Compressive sensing (CS) exploits sparsity to recover sparse or compressible signals from dimensionality reducing, non-adaptive sensing mechanisms. Sparsity is also used to enhance interpretability in machine learning and statistics applications: While the ambient dimension is vast in modern data analysis problems, the relevant information therein typically resides in a much lower dimensional space. However, many solutions proposed nowadays do not leverage the true underlying structure. Recent results in CS extend the simple sparsity idea to more sophisticated {\em structured} sparsity models, which describe the interdependency between the nonzero components of a signal, allowing to increase the interpretability of the results and lead to better recovery performance. In order to better understand the impact of structured sparsity, in this chapter we analyze the connections between the discrete models and their convex relaxations, highlighting their relative advantages. We start with the general group sparse model and then elaborate on two important special cases: the dispersive and the hierarchical models. For each, we present the models in their discrete nature, discuss how to solve the ensuing discrete problems and then describe convex relaxations. We also consider more general structures as defined by set functions and present their convex proxies. Further, we discuss efficient optimization solutions for structured sparsity problems and illustrate structured sparsity in action via three applications.
In oversampled adaptive sensing (OAS), noisy measurements are collected in multiple subframes. The sensing basis in each subframe is adapted according to some posterior information exploited from previous measurements. The framework is shown to significantly outperform the classic non-adaptive compressive sensing approach. This paper extends the notion of OAS to signals with structured sparsity. We develop a low-complexity OAS algorithm based on structured orthogonal sensing. Our investigations depict that the proposed algorithm outperforms the conventional non-adaptive compressive sensing fra…
Krannert School of Management（克兰纳特管理学院）；Purdue University（普渡大学）；Department of Computer Science（计算机科学系）
We analyze the necessary number of samples for sparse vector recovery in a noisy linear prediction setup. This model includes problems such as linear regression and classification. We focus on structured graph models. In particular, we prove that sufficient number of samples for the weighted graph model proposed by Hegde and others is also necessary. We use the Fano's inequality on well constructed ensembles as our main tool in establishing information theoretic lower bounds.
Reconfigurable intelligent surface (RIS) can manipulate the wireless communication environment by controlling the coefficients of RIS elements. However, due to the large number of passive RIS elements without signal processing capability, channel estimation in RIS assisted wireless communication system requires high pilot overhead. In the second part of this invited paper, we propose to exploit the double-structured sparsity of the angular cascaded channels among users to reduce the pilot overhead. Specifically, we first reveal the double-structured sparsity, i.e., different angular cascaded c…
Compressed Sensing (CS) is an appealing framework for applications such as Magnetic Resonance Imaging (MRI). However, up-to-date, the sensing schemes suggested by CS theories are made of random isolated measurements, which are usually incompatible with the physics of acquisition. To reflect the physical constraints of the imaging device, we introduce the notion of blocks of measurements: the sensing scheme is not a set of isolated measurements anymore, but a set of groups of measurements which may represent any arbitrary shape (parallel or radial lines for instance). Structured acquisition wit…
Structured-Sparsity-Aware Joint User Activity Detection and Channel Estimation for OTFS-Based Grant-Free Random Access
Nanyang Technological University（南洋理工大学）；State Key Laboratory of Integrated Services Networks, Xidian University（西安电子科技大学集成服务网络国家重点实验室）；AUMOVIO Germany GmbH（奥莫维欧德国有限公司）；University of California at Davis（加州大学戴维斯分校）
In this work we consider numerical efficiency and convergence rates for solvers of non-convex multi-penalty formulations when reconstructing sparse signals from noisy linear measurements. We extend an existing approach, based on reduction to an augmented single-penalty formulation, to the non-convex setting and discuss its computational intractability in large-scale applications. To circumvent this limitation, we propose an alternative single-penalty reduction based on infimal convolution that shares the benefits of the augmented approach but is computationally less dependent on the problem si…
Reconfigurable intelligent surface (RIS) has been recognized as a potential technology for 5G beyond and attracted tremendous research attention. However, channel estimation in RIS-aided system is still a critical challenge due to the excessive amount of parameters in cascaded channel. The existing compressive sensing (CS)-based RIS estimation schemes only adopt incomplete sparsity, which induces redundant pilot consumption. In this paper, we exploit the specific triple-structured sparsity of the cascaded channel, i.e., the common column sparsity, structured row sparsity after offset compensat…
Convex optimization recently emerges as a compelling framework for performing super resolution, garnering significant attention from multiple communities spanning signal processing, applied mathematics, and optimization. This article offers a friendly exposition to atomic norm minimization as a canonical convex approach to solve super resolution problems. The mathematical foundations and performances guarantees of this approach are presented, and its application in super resolution image reconstruction for single-molecule fluorescence microscopy are highlighted.
Orthogonal time frequency space (OTFS) modulation outperforms orthogonal frequency division multiplexing (OFDM) in high-mobility scenarios. One challenge for OTFS massive MIMO is downlink channel estimation due to the large number of base station antennas. In this paper, we propose a 3D structured orthogonal matching pursuit algorithm based channel estimation technique to solve this problem. First, we show that the OTFS MIMO channel exhibits 3D structured sparsity: normal sparsity along the delay dimension, block sparsity along the Doppler dimension, and burst sparsity along the angle dimensio…
In this paper, we analyze the information theoretic lower bound on the necessary number of samples needed for recovering a sparse signal under different compressed sensing settings. We focus on the weighted graph model, a model-based framework proposed by Hegde et al. (2015), for standard compressed sensing as well as for one-bit compressed sensing. We study both the noisy and noiseless regimes. Our analysis is general in the sense that it applies to any algorithm used to recover the signal. We carefully construct restricted ensembles for different settings and then apply Fano's inequality to …
It is known that certain structures of the signal in addition to the standard notion of sparsity (called structured sparsity) can improve the sample complexity in several compressive sensing applications. Recently, Hegde et al. proposed a framework, called approximation-tolerant model-based compressive sensing, for recovering signals with structured sparsity. Their framework requires two oracles, the head- and the tail-approximation projection oracles. The two oracles should return approximate solutions in the model which is closest to the query signal. In this paper, we consider two structure…
We consider the problem of determining the asymptotic order of the Gelfand numbers of mixed-(quasi-)norm embeddings $\ell^b_p(\ell^d_q) \hookrightarrow \ell^b_r(\ell^d_u)$ given that $p \leq r$ and $q \leq u$, with emphasis on cases with $p\leq 1$ and/or $q\leq 1$. These cases turn out to be related to structured sparsity. We obtain sharp bounds in a number of interesting parameter constellations. Our new matching bounds for the Gelfand numbers of the embeddings of $\ell_1^b(\ell_2^d)$ and $\ell_2^b(\ell_1^d)$ into $\ell_2^b(\ell_2^d)$ imply optimality assertions for the recovery of block-spar…
Sparsity-based methods are widely used in machine learning, statistics, and signal processing. There is now a rich class of structured sparsity approaches that expand the modeling power of the sparsity paradigm and incorporate constraints such as group sparsity, graph sparsity, or hierarchical sparsity. While these sparsity models offer improved sample complexity and better interpretability, the improvements come at a computational cost: it is often challenging to optimize over the (non-convex) constraint sets that capture various sparsity structures. In this paper, we make progress in this di…
Cornell University（康奈尔大学）；University of Sydney（悉尼大学）；Google Research（谷歌研究院）；Indian Institute of Science（印度科学学院）
We study high-dimensional sparse estimation under three natural constraints: communication constraints, local privacy constraints, and linear measurements (compressive sensing). Without sparsity assumptions, it has been established that interactivity cannot improve the minimax rates of estimation under these information constraints. The question of whether interactivity helps with natural inference tasks has been a topic of active research. We settle this question in the affirmative for the prototypical problems of high-dimensional sparse mean estimation and compressive sensing, by demonstrati…
The problem of recovering signals of high complexity from low quality sensing devices is analyzed via a combination of tools from signal processing and harmonic analysis. By using the rich structure offered by the recent development in fusion frames, we introduce a compressed sensing framework in which we split the dense information into sub-channel or local pieces and then fuse the local estimations. Each piece of information is measured by potentially low quality sensors, modeled by linear matrices and recovered via compressed sensing -- when necessary. Finally, by a fusion process within th…
