{"id":990,"date":"2025-01-06T07:03:09","date_gmt":"2025-01-06T07:03:09","guid":{"rendered":"https:\/\/mailitics.com\/index.php\/2025\/01\/06\/2501-01696\/"},"modified":"2025-01-06T07:03:09","modified_gmt":"2025-01-06T07:03:09","slug":"2501-01696","status":"publish","type":"post","link":"https:\/\/mailitics.com\/index.php\/2025\/01\/06\/2501-01696\/","title":{"rendered":"Guaranteed Nonconvex Low-Rank Tensor Estimation via Scaled Gradient Descent"},"content":{"rendered":"

Guaranteed Nonconvex Low-Rank Tensor Estimation via Scaled Gradient Descent
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arXiv:2501.01696v1 Announce Type: new
\nAbstract: Tensors, which give a faithful and effective representation to deliver the intrinsic structure of multi-dimensional data, play a crucial role in an increasing number of signal processing and machine learning problems. However, tensor data are often accompanied by arbitrary signal corruptions, including missing entries and sparse noise. A fundamental challenge is to reliably extract the meaningful information from corrupted tensor data in a statistically and computationally efficient manner. This paper develops a scaled gradient descent (ScaledGD) algorithm to directly estimate the tensor factors with tailored spectral initializations under the tensor-tensor product (t-product) and tensor singular value decomposition (t-SVD) framework. In theory, ScaledGD achieves linear convergence at a constant rate that is independent of the condition number of the ground truth low-rank tensor for two canonical problems — tensor robust principal component analysis and tensor completion — as long as the level of corruptions is not too large and the sample size is sufficiently large, while maintaining the low per-iteration cost of gradient descent. To the best of our knowledge, ScaledGD is the first algorithm that provably has such properties for low-rank tensor estimation with the t-SVD decomposition. Finally, numerical examples are provided to demonstrate the efficacy of ScaledGD in accelerating the convergence rate of ill-conditioned low-rank tensor estimation in these two applications.<\/div>\n

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\n Tong Wu
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Guaranteed Nonconvex Low-Rank Tensor Estimation via Scaled Gradient Descent arXiv:2501.01696v1 Announce Type: new Abstract: Tensors, which give a faithful and effective representation to deliver the intrinsic structure of multi-dimensional data, play a crucial role in an increasing number of signal processing and machine learning problems. However, tensor data are often accompanied by arbitrary signal corruptions, […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[62,414,113,415,112],"tags":[588,589,612],"class_list":["post-990","post","type-post","status-publish","format-standard","hentry","category-aimldsaimlds","category-cs-it","category-cs-lg","category-math-it","category-stat-ml","tag-low","tag-rank","tag-tensor"],"_links":{"self":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/990"}],"collection":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/comments?post=990"}],"version-history":[{"count":0,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/990\/revisions"}],"wp:attachment":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/media?parent=990"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/categories?post=990"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/tags?post=990"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}