Rui M. Castro

Distilled Sensing: Adaptive Sampling for Sparse Detection and Estimation

Adaptive sampling results in significant improvements in the recovery of sparse signals in white Gaussian noise. A sequential adaptive sampling-and-refinement procedure called Distilled Sensing (DS) is proposed and analyzed. DS is a form of multistage experimental design and testing. Because of the adaptive nature of the data collection, DS can detect and localize far weaker signals than possible from non-adaptive measurements. In particular, reliable detection and localization (support estimation) using non-adaptive samples is possible only if the signal amplitudes grow logarithmically with the problem dimension. Here it is shown that using adaptive sampling, reliable detection is possible provided the amplitude exceeds a constant, and localization is possible when the amplitude exceeds any arbitrarily slowly growing function of the dimension.

Likelihood based hierarchical clustering

This paper develops a new method for hierarchical clustering. Unlike other existing clustering schemes, our method is based on a generative, tree-structured model that represents relationships between the objects to be clustered, rather than directly modeling properties of objects themselves. In certain problems, this generative model naturally captures the physical mechanisms responsible for relationships among objects, for example, in certain evolutionary tree problems in genetics and communication network topology identification. The paper examines the networking problem in some detail to illustrate the new clustering method. More broadly, the generative model may not reflect actual physical mechanisms, but it nonetheless provides a means for dealing with errors in the similarity matrix, simultaneously promoting two desirable features in clustering: intraclass similarity and interclass dissimilarity.

Compressive distilled sensing: Sparse recovery using adaptivity in compressive measurements

The recently-proposed theory of distilled sensing establishes that adaptivity in sampling can dramatically improve the performance of sparse recovery in noisy settings. In particular, it is now known that adaptive point sampling enables the detection and/or support recovery of sparse signals that are otherwise too weak to be recovered using any method based on non-adaptive point sampling. In this paper the theory of distilled sensing is extended to highly-undersampled regimes, as in compressive sensing. A simple adaptive sampling-and-refinement procedure called compressive distilled sensing is proposed, where each step of the procedure utilizes information from previous observations to focus subsequent measurements into the proper signal subspace, resulting in a significant improvement in effective measurement SNR on the signal subspace. As a result, for the same budget of sensing resources, compressive distilled sensing can result in significantly improved error bounds compared to those for traditional compressive sensing.

Sequentially designed compressed sensing

A sequential adaptive compressed sensing procedure for signal support recovery is proposed and analyzed. The procedure is based on the principle of distilled sensing, and makes used of sparse sensing matrices to perform sketching observations able to quickly identify irrelevant signal components. It is shown that adaptive compressed sensing enables recovery of weaker sparse signals than those that can be recovered using traditional non-adaptive compressed sensing approaches.

Adaptive Sensing for Sparse Signal Recovery

The theory of compressed sensing shows that sparse signals in high-dimensional spaces can be recovered from a relatively small number of samples in the form of random projections. However, in severely resource-constrained settings even CS techniques may fail, and thus, a less aggressive goal of partial signal recovery is reasonable. This paper describes a simple data-adaptive procedure that efficiently utilizes information from previous observations to focus subsequent measurements into subspaces that are increasingly likely to contain true signal components. The procedure is analyzed in a simple setting, and more generally, shown experimentally to be more effective than methods based on traditional (non-adaptive) random projections for partial signal recovery.

Efficient Decentralized Approximation via Selective Gossip

Recently, gossip algorithms have received much attention from the wireless sensor network community due to their simplicity, scalability and robustness. Motivated by applications such as compression and distributed transform coding, we propose a new gossip algorithm called Selective Gossip. Unlike traditional randomized gossip which computes the average of scalar values, we run gossip algorithms in parallel on the elements of a vector. The goal is to compute only the entries which are above a defined threshold in magnitude, i.e., significant entries. Nodes adaptively approximate the significant entries while abstaining from calculating the insignificant ones. Consequently, network lifetime and bandwidth are preserved. We show that with the proposed algorithm nodes reach consensus on the values of the significant entries and on the indices of insignificant ones. We illustrate the performance of our algorithm with a field estimation application. For regular topologies, selective gossip computes an approximation of the field using the wavelet transform. For irregular network topologies, we construct an orthonormal transform basis using eigenvectors of the graph Laplacian. Using two real sensor network datasets we show substantial communication savings over randomized gossip. We also propose a decentralized adaptive threshold mechanism such that nodes estimate the threshold while approximating the entries of the vector for computing the best m -term approximation of the data.

Adaptive sensing performance lower bounds for sparse signal detection and support estimation

This paper gives a precise characterization of the fundamental limits of adaptive sensing for diverse estimation and testing problems concerning sparse signals. We consider in particular the setting introduced in (IEEE Trans. Inform. Theory 57 (2011) 6222–6235) and show necessary conditions on the minimum signal magnitude for both detection and estimation: if x ? R^n is a sparse vector with s non-zero components then it can be reliably detected in noise provided the magnitude of the non-zero components exceeds v 2/s . Furthermore, the signal support can be exactly identified provided the minimum magnitude exceedsv 2 log s . Notably there is no dependence on n , the extrinsic signal dimension. These results show that the adaptive sensing methodologies proposed previously in the literature are essentially optimal, and cannot be substantially improved. In addition, these results provide further insights on the limits of adaptive compressive sensing.

Adaptive Sensing of Congested Spectrum Bands

Cognitive radios process their sensed information collectively in order to opportunistically identify and access underutilized spectrum segments (spectrum holes). Due to the transient and rapidly varying nature of the spectrum occupancy, the cognitive radios (secondary users) must be agile in identifying the spectrum holes in order to enhance their spectral efficiency. We propose a novel adaptive procedure to reinforce the agility of the secondary users for identifying multiple spectrum holes simultaneously over a wide spectrum band. This is accomplished by successively exploring the set of potential spectrum holes and progressively allocating the sensing resources to the most promising areas of the spectrum. Such exploration and resource allocation results in conservative spending of the sensing resources and translates into very agile spectrum monitoring. The proposed successive and adaptive sensing procedure is in contrast to the more conventional approaches that distribute the sampling resources equally over the entire spectrum. Besides improved agility, the adaptive procedure requires less-stringent constraints on the power of the primary users to guarantee that they remain distinguishable from the environment noise and renders more reliable spectrum hole detection.

Adaptive discovery of sparse signals in noise

A multi-step adaptive resampling procedure is proposed, and shown to be an effective approach when detecting high-dimensional sparse signals in noise. Each step of the proposed procedure refines an estimate of the true signal subspace, allowing sensing energy to be focused more directly into the subspace of interest and significantly improving the performance of the final detection test. Large-sample analysis shows that for the sparse signal detection problems considered, the proposed adaptive sensing procedure outperforms the best possible detection methods based on non-adaptive sensing, allowing for the detection of signals that are exponentially weaker than what can be detected using non-adaptive samples.

Compressed Sensing Vs. Active Learning

Compressive sampling (CS), or compressed sensing, has generated a tremendous amount of excitement in the signal processing community. Compressive sampling, which involves non-traditional samples in the form of randomized projections, can capture most of the salient information in a signal with a relatively small number of samples, often far fewer samples than required using traditional sampling schemes. Adaptive sampling (AS), also called active learning, uses information gleaned from previous observations (e.g., feedback) to focus the sampling process. Theoretical and experimental results have shown that adaptive sampling can dramatically outperform conventional (non-adaptive) sampling schemes. This paper compares the theoretical performance of compressive and adaptive sampling in noisy conditions, and it is shown that for certain classes of piecewise constant signals and high SNR regimes both CS and AS are near-optimal. This result is remarkable since it is the first evidence that shows that compressive sampling, which is non-adaptive, cannot be significantly outperformed by any other method (including adaptive sampling procedures), even in presence of noise