Tensor Completion via Integer Optimization

• URL: http://arxiv.org/abs/2402.05141v2
• Date: Wed, 3 Apr 2024 21:25:49 GMT
• Title: Tensor Completion via Integer Optimization
• Authors: Xin Chen, Sukanya Kudva, Yongzheng Dai, Anil Aswani, Chen Chen,
• Abstract summary: The main challenge with the tensor completion problem is a fundamental tension between computation power and the information-theoretic sample complexity rate. Past approaches either achieve the information-theoretic rate but lack practical algorithms to compute the corresponding solution. This paper develops a novel tensor completion algorithm that resolves this tension by achieving both provable convergence (in numerical tolerance) in a linear number of oracle steps and the information-theoretic rate.
• Score: 7.813563137863005
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The main challenge with the tensor completion problem is a fundamental tension between computation power and the information-theoretic sample complexity rate. Past approaches either achieve the information-theoretic rate but lack practical algorithms to compute the corresponding solution, or have polynomial-time algorithms that require an exponentially-larger number of samples for low estimation error. This paper develops a novel tensor completion algorithm that resolves this tension by achieving both provable convergence (in numerical tolerance) in a linear number of oracle steps and the information-theoretic rate. Our approach formulates tensor completion as a convex optimization problem constrained using a gauge-based tensor norm, which is defined in a way that allows the use of integer linear optimization to solve linear separation problems over the unit-ball in this new norm. Adaptations based on this insight are incorporated into a Frank-Wolfe variant to build our algorithm. We show our algorithm scales-well using numerical experiments on tensors with up to ten million entries.

Related papers

• A Sample Efficient Alternating Minimization-based Algorithm For Robust Phase Retrieval [56.67706781191521]
  In this work, we present a robust phase retrieval problem where the task is to recover an unknown signal. Our proposed oracle avoids the need for computationally spectral descent, using a simple gradient step and outliers.
  arXiv  Detail & Related papers  (2024-09-07T06:37:23Z)
• Linearization Algorithms for Fully Composite Optimization [61.20539085730636]
  This paper studies first-order algorithms for solving fully composite optimization problems convex compact sets. We leverage the structure of the objective by handling differentiable and non-differentiable separately, linearizing only the smooth parts.
  arXiv  Detail & Related papers  (2023-02-24T18:41:48Z)
• Optimal Algorithms for the Inhomogeneous Spiked Wigner Model [89.1371983413931]
  We derive an approximate message-passing algorithm (AMP) for the inhomogeneous problem. We identify in particular the existence of a statistical-to-computational gap where known algorithms require a signal-to-noise ratio bigger than the information-theoretic threshold to perform better than random.
  arXiv  Detail & Related papers  (2023-02-13T19:57:17Z)
• Accelerated Nonnegative Tensor Completion via Integer Programming [7.3149416054553065]
  We present an approach to nonnegative tensor completion based on integer programming. We explore several variants that can maintain the same theoretical guarantees as the algorithm, but offer potentially faster computation.
  arXiv  Detail & Related papers  (2022-11-28T21:00:25Z)
• Nonnegative Tensor Completion via Integer Optimization [5.932152752097064]
  We prove that our algorithm converges in a linear (in numerical tolerance) number of oracle steps, while achieving the information-theoretic rate. Because the norm is defined using a 0-1 polytope, this means we can use integer linear programming to solve linear separation problems over the polytope.
  arXiv  Detail & Related papers  (2021-11-08T15:43:19Z)
• Robust M-estimation-based Tensor Ring Completion: a Half-quadratic Minimization Approach [14.048989759890475]
  We develop a robust approach to tensor ring completion that uses an M-estimator as its error statistic. We present two HQ-based algorithms based on truncated singular value decomposition and matrix factorization.
  arXiv  Detail & Related papers  (2021-06-19T04:37:50Z)
• New Riemannian preconditioned algorithms for tensor completion via polyadic decomposition [10.620193291237262]
  These algorithms exploit a non-Euclidean metric on the product space of the factor matrices of the low-rank tensor in the polyadic decomposition form. We prove that the proposed gradient descent algorithm globally converges to a stationary point of the tensor completion problem. Numerical results on synthetic and real-world data suggest that the proposed algorithms are more efficient in memory and time compared to state-of-the-art algorithms.
  arXiv  Detail & Related papers  (2021-01-26T22:11:06Z)
• Single-Timescale Stochastic Nonconvex-Concave Optimization for Smooth Nonlinear TD Learning [145.54544979467872]
  We propose two single-timescale single-loop algorithms that require only one data point each step. Our results are expressed in a form of simultaneous primal and dual side convergence.
  arXiv  Detail & Related papers  (2020-08-23T20:36:49Z)
• Boosting Data Reduction for the Maximum Weight Independent Set Problem Using Increasing Transformations [59.84561168501493]
  We introduce new generalized data reduction and transformation rules for the maximum weight independent set problem. Surprisingly, these so-called increasing transformations can simplify the problem and also open up the reduction space to yield even smaller irreducible graphs later in the algorithm. Our algorithm computes significantly smaller irreducible graphs on all except one instance, solves more instances to optimality than previously possible, is up to two orders of magnitude faster than the best state-of-the-art solver, and finds higher-quality solutions than solvers DynWVC and HILS.
  arXiv  Detail & Related papers  (2020-08-12T08:52:50Z)
• Optimal Randomized First-Order Methods for Least-Squares Problems [56.05635751529922]
  This class of algorithms encompasses several randomized methods among the fastest solvers for least-squares problems. We focus on two classical embeddings, namely, Gaussian projections and subsampled Hadamard transforms. Our resulting algorithm yields the best complexity known for solving least-squares problems with no condition number dependence.
  arXiv  Detail & Related papers  (2020-02-21T17:45:32Z)
• Grassmannian Optimization for Online Tensor Completion and Tracking with the t-SVD [10.137631021498109]
  We show the t-SVD is a specialization of the well-studied block-term decomposition for third-order tensors. We present an algorithm under this model that can track changing free submodules from incomplete streaming 2-D data. Our results are competitive in accuracy but much faster in compute time than state-of-the-art tensor completion algorithms on real applications.
  arXiv  Detail & Related papers  (2020-01-30T15:56:14Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.