the previous idea for me to understand that. But you know pandemic did not even know this is one stage, but actually there is a lot more that happened because of the pandemic supply chains got disrupted significantly. So in some of our companies which had let's say things coming from China, we got an inkling about this problem much earlier because you know they were China was shut down for holiday season for the Chinese New Year after that supplies did not start immediately. So, in February, we started looking at some of the companies saying that look how much inventory you have if you continue to grow then how are you going to manage this. Those were actually and how to build resilience in the supply chain side, which many people did not think you know everybody including most of us company, but dependent on single source suppliers and single country. And then when things could not move and they nothing came in as a supply. Right. So how to create second lines. What is the additional cost that you are willing to bear. Right, because China was the cheapest in most cases. But, you know, how do you go to a Taiwan or Thailand or Vietnam to create supplies or find Indian suppliers. Right. To some extent, it was part of it also started coming in people started finding out in suppliers and there is certain additional cost associated because of scale benefits are not available to Indian suppliers. Those are some of the things which are big learnings for companies. Now, I think most of the companies will not be dependent on a single vendor or a single country. So before we start taking some questions that we have Praveen I'll come to you and you know if I have to ask you one anecdote that you know you heard about an investor and you know founder where it seems like you know there was some love loss but the confidence is back is there anything very interesting something that you came across during this past four, five, six months, whatever. From the point of view of that, you know any interesting any any interesting anecdote anecdote you know if you want to share before we go to take up questions. I think one of the things that I would like to highlight is I was speaking to a friend of mine, who is also a big time investor was a professor. And he told me Praveen if in this time, if your models are right your business model is model is right, then products will join the journey so focus on the model, not on the products, the companies that focus too much on the product. And whenever there were ups and downs that kind of failed so if you look at the journey of companies who have survived over the last two three down turns in the economy are the companies who had the right models in place. I think that was a very interesting insight. And I look back now, what we started doing is our model, we started correcting we said, one of the things that we will work is how to make employers more successful or be more efficient with your resources, that is our business model. So the whole idea turned from saying I have a solution a or module B or module C, it is about ensuring that they have an efficient workforce management right now when we started thinking that way we started creating new solutions new ideas. And TCA mentioned about the company started doing supplies the medical supplies, perhaps that was part of the model not a product focus right otherwise you start developing many features in the product, but that does not fit into the model that you want to solve. I think that was a very interesting insight that we got during this time. And, and perhaps, not only in pandemic but in any case, this is a learning that everybody should apply in their journey of creating big businesses is what I strongly believe in. Yeah, very interesting, very interesting. So, you know, we have just a few minutes left so we have some questions, and we'll take a couple of them. So this one for you TCM, you know, so off late we have seen a lot of non Chinese foreign investments coming in. So, so what. So, you know, basically I think what the person wants to ask is that a lot of non, right now the Chinese investment obviously is not allowed so is the non non Chinese foreign investment taking their place or will they be able to fill that gap that we were expecting to be filled by Indian investors. I doubt it. You know, not in the short term to short term. Nobody knows about the medium term. But the reason is simple right Chinese strategic investors and some P investors were willing to write much larger checks. And so companies in series C or plus, which would have been easily able to raise money with with investors strategic investors like a 10 cent or an Ali Baba or a four soon. They possibly their potential definitely got affected. Investors who are now stepping in obviously from Western hemisphere, but that would have actually added to this. So, now it is only replacing it with more and more Western hemisphere investors looking at Indian market was a natural progression with us having 500 million internet users potentially going to one billion the next five, seven years is a natural progression that would have anyway come in. Right, but a big chunk has gone away but that is something that companies have to start adjusting their expectation. It has a very positive side effects also when you know capital is available in a very unlimited way. The rationality in business goes away. Right. It's a question of how long you can discount how can you how long you can continue to cut each other's leg. And you can now hear both. Taking one category example where both are very well funded both are the unicorns like as some act on Swiggy in the food delivery side. Both of them are talking about unit economics, right, which they never bothered about the pandemic because capital was always available. So that's a positive side effect I would say that people are looking at unit economics, building sustainable and profitable businesses. That's a good thing. But also, India can absorb a lot of capital. So to that extent, there is some sort of negative impact. So as an investor I'm not complaining because valuations are not going up insanely. Yes, so what do you mean is that instead of irrational capital whatever foreign funds that are coming in is actually coming in very rational doses and which is keeping the sanity of the market center. I won't call it only rational capital if it is available in unlimited way. It drives entrepreneurs to grow to justify the type of money use of money and the valuation that is given. Correct. And that itself is, then there is more question in terms of, okay, I need to manage it less capital that drives a lot of discipline. Okay, okay. There's another question for the entrepreneurs here is that there will be a suggestion to those entrepreneurs who are trying to raise funds but fail because of pandemic what according to you should be done to win investors confidence. I can share my last six months where we raised and we just made the announcement of our race with jungle ventures. I think the focus shall be on creating a business and and and not on capital. You are you are not raising capital to survive the business.  And if you can give confidence to your investors that that at this stage, I'm not talking about a run rate of 12 months to 24 months and hence I look for a capital. I'm raising capital because there is a clear opportunity. I have a running business. This is sustainable. And this additional capital will help me to expand it to x 3 x 10 x or whatever. I think that has to be absolutely clear in my mind as an entrepreneur and my communication to the potential investors. If those two things are clear, I think it should be an easy game, but at DCM can add something on that because he keeps sitting on the other side.  Praveen's point, it really depends on the market that you're in.  I mean, I just read in September, there was funding of over 550 million happened in India. But I'm assuming the question is from someone who's there's still uncertainties in the market. So at that time, the only thing you can do is Praveen rightly said is run. I mean, become a better company and focus on stretching your runway. Focus has to be on unit economics, freezing capex or unnecessary headcount and like cutting the general administrative fact. And another thing I would advise is don't fixate too much on valuations at this situation. You have to discount to valuations because 10, 20% more of zero is still zeros. Yeah, so I think from an investor perspective, who's an existing investor in a company, we have always advised companies, take money and worry less about valuation. So because I think anybody who survives through this difficult period and comes out standing at the end of this will have success written all over them. We have seen tech bust exactly the same thing happened 8090% companies vanish. Those who survived actually went on to become, you know, big companies at multiple times we have seen it. So that is one advice we tell the entrepreneur, don't worry too much about this thing. Second thing, when we are looking at new investments, obviously, many of the VCs in India have got a lot of dry powder, right, we have raised funds in the last two years time. And so we are continuing to actively look at companies. And one of the important things we look at is unit economics, and then, you know, how efficiently they can grow. Not just how fast they can grow. I think capital efficiency is coming back in the criteria also because like I cannot say whether lot more capital will be available in the next few rounds right like doing a 50 million 100 million 200 million round may take longer. So, to that extent there to plan for being capital efficient. All right, so you know we have run out of time but just before we sign out you know very quickly probably in the show when it comes to all together. So is the confidence back in the market between the founders and investors for our listeners who are here so you know so they take away that is the confidence back if they have an idea which they have been working on and you know they think it will work. You know, is this the time when they can unleash their ideas and start working towards you know, go ahead if they had stopped at all. So my my take would be that there is never a good or a bad time for an idea and idea can always work right and it depends on the situation depends on the opportunity and and sometimes the bad times actually through more opportunities than less. So there are new ways of doing things that perhaps people never thought of in past right and this is the time to make use of that. Either the companies who are doing it just keep growing on that spare or look left and right and add more or a new situation that might pop up and just take the advantage and take it forward and if there are good ideas I'm sure like there is dry powder all across the investor circle and people would be willing to fund this and if you open newspaper at least I see some company or the other getting some investment every now and then so it is just that you have to create the right opportunity. I think the check sizes have become smaller but funding is happening in various and I am earlier it was maybe restricted to ed tech which was seeing a lot of movement but I now I see that it's across across sectors as well. Thank you gentlemen and we have run out of time to my team is asking me to sign out. So thank you so much and I will see that you know the bottom is still there and the confidence is there. And as Praveen also said Rishabh said that you know they have already reached almost reached the pre-peak peak level so hopefully things are looking up and despite the 24% contraction we are hopeful that you know India has found resilience and the startup ecosystem as well. So we will be resilient and we will come out of this one. Thank you gentlemen. Thank you sir. Thank you. Bye bye.