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

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Non-invasive measure of brain activity

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Non-invasive measure of brain activity

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Objective: predict a variable from M/EEG brain signals

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Objective: predict a variable from M/EEG brain signals

We want to predict a continuous variable From M/EEG brain signals

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Objective: predict a variable from M/EEG brain signals

We want to predict a continuous variable From M/EEG brain signals

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Model: generative model of M/EEG data

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Model: generative model of M/EEG data

We measure M/EEG signal of subject i = 1 . . . N on P channels: xi(t) = A si(t) + ni(t) ∈ RP t = 1 . . . T mixing matrix A = [a1, . . . , aQ] ∈ RP×Q fixed accross subjects source patterns aj ∈ RP, j = 1 . . . Q with Q < P source vector si(t) ∈ RQ noise ni(t) ∈ RP

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Model: generative model of M/EEG data

We measure M/EEG signal of subject i = 1 . . . N on P channels: xi(t) = A si(t) + ni(t) ∈ RP t = 1 . . . T mixing matrix A = [a1, . . . , aQ] ∈ RP×Q fixed accross subjects source patterns aj ∈ RP, j = 1 . . . Q with Q < P source vector si(t) ∈ RQ noise ni(t) ∈ RP Under stationnarity and gaussianity assumptions, we can represent band-pass filtered signal by its second-order statistics C i = E[xi(t)xi(t)⊤] ≃ X iX ⊤

i

T ∈ RP×P with X i ∈ RP×T

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Model: generative model of target variable

We want to predict a continuous variable: yi =

Q

• j=1

αjf (pi,j) ∈ R with pi,j = Et[s2

i,j(t)] band-power of sources

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Model: generative model of target variable

We want to predict a continuous variable: yi =

Q

• j=1

αjf (pi,j) ∈ R with pi,j = Et[s2

i,j(t)] band-power of sources

linear yi = Q

j=1 αjpi,j

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Model: generative model of target variable

We want to predict a continuous variable: yi =

Q

• j=1

αjf (pi,j) ∈ R with pi,j = Et[s2

i,j(t)] band-power of sources

linear yi = Q

j=1 αjpi,j

Euclidean vectorization leads to consistent model yi =

k≤l Θk,lC i(k, l)

i.e. yi is linear in coeff. of Upper(C i)

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Model: generative model of target variable

We want to predict a continuous variable: yi =

Q

• j=1

αjf (pi,j) ∈ R with pi,j = Et[s2

i,j(t)] band-power of sources

linear yi = Q

j=1 αjpi,j

Euclidean vectorization leads to consistent model yi =

k≤l Θk,lC i(k, l)

i.e. yi is linear in coeff. of Upper(C i) log linear yi = Q

j=1 αj log (pi,j)

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Model: generative model of target variable

We want to predict a continuous variable: yi =

Q

• j=1

αjf (pi,j) ∈ R with pi,j = Et[s2

i,j(t)] band-power of sources

linear yi = Q

j=1 αjpi,j

Euclidean vectorization leads to consistent model yi =

k≤l Θk,lC i(k, l)

i.e. yi is linear in coeff. of Upper(C i) log linear yi = Q

j=1 αj log (pi,j)

C i live on a Riemannian manifold so can’t be naively vectorized

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Riemannian matrix manifolds (in a nutshell)

ξ

M

T M

M

M

M'

LogM Exp

M

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Riemannian matrix manifolds (in a nutshell)

ξ

M

T M

M

M

M'

LogM Exp

M

[Absil & al. Optimization algorithms on matrix manifolds. 2009]

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Riemannian matrix manifolds (in a nutshell)

ξ

M

T M

M

M

M'

LogM Exp

M

Vectorization operator: PM(M′) = φM(LogM(M′)) ≃ Upper(LogM(M′)) d(M, M′) = PM(M′)2 + o(PM(M′)2) d(Mi, Mj) ≃ PM(Mi) − PM(Mj)2 Vectorization operator key for ML

[Absil & al. Optimization algorithms on matrix manifolds. 2009]

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Regression on matrix manifolds

Given a training set of samples M1, . . . , MN ∈ M and target continuous variables y1, . . . , yN ∈ R:

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Regression on matrix manifolds

Given a training set of samples M1, . . . , MN ∈ M and target continuous variables y1, . . . , yN ∈ R: compute the mean of the samples M = Meand(M1, . . . , MN)

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Regression on matrix manifolds

Given a training set of samples M1, . . . , MN ∈ M and target continuous variables y1, . . . , yN ∈ R: compute the mean of the samples M = Meand(M1, . . . , MN) compute the vectorization of the samples w.r.t. this mean: v 1, . . . , v N ∈ RK as v i = PM(Mi)

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Regression on matrix manifolds

Given a training set of samples M1, . . . , MN ∈ M and target continuous variables y1, . . . , yN ∈ R: compute the mean of the samples M = Meand(M1, . . . , MN) compute the vectorization of the samples w.r.t. this mean: v 1, . . . , v N ∈ RK as v i = PM(Mi) use those vectors as features in regularized linear regression algorithm (e.g. ridge regression) with parameters β ∈ RK assuming that yi ≃ v ⊤

i β

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Distance and invariance on positive matrix manifolds

[Bhatia. Positive Definite Matrices. Princeton University Press, 2007]

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Distance and invariance on positive matrix manifolds

Manifold of positive definite matrices: Mi = C i ∈ S++

P

[Bhatia. Positive Definite Matrices. Princeton University Press, 2007]

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Distance and invariance on positive matrix manifolds

Manifold of positive definite matrices: Mi = C i ∈ S++

P

Geometric distance: dG(S, S′) = log(S−1S′)F =

• P

i=1 log2 λk

1

2

where λk, k = 1 . . . P are the real eigenvalues of S−1S′. Tangent Space Projection: PS(S′) = Upper(log(S− 1

2 S′S− 1 2 ))

[Bhatia. Positive Definite Matrices. Princeton University Press, 2007]

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Distance and invariance on positive matrix manifolds

Manifold of positive definite matrices: Mi = C i ∈ S++

P

Geometric distance: dG(S, S′) = log(S−1S′)F =

• P

i=1 log2 λk

1

2

where λk, k = 1 . . . P are the real eigenvalues of S−1S′. Tangent Space Projection: PS(S′) = Upper(log(S− 1

2 S′S− 1 2 ))

Affine invariance property: For W invertible, dG(W ⊤SW , W ⊤S′W ) = dG(S, S′)

[Bhatia. Positive Definite Matrices. Princeton University Press, 2007]

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Distance and invariance on positive matrix manifolds

Manifold of positive definite matrices: Mi = C i ∈ S++

P

Geometric distance: dG(S, S′) = log(S−1S′)F =

• P

i=1 log2 λk

1

2

where λk, k = 1 . . . P are the real eigenvalues of S−1S′. Tangent Space Projection: PS(S′) = Upper(log(S− 1

2 S′S− 1 2 ))

Affine invariance property: For W invertible, dG(W ⊤SW , W ⊤S′W ) = dG(S, S′) Affine invariance is key: working with C i is then equivalent to working with covariance matrices of sources si

[Bhatia. Positive Definite Matrices. Princeton University Press, 2007]

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Consistency of linear regression in tangent space of S++

P

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Consistency of linear regression in tangent space of S++

P

Geometric vectorization Assume yi = Q

j=1 αj log(pi,j). Denote C = MeanG(C 1, . . . , C N)

and v i = PC(C i). Then, the relationship between yi and v i is linear.

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Consistency of linear regression in tangent space of S++

P

Geometric vectorization Assume yi = Q

j=1 αj log(pi,j). Denote C = MeanG(C 1, . . . , C N)

and v i = PC(C i). Then, the relationship between yi and v i is linear. We generate i.i.d. samples following the log linear generative

• model. A = exp(µB) with B ∈ RP×P random
• chance level

0.00 0.25 0.50 0.75 1.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0

µ Normalized MAE

• log−diag

Wasserstein geometric

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And in the real world ?

We investigated 3 violations from previous idealized model:

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And in the real world ?

We investigated 3 violations from previous idealized model: Noise in target variable yi =

• j

αj log(pij) + εi , with εi ∼ N(0, σ2)

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And in the real world ?

We investigated 3 violations from previous idealized model: Noise in target variable yi =

• j

αj log(pij) + εi , with εi ∼ N(0, σ2) Subject-dependent mixing matrix Ai = A + E i , with entries of E i ∼ N(0, σ2)

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And in the real world ?

We investigated 3 violations from previous idealized model: Noise in target variable yi =

• j

αj log(pij) + εi , with εi ∼ N(0, σ2) Subject-dependent mixing matrix Ai = A + E i , with entries of E i ∼ N(0, σ2)

• chance level

0.00 0.25 0.50 0.75 1.00 0.01 0.10 1.00 10.00

σ Normalized MAE

• log−diag

Wasserstein geometric

• chance level

0.00 0.25 0.50 0.75 1.00 0.01 0.03 0.10 0.30

σ Normalized MAE

• log−diag
• sup. log−diag

Wasserstein geometric

• sup. geometric

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And in the real world ?

Rank-deficient signals (e.g. cleaning process): C i ∈ S+

P,R

[Massart, Absil. Quotient geometry with simple geodesics for the manifold

• f fixed-rank positive-semidefinite matrices. Technical report, UCLouvain, 2018] 12 / 14

And in the real world ?

Rank-deficient signals (e.g. cleaning process): C i ∈ S+

P,R

We can manipulate them in their native manifolds S+

P,R:

Wasserstein distance: dW (S, S′) =

• Tr(S) + Tr(S′) − 2Tr((S

1 2 S′S 1 2 ) 1 2 )

1

2

Tangent Space Projection: PY Y ⊤(Y ′Y ′⊤) = vect(Y ′Q∗ − Y ) ∈ RPR where UΣV ⊤ = Y ⊤Y ′, Q∗ = V U⊤ Orthogonal invariance property: For W orthogonal, dW (W ⊤SW , W ⊤S′W ) = dW (S, S′)

[Massart, Absil. Quotient geometry with simple geodesics for the manifold

• f fixed-rank positive-semidefinite matrices. Technical report, UCLouvain, 2018] 12 / 14

And in the real world ?

Rank-deficient signals (e.g. cleaning process): C i ∈ S+

P,R

We can manipulate them in their native manifolds S+

P,R:

Wasserstein distance: dW (S, S′) =

• Tr(S) + Tr(S′) − 2Tr((S

1 2 S′S 1 2 ) 1 2 )

1

2

Tangent Space Projection: PY Y ⊤(Y ′Y ′⊤) = vect(Y ′Q∗ − Y ) ∈ RPR where UΣV ⊤ = Y ⊤Y ′, Q∗ = V U⊤ Orthogonal invariance property: For W orthogonal, dW (W ⊤SW , W ⊤S′W ) = dW (S, S′) Wasserstein vectorization Assume yi = Q

j=1 αj√pi,j and A orthogonal. Denote

C = MeanW (C 1, . . . , C N) and v i = PC(C i). Then the relationship between yi and v i is linear.

[Massart, Absil. Quotient geometry with simple geodesics for the manifold

• f fixed-rank positive-semidefinite matrices. Technical report, UCLouvain, 2018] 12 / 14

Experiment: predict age from MEG data

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Experiment: predict age from MEG data

Task-free MEG recordings from Cam-CAN dataset

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Experiment: predict age from MEG data

Task-free MEG recordings from Cam-CAN dataset Age is a dominant driver of cross-person variance in neuroscience data

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Experiment: predict age from MEG data

Task-free MEG recordings from Cam-CAN dataset Age is a dominant driver of cross-person variance in neuroscience data Dimension: N = 595, P = 102, T ≃ 520, 000, 65 ≤ Ri ≤ 73

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Experiment: predict age from MEG data

Task-free MEG recordings from Cam-CAN dataset Age is a dominant driver of cross-person variance in neuroscience data Dimension: N = 595, P = 102, T ≃ 520, 000, 65 ≤ Ri ≤ 73 Signals is filtered into 9 frequency bands

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Experiment: predict age from MEG data

Task-free MEG recordings from Cam-CAN dataset Age is a dominant driver of cross-person variance in neuroscience data Dimension: N = 595, P = 102, T ≃ 520, 000, 65 ≤ Ri ≤ 73 Signals is filtered into 9 frequency bands

biophysics unsupervised identity supervised identity 6 7 8 9 10 11

mean absolute error (years)

log−diag Wasserstein geometric MNE

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Conclusion

Proposed Riemannian method for regression from M/EEG data

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Conclusion

Proposed Riemannian method for regression from M/EEG data With theoritical guarantees and empirical robustness

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Conclusion

Proposed Riemannian method for regression from M/EEG data With theoritical guarantees and empirical robustness Working to translate proposed method into the clinic

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Conclusion

Proposed Riemannian method for regression from M/EEG data With theoritical guarantees and empirical robustness Working to translate proposed method into the clinic [Sabbagh, Ablin, Varoquaux, Gramfort, Engemann (2019), Manifold-regression to predict from MEG/EEG brain signals without source modeling, Proc. NeurIPS 2019] https://arxiv.org/abs/1906.02687

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