arXiv subjects

Myung Cho

Publications and source records attributed to Myung Cho.

23 records · Page 2Linked to original sources

arXiv2013

Off-The-Grid Spectral Compressed Sensing With Prior Information

Recent research in off-the-grid compressed sensing (CS) has demonstrated that, under certain conditions, one can successfully recover a spectrally sparse signal from a few time-domain samples even though the dictionary is continuous. In this paper, we extend off-the-grid CS to applications where some prior information about spectrally sparse signal is known. We specifically consider cases where a few contributing frequencies or poles, but not their amplitudes or phases, are known a priori. Our results show that equipping off-the-grid CS with the known-poles algorithm can increase the probability of recovering all the frequency components.

cs.IT↗

arXiv2013

Precisely Verifying the Null Space Conditions in Compressed Sensing: A Sandwiching Algorithm

In this paper, we propose new efficient algorithms to verify the null space condition in compressed sensing (CS). Given an $(n-m) \times n$ ($m>0$) CS matrix $A$ and a positive $k$, we are interested in computing $\displaystyle α_k = \max_{\{z: Az=0,z\neq 0\}}\max_{\{K: |K|\leq k\}}$ ${\|z_K \|_{1}}{\|z\|_{1}}$, where $K$ represents subsets of $\{1,2,...,n\}$, and $|K|$ is the cardinality of $K$. In particular, we are interested in finding the maximum $k$ such that $α_k < {1}{2}$. However, computing $α_k$ is known to be extremely challenging. In this paper, we first propose a series of new polynomial-time algorithms to compute upper bounds on $α_k$. Based on these new polynomial-time algorithms, we further design a new sandwiching algorithm, to compute the \emph{exact} $α_k$ with greatly reduced complexity. When needed, this new sandwiching algorithm also achieves a smooth tradeoff between computational complexity and result accuracy. Empirical results show the performance improvements of our algorithm over existing known methods; and our algorithm outputs precise values of $α_k$, with much lower complexity than exhaustive search.

cs.IT↗

arXiv2013

Universally Elevating the Phase Transition Performance of Compressed Sensing: Non-Isometric Matrices are Not Necessarily Bad Matrices

In compressed sensing problems, $\ell_1$ minimization or Basis Pursuit was known to have the best provable phase transition performance of recoverable sparsity among polynomial-time algorithms. It is of great theoretical and practical interest to find alternative polynomial-time algorithms which perform better than $\ell_1$ minimization. \cite{Icassp reweighted l_1}, \cite{Isit reweighted l_1}, \cite{XuScaingLaw} and \cite{iterativereweightedjournal} have shown that a two-stage re-weighted $\ell_1$ minimization algorithm can boost the phase transition performance for signals whose nonzero elements follow an amplitude probability density function (pdf) $f(\cdot)$ whose $t$-th derivative $f^{t}(0) \neq 0$ for some integer $t \geq 0$. However, for signals whose nonzero elements are strictly suspended from zero in distribution (for example, constant-modulus, only taking values `$+d$' or `$-d$' for some nonzero real number $d$), no polynomial-time signal recovery algorithms were known to provide better phase transition performance than plain $\ell_1$ minimization, especially for dense sensing matrices. In this paper, we show that a polynomial-time algorithm can universally elevate the phase-transition performance of compressed sensing, compared with $\ell_1$ minimization, even for signals with constant-modulus nonzero elements. Contrary to conventional wisdoms that compressed sensing matrices are desired to be isometric, we show that non-isometric matrices are not necessarily bad sensing matrices. In this paper, we also provide a framework for recovering sparse signals when sensing matrices are not isometric.

cs.IT↗

arXiv2013

Outliers and Random Noises in System Identification: a Compressed Sensing Approach

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 both outliers and observation noise are present, we have shown that the estimation error goes to 0 asymptotically as long as the probability density function for observation noise is not "vanishing" around 0. No probabilistic assumptions are imposed on the outliers.

cs.IT↗

arXiv2012

Toeplitz Matrix Based Sparse Error Correction in System Identification: Outliers and Random Noises

In this paper, we consider robust system identification under sparse outliers and random noises. In our 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 also established. When both outliers and observation noise are present, we have shown that the estimation error goes to 0 asymptotically as long as the probability density function for observation noise is not "vanishing" around 0.

cs.IT↗