New directions in sparse sampling and estimation for underdetermined systems

A central objective in signal processing is to infer meaningful information from a set of measurements or data. While most signal models have an overdetermined structure (the number of unknowns less than the number of equations), traditionally very few statistical estimation problems have considered a data model which is underdetermined (number of unknowns more than the number of equations). However, in recent times, an explosion of theoretical and computational methods have been developed primarily to study underdetermined systems by imposing sparsity on the unknown variables. This is motivated by the observation that inspite of the huge volume of data that arises in sensor networks, genomics, imaging, particle physics, web search etc., their information content is often much smaller compared to the number of raw measurements. This has given rise to the possibility of reducing the number of measurements by down sampling the data, which automatically gives rise to underdetermined systems.

In this thesis, we provide new directions for estimation in an underdetermined system, both for a class of parameter estimation problems and also for the problem of sparse recovery in compressive sensing. There are two main contributions of the thesis: design of new sampling and statistical estimation algorithms for array processing, and development of improved guarantees for sparse reconstruction by introducing a statistical framework to the recovery problem.

We consider underdetermined observation models in array processing where the number of unknown sources simultaneously received by the array can be considerably larger than the number of physical sensors. We study new sparse spatial sampling schemes (array geometries) as well as propose new recovery algorithms that can exploit priors on the unknown signals and unambiguously identify all the sources. The proposed sampling structure is generic enough to be extended to multiple dimensions as well as to exploit different kinds of priors in the model such as correlation, higher order moments, etc.

Recognizing the role of correlation priors and suitable sampling schemes for underdetermined estimation in array processing, we introduce a correlation aware framework for recovering sparse support in compressive sensing. We show that it is possible to strictly increase the size of the recoverable sparse support using this framework provided the measurement matrix is suitably designed. The proposed nested and coprime arrays are shown to be appropriate candidates in this regard. We also provide new guarantees for convex and greedy formulations of the support recovery problem and demonstrate that it is possible to strictly improve upon existing guarantees.

This new paradigm of underdetermined estimation that explicitly establishes the fundamental interplay between sampling, statistical priors and the underlying sparsity, leads to exciting future research directions in a variety of application areas, and also gives rise to new questions that can lead to stand-alone theoretical results in their own right.

Neural Prosthetics and Parietal Cortex

In the last decade, research efforts into directly interfacing with the neurons of individuals with motor deficits have increased. The goal of such research is clear: Enable individuals affected by paralysis or amputation to regain control of their environments by manipulating external devices with thought alone. Though the motor cortices are the usual brain areas upon which neural prosthetics depend, research into the parietal lobe and its subregions, primarily in non-human primates, has uncovered alternative areas that could also benefit neural interfaces. Similar to the motor cortical areas, parietal regions can supply information about the trajectories of movements. In addition, the parietal lobe also contains cognitive signals like movement goals and intentions. But, these areas are also known to be tuned to saccadic eye movements, which could interfere with the function of a prosthetic designed to capture motor intentions only. In this thesis, we develop and examine the functionality of a neural prosthetic with a non-human primate model using the superior parietal lobe to examine the effectiveness of such an interface and the effects of unconstrained eye movements in a task that more closely simulates clinical applications. Additionally, we examine methods for improving usability of such interfaces.

The parietal cortex is also believed to contain neural signals relating to monitoring of the state of the limbs through visual and somatosensory feedback. In one of the world’s first clinical neural prosthetics based on the human parietal lobe, we examine the extent to which feedback regarding the state of a movement effector alters parietal neural signals and what the implications are for motor neural prosthetics and how this informs our understanding of this area of the human brain.

Wnt and FGF signaling in C. elegans vulval cell lineage polarity

The interpretation of extracellular cues leading to the polarization of intracellular components and asymmetric cell divisions is a fundamental part of metazoan organogenesis. The C. elegans vulva, with its invariant cell lineage and interaction of multiple cell signaling pathways, provides an excellent model for the study of cell polarity within an organized epithelial tissue. Herein I discuss the interaction of Wnt and FGF signaling in controlling vulval cell lineage polarity with emphasis on the posterior-most cell that forms the vulva, P7.p.

The mirror symmetry of the C. elegans vulva is achieved by the opposite division orientation of the vulval precursor cells (VPCs) flanking the axis of symmetry. Opposing Wnt signals control the division patterns of the VPCs by controlling the localization of SYS-1/ β-catenin toward the direction of the Wnt gradient. Multiple Wnt signals, expressed at the axis of symmetry, promote the wild-type, anterior-facing, P7.p orientation, whereas Wnts EGL-20 and CWN-1 from the tail and posterior body wall muscle, respectively, promote the daughter cells of P7.p to face the posterior. EGL-20 acts through a member of the LDL receptor superfamily, LRP-2, along with Ror/CAM-1 and Van Gogh/VANG-1. All three transmembrane proteins control orientation through the localization of the SYS-1.

The Fibroblast Growth Factor (FGF) pathway acts in concert with LIN-17/Frizzled to regulate the localization of SYS-1. The source of the FGF ligand is the 1° VPC, P6.p, which controls the polarity of the neighboring 2° VPC, P7.p, by signaling through the sex myoblasts (SMs), activating the FGF pathway. The Wnt, cwn-1, is expressed in the posterior body wall muscle of the worm as well as the SMs, making it the only Wnt expressed on the posterior and anterior sides of P7.p at the time of the polarity decision. Both sources of cwn-1 act instructively to influence P7.p polarity in the direction of the Wnt gradient. The FGF pathway leads to the regulation of cwn-1 transcripts in the SMs. These results illustrate the first evidence of the interaction between FGF and Wnt in C. elegans development and vulval cell lineage polarity as well as highlight the promiscuous nature of Wnt signaling within C. elegans.