Manifold-regression to predict from MEG/EEGbrain 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

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

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

We measure M/EEG signal of subject i = 1 . . .N on P channels:

x i (t) = A s i (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 s i (t) ∈ RQ

noise ni (t) ∈ RP

Under stationnarity and gaussianity assumptions, we can representband-pass filtered signal by its second-order statistics

C i = E[x i (t)x i (t)>] ' X iX>iT

∈ RP×P with X i ∈ RP×T

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) + 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 s i (t) ∈ RQ

noise ni (t) ∈ RP

Under stationnarity and gaussianity assumptions, we can representband-pass filtered signal by its second-order statistics

C i = E[x i (t)x i (t)>] ' X iX>iT

∈ RP×P with X i ∈ RP×T

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) + 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 s i (t) ∈ RQ

noise ni (t) ∈ RP

Under stationnarity and gaussianity assumptions, we can representband-pass filtered signal by its second-order statistics

C i = E[x i (t)x i (t)>] ' X iX>iT

∈ RP×P with X i ∈ RP×T

4 / 14

Model: generative model of target variable

We want to predict a continuous variable: yi =Q∑j=1

αj f (pi ,j) ∈ R

with pi ,j = Et [s2i ,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 naivelyvectorized

5 / 14

Model: generative model of target variable

We want to predict a continuous variable: yi =Q∑j=1

αj f (pi ,j) ∈ R

with pi ,j = Et [s2i ,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 naivelyvectorized

5 / 14

Model: generative model of target variable

We want to predict a continuous variable: yi =Q∑j=1

αj f (pi ,j) ∈ R

with pi ,j = Et [s2i ,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 naivelyvectorized

5 / 14

Model: generative model of target variable

We want to predict a continuous variable: yi =Q∑j=1

αj f (pi ,j) ∈ R

with pi ,j = Et [s2i ,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 naivelyvectorized

5 / 14

Model: generative model of target variable

We want to predict a continuous variable: yi =Q∑j=1

αj f (pi ,j) ∈ R

with pi ,j = Et [s2i ,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 naivelyvectorized

5 / 14

Riemannian matrix manifolds (in a nutshell)

ξ MT

M

M M

M'

LogM

ExpM

6 / 14

Riemannian matrix manifolds (in a nutshell)

ξ MT

M

M M

M'

LogM

ExpM

Vectorization operator:PM(M ′) = φM(LogM(M ′)) ' Upper(LogM(M ′))d(M ,M ′) = ‖PM(M ′)‖2 + o(‖PM(M ′)‖2)d(M i ,M j) ' ‖PM(M i )− PM(M j)‖2Vectorization operator key for ML

[Absil & al. Optimization algorithms on matrix manifolds. 2009]7 / 14

Riemannian matrix manifolds (in a nutshell)

ξ MT

M

M M

M'

LogM

ExpM

Vectorization operator:PM(M ′) = φM(LogM(M ′)) ' Upper(LogM(M ′))d(M ,M ′) = ‖PM(M ′)‖2 + o(‖PM(M ′)‖2)d(M i ,M j) ' ‖PM(M i )− PM(M j)‖2Vectorization operator key for ML

[Absil & al. Optimization algorithms on matrix manifolds. 2009]7 / 14

Regression on matrix manifolds

Given a training set of samples M1, . . . ,MN ∈M and targetcontinuous 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:v1, . . . , vN ∈ RK as v i = PM(M i )

use those vectors as features in regularized linear regressionalgorithm (e.g. ridge regression) with parameters β ∈ RK

assuming that yi ' v>i β

8 / 14

Regression on matrix manifolds

Given a training set of samples M1, . . . ,MN ∈M and targetcontinuous 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:v1, . . . , vN ∈ RK as v i = PM(M i )

use those vectors as features in regularized linear regressionalgorithm (e.g. ridge regression) with parameters β ∈ RK

assuming that yi ' v>i β

8 / 14

Regression on matrix manifolds

Given a training set of samples M1, . . . ,MN ∈M and targetcontinuous 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:v1, . . . , vN ∈ RK as v i = PM(M i )

use those vectors as features in regularized linear regressionalgorithm (e.g. ridge regression) with parameters β ∈ RK

assuming that yi ' v>i β

8 / 14

Regression on matrix manifolds

Given a training set of samples M1, . . . ,MN ∈M and targetcontinuous 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:v1, . . . , vN ∈ RK as v i = PM(M i )

use those vectors as features in regularized linear regressionalgorithm (e.g. ridge regression) with parameters β ∈ RK

assuming that yi ' v>i β

8 / 14

Distance and invariance on positive matrix manifolds

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

Geometric distance:

dG (S ,S ′) = ‖ log(S−1S ′)‖F =

[∑Pi=1 log2 λk

] 12

where λk , k = 1 . . .P are the real eigenvalues of S−1S ′.

Tangent Space Projection:

PS(S ′) = Upper(log(S−12S ′S−

12 ))

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 toworking with covariance matrices of sources s i

[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:

dG (S ,S ′) = ‖ log(S−1S ′)‖F =

[∑Pi=1 log2 λk

] 12

where λk , k = 1 . . .P are the real eigenvalues of S−1S ′.

Tangent Space Projection:

PS(S ′) = Upper(log(S−12S ′S−

12 ))

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 toworking with covariance matrices of sources s i

[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:

dG (S ,S ′) = ‖ log(S−1S ′)‖F =

[∑Pi=1 log2 λk

] 12

where λk , k = 1 . . .P are the real eigenvalues of S−1S ′.

Tangent Space Projection:

PS(S ′) = Upper(log(S−12S ′S−

12 ))

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 toworking with covariance matrices of sources s i

[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:

dG (S ,S ′) = ‖ log(S−1S ′)‖F =

[∑Pi=1 log2 λk

] 12

where λk , k = 1 . . .P are the real eigenvalues of S−1S ′.

Tangent Space Projection:

PS(S ′) = Upper(log(S−12S ′S−

12 ))

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 toworking with covariance matrices of sources s i

[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:

dG (S ,S ′) = ‖ log(S−1S ′)‖F =

[∑Pi=1 log2 λk

] 12

where λk , k = 1 . . .P are the real eigenvalues of S−1S ′.

Tangent Space Projection:

PS(S ′) = Upper(log(S−12S ′S−

12 ))

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 toworking 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

Geometric vectorization

Assume yi =∑Q

j=1 αj log(pi ,j). Denote C = MeanG (C 1, . . . ,CN)and v i = PC (C i ). Then, the relationship between yi and v i islinear.

We generate i.i.d. samples following the log linear generativemodel. 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 3.0µ

Nor

mal

ized

MA

E

● ● ●log−diag Wasserstein geometric

10 / 14

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, . . . ,CN)and v i = PC (C i ). Then, the relationship between yi and v i islinear.

We generate i.i.d. samples following the log linear generativemodel. 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 3.0µ

Nor

mal

ized

MA

E

● ● ●log−diag Wasserstein geometric

10 / 14

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, . . . ,CN)and v i = PC (C i ). Then, the relationship between yi and v i islinear.

We generate i.i.d. samples following the log linear generativemodel. 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 3.0µ

Nor

mal

ized

MA

E

● ● ●log−diag Wasserstein geometric

10 / 14

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 10.00σ

Nor

mal

ized

MA

E

● ● ●log−diag Wasserstein geometric

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●●

●●

●

●●●●

●

●●

●●

●●●●●

●●●●●

chance level

0.00

0.25

0.50

0.75

1.00

0.01 0.03 0.10 0.30σ

Nor

mal

ized

MA

E

●

●

●

●

●log−diagsup. log−diag

Wassersteingeometric

sup. geometric

11 / 14

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 10.00σ

Nor

mal

ized

MA

E

● ● ●log−diag Wasserstein geometric

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●●

●●

●

●●●●

●

●●

●●

●●●●●

●●●●●

chance level

0.00

0.25

0.50

0.75

1.00

0.01 0.03 0.10 0.30σ

Nor

mal

ized

MA

E

●

●

●

●

●log−diagsup. log−diag

Wassersteingeometric

sup. geometric

11 / 14

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 10.00σ

Nor

mal

ized

MA

E

● ● ●log−diag Wasserstein geometric

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●●

●●

●

●●●●

●

●●

●●

●●●●●

●●●●●

chance level

0.00

0.25

0.50

0.75

1.00

0.01 0.03 0.10 0.30σ

Nor

mal

ized

MA

E

●

●

●

●

●log−diagsup. log−diag

Wassersteingeometric

sup. geometric

11 / 14

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 10.00σ

Nor

mal

ized

MA

E

● ● ●log−diag Wasserstein geometric

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●●

●●

●

●●●●

●

●●

●●

●●●●●

●●●●●

chance level

0.00

0.25

0.50

0.75

1.00

0.01 0.03 0.10 0.30σ

Nor

mal

ized

MA

E

●

●

●

●

●log−diagsup. log−diag

Wassersteingeometric

sup. geometric

11 / 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

12S ′S

12 )

12 )] 1

2

Tangent Space Projection:PYY>(Y ′Y ′>) = vect(Y ′Q∗ − Y ) ∈ RPR

where UΣV> = Y>Y ′, Q∗ = VU>

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, . . . ,CN) 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 manifoldof 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

12S ′S

12 )

12 )] 1

2

Tangent Space Projection:PYY>(Y ′Y ′>) = vect(Y ′Q∗ − Y ) ∈ RPR

where UΣV> = Y>Y ′, Q∗ = VU>

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, . . . ,CN) 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 manifoldof 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

12S ′S

12 )

12 )] 1

2

Tangent Space Projection:PYY>(Y ′Y ′>) = vect(Y ′Q∗ − Y ) ∈ RPR

where UΣV> = Y>Y ′, Q∗ = VU>

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, . . . ,CN) 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 manifoldof fixed-rank positive-semidefinite matrices. Technical report, UCLouvain, 2018] 12 / 14

Experiment: predict age from MEG data

Task-free MEG recordings from Cam-CAN dataset

Age is a dominant driver of cross-person variance inneuroscience 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 11mean absolute error (years)

log−diag Wasserstein geometric MNE

13 / 14

Experiment: predict age from MEG data

Task-free MEG recordings from Cam-CAN dataset

Age is a dominant driver of cross-person variance inneuroscience 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 11mean absolute error (years)

log−diag Wasserstein geometric MNE

13 / 14

Experiment: predict age from MEG data

Task-free MEG recordings from Cam-CAN dataset

Age is a dominant driver of cross-person variance inneuroscience 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 11mean absolute error (years)

log−diag Wasserstein geometric MNE

13 / 14

Experiment: predict age from MEG data

Task-free MEG recordings from Cam-CAN dataset

Age is a dominant driver of cross-person variance inneuroscience 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 11mean absolute error (years)

log−diag Wasserstein geometric MNE

13 / 14

Experiment: predict age from MEG data

Task-free MEG recordings from Cam-CAN dataset

Age is a dominant driver of cross-person variance inneuroscience 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 11mean absolute error (years)

log−diag Wasserstein geometric MNE

13 / 14

Experiment: predict age from MEG data

Task-free MEG recordings from Cam-CAN dataset

Age is a dominant driver of cross-person variance inneuroscience 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 11mean absolute error (years)

log−diag Wasserstein geometric MNE

13 / 14

Conclusion

Proposed Riemannian method for regression from M/EEGdata

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 signalswithout source modeling, Proc. NeurIPS 2019]

https://arxiv.org/abs/1906.02687

14 / 14

Conclusion

Proposed Riemannian method for regression from M/EEGdata

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 signalswithout source modeling, Proc. NeurIPS 2019]

https://arxiv.org/abs/1906.02687

14 / 14

Conclusion

Proposed Riemannian method for regression from M/EEGdata

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 signalswithout source modeling, Proc. NeurIPS 2019]

https://arxiv.org/abs/1906.02687

14 / 14

Conclusion

Proposed Riemannian method for regression from M/EEGdata

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 signalswithout source modeling, Proc. NeurIPS 2019]

https://arxiv.org/abs/1906.02687

14 / 14