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Abstract:
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Recent years have seen exciting progress in our understanding of spectral algorithms for high dimensional data analysis in the presence of a low-rank signal observed in measurement noise. Classics such as Singular Value Decomposition, Multidimensional Scaling, Principal Component Analysis, Spectral Synchronization and manifold learning using the Mahalanobis Distance are now amenable to analysis under a more-or-less unified theoretical framework. In all these cases, the theory explains when these algorithms fail, when they work and how well they perform (with asymptotically exact predictions); and provides a simple recipe for carefully tuning the spectral content of the data matrix in order to achieve optimal performance. Also, long-standing conundrums such as how to select the number of singular values or principal components have arguably satisfying answers now. I will attempt to draw a unified theoretical picture, summarize what is known and what's still to be discovered in this area. Based on joint papers with subsets of {Erez Peterfreund, Danny Barash, Elad Romanov, Ronen Talmon, Hau-tieng Wu, Pei-Chun Su, Iain Johnstone, David Donoho}.
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