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### 储德林教授讲学预告

https://meeting.tencent.com/s/k0gjm3GAI1BC

Nonnegative matrix factorization (NMF) is a prominent technique for data dimensionality reduction. In this talk, a framework called ARkNLS (Alternating Rank-k Nonnegativity constrained Least Squares) is proposed for computing NMF. First, a recursive formula for the solution of the rank-k nonnegativity-constrained least squares (NLS) is established. This recursive formula can be used to derive the closed-form solution for the Rank-k NLS problem for any integer $k\geq 1$. As a result, each subproblem for an alternating \emph{rank-k} nonnegative least squares framework can be obtained based on this closed form solution. Assuming that all matrices involved in {rank-k NLS in the context of NMF computation are of full rank, two of the currently best NMFalgorithms HALS (hierarchical alternating least squares) and ANLS-BPP (Alternating NLS based on Block Principal Pivoting) can be considered as special cases of ARkNLS with $k=1$ and $k=r$ for rank $r$ NMF, respectively.
This talk is then focused on the framework with $k=3$, which leads to a new algorithm for NMF via the closed-form solution of the rank-3 NLS problem. Furthermore, a new strategy that efficiently overcomes the potential singularity problem in rank-3 NLS within the context of NMF computation is also presented. Extensive numerical comparisons using real and synthetic data sets demonstrate that the proposed algorithm provides state-of-the-art performance in terms of computational accuracy and cpu time.

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