Manifold-regression to predict from MEG/EEG brain signals without - PowerPoint PPT Presentation

Manifold-regression to predict from MEG/EEG brain signals without source modeling D. Sabbagh, P. Ablin, G. Varoquaux, A. Gramfort, D.Engemann NeurIPS 2019 1 / 14

Non-invasive measure of brain activity 2 / 14

Non-invasive measure of brain activity 2 / 14

Objective: predict a variable from M/EEG brain signals 3 / 14

Objective: predict a variable from M/EEG brain signals We want to predict a continuous variable From M/EEG brain signals 3 / 14

Objective: predict a variable from M/EEG brain signals We want to predict a continuous variable From M/EEG brain signals 3 / 14

Model: generative model of M/EEG data 4 / 14

Model: generative model of M/EEG data We measure M/EEG signal of subject i = 1 . . . N on P channels: x i ( t ) = A s i ( t ) + n i ( t ) ∈ R P t = 1 . . . T mixing matrix A = [ a 1 , . . . , a Q ] ∈ R P × Q fixed accross subjects source patterns a j ∈ R P , j = 1 . . . Q with Q < P source vector s i ( t ) ∈ R Q noise n i ( t ) ∈ R P 4 / 14

Model: generative model of M/EEG data We measure M/EEG signal of subject i = 1 . . . N on P channels: x i ( t ) = A s i ( t ) + n i ( t ) ∈ R P t = 1 . . . T mixing matrix A = [ a 1 , . . . , a Q ] ∈ R P × Q fixed accross subjects source patterns a j ∈ R P , j = 1 . . . Q with Q < P source vector s i ( t ) ∈ R Q noise n i ( t ) ∈ R P Under stationnarity and gaussianity assumptions, we can represent band-pass filtered signal by its second-order statistics C i = E [ x i ( t ) x i ( t ) ⊤ ] ≃ X i X ⊤ i ∈ R P × P X i ∈ R P × T with T 4 / 14

Model: generative model of target variable Q � We want to predict a continuous variable: y i = α j f ( p i , j ) ∈ R j =1 with p i , j = E t [ s 2 i , j ( t )] band-power of sources 5 / 14

Model: generative model of target variable Q � We want to predict a continuous variable: y i = α j f ( p i , j ) ∈ R j =1 with p i , j = E t [ s 2 i , j ( t )] band-power of sources linear y i = � Q j =1 α j p i , j 5 / 14

Model: generative model of target variable Q � We want to predict a continuous variable: y i = α j f ( p i , j ) ∈ R j =1 with p i , j = E t [ s 2 i , j ( t )] band-power of sources linear y i = � Q j =1 α j p i , j Euclidean vectorization leads to consistent model y i = � k ≤ l Θ k , l C i ( k , l ) i.e. y i is linear in coeff. of Upper( C i ) 5 / 14

Model: generative model of target variable Q � We want to predict a continuous variable: y i = α j f ( p i , j ) ∈ R j =1 with p i , j = E t [ s 2 i , j ( t )] band-power of sources linear y i = � Q j =1 α j p i , j Euclidean vectorization leads to consistent model y i = � k ≤ l Θ k , l C i ( k , l ) i.e. y i is linear in coeff. of Upper( C i ) log linear y i = � Q j =1 α j log ( p i , j ) 5 / 14

Model: generative model of target variable Q � We want to predict a continuous variable: y i = α j f ( p i , j ) ∈ R j =1 with p i , j = E t [ s 2 i , j ( t )] band-power of sources linear y i = � Q j =1 α j p i , j Euclidean vectorization leads to consistent model y i = � k ≤ l Θ k , l C i ( k , l ) i.e. y i is linear in coeff. of Upper( C i ) log linear y i = � Q j =1 α j log ( p i , j ) C i live on a Riemannian manifold so can’t be naively vectorized 5 / 14

Riemannian matrix manifolds (in a nutshell) M T M ξ M Log M Exp M M' M 6 / 14

Riemannian matrix manifolds (in a nutshell) M T M ξ M Log M Exp M M' M [Absil & al. Optimization algorithms on matrix manifolds. 2009] 7 / 14

Riemannian matrix manifolds (in a nutshell) M T M ξ M Vectorization operator: Log M P M ( M ′ ) = φ M (Log M ( M ′ )) ≃ Upper(Log M ( M ′ )) d ( M , M ′ ) = �P M ( M ′ ) � 2 + o ( �P M ( M ′ ) � 2 ) Exp M d ( M i , M j ) ≃ �P M ( M i ) − P M ( M j ) � 2 M' Vectorization operator key for ML M [Absil & al. Optimization algorithms on matrix manifolds. 2009] 7 / 14

Regression on matrix manifolds Given a training set of samples M 1 , . . . , M N ∈ M and target continuous variables y 1 , . . . , y N ∈ R : 8 / 14

Regression on matrix manifolds Given a training set of samples M 1 , . . . , M N ∈ M and target continuous variables y 1 , . . . , y N ∈ R : compute the mean of the samples M = Mean d ( M 1 , . . . , M N ) 8 / 14

Regression on matrix manifolds Given a training set of samples M 1 , . . . , M N ∈ M and target continuous variables y 1 , . . . , y N ∈ R : compute the mean of the samples M = Mean d ( M 1 , . . . , M N ) compute the vectorization of the samples w.r.t. this mean: v 1 , . . . , v N ∈ R K as v i = P M ( M i ) 8 / 14

Regression on matrix manifolds Given a training set of samples M 1 , . . . , M N ∈ M and target continuous variables y 1 , . . . , y N ∈ R : compute the mean of the samples M = Mean d ( M 1 , . . . , M N ) compute the vectorization of the samples w.r.t. this mean: v 1 , . . . , v N ∈ R K as v i = P M ( M i ) use those vectors as features in regularized linear regression algorithm (e.g. ridge regression) with parameters β ∈ R K assuming that y i ≃ v ⊤ i β 8 / 14

Distance and invariance on positive matrix manifolds [Bhatia. Positive Definite Matrices. Princeton University Press, 2007] 9 / 14

Distance and invariance on positive matrix manifolds Manifold of positive definite matrices : M i = C i ∈ S ++ P [Bhatia. Positive Definite Matrices. Princeton University Press, 2007] 9 / 14

Distance and invariance on positive matrix manifolds Manifold of positive definite matrices : M i = C i ∈ S ++ P Geometric distance : � 1 � 2 i =1 log 2 λ k � P d G ( S , S ′ ) = � log( S − 1 S ′ ) � F = where λ k , k = 1 . . . P are the real eigenvalues of S − 1 S ′ . Tangent Space Projection : P S ( S ′ ) = Upper(log( S − 1 2 S ′ S − 1 2 )) [Bhatia. Positive Definite Matrices. Princeton University Press, 2007] 9 / 14

Distance and invariance on positive matrix manifolds Manifold of positive definite matrices : M i = C i ∈ S ++ P Geometric distance : � 1 � 2 i =1 log 2 λ k � P d G ( S , S ′ ) = � log( S − 1 S ′ ) � F = where λ k , k = 1 . . . P are the real eigenvalues of S − 1 S ′ . Tangent Space Projection : P S ( S ′ ) = Upper(log( S − 1 2 S ′ S − 1 2 )) Affine invariance property : For W invertible, d G ( W ⊤ SW , W ⊤ S ′ W ) = d G ( S , S ′ ) [Bhatia. Positive Definite Matrices. Princeton University Press, 2007] 9 / 14

Distance and invariance on positive matrix manifolds Manifold of positive definite matrices : M i = C i ∈ S ++ P Geometric distance : � 1 � 2 i =1 log 2 λ k � P d G ( S , S ′ ) = � log( S − 1 S ′ ) � F = where λ k , k = 1 . . . P are the real eigenvalues of S − 1 S ′ . Tangent Space Projection : P S ( S ′ ) = Upper(log( S − 1 2 S ′ S − 1 2 )) Affine invariance property : For W invertible, d G ( W ⊤ SW , W ⊤ S ′ W ) = d G ( S , S ′ ) Affine invariance is key: working with C i is then equivalent to working with covariance matrices of sources s i [Bhatia. Positive Definite Matrices. Princeton University Press, 2007] 9 / 14

Consistency of linear regression in tangent space of S ++ P 10 / 14

Consistency of linear regression in tangent space of S ++ P Geometric vectorization Assume y i = � Q j =1 α j log( p i , j ). Denote C = Mean G ( C 1 , . . . , C N ) and v i = P C ( C i ). Then, the relationship between y i and v i is linear. 10 / 14

Consistency of linear regression in tangent space of S ++ P Geometric vectorization Assume y i = � Q j =1 α j log( p i , j ). Denote C = Mean G ( C 1 , . . . , C N ) and v i = P C ( C i ). Then, the relationship between y i and v i is linear. We generate i.i.d. samples following the log linear generative model. A = exp( µ B ) with B ∈ R P × P random ● log−diag ● Wasserstein ● geometric chance level ● 1.00 ● ● ● ● Normalized MAE ● 0.75 ● 0.50 ● ● ● ● ● ● ● ● ● ● ● 0.25 ● ● ● ● ● ● ● ● ● ● ● ● 0.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 µ 10 / 14

And in the real world ? We investigated 3 violations from previous idealized model: 11 / 14

And in the real world ? We investigated 3 violations from previous idealized model: Noise in target variable � ε i ∼ N (0 , σ 2 ) y i = α j log( p ij ) + ε i , with j 11 / 14

And in the real world ? We investigated 3 violations from previous idealized model: Noise in target variable � ε i ∼ N (0 , σ 2 ) y i = α j log( p ij ) + ε i , with j Subject-dependent mixing matrix E i ∼ N (0 , σ 2 ) A i = A + E i , with entries of 11 / 14