Compressed sensing seeks to recover a sparse vector from a small number of linear and non-adaptive measurements. While most work so far focuses on Gaussian or Bernoulli random measurements we investigate the use of partial random circulant and Toeplitz matrices in connection with recovery by $\ell_1$-minization. In contrast to recent work in this direction we allow the use of an arbitrary subset of rows of a circulant and Toeplitz matrix. Our recovery result predicts that the necessary number of measurements to ensure sparse reconstruction by $\ell_1$-minimization with random partial circulant or Toeplitz matrices scales linearly in the sparsity up to a $\log$-factor in the ambient dimension. This represents a significant improvement over previous recovery results for such matrices. As a main tool for the proofs we use a new version of the non-commutative Khintchine inequality.
Any sequence of uniformly bounded $N\times N$ Hermitian Toeplitz matrices $\{\boldsymbol{H}_N\}$ is asymptotically equivalent to a certain sequence of $N\times N$ circulant matrices $\{\boldsymbol{C}_N\}$ derived from the Toeplitz matrices in the sense that $\left\| \boldsymbol{H}_N - \boldsymbol{C}_N \right\|_F = o(\sqrt{N})$ as $N\rightarrow \infty$. This implies that certain collective behaviors of the eigenvalues of each Toeplitz matrix are reflected in those of the corresponding circulant matrix and supports the utilization of the computationally efficient fast Fourier transform (instead …
Double Toeplitz (shortly DT) codes are introduced here as a generalization of double circulant codes. We show that such a code is isodual, hence formally self-dual. Self-dual DT codes are characterized as double circulant or double negacirculant. Likewise, even DT binary codes are characterized as double circulants. Numerical examples obtained by exhaustive search show that the codes constructed have best-known minimum distance, up to one unit, amongst formally self-dual codes, and sometimes improve on the known values. Over $\F_4$ an explicit construction of DT codes, based on quadratic resid…
In this paper we consider memoryless one-bit compressed sensing with randomly subsampled Gaussian circulant matrices. We show that in a small sparsity regime and for small enough accuracy $δ$, $m\sim δ^{-4} s\log(N/sδ)$ measurements suffice to reconstruct the direction of any $s$-sparse vector up to accuracy $δ$ via an efficient program. We derive this result by proving that partial Gaussian circulant matrices satisfy an $\ell_1/\ell_2$ RIP-property. Under a slightly worse dependence on $δ$, we establish stability with respect to approximate sparsity, as well as full vector recovery results.
How to construct a suitable measurement matrix is still an open question in compressed sensing. A significant part of the recent work is that the measurement matrices are not completely random on the entries but exhibit considerable structure. In this paper, we proved that the symmetric Toeplitz matrix and its transforms can be used as measurement matrix and recovery signal with high probability. Compared with random matrices (e.g. Gaussian and Bernullio matrices) and some structured matrices (e.g. Toeplitz and circulant matrices), we need to generate fewer independent entries to obtain the me…
Recently, the statistical restricted isometry property (RIP) has been formulated to analyze the performance of deterministic sampling matrices for compressed sensing. In this paper, we propose the usage of orthogonal symmetric Toeplitz matrices (OSTM) for compressed sensing and study their statistical RIP by taking advantage of Stein's method. In particular, we derive the statistical RIP performance bound in terms of the largest value of the sampling matrix and the sparsity level of the input signal. Based on such connections, we show that OSTM can satisfy the statistical RIP for an overwhelmi…
We present optimal sample complexity estimates for one-bit compressed sensing problems in a realistic scenario: the procedure uses a structured matrix (a randomly sub-sampled circulant matrix) and is robust to analog pre-quantization noise as well as to adversarial bit corruptions in the quantization process. Our results imply that quantization is not a statistically expensive procedure in the presence of nontrivial analog noise: recovery requires the same sample size one would have needed had the measurement matrix been Gaussian and the noisy analog measurements been given as data.
In this paper we present a new family of discrete sequences having "random like" uniformly decaying auto-correlation properties. The new class of infinite length sequences are higher order chirps constructed using irrational numbers. Exploiting results from the theory of continued fractions and diophantine approximations, we show that the class of sequences so formed has the property that the worst-case auto-correlation coefficients for every finite length sequence decays at a polynomial rate. These sequences display doppler immunity as well. We also show that Toeplitz matrices formed from suc…
Recent work in compressed sensing theory shows that $n\times N$ independent and identically distributed (IID) sensing matrices whose entries are drawn independently from certain probability distributions guarantee exact recovery of a sparse signal with high probability even if $n\ll N$. Motivated by signal processing applications, random filtering with Toeplitz sensing matrices whose elements are drawn from the same distributions were considered and shown to also be sufficient to recover a sparse signal from reduced samples exactly with high probability. This paper considers Toeplitz block mat…
In this paper, we consider robust system identification under sparse outliers and random noises. In this problem, system parameters are observed through a Toeplitz matrix. All observations are subject to random noises and a few are corrupted with outliers. We reduce this problem of system identification to a sparse error correcting problem using a Toeplitz structured real-numbered coding matrix. We prove the performance guarantee of Toeplitz structured matrix in sparse error correction. Thresholds on the percentage of correctable errors for Toeplitz structured matrices are established. When bo…
Compressed sensing (CS) is a signal acquisition paradigm to simultaneously acquire and reduce dimension of signals that admit sparse representation. This is achieved by collecting linear, non-adaptive measurements of a signal, which can be formalized as multiplying the signal with a "measurement matrix". Most of matrices used in CS are random matrices as they satisfy the restricted isometry property (RIP) in an optimal regime of number of measurements with high probability. However, these matrices have their own caveats and for this reason, deterministic measurement matrices have been proposed…
Compressive sensing achieves effective dimensionality reduction of signals, under a sparsity constraint, by means of a small number of random measurements acquired through a sensing matrix. In a signal processing system, the problem arises of processing the random projections directly, without first reconstructing the signal. In this paper, we show that circulant sensing matrices allow to perform a variety of classical signal processing tasks such as filtering, interpolation, registration, transforms, and so forth, directly in the compressed domain and in an exact fashion, \emph{i.e.}, without…
In the theory of compressed sensing, restricted isometry analysis has become a standard tool for studying how efficiently a measurement matrix acquires information about sparse and compressible signals. Many recovery algorithms are known to succeed when the restricted isometry constants of the sampling matrix are small. Many potential applications of compressed sensing involve a data-acquisition process that proceeds by convolution with a random pulse followed by (nonrandom) subsampling. At present, the theoretical analysis of this measurement technique is lacking. This paper demonstrates that…
We study the problem of approximately recovering signals on a manifold from one-bit linear measurements drawn from either a Gaussian ensemble, partial circulant ensemble, or bounded orthonormal ensemble and quantized using Sigma-Delta or distributed noise shaping schemes. We assume we are given a Geometric Multi-Resolution Analysis, which approximates the manifold, and we propose a convex optimization algorithm for signal recovery. We prove an upper bound on the recovery error which outperforms prior works that use memoryless scalar quantization, requires a simpler analysis, and extends the cl…
This paper proposes a set of piecewise Toeplitz matrices as the linear mapping/sensing operator $\mathcal{A}: \mathbf{R}^{n_1 \times n_2} \rightarrow \mathbf{R}^M$ for recovering low rank matrices from few measurements. We prove that such operators efficiently encode the information so there exists a unique reconstruction matrix under mild assumptions. This work provides a significant extension of the compressed sensing and rank minimization theory, and it achieves a tradeoff between reducing the memory required for storing the sampling operator from $\mathcal{O}(n_1n_2M)$ to $\mathcal{O}(\max…
We study how to estimate a nearly low-rank Toeplitz covariance matrix $T$ from compressed measurements. Recent work of Qiao and Pal addresses this problem by combining sparse rulers (sparse linear arrays) with frequency finding (sparse Fourier transform) algorithms applied to the Vandermonde decomposition of $T$. Analytical bounds on the sample complexity are shown, under the assumption of sufficiently large gaps between the frequencies in this decomposition. In this work, we introduce random ultra-sparse rulers and propose an improved approach based on these objects. Our random rulers effecti…
Radar imaging systems transmit modulated wideband waveform to achieve high range resolution resulting in high sampling rates at the receiver proportional to the bandwidth of the transmit waveform. Analog processing techniques can be used on receive to reduce the number of measurements to N, the number of potential delay bins. If the scene interrogated by the radar is assumed to be sparse consisting of K point targets, results from compressive sensing suggest that number of measurements can be further reduced to scale with K logN for stable recovery of a sparse scene from measurements with addi…
This paper deals with two related problems, namely distance-preserving binary embeddings and quantization for compressed sensing . First, we propose fast methods to replace points from a subset $\mathcal{X} \subset \mathbb{R}^n$, associated with the Euclidean metric, with points in the cube $\{\pm 1\}^m$ and we associate the cube with a pseudo-metric that approximates Euclidean distance among points in $\mathcal{X}$. Our methods rely on quantizing fast Johnson-Lindenstrauss embeddings based on bounded orthonormal systems and partial circulant ensembles, both of which admit fast transforms. Our…
In this paper, a new class of circulant matrices built from deterministic sequences is proposed for convolution-based compressed sensing (CS). In contrast to random convolution, the coefficients of the underlying filter are given by the discrete Fourier transform of a deterministic sequence with good autocorrelation. Both uniform recovery and non-uniform recovery of sparse signals are investigated, based on the coherence parameter of the proposed sensing matrices. Many examples of the sequences are investigated, particularly the Frank-Zadoff-Chu (FZC) sequence, the \textit{m}-sequence and the …
Compressed sensing and its extensions have recently triggered interest in randomized signal acquisition. A key finding is that random measurements provide sparse signal reconstruction guarantees for efficient and stable algorithms with a minimal number of samples. While this was first shown for (unstructured) Gaussian random measurement matrices, applications require certain structure of the measurements leading to structured random measurement matrices. Near optimal recovery guarantees for such structured measurements have been developed over the past years in a variety of contexts. This arti…
We study the problem of recovering an $s$-sparse signal $\mathbf{x}^{\star}\in\mathbb{C}^n$ from corrupted measurements $\mathbf{y} = \mathbf{A}\mathbf{x}^{\star}+\mathbf{z}^{\star}+\mathbf{w}$, where $\mathbf{z}^{\star}\in\mathbb{C}^m$ is a $k$-sparse corruption vector whose nonzero entries may be arbitrarily large and $\mathbf{w}\in\mathbb{C}^m$ is a dense noise with bounded energy. The aim is to exactly and stably recover the sparse signal with tractable optimization programs. In this paper, we prove the uniform recovery guarantee of this problem for two classes of structured sensing matric…
We introduce a general framework to deterministically construct binary measurement matrices for compressed sensing. The proposed matrices are composed of (circulant) permutation submatrix blocks and zero submatrix blocks, thus making their hardware realization convenient and easy. Firstly, using the famous Johnson bound for binary constant weight codes, we derive a new lower bound for the coherence of binary matrices with uniform column weights. Afterwards, a large class of binary base matrices with coherence asymptotically achieving this new bound are presented. Finally, by choosing proper ro…
Suppose the signal x is realized by driving a k-sparse signal u through an arbitrary unknown stable discrete-linear time invariant system H. These types of processes arise naturally in Reflection Seismology. In this paper we are interested in several problems: (a) Blind-Deconvolution: Can we recover both the filter $H$ and the sparse signal $u$ from noisy measurements? (b) Compressive Sensing: Is x compressible in the conventional sense of compressed sensing? Namely, can x, u and H be reconstructed from a sparse set of measurements. We develop novel L1 minimization methods to solve both cases …
Compressed sensing is a scheme that allows for sparse signals to be acquired, transmitted and stored using far fewer measurements than done by conventional means employing Nyquist sampling theorem. Since many naturally occurring signals are sparse (in some domain), compressed sensing has rapidly seen popularity in a number of applied physics and engineering applications, particularly in designing signal and image acquisition strategies, e.g., magnetic resonance imaging, quantum state tomography, scanning tunneling microscopy, analog to digital conversion technologies. Contemporaneously, causal…
Compressed sensing provided a data-acquisition paradigm for sparse signals. Remarkably, it has been shown that practical algorithms provide robust recovery from noisy linear measurements acquired at a near optimal sampling rate. In many real-world applications, a signal of interest is typically sparse not in the canonical basis but in a certain transform domain, such as wavelets or the finite difference. The theory of compressed sensing was extended to the analysis sparsity model but known extensions are limited to specific choices of sensing matrix and sparsifying transform. In this paper, we…
Dimensionality reduction is a popular approach to tackle high-dimensional data with low-dimensional nature. Subspace Restricted Isometry Property, a newly-proposed concept, has proved to be a useful tool in analyzing the effect of dimensionality reduction algorithms on subspaces. In this paper, we provide a characterization of subspace Restricted Isometry Property, asserting that matrices which act as a near-isometry on low-dimensional subspaces possess subspace Restricted Isometry Property. This points out a unified approach to discuss subspace Restricted Isometry Property. Its power is furth…
