Qiuwei Li

Alternating Optimization of Sensing Matrix and Sparsifying Dictionary for Compressed Sensing

This paper deals with alternating optimization of sensing matrix and sparsifying dictionary for compressed sensing systems. Under the same framework proposed by J. M. Duarte-Carvajalino and G. Sapiro, a novel algorithm for optimal sparsifying dictionary design is derived with an optimized sensing matrix embedded. A closed-form solution to the optimal dictionary design problem is obtained. A new measure is proposed for optimizing sensing matrix and an algorithm is developed for solving the corresponding optimization problem. Experiments are carried out with synthetic data and real images, which demonstrate promising performance of the proposed algorithms and superiority of the CS system designed with the optimized sensing matrix and dictionary to existing ones in terms of signal reconstruction accuracy. Particularly, the proposed CS system yields in general a much improved performance than those designed using previous methods in terms of peak signal-to-noise ratio for the application to image compression.

The non-convex geometry of low-rank matrix optimization

This work considers the minimization of a general convex function f (X) over the cone of positive semi-definite matrices whose optimal solution X∗ is of low-rank. Standard first-order convex solvers require performing an eigenvalue decomposition in each iteration, severely limiting their scalability. A natural nonconvex reformulation of the problem factors the variable X into the product of a rectangular matrix with fewer columns and its transpose. For a special class of matrix sensing and completion problems with quadratic objective functions, local search algorithms applied to the factored problem have been shown to be much more efficient and, in spite of being nonconvex, to converge to the global optimum. The purpose of this work is to extend this line of study to general convex objective functions f (X) and investigate the geometry of the resulting factored formulations. Specifically, we prove that when f (X) satisfies the restricted well-conditioned assumption, each critical point of the factored problem either corresponds to the optimal solution X∗ or a strict saddle where the Hessian matrix has a strictly negative eigenvalue. Such a geometric structure of the factored formulation ensures that many local search algorithms can converge to the global optimum with random initializations.

Global optimality in low-rank matrix optimization

This paper considers the minimization of a general objective function f (X) over the set of non-square n × m matrices where the optimal solution X∗ is low-rank. To reduce the computational burden, we factorize the variable X into a product of two smaller matrices and optimize over these two matrices instead of X. We analyze the global geometry for a general and yet well-conditioned objective function f (X) whose restricted strong convexity and restricted strong smoothness constants are comparable. In particular, we show that the reformulated objective function has no spurious local minima and obeys the strict saddle property. These geometric properties imply that a number of iterative optimization algorithms (such as gradient descent) can provably solve the factored problem with global convergence.

Projection matrix optimization for block-sparse compressive sensing

Traditionally, the projection matrix in compressive sensing (CS) is chosen as a random matrix. In recent years, we have seen that the performance of CS systems can be improved by using a carefully designed projection matrix rather than a random one. In particular, we can reduce the coherence between the columns of the equivalent dictionary thanks to a well-designed projection matrix. Then, we can get a lower reconstruction error and a higher successful reconstruction rate. In some applications, the signals of interest have nonzero entries occurring in clusters - i.e., block-sparse signals. In this paper, we use the equiangular tight frame (ETF) to approach the Gram matrix of equivalent dictionary rather than the identity matrix used in [1]. Then, we minimize a weighted sum of the subblock coherence and the interblock coherence of the equivalent dictionary. The simulation results show that our novel method for projection matrix optimization significantly improves the ability of block-sparse approximation techniques to reconstruct and classify signals than the method proposed by Lihi Zelnik-Manor (LZM) [1].

Overcomplete tensor decomposition via convex optimization

This work develops theories and computational methods for overcomplete, non-orthogonal tensor decomposition using convex optimization. Under an incoherence condition of the rank-one factors, we show that one can retrieve tensor decomposition by solving a convex, infinite-dimensional analog of ℓ1 minimization on the space of measures. The optimal value of this optimization defines the tensor nuclear norm. Two computational schemes are proposed to solve the infinite-dimensional optimization: semidefinite programs based on sum-of-squares relaxations and nonlinear programs that are an exact reformulation of the tensor nuclear norm. The latter exhibits superior performance compared with the state-of-the-art tensor decomposition methods.

Approximate support recovery of atomic line spectral estimation: A tale of resolution and precision

This work investigates the parameter estimation performance of line spectral estimation/super-resolution using atomic norm minimization. The focus is on analyzing the algorithms accuracy of inferring the frequencies and complex magnitudes from noisy observations. When the Signal-to-Noise Ratio is reasonably high and the true frequencies are separated by O(1/n), the atomic norm estimator is shown to localize the correct number of frequencies, each within a neighborhood of size O(√log n/n3 σ) of one of the true frequencies. Here n is half the number of temporal samples and σ2 is the Gaussian noise variance. The analysis is based on a primal-dual witness construction procedure. The error bound obtained matches the Cramer-Rao lower bound up to a logarithmic factor. The relationship between resolution (separation of frequencies) and precision/accuracy of the estimator is highlighted.

On Collaborative Compressive Sensing Systems: The Framework, Design, and Algorithm

We propose a collaborative compressive sensing (CCS) framework consisting of a bank of

Simultaneous Sensing Matrix and Sparsifying Dictionary Optimization for Block-sparse Compressive Sensing

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Jazz: A companion to music for frequency estimation with missing data

compressive sensing (CS) systems that share the same sensing matrix but have different sparsifying dictionaries. This CCS system is guaranteed to yield better performance than each individual CS system in a statistical sense, while with the parallel computing strategy, it requires the same time as that needed for each individual CS system to conduct compression and signal recovery. We then provide an approach to designing optimal CCS systems by utilizing a measure that involves both the sensing matrix and dictionaries and hence allows us to simultaneously optimize the sensing matrix and all the

Iteratively reweighted least squares for block-sparse recovery

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