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Compressed sensing is a novel technique where one can recover sparse signals from the undersampled measurements. In this paper, a $K \times N$ measurement matrix for compressed sensing is deterministically constructed via multiplicative…
The problems of optimal recovery of unbounded operators are studied. Optimality means the highest possible accuracy and the minimal amount of discrete information involved. It is established that the truncation method, when certain…
Exact recovery of $K$-sparse signals $x \in \mathbb{R}^{n}$ from linear measurements $y=Ax$, where $A\in \mathbb{R}^{m\times n}$ is a sensing matrix, arises from many applications. The orthogonal matching pursuit (OMP) algorithm is widely…
Measurement samples are often taken in various monitoring applications. To reduce the sensing cost, it is desirable to achieve better sensing quality while using fewer samples. Compressive Sensing (CS) technique finds its role when the…
This paper presents a new analysis for the orthogonal matching pursuit (OMP) algorithm. It is shown that if the restricted isometry property (RIP) is satisfied at sparsity level $O(\bar{k})$, then OMP can recover a $\bar{k}$-sparse signal…
Auto-Encoders are unsupervised models that aim to learn patterns from observed data by minimizing a reconstruction cost. The useful representations learned are often found to be sparse and distributed. On the other hand, compressed sensing…
There is a recent surge of interest in developing algorithms for finding sparse solutions of underdetermined systems of linear equations $y = \Phi x$. In many applications, extremely large problem sizes are envisioned, with at least tens of…
Due to excessive need for faster propagations of signals and necessity to reduce number of measurements and rapidly increase efficiency, new sensing theories have been proposed. Conventional sampling approaches that follow Shannon-Nyquist…
We study the sparse phase retrieval problem, which seeks to recover a sparse signal from a limited set of magnitude-only measurements. In contrast to prevalent sparse phase retrieval algorithms that primarily use first-order methods, we…
Magnetic resonance imaging (MRI) is an essential medical tool with inherently slow data acquisition process. Slow acquisition process requires patient to be long time exposed to scanning apparatus. In recent years significant efforts are…
The goal in signal compression is to reduce the size of the input signal without a significant loss in the quality of the recovered signal. One way to achieve this goal is to apply the principles of compressive sensing, but this has not…
We consider the problem of clustering mixtures of mean-separated Gaussians in high dimensions. We are given samples from a mixture of $k$ identity covariance Gaussians, so that the minimum pairwise distance between any two pairs of means is…
All Lossy compression algorithms employ similar compression schemes -- frequency domain transform followed by quantization and lossless encoding schemes. They target tradeoffs by quantizating high frequency data to increase compression…
This article derives lower bounds on the convergence rate of continuous-time gradient-based optimization algorithms. The algorithms are subjected to a time-normalization constraint that avoids a reparametrization of time in order to make…
The goal of (stable) sparse recovery is to recover a $k$-sparse approximation $x*$ of a vector $x$ from linear measurements of $x$. Specifically, the goal is to recover $x*$ such that ||x-x*||_p <= C min_{k-sparse x'} ||x-x'||_q for some…
Clustering algorithms remain valuable tools for grouping and summarizing the most important aspects of data. Example areas where this is the case include image segmentation, dimension reduction, signals analysis, model order reduction,…
Solving inverse problems with iterative algorithms is popular, especially for large data. Due to time constraints, the number of possible iterations is usually limited, potentially affecting the achievable accuracy. Given an error one is…
Orthogonal least squares (OLS) is a classic algorithm for sparse recovery, function approximation, and subset selection. In this paper, we analyze the performance guarantee of the OLS algorithm. Specifically, we show that OLS guarantees the…
In this paper, we study the sample complexity and develop efficient optimal algorithms for 1-bit phase retrieval: recovering a signal $\mathbf{x}\in\mathbb{R}^n$ from $m$ phaseless bits…
Applying the theory of compressive sensing in practice always takes different kinds of perturbations into consideration. In this paper, the recovery performance of greedy pursuits with replacement for sparse recovery is analyzed when both…