On Tensors, Sparsity, and Nonnegative Factorizations

By Eric C. Chi, Tamara G. Kolda. 2011

316 citations recorded by the source corpus.

Tensors have found application in a variety of fields, ranging from chemometrics to signal processing and beyond. In this paper, we consider the problem of multilinear modeling of sparse count data. Our goal is to develop a descriptive tensor factorization model of such data, along with appropriate algorithms and theory. To do so, we propose that the random variation is best described via a Poisson distribution, which better describes the zeros observed in the data as compared to the typical assumption of a Gaussian distribution. Under a Poisson assumption, we fit a model to observed data using the negative log-likelihood score. We present a new algorithm for Poisson tensor factorization called CANDECOMP--PARAFAC alternating Poisson regression (CP-APR) that is based on a majorization-minimization approach. It can be shown that CP-APR is a generalization of the Lee--Seung multiplicative updates. We show how to prevent the algorithm from converging to non-KKT points and prove convergence of CP-APR under mil...

Subfields: Computational Mathematics, Computational Mechanics

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Citations in the corpus

Privacy-Preserving Tensor Factorization for Collaborative Health Data Analysis (2019, methodology): “We use the factor match score (FMS) [5] to compare the similarity between the phenotype Session: Long - Privacy CIKM ’19, November 3–7, 2019, Beijing, China”
Approximate PCFG Parsing Using Tensor Decomposition (2013, methodology): “In our experiments, we use the method described in Chi and Kolda (2011) for tensor decomposition.”
Limestone: High-throughput candidate phenotype generation via tensor factorization (2014, methodology): “The details of the subproblem solver and the overall CP-APR algorithm can be found in the paper [40].”
Time-Sensitive Recommendation From Recurrent User Activities (2015, methodology): “We compare with the stateof-art method [3] which considers poisson regression as the loss function to fit the number of events in each discretized time slot and shows better performance compared to other alternatives with the squared loss [25, 13, 22, 21].”
The Why and How of Nonnegative Matrix Factorization (2014, methodology): “A simpler and nice way to guarantee convergence of the MU to a stationary point is proposed in [23]: use the original updates (5) while reinitializing zero entries of W to a small positive constant when their partial derivatives become negative.”
Marble: high-throughput phenotyping from electronic health records via sparse nonnegative tensor factorization (2014, methodology): “Our model extends the non-negative CANDECOMP/PARAFAC (CP) Poisson tensor decomposition [6] from two aspects: (i) constraints on the factor matrices to minimize the number of non-zero elements, and (ii) augmentation of the tensor approximation.”
Rubik: Knowledge Guided Tensor Factorization and Completion for Health Data Analytics (2015, methodology): “Chi and Kolda [9] proposed nonnegative CP alternation Possion regression (CP-APR) model.”
On the Complexity of Robust PCA and ℓ1-norm Low-Rank Matrix Approximation (2015, unknown): “, the discussion in [9] and the references therein.”
Unsupervised Discovery of Demixed, Low-Dimensional Neural Dynamics across Multiple Timescales through Tensor Component Analysis (2017, methodology): “Both PCA and TCA can be extended to incorporate different loss functions, such as a Poisson negative log-likelihood 625 [100], however we do not consider these models in this paper.”