Recovery of low-rank matrices from a small number of linear measurements is now well-known to be possible under various model assumptions on the measurements. Such results demonstrate robustness and are backed with provable theoretical guarantees. However, extensions to tensor recovery have only recently began to be studied and developed, despite an abundance of practical tensor applications. Recently, a tensor variant of the Iterative Hard Thresholding method was proposed and theoretical results were obtained that guarantee exact recovery of tensors with low Tucker rank. In this paper, we utilize the same tensor version of the Restricted Isometry Property (RIP) to extend these results for tensors with low CANDECOMP/PARAFAC (CP) rank. In doing so, we leverage recent results on efficient approximations of CP decompositions that remove the need for challenging assumptions in prior works. We complement our theoretical findings with empirical results that showcase the potential of the approach.
We consider linear sparse recovery problems where additional structure regarding the support of the solution is known. The form of the structure considered is non-overlapping sets of indices that each contain part of the support. An algorithm based on iterative hard thresholding is proposed to solve this problem. The convergence and error of the method are analyzed with respect to mutual coherence. Numerical simulations are examined in the context of an inverse source problem, including modifications for off-grid recovery
Low-rank tensor recovery problems have been widely studied in many applications of signal processing and machine learning. Tucker decomposition is known as one of the most popular decompositions in the tensor framework. In recent years, researchers have developed many state-of-the-art algorithms to address the problem of low-Tucker-rank tensor recovery. Motivated by the favorable properties of the stochastic algorithms, such as stochastic gradient descent and stochastic iterative hard thresholding, we aim to extend the well-known stochastic iterative hard thresholding algorithm to the tensor f…
In this paper, we consider the $\ell_0$ minimization problem whose objective function is the sum of $\ell_0$-norm and convex differentiable function. A variable metric type method which combines the PIHT method and the skill in quasi-newton method, named variable metric extrapolation proximal iterative hard thresholding (VMEPIHT) method, is proposed. Then we analyze its convergence, linear convergence rate and superlinear convergence rate under appropriate assumptions. Finally, we conduct numerical experiments on compressive sensing problem and CT image reconstruction problem to confirm VMPIHT…
In this paper, we present modifications of the iterative hard thresholding (IHT) method for recovery of jointly row-sparse and low-rank matrices. In particular a Riemannian version of IHT is considered which significantly reduces computational cost of the gradient projection in the case of rank-one measurement operators, which have concrete applications in blind deconvolution. Experimental results are reported that show near-optimal recovery for Gaussian and rank-one measurements, and that adaptive stepsizes give crucial improvement. A Riemannian proximal gradient method is derived for the spe…
We analyze sparse frame based regularization of inverse problems by means of a diagonal frame decomposition (DFD) for the forward operator, which generalizes the SVD. The DFD allows to define a non-iterative (direct) operator-adapted frame thresholding approach which we show to provide a convergent regularization method with linear convergence rates. These results will be compared to the well-known analysis and synthesis variants of sparse $\ell^1$-regularization which are usually implemented thorough iterative schemes. If the frame is a basis (non-redundant case), the three versions of sparse…
Compressed sensing is a technique to sample compressible signals below the Nyquist rate, whilst still allowing near optimal reconstruction of the signal. In this paper we present a theoretical analysis of the iterative hard thresholding algorithm when applied to the compressed sensing recovery problem. We show that the algorithm has the following properties (made more precise in the main text of the paper) - It gives near-optimal error guarantees. - It is robust to observation noise. - It succeeds with a minimum number of observations. - It can be used with any sampling operator for which the …
Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences（奥地利科学院约翰·拉顿计算与应用数学研究所）；Program in Applied and Computational Mathematics, Princeton University（普林斯顿大学应用与计算数学项目）
We investigate the problem of recovering jointly $r$-rank and $s$-bisparse matrices from as few linear measurements as possible, considering arbitrary measurements as well as rank-one measurements. In both cases, we show that $m \asymp r s \ln(en/s)$ measurements make the recovery possible in theory, meaning via a nonpractical algorithm. In case of arbitrary measurements, we investigate the possibility of achieving practical recovery via an iterative-hard-thresholding algorithm when $m \asymp r s^γ\ln(en/s)$ for some exponent $γ> 0$. We show that this is feasible for $γ= 2$, and that the propo…
Indian Institute of Technology, Kharagpur（印度理工学院卡拉格普尔分校）
Hard thresholding pursuit (HTP) is a recently proposed iterative sparse recovery algorithm which is a result of combination of a support selection step from iterated hard thresholding (IHT) and an estimation step from the orthogonal matching pursuit (OMP). HTP has been seen to enjoy improved recovery guarantee along with enhanced speed of convergence. Much of the success of HTP can be attributed to its improved support selection capability due to the support selection step from IHT. In this paper, we propose a generalized HTP algorithm, called regularized HTP (RHTP), where the support selectio…
