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1. HOSVD-Based Algorithm for Weighted Tensor Completion

   https://doi.org/10.3390/jimaging7070110  

   Chao, Zehan; Huang, Longxiu; Needell, Deanna (July 2021, Journal of Imaging)

   null (Ed.)

   Matrix completion, the problem of completing missing entries in a data matrix with low-dimensional structure (such as rank), has seen many fruitful approaches and analyses. Tensor completion is the tensor analog that attempts to impute missing tensor entries from similar low-rank type assumptions. In this paper, we study the tensor completion problem when the sampling pattern is deterministic and possibly non-uniform. We first propose an efficient weighted Higher Order Singular Value Decomposition (HOSVD) algorithm for the recovery of the underlying low-rank tensor from noisy observations and then derive the error bounds under a properly weighted metric. Additionally, the efficiency and accuracy of our algorithm are both tested using synthetic and real datasets in numerical simulations. 

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2. Tensor Methods for Nonlinear Matrix Completion

   Ongie, Greg; Pimentel-Alarcon, Daniel; Nowak, Robert; Willett, Rebecca; Balzano, Laura (September 2020, SIAM journal on mathematics of data science)

   null (Ed.)

   In the low-rank matrix completion (LRMC) problem, the low-rank assumption means that the columns (or rows) of the matrix to be completed are points on a low-dimensional linear algebraic variety. This paper extends this thinking to cases where the columns are points on a low-dimensional nonlinear algebraic variety, a problem we call Low Algebraic Dimension Matrix Completion (LADMC). Matrices whose columns belong to a union of subspaces are an important special case. We propose a LADMC algorithm that leverages existing LRMC methods on a tensorized representation of the data. For example, a second-order tensorized representation is formed by taking the Kronecker product of each column with itself, and we consider higher order tensorizations as well. This approach will succeed in many cases where traditional LRMC is guaranteed to fail because the data are low-rank in the tensorized representation but not in the original representation. We also provide a formal mathematical justi cation for the success of our method. In particular, we give bounds of the rank of these data in the tensorized representation, and we prove sampling requirements to guarantee uniqueness of the solution. We also provide experimental results showing that the new approach outperforms existing state-of-the-art methods for matrix completion under a union of subspaces model. 

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3. Node-Polysemy Aware Recommendation by Matrix Completion with Side Information

   https://doi.org/10.1109/BigData52589.2021.9672000  

   Hui, Bo; Yan, Da; Ku, Wei-Shinn (January 2021, 2021 IEEE International Conference on Big Data (Big Data))

   Matrix completion is a well-known approach for recommender systems. It predicts the values of the missing entries in a sparse user-item interaction matrix, based on the low-rank structure of the rating matrix. However, existing matrix completion methods do not take node polysemy and side information of social relationships into consideration, which can otherwise further improve the performance. In this paper, we propose a novel matrix completion method that employs both users’ friendships and rating entries to predict the missing values in a user-item matrix. Our approach adopts a graph-based modeling where nodes are users and items, and two types of edges are considered: user friendships and user-item interactions. Polysemy-aware node features are extracted from this heterogeneous graph through a graph convolution network by considering the multifaceted factors for edge formation, which are then connected to a hybrid loss function with two heads: (1) a social-homophily head to address node polysemy, and (2) an error head for user-item rating regression. The latter is formulated on all matrix entries to combat the sensitivity of negative sampling of the vast majority of missing entries during training, with a smart technique to reduce the time complexity. Extensive experiments over real datasets verify that our model outperforms the state-of-the-art matrix completion methods by a significant margin. 

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4. Robust Matrix Sensing in the Semi-Random Model

   Gao, Xing; Cheng, Yu (December 2023, Proceedings of the 37th Conference on Neural Information Processing Systems)

   Low-rank matrix recovery is a fundamental problem in machine learning with numerous applications. In practice, the problem can be solved by convex optimization namely nuclear norm minimization, or by non-convex optimization as it is well-known that for low-rank matrix problems like matrix sensing and matrix completion, all local optima of the natural non-convex objectives are also globally optimal under certain ideal assumptions. In this paper, we study new approaches for matrix sensing in a semi-random model where an adversary can add any number of arbitrary sensing matrices. More precisely, the problem is to recover a low-rank matrix $$X^\star$$ from linear measurements $$b_i = \langle A_i, X^\star \rangle$$, where an unknown subset of the sensing matrices satisfies the Restricted Isometry Property (RIP) and the rest of the $$A_i$$'s are chosen adversarially. It is known that in the semi-random model, existing non-convex objectives can have bad local optima. To fix this, we present a descent-style algorithm that provably recovers the ground-truth matrix $$X^\star$$. For the closely-related problem of semi-random matrix completion, prior work [CG18] showed that all bad local optima can be eliminated by reweighting the input data. However, the analogous approach for matrix sensing requires reweighting a set of matrices to satisfy RIP, which is a condition that is NP-hard to check. Instead, we build on the framework proposed in [KLL$^+$23] for semi-random sparse linear regression, where the algorithm in each iteration reweights the input based on the current solution, and then takes a weighted gradient step that is guaranteed to work well locally. Our analysis crucially exploits the connection between sparsity in vector problems and low-rankness in matrix problems, which may have other applications in obtaining robust algorithms for sparse and low-rank problems. 

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5. High-Rank Matrix Completion with Side Information

   Wang, Y.; Elhamifar, E. (January 2018, AAAI Conference on Artificial Intelligence)

   We address the problem of high-rank matrix completion with side information. In contrast to existing work dealing with side information, which assume that the data matrix is low-rank, we consider the more general scenario where the columns of the data matrix are drawn from a union of low-dimensional subspaces, which can lead to a high rank matrix. Our goal is to complete the matrix while taking advantage of the side information. To do so, we use the self-expressive property of the data, searching for a sparse representation of each column of matrix as a combination of a few other columns. More specifically, we propose a factorization of the data matrix as the product of side information matrices with an unknown interaction matrix, under which each column of the data matrix can be reconstructed using a sparse combination of other columns. As our proposed optimization, searching for missing entries and sparse coefficients, is non-convex and NP-hard, we propose a lifting framework, where we couple sparse coefficients and missing values and define an equivalent optimization that is amenable to convex relaxation. We also propose a fast implementation of our convex framework using a Linearized Alternating Direction Method. By extensive experiments on both synthetic and real data, and, in particular, by studying the problem of multi-label learning, we demonstrate that our method outperforms existing techniques in both low-rank and high-rank data regimes. 

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