very happy to be here and I thank Dima for inviting me so my talk is maybe a little bit off the main topic of this workshop I would have liked to talk about networked economies that would have been maybe a little more appropriate but this work is not finished yet so I'd like to talk about something that we worked on for the last few years with Hall-Maalez who was a PhD student and Mark Potters and I hope that you'll find it interesting and I think that you know the topic is pretty general and has a lot of different applications and I'll show at the end one application to financial data okay so in this talk I'm going to look at random matrices that are constructed from something that I'll generically call a signal and something that I'm going to call some noise okay so so so these are randomly perturbed matrices and I'll be more precise about these two components later and the questions I'm going to try to address in this talk is not about eigenvalues that have been really studied a lot in the last I don't know 60 or 70 years but but about eigenvectors and in particular I'm going to ask how similar are the eigen vectors of the pure matrix the signal C and those of the noisy observation of C which is M so similarity of the eigenvectors and another question that's as you will see very interesting is how similar are the eigenvectors of two independent noisy observations of C so the C will be the same but the noisy part will be drawn from the same ensemble but another sample and you can ask how the eigenvectors between these two noisy observations are or not similar and then I'll end by some applications okay so in order to be slightly more precise about the type of random matrices I'm going to consider one I've already written and I'm going to call it the free additive noise case so in a generic case I'm going to consider the C as the arbitrary matrix of course of large size so all these results will in principle be that be valid when the size of these matrices go to infinity and then the noise component is made of a sudden diagonal matrix B with the arbitrary spectrum and and some random rotations or transpose okay and you know in a very loose way this is what defines freeness between matrices two matrices are free if you know in a sense the basis in which they are diagonal are randomly rotated with respect to one another and another case which as you'll see is very important is what I'm going to call free multiplicative noise which in this case is considering this matrix C again to be a positive definite matrix and M is now the square root of C the same OB O transpose square root of C so the signal is here and this is this is the noise okay so there are many examples of such problems one is an inference problem you observe M and you would like to reconstruct C as best as you can and a very usual case in this respect is when B is a Vignard matrix in which case OB O transpose is also a Vignard matrix so Vignard matrix means that all the elements are Gaussian independent so this is the most natural noisy case where you add to a sudden deterministic matrix random noise with a randomness on all entries you can think of this perturbation as in quantum mechanics like adding a kind of time-dependent random perturbation to the system and this is considered in in models of quantum dissipations or things like that and then there's the well-known Dyson Brownian motion where OBO transpose poses actually a Brownian noise and in this case you try to study the stochastic evolution of the eigenvalues that's the typical problem that Dyson looked at but it's also interesting to look at the eigenvectors so the arch example in the multiplicative case is when C is a covariance matrix okay so you have observables for example n time series of of length T so as I'm going to show later for financial markets imagine that you have n stocks say n equal 500 if you look at the S&P 500 and you have a certain number of years of data so typically there's 250 days in a year so if you have 10 years there 2500 entries so what you're looking at are returns of these stocks and you're trying to infer from these returns the covariance matrix and so of course what you have is a sudden realization of these returns drawn from a putative true covariance matrix and you try to reconstruct this covariance matrix from observation and so in this case what you observe is the empirical covariance matrix this you don't know and this is usually model as what's called a wichord noise where OBO transform is xx transpose where x is an n by T white noise maybe Gaussian maybe not matrix okay so as I said the problem of knowing the eigenvalue spectrum of these matrices been beaten to death something that's less studied is is the overlap problem and so what I'm going to focus on is this object phi of lambda i Cj which is defined as follows as n times and I'm going to discuss why the normalization in the second the expectation and again I'm going to discuss why we need an expectation here of the dot product or in a bracket quantum mechanics notation ui vj where ui is the eigenvector of m of the noisy matrix associated to eigenvalue lambda i and vj is the eigenvector of c associated to eigenvalue cj so I'm going to consider very large matrices as I've said n is much larger than one and if you really look at this concy for a given lambda and a given c this without averaging this doesn't convert to anything in the large n limit so it's better to actually average these overlaps I forgot the square here so it's the overlap squared that I'm going to call an overlap by the views of language and so this thing if you actually average over small intervals say around a given value of lambda you can fix cj and you average over certain small interval of lambda of width of small very small width but still much larger than one of red in such case there is there are still a lot of eigenvalues in these small intervals then this converges with something that we're going to compute the n factor here means actually that the overlaps are very quickly of all the one over n so if you want as soon as you put up a major a large matrix the overlap with the initial direction gets very very rapidly lost and so it becomes of all the one of ren and you have to look at at this scale one of rent to see something non-trivial and the reason why these overlaps are very quickly small can be into it I mean you can build an intuition of this phenomenon using this Dyson Brownian motion picture that I alluded to because if you look for example that's very easy to derive it's a second-order perturbation theory if you want the evolution of a certain vector when you add a small perturbation to a matrix that that's an eigenvector there's a term here which is deterministic but there's a random term and you see there's a lambda i minus lambda j here which hybridizes with all the other eigenvectors of different indices and so this is a very efficient way to mix and hybridize as I said the eigenvectors so very quickly at each collision of these eigenvalues you lose some coherence and and the the the structure of the eigenvector becomes spread over a finite interval and hence of order n eigenvectors so that's really why you need an end here okay so before telling you the results that that we obtained I need to introduce some pretty standard I mean most of you probably know these objects but some of them are maybe less well known so I'm going to spend a little time on yes I'm averaging over a certain interval I'm not discretizing okay so lambda i lambda j can be as close as the one no so so here what I'm sorry what I'm going to compute is this object here where lambda i I mean in the last slide oh I'm sorry so this is just an illustration of this mechanism of why you lose coherence so this is an exact result but if you look at the equation for lambda i then what you find is that collisions are are avoided so that's that's the you know repulsion of levels but but they can come very close together and when they come close together you hybridize with the rest okay so so the most used tools in random matrix theory is is the so-called resolvent or or many names for this object but the resolvent is just the inverse so m is again our noisy matrix z is a complex number and and so you consider this matrix z times the identity minus m minus 1 so the inverse of z i minus m and from the knowledge of this object you can for example recover the spectral density or the eigenvalue distribution depending on the background physicist or math that you prefer so if you take the imaginary part of the normalized trace of this resolvent computed at lambda minus a small imaginary part and you take the limit where eta goes to zero then this gives you pi times the the density of eigenvalues that around lambda and interestingly if you are interested in the overlaps then it's very easy to show the similar formula if you take the if you bracket the imaginary part of the full matrix now between vi and vi where again vi is the the vector the eigenvector associated to eigenvalue ci then again when you take eta goes to zero you get this quantity becomes pi times the density of states times the overlap the normalized overlap that I talked about so in practice again it's very important to understand that eta is the resolution scale at which you look at your eigenvalue spectrum and so in order for all this thing to make sense and in particular in order to in a sense automatically compute these averages over small intervals around lambda you should take eta to be small formally to go to zero but still within mind the idea that it must be much larger than one of rent so that you know these these these objects actually in a way are microscopes that zoom around some lambdas but with some finite resolutions and this resolution should not be too strong so you shouldn't be able to look at individual eigenvalues but some kind of coarse-grained version of them okay once you get once you have these the still just transform which is again normalized trace you can define more objects so for example you can define so so G is often also called a green function so Tony Z introduced a name for the in functional inverse of G which is the blue function so the blue function actually acting on the G fun on the green function is identity and with the blue function you can construct what's called the R transform which is the blue function minor minus one over Z and you'll see why this is an interesting object and for example if you compute the R transform of a Wigner matrix it's very simple it's just a linear function of Z you can also introduce another transform that's called the S transform again it looks very abstract at this point but you'll see why these objects are interesting so you know it's a little bit more involved you have to define this quantity and then from this T transform there's a formula for the S transform and you remember I talked about Wishart matrices so Wishart matrices are defined if you want as the empirical covariance matrix you would measure on white noise so if you have time series and time series of lengths capital T so in principle the true covariance matrix is identity there's no structure but if you measure on a finite data set this covariance matrix you'll find something different from identity and what you find is is a member of the Wishart ensemble and so the S transform of a Wishart matrix is given by one over one plus QZ where Q is a very important quantity that will come back later is the ratio of the size of the matrix to the lengths of the time series and so for example when T goes to infinity at finite n it means that you have a lot more data than you have time series and therefore in this limit you will recover the fact that the empirical covariance matrix is the identity which is the true covariance matrix so when Q goes to zero it means that you remove the noise and when Q is large there's a lot of noise okay so the main theoretical result that we got with the collaborators I alluded to in my first slide that's published in IEE 2016 is the following is that you can actually compute in the large n limit the green function the matrix green function the resolvent for this noisy matrix M and express it in terms of the resolvent of the pure matrix C and the point is that in order to do that actually there you know you can forget about brackets in the large n limit this becomes true self averaging and but the point is that you're not computing these objects at the same point in the complex plane and this is really what makes the whole story interesting as you'll see in a minute so for example in the additive case so you remember the additive case is this one so it's described by the Sun diagonal matrix B and then these random rotations then capital Z of small Z is given by a self consistent formula which is that capital Z of small Z is Z minus the odd transform of the B matrix computed at point GM of Z but of course GM of Z is going to be the trace of this matrix so this is a self consistent equation defining capital Z of Z in the multiplicative noise case there's a similar formula that involves the S transform of the matrix B so how do we get these results well you can use you know what physicists like to do that is a replica calculation I won't give you details but if you use the replica representation of the resolvent which is a kind of Gaussian integral with zero dimension and use the Harish Andra its X and Zuber formula in the low-rank case then you can pretty easily obtain these formulas what's easy to check is that if you take the trace of these matrix equalities then you recover what are called the free convolution rules and maybe some of you are familiar with with this quite amazing result I think is that if you have free the sum of two free matrices then the spectrum obeys this sum rule in a way convolution rule which is the R transform of the sum is the sum of the R transform and similar in the multiplicative case the S transform of the product is given by the product of the S transform so this is just a consequence of this more general result that holds true for in a matrix sense okay so now we have all the ingredients we need to go back to the main topic of my talk which is the computing this object and as I've said this object is actually hidden in this this dot product of G Vi with Vi itself so okay you turn the crank and one gets explicit formulas for these functions Vi of lambda and C which are though thus these kind of coarse grain average of the overlap squared of the corresponding eigenvectors within mind again the factor n that I emphasized a lot at the beginning so all these curves should you should think of them as is actually very small they're over the over the 1 over n but we blow up to see this fraction so for example if you have a matrix to which you add a random term so for example if B is a Wigner spectrum so it means that the the noise that you add to C is say independent entries with with a second moment that's finite they don't necessarily need to be Gaussian then there's a lot of universality in this problem and what you get is that the overlap is given by this expression so what you see is that there's a C minus lambda squared here so this is a Cauchy like formula as a fun for a given lambda as a function of C or for a given C as a function of lambda with a sun width which is given by this this object so so this is the shape of these functions say for a given lambda as a function of C it peaks around the the unperturbed eigenvalue but there's a there's a dispersion and what you see here are the blue lines are the exact formulas and the dots are numerical simulations to check so you can check for example when when sigma goes to zero so when there's no noise you should expect this phi of lambda C to become a delta function at lambda equal C because in this case you don't put up at all the eigenvectors and that's that's what you recover you you can see that when sigma goes to zero you have a Cauchy distribution of width zero and and the the shift is also zero so this Cauchy distribution becomes a delta function but the Cauchy distribution has also this interesting property that it decays very slowly as a function of the distance between C and lambda and in particular it decays as a parallel so it means that the initial eigenvector actually spreads out very far in that with a with a very slowly decaying tail and this is again related to this kind of Dyson mixing that I that I alluded to before so this formula as I said it is true for all Wigner like matrices it's you don't necessarily have to perturb with a Gaussian noise provided the noise as a second moment this is this is a universal result which is interesting actually there's a similar result for multiplicative noise and for the case of empirical covariance matrices which is the most relevant example then you get again an explicit formula where you see appearing explicitly in the formula the Q factor so you remember the Q factor is the ratio of the size of the matrix the the number of time series that you want to correlate with the number of time steps that you have to measure them and again what you see is that when Q goes to zero which means that you have a lot more data points than you have time series then again this this thing tends to a delta function which is what you expect and this result this explicit result was actually obtained in 2011 in a beautiful paper by Ludwig and Pichier what are your assumptions on the matrix C on the no no assumption no assumption no are we okay well in the identity there's a degeneracy so that there's a problem there but if you have if you have you know generic matrix with non-degenerate spectrum and this is just your slide before just before concern the result itself not not the trace no exactly this is a matrix identity from which you can get a trace result and the trace result are these ones so the trace are for the spectrum and but what's interesting is that if you keep the full matrix structure then you get information on the eigenvectors okay so there's an example which is well maybe a little bit related to your question which is interesting which is the case where C is actually of rank one so there's a unique non-zero eigenvalue gamma all the other ones are zero and you put up this with a Wigner sorry yes a Wigner a Brownian motion WFT a Brownian matrix noise and so what's interesting is that in principle all these strictly speaking all these results are true in the end go to infinity limit but but they still make sense to order one over M and so you can still use this formalism and if you are careful keeping terms to order one over M then you get exact results that have been obtained using different techniques and one of them that I want to emphasize is the so-called by the narrow especially transition which is what happens in this case so again you start from the spectrum where there are n minus one eigenvalues equal to zero and one equal to gamma which is equal to five here so that's at time t equals zero if you want and then you run this Brownian noise WFT and you see how the spectrum evolves and what happens so if you do that without this outlier so called outlier then it's clear that what you have is zero plus a Wigner matrix so you have a semicircle law that appears and the width of the semicircle grows with time and so so this is this is what you see you see actually lines this is a really numerical simulation so that all these lines here they repel each other and they occupy a parabola and inside the parabola the spectrum would be a semicircle so in this parabola you have for each time a Wigner semicircle of radius 2 sigma square root of t and there's something happening to the isolated eigenvalue as well and again this you can grab using the the previous results to all the one over n and you find that the outlier has another dynamics it evolves as gamma plus sigma square t over gamma so it's a small less a straight line and this is true as long as t is smaller than a sudden critical time which in this case is gamma over sigma squared and what happens when t reaches t star is is this star here where the red line crosses the boundary of the Wigner semicircle and in this case the outlier just disappears it's eaten by the by the Wigner C and it's it's in a sense a true phase transition that is called now the BBB transition after Ben-Aruz and Pichet so for t greater than t star there's no isolated eigenvalue anymore it's just the the semicircle so you can ask also so this as I said you get from the r transform formalism to order one over n but you can also ask what happens for the overlaps and for the overlaps what you find is that the overlap of this isolated eigenvector with itself with what it was at time t equals 0 is equal to 1 minus t over t star so you see when t is 0 the overlap is 1 because you haven't done anything and when t goes to t star then it's the last moment where the isolated eigenvector remembers the structure it had at time t equals 0 after that it's completely lost in the Wigner C there's actually in an informatic information theoretic way there's no way of recovering the information contained in the eigenvector in the initial eigenvector after this time or when the the amount of noise is beyond some critical value what happens just at t star is to my knowledge not known or not written in one I mean if we do hand waving arguments you can conjecture that precisely at t star the overlap goes down to 0 as n to the minus one-third okay so now I'm going to try to show how you can use these results for something concrete and the question I'm going to ask is an inference problem is you observe a random realization of this noisy observation m and what you would like is to have the best estimate of c of the true signal c for example the true covariance matrix or the true matrix to which you add some noise knowing m and what I'm going to assume here is that the problem is rotationally invariant in the sense that there's no information that you have about the problem that allows you to privilege some directions in in space so you have no idea of where the true eigen vectors of c are pointing in space and so in this case the best you can do is to assume that your estimator is going to have the same eigenvectors as the noisy as your observation because this is the only thing that breaks the invariance is the observation of the covariance matrix or the noisy matrix m so you're stuck with these directions if you want there's nothing else you can do except if you have some information about the problem of course you might have some information about the problem that tells you that some directions should exist for symmetry reasons or you know but here we're assuming that there's nothing that the prior that you have there's no information about the directions in the n-dimensional space so the best you can do the only thing you can do is to write that your best estimator is going to be the sum over the eigenvectors of the noisy observation again that's the only thing you can choose times some pseudo or a dressed eigenvalues if you want psi i that you have to determine okay so now if you if you try to minimize the distance between the L2 distance between psi and and c so how you know what's was the closest you can get to see using this representation then you find conditions that fix these psi i and these psi i's are given by this equation so it's a weighted average of the true eigenvalues of c multiplied by the overlaps but you stare at this formula and you you see that's there's something completely stupid about this formula which is that I want to know c so if I express psi hat as a function of things that I don't know this is called you know where the oracle estimator you've done nothing useful because you want to have this best estimator psi hat as a function of things that if you knew them you would not need to estimate them so this formula looks to be empty and what I'm going to show you is that for some kind of miraculous reason in the large end limit you can estimate these objects without knowing see just knowing what you observe okay so here there's a little bit of transformations so this formula you first re-express it as in the large end limit as an integral rather than a sum and so these overlaps appear like like like the object I've defined before and it so happens that you can re-express this integral like this and by using the rules that I've written here in the end you get a result that where c has disappeared only m appears so for example in the additive case you get that what you should do is you take the empirical eigenvalue that's lambda i you've observed this is the eigenvalue of m and you transform it using a function f1 in order to get this this dressed eigenvalue psi hat and this function only depends on objects that are known once you've observed m and you know the structure of the noise so for example here is the the real part of the of the still just transform the hillbett transform of the spectrum and and you see alpha 1 and off and beta 1 are kind of complicated functions but they only involve m and and the noise rb is again the r transform of the of the noise matrix so everything only depends on on m and of course you have to know something about the noise so every if everything is Gaussian what you get is the well-known formula that when you observe Gaussian so one-dimensional Gaussian variables if you know that you're observing a Gaussian variable with noise the best estimator of the of the variable itself is the variable with you've observed times the signal divided by signal plus noise so so this is the one-dimensional version of this complicated formula if you want in the Gaussian case so in the multiplicative case there's a similar formula that again only involves m and the structure of the noise so gamma b and omega b are the equivalent of these transforms here and so in the empirical a covariance matrix case you get this explicit formula f2 of lambda how you should renormalize the empirical value to get the the estimator is lambda divided by this object so again you see that when q goes to zero you don't have to do anything because you observe perfectly the matrix and if q is non-zero you know what to do so this is an example of this function f2 this is the observed lambda this is how you should change the lambda to get the best estimator it looks very close to a straight line which in this context is called the linear shrinkage so for a long time people have proposed to transform the eigenvalues in a very simple way just in a linear way in this kind of very simple fashion but that you see that in it's actually in general more more complicated okay now what's I think maybe even more interesting for some application is that you can extend the techniques I've explained to you for another problem which is not the overlap of the matrix of the eigenvectors of m with those of c but the eigenvalue vectors of m with those of another realization of this random matrix problem m prime so here w and w prime are two independent realization of the noise and what I'm looking at here is this different overlap phi between the vector close to the eigenvalue lambda of the first realization and close to the eigenvalue lambda twiddle of the second realization okay so I think it's pretty explicit and again using the kind of techniques that I've told you about you can actually get rather cumbersome I haven't even there to write the full formulas but explicit formula for this five lambda lambda twiddle for the additive case or form of multiplicative case so for example you know it's even if alpha and beta I've not shown them but there's a function phi of q q twiddle these two matrices may not even have the same signal to noise ratio and n over t ratio in the multiplicative case so so there's a formula I mean which depends on on the parameters of the problem and if you want to imagine what these formulas look like then this is for a fixed lambda twiddle the overlap of the eigenvectors of one matrix compared to those of the second one it peaks around lambda twiddle but it has a sign again a kind of cushy shape so the red line is again the theoretical formula and the dots are numerical experiments but what's really cute I think is that this formula this that I haven't shown this phi of q q twiddle in the multiplicative case or the analog formula for the additive case it doesn't depend on C you just need to know mmm prime which is your which you observe and from mmm prime from the knowledge of mmm prime the the spectrum you can calculate what you expect to be the overlap between lambda and lambda twiddle I mean the overlap between the eigenvectors corresponding to lambda and lambda twiddle assuming that they both come from the same ensemble so assuming they come from the same ensemble then you're able to compute this function without knowing C so it's a kind of strange result right you're trying to test whether m and m prime are coming from the same ensemble but you don't know this and you don't know the generating matrix C so it can be used to test actually whether your two observations originate or not from the same unknown C and the formulas that you get are interesting because they are universal you don't have to assume much about the noise so for example if everything was Gaussian then you could have other ways of testing this hypothesis but what makes the result I think quite interesting is that you don't have to have a lot of knowledge about the structure of the noise to get in the large and limit these universal formulas so okay so it's interesting to apply these this idea to financial time series because it's very hard to imagine a priori that the financial markets are the same between say 1990 and 2000 and between 2000 and 2010 so so it's interesting to see whether if you measure empirically the covariance matrix on say 10 years and then on 10 subsequent years can you test whether the true covariance matrix that you don't know you only know the observation and by the way if you see that this quantity Q which determines the amount of noise the the lack of knowledge that you have about your problem it's very very difficult to have small values of Q again Q goes to zero is the perfect signal case but in this present era of big data say if you know you consider say I said 500 stocks in a in a portfolio of 500 stocks in the S&P 500 and I said 10 years 10 years is 2500 points with daily data and so this ratio is point two so it's not very small so you're clearly in a in a high noise regime because because you have a lot of objects so so this is the overlap that I just talked about at the same value of lambda in principle you could make tests for different values lambda and lambda twiddle I'm specializing this formula to the case where lambda equals lambda twiddle and Q equals Q twiddle and in this case what you get is so if you zoom on the part of the spectrum where most eigenvalues of the covariance matrix lie you get the empirical points are the green points and the prediction okay don't worry about the the fact that there are two lines there's some slight subtlety here that maybe if you're interested I can tell you about but look at the red line the red line is is not bad it's not bright it's not exactly on the data points but as I said it is a little bit of ambiguity here on the value of Q that you should use and again for reason I can explain but suddenly if you zoom out and look at the largest eigenvalues of the covariance matrix the predictions the theoretical prediction if the covariance is the same in the two time periods so I don't remember here if it's probably it's 2000 2010 and 2010 and beyond then you see that there's a very strong discrepancy the overlap is much smaller than what it should be if the true covariance matrix hadn't changed so the conclusion is that the large eigenvectors are unstable in a sense it's not a problem of noise it's not a problem of observation it's really the structure of the underlying problem that makes the eigenvectors evolve with time and this is a conclusion that we had obtained already we're using different methods with hall-mallez and what you have to realize is that this is extremely important for portfolio optimization the reason is that these large eigenvalues in terms of risk they correspond to portfolios portfolios define direction in space and these directions are the most risky ones this is the risk associated with the corresponding portfolios and so if you think that you're neutral in the direction of largest risk but these directions of largest risk shift with time then you're exposed to kind of unknown risk that were not in your risk model so here you know I've shown you a kind of eyeballing test it's just by the eye but it would be interesting to turn this into a true statistical test and this we haven't done and in particular there are subtleties about the hypothesis this change of value of Q comes from the fact that there are it's it's not a good it's not a very good model to assume that the data is completely white noise okay so I told you that this overlap function between two realization of the same realization of the same ensemble is an ugly formula that I haven't written but there's a simple interpretation of this formula which you you probably will understand intuitively so remember what I call now phi 0 phi 0 is the overlap between m and c so this is these are the objects that are considered in the first part of the talk phi now is the overlap between two different realizations and so what you can write by definition is that the eigenvalues of the noisy matrix is a superposition of the sorry the eigenvectors of the noisy matrix are a superposition of the eigenvectors of the the underlying pure matrix v mu with the square root of the overlaps which gives you the weight of u lambda on v mu and and random sine factors epsilon mu lambda and then you have to integrate over all muses with the corresponding eigenvalue density so if you compute the overlaps between two eigenvectors corresponding to different and realization of the same problem then you get something like this and now if you if you make what's called in this field the an organic hypothesis that is that all these phase factors all these signs are completely random as soon as mu and lambda are different then by by taking the average over this you get this kind of triangular convolution formula which is that the overlap between eigenvectors at lambda and lambda prime for the problem of MMM prime is a convolution of the overlaps of the pure matrix with one of them and the pure matrix of the other which I think is kind of triangular equality that that is intuitive but if you use this formula then then it appears to depend on c because for example there's this explicit density of eigenvalues rho c but as I've shown you the in the end the dependence of c on c disappears for five okay there's maybe I should do how long do I have okay okay including questions okay so I have time for so this formula here as has an interesting consequence for applications which is the following so I'm again going to reconsider this problem of of the estimation of my random matrix again I'm coming back to this problem here okay and this formula here and what I'm going to show is that using this formula you get a representation that's very useful empirically and that makes again a lot of sense so the idea is to consider a first realization of your problem with eigenvectors ui and then you take a second realization independent realization with I'm sorry I should have called it M here so this is M prime which is an independent realization of the of the covariance matrix and what I'm defining are these object new I of q which are the projection of this independent realization on the vector corresponding to the first realization bracket bracketed with ui itself so again all this can be measured empirically and what you get using the convolution formula is that this new I of q is actually exactly given recovering psi I hat so this is called a kind of cross validation or out of sample estimator of psi hat where in a sense you don't even need any formula you just compute this with an out of sample or a cross validation a period to determine the covariance matrix C and so that's how it looks like if you do this experiment numerically this is real financial data and and so all these points here are point-wise estimation of this formula but when you average what we call here the this oracle estimator you get the the the blue triangles and the formula the analytical formula that I told you about is the is the green one so this is this is an easy way if you want to estimate these psi hat that you need to use in your L2 estimator of the covariance matrix okay so let me conclude and and give a few problems that I you know at least I find them interesting so I try to tell you that the free random matrix results on the R transform and S transform for the still just transform can be extended to the full resolvent matrix and this gives access to overlaps and these overlaps formulas lead to what I've called here large-dimensional miracles that the the so-called oracle estimator that when you write it down you have the impression that you're writing something silly because you're expressing things that you would like to know in terms of things that you don't know so but but in the larger limit it turns out that you it can be estimated and in particular this hypothesis large matrices are generated from the same underlying matrix C without knowing see this can be tested and as I told you it's very useful in some cases you have a physical model so you you may have some information about C but in the case of financial markets for example there's nothing really that you can use except the data itself so so as I said it's an eyeballing test at this stage I think it would be interesting to turn this into a true statistical test large n something interesting that we've done recently with the floor on the nation and mark borders is to extend this to the SVD of more complicated covariance matrices so in the present case I've only considered square symmetric matrices where you correlate n objects with themselves but it's clear that in many empirical problems you want to correlate n objects with m other objects that may have absolutely nothing to do so for example maybe you've heard about this Sunspot problem where stock markets are supposed to be correlated with the activity of the Sun so you could you know correlate anything with anything else so in principle in general cross correlations have n rows and m columns and it's interesting to ask the same questions when n and m goes to infinity and in particular how to clean these cross correlation matrices so since these are rectangular matrices you don't you shouldn't you I mean the notion of eigenvector doesn't eigenvalues and eigenvectors don't make sense directly so you have to use an SVD representation but the this are the rotation invariant estimator that I told you about I said en route but actually now we've done it so we have an explicit formula that generalizes by f1 and f2 formulas these ones for the case of the singular values of more generate general correlation matrices what's interesting is that I've insisted on the fact that there's a Dyson motion description of the additive case we believe that there's also a Dyson motion description of the of the multiplicative case but for the moment where we haven't been able to I mean again there are leads but we haven't completed that project yet something that I find interesting as well maybe there's no nice solution but it seems that it's possible at least in some sense I mean there are already results showing that some of the free algebra cannot be generalized but I think there might be interesting interpolation schemes between the trivial case where you add two matrices that commute in this case it's obvious that you don't change the eigenvectors and you just add the eigenvalues and the free case where you have another rule for addition which is more complex but explicit and and you see I've tried to insist on the fact that freeness means that the the direction of the eigen space are completely independent from one another maybe there are ways to introduce some correlations between these directions that make some some generalizations interesting but although we have very preliminary ideas I don't have much to say I think just an interesting question and an obvious thing also that you can ask is in some cases there is a prior information on eigenvectors how can one use it to go beyond this this rotational invariant estimator for example in the case of financial markets it is clear that and actually Mateo worked on that a long time ago that the largest eigenvalue the one corresponding to you know actually in in this scale the largest eigenvalue is out of the window here and it corresponds to what people call the market mode that is a portfolio made of say one over square root of n on each on each stock so you take an exposure to the market to the market with an equal weight so there's a prior which is just you know one one one one the largest eigenvalue should be around this direction and and and those those those here are related I mean you can read some economic structure information in these in the structure of these ones related to economic sectors for example you know energy or banks or things like that so so it's not true that we have no prior information but the way to marry this prior information with this idea of the rotation the invariant estimators is I think to be worked out and on this side finish thank you questions yes okay so so for example something that you can do is to extend the wishart ensemble in cases where you have parallel distribution and what you see is not it's pretty obvious that you can be dominated if you compute a covariance and that the returns are very large on some days then even if there's no correlation but these two objects happen to go in the same direction the day when things move a lot then you're going to find some spurious correlation because everything is coming dominated by this so there are priors that you that you can use to account for the tails or account for the fact that the volatility of the markets is itself changing in time there are periods where the velocity is large and periods where it's smaller but you can extend these wishart ensembles to account for for these non-stationarity if you want on this so you use maximum estimators for the true eigenvalues of C that exactly coincide with the eigenvectors of C which coincide with the eigenvectors of M yes but then you show that there is a huge utilization so what I mean the question is that's why because of this arbitration that's why you you have a non-trivial result folks I had yeah so and this is exactly what in the end tells you that the best estimator is not to take the empirical matrix that you've observed it's to take only its eigenvectors but you have to shift the eigenvalues by some quantity because of this because of this mixing so essentially so this mixing tells you that so eigenvalues have satisfied some laws of large numbers yes yes yes absolutely but still you have you are choosing essentially the same eigenvectors so yes because because there's no prior because you can't do anything else because you have no prior that's the hypothesis of this rotation invariant prior what else can you do you know imagine what you get the same you know the Bayesian approach tells you that the only thing you can do is to take you as the as your prior for the eigenvectors of C again as I said there's nothing else you can do the only thing you can play with is not the eigenvectors is the eigenvalues that's the only thing you can do because of the rotation invariant assumption and that's what I was saying in the end if you have more information then the problem becomes maybe more interesting but also more complicated because you break the rotation of rotational symmetry but that's that's exactly the point and you're right I mean there's a law of large numbers that means that in the end you only if I mean that what's really nice is that M the noisy version of C contains enough information to compute these overlaps even if you don't know see but that's because all these results are true in the large n limit they're true only so you have a lot of information actually so couldn't you modify a bit your your prior I mean this this formula for potato I mean the previous page by mixing by mixing a bit the eigenvectors ui which are close which have closed by your eigenvalue somehow this here is I mean this this estimator is diagonal in this basis diagonal matrix in this basis of the UIs could choose somehow mix a bit include off diagonal terms but only near which are close concerns eigenvectors which are closed by energy well I think that in this case it means that you're you have a prior that that is not rotation invariant so that's what I said that if you have a rotation invariant prior then the maximum likelihood is is diagonal but I'm not saying that in concrete cases this my this could be a good idea but I think that's what's interesting in my opinion about what I said at the end trying to extend this to non-rotational invariant ensembles and see what you can do so yeah so here there's a little bit of numerical experiments to optimize this what we found is that so I said it should be large compared to one over n but small and so one of the square root of n seems to be a good compromise can we expect that each of the component is converging to some distribution to some Gaussian or as it is for the the eigenvectors of GUE I think in the in this rotation invariant case yes


Welcome viewers to our ongoing program nuclear free future conversation coming to you from Burlington, Vermont, from Tom meeting TV channel 17 Center for media and democracy. I'm your host Margaret Harrington, and viewers this is part two of our program 75 years of nuclear fallout from Hiroshima to now. And let's welcome back our guests Alfred see Meyer from positions for social responsibility, and Kevin camps of beyond nuclear. Welcome back, Alfred and Kevin. I'm. Yes. So we're continuing this, this conversation about the nuclear fall out from 19 from August 6 and August 9, 1945. So can you just give us a little recap of what we were talking about in part one, and we'll continue on the subject. Part one, we began to tell the story of how, starting with the mining of uranium in the Belgian Congo, great harm was done to the miners, great health problems for the communities and the people in them. And that this led to this huge industrial endeavor, which led to the bombing, the incineration of two Japanese cities, Hiroshima and Nagasaki. Kevin mentioned that initially, these explosions like the very first atomic weapon or atomic explosion the Trinity test. The Trinity test was used to be listed as the first test in the second and third tests where the two bombs dropped on Japan, because the first one in Hiroshima was a uranium bomb. One kind of design and the second one was a plutonium bomb which is entirely different kind of mechanism. So we were field testing these weapons. And we talked about that there are strong arguments that say they were not needed to end the war, that Japan was crying to surrender. The sticking point was whether the emperor could have any rights or respect. And then also we wanted to rush the peace process to keep the Russians from taking part in it. The Russians had a lot of scores to settle with Japan they very much wanted to be there, but we saw Russia as a real enemy so using those weapons, in one sense was a real message to Russia. And we talked about some of the accidents, like the mind tailings breach of the dam in Church Rock, New Mexico, which was a the biggest release of atomic material in this country. But this whole endeavor is very costly and very damaging, even if we never use another weapon. And we now we also mentioned that after going through a period, well we didn't talk about this but I'll mention it now. At one point, you know, we had the Cold War and at one point between Russia and the US there were 72,000 nuclear warhead. And then we realized that this is horribly dangerous. There are many treaties signed international treaties bilateral treaties summer multilateral the nuclear non proliferation treaty, the anti ballistic missile treaty, the comprehensive test man treaty, all these different means to try to control this genie to kind of at least reduce its size even if we can't put it back in the bottle. Starting with Reagan Reagan and Gorbachev and Reykjavik almost got rid of nuclear weapons and it's to me so tragic because the sticking point was Reagan insisted on his missile defense the so called Star Wars program which Russia I think rightly saw as an offensive missile, you know, weapons system, but because of that we did not get rid of these things and so now we're at a point where Trump is pulling us out of treaties and pushing for a huge new nuclear arms race. So not only, and to show you how insidious this is I want to mention that when President Obama was in office. He worked very hard to get the start to treaty pass strategic arms. Reduction. New starts. Yeah, the new start. And he was successful in that but there was a price he had to pay and the price was agreeing to a $1.7 trillion, he isn't trillion dollar refurbishment of our entire nuclear weapons complex. And to me if you're refurbishing your entire nuclear weapons complex, I don't think you're getting rid of nuclear weapons, even though a recent US Department of Energy report says exactly that it says that the only way we can have non proliferation is if we are dominant in nuclear weapons. And if I can't think of anything much more or well Ian and double speak ish than that. So we're at the precipice of this huge dive into new weapons, more weapons that there's not the threats, you know, the wrong instruments. Kevin, do you want to add anything else to kind of summarize where we were and then the point I would make in handing it over to you is to say that if we're going to build all sorts of new weapons, more usable weapons, oddly enough, that it's going to make a lot more nuclear waste and we'd have a clue as to what to really do with the waste we already have as Kevin always points out the very first couple of nuclear waste created under stag field at the University of Chicago in 1942, still has not been disposed of properly. I'm just reminded of the Hopi prophecy. We've spoken a lot about New Mexico and the Hopi have ancestral lands in New Mexico. In fact, the whole tech international dump site targeted at southeastern New Mexico, the Hopi have land claims down there, even though it's far removed from their current reservation and in Arizona. The Hopi prophecy is quite incredible. It came from the creator and the creators original instructions to the Hopi. And the atomic bombings of Hiroshima and Nagasaki were assigned to the Hopi to share this warning with the world from their ancient teachings from the creator. And in their prophecy, the atomic bombs were referred to as a gourd full of ashes because they had no concept of, you know, nuclear weapons per se, but a gourd full of ashes falling from the sky, turning all the land into ashes. And on part one, Alfred spoke about the radioactive ash that fell for days after the Trinity blast in south central New Mexico. So the Hopi warned beginning in 1947. They warned the world. They attempted to warn the United Nations General Assembly in their prophecy. It was called the House of Micah because the Hopi didn't even have glass. They knew of this House of Micah, this glass building in New York City and their ambassador, their spiritual ambassador for the spiritual elders, Thomas Baniakya Sr. Traveled to the UN time and time again, knocking on the door of the House of Micah, trying to speak to the General Assembly. A warning from the Hopi, actually a warning from the creator that you human beings are at a crossroads. And if you take this one road, it leads to chaos, catastrophe and an end. And that is the nuclear road. If you take this other road, it's a better outcome. And the creator would like us to take that other road. So it's just incredible wisdom and a warning to the world that we had stopped, we'd better stop going down this road. And Alfred mentioned this essentially $2 trillion US nuclear weapons renovation plan. It's madness. I know there's a lot of folks addicted to this nuclear money. New Mexico itself is addicted to this nuclear money. Los Alamos National Laboratory, Sandia National Laboratory are addicted to billions of dollars per year of not cleanup money necessarily. They have a big mess to clean up out there. They are addicted to nuclear weapons, manufacture money. So unfortunately, the Trump administration is planning plutonium pit production at Los Alamos National Lab. It's new weapons for new military purposes, a violation of US commitments under the non-proliferation treaty to disarm, to abolish its nuclear weapons. The NPT is 50 years old this year. It sure has taken us a while to abolish our nuclear arsenal. We're not the only ones, though. Russia is still gung-ho about its nuclear arsenal. They are making insane advances, so-called, in nuclear-powered, nuclear-tipped cruise missiles. And they had an accident with one in August of 2019 that was a significant nuclear accident, a significant radiological release. People were killed, and we had better learn as the Hopi have warned. We had better learn before it's too late. And don't take it from me. Take it from Henry Kissinger of all people who got together with George Schultz and some Democratic colleagues, former secretaries of state, William Perry, former Pentagon chiefs, warning back in 2007 and numerous times since, they're called the Four Horsemen of the Nuclear Apocalypse. These were nuclear war chiefs for the United States. These were the top officials in the U.S. government who relied on this so-called deterrent policy to assert U.S. interests around the world for decades on end. They warned that there is one existential threat to the United States. It is nuclear weapons. It is nuclear weapons proliferation. Better abolish, we had better abolish nuclear weapons before they abolish us to echo John Kennedy. Well, then talk about the treaties like the non-proliferation treaty that was trashed by President Bush in 2004 with handing over nuclear power information to India, which never signed the nuclear non-proliferation treaty. And that's another subject about the nuclear powers which include India and Pakistan and the fact that India itself never signed the nuclear non-proliferation treaty. Margaret, I'll take a moment to mention a study that PSR did a few years ago called Nuclear Famine. And it is a book about what would happen if India and Pakistan, for instance, had a limited small exchange of nuclear weapons. Nothing like the vast stockpiles that the U.S. and Russia have, but there's just a little nuclear war. This would not only cause great devastation if the people were bombed and irradiated, but it would loft so much dust into the atmosphere from the fires and the explosions that it would reduce agricultural output worldwide by 10% for at least a decade. So more than two billion people are likely to starve to death should there be a small nuclear conflagration. And the treaty is even like the NPT. I mean, it's such a fraught topic in my mind because the NPT treaty is about the only one left and at least has people talking in some degree of oversight and exchange of information. But it has the provision which allows for civilian nuclear power. And in part one, we talked about how civilian nuclear power is essential for nuclear weapons. And to illustrate that point, I would look at a while you could you mentioned India. They had a Canadian can do reactor and for peaceful purposes. And that's how they made their first weapons. So it's, you know, saying that the countries have the inalienable right to nuclear power. And another example is Iran. Iran's just claiming we were just taking our, you know, treaty given right there. But why are we so upset? It's because this is the path to nuclear weapons. It's the same infrastructure. It's the same science, the same technical capacity. There's just slightly different paths right at the end. And Alfred mentioned in part one that Eisenhower gave his Adams for peace speech in 1953. And another amazing book by Arjun Makajani is called the nuclear power deception from 1999. And the thesis from my read of that book is that this Adams for peace speech the smiley face propaganda was really a cover for nuclear weapons development because if you think about it when he gave the speech in 1953 there were no atomic reactors for electricity in the United States. Yes, the US nuclear Navy under Rick over would build one in shipping port Pennsylvania in 1957 the first so called civilian reactor built by the US nuclear Navy, a submarine reactor designed built on land. But reactors didn't start to come online until the early 1960s really they didn't come online and any numbers until the 1970s. Where was all that uranium going while it was going into the arms race of the United States it was the nuclear weapons build up with a Adams for peace cover story. And here we go again I mean Alfred just mentioned it. If you know if thorium reactors proliferate internationally if small modular reactors proliferate internationally the new catch words of the industry. It's going to give nuclear weapons capability to these countries and Saudi Arabia is a really concerning one right now. The crown prince on 60 minutes said right out loud if Iran develops a nuclear weapon will be right on their heels doing it ourselves. They would use American or Korean nuclear power technology, whether uranium enrichment or plutonium reprocessing to build a Saudi nuclear arsenal. Talk about a rough neighborhood I mean Israel already has nuclear weapons. Iran has said it does not intend to build nuclear weapons but with the technology could, if it chose to Saudi Arabia could do the thing. The same thing. Alfred mentioned another tough neighborhood in the world India Pakistan and that study that PSR and IPP and W did was about the exchange of 100 Nagasaki sized bombs, causing a nuclear famine that would kill 2 billion people from starvation. So, we're really playing with fire and I mentioned those cold warriors like Kissinger have said we need to abolish these things before they abolish us. Kevin, show me what shirt you have on there and what is the significance of that. Yeah, I wore it when Diane Drigo and I were on your show this is my whole whole tech t shirt. And it is the latest environmental injustice the latest nuclear racism to afflict New Mexico. A majority minority state New Mexico is a majority of Hispanics and Native Americans and the white population is the minority in New Mexico. So guess what the state that's already suffered so much from nuclear since 1943 when Los Alamos moved in now is being targeted for a high level radioactive waste dump. This company called whole tech international headquartered in New Jersey wants to dump 100,000 metric tons or even up to 173,000 metric tons commercial radiated nuclear fuel in southeastern New Mexico in a Hispanic area that as I mentioned has Hopi connections, the Mescalero Apache are very nearby. In fact, the Mescalero Apache 25 years ago were targeted for this very dump. They didn't get away with it that time because they were stopped by traditionals like Rufina Marie Laws and Joe Geronimo who led the anti dump effort. So this time around groups like this t shirt Alliance for environmental strategies, a Hispanic environmental justice group are fighting with everything they have joined by groups across New Mexico and around the country to try to stop this dump. And the fight is very much on. We have a public comment deadline on the draft environmental impact statement of September 22. And I should hasten to add that Rose Gardner, the co founder of Alliance for environmental strategies lives in Eunice New Mexico, which is right on the Texas border. So this whole tech dump is about 3540 miles from rose, the Texas dump, another consolidated interim storage facility proposal for high level waste is about five miles from rose at waste control specialists in Texas. It's also called interim storage partners. So that neck of the woods within 40 miles to of these high level radioactive waste dumps that could easily become not temporary or interim but de facto permanent are targeted at this area. And we have a national coalition of environmental and environmental justice and anti nuclear groups fighting with everything we have to try to stop this major radioactive racism that is targeted at that area. And part of the problem with what the waste situation I would argue is the lack of congressional respect for science because Yucca mountain, which was slated to be our geologic repository was chosen, not through the original scientific process outlined in the 1982 Nuclear Waste Policy Act, which called for an assessment of 10 sites and one east of the Mississippi one west of the Mississippi in 1987 when Congress amended that act. They just simply up and chose Yucca mountain is the place and Yucca mountain is an area that's built out of volcanic tough. It's a porous rock. It's in an area of seismic activity. It's in an area of volcanic activity, and it's over an aquifer. I mean, it's ludicrous that this would be chosen as a place to keep something safe for millions of years. And Nevada had little political power back then, and it was, they just were chosen by the wisdom of the congressional votes, not by science. So really Yucca mountain I think will never open. But that is relevant to what Kevin's talking about about centralized interim storage, because how do you define interim. There's, if there's no other place to put it. It's, as Kevin mentioned, de facto permanent. So, and then we so people really should follow the web links to letters and how to make comments and just express yourselves because it really is, you know, the few, the many of us against the few of them who are in control. And if we're not active will soon be radioactive. The other thing to bring up is that as if this isn't bad enough. And this, my hope is that maybe the ludicrousness of this proposal will actually cause action about it. But the Nuclear Regulatory Commission is in the process of in the back room rewriting a little regulatory rule, which would basically allow vast amounts of radioactive materials being left behind when you shut down a nuclear reactor nuclear power plant. And instead of making it go to one of these designated regulated dumps, like Yucca mountain, or even these consolidate interim sites that this radioactive material could be put virtually any place it could be put in your town's transfer station just down the road. Now clearly this is a great boon to the industry. The industry was supposed to be saving decommissioning money. So they have huge pots of money that's going to be used for this very expensive complicated thing. But if this very low level waste rule is changed. This is the entire game. The industry can just dump whatever it wants wherever it wants. And as mentioned before, radioactivity is tasteless. It's odorless. You need highly sophisticated equipment to measure it so how do you know what's in your landfill down the road. This is just, you know, it's part of Trump's effort to completely deregulate things and just dump whatever you want where you want. And this is another issue that we need to be in touch with our state attorney generals. Our state political parties are federal legislators, legislators and tell them do not do this. We must control this matter as much as we can. We've already greatly solid our home, the planet Earth. Let's let's not go whole hog on it. And just this morning I heard from here in Vermont from representative Welsh and from Senator Sanders, and it was about the, the, what I signed from the, the resource services the diandere goes organization, nuclear information and resource. And near near is net right and IRS net. And that was a response from from both of them from Congressman Welsh and Senator Sanders, but it was not a definitive condemnation from either one about the, the putting of very low level nuclear waste into our local landfills. I'm sorry to say. I'd encourage your listeners and viewers to attend a meeting of the Vermont nuclear decommissioning community advisory panel that is scheduled for Monday, September 1 21 Monday, September 21 beginning at 6pm. This is a very important meeting of the end of cap. It has to do with Vermont Yankees decommissioning and these issues will come up very low level waste being allowed to be dumped in local landfills with with no regulatory control, but also these issues of consolidated interim storage facilities, whether in New Mexico or Texas, the Yucca Mountain dump, all of these issues will likely be discussed, because unfortunately, the end of cap back in 2015, five years ago, took a position in favor of consolidated interim storage facilities. And they've never corrected that position. So, as we've discussed, this is an environmental injustice, consolidated interim storage in Texas and New Mexico, and they really need to correct their position because they are speaking on behalf of Vermonters. And already the Texas dump is a national low level radioactive waste dump located above the Oglala aquifer. Vermont waste is going there. In fact, Vermont and Texas are compact. They really run the show out there, but they've opened the doors to the entire country. And it's already an environmental injustice that already threatens the Oglala. But this high level waste would make the situation much worse, as would very low level waste being dumped in local landfills or incinerated or even recycled or poured down the sewer. And companies like North Star at Vermont Yankee, which is closely connected in fact is the same consortium as runs waste control specialists would very much like to save a lot of money on disposal costs and simply pocket the leftover money as pure profit. That is their business model so it's going to take the people of Vermont, it's going to take the people of the United States stopping these insane proposals that put public health and safety in the environment at risk we have to get very active. And as Alfred said contacts your elected officials at all levels federal state and local. Okay, gentlemen, are we are we ready to wrap up part two of our discussion. I with I am. I'm filled with a fear of the future, given our past. And I am very grateful grateful to both of you and the organizations that you represent for bringing your message here to Vermont and because we are online, of course, we're all remote at the moment, but we are online so we have an international audience to so. Thank you very much and and please would I'll give you both the floor to to to say, you know, to wrap up the show and we'll look forward to part three at an at another time but meanwhile we'll go through the 75th anniversary of Hiroshima and Nagasaki 75 years. So thank you, Kevin Alfred, and I'll give you both the last words. Thank you, Margaret I am empathetic with your sense of foreboding and fear and looking at the horrible parts of our past history, but I take great support or encouragement from the current huge social justice movement that's a foot in this country. And that's why I've tried to say how much so much of this atomic effort the nuclear weapons nuclear power is similar to racism and and is in itself racist. And that it's again it's a call for the majority of us is this like to take action to stop a tiny minority who are endangering all of us terrifically damaging all of us terrifically and making a pile of money to boot that you know this is a very bountiful world. And we should be able to provide housing and health care and food for all. You know, there can still be some rich people they won't be quite as rich, but you know that I think it's the time for us to act and I feel personally that the only reason we're able to be here to talk today is that the vast majority of people, in fact are very good, honest and honorable people. The nuclear world is filled with stories of near misses of people who didn't do what they're supposed to do because they didn't want to blow up the world. And that's why we're here so it's nip and tuck. And you know with all these weapons and high alert it could take just a moment to happen. But I also have great hope that we all through the like your efforts of putting this show on the air. And getting this message out that the people who are listening to it will take action and we will make these changes. Thank you, Margaret. Thank you. Thank you. Kevin. Well, Margaret, I noticed the Helen Caldicott book behind you. And I remember hearing her speak once and she quoted a founder of the civil rights movement who said, when you see a good fight, jump in it. And like Alfred said, we are at a moment of national change for the better with the George Floyd movement. So, you know, with Trump proposing a resumption of nuclear weapons testing on Western Shoshone land in Nevada for the first time since 1992, full scale nuclear weapons testing. With all these dump fights, whether it's Yucca on Shoshone land or these dumps in New Mexico in Texas. We have to, you know, folks in Vermont have to get clearly through to Peter Welsh that voting in favor of these dumps in New Mexico, Texas and Nevada is not okay. It's an environmental injustice and on that reason alone is not okay. He needs to stop doing that. So, Vermonters have a real political power right now with Bernie Sanders advising Joe Biden closely. So, on nuclear weapons issues as Alfred has said defunding nuclear weapons like defunding the police and redirecting that funding to human needs and to justice and to reparations. And not being in touch with their congressional delegation right now would make a big difference in a lot of ways. So I just encourage people that when you see a good fight, jump in it. So be there for the end of cat meeting on September 21st, get those comments in on whole tech by September 22nd, get those comments in on Texas by November 3rd, and change the world. That's what we got to do, change the world for the better. Thank you.


Hello and welcome to the Grand Line review, your source for everything on peace, except for maybe today, because we are celebrating another monumentous occasion. As for whatever reason, a mighty 200,000 of you have ended up subscribing to this channel, which to be fair, does inject regular on peace content straight into your YouTube feed. And if you have not already hit that button and that bell thing, then do it right now. What have you got to lose? Nothing, that's what. What have you got to gain? Everything. But I'm incredibly overwhelmed because to be perfectly frank, I never thought we'd get anywhere near this far into the whole YouTube game, especially with such a focused niche. So as a result, we are going to be breaking our one piece focus for this video. Don't worry though, it will be back soon. And in copious amounts, I assure you that. But to mark this occasion, I wanted to let you all in on the back end of this channel for a bit. And quite specifically, I am going to be going through some of my biggest YouTube regrets, which I know doesn't sound particularly celebratory, but I wanted to do something a bit different than just like a thank you video or a Q&A. And maybe if some of you were considering starting a channel or even have channels already, then this may help you not fall into some of the many, many, many holes that I did. And also stay tuned because after I go through my biggest regrets on this platform, I'm also going to drop what I would describe as the one crowning feature of this channel success. Something I did that a lot of my contemporaries will disagree with, but I just don't think I would be here today without it. So look forward to that. All right, long rambling intro over. Let's get into my first biggest regret, which is simply not starting YouTube earlier. Now I'm sure many of you all have heard this from other creators, and that's because with the benefit of hindsight, it's an extraordinarily easy thing to say. Looking at where this channel is now, it's a no brainer to go, oh man, I really should have been working towards this ever since like 2006 or something. But it's a really tricky one because who knows what would have happened. If I'd started this channel way earlier without the various bits of life experience I acquired, then maybe things would have just crashed and burned or not even taken off to begin with. However, in this case, I do have a very specific regret because if you look at when Grand Line Review was created, that date would be January 11th, 2015. That is when I first had the desire to start making one piece based content on YouTube. So I made the channel, and then I did absolutely nothing with it for like a year. And then I did actually start making videos but gave up after a couple of months. And it was really only in 2017 that I began to properly invest my time and energy into making videos. And I can't help but strongly wish that I had that desire when I first came up with the idea in 2015. And here's why I didn't do it summarized into one lovely yet annoying word, excuses. I didn't think I was capable of engaging in a time consuming hobby like YouTube, along with working full time and attempting to have some kind of social life, not a great social life because I'm a profound introvert, but still. And I know for a fact that it was just a pile of excuses now because I have done it. Prior to the global apocalypse thing, I was working full time and managing the Grand Line Review, which isn't to say that I'm some kind of amazing being, it's actually quite the opposite. It's more to say that if even someone like me can do it, then you absolutely can. If you have the drive to start a YouTube channel or you know what, if you have the drive to do anything, then just get over the excuses. The human mind and body are capable of so much more than it gets credit for. So put it to use and just do it. Do not be like me and procrastinate for a total of two years before finally having the determination to go for it. All right, cool. Now that that's out of the way, another huge regret I have is not doing enough research. And that might sound kind of vague and that's because it is. There is a phenomenal amount to consider when it comes to making even the simplest of videos. And one of my biggest issues was definitely not watching enough videos made by people who were and are currently sharing my niche at the time. Other One Piece and or General Anime YouTubers basically. And this is a trap I fell into because part of the desire to begin the Grand Line Review spawned from the idea that I wasn't interested in watching other One Piece videos.  So in response, I quite foolishly decided that I had no need to really research them because I did not want to be like them. However, as a result, I missed out on a lot of really basic information that I could have easily gathered. Really simple stuff like acceptable audio quality are allowed to make your music. What kind of topics generate the most interest for viewers? What kind of video techniques were being utilized at OneFaceCams? How people were crafting their thumbnails? All of this information was readily available and I ignored it. And so I began creating videos with no standard of comparison. So of course, in my early days, this led to a lot of issues like my music being far too loud, my voice being too slow and droney and just boring. My thumbnails being the epitome of abysmal and yeah. Honestly, there are just not a lot of redeeming factors about Early Grand Line Review, which is a stage I feel like I could have avoided completely with just a little bit of willingness to cast my gaze a bit further into the realm of research. And to be clear, that doesn't mean copying what other people do. It just means critically analyzing their work. Ask yourself what they do, how they do it and how that affects people watching. And then even if you end up doing something completely different, then at least you have a basis for something rather than just blindly wandering around in the dark like I did. Now my third most profound regret on YouTube is something I still do to this day, which is saying it's good enough. If you have ever mentally or verbally told yourself that something you've made as quote unquote, good enough, then the overwhelming probability is that it is not in fact, good enough. Because saying that to me is like admitting defeat. It's like stating, yeah, this could be better, but I don't have the time or energy to make it better. And I do that a lot more than I'd like to admit, especially when it comes to thumbnails. In my process, which is changing now, thumbnails were typically the last thing I tended to make for a video, which is utter madness because they're usually far more important than the actual video itself.  And I try not to slip into that mindset these days and just keep pushing until I land on a thumbnail that I feel would entice me to click.  I mean, I've recently changed the thumbnails for all of these, but thank you to everyone who clicked on these early thumbnails because when I look back at these, I just go, oh, who in their right mind would click on that? They're cluttered, the font is weird and there's nothing to like really grip you. And that's because I made these and I said good enough, then proceeded to pump them out. The mistake which I think I began to rectify near the end of the Sargas in Minutes series where I just enacted the mindset of don't produce thumbnail that you yourself would not click on. But even then, there are still times when I go good enough in every aspect of video making and I suppose just be aware of any time you say or think that to yourself because it is a huge, huge, huge creative red flag. As for my fourth mega regret in this film, it would be not investing in decent equipment earlier, quite specifically a microphone and some acoustic foam. Because the truth is you don't really need a whole lot of fancy stuff to do what I do on this channel, but when I began, I didn't even have that. And you know what?  And you know what? Here's a fun story that relates to this topic pretty heavily which is that one of my best friends one day decided to begin a YouTube channel of his own focusing on Magic the Gathering. And the general concept is that he would go through deck ideas and commentate on his games which he was good at. He was actually really good at it because he's insanely intelligent. He had his pulse on this very niche topic and he even had a side of wit to add on top of it. But he was using an inbuilt microphone to record his audio. And as a result, no matter what substance was coming out of his mouth, it was frequently peaking, highly distorted and it just didn't sound good. So I told him that the one thing he really needed to do immediately was invest in a proper microphone. And his response was, yeah, I'll buy a microphone if this channel starts to become successful.  Like I can practically guarantee anybody that in this day and age on YouTube, you will not become quote unquote successful without at least, at least an entry level microphone. Not doing the sort of thing we were doing anyway. But at the same time, I'm also a bit of a hypocrite there because that was my mindset as well. In the very early days of Grand Line Review, I thought to myself, yeah, I'm just making these videos as a hobby. And if they take off, then that's great. That's when I'll invest in something. Which is fine, but I really do wish that I'd not wasted like a whole year or so in this channel using a pathetic microphone because when I did finally switch to something halfway decent, that's when the Grand Line Review really started to explode and became much more popular. Probably because people could actually stand listening to me now. And so if you think that your YouTube videos are just a hobby, I understand that. But I would strongly encourage you to invest in your hobby. You know, it's pretty difficult to collect Pokemon cards without buying the cards or engaging in woodwork without woodworking tools. Similarly, it's very hard to make YouTube videos without proper recording gear, for example. So even if it's just something you do for fun, make it more fun for yourself by getting some reasonably priced equipment to help you. All right, that turned into a long and rambling one. So my next regret, regret number five, if you will, would be that very brief period where I tried to do videos seven days a week on Grand Line Review. And I don't know how many of you will actually remember this time. It was about a year and a half ago now where I published one video every day for about a month and a half. And the theory behind it was my videos were doing great. On average, they were bringing in around 24,000 views in 24 hours, which was a nice handy metric I used. But I was only posting between three to four videos a week. So in my rudimentary thinking, I figured that if I doubled the amount of videos I posted, then that would effectively double my views, which would double my revenue, right? Well, surprisingly, not only was it not right, it was actually very, very wrong. After a couple of weeks of implementing the strategy, I noticed that my average was sinking to around 18,000 views in 24 hours. And while the money I was earning did go up, it was a very, very incremental amount of growth. And after speaking to a fair few people, it became apparent to me that I was flooding my audience with content, which was probably great for the more core viewers of the channel, but it meant that there was so much coming out that people simply could not keep up with it all. Which had some more long-term ramifications because if someone doesn't click your video a couple of times in a row, then YouTube will be far less likely to recommend those videos to them in the future. So I made the kind of painful decision to go back to publishing three to four times a week, but the damage had already been done by then. And it was quite a long journey to get things back to where they were before I changed up the schedule. So I discovered what the critical mass for my videos were the hard way. And then instead of making stuff seven days a week on Grand Line Review, I made a new channel for other series in general called New World Review. So I do still effectively publish seven days a week just over two channels, which keeps them both healthy, but not overly saturated. And speaking of my final regret for this video has to do with my second channel, which was the decision to name it New World Review. So to all of you, the reasoning behind that name is probably pretty obvious. I mean, when I was thinking about how to expand this brand, it made all the sense in the world to name a second channel after the second half of the Grand Line. It's kind of perfect, isn't it? However, what I really, really should have thought more about was the idea that this channel was designed to cover literally any series that wasn't One Piece. So New World Review is actually an incredibly meaningless name for anyone who goes there without any knowledge of One Piece. But the worst part is, even if they are a fan of One Piece, it's very confusing because it has a One Piece based name, but no One Piece content. So it's bad all around. So as much as I do like the name and how it blends with Grand Line Review, it was far from the smartest idea. And I do wish that I could go back in time and change it.  The one thing I feel like I really nailed with the whole YouTube thing, which I suspect many other YouTubers will disagree with, but that is keeping the Grand Line Review focused exclusively on One Piece. Over the years, I've had a lot of people ask me why I made a second channel for other series when I already had an established audience here. And that's because I really believe in the power of niche. I really like the idea that everyone who comes to this channel will potentially be interested in every single video it features because it's all One Piece based. I think it's a much more solid proposition than trying to cover multiple series like if the Grand Line Review featured One Piece, Bleach, Hunter, Hunter, whatever, it would become far more general. But I think that this laser-like focus on One Piece was super valuable in the early days because despite the fact that I was making bad videos, some amazingly rare people were finding and watching them anyway. And those people would then go through the channel library, which helped push out videos to more people, whereas if I'd split my focus, that growth may not have happened. Someone may have come to the channel and enjoyed a One Piece based video, but not clicked on any of the others because they weren't keen for the other series being featured or even not subscribed because they didn't want, say, Hunter, Hunter content, clogging up their recommendations when all they came to me for was One Piece. I wholeheartedly believe that it made it a lot easier for this channel to grow. And I'm sure a lot of people will point to figures like, say, Tekking or NoxTaku, who have gigantic audiences with a base in covering multiple series, but it's not a direct comparison because they can drive that kind of engine based on pure personality. It's a very different value proposition. And while the reality of that situation means that a channel like Groundline Review will be highly unlikely to reach their levels of views and subscribers, I honestly don't think I'd even be here today if I'd spread my focus across multiple series instead of just providing a solid injection of pure One Piece onto this platform. So I think that was absolutely the greatest decision that I've made in regards to the Groundline Review, accompanied by very, very many regrets, but there's nothing wrong with that. Very few people conduct themselves in any first-time venture with flawless execution and learning a lot of these lessons the very hard way will ensure that at the very least, I never forget them. So take my mistakes and hopefully it's provided some insight for your own endeavors. But what do you guys think? Please leave your thoughts in the comments below or even join my Discord server. And if you'd like to see more videos like this, except probably not exactly like this, then please do go and check out some of my other content or even subscribe to the channel for more glorious One Piece business uploaded straight into your YouTube feed. But for now, this has been the Groundline Review and I'll see you next time.


For more videos on People's Struggles, please subscribe to our YouTube channel. On June 6th, 25 million Peruvians will elect their next president and two vice presidents for the period 2021-2026. Leftist candidate Pedro Castillo of the Free Peru Political Party will face off against far-right candidate Keiko Fujimori of the Popular Forest Party. All major polls have predicted a victory for Castillo. According to the latest opinion poll conducted by Ipsos Perú pollster, Castillo is leading the voter intention with 51% of the votes, while his opponent Keiko Fujimori of the far-right Popular Forest Party follows him with 49% of the votes. The Institute for Peruvian Studies Research Group carried out its voter intention survey giving Pedro Castillo 40.3% and Keiko Fujimori 38.3% of the votes. Castillo is a primary school teacher and union leader. He has expressed himself in favor of rewriting the country's constitution with the aim of giving the state more control and a leading role in the economy to compete with private companies. He has promised an anti-neoliberal economic model to address the crisis facing Peru. He has also promised to nationalize natural resources and strategic industries such as mining, oil, hydropower, gas, and communications. Meanwhile, Fujimori, the daughter of former dictator Alberto Fujimori, has expressed herself in support of preserving the current constitution which promotes a neoliberal capitalist economic model and deepens inequality. It was drafted during her father's dictatorship. Meanwhile, disconsent has been mounting against Fujimori. Peru witnessed a massive demonstration with close to 50,000 taking part against her candidacy on June 1st. With the slogan, Fujimori never again, now and forever, thousands of protesters carried photos of victims of massacres carried out during the presidency of Alberto Fujimori. Fujimori is serving a 25-year sentence for crimes against humanity, including his role in the massacres of Varios Altos and La Cantunta. Peru has been devastated by one of the world's worst coronavirus outbreaks, mounting poverty and unemployment, as well as corruption in recent months. Over 10,000 Peruvians who had lost their livelihoods during the COVID-19 pandemic had sought refuge in temporary camps set up on abandoned land in the Lomo de Corvina area and the Villa del Salvalo district, situated on the outskirts of the capital Lima. On April 28, the Peruvian national police violently evicted the hundreds of homeless families living in temporary camps. 70% of Peru's workforce is employed in the informal sector and depends on daily income to survive. Decades of neoliberal policies and chronic underfunding of the public sectors have deepened social inequality in Peru. Last year, November 2020, the country saw widespread protests demanding systemic change and against corruption. One of the key demands was precisely for a new inclusive constitution to replace the current one. Within the candidacy and project of Pedro Castillo and the Free Peru Party, many of the historically marginalized sectors are finding newfound hope and inspiration. Despite a vicious campaign of attacks and fake news against Castillo, his message continues to ring true for millions of Peruvians.


Last month Johnson County Animal Shelter revealed that as the season transitions into spring, shelters become overcrowded due to families surrendering animals. In recent weeks, the shelter has increased pet space, however help is still needed. Of course, the last time we spoke, the shelter was, we were over capacity, honestly, because we do have 32 kennels inside, particularly for our dogs. And you like to keep those numbers a little lower than your 32 kennels because you have to be able to clean adequately. And when you're full, it is hard to do that and care for your animals when you have limited staff, which is very limited and volunteers. But I'm happy to say that we've been able to get those numbers down somewhat. We just had, we had seven go out and that means going to transport, leaving on transport, which opened up some kennels. I have five that will be leaving in the next, should be in the next week, that have been adopted, so they've already been adopted. So they're going straight into adoptive homes, which is excellent. So that is a very positive thing. At Pikeville Medical Center, we believe that together we can make a difference in the fight against cancer. We are proud to have patients who have experienced successful outcomes over many years. As always, we believe in the power of faith. Pikeville Medical Center, where cancer can be defeated. And while shelter staff continues to find homes for pets, Robert says they are still in need of supplies. We always need the kitty litter. Meow Mix is what we use in kitten chow and alpo dog food because we like to keep the animals on the same food. And that just helps with their digestion and things of that nature. It just keeps them steady and healthy. So we like to keep them on the same if possible. I mean, we appreciate anything. If you would like to adopt or donate, you can contact the shelter at WowAnimalShelter at gmail.com. You can also reach the shelter at<PHONE_NUMBER> or pets. For Mountain Top News, I'm Joel Porchal.


Hello and welcome to the session. I am Alka. We are going to discuss matrices. Our given question is, if A equal to matrix 1, 2, minus 3, 5, 0, 2, 1, minus 1, 1, B equal to matrix 3, minus 1, 2, 4, 2, 5, 2, 0, 3, and C equal to matrix 4, 1, 2, 0, 3, 2, 1, minus 2, 3, then compute A plus B and B minus C. Also verify that A plus B minus C equal to A plus B minus C. Now, let's start with a solution. So, first of all according to our question, we'll calculate the value of A plus B. A plus B equal to 1 plus 3, 2 plus minus 1, minus 3, plus 2. Similarly, we'll write for the second row and for third row.  This can be written as 4, 1, minus 1, 9, 2, 7, 3, minus 1, 4. Now, we have to calculate B minus C. For that, we need negative of C. So, first of all we'll calculate minus C. A C is C equal to matrix 4, 1, 2, 0, 3, 2, 1, minus 2, 3. Therefore, minus C equal to minus 4, minus 1, minus 2, 0, minus 3, minus 2, minus 1, plus 2, minus 3. Now, we'll calculate the value of B minus C which can be written as B plus minus C. Now, on placing the value of B and C, we get B plus minus C equal to 3, minus 4, 1, minus 1, minus 1, minus 2, 2, minus 2, 0. And similarly, we'll write for the second row and for third row. So, the value of B minus C is equal to 1, minus 2, 0, 4, minus 1, 3, 1, 2, 0. Now, we'll verify that A plus B minus C equal to A plus B minus C. To verify this equation, first of all let us take LHS and then we'll calculate RHS and then we'll verify. So, LHS is A plus B minus C. We know the value of B minus C as we have calculated above. So, A plus B minus C can be written as this is equal to 1 plus minus 1, then 2 plus minus 2, minus 3, plus 0, 5 plus 4, 2 plus minus 1, 2 plus 3. Then 1 plus 1, minus 1, plus 2, 1 plus 0. This is equal to 0, 0, minus 3, 9, minus 1, 5, 2, 1, 1. So, this is our first equation. Now, we'll see the LHS. A plus B, we already have calculated the value of A plus B, which is equal to 4, 1, minus 1, 9, 2, 7, 3, minus 1, 4. And we also know the value of minus C, which is equal to minus 4, minus 1, minus 2, 0, minus 3, minus 2, minus 1, plus 2, minus 3. Therefore, A plus B minus C equal to matrix 4, 1, minus 1, 9, 2, 7, 3, 1, 4, plus matrix minus 4, minus 1, minus 2, 0, minus 3, minus 2, minus 1, 2, minus 3. This can be written as, on adding, this is equal to 4, plus, minus 4, 1, plus, minus 1, minus 1, plus, minus 2. Similarly, we'll write for second row and for third row.  This is equal to 0, 0, minus 3, 9, minus 1, 5, 2, 1, and 1. This is our second matrix. Now, on comparing first and second, we know, we came to know that A plus B minus C equal to A plus B minus C. So, this is verified. A plus B minus C equal to A plus B minus C, hence verified. Hope you understood it and enjoyed the session. Goodbye and take care.