Talk given at the OHBM 2017 education course. I present the challenges and techniques to estimating meaningful brain functional connectomes from fMRI: why sparsity in inverse covariance leads to models that can interpreted as interactions between regions. Then I discuss the limitations of sparse estimators and introduce shrinkage as an alternative. Finally, I discuss how to compare multiple functional connectomes.

1. Estimating Functional Connectomes:
Sparsity’s Strength and Limitations
Ga¨el Varoquaux Ssssssskeptical

2. Graphical models in cognitive neuroscience
G Varoquaux 2

3. Functional connectome analysis
Functional regions
G Varoquaux 3

4. Functional connectome analysis
Functional regions
Functional connections
G Varoquaux 3

5. Functional connectome analysis
Functional regions
Functional connections
Variations in connections
G Varoquaux 3

6. Outline
1 Estimating connectomes
2 Comparing connectomes
G Varoquaux 4

7. 1 Estimating connectomes
Functional connectome
Graph of interactions between regions
[Varoquaux and Craddock 2013]
G Varoquaux 5

8. 1 Graphical model in cognitive neuroscience
Whish list
Causal links
Directed model:
IPS = V 2 + MT
FEF = IPS + ACC
G Varoquaux 6

9. 1 Graphical model in cognitive neuroscience
Whish list
Causal links
Directed model:
IPS = V 2 + MT
FEF = IPS + ACC
Unreliable delays (HRF)
Few samples
× many signals
Heteroscedastic noise
G Varoquaux 6

10. 1 Graphical model in cognitive neuroscience
Whish list
Causal links
Directed model:
IPS = V 2 + MT
FEF = IPS + ACC
Unreliable delays (HRF)
Few samples
× many signals
Heteroscedastic noise
Independence structure
Knowing IPS, FEF is independent of V2 and MT
G Varoquaux 6

11. 1 From correlations to connectomes
Conditional independence structure?
G Varoquaux 7

12. 1 Probabilistic model for interactions
Simplest data generating process
= multivariate normal:
P(X) ∝ |Σ−1|e−1
2XT Σ−1X
Model parametrized by inverse covariance matrix,
K = Σ−1
: conditional covariances
Goodness of fit:
likelihood of observed covariance ˆΣ in model Σ
L( ˆΣ|K) = log |K| − trace( ˆΣ K)
G Varoquaux 8

13. 1 Graphical structure from correlations
Observations
Covariance
0
1
2
3
4
Diagonal:
signal variance
Direct connections
Inverse covariance
0
1
2
3
4
Diagonal:
node innovation
G Varoquaux 9

14. 1 Independence structure (Markov graph)
Zeros in partial correlations
give conditional independence
Reflects the large-scale
brain interaction structure
G Varoquaux 10

15. 1 Independence structure (Markov graph)
Zeros in partial correlations
give conditional independence
Ill-posed problem:
multi-collinearity
⇒ noisy partial correlations
Independence between nodes makes estimation
of partial correlations well-conditionned.
Chicken and egg problem
G Varoquaux 10

16. 1 Independence structure (Markov graph)
Zeros in partial correlations
give conditional independence
Ill-posed problem:
multi-collinearity
⇒ noisy partial correlations
Independence between nodes makes estimation
of partial correlations well-conditionned.
0
1
2
3
4
0
1
2
3
4
+
Joint estimation:
Sparse inverse covariance
G Varoquaux 10

17. 1 Sparse inverse covariance: penalization
[Friedman... 2008, Varoquaux... 2010b, Smith... 2011]
Maximum a posteriori:
Fit models with a penalty
Sparsity ⇒ Lasso-like problem: 1 penalization
K = argmin
K 0
L( ˆΣ|K) + λ 1(K)
Data fit,
Likelihood
Penalization,
x2
x1
G Varoquaux 11

18. 1 Sparse inverse covariance: penalization
[Varoquaux... 2010b]
ˆΣ−1 Sparse
inverse
Likelihood of new data (cross-validation)
Subject data, Σ−1
-57.1
Subject data, sparse inverse 43.0
G Varoquaux 12

19. 1 Limitations of sparsity Sssssskeptical
Theoretical limitation to sparse recovery
Number of samples for s edges, p nodes:
n = O (s + p) log p [Lam and Fan 2009]
High-degree nodes fail [Ravikumar... 2011]
Empirically
Optimal graph
almost dense
2.5 3.0 3.5 4.0
−log10λ
Test-datalikelihood
Sparsity
[Varoquaux... 2012]
Very sparse graphs
don’t fit the data
G Varoquaux 13

20. 1 Multi-subject to overcome subject data scarsity
[Varoquaux... 2010b]
ˆΣ−1 Sparse
inverse
Sparse group
concat
Likelihood of new data (cross-validation)
Subject data, Σ−1
-57.1
Subject data, sparse inverse 43.0
Group concat data, Σ−1
40.6
Group concat data, sparse inverse 41.8
Inter-subject variability
G Varoquaux 14

21. 1 Multi-subject sparsity
[Varoquaux... 2010b]
Common independence structure but different
connection values
{Ks
} = argmin
{Ks 0} s
L( ˆΣs
|Ks
) + λ 21({Ks
})
Multi-subject data fit,
Likelihood
Group-lasso penalization
G Varoquaux 15

22. 1 Multi-subject sparsity
[Varoquaux... 2010b]
Common independence structure but different
connection values
{Ks
} = argmin
{Ks 0} s
L( ˆΣs
|Ks
) + λ 21({Ks
})
Multi-subject data fit,
Likelihood
1 on the connections of
the 2 on the subjects
G Varoquaux 15

23. 1 Multi-subject sparse graphs perform better
[Varoquaux... 2010b]
ˆΣ−1 Sparse
inverse
Population
prior
Likelihood of new data (cross-validation) sparsity
Subject data, Σ−1
-57.1
Subject data, sparse inverse 43.0 60% full
Group concat data, Σ−1
40.6
Group concat data, sparse inverse 41.8 80% full
Group sparse model 45.6 20% full
G Varoquaux 16

24. 1 Independence structure of brain activity
Subject-sparse
estimate
G Varoquaux 17

25. 1 Independence structure of brain activity
Population-
sparse estimate
G Varoquaux 17

26. 1 Large scale organization: communities
Graph communities
[Eguiluz... 2005]
Non-sparse
Neural communities
G Varoquaux 18

27. 1 Large scale organization: communities
Graph communities
[Eguiluz... 2005]
Group-sparse
Neural communities
= large known functional networks
[Varoquaux... 2010b]
G Varoquaux 18

28. 1 Giving up on sparsity?
Sparsity is finicky
Sensitive hyper-parameter
Slow and unreliable convergence
Unstable set of selected edges
Shrinkage
Softly push partial correlations to zero
ΣShrunk = (1 − λ)ΣMLE + λId
Ledoit-Wolf oracle to set λ
[Ledoit and Wolf 2004]
G Varoquaux 19

29. 2 Comparing connectomes
Functional biomarkers
Population imaging
G Varoquaux 20

30. 2 Failure of univariate approach on correlations
Subject variability spread across correlation matrices
0 5 10 15 20 25
0
5
10
15
20
25 Control
0 5 10 15 20 25
0
5
10
15
20
25 Control
0 5 10 15 20 25
0
5
10
15
20
25 Control
0 5 10 15 20 25
0
5
10
15
20
25Large lesion
dΣ = Σ2 − Σ1 is not definite positive
⇒ not a covariance
Σ does not live in a vector space
G Varoquaux 21

31. 2 Inverse covariance very noisy
Partial correlations are hard to estimate
0 5 10 15 20 25
0
5
10
15
20
25 Control
0 5 10 15 20 25
0
5
10
15
20
25 Control
0 5 10 15 20 25
0
5
10
15
20
25 Control
0 5 10 15 20 25
0
5
10
15
20
25Large lesion
G Varoquaux 22

32. 2 A toy model of differences in connectivity
Two processes with different partial correlations
K1: K1 − K2: Σ1: Σ1 − Σ2:
+ jitter in observed covariance
MSE(K1 − K2): MSE(Σ1 − Σ2):
Non-local effects and non homogeneous noise
G Varoquaux 23

33. 2 Theory: error geometry
Disentangle parameters (edge-level connectivities)
Connectivity matrices form a manifold
⇒ project to tangent space
θ¹
θ²
( )θ¹I
-1
( )θ²I
-1
Estimation error of covariances
Assymptotics given by Fisher matrix [Rao 1945]
Cramer-Rao bounds
G Varoquaux 24

34. 2 Theory: error geometry
Disentangle parameters (edge-level connectivities)
Connectivity matrices form a manifold
⇒ project to tangent space
M
anifold
[Varoquaux... 2010a]
Estimation error of covariances
Assymptotics given by Fisher matrix [Rao 1945]
Defines a metric on a manifold of models
With covariances: Lie-algebra structure [Lenglet... 2006]
G Varoquaux 24

35. 2 Reparametrization for uniform error geometry
Disentangle parameters (edge-level connectivities)
Connectivity matrices form a manifold
⇒ project to tangent space
Controls
Patient
dΣ
M
anifold
Tangent
dΣ = Σ
− 1
/2
Ctrl ΣPatientΣ
− 1
/2
Ctrl
[Varoquaux... 2010a]
G Varoquaux 24

36. 2 Reparametrization for uniform error geometry
The simulations
K1 − K2: Σ1 − Σ2: dΣ: MSE(dΣ):
Semi-local effects and homogeneous noise
G Varoquaux 25

37. 2 Residuals
Correlation matrices: Σ -1.0 0.0 1.0
0 5 10 15 20 25
0
5
0
5
0
5
0 5 10 15 20 25
0
5
10
15
20
25
0 5 10 15 20 25
0
5
10
15
20
25
0 5 10 15 20 25
0
5
10
15
20
25
Residuals: dΣ -1.0 0.0 1.0
0 5 10 15 20 25
0
5
0
5
0
5
Control
0 5 10 15 20 25
0
5
10
15
20
25
Control
0 5 10 15 20 25
0
5
10
15
20
25
Control
0 5 10 15 20 25
0
5
10
15
20
25
Large lesion
G Varoquaux 26

38. 2 Post-stroke covariance modifications
p-value: 5·10−2
Bonferroni-corrected
G Varoquaux 27

39. 2 Prediction from connectomes
RS-fMRI
Functional
connectivity
Time series
2
4
3
1
Diagnosis
ROIs
G Varoquaux 28

40. 2 Prediction from connectomes
Time series
2
RS-fMRI
41
Diagnosis
ROIs Functional
connectivity
3
Connectivity matrix
Correlation
Partial correlations
Tangent space
G Varoquaux 28

41. 2 Prediction from connectomes
Time series
2
RS-fMRI
41
Diagnosis
ROIs Functional
connectivity
3
Connectivity matrix
Correlation
Partial correlations
Tangent space
Prediction accuracy
Autism
[Abraham2016]
[K. Reddy, Poster 3916]
G Varoquaux 28

42. @GaelVaroquaux
Estimation functional connectomes:
sparsity and beyond
Zeros in inverse covariance give
conditional independance
⇒ sparsity
Shrinkage: simpler, faster
(Ledoit-Wolf)
Tangent space
for comparisons
Controls
Patient
Controls
Patient
Software:
http://nilearn.github.io/ ni

43. References I
V. M. Eguiluz, D. R. Chialvo, G. A. Cecchi, M. Baliki, and
A. V. Apkarian. Scale-free brain functional networks.
Physical review letters, 94:018102, 2005.
J. Friedman, T. Hastie, and R. Tibshirani. Sparse inverse
covariance estimation with the graphical lasso. Biostatistics,
9:432, 2008.
C. Lam and J. Fan. Sparsistency and rates of convergence in
large covariance matrix estimation. Annals of statistics, 37
(6B):4254, 2009.
O. Ledoit and M. Wolf. A well-conditioned estimator for
large-dimensional covariance matrices. J. Multivar. Anal.,
88:365, 2004.

44. References II
C. Lenglet, M. Rousson, R. Deriche, and O. Faugeras.
Statistics on the manifold of multivariate normal
distributions: Theory and application to diffusion tensor
MRI processing. Journal of Mathematical Imaging and
Vision, 25:423, 2006.
C. Rao. Information and accuracy attainable in the estimation
of statistical parameters. Bull. Calcutta Math. Soc., 37:81,
1945.
P. Ravikumar, M. J. Wainwright, G. Raskutti, B. Yu, ...
High-dimensional covariance estimation by minimizing
1-penalized log-determinant divergence. Electronic Journal
of Statistics, 5:935–980, 2011.
S. Smith, K. Miller, G. Salimi-Khorshidi, M. Webster,
C. Beckmann, T. Nichols, J. Ramsey, and M. Woolrich.
Network modelling methods for fMRI. Neuroimage, 54:875,
2011.

45. References III
G. Varoquaux and R. C. Craddock. Learning and comparing
functional connectomes across subjects. NeuroImage, 80:
405, 2013.
G. Varoquaux, F. Baronnet, A. Kleinschmidt, P. Fillard, and
B. Thirion. Detection of brain functional-connectivity
difference in post-stroke patients using group-level
covariance modeling. In MICCAI. 2010a.
G. Varoquaux, A. Gramfort, J. B. Poline, and B. Thirion.
Brain covariance selection: better individual functional
connectivity models using population prior. In NIPS. 2010b.
G. Varoquaux, A. Gramfort, J. B. Poline, and B. Thirion.
Markov models for fMRI correlation structure: is brain
functional connectivity small world, or decomposable into
networks? Journal of Physiology - Paris, 106:212, 2012.