{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/fourier-pca-and-robust-tensor-decomposition","title":"Fourier PCA and Robust Tensor Decomposition","arxiv_id":"1306.5825","date":"2013-06-25","proceeding":null,"authors":["Navin Goyal","Santosh Vempala","Ying Xiao"],"abstract":"Fourier PCA is Principal Component Analysis of a matrix obtained from higher\norder derivatives of the logarithm of the Fourier transform of a\ndistribution.We make this method algorithmic by developing a tensor\ndecomposition method for a pair of tensors sharing the same vectors in rank-$1$\ndecompositions. Our main application is the first provably polynomial-time\nalgorithm for underdetermined ICA, i.e., learning an $n \\times m$ matrix $A$\nfrom observations $y=Ax$ where $x$ is drawn from an unknown product\ndistribution with arbitrary non-Gaussian components. The number of component\ndistributions $m$ can be arbitrarily higher than the dimension $n$ and the\ncolumns of $A$ only need to satisfy a natural and efficiently verifiable\nnondegeneracy condition. As a second application, we give an alternative\nalgorithm for learning mixtures of spherical Gaussians with linearly\nindependent means. These results also hold in the presence of Gaussian noise.","url_abs":"http://arxiv.org/abs/1306.5825v5","url_pdf":"http://arxiv.org/pdf/1306.5825v5.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"fourier-pca-and-robust-tensor-decomposition","repo_url":"https://github.com/yingusxiaous/libFPCA","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"BSD-2-Clause"}}],"tasks":[{"task_slug":"tensor-decomposition","task_name":"Tensor Decomposition"}],"methods":[{"method_slug":"ica","method_name":"ICA"},{"method_slug":"pca","method_name":"PCA"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1306.5825","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}