Timezone: »
Tensor train decomposition is widely used in machine learning and quantum physics due to its concise representation of high-dimensional tensors, overcoming the curse of dimensionality. Cross approximation---originally developed for representing a matrix from a set of selected rows and columns---is an efficient method for constructing a tensor train decomposition of a tensor from few of its entries. While tensor train cross approximation has achieved remarkable performance in practical applications, its theoretical analysis, in particular regarding the error of the approximation, is so far lacking. To our knowledge, existing results only provide element-wise approximation accuracy guarantees, which lead to a very loose bound when extended to the entire tensor. In this paper, we bridge this gap by providing accuracy guarantees in terms of the entire tensor for both exact and noisy measurements. Our results illustrate how the choice of selected subtensors affects the quality of the cross approximation and that the approximation error caused by model error and/or measurement error may not grow exponentially with the order of the tensor. These results are verified by numerical experiments, and may have important implications for the usefulness of cross approximations for high-order tensors, such as those encountered in the description of quantum many-body states.
Author Information
Zhen Qin (Ohio State University, Columbus)
Alexander Lidiak (Colorado School of Mines)
Zhexuan Gong (Colorado School of Mines)
Gongguo Tang
Michael B Wakin (Colorado School of Mines)
Zhihui Zhu (The Ohio State University)
More from the Same Authors
-
2021 Spotlight: A Geometric Analysis of Neural Collapse with Unconstrained Features »
Zhihui Zhu · Tianyu Ding · Jinxin Zhou · Xiao Li · Chong You · Jeremias Sulam · Qing Qu -
2022 Poster: Neural Collapse with Normalized Features: A Geometric Analysis over the Riemannian Manifold »
Can Yaras · Peng Wang · Zhihui Zhu · Laura Balzano · Qing Qu -
2022 Poster: Are All Losses Created Equal: A Neural Collapse Perspective »
Jinxin Zhou · Chong You · Xiao Li · Kangning Liu · Sheng Liu · Qing Qu · Zhihui Zhu -
2022 Poster: Revisiting Sparse Convolutional Model for Visual Recognition »
xili dai · Mingyang Li · Pengyuan Zhai · Shengbang Tong · Xingjian Gao · Shao-Lun Huang · Zhihui Zhu · Chong You · Yi Ma -
2021 Poster: A Geometric Analysis of Neural Collapse with Unconstrained Features »
Zhihui Zhu · Tianyu Ding · Jinxin Zhou · Xiao Li · Chong You · Jeremias Sulam · Qing Qu -
2021 Poster: Only Train Once: A One-Shot Neural Network Training And Pruning Framework »
Tianyi Chen · Bo Ji · Tianyu Ding · Biyi Fang · Guanyi Wang · Zhihui Zhu · Luming Liang · Yixin Shi · Sheng Yi · Xiao Tu -
2021 Poster: Rank Overspecified Robust Matrix Recovery: Subgradient Method and Exact Recovery »
Lijun Ding · Liwei Jiang · Yudong Chen · Qing Qu · Zhihui Zhu -
2021 Poster: Convolutional Normalization: Improving Deep Convolutional Network Robustness and Training »
Sheng Liu · Xiao Li · Simon Zhai · Chong You · Zhihui Zhu · Carlos Fernandez-Granda · Qing Qu -
2020 Poster: Robust Recovery via Implicit Bias of Discrepant Learning Rates for Double Over-parameterization »
Chong You · Zhihui Zhu · Qing Qu · Yi Ma -
2020 Spotlight: Robust Recovery via Implicit Bias of Discrepant Learning Rates for Double Over-parameterization »
Chong You · Zhihui Zhu · Qing Qu · Yi Ma -
2019 Poster: Distributed Low-rank Matrix Factorization With Exact Consensus »
Zhihui Zhu · Qiuwei Li · Xinshuo Yang · Gongguo Tang · Michael B Wakin -
2019 Poster: A Nonconvex Approach for Exact and Efficient Multichannel Sparse Blind Deconvolution »
Qing Qu · Xiao Li · Zhihui Zhu -
2019 Spotlight: A Nonconvex Approach for Exact and Efficient Multichannel Sparse Blind Deconvolution »
Qing Qu · Xiao Li · Zhihui Zhu -
2019 Poster: A Linearly Convergent Method for Non-Smooth Non-Convex Optimization on the Grassmannian with Applications to Robust Subspace and Dictionary Learning »
Zhihui Zhu · Tianyu Ding · Daniel Robinson · Manolis Tsakiris · René Vidal -
2019 Poster: The Landscape of Non-convex Empirical Risk with Degenerate Population Risk »
Shuang Li · Gongguo Tang · Michael B Wakin -
2018 Poster: Dual Principal Component Pursuit: Improved Analysis and Efficient Algorithms »
Zhihui Zhu · Yifan Wang · Daniel Robinson · Daniel Naiman · René Vidal · Manolis Tsakiris -
2018 Poster: Dropping Symmetry for Fast Symmetric Nonnegative Matrix Factorization »
Zhihui Zhu · Xiao Li · Kai Liu · Qiuwei Li -
2009 Workshop: Manifolds, sparsity, and structured models: When can low-dimensional geometry really help? »
Richard Baraniuk · Volkan Cevher · Mark A Davenport · Piotr Indyk · Bruno Olshausen · Michael B Wakin