We consider supervised learning problems where the features are embedded in a
graph, such as gene expressions in a gene network. In this context, it is of
much interest to take into account the problem structure, and automatically
select a subgraph with a small number of connected components. By exploiting
prior knowledge, one can indeed improve the prediction performance and/or
obtain better interpretable results. Regularization or penalty functions for
selecting features in graphs have recently been proposed but they raise new
algorithmic challenges.

Functional MRI (fMRI) has become the most common method for investigating the
human brain. However, fMRI data present some complications for statistical
analysis and modeling. One recently developed approach to these data focuses on
estimation of computational encoding models that describe how stimuli are
transformed into brain activity measured in individual voxels. Here we aim at
building encoding models for fMRI signals recorded in primary visual cortex of
the human brain. We use residual analyses to reveal systematic nonlinearity
across voxels not taken into account by previous models.

The performance of the Lasso is well understood under the assumptions of the
standard linear model with homoscedastic noise. However, in several
applications, the standard model does not describe the important features of
the data. This paper examines how the Lasso performs on a non-standard model
that is motivated by medical imaging applications. In these applications, the
variance of the noise scales linearly with the expectation of the observation.
Like all heteroscedastic models, the noise terms in this Poisson-like model are
\textit{not} independent of the design matrix.

Sparse additive models are families of $d$-variate functions that have the
additive decomposition \mbox{$f^* = \sum_{j \in S} f^*_j$,} where $S$ is a
unknown subset of cardinality $s \ll d$. We consider the case where each
component function $f^*_j$ lies in a reproducing kernel Hilbert space, and
analyze a simple kernel-based convex program for estimating the unknown
function $f^*$. Working within a high-dimensional framework that allows both
the dimension $d$ and sparsity $s$ to scale, we derive convergence rates in the
$L^2(\mathbb{P})$ and $L^2(\mathbb{P}_n)$ norms.

This paper focuses on obtaining clustering information about a distribution
from its i.i.d. samples. We develop theoretical results to understand and use
clustering information contained in the eigenvectors of data adjacency matrices
based on a radial kernel function with a sufficiently fast tail decay. In
particular, we provide population analyses to gain insights into which
eigenvectors should be used and when the clustering information for the
distribution can be recovered from the sample.

Consider the standard linear regression model $\y = \Xmat \betastar + w$,
where $\y \in \real^\numobs$ is an observation vector, $\Xmat \in
\real^{\numobs \times \pdim}$ is a design matrix, $\betastar \in \real^\pdim$
is the unknown regression vector, and $w \sim \mathcal{N}(0, \sigma^2 I)$ is
additive Gaussian noise. This paper studies the minimax rates of convergence
for estimation of $\betastar$ for $\ell_\rpar$-losses and in the
$\ell_2$-prediction loss, assuming that $\betastar$ belongs to an
$\ell_{\qpar}$-ball $\Ballq(\myrad)$ for some $\qpar \in [0,1]$.