Noise-Augmented $\ell_0$ Regularization of Tensor Regression with Tucker Decomposition

• URL: http://arxiv.org/abs/2302.10775v1
• Date: Sun, 19 Feb 2023 02:40:35 GMT
• Title: Noise-Augmented $\ell_0$ Regularization of Tensor Regression with Tucker Decomposition
• Authors: Tian Yan, Yinan Li, Fang Liu
• Abstract summary: Low-rank decomposition-based regression methods with tensor predictors exploit the structural information in tensor predictors. We propose a method named NA$$CT$2$ to regularize the parameters in tensor regression.
• Score: 13.38920246791158
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Tensor data are multi-dimension arrays. Low-rank decomposition-based regression methods with tensor predictors exploit the structural information in tensor predictors while significantly reducing the number of parameters in tensor regression. We propose a method named NA$_0$CT$^2$ (Noise Augmentation for $\ell_0$ regularization on Core Tensor in Tucker decomposition) to regularize the parameters in tensor regression (TR), coupled with Tucker decomposition. We establish theoretically that NA$_0$CT$^2$ achieves exact $\ell_0$ regularization in linear TR and generalized linear TR on the core tensor from the Tucker decomposition. To our knowledge, NA$_0$CT$^2$ is the first Tucker decomposition-based regularization method in TR to achieve $\ell_0$ in core tensor. NA$_0$CT$^2$ is implemented through an iterative procedure and involves two simple steps in each iteration -- generating noisy data based on the core tensor from the Tucker decomposition of the updated parameter estimate and running a regular GLM on noise-augmented data on vectorized predictors. We demonstrate the implementation of NA$_0$CT$^2$ and its $\ell_0$ regularization effect in both simulation studies and real data applications. The results suggest that NA$_0$CT$^2$ improves predictions compared to other decomposition-based TR approaches, with or without regularization and it also helps to identify important predictors though not designed for that purpose.

Related papers

• Highly Adaptive Ridge [84.38107748875144]
  We propose a regression method that achieves a $n-2/3$ dimension-free L2 convergence rate in the class of right-continuous functions with square-integrable sectional derivatives. Har is exactly kernel ridge regression with a specific data-adaptive kernel based on a saturated zero-order tensor-product spline basis expansion. We demonstrate empirical performance better than state-of-the-art algorithms for small datasets in particular.
  arXiv  Detail & Related papers  (2024-10-03T17:06:06Z)
• A Novel Tensor Factorization-Based Method with Robustness to Inaccurate Rank Estimation [9.058215418134209]
  We propose a new tensor norm with a dual low-rank constraint, which utilizes the low-rank prior and rank information at the same time. It is proven theoretically that the resulting tensor completion model can effectively avoid performance degradation caused by inaccurate rank estimation. Based on this, the total cost at each iteration of the optimization algorithm is reduced to $mathcalO(n3log n +kn3)$ from $mathcalO(n4)$ achieved with standard methods.
  arXiv  Detail & Related papers  (2023-05-19T06:26:18Z)
• Decomposable Sparse Tensor on Tensor Regression [1.370633147306388]
  We consider the sparse low rank tensor on tensor regression where predictors $mathcalX$ and responses $mathcalY$ are both high-dimensional tensors. We propose a fast solution based on stagewise search composed by contraction part and generation part which are optimized alternatively.
  arXiv  Detail & Related papers  (2022-12-09T18:16:41Z)
• Near-Linear Time and Fixed-Parameter Tractable Algorithms for Tensor Decompositions [51.19236668224547]
  We study low rank approximation of tensors, focusing on the tensor train and Tucker decompositions. For tensor train decomposition, we give a bicriteria $(1 + eps)$-approximation algorithm with a small bicriteria rank and $O(q cdot nnz(A))$ running time. In addition, we extend our algorithm to tensor networks with arbitrary graphs.
  arXiv  Detail & Related papers  (2022-07-15T11:55:09Z)
• 2D+3D facial expression recognition via embedded tensor manifold regularization [16.98176664818354]
  A novel approach via embedded tensor manifold regularization for 2D+3D facial expression recognition (FERETMR) is proposed. We establish the first-order optimality condition in terms of stationary points, and then design a block coordinate descent (BCD) algorithm with convergence analysis. Numerical results on BU-3DFE database and Bosphorus databases demonstrate the effectiveness of our proposed approach.
  arXiv  Detail & Related papers  (2022-01-29T06:11:00Z)
• When Random Tensors meet Random Matrices [50.568841545067144]
  This paper studies asymmetric order-$d$ spiked tensor models with Gaussian noise. We show that the analysis of the considered model boils down to the analysis of an equivalent spiked symmetric textitblock-wise random matrix.
  arXiv  Detail & Related papers  (2021-12-23T04:05:01Z)
• Fast Low-Rank Tensor Decomposition by Ridge Leverage Score Sampling [5.740578698172382]
  We study Tucker decompositions and use tools from randomized numerical linear algebra called ridge leverage scores. We show how to use approximate ridge leverage scores to construct a sketched instance for any ridge regression problem. We demonstrate the effectiveness of our approximate ridge regressioni algorithm for large, low-rank Tucker decompositions on both synthetic and real-world data.
  arXiv  Detail & Related papers  (2021-07-22T13:32:47Z)
• MTC: Multiresolution Tensor Completion from Partial and Coarse Observations [49.931849672492305]
  Existing completion formulation mostly relies on partial observations from a single tensor. We propose an efficient Multi-resolution Completion model (MTC) to solve the problem.
  arXiv  Detail & Related papers  (2021-06-14T02:20:03Z)
• Online nonparametric regression with Sobolev kernels [99.12817345416846]
  We derive the regret upper bounds on the classes of Sobolev spaces $W_pbeta(mathcalX)$, $pgeq 2, beta>fracdp$. The upper bounds are supported by the minimax regret analysis, which reveals that in the cases $beta> fracd2$ or $p=infty$ these rates are (essentially) optimal.
  arXiv  Detail & Related papers  (2021-02-06T15:05:14Z)
• Beyond Lazy Training for Over-parameterized Tensor Decomposition [69.4699995828506]
  We show that gradient descent on over-parametrized objective could go beyond the lazy training regime and utilize certain low-rank structure in the data. Our results show that gradient descent on over-parametrized objective could go beyond the lazy training regime and utilize certain low-rank structure in the data.
  arXiv  Detail & Related papers  (2020-10-22T00:32:12Z)
• Sparse and Low-Rank High-Order Tensor Regression via Parallel Proximal Method [6.381138694845438]
  We propose the Sparse and Low-rank Regression model for large-scale data with high-order structures. Our model enforces sparsity and low-rankness of the tensor coefficient. Our model's predictions exhibit meaningful interpretations on the video dataset.
  arXiv  Detail & Related papers  (2019-11-29T06:25:36Z)

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.