Matrix completion and approximation are popular tools to capture a user's preferences for recommendation and to approximate missing data. Instead of using low-rank factorization we take a drastically different approach, based on the simple insight that an additive model of co-clusterings allows one to approximate matrices efficiently. This allows us to build a concise model that, per bit of model learned, significantly beats all factorization approaches to matrix approximation. Even more surprisingly, we find that summing over small co-clusterings is more effective in modeling matrices than classic co-clustering, which uses just one large partitioning of the matrix. Following Occam's razor principle suggests that the simple structure induced by our model better captures the latent preferences and decision making processes present in the real world than classic co-clustering or matrix factorization. We provide an iterative minimization algorithm, a collapsed Gibbs sampler, theoretical guarantees for matrix approximation, and excellent empirical evidence for the efficacy of our approach. We achieve state-of-the-art results on the Netflix problem with a fraction of the model complexity.
Matrix Approximation There exists a large body of work on matrix approximation in the theoretical computer science community. They focus mainly on efficient low-rank approximations, e.g. by projection or by interpolation. Examples of the projection based strategy are @cite_3 @cite_14 . Essentially one aims to find a general low-rank approximation of the matrix, as is common in most recommender models.
@cite_14 @cite_3
2169661502 2336195973 1228575997 2124584309
Matrix approximation is a common tool in recommendation systems, text mining, and computer vision. A prevalent assumption in constructing matrix approximations is that the partially observed matrix is of low-rank. We propose a new matrix approximation model where we assume instead that the matrix is locally of low-rank, leading to a representation of the observed matrix as a weighted sum of low-rank matrices. We analyze the accuracy of the proposed local low-rank modeling. Our experiments show improvements in prediction accuracy over classical approaches for recommendation tasks. Matrix approximation is a common tool in recommendation systems, text mining, and computer vision. A prevalent assumption in constructing matrix approximations is that the partially observed matrix is low-rank. In this paper, we propose, analyze, and experiment with two procedures, one parallel and the other global, for constructing local matrix approximations. The two approaches approximate the observed matrix as a weighted sum of low-rank matrices. These matrices are limited to a local region of the observed matrix. We analyze the accuracy of the proposed local low-rank modeling. Our experiments show improvements in prediction accuracy over classical approaches for recommendation tasks. Matrix approximation is a common model-based approach to collaborative filtering in recommender systems. Many relevant algorithms that fuse social contextual information have been proposed. They mainly focus on using the latent factor vectors of users and items to construct a low-rank approximation of the user-item rating matrix. However, due to data sparsity, it is difficult for current approaches to accurately learn every user item vector, which may cause low-quality recommendations for some users or items. But we implicitly detect some similar users or items based on the distance among the vectors. In this paper, we take advantage of the implicit similarity to improve matrix approximation. Borrowing parts of ideas from CodeBook Transfer, we propose a reconstructive method that compresses low-rank approximation into a cluster-level rating-pattern referred to as a codebook, and then constructs an improved approximation by expending the codebook. Experiments on real life datasets demonstrate our method improves the prediction accuracy of the state-of-the-art matrix factorization and social recommendation models. The calculation of a low-rank approximation of a matrix is a fundamental operation in many computer vision applications. The workhorse of this class of problems has long been the Singular Value Decomposition. However, in the presence of missing data and outliers this method is not applicable, and unfortunately, this is often the case in practice. In this paper we present a method for calculating the low-rank factorization of a matrix which minimizes the L 1 norm in the presence of missing data. Our approach represents a generalization the Wiberg algorithm of one of the more convincing methods for factorization under the L 2 norm. By utilizing the differentiability of linear programs, we can extend the underlying ideas behind this approach to include this class of L 1 problems as well. We show that the proposed algorithm can be efficiently implemented using existing optimization software. We also provide preliminary experiments on synthetic as well as real world data with very convincing results.