arXiv subjects

Huanmin Ge

Publications and source records attributed to Huanmin Ge.

10 recordsLinked to original sources

arXiv2026

Stable Image Reconstruction via Two-Parameter Power-Scale Variation Minimization

In this article, we introduce a power-scale variation (PSV$_{a,p}$) with two tunable parameters: the sparsity-inducing exponent $p\in(0,1]$ and the scaling factor $a\in(0,\infty)$. By minimizing the PSV$_{a,p}$, we establish stable reconstructions in both the gradient and the image domains under the restricted isometry property (RIP) framework. Furthermore, we design an iteratively re-weighted least squares algorithm IRLSPSV to solve the unconstrained PSV$_{a,p}$ minimization. Numerical experiments demonstrate its superior performance and broad applicability. The main novelties are: (i) the PSV$_{a,p}$ minimization enjoys great flexibility and wide applicability due to its two tunable parameters $a$ and $p$, (ii) as $a\to\infty$, the PSV$_{a,p}$ minimization reduces to the $p$-th power total variation (TV$_p$) minimization and, even in this limiting case, the established RIP condition for image reconstruction is also new, (iii) the derived RIP upper bound $\overline{\delta}$ is proved to be asymptotically optimal in $a$ for gradient recovery, (iv) sensitivity analysis confirms the distinct roles of $a$ and $p$, thereby motivating a practical parameter tuning scheme for the proposed model.

cs.IT↗

arXiv2026

A Novel Two-Parameter Penalty: Relaxation Degree Analysis and Sparse Signal Recovery

In this article, we introduce a nonconvex two-parameter penalty function $P_{a,p}$, parameterized by $a\in(0,\infty)$ and $p\in(0,1]$, and the relaxation degree RD$_P$ for a separable nonconvex penalty function $P$. Based on $P_{a,p}$, we further propose the $P_{a,p}$ minimization framework for sparse signal recovery. This framework generalizes the TL1 minimization model established by S. Zhang and J. Xin (corresponding to the special case $p=1$) and provides a unified and flexible family of nonconvex penalty functions for sparse signal recovery. Using the sparse convex-combination technique, we establish both exact and stable sparse signal recovery under the restricted isometry property (RIP). To efficiently solve the resulting nonconvex optimization problem, we apply a modified iteratively re-weighted least squares method and the difference of convex functions algorithm (DCA) to develop the IRLSTLp algorithm for unconstrained $P_{a,p}$ minimization and prove some convergence results. Finally, some numerical experiments are conducted to show the flexibility of the $P_{a,p}$ minimization framework, the robustness of the IRLSTLp, and also the utility of the relaxation degree.

math.FA↗

arXiv2025

On the Performance of Amplitude-Based Models for Low-Rank Matrix Recovery

In this paper, we focus on low-rank phase retrieval, which aims to reconstruct a matrix $\mathbf{X}_0\in \mathbb{R}^{n\times m}$ with ${\mathrm{ rank}}(\mathbf{X}_0)\le r$ from noise-corrupted amplitude measurements $\mathbf{y}=|\mathcal{A}(\mathbf{X}_0)|+\boldsymbol{\eta}$, where $\mathcal{A}:\mathbb{R}^{n\times m}\rightarrow \mathbb{R}^{p}$ is a linear map and $\boldsymbol{\eta}\in \mathbb{R}^p$ is the noise vector. We first examine the rank-constrained nonlinear least-squares model $\hat{\mathbf{X}}\in \mathop{\mathrm{argmin}}\limits_{\substack{\mathbf{X}\in \mathbb{R}^{n\times m},\mathrm{rank}(\mathbf{X})\le r}}\||\mathcal{A}(\mathbf{X})|-\mathbf{y}\|_2^2$ to estimate $\mathbf{X}_0$, and demonstrate that the reconstruction error satisfies $\min\{\|\hat{\mathbf{X}}-\mathbf{X}_0\|_F, \|\hat{\mathbf{X}}+\mathbf{X}_0\|_F\}\lesssim \frac{\|\boldsymbol{\eta}\|_2}{\sqrt{p}}$ with high probability, provided $\mathcal{A}$ is a Gaussian measurement ensemble and $p\gtrsim (m+n)r$. We also prove that the error bound $\frac{\|\boldsymbol{\eta}\|_2}{\sqrt{p}}$ is tight up to a constant. Furthermore, we relax the rank constraint to a nuclear-norm constraint. Hence, we propose the Lasso model for low-rank phase retrieval, i.e., the constrained nuclear-norm model and the unconstrained version. We also establish comparable theoretical guarantees for these models. To achieve this, we introduce a strong restricted isometry property (SRIP) for the linear map $\mathcal{A}$, analogous to the strong RIP in phase retrieval. This work provides a unified treatment that extends existing results in both phase retrieval and low-rank matrix recovery from rank-one measurements.

cs.IT↗

arXiv2021

Signal and Image Reconstruction with Tight Frames via Unconstrained $\ell_1-α\ell_2$-Analysis Minimizations

In the paper, we introduce an unconstrained analysis model based on the $\ell_{1}-α\ell_{2}$ $(0< α\leq1)$ minimization for the signal and image reconstruction. We develop some new technology lemmas for tight frame, and the recovery guarantees based on the restricted isometry property adapted to frames. The effective algorithm is established for the proposed nonconvex analysis model. We illustrate the performance of the proposed model and algorithm for the signal and compressed sensing MRI reconstruction via extensive numerical experiments. And their performance is better than that of the existing methods.

cs.IT↗

arXiv2021

The Dantzig selector: Recovery of Signal via $\ell_1-α\ell_2$ Minimization

In the paper, we proposed the Dantzig selector based on the $\ell_{1}-α\ell_{2}$~$(0< α\leq1)$ minimization for the signal recovery. In the Dantzig selector, the constraint $\|{\bf A}^{\top}({\bf b}-{\bf A}{\bf x})\|_\infty \leq η$ for some small constant $η>0$ means the columns of ${\bf A}$ has very weakly correlated with the error vector ${\bf e}={\bf A}{\bf x}-{\bf b}$. First, recovery guarantees based on the restricted isometry property (RIP) are established for signals. Next, we propose the effective algorithm to solve the proposed Dantzig selector. Last, we illustrate the proposed model and algorithm by extensive numerical experiments for the recovery of signals in the cases of Gaussian, impulsive and uniform noise. And the performance of the proposed Dantzig selector is better than that of the existing methods.

cs.IT↗

arXiv2018

Efficient and Robust Recovery of Signal and Image in Impulsive Noise via $\ell_1-α\ell_2$ Minimization

In this paper, we consider the efficient and robust reconstruction of signals and images via $\ell_{1}-α\ell_{2}~(0<α\leq 1)$ minimization in impulsive noise case. To achieve this goal, we introduce two new models: the $\ell_1-α\ell_2$ minimization with $\ell_1$ constraint, which is called $\ell_1-α\ell_2$-LAD, the $\ell_1-α\ell_2$ minimization with Dantzig selector constraint, which is called $\ell_1-α\ell_2$-DS. We first show that sparse signals or nearly sparse signals can be exactly or stably recovered via $\ell_{1}-α\ell_{2}$ minimization under some conditions based on the restricted $1$-isometry property ($\ell_1$-RIP). Second, for $\ell_1-α\ell_2$-LAD model, we introduce unconstrained $\ell_1-α\ell_2$ minimization model denoting $\ell_1-α\ell_2$-PLAD and propose $\ell_1-α\ell_2$LA algorithm to solve the $\ell_1-α\ell_2$-PLAD. Last, numerical experiments %on success rates of sparse signal recovery demonstrate that when the sensing matrix is ill-conditioned (i.e., the coherence of the matrix is larger than 0.99), the $\ell_1-α\ell_2$LA method is better than the existing convex and non-convex compressed sensing solvers for the recovery of sparse signals. And for the magnetic resonance imaging (MRI) reconstruction with impulsive noise, we show that the $\ell_1-α\ell_2$LA method has better performance than state-of-the-art methods via numerical experiments.

math.OC↗

arXiv2017

A sharp recovery condition for sparse signals with partial support information via orthogonal matching pursuit

This paper considers the exact recovery of $k$-sparse signals in the noiseless setting and support recovery in the noisy case when some prior information on the support of the signals is available. This prior support consists of two parts. One part is a subset of the true support and another part is outside of the true support. For $k$-sparse signals $\mathbf{x}$ with the prior support which is composed of $g$ true indices and $b$ wrong indices, we show that if the restricted isometry constant (RIC) $δ_{k+b+1}$ of the sensing matrix $\mathbf{A}$ satisfies \begin{eqnarray*} δ_{k+b+1}<\frac{1}{\sqrt{k-g+1}}, \end{eqnarray*} then orthogonal matching pursuit (OMP) algorithm can perfectly recover the signals $\mathbf{x}$ from $\mathbf{y}=\mathbf{Ax}$ in $k-g$ iterations. Moreover, we show the above sufficient condition on the RIC is sharp. In the noisy case, we achieve the exact recovery of the remainder support (the part of the true support outside of the prior support) for the $k$-sparse signals $\mathbf{x}$ from $\mathbf{y}=\mathbf{Ax}+\mathbf{v}$ under appropriate conditions. For the remainder support recovery, we also obtain a necessary condition based on the minimum magnitude of partial nonzero elements of the signals $\mathbf{x}$.

cs.IT↗

arXiv2017

Recovery of signals by a weighted $\ell_2/\ell_1$ minimization under arbitrary prior support information

In this paper, we introduce a weighted $\ell_2/\ell_1$ minimization to recover block sparse signals with arbitrary prior support information. When partial prior support information is available, a sufficient condition based on the high order block RIP is derived to guarantee stable and robust recovery of block sparse signals via the weighted $\ell_2/\ell_1$ minimization. We then show if the accuracy of arbitrary prior block support estimate is at least $50\%$, the sufficient recovery condition by the weighted $\ell_2/\ell_{1}$ minimization is weaker than that by the $\ell_2/\ell_{1}$ minimization, and the weighted $\ell_2/\ell_{1}$ minimization provides better upper bounds on the recovery error in terms of the measurement noise and the compressibility of the signal. Moreover, we illustrate the advantages of the weighted $\ell_2/\ell_1$ minimization approach in the recovery performance of block sparse signals under uniform and non-uniform prior information by extensive numerical experiments. The significance of the results lies in the facts that making explicit use of block sparsity and partial support information of block sparse signals can achieve better recovery performance than handling the signals as being in the conventional sense, thereby ignoring the additional structure and prior support information in the problem.

cs.IT↗

arXiv2016

A sharp recovery condition for block sparse signals by block orthogonal multi-matching pursuit

We consider the block orthogonal multi-matching pursuit (BOMMP) algorithm for the recovery of block sparse signals. A sharp bound is obtained for the exact reconstruction of block $K$-sparse signals via the BOMMP algorithm in the noiseless case, based on the block restricted isometry constant (block-RIC). Moreover, we show that the sharp bound combining with an extra condition on the minimum $\ell_2$ norm of nonzero blocks of block $K-$sparse signals is sufficient to recover the true support of block $K$-sparse signals by the BOMMP in the noise case. The significance of the results we obtain in this paper lies in the fact that making explicit use of block sparsity of block sparse signals can achieve better recovery performance than ignoring the additional structure in the problem as being in the conventional sense.

cs.IT↗

arXiv2016

A sharp bound on RIC in generalized orthogonal matching pursuit

Generalized orthogonal matching pursuit (gOMP) algorithm has received much attention in recent years as a natural extension of orthogonal matching pursuit. It is used to recover sparse signals in compressive sensing. In this paper, a new bound is obtained for the exact reconstruction of every $K$-sparse signal via the gOMP algorithm in the noiseless case. That is, if the restricted isometry constant (RIC) $δ_{NK+1}$ of the sensing matrix $A$ satisfies \begin{eqnarray*} δ_{NK+1}<\frac{1}{\sqrt{\frac{K}{N}+1}}, \end{eqnarray*} then the gOMP can perfectly recover every $K$-sparse signal $x$ from $y=Ax$. Furthermore, the bound is proved to be sharp in the following sense. For any given positive integer $K$, we construct a matrix $A$ with the RIC \begin{eqnarray*} δ_{NK+1}=\frac{1}{\sqrt{\frac{K}{N}+1}} \end{eqnarray*} such that the gOMP may fail to recover some $K$-sparse signal $x$. In the noise case, an extra condition on the minimum magnitude of the nonzero components of every $K-$sparse signal combining with the above bound on RIC of the sensing matrix $A$ is sufficient to recover the true support of every $K$-sparse signal by the gOMP.

cs.IT↗