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We discuss how to evaluate the proximal operator of a convex and increasing function of a nuclear norm, which forms the key computational step in several first-order optimization algorithms such as (accelerated) proximal gradient descent and ADMM. Various special cases of the problem arise in low-rank matrix completion, dropout training in deep learning and high-order low-rank tensor recovery, although they have all been solved on a case-by-case basis. We provide an unified and efficiently computable procedure for solving this problem.
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The problem of recovering a low-rank matrix from the linear constraints, known as affine matrix rank minimization problem, has been attracting extensive attention in recent years. In general, affine matrix rank minimization problem is a NP-hard. In our latest work, a non-convex fraction function is studied to approximate the rank function in affine matrix rank minimization problem and translate the NP-hard affine matrix rank minimization problem into a transformed affine matrix rank minimization problem. A scheme of iterative singular value thresholding algorithm is generated to solve the regularized transformed affine matrix rank minimization problem. However, one of the drawbacks for our iterative singular value thresholding algorithm is that the parameter $a$, which influences the behaviour of non-convex fraction function in the regularized transformed affine matrix rank minimization problem, needs to be determined manually in every simulation. In fact, how to determine the optimal parameter $a$ is not an easy problem. Here instead, in this paper, we will generate an adaptive iterative singular value thresholding algorithm to solve the regularized transformed affine matrix rank minimization problem. When doing so, our new algorithm will be intelligent both for the choice of the regularized parameter $lambda$ and the parameter $a$.
We derive a formula for optimal hard thresholding of the singular value decomposition in the presence of correlated additive noise; although it nominally involves unobservables, we show how to apply it even where the noise covariance structure is not a-priori known or is not independently estimable. The proposed method, which we call ScreeNOT, is a mathematically solid alternative to Cattells ever-popular but vague Scree Plot heuristic from 1966. ScreeNOT has a surprising oracle property: it typically achieves exactly, in large finite samples, the lowest possible MSE for matrix recovery, on each given problem instance - i.e. the specific threshold it selects gives exactly the smallest achievable MSE loss among all possible threshold choices for that noisy dataset and that unknown underlying true low rank model. The method is computationally efficient and robust against perturbations of the underlying covariance structure. Our results depend on the assumption that the singular values of the noise have a limiting empirical distribution of compact support; this model, which is standard in random matrix theory, is satisfied by many models exhibiting either cross-row correlation structure or cross-column correlation structure, and also by many situations where there is inter-element correlation structure. Simulations demonstrate the effectiveness of the method even at moderate matrix sizes. The paper is supplemented by ready-to-use software packages implementing the proposed algorithm.
Affine rank minimization problem is the generalized version of low rank matrix completion problem where linear combinations of the entries of a low rank matrix are observed and the matrix is estimated from these measurements. We propose a trainable deep neural network by unrolling a popular iterative algorithm called the singular value thresholding (SVT) algorithm to perform this generalized matrix completion which we call Learned SVT (LSVT). We show that our proposed LSVT with fixed layers (say T) reconstructs the matrix with lesser mean squared error (MSE) compared with that incurred by SVT with fixed (same T) number of iterations and our method is much more robust to the parameters which need to be carefully chosen in SVT algorithm.
This correspondence disproves the main results in the paper Design of Asymmetric Shift Operators for Efficient Decentralized Subspace Projection by S. Mollaebrahim and B. Beferull-Lozano. Counterexamples and counterproofs are provided when applicable. When those problems can be amended, a correction is suggested. However, in most cases, no correction may be possible since the problem addressed by the aforementioned paper is for the most part intractable.
In this paper, we have developed a parallel branch and bound algorithm which computes the maximal structured singular value $mu$ without tightly bounding $mu$ for each frequency and thus significantly reduce the computational complexity.
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