A hybrid DFT-LDPC framework for fast, efficient and robust compressive sensing

Pawar, Sameer; Ramchandran, Kannan

doi:10.1109/allerton.2012.6483460

Cited by 16 publications

(27 citation statements)

References 22 publications

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“…As can be inferred from the name, this database query property is more often considered in the database community, for instance in the work on IBLTs [30]. 4 While writing this paper, we became aware of a parallel work by Pawar and Ramchandran [33] that seems to achieve similar performance. However, at the time of submission, a preprint of this work was not available for us to compare the two works.…”

Section: A Our Contributionsmentioning

confidence: 98%

SHO-FA: Robust compressive sensing with order-optimal complexity, measurements, and bits

Bakshi

Jaggi

Cai

et al. 2012

2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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Suppose x is any exactly k-sparse vector in n . We present a class of "sparse" matrices A, and a corresponding algorithm that we call SHO-FA (for Short and Fast 1 ) that, with high probability over A, can reconstruct x from Ax. The SHO-FA algorithm is related to the Invertible Bloom Lookup Tables (IBLTs) recently introduced by Goodrich et al., with two important distinctions -SHO-FA relies on linear measurements, and is robust to noise and approximate sparsity. The SHO-FA algorithm is the first to simultaneously have the following properties: (a) it requires only O(k) measurements, (b) the bitprecision of each measurement and each arithmetic operation is O (log(n) + P ) (here 2 −P corresponds to the desired relative error in the reconstruction of x), (c) the computational complexity of decoding is O(k) arithmetic operations, and (d) if the reconstruction goal is simply to recover a single component of x instead of all of x, with high probability over A this can be done in constant time. All constants above are independent of all problem parameters other than the desired probability of success. For a wide range of parameters these properties are information-theoretically order-optimal. In addition, our SHO-FA algorithm is robust to random noise, and (random) approximate sparsity for a large range of k. In particular, suppose the measured vector equals A(x + z) + e, where z and e correspond respectively to the source tail and measurement noise. Under reasonable statistical assumptions on z and e our decoding algorithm reconstructs x with an estimation error of O(||z||1 + (log k) 2 ||e||1). The SHO-FA algorithm works with high probability over A, z, and e, and still requires only O(k) steps and O(k) measurements over O(log(n))-bit numbers. This is in contrast to most existing algorithms which focus on the "worst-case" z model, where it is known Ω(k log(n/k)) measurements over O(log(n))-bit numbers are necessary.

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Section: A Our Contributionsmentioning

confidence: 98%

SHO-FA: Robust compressive sensing with order-optimal complexity, measurements, and bits

Bakshi

Jaggi

Cai

et al. 2012

2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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“…developed a very low-complexity algorithm for computing the 2D-DFT of a √ N × √ N signal by extending the results of [12]. In a similar line of work, Pawar et al [14], [15] used the subsampling property of the DFT to develop a lowcomplexity algorithm for recovering the nonzero frequencydomain components of a signal by using ideas from sparsegraph codes [16].…”

Section: Introductionmentioning

confidence: 98%

A Fast Hadamard Transform for Signals With Sublinear Sparsity in the Transform Domain

Scheibler

Haghighatshoar

Vetterli

2015

IEEE Trans. Inform. Theory

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In this paper, we design a new iterative lowcomplexity algorithm for computing the Walsh-Hadamard transform (WHT) of an N dimensional signal with a K-sparse WHT. We suppose that N is a power of two and K = O(N α ), scales sub-linearly in N for some α ∈ (0, 1). Assuming a random support model for the nonzero transform-domain components, our algorithm reconstructs the WHT of the signal with a sample complexity O(K log 2 ( N K )) and a computational complexity O(K log 2 (K) log 2 ( N K )). Moreover, the algorithm succeeds with a high probability approaching 1 for large dimension N .Our approach is mainly based on the subsampling (aliasing) property of the WHT, where by a carefully designed subsampling of the time-domain signal, a suitable aliasing pattern is induced in the transform domain. We treat the resulting aliasing patterns as parity-check constraints and represent them by a bipartite graph. We analyze the properties of the resulting bipartite graphs and borrow ideas from codes defined over sparse bipartite graphs to formulate the recovery of the nonzero spectral values as a peeling decoding algorithm for a specific sparse-graph code transmitted over a binary erasure channel (BEC). This enables us to use tools from coding theory (belief-propagation analysis) to characterize the asymptotic performance of our algorithm in the very sparse (α ∈ (0, 1 3 ]) and the less sparse (α ∈ ( 1 3 , 1)) regime. Comprehensive simulation results are provided to assess the empirical performance of the proposed algorithm.

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“…The scheme This work was supported in part by the National Science Foundation under Grant No. CCF-1423040. adopts the sublinear recovery algorithm framework in [11]. For the measurement matrix design, the scheme also adopts the LDPC structure to disperse the signal into measurement bins.…”

Section: Introductionmentioning

confidence: 99%

A generalized LDPC framework for robust and sublinear compressive sensing

Chen

Guo

2016

2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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Abstract-Compressive sensing aims to recover a high-dimensional sparse signal from a relatively small number of measurements. In this paper, a novel design of the measurement matrix is proposed. The design is inspired by the construction of generalized low-density parity-check codes, where the capacity-achieving point-to-point codes serve as subcodes to robustly estimate the signal support. In the case that each entry of the n-dimensional k-sparse signal lies in a known discrete alphabet, the proposed scheme requires only O(k log n) measurements and arithmetic operations. In the case of arbitrary, possibly continuous alphabet, an error propagation graph is proposed to characterize the residual estimation error. With O(k log 2 n) measurements and computational complexity, the reconstruction error can be made arbitrarily small with high probability.

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A hybrid DFT-LDPC framework for fast, efficient and robust compressive sensing

Cited by 16 publications

References 22 publications

SHO-FA: Robust compressive sensing with order-optimal complexity, measurements, and bits

SHO-FA: Robust compressive sensing with order-optimal complexity, measurements, and bits

A Fast Hadamard Transform for Signals With Sublinear Sparsity in the Transform Domain

A generalized LDPC framework for robust and sublinear compressive sensing

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