We propose a novel convex relaxation of sparse principal subspace estimation based on the convex hull of rank-$d$ projection matrices (the Fantope). The convex problem can be solved efficiently using alternating direction method of multipliers (ADMM). We establish a near-optimal convergence rate, in terms of the sparsity, ambient dimension, and sample size, for estimation of the principal subspace of a general covariance matrix without assuming the spiked covariance model. In the special case of $d=1$, our result implies the near- optimality of DSPCA even when the solution is not rank 1. We also provide a general theoretical framework for analyzing the statistical properties of the method for arbitrary input matrices that extends the applicability and provable guarantees to a wide array of settings. We demonstrate this with an application to Kendall's tau correlation matrices and transelliptical component analysis.

Information theory provides an attractive framework for attacking the neural coding problem. This entails estimating information theoretic quantities from neural spike train data. This paper highlights two issues that may arise: non-parametric entropy estimation and non-stationarity. It gives an overview of these issues and some of the progress that has been made.

Information estimates such as the direct method of Strong, Koberle, de Ruyter van Steveninck, and Bialek (1998) sidestep the difficult problem of estimating the joint distribution of response and stimulus by instead estimating the difference between the marginal and conditional entropies of the response. While this is an effective estimation strategy, it tempts the practitioner to ignore the role of the stimulus and the meaning of mutual information. We show here that as the number of trials increases indefinitely, the direct (or plug-in) estimate of marginal entropy converges (with probability 1) to the entropy of the time-averaged conditional distribution of the response, and the direct estimate of the conditional entropy converges to the time-averaged entropy of the conditional distribution of the response. Under joint stationarity and ergodicity of the response and stimulus, the difference of these quantities converges to the mutual information. When the stimulus is deterministic or nonstationary the direct estimate of information no longer estimates mutual information, which is no longer meaningful, but it remains a measure of variability of the response distribution across time.

We propose a novel hierarchical, nonlinear model that predicts brain activity in area V1 evoked by natural images. In the study reported here brain activity was measured by means of functional magnetic resonance imaging (fMRI), a noninvasive technique that provides an indirect measure of neural activity pooled over a small volume (~ 2mm cube) of brain tissue. Our model, which we call the SpAM V1 model, is based on the reasonable assumption that fMRI measurements reflect the (possibly nonlinearly) pooled, rectified output of a large population of simple and complex cells in V1. It has a hierarchical filtering stage that consists of three layers: model simple cells, model complex cells, and a third layer in which the complex cells are linearly pooled (called “pooled-complex” cells). The pooling stage then obtains the measured fMRI signals as a sparse additive model (SpAM) in which a sparse nonparametric (nonlinear) combination of model complex cell and model pooled-complex cell outputs are summed. Our results show that the SpAM V1 model predicts fMRI responses evoked by natural images better than a benchmark model that only provides linear pooling of model complex cells. Furthermore, the spatial receptive fields, frequency tuning and orientation tuning curves of the SpAM V1 model estimated for each voxel appears to be consistent with the known properties of V1, and with previous analyses of this data set. A visualization procedure applied to the SpAM V1 model shows that most of the nonlinear pooling consists of simple compressive or saturating nonlinearities.

The alarm that worms start to spread on increasingly popular mobile devices calls for an in-depth investigation of their propagation dynamics. In this paper, we study how mobility patterns affect Bluetooth worm spreading speeds. We find that the impact of mobility patterns is substantial over a large set of of changing Bluetooth and worm parameters. For instance, a mobility model under which devices move among a fixed set of activity locations can result in worm propagation speeds four times faster than a classical mobility model such as the random walk model. Our investigation reveals that the key factors affecting Bluetooth worm propagation speeds include spatial distributions of nodes, link duration distributions, degrees to which devices are mixed together, and even the burstiness of successive links.