Projected Tensor-Tensor Products for Efficient Computation of Optimal Multiway Data Representations

• URL: http://arxiv.org/abs/2409.19402v1
• Date: Sat, 28 Sep 2024 16:29:54 GMT
• Title: Projected Tensor-Tensor Products for Efficient Computation of Optimal Multiway Data Representations
• Authors: Katherine Keegan, Elizabeth Newman,
• Abstract summary: We propose a new projected tensor-tensor product that relaxes the invertibility restriction to reduce computational overhead. We provide theory to prove the matrix mimeticity and the optimality of compressed representations within the projected product framework.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Tensor decompositions have become essential tools for feature extraction and compression of multiway data. Recent advances in tensor operators have enabled desirable properties of standard matrix algebra to be retained for multilinear factorizations. Behind this matrix-mimetic tensor operation is an invertible matrix whose size depends quadratically on certain dimensions of the data. As a result, for large-scale multiway data, the invertible matrix can be computationally demanding to apply and invert and can lead to inefficient tensor representations in terms of construction and storage costs. In this work, we propose a new projected tensor-tensor product that relaxes the invertibility restriction to reduce computational overhead and still preserves fundamental linear algebraic properties. The transformation behind the projected product is a tall-and-skinny matrix with unitary columns, which depends only linearly on certain dimensions of the data, thereby reducing computational complexity by an order of magnitude. We provide extensive theory to prove the matrix mimeticity and the optimality of compressed representations within the projected product framework. We further prove that projected-product-based approximations outperform a comparable, non-matrix-mimetic tensor factorization. We support the theoretical findings and demonstrate the practical benefits of projected products through numerical experiments on video and hyperspectral imaging data.

Related papers

• Learning Linear Attention in Polynomial Time [115.68795790532289]
  We provide the first results on learnability of single-layer Transformers with linear attention. We show that linear attention may be viewed as a linear predictor in a suitably defined RKHS. We show how to efficiently identify training datasets for which every empirical riskr is equivalent to the linear Transformer.
  arXiv  Detail & Related papers  (2024-10-14T02:41:01Z)
• Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry [63.694184882697435]
  Global Covariance Pooling (GCP) has been demonstrated to improve the performance of Deep Neural Networks (DNNs) by exploiting second-order statistics of high-level representations.
  arXiv  Detail & Related papers  (2024-07-15T07:11:44Z)
• Optimal Matrix-Mimetic Tensor Algebras via Variable Projection [0.0]
  Matrix mimeticity arises from interpreting tensors as operators that can be multiplied, factorized, and analyzed analogous to matrices. We learn optimal linear mappings and corresponding tensor representations without relying on prior knowledge of the data. We provide original theory of uniqueness of the transformation and convergence analysis of our variable-projection-based algorithm.
  arXiv  Detail & Related papers  (2024-06-11T04:52:23Z)
• Large-scale gradient-based training of Mixtures of Factor Analyzers [67.21722742907981]
  This article contributes both a theoretical analysis as well as a new method for efficient high-dimensional training by gradient descent. We prove that MFA training and inference/sampling can be performed based on precision matrices, which does not require matrix inversions after training is completed. Besides the theoretical analysis and matrices, we apply MFA to typical image datasets such as SVHN and MNIST, and demonstrate the ability to perform sample generation and outlier detection.
  arXiv  Detail & Related papers  (2023-08-26T06:12:33Z)
• Bounding Real Tensor Optimizations via the Numerical Range [0.0]
  We show how the numerical range of a matrix can be used to bound the optimal value of certain optimization problems over real tensor product vectors. Our bound is stronger than the trivial bounds based on eigenvalues, and can be computed significantly faster than bounds provided by semidefinite programming relaxations.
  arXiv  Detail & Related papers  (2022-12-24T20:03:06Z)
• Accelerated structured matrix factorization [0.0]
  Matrix factorization exploits the idea that, in complex high-dimensional data, the actual signal typically lies in lower-dimensional structures. By exploiting Bayesian shrinkage priors, we devise a computationally convenient approach for high-dimensional matrix factorization. The dependence between row and column entities is modeled by inducing flexible sparse patterns within factors.
  arXiv  Detail & Related papers  (2022-12-13T11:35:01Z)
• Learning Graphical Factor Models with Riemannian Optimization [70.13748170371889]
  This paper proposes a flexible algorithmic framework for graph learning under low-rank structural constraints. The problem is expressed as penalized maximum likelihood estimation of an elliptical distribution. We leverage geometries of positive definite matrices and positive semi-definite matrices of fixed rank that are well suited to elliptical models.
  arXiv  Detail & Related papers  (2022-10-21T13:19:45Z)
• Adversarially-Trained Nonnegative Matrix Factorization [77.34726150561087]
  We consider an adversarially-trained version of the nonnegative matrix factorization. In our formulation, an attacker adds an arbitrary matrix of bounded norm to the given data matrix. We design efficient algorithms inspired by adversarial training to optimize for dictionary and coefficient matrices.
  arXiv  Detail & Related papers  (2021-04-10T13:13:17Z)
• Non-negative matrix and tensor factorisations with a smoothed Wasserstein loss [0.0]
  We introduce a general mathematical framework for computing non-negative factorisations of matrices and tensors with respect to an optimal transport loss. We demonstrate the applicability of this approach with several numerical examples.
  arXiv  Detail & Related papers  (2021-04-04T22:42:21Z)
• Generative Flows with Matrix Exponential [25.888286821451562]
  Generative flows models enjoy the properties of tractable exact likelihood and efficient sampling. We incorporate matrix exponential into generative flows. Our model achieves great performance on density estimation amongst generative flows models.
  arXiv  Detail & Related papers  (2020-07-19T11:18:47Z)
• Supervised Quantile Normalization for Low-rank Matrix Approximation [50.445371939523305]
  We learn the parameters of quantile normalization operators that can operate row-wise on the values of $X$ and/or of its factorization $UV$ to improve the quality of the low-rank representation of $X$ itself. We demonstrate the applicability of these techniques on synthetic and genomics datasets.
  arXiv  Detail & Related papers  (2020-02-08T21:06:02Z)

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.