The Canonical Polyadic Decomposition (CPD) is the tensor analog of the Singular Value Decomposition (SVD) for a matrix and has many data science applications including signal processing and machine learning. For the CPD, the Alternating Least Squares (ALS) algorithm has been used extensively. Although the ALS algorithm is simple, it is sensitive to a noise of a data tensor in the applications. In this paper, we propose a novel strategy to realize the noise suppression for the CPD. The proposed strategy is decomposed into two steps: (Step 1) denoising the given tensor and (Step 2) solving the exact CPD of the denoised tensor. Step 1 can be realized by solving a structured low-rank approximation with the Douglas-Rachford splitting algorithm and then Step 2 can be realized by solving the simultaneous diagonalization of a matrix tuple constructed by the denoised tensor with the DODO method. Numerical experiments show that the proposed algorithm works well even in typical cases where the ALS algorithm suffers from the so-called bottleneck/swamp effect.

Approximate Simultaneous Diagonalization (ASD) is a problem to find a common similarity transformation which approximately diagonalizes a given square-matrix tuple. Many data science problems have been reduced into ASD through ingenious modelling. For ASD, the so-called Jacobi-like methods have been extensively used. However, the methods have no guarantee to suppress the magnitude of off-diagonal entries of the transformed tuple even if the given tuple has an exact common diagonalizer, i.e., the given tuple is simultaneously diagonalizable. In this paper, to establish an alternative powerful strategy for ASD, we present a novel two-step strategy, called Approximate-Then-Diagonalize-Simultaneously (ATDS) algorithm. The ATDS algorithm decomposes ASD into (Step 1) finding a simultaneously diagonalizable tuple near the given one; and (Step 2) finding a common similarity transformation which diagonalizes exactly the tuple obtained in Step 1. The proposed approach to Step 1 is realized by solving a Structured Low-Rank Approximation (SLRA) with Cadzow's algorithm. In Step 2, by exploiting the idea in the constructive proof regarding the conditions for the exact simultaneous diagonalizability, we obtain an exact common diagonalizer of the obtained tuple in Step 1 as a solution for the original ASD. Unlike the Jacobi-like methods, the ATDS algorithm has a guarantee to find an exact common diagonalizer if the given tuple happens to be simultaneously diagonalizable. Numerical experiments show that the ATDS algorithm achieves better performance than the Jacobi-like methods.

Recovery of low-rank matrices has seen significant activity in many areas of science and engineering, motivated by theoretical results for exact reconstruction guarantees and interesting practical applications. Recently, numerous methods incorporated the nuclear norm to pursue the convexity of the optimization. However, this greatly restricts its capability and flexibility in dealing with many practical problems, where the singular values have clear physical meanings. This paper studies a generalized non-convex low-rank approximation, where the singular values are in lp-heuristic. Then specific results are derived for image restoration, including denoising and deblurring. Extensive experimental results on natural images demonstrate the improvement of the proposed method over the recent image restoration methods.

Kernel methods such as the support vector machine are one of the most successful algorithms in modern machine learning. Their advantage is that linear algorithms are extended to non-linear scenarios in a straightforward way by the use of the kernel trick. However, naive use of kernel methods is computationally expensive since the computational complexity typically scales cubically with respect to the number of training samples. In this article, we review recent advances in the kernel methods, with emphasis on scalability for massive problems.