demo_data/nips-2021/25957/transcript_whisper_large-v2.vtt

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Hi, I'm Hugo Richard, I'm a third year PhD student at Université Paris-Saclay.

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I'm in the INRIA Paris et Alpes team and my supervisor is Bertrand Thirion.

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Today I'll talk about shared independent component analysis for multi-subject neuroimaging.

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This is a joint work with Pierre Abelin, Alexandre Grandfort, Bertrand Thirion and Anna Pouy-Varine.

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First let us consider two sources that are emitting a signal that is recorded by two

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sensors.

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This can be seen as a simplified model of magnetoencephalography where brain sources

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are recorded by magnetometers.

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Because propagation time can be neglected, the signal recorded by the sensors can be

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seen as a linear mixture of the signal emitted by the sources.

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S is a set of sources that are assumed to be independent.

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X are the recordings and A describes how the sources are mixed to produce the recordings.

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At first sight this model may seem ill-defined because if we permute two columns in A and

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permute the corresponding sources in S, we'll get a new set of sources S' and a new mixing

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matrix A' that describes X just as well as A and S.

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And similarly if we scale the column of A by some constant, one column of A by some

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constant and the corresponding source by the same constant, we'll also get an equivalent

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description of X.

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However, these scale and permutation indeterminacies are the only one if the sources contain at

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most one Gaussian component.

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Let us consider the more general problem where you have multiple subjects that are exposed

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to the same stimuli.

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We have two subjects, X1 and X2, and they have different mixing matrices, A1 and A2,

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and different noise levels, N1 and N2.

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The interpretation is that they have shared sources because they have shared connective

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processes.

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They have different mixing matrices because they have different spatial topography.

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And they have different noises because we want to model inter-subject variability.

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This model is called group ICA.

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There are many methods to provide a solution for the group ICA problem.

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A very popular one introduced by Calhoun in 2001 is to just stack the data of all subjects

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feature-wise and then perform a PCA, a principal component analysis, on the stacked data.

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And therefore you obtain reduced data and apply independent component analysis on the

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reduced data to obtain a set of sources.

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Another formulation is introduced by Varoko in 2010 and is called K-NICA.

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You just replace the principal component analysis with a multiset CCA, so a multiset canonical

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correlation analysis, where you have to solve a generalized eigenvalue problem.

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There are many different formulations of multiset CCA, but this one with a generalized eigenvalue

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problem is the fastest to solve.

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KNICA and Cut-ICA have a lot of advantages.

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First, they are very fast to fit.

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And second, they are simple to implement.

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These are the two reasons why they are so popular in neuroimaging.

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However, they do not optimize the proper likelihood.

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So therefore they do not benefit from advantages of such estimators such as asymptotic efficiency.

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There are a lot of other related work that do optimize the proper likelihood.

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I want to mention the independent vector analysis, which is a very powerful framework introduced

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by Li in 2008.

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So unified approach of Guo in 2008 that we will also mention and talk about later.

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The approach of Shen in 2015 that also allows to perform dimension reduction.

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And the multi-view ICA that was introduced by our team last year.

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I want to quickly say that it's not obvious to design a likelihood-based approach that

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is tractable.

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And with this example of the Gaussian mixture noisy ICA by Bermond and Cardozo, we'll see

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that standard approach leads to intractable algorithms.

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The model we take here is the same as the group ICA, but we assume that the noise is

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Gaussian with the same variance for all subjects.

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We'll also assume that the sources follow a Gaussian mixture model.

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And we further assume that the weights of the Gaussian mixtures are known.

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We can solve such model via expectation maximization.

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And if we write the E-step, we'll get a closed form that involves a large sum.

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Because of this large size, this sum, and therefore the M algorithm is intractable whenever

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Q and K are large.

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Our contribution is shared ICA, what we call Shikha for short, where the data of subject

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i are assumed as a linear mixture of noisy sources, and the noise here is not on the

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sensor, but on the sources.

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The noise is Gaussian with a variance that can be different for each subject and different

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for each component.

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S are assumed to be independent, but in contrast to almost all existing work, some components

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can be Gaussian.

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We have a few blanket assumptions.

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We assume that the data are centered, that the mixing metrics are invertible, that the

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sources have identical variance, and that the number of subjects is greater than 3.

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We have two algorithms to solve the Shikha model.

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We have ShikhaJ, that is a FAS algorithm that is based on multiset CCA, and ShikhaML, a

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maximum likelihood approach.

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In Shikha, there are two ways to recover the parameters.

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Either the source are non-Gaussian, in which case we can use classical ICA results to recover

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the unmixing matrices.

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When the components are Gaussian, then we need something else, and what we use here

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is noise diversity.

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When the noise is sufficiently diverse, then it's possible to recover the unmixing matrix

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and the noise covariance up to a permutation and sign indeterminacy.

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Note that the noise diversity in Gaussian components is also a necessary condition.

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If it does not hold, then Shikha cannot be identified.

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Let us now focus on this theorem that is at the core of the ShikhaJ algorithm.

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Namely it shows that we can solve group ICA with multiset CCA.

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So assume the data follows the Shikha model, and consider the multiset CCA framed as a

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generalized eigenvalue problem.

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This generalized eigenvalue problem relies on two matrices, C and D. So C is formed by

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second-order statistics, and D is formed by the diagonal blocks in C.

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And so if we solve this eigenvalue problem and take the first k leading eigenvectors,

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we can recover the correct unmixing matrix from them, up to a permutation and a scaling.

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And this can only be done if the k first eigenvalues are distinct.

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Note that the distinct eigenvalue condition is also necessary.

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If two eigenvalues are the same, then this adds the need to determine IC, and therefore

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we cannot solve group IC.

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Note also that the condition that some eigenvalues need to be distinct is stronger than the noise

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diversity condition we have in the identifiability theorem.

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And therefore we can exhibit an example which is identifiable, but on which multiset CCA

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will fail.

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And I refer you to the paper for more details on this.

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So in our theorem, in order to recover the correct unmixing matrix, we need to have access

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to the second-order statistics.

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However, in practice, we only have access to them, up to some sampling noise.

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And because the mapping from matrices to eigenvectors is highly non-smooth, a small deviation in

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the second-order statistics can lead to a high deviation of the recovered unmixing matrix.

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Now to show this in practice, we take three subjects, two components, and noise covariance

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matrices with two values, lambda1 and lambda2, that are separated by an eigengap epsilon.

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And we compare the solution of multiset CCA on the true covariance matrices and on the

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perturbed covariance matrix, where the perturbation scale is given by delta.

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And for different values of epsilon, 10-4, 10-3, 10-2, 10-1, we show how the performance

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of the algorithm, so the M-ary distance between the true unmixing matrix and the estimated

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unmixing matrix, varies when the perturbation scale increases.

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And we see that when the eigengap is very close, so 10-4, the violet curve, then even

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with a very small perturbation, you can get to a very bad M-ary distance.

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So the black dashed curve is a performance of chance.

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Luckily, there is a large gap between the k-th eigenvalues and the k plus 1.

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This means that in practice, the span of the p-leading eigenvectors is approximately preserved.

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We can recover the true unmixing matrix from the unmixing matrix estimated by multiset

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CCA, just by multiplying by a matrix Q.

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And in order to estimate Q, we make use of the fact that the unmixed data should have

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a diagonal covariance.

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This leads us to a joint diagonalization problem that we can solve efficiently.

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So if we take the experiments we've done on the previous slide, the results are still

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shown here.

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You can see the violet curves, and that is very sensitive to perturbation.

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And so if we apply joint diagonalization, all these curves move, and they join the dashed

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curve on the bottom.

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And therefore, it's much better, because now the new curves that are represented by the

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dashed line are less sensitive to perturbations.

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So now we've obtained the correct unmixing matrix, but up to a scaling.

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And so we need an additional step to find the correct scaling, and another one to find

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the other parameter that is still unestimated, which are the noise covariance.

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And luckily, it's very easy to find the noise covariance.

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We can do this via an EM algorithm.

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The E-step and the M-step are in closed form, and this yields a very fast algorithm.

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But the Shikha-J is not a maximum likelihood estimator.

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So now we will focus on Shikha-ML, which is our maximum likelihood estimator.

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So I won't go too much into details on this, but we optimize this via an EM using a Gaussian

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mixture assumption as a source.

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We assume that the weights are known.

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What I just want to showcase here is that the E-step of the algorithm, the one that

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gives you the expectation of the sources given the data, and the variance of the sources

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given the data, only involves the sum of size 2.

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So previously we had a sum that had an exponential number of terms, and here we don't have that

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anymore.

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So the E-step is much faster than what we had before, and therefore the EM algorithm

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here is tractable, whereas it was not the case before.

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I first want to present our synthetic experiment where we generate data according to the Shikha-ML

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and Shikha-J model.

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In case A, we have only Gaussian components, but we have noise diversity, and therefore

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methods that use noise diversity to recover the sources such as Shikha-ML and Shikha-J

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perform best.

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In the second case, we have only non-Gaussian components and no noise diversity, so methods

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that use non-Gaussianity perform well such as Kana-ICA, Shikha-ML, or MultiView-ICA.

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And the last case, half of the components are Gaussian with noise diversity, and the

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other half are non-Gaussian but without noise diversity.

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And in this case, only Shikha-ML is able to correctly recover the sources.

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MV-ICA doesn't do that, but it's not as good as Shikha-ML.

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Let us now talk about our experiments on real data.

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We have this reconstruction experiment on fMRI data where subjects are exposed to a

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naturalistic stimuli such as movie watching.

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We use 80% of the movie to learn the unmixing matrices of all subjects, and then on the

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20% left of the movie, we compute the common sources, and from these common sources computed

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using 80% of the subject, we try to reconstruct the data of the 20% left of the subject.

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We compute the R2 score within regions of interest between the reconstructed data and

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the true data, and plot them as a function of the number of components used.

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As we see, Shikha-ML outperforms all of the methods.

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As a take-home message, Shikha is a powerful framework to extract shared sources.

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Shikha-J is a fast approach to fit the model, but it only uses second-order information.

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In contrast, Shikha-ML is a bit slower, but is able to use non-gaussianity in addition

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to second-order information.

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In practice, Shikha-ML yields the best results.

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The methods we've introduced work on reduced data.

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It would be interesting to know how to reduce the data so that they perform optimally.

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Another way to improve our results would be to learn the density of the shared sources

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in Shikha-ML instead of having them fixed.

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Thanks for listening, and have a good day!