TechTalks from event: ICML 2011
Recommendation and Matrix Factorization
GoDec: Randomized Low-rank & Sparse Matrix Decomposition in Noisy CaseLow-rank and sparse structures have been profoundly studied in matrix completion and compressed sensing. In this paper, we develop ``Go Decomposition'' (GoDec) to efficiently and robustly estimate the low-rank part $L$ and the sparse part $S$ of a matrix $X=L+S+G$ with noise $G$. GoDec alternatively assigns the low-rank approximation of $X-S$ to $L$ and the sparse approximation of $X-L$ to $S$. The algorithm can be significantly accelerated by bilateral random projections (BRP). We also propose GoDec for matrix completion as an important variant. We prove that the objective value $|X-L-S|_F^2$ converges to a local minimum, while $L$ and $S$ linearly converge to local optimums. Theoretically, we analyze the influence of $L$, $S$ and $G$ to the asymptotic/convergence speeds in order to discover the robustness of GoDec. Empirical studies suggest the efficiency, robustness and effectiveness of GoDec comparing with representative matrix decomposition and completion tools, e.g., Robust PCA and OptSpace.
Large-Scale Convex Minimization with a Low-Rank ConstraintWe address the problem of minimizing a convex function over the space of large matrices with low rank. While this optimization problem is hard in general, we propose an efficient greedy algorithm and derive its formal approximation guarantees. Each iteration of the algorithm involves (approximately) finding the left and right singular vectors corresponding to the largest singular value of a certain matrix, which can be calculated in linear time. This leads to an algorithm which can scale to large matrices arising in several applications such as matrix completion for collaborative filtering and robust low rank matrix approximation.
Linear Regression under Fixed-Rank Constraints: A Riemannian ApproachIn this paper, we tackle the problem of learning a linear regression model whose parameter is a fixed-rank matrix. We study the Riemannian manifold geometry of the set of fixed-rank matrices and develop efficient line-search algorithms. The proposed algorithms have many applications, scale to high-dimensional problems, enjoy local convergence properties and confer a geometric basis to recent contributions on learning fixed-rank matrices. Numerical experiments on benchmarks suggest that the proposed algorithms compete with the state-of-the-art, and that manifold optimization offers a versatile framework for the design of rank-constrained machine learning algorithms.
Clustering by Left-Stochastic Matrix FactorizationWe propose clustering samples given their pairwise similarities by factorizing the similarity matrix into the product of a cluster probability matrix and its transpose. We propose a rotation-based algorithm to compute this left-stochastic decomposition (LSD). Theoretical results link the LSD clustering method to a soft kernel k-means clustering, give conditions for when the factorization and clustering are unique, and provide error bounds. Experimental results on simulated and real similarity datasets show that the proposed method reliably provides accurate clusterings.