arXiv subjects

Erin E. Tripp

Publications and source records attributed to Erin E. Tripp.

7 recordsLinked to original sources

arXiv2024

Sparse Recovery: The Square of $\ell_1/\ell_2$ Norms

This paper introduces a nonconvex approach for sparse signal recovery, proposing a novel model termed the $τ_2$-model, which utilizes the squared $\ell_1/\ell_2$ norms for this purpose. Our model offers an advancement over the $\ell_0$ norm, which is often computationally intractable and less effective in practical scenarios. Grounded in the concept of effective sparsity, our approach robustly measures the number of significant coordinates in a signal, making it a powerful alternative for sparse signal estimation. The $τ_2$-model is particularly advantageous due to its computational efficiency and practical applicability. We detail two accompanying algorithms based on Dinkelbach's procedure and a difference of convex functions strategy. The first algorithm views the model as a linear-constrained quadratic programming problem in noiseless scenarios and as a quadratic-constrained quadratic programming problem in noisy scenarios. The second algorithm, capable of handling both noiseless and noisy cases, is based on the alternating direction linearized proximal method of multipliers. We also explore the model's properties, including the existence of solutions under certain conditions, and discuss the convergence properties of the algorithms. Numerical experiments with various sensing matrices validate the effectiveness of our proposed model.

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arXiv2023

Minimax Concave Penalty Regularized Adaptive System Identification

We develop a recursive least square (RLS) type algorithm with a minimax concave penalty (MCP) for adaptive identification of a sparse tap-weight vector that represents a communication channel. The proposed algorithm recursively yields its estimate of the tap-vector, from noisy streaming observations of a received signal, using expectation-maximization (EM) update. We prove the convergence of our algorithm to a local optimum and provide bounds for the steady state error. Using simulation studies of Rayleigh fading channel, Volterra system and multivariate time series model, we demonstrate that our algorithm outperforms, in the mean-squared error (MSE) sense, the standard RLS and the $\ell_1$-regularized RLS.

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arXiv2023

Semantic Communication with Conceptual Spaces

Despite the fact that Shannon and Weaver's Mathematical Theory of Communication was published over 70 years ago, all communication systems continue to operate at the first of three levels defined in this theory: the technical level. In this letter, we argue that a transition to the semantic level embodies a natural, important step in the evolution of communication technologies. Furthermore, we propose a novel approach to engineering semantic communication using conceptual spaces and functional compression. We introduce a model of semantic communication utilizing this approach, and simulate communication of image semantics demonstrating a 99.79% reduction in rate.

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arXiv2022

Asynchronous Multiuser Detection for SCMA with Unknown Delays: A Compressed Sensing Approach

Despite being the subject of a growing body of research, non-orthogonal multiple access has failed to garner sufficient support to be included in modern standards. One of the more promising approaches to non-orthogonal multiple access is sparse code multiple access, which seeks to utilize non-orthogonal, sparse spreading codes to share bandwidth among users more efficiently than traditional orthogonal methods. Nearly all of the studies regarding sparse code multiple access assume synchronization at the receiver, which may not always be a practical assumption. In this work, we aim to bring this promising technology closer to a practical realization by dropping the assumption of synchronization. We therefore propose a compressed sensing-based delay estimation technique developed specifically for an uplink sparse code multiple access system. The proposed technique can be used with nearly all of the numerous decoding algorithms proposed in the existing literature, including the popular message passing approach. Furthermore, we derive a theoretical bound regarding the recovery performance of the proposed technique, and use simulations to demonstrate its viability in a practical uplink system.

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arXiv2021

The Proximity Operator of the Log-Sum Penalty

The log-sum penalty is often adopted as a replacement for the $\ell_0$ pseudo-norm in compressive sensing and low-rank optimization. The hard-thresholding operator, i.e., the proximity operator of the $\ell_0$ penalty, plays an essential role in applications; similarly, we require an efficient method for evaluating the proximity operator of the log-sum penalty. Due to the nonconvexity of this function, its proximity operator is commonly computed through the iteratively reweighted $\ell_1$ method, which replaces the log-sum term with its first-order approximation. This paper reports that the proximity operator of the log-sum penalty actually has an explicit expression. With it, we show that the iteratively reweighted $\ell_1$ solution disagrees with the true proximity operator of the log-sum penalty in certain regions. As a by-product, the iteratively reweighted $\ell_1$ solution is precisely characterized in terms of the chosen initialization. We also give the explicit form of the proximity operator for the composition of the log-sum penalty with the singular value function, as seen in low-rank applications. These results should be useful in the development of efficient and accurate algorithms for optimization problems involving the log-sum penalty.

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arXiv2019

Algorithmic Versatility of SPF-regularization Methods

Sparsity promoting functions (SPFs) are commonly used in optimization problems to find solutions which are assumed or desired to be sparse in some basis. For example, the l1-regularized variation model and the Rudin-Osher-Fatemi total variation (ROF-TV) model are some of the most well-known variational models for signal and image denoising, respectively. However, recent work demonstrates that convexity is not always desirable in sparsity promoting functions. In this paper, we replace convex SPFs with their induced nonconvex SPFs and develop algorithms for the resulting model by exploring the intrinsic structures of the nonconvex SPFs. We also present simulations illustrating the performance of the SPF and the developed algorithms in image denoising.

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arXiv2018

Structured Sparsity Promoting Functions

Motivated by the minimax concave penalty based variable selection in high-dimensional linear regression, we introduce a simple scheme to construct structured semiconvex sparsity promoting functions from convex sparsity promoting functions and their Moreau envelopes. Properties of these functions are developed by leveraging their structure. In particular, we provide sparsity guarantees for the general family of functions. We further study the behavior of the proximity operators of several special functions including indicator functions of closed convex sets, piecewise quadratic functions, and the linear combinations of them. To demonstrate these properties, several concrete examples are presented and existing instances are featured as special cases.

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