Stochastic Alternating Least Squares (SALS) is a method that approximates the canonical decomposition of averages of sampled random tensors. Its simplicity and efficient memory usage make SALS an ideal tool for decomposing tensors in an online setting. We show, under mild regularization and readily verifiable assumptions on the boundedness of the data, that the SALS algorithm is globally convergent. Numerical experiments validate our theoretical findings and demonstrate the algorithm's performance and complexity.
The alternating least squares algorithm for CP and Tucker decomposition is dominated in cost by the tensor contractions necessary to set up the quadratic optimization subproblems. We introduce a novel family of algorithms that uses perturbative corrections to the subproblems rather than recomputing the tensor contractions. This approximation is accurate when the factor matrices are changing little across iterations, which occurs when alternating least squares approaches convergence. We provide a theoretical analysis to bound the approximation error. Our numerical experiments demonstrate that t…
The CP tensor decomposition is used in applications such as machine learning and signal processing to discover latent low-rank structure in multidimensional data. Computing a CP decomposition via an alternating least squares (ALS) method reduces the problem to several linear least squares problems. The standard way to solve these linear least squares subproblems is to use the normal equations, which inherit special tensor structure that can be exploited for computational efficiency. However, the normal equations are sensitive to numerical ill-conditioning, which can compromise the results of t…
Alternating least squares as moving subspace correction
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）；ETH Zurich（苏黎世联邦理工学院）；University of Bonn（波恩大学）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学促进学会数理研究所）
A few iterations of alternating least squares with a random starting point provably suffice to produce nearly optimal spectral- and Frobenius-norm accuracies of low-rank approximations to a matrix; iterating to convergence is unnecessary. Thus, software implementing alternating least squares can be retrofitted via appropriate setting of parameters to calculate nearly optimally accurate low-rank approximations highly efficiently, with no need for convergence.
The epsilon alternating least squares ($ε$-ALS) is developed and analyzed for canonical polyadic decomposition (approximation) of a higher-order tensor where one or more of the factor matrices are assumed to be columnwisely orthonormal. It is shown that the algorithm globally converges to a KKT point for all tensors without any assumption. For the original ALS, by further studying the properties of the polar decomposition, we also establish its global convergence under a reality assumption not stronger than those in the literature. These results completely address a question concerning the glo…
In this work, we study the tensor ring decomposition and its associated numerical algorithms. We establish a sharp transition of algorithmic difficulty of the optimization problem as the bond dimension increases: On one hand, we show the existence of spurious local minima for the optimization landscape even when the tensor ring format is much over-parameterized, i.e., with bond dimension much larger than that of the true target tensor. On the other hand, when the bond dimension is further increased, we establish one-loop convergence for alternating least square algorithm for tensor ring decomp…
Alternating least squares is the most widely used algorithm for CP tensor decomposition. However, alternating least squares may exhibit slow or no convergence, especially when high accuracy is required. An alternative approach is to regard CP decomposition as a nonlinear least squares problem and employ Newton-like methods. Direct solution of linear systems involving an approximated Hessian is generally expensive. However, recent advancements have shown that use of an implicit representation of the linear system makes these methods competitive with alternating least squares. We provide the fir…
Tensor ring (TR) decomposition has been widely applied as an effective approach in a variety of applications to discover the hidden low-rank patterns in multidimensional data. A well-known method for TR decomposition is the alternating least squares (ALS). However, it often suffers from the notorious intermediate data explosion issue, especially for large-scale tensors. In this paper, we provide two strategies to tackle this issue and design three ALS-based algorithms. Specifically, the first strategy is used to simplify the calculation of the coefficient matrices of the normal equations for t…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
CP decomposition (CPD) is prevalent in chemometrics, signal processing, data mining and many more fields. While many algorithms have been proposed to compute the CPD, alternating least squares (ALS) remains one of the most widely used algorithm for computing the decomposition. Recent works have introduced the notion of eigenvalues and singular values of a tensor and explored applications of eigenvectors and singular vectors in areas like signal processing, data analytics and in various other fields. We introduce a new formulation for deriving singular values and vectors of a tensor by consider…
Nonnegative matrix factorization (NMF) is a prominent technique for data dimensionality reduction that has been widely used for text mining, computer vision, pattern discovery, and bioinformatics. In this paper, a framework called ARkNLS (Alternating Rank-k Nonnegativity constrained Least Squares) is proposed for computing NMF. First, a recursive formula for the solution of the rank-k nonnegativity-constrained least squares (NLS) is established. This recursive formula can be used to derive the closed-form solution for the Rank-k NLS problem for any integer k $\ge$ 1. As a result, each subprobl…
RWTH Aachen University（亚琛工业大学）；Umeå Universitet（于默奥大学）；University of Copenhagen（哥本哈根大学）
Tensor decompositions, such as CANDECOMP/PARAFAC (CP), are widely used in a variety of applications, such as chemometrics, signal processing, and machine learning. A broadly used method for computing such decompositions relies on the Alternating Least Squares (ALS) algorithm. When the number of components is small, regardless of its implementation, ALS exhibits low arithmetic intensity, which severely hinders its performance and makes GPU offloading ineffective. We observe that, in practice, experts often have to compute multiple decompositions of the same tensor, each with a small number of c…
Monash University（莫纳什大学）；University of Queensland（昆士兰大学）；University of Waterloo（滑铁卢大学）
We present Nesterov-type acceleration techniques for Alternating Least Squares (ALS) methods applied to canonical tensor decomposition. While Nesterov acceleration turns gradient descent into an optimal first-order method for convex problems by adding a momentum term with a specific weight sequence, a direct application of this method and weight sequence to ALS results in erratic convergence behaviour. This is so because the tensor decomposition problem is non-convex and ALS is accelerated instead of gradient descent. Instead, we consider various restart mechanisms and suitable choices of mome…
The affine rank minimization (ARM) problem is well known for both its applications and the fact that it is NP-hard. One of the most successful approaches, yet arguably underrepresented, is iteratively reweighted least squares (IRLS), more specifically $\mathrm{IRLS}$-$0$. Despite comprehensive empirical evidence that it overall outperforms nuclear norm minimization and related methods, it is still not understood to a satisfying degree. In particular, the significance of a slow decrease of the therein appearing regularization parameter denoted $γ$ poses interesting questions. While commonly equ…
Multiway data often naturally occurs in a tensorial format which can be approximately represented by a low-rank tensor decomposition. This is useful because complexity can be significantly reduced and the treatment of large-scale data sets can be facilitated. In this paper, we find a low-rank representation for a given tensor by solving a Bayesian inference problem. This is achieved by dividing the overall inference problem into sub-problems where we sequentially infer the posterior distribution of one tensor decomposition component at a time. This leads to a probabilistic interpretation of th…
A popular method for finding the projection onto the intersection of two closed convex subsets in Hilbert space is Dykstra's algorithm. In this paper, we provide sufficient conditions for Dykstra's algorithm to converge rapidly, in finitely many steps. We also analyze the behaviour of Dykstra's algorithm applied to a line and a square. This case study reveals stark similarities to the method of alternating projections. Moreover, we show that Dykstra's algorithm may stall for an arbitrarily long time. Finally, we present some open problems.
Peking University（北京大学）；Chinese Academy of Sciences（中国科学院）
The low multilinear rank approximation, also known as the truncated Tucker decomposition, has been extensively utilized in many applications that involve higher-order tensors. Popular methods for low multilinear rank approximation usually rely directly on matrix SVD, therefore often suffer from the notorious intermediate data explosion issue and are not easy to parallelize, especially when the input tensor is large. In this paper, we propose a new class of truncated HOSVD algorithms based on alternating least squares (ALS) for efficiently computing the low multilinear rank approximation of ten…
We study a natural alternating method of Schwarz type (domain decomposition) for certain class of couplings between local and nonlocal operators. We show that our method fits into Lion's framework and prove, as a consequence, convergence in both, the continuous and the discrete settings.
Affine sum-of-ranks minimization (ASRM) generalizes the affine rank minimization (ARM) problem from matrices to tensors. Here, the interest lies in the ranks of a family $\mathcal{K}$ of different matricizations. Transferring our priorly discussed results on asymptotic log-det rank minimization, we show that iteratively reweighted least squares with weight strength $p = 0$ remains a, theoretically and practically, particularly viable method denoted as $\mathrm{IRLS}$-$0\mathcal{K}$. As in the matrix case, we prove global convergence of asymptotic minimizers of the log-det sum-of-ranks function…
Accelerating the Canonical Polyadic Alternating Least Squares Optimization via a Randomized Interpolative Decomposition
PSI Center for Scientific Computing, Theory and Data（PSI科学计算、理论与数据中心）；Institute of Mathematics, EPFL（洛桑联邦理工学院数学研究所）；The University of Maryland, College Park（马里兰大学帕克分校）
Alternating minimization algorithms for graph regularized tensor completion
Huawei European Research Institute（华为欧洲研究院）；LISN, INRIA, Université Paris-Saclay（法国国家信息与自动化研究所，巴黎萨克雷大学）；LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院科学与工程计算国家重点实验室）；Department of INMA, Université catholique de Louvain（鲁汶天主教大学）
This article presents matrix backpropagation algorithms for the QR decomposition of matrices $A_{m, n}$, that are either square (m = n), wide (m < n), or deep (m > n), with rank $k = min(m, n)$. Furthermore, we derive novel matrix backpropagation results for the pivoted (full-rank) QR decomposition and for the LQ decomposition of deep input matrices. Differentiable QR decomposition offers a numerically stable, computationally efficient method to solve least squares problems frequently encountered in machine learning and computer vision. Other use cases such as graph learning and network compre…
Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；UC Berkeley（加州大学伯克利分校）
We show how to develop sampling-based alternating least squares (ALS) algorithms for decomposition of tensors into any tensor network (TN) format. Provided the TN format satisfies certain mild assumptions, resulting algorithms will have input sublinear per-iteration cost. Unlike most previous works on sampling-based ALS methods for tensor decomposition, the sampling in our framework is done according to the exact leverage score distribution of the design matrices in the ALS subproblems. We implement and test two tensor decomposition algorithms that use our sampling framework in a feature extra…
