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- 1. Brain reading: Compressive sensing, fMRI, and statistical learning in Python Ga¨l Varoquaux e INRIA/Parietal
- 2. 1 Brain reading: predictive models 2 Sparse recovery with correlated de- signs 3 Having an impact: softwareG Varoquaux 2
- 3. 1 Brain reading: predictive models Functional brain imaging: Study of human cognitionG Varoquaux 3
- 4. 1 Brain imaging fMRI data > 50 000 voxels stimuliG Varoquaux 4
- 5. 1 Brain imaging fMRI data > 50 000 voxels stimuli Standard analysis Detect voxels that correlate to the stimuliG Varoquaux 4
- 6. 1 Brain reading Predicting the object category viewed [Haxby 2001, Distributed and Overlapping Representations of Faces and Objects in Ventral Temporal Cortex ] Supervised learning taskG Varoquaux 5
- 7. 1 Brain reading Predicting the object category viewed [Haxby 2001, Distributed and Overlapping Representations of Faces and Objects in Ventral Temporal Cortex ]combinations of voxels to Find predict the stimuli Supervised learning task Multi-variate statisticsG Varoquaux 5
- 8. 1 Linear model for fMRI sign(X w + e) = y Target Design matrix × Coeﬃcients = Problem size: p > 50 000 n∼100 per categoryG Varoquaux 6
- 9. 1 Estimation: statistical learning Inverse problem Minimize an error term: w = argmin l(y − X w) ˆ w Ill-posed: X is not full rank Inject prior: regularize w = argmin l(y − X w) + p(w) ˆ wG Varoquaux 7
- 10. 1 Estimation: statistical learning Inverse problem Minimize an error term: w = argmin l(y − X w) ˆ w Example: Lasso = sparseis not full rank Ill-posed: X regression w = argmin y − X w 2 + 1 (w) ˆ 2 w 1 (w) = i |wi |Inject prior: regularize w = argmin l(y − X w) + p(w) ˆ wG Varoquaux 7
- 11. 1 TV-penalization to promote regions [Haxby 2001] Neuroscientists think in terms of brain regions Total-variation penalization Impose sparsity on the gradient of the image: p(w) = 1( w) [Michel TMI 2011]G Varoquaux 8
- 12. 1 Prediction with logistic regression - TV w = argmin l(y − X w) + p(w) ˆ w l: least-square or logistic-regression p: TV Optimization: proximal gradient (FISTA) - Gradient descent on l (smooth term) - Projections on TV Prediction performance: Feature screening + SVC 0.77 Sparse regression 0.78 Total Variation 0.84 (explained variance)G Varoquaux 9
- 13. 1 Prediction with logistic regression - TV wStandard l(y − X w) + p(w) ˆ = argmin analysis w l: least-square or logistic-regression p: TV Optimization: proximal gradient (FISTA) - Gradient descent on l (smooth term) - Projections on TV Prediction performance: Feature screening + SVC 0.77 Sparse regression 0.78 Total Variation 0.84 (explained variance)G Varoquaux 9
- 14. 1 Standard analysis or predictive modeling? Predicting the object category viewed [Haxby 2001, Distributed and Overlapping Representations of Faces and Objects in Ventral Temporal Cortex ] Take home message: brain regions, not predictionG Varoquaux 10
- 15. 1 Standard analysis or predictive modeling? Recovery rather than predictionG Varoquaux 11
- 16. 1 Good prediction = good recovery Ground truth Simulations Ground truth Lasso Prediction: 0.78 Recovery: 0.429 SVM Prediction: 0.71 Recovery: 0.486 Need a method suited for recoveryG Varoquaux 12
- 17. 1 Brain mapping: a statistical perspective Small sample linear model estimation Random correlated design × = Problem size: p > 50 000 n∼100 per categoryG Varoquaux 13
- 18. 1 Brain mapping: a statistical perspective Small sample linear model estimation Random correlated design Estimation strategy Standard approach: univariate statistics × Multiple comparisons problem ⇒ statistical power ∝ 1/p = We want sub-linear sample complexity ⇒ non-rotationally-invariant estimators e.g. 1 penalization [ Ng, 2004 Feature selection, 1 vs. 2 regularization, and rotational invariance ]G Varoquaux 13
- 19. 1 Brain mapping as a sparse recovery task Recovering brain regionsG Varoquaux 14
- 20. 1 Brain mapping as a sparse recovery task Recovering k non-zero coeﬃcients nmin ∼ 2 k log p Restricted-isometry-like property: The design matrix is well-conditioned [Candes 2006] on sub-matrices of size > k [Tropp 2004] [Wainwright 2009] Mutual incoherence: Relevant features S and irrelevant ones S are not too correlated Violated by spatial correlations in our design lasso: 23 non-zerosG Varoquaux 14
- 21. 1 Randomized sparsity [Meinshausen and Buhlmann 2010, Bach 2008] Perturb the design matrix: Subsample the data Randomly rescale features + Run sparse estimator Keep features that are often selected ⇒ Good recovery without mutual incoherence But RIP-like condition Cannot recover large correlated groups For m correlated features, selection frequency divided by mG Varoquaux 15
- 22. 2 Sparse recovery with correlated designs Not enough samples: nmin ∼ 2 k log p Spatial correlationsG Varoquaux 16
- 23. 2 Sparse recovery with correlated designs Combining Clustering Sparsity [Varoquaux ICML 2012]G Varoquaux 16
- 24. 2 Brain parcellations Spatially-connected hierarchical clustering ⇒ reduces voxel numbers [Michel Pat Rec 2011] Replace features by corresponding cluster average + Use a supervised learner on reduced problem Cluster choice sub-optimal for regressionG Varoquaux 17
- 25. 2 Brain parcellations + sparsity Hypothesis: clustering compatible with support(w) Beneﬁts of clustering Reduced k and p ⇒ n > nmin : good side of the “sharp threshold” Cluster together correlated features ⇒ Improves RIP-like conditions Recovery possible on reduced featuresG Varoquaux 18
- 26. 2 Randomized parcellations + sparsity Randomization + Stability scores Marginalize the cluster choice Relaxes mutual incoherence requirementG Varoquaux 19
- 27. 2 Algorithm 1 set n clusters and sparsity by cross-validation 2 loop: perturb randomly data 3 clustering to form reduced features 4 sparse linear model on reduced features 5 accumulate non-zero features 6 threshold map of apparition countsG Varoquaux 20
- 28. 2 Simulations p = 2048, k = 64, n = 256 (nmin > 1000) Weights w: patches of varying size Design matrix: 2D Gaussian random images of varying smoothness Estimators Randomized lasso Our approach Elastic Net Univariate F test Parameters set by cross-validation Performance metric Recovery seen as a 2-class problem ⇒ Report AUC of the precision-recall curveG Varoquaux 21
- 29. 2 When can we recover patches? Smoothness helps (reduces noise degrees of freedom) Small patches are hard to recoverG Varoquaux 22
- 30. 2 What is the best method for patch recovery? For small patches: elastic net For large patches: randomized-clustered sparsity Large patches and very smooth images: F-testG Varoquaux 23
- 31. 2 Randomizing clusters matters! Non-random (Ward) clustering ineﬃcient Fully-random performs quite well Randomized Ward gives an extra gainG Varoquaux 24
- 32. 2 Randomizing clusters matters! Degenerate family of cluster assignements Non-random (Ward) clustering ineﬃcient Fully-random performs quite well Randomized Ward gives an extra gainG Varoquaux 24
- 33. 2 fMRI: face vs house discrimination [Haxby 2001] F-scores L R L Ry=-31 x=17 z=-17 G Varoquaux 25
- 34. 2 fMRI: face vs house discrimination [Haxby 2001] 1 Logistic L R L Ry=-31 x=17 z=-17 G Varoquaux 25
- 35. 2 fMRI: face vs house discrimination [Haxby 2001] Randomized 1 Logistic L R L Ry=-31 x=17 z=-17 G Varoquaux 25
- 36. 2 fMRI: face vs house discrimination [Haxby 2001] Randomized Clustered 1 Logistic L R L Ry=-31 x=17 z=-17 G Varoquaux 25
- 37. 2 fMRI: face vs house discrimination [Haxby 2001] F-scores L R L Ry=-31 x=17 z=-17 G Varoquaux 25
- 38. 2 Predictive model on selected features Object recognition [Haxby 2001] Using recovered features improves predictionG Varoquaux 26
- 39. Small-sample brain mapping Sparse recovery of patches on spatially-correlated designs Ingredients: Clustering + Randomization ⇒ Reduced feature set compatible with recovery: matches sparsity pattern + recovery conditions Compressive sensing questions Can we recover k > n, in the case of large patches? When do we loose sub-linear sample-complexity?G Varoquaux 27
- 40. 3 Having an impact: software How to we reach our target audience (neuroscientists)? How do we disseminate our ideas? How do we facilitate new ideas?G Varoquaux 28
- 41. 3 Python as a scientiﬁc environment General purpose Easy, readable syntax Interactive (ipython) Great scientiﬁc libraries (numpy, scipy, matplotlib...)G Varoquaux 29
- 42. 3 Growing a software stack Code lines are costly ⇒ Open source + community driven Need quality and impact ⇒ Focus on the general purpose libraries ﬁrst Scikit-learn: machine learning in Python http://scikit-learn.orgG Varoquaux 30
- 43. 3 scikit-learn: machine learning in PythonTechnical choices Prefer Python or Cython, focus on readability Documentation and examples are paramount Little object-oriented design. Opt for simplicity Prefer algorithms to framework Code quality: consistency and testingG Varoquaux 31
- 44. 3 scikit-learn: machine learning in PythonAPI Inputs are numpy arrays Learn a model from the data: estimator.fit(X train, Y train) Predict using learned model estimator.predict(X test) Test goodness of ﬁt estimator.score(X test, y test) Apply change of representation estimator.transform(X, y)G Varoquaux 32
- 45. 3 scikit-learn: machine learning in PythonComputational performance scikit-learn mlpy pybrain pymvpa mdp shogunSVM 5.2 9.47 17.5 11.52 40.48 5.63LARS 1.17 105.3 - 37.35 - -Elastic Net 0.52 73.7 - 1.44 - -kNN 0.57 1.41 - 0.56 0.58 1.36PCA 0.18 - - 8.93 0.47 0.33k-Means 1.34 0.79 ∞ - 35.75 0.68 Algorithms rather than low-level optimization convex optimization + machine learning Avoid memory copiesG Varoquaux 33
- 46. 3 scikit-learn: machine learning in PythonCommunity 163 contributors since 2008, 397 github forks 25 contributors in latest release (3 months span)Why this success? Trendy topic? Low barrier of entry Friendly and very skilled mailing list Credit to peopleG Varoquaux 34
- 47. 3 Research code = software library Factor 10 in time investment Corner cases in algorithm (numerical stability) Multiple platforms and library versions (Blas ) Documentation Making it simpler (and get less educated users) User and developer support ( ∼ 100 mails/day) Exhausting, but has impact on science and societyG Varoquaux 35
- 48. 3 Research code = software library Factor 10 in time investment CornerTechnical + scientiﬁc tradeoﬀs cases in algorithm (numerical stability) Multiple platforms andof use rather than speed Ease of install/ease library versions (Blas ) Documentation science” Focus on “old Making it simpler (and get less educatednot Nice publications and theorems are users) a recipe for useful code User and developer support ( ∼ 100 mails/day) Exhausting, but has impact on science and societyG Varoquaux 35
- 49. Statistical learning to study brain function Spatial regularization for predictive models Total variation Compressive-sensing approach Sparsity + randomized clustering for correlated designs Machine learning in Python Huge impact Post-doc positions availableG Varoquaux 36
- 50. Bibliography[Michel TMI 2011] V. Michel, et al., Total variation regularization forfMRI-based prediction of behaviour, IEEE Transactions in medicalimaging (2011)http://hal.inria.fr/inria-00563468/en[Varoquaux ICML 2012] G. Varoquaux, A. Gramfort, B. ThirionSmall-sample brain mapping: sparse recovery on spatially correlateddesigns with randomization and clustering, ICML (2012)http://hal.inria.fr/hal-00705192/en[Michel Pat Rec 2011] V. Michel, et al., A supervised clustering approachfor fMRI-based inference of brain states, Pattern Recognition (2011)http://hal.inria.fr/inria-00589201/en[Pedregosa ICML 2011] F. Pedregosa, el al., Scikit-learn: machinelearning in Python, JMRL (2011)http://hal.inria.fr/hal-00650905/enG Varoquaux 37

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