Sparse Model Estimation: Parametric and Nonparametric Settings
                                           Pradeep Ravikumar
                                  University of California, Berkeley

A common approach in settings with high-dimensional data has been to 
estimate models that are ``sparse,'' in the sense that an index set of
relevant model components has small cardinality. In this talk I will cover two instances,
one parametric and the other nonparametric, of sparse model estimation.

The first part of the talk considers the task of estimating the covariance
and inverse covariance, or concentration, matrices of a random vector 
from i.i.d. observations. We study an estimator based on minimizing an 
l1-penalized log-determinant Bregman divergence, that is equivalent to the usual 
l1-regularized maximum likelihood estimator when the random vector is 
multivariate Gaussian. We analyze the performance of this estimator under
high-dimensional scaling, in which the number of variables and other
model parameters are allowed to grow as a function of the sample size. 
Our analysis identifies key players affecting the convergence rates
in various norms as well as the model selection consistency of the estimator.

The second part of the talk considers the task of encoding fMRI
signals from the primary visual cortex, also called area V1, of the brain 
in response to natural image stimuli; as well as to identify potential 
features of images that drive the neural activity. Our method is based on the 
understanding that the fMRI signal reflects the pooled, and potentially 
nonlinearly transformed output of a large 
population of neurons in area V1. Our class of models, 
which we call the V-SPAM framework, mimics this with an initial 
hierarchical filtering stage that consists of three layers of artificial 
neuronal cells, and a final nonparametric pooling stage which learns 
nonparametric transformations of a sparse set of neuronal filters.


This is joint work with Garvesh Raskutti, Vincent Vu, Martin Wainwright, Bin Yu,
and the Jack Gallant lab at UC Berkeley; Kendrick Kay, Thomas Naselaris and Jack Gallant.