Random split (training/validating): $n_\mbox{tr} \asymp n_\mbox{val} \asymp n$
Select tuning $\lambda_i$ on a grid (size $N$) to min loss $$\hat{R}(\lambda_i)= \frac{1}{2}(\hat{\beta}^{\mbox{tr}}(\lambda_i))^{T}\hat{\Sigma}_{\mbox{val}} \hat{\beta}^{\mbox{tr}}(\lambda_i)-e^{T}\hat{\beta}^{\mbox{tr}}(\lambda_i)$$
Use selected $\hat{\lambda}$ above for SCIO estimator $\hat{\Omega}_{\mbox{cv}}$
Theorem. Under regularity conditions, $\log N = O(\log p)$, $n^{1/2} \log^{-1/2} p = o(N)$, as $n,\,p\rightarrow \infty$,
$$\frac{1}{p} \left\| \hat{\Omega}_{\mbox{cv}} - \Omega \right\|_F^2 = O_p \left(\frac{\log p}{n} \right)$$
Simulations
Matrix Loss Comparison
SCIO has
samller losses for most scenarios
Network Recovery Comparison
Truth
SCIO
Glasso
Heatmaps: black-nonzero over 100% runs; white-100% zero.
SCIO closer to the truth
fMRI Simulation
GM is among the
top 3 methods of 30+ by massive dynamic simulations (600+ citations) Smith et al, 10
Using their data,
SCIO has
better ROC of recovering the connections (non-zero $\Omega$), vs penalized likelihood (GLASSO)
Real Data
ADHD
ADHD affects about 10% children in US
Data from the ADHD-200 project
fMRI data from 61 Healthy, 22 ADHD cases
116 brain regions (AAL), 148 observations
Heatmaps: black-nonzero over 100% subs; white-100% zero.
SCIO:
clearer contrasts between healthy and ADHD
SCIO:
faster computation and
better scaling
Another Data Example: HIV
Predicting HIV/non-HIV brains using gene exprs using LDA ($\Omega$)
SCIO:
higher pred accuracy
Summary
New loss functions without likelihood
Improved accuracy and theory
Fast computation
Optimization: build methods to recover patterns
A step for big and complex (network) data
Accurate network recovery for brain networks
For a broad range of distributions and data
Utility: diagnosis and personalized medicine
Limitations and future work: complex models, faithful dimension reduction, implementation
Collaborators
Tony Cai
Univ of Penn
Weidong Liu
Shanghai JiaoTong Univ
References
Bunea, Giraud, X Luo. Community Estimation in G-models via CORD. Submitted to Annals Stat
Luo. A Hierarchical Graphical Model for Big Inverse Covariance Estimation with an Application to fMRI. Revision for Biostat
Luo, Gee, Sohal, Small (2016). A Point-process Response Model for Optogenetics Experiments on Neural Circuits. Stat Med.
Liu, Luo (2015). Fast and Adaptive Sparse Precision Matrix Estimation in High Dimensions. J Multivariate Analysis.
Luo et al (2013) Cognitive control and gender specific neural predictors of relapse in cocaine dependence. Brain
Luo, Small, Li, Rosenbaum (2012). Inference with Interference between Units in an fMRI Experiment of Motor Inhibition. JASA
Cai, Liu, Luo (2011). A Constrained $\ell_1$ Minimization Approach to Sparse Precision Matrix Estimation. JASA