Tensor-on-Tensor Regression: Riemannian Optimization, Over-parameterization, Statistical-computational Gap, and Their Interplay
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| Format: | Preprint |
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2022
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| author | Luo, Yuetian Zhang, Anru R. |
| author_facet | Luo, Yuetian Zhang, Anru R. |
| contents | We study the tensor-on-tensor regression, where the goal is to connect tensor responses to tensor covariates with a low Tucker rank parameter tensor/matrix without the prior knowledge of its intrinsic rank. We propose the Riemannian gradient descent (RGD) and Riemannian Gauss-Newton (RGN) methods and cope with the challenge of unknown rank by studying the effect of rank over-parameterization. We provide the first convergence guarantee for the general tensor-on-tensor regression by showing that RGD and RGN respectively converge linearly and quadratically to a statistically optimal estimate in both rank correctly-parameterized and over-parameterized settings. Our theory reveals an intriguing phenomenon: Riemannian optimization methods naturally adapt to over-parameterization without modifications to their implementation. We also prove the statistical-computational gap in scalar-on-tensor regression by a direct low-degree polynomial argument. Our theory demonstrates a "blessing of statistical-computational gap" phenomenon: in a wide range of scenarios in tensor-on-tensor regression for tensors of order three or higher, the computationally required sample size matches what is needed by moderate rank over-parameterization when considering computationally feasible estimators, while there are no such benefits in the matrix settings. This shows moderate rank over-parameterization is essentially "cost-free" in terms of sample size in tensor-on-tensor regression of order three or higher. Finally, we conduct simulation studies to show the advantages of our proposed methods and to corroborate our theoretical findings. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2206_08756 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Tensor-on-Tensor Regression: Riemannian Optimization, Over-parameterization, Statistical-computational Gap, and Their Interplay Luo, Yuetian Zhang, Anru R. Statistics Theory Machine Learning Optimization and Control Methodology We study the tensor-on-tensor regression, where the goal is to connect tensor responses to tensor covariates with a low Tucker rank parameter tensor/matrix without the prior knowledge of its intrinsic rank. We propose the Riemannian gradient descent (RGD) and Riemannian Gauss-Newton (RGN) methods and cope with the challenge of unknown rank by studying the effect of rank over-parameterization. We provide the first convergence guarantee for the general tensor-on-tensor regression by showing that RGD and RGN respectively converge linearly and quadratically to a statistically optimal estimate in both rank correctly-parameterized and over-parameterized settings. Our theory reveals an intriguing phenomenon: Riemannian optimization methods naturally adapt to over-parameterization without modifications to their implementation. We also prove the statistical-computational gap in scalar-on-tensor regression by a direct low-degree polynomial argument. Our theory demonstrates a "blessing of statistical-computational gap" phenomenon: in a wide range of scenarios in tensor-on-tensor regression for tensors of order three or higher, the computationally required sample size matches what is needed by moderate rank over-parameterization when considering computationally feasible estimators, while there are no such benefits in the matrix settings. This shows moderate rank over-parameterization is essentially "cost-free" in terms of sample size in tensor-on-tensor regression of order three or higher. Finally, we conduct simulation studies to show the advantages of our proposed methods and to corroborate our theoretical findings. |
| title | Tensor-on-Tensor Regression: Riemannian Optimization, Over-parameterization, Statistical-computational Gap, and Their Interplay |
| topic | Statistics Theory Machine Learning Optimization and Control Methodology |
| url | https://arxiv.org/abs/2206.08756 |