All-at-once Optimization for Coupled Matrix and Tensor Factorizations

By Evrim Acar, Tamara G. Kolda, Daniel M. Dunlavy. arXiv.org, 2011

310 citations recorded by the source corpus.

Joint analysis of data from multiple sources has the potential to improve our understanding of the underlying structures in complex data sets. For instance, in restaurant recommendation systems, recommendations can be based on rating histories of customers. In addition to rating histories, customers' social networks (e.g., Facebook friendships) and restaurant categories information (e.g., Thai or Italian) can also be used to make better recommendations. The task of fusing data, however, is challenging since data sets can be incomplete and heterogeneous, i.e., data consist of both matrices, e.g., the person by person social network matrix or the restaurant by category matrix, and higher-order tensors, e.g., the "ratings" tensor of the form restaurant by meal by person. In this paper, we are particularly interested in fusing data sets with the goal of capturing their underlying latent structures. We formulate this problem as a coupled matrix and tensor factorization (CMTF) problem where heterogeneous data sets are modeled by fitting outer-product models to higher-order tensors and matrices in a coupled manner. Unlike traditional approaches solving this problem using alternating algorithms, we propose an all-at-once optimization approach called CMTF-OPT (CMTF-OPTimization), which is a gradient-based optimization approach for joint analysis of matrices and higher-order tensors. We also extend the algorithm to handle coupled incomplete data sets. Using numerical experiments, we demonstrate that the proposed all-at-once approach is more accurate than the alternating least squares approach.

Subfields: Computational Mathematics

View Paper: https://www.semanticscholar.org/paper/18977c6f7abb245691f4268ccd116036bd2391f0

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In Computational Mathematics

Citations in the corpus

Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis (2014, methodology): “As an alternative to Tucker decompositions, coupled tensor decompositions may be of a polyadic or even block term type [89], [109].”
On Approximate Reasoning Capabilities of Low-Rank Vector Spaces (2015, methodology): “The main difference between the approaches is the way the multi-linear dot-product is deﬁned, including Tucker decomposition (Acar, Kolda, and Dunlavy 2011), Rescal model (Nickel, Tresp, and Kriegel 2011), or the simpler but more scalable CP/Parafac model (Singh and Gordon 2008).”
DFacTo: Distributed Factorization of Tensors (2014, methodology): “In fact, we believe that this joint matrix completion and tensor factorization model by itself is somewhat new and interesting in its own right, despite its resemblance to other joint models including tensor factorization such as [14].”
Rubik: Knowledge Guided Tensor Factorization and Completion for Health Data Analytics (2015, background): “However, as labeled data are diﬃcult to obtain, eﬃcient unsupervised phenotyping approaches are needed to leverage the vast amount of Property Marble [15] FaLRTC [22] CTMF [2] TF-BPP [18] WCP [1] NETWORK [10] Rubik Factorization Completion Non-negativity Guidance Sparse solution Scalability Table 2:”
Limestone: High-throughput candidate phenotype generation via tensor factorization (2014, methodology): “FMS is [40,78,79] commonly used to compare two tensor factorization results, quantifying the closeness via a single number between 1⁄20;1 .”
Tensors for Data Mining and Data Fusion (2016, methodology): “…et al. 2009; Zheng et al. 2010; Acar et al. 2011, 2012; Narita et al. 2012; Yokota et al. 2012; Wang et al. 2014; Ermis¸ et al. 2015] One of the most popular models is the so-called Coupled Matrix-Tensor Factorization (CMTF) [Acar et al. 2011], where one or more matrices are coupled with a tensor.”