19/01/2018 Bertrand Thirion – Statistics/Learning at Paris-Saclay 1

Toward a rigorous statistical framework for brain mapping

Bertrand Thirion, bertrand.thirion@inria.fr

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Cognitive neuroscience

How are cognitive activities affected or controlled by neural circuits in the brain ?

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The brain, the mind and the scanner

Cognitive theories

Brain

Sca

nn

er

FMRI data

Brain mapping

Experimental paradigm

stimuli

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The brain, the mind and the scanner

Cognitive theories

Brain

Sca

nn

er

FMRI data

Decoding

Encoding

Brain mapping

Experimental paradigm

stimuli

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Mapping cognitive functions to brain activity

right hand-left hand

listen-read

Button press - reading

Computation - instructions

expression - intention

Guess the gender

Face trustworthiness

false belief – mechanistic(auditory) False belief –

mechanistic(visual)

Grasping-orientation judgement

Hand – side judgement

Intention - random

Sentence - checkerboards

saccades - fixation

Speech-non speech

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Resolution increases

2007: 3 mm

2014: 1.5 mm

2020: 0.5 mm ?

p = 50,000 p = 400,000 p = 107

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better estimators for large-scale brain imaging

● Massive online dictionary learning● Dimension reduction for images● Fast regularized ensembles of models● Statistical inference for high-dimensional models

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fMRI datasets are feature-rich

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Discovering structure in fMRI

Can be captured by dictionary learning / sparse coding [Olshausen Nature 1996]→ Use of sparse PCA

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High-dimensional fMRI

● n = number of samples, 102 to 106

● p = number of voxels, 105-106

“Has anyone on the ML run group-wise analysis on the HCP resting state data, and if so what tools did you use?

I am having memory issues when running more than 10 subjects and I was wondering if anyone has a way of getting around the large memory requirements when concatenating in time.”

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Huge ?

● Human Connectome project n=2.106, p=2.105, 2TB of data

● Online dictionary learning [Mairal et al. ICML 2009]

● Constrained rather than penalized formulation ● How to go faster ?

– Work on batches of images and voxels● Online method in both samples and feature dimensions

[Mensch et al. ICML 2016, IEEE TSP 2018]

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Stochastic gradient approacheshttp://amensch.fr/research/2016/06/10/modl.html

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Stochastic gradient approaches

10-fold gain in CPU time without loss in accuracy

Can be used for recommender systems

http://amensch.fr/research/2016/06/10/modl.html

[Mensch et al. ICML 2016,

IEEE TSP 2018]

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Brain atlases

[Mensch et al. ICML 2016 IEEE TSP 2018]

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Leveraging rest data for brain decoding

Different datasets share some common patterns

Different datasets share some common representations[Bzdok et al. Plos Comp Biol 2016, Mensch et al NIPS 2017]

Cognitive representation

Spatial representation

FACES

VIDEO ONLY

HORIZONTAL CHECKERBOARD

HCP

Archi

CamCan

Joint training

Task fMRI study

Z-score images

Cognitive-aware projection

Per-dataset classification

Sparse decompositions

Resting-state data

~ 2

00

00

0

~ 3

36

~ 5

0

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Advantage of large-scale analysis

Information transferred from large datasets (HCP) to smaller ones increases classification accuracy[Mensch et al NIPS 2017]

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Outline

● Massive online dictionary learning● Dimension reduction for images● Fast regularized ensembles of Models● Statistical inference for high-dimensional

models

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Compression in the image domain

● Reduce the complexity of learning algorithms: p→k ≪ p

● Random projections = fast generic solution, but – Sub-optimal for structured signals

– Not invertible when p and k are large

● Local redundancy → feature grouping strategies / clustering: “super-pixels”– Fast clustering procedures needed (large k regime)

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Compression by feature grouping

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Crafting good image compression

● Key assumption: signal of interest L-Lipschitz

● Feature grouping matrix

● almost trivially:

● And

● Worst case

Need a fast method to learn balanced clusters

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Denoising properties

● Noisy signal model

● Denoising

● Equal-size clusters

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Recursive nearest neighbor

Based on local decisions = fast (linear time) – avoid percolation

[Thirion et al. Stamlins 2015]

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Effect on data analysis tasks

Impressive speed-up and increased accuracy with respect to non-compressed representation

– Clustering has a denoising effect

[Hoyos Idrobo IEEE PAMI under revision]

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More results

[Hoyos Idrobo IEEE PAMI under revision]

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Outline

● Massive online dictionary learning● Dimension reduction for images● Fast regularized ensembles of Models● Statistical inference for high-dimensional

models

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Brain activity decoding

X1

X2

Xp

... y

● behavior = f (brain activity)

w

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Penalized linear regression

> convex optimization> set hyperparameters by cross-validation

Ridge (shrinkage)

Lasso (very sparse)

Elastic net (sparsity + grouping)

Smooth lasso (sparsity + smoothness)

Total variation (piecewise sparsity)

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Structure-inducing priors

● Large p → redundancy, latent structure● Brain imaging: spatial regularity → small total

variationHCP dataset: “shape” versus “face” contrast across subjects

[Michel et al TMI 2011, Gramfort et al. 2013 Eickenberg et al. MICCAI 2015, Dohmatob et al. PRNI 2014, 2015]

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Optimizing TV takes time

[Varoquaux et al. GRETSI 2015]

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Bagging of sparse clustered models

X

Clustering(create contiguous regions)

y

Solve Lasso

on cluster-based

representation

average

...

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Computationally efficient structure

State of the art solution: not very stable, but cheap

“fast regularized ensembles of models”

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Computationally efficient structure

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Computationally efficient structure

“fast regularized ensembles of models”

[Hoyos Idrobo et al PRNI 2015, Neuroimage 2017, PAMI under review]

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Computationally efficient structure

[Hoyos Idrobo et al PRNI 2015, Neuroimage 2017, PAMI under review]

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Computationally efficient structure

[Hoyos Idrobo et al PRNI 2015, Neuroimage 2017, PAMI under review]

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Benchmark

HCP dataset: “shape” versus “face” contrast across subjects

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Outline

● Massive online dictionary learning● Dimension reduction for images● Fast regularized ensembles of Models● Statistical inference for high-dimensional

models

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Statistical inference on w

● Standard solutions for high-dimensional linear models (p > n)– Corrected ridge

– Desparsified Lasso

● Adaptation to brain imaging (p≫n)

● Inference: find {j: wj > 0} with some statistical guarantees

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Desparsified Lasso

[Zhang & Zhang 2014 Series B Stat Meth]

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Desparsified Lasso

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Desparsified Lasso: which λ

[zhang & zhang 2014 Series B Stat Meth]

● For each j,

– ηj should be as small as possible

● keep λ small

– τj should be as high as possible

● λ not too small

Evaluating η and τ for many λ’s is expensive

→ We choose λ = .03 λmax

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Preliminary assessment

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Preliminary assessment

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Preliminary assessment

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Adaptation to brain imaging

Step 1: clustering

Step 2: inference on compressed representations

Step 3: repeat on different parcellations and aggregate the p-values(FReM-like approach)

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Some initial results

DL p-values from different clusterings

aggregation

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Some initial results

DL p-values from different clusterings

aggregation

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Conclusion

● Large-p data bring challenges:– Computation cost

– Overfit

– Difficulty of statistical inference

● Solutions: online learning, subsampling, compression

● Ensembling improves estimators

● Open frontiers: statistical inference

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From good ideas to good practices: software

● Machine learning in Python

● Machine learning for neuroimaging http://nilearn.github.io

● BSD, Python, OSS

– Classification of (neuroimaging) data

– Network analysis

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AcknowledgementsParietalG. Varoquaux,A. Gramfort,P. Ciuciu,D. Wassermann,D. Engemann,A. Manoel,D. ChyzhykA.L. Grilo Pinho,E. Dohmatob,A. Mensch,J.A. Chevalier,A. Hoyos idrobo,D. Bzdok,J. Dockès,P. Cerda,C. LazarusD. La RoccaG. LemaitreL. El GueddariO. GriselM. MassiasP. AblinH. JanatiJ. MassichK. DadiC. Petitot

Other collaborators (thanks for the data)S. Dehaene R. Poldrack, J. HaxbyC. F. GorgolevskiJ. Salmon