ADMM-MM Algorithm for General Tensor Decomposition

In this paper, we propose a new unified optimization algorithm for general tensor decomposition which is formulated as an inverse problem for low-rank tensors in the general linear observation models. The proposed algorithm supports three basic loss functions ($\ell_2$-loss, $\ell_1$-loss and KL divergence) and various low-rank tensor decomposition models (CP, Tucker, TT, and TR decompositions). We derive the optimization algorithm based on hierarchical combination of the alternating direction method of multiplier (ADMM) and majorization-minimization (MM). We show that wide-range applications can be solved by the proposed algorithm, and can be easily extended to any established tensor decomposition models in a {plug-and-play} manner

Exploring Numerical Priors for Low-Rank Tensor Completion with Generalized CP Decomposition

Tensor completion is important to many areas such as computer vision, data analysis, and signal processing. Enforcing low-rank structures on completed tensors, a category of methods known as low-rank tensor completion has recently been studied extensively. While such methods attained great success, none considered exploiting numerical priors of tensor elements. Ignoring numerical priors causes loss of important information regarding the data, and therefore prevents the algorithms from reaching optimal accuracy. This work attempts to construct a new methodological framework called GCDTC (Generalized CP Decomposition Tensor Completion) for leveraging numerical priors and achieving higher accuracy in tensor completion. In this newly introduced framework, a generalized form of CP Decomposition is applied to low-rank tensor completion. This paper also proposes an algorithm known as SPTC (Smooth Poisson Tensor Completion) for nonnegative integer tensor completion as an instantiation of the GCDTC framework. A series of experiments on real-world data indicated that SPTC could produce results superior in completion accuracy to current state-of-the-arts.Comment: 11 pages, 4 figures, 3 pseudocode algorithms, and 1 tabl

CP decomposition and low-rank approximation of antisymmetric tensors

For the antisymmetric tensors the paper examines a low-rank approximation which is represented via only three vectors. We describe a suitable low-rank format and propose an alternating least squares structure-preserving algorithm for finding such approximation. The case of partial antisymmetry is also discussed. The algorithms are implemented in Julia programming language and their numerical performance is discussed.Comment: 16 pages, 4 table

Spatiotemporal Tensor Completion for Improved Urban Traffic Imputation

Effective management of urban traffic is important for any smart city initiative. Therefore, the quality of the sensory traffic data is of paramount importance. However, like any sensory data, urban traffic data are prone to imperfections leading to missing measurements. In this paper, we focus on inter-region traffic data completion. We model the inter-region traffic as a spatiotemporal tensor that suffers from missing measurements. To recover the missing data, we propose an enhanced CANDECOMP/PARAFAC (CP) completion approach that considers the urban and temporal aspects of the traffic. To derive the urban characteristics, we divide the area of study into regions. Then, for each region, we compute urban feature vectors inspired from biodiversity which are used to compute the urban similarity matrix. To mine the temporal aspect, we first conduct an entropy analysis to determine the most regular time-series. Then, we conduct a joint Fourier and correlation analysis to compute its periodicity and construct the temporal matrix. Both urban and temporal matrices are fed into a modified CP-completion objective function. To solve this objective, we propose an alternating least square approach that operates on the vectorized version of the inputs. We conduct comprehensive comparative study with two evaluation scenarios. In the first one, we simulate random missing values. In the second scenario, we simulate missing values at a given area and time duration. Our results demonstrate that our approach provides effective recovering performance reaching 26% improvement compared to state-of-art CP approaches and 35% compared to state-of-art generative model-based approaches

SWIFT: Scalable Wasserstein Factorization for Sparse Nonnegative Tensors

Existing tensor factorization methods assume that the input tensor follows some specific distribution (i.e. Poisson, Bernoulli, and Gaussian), and solve the factorization by minimizing some empirical loss functions defined based on the corresponding distribution. However, it suffers from several drawbacks: 1) In reality, the underlying distributions are complicated and unknown, making it infeasible to be approximated by a simple distribution. 2) The correlation across dimensions of the input tensor is not well utilized, leading to sub-optimal performance. Although heuristics were proposed to incorporate such correlation as side information under Gaussian distribution, they can not easily be generalized to other distributions. Thus, a more principled way of utilizing the correlation in tensor factorization models is still an open challenge. Without assuming any explicit distribution, we formulate the tensor factorization as an optimal transport problem with Wasserstein distance, which can handle non-negative inputs. We introduce SWIFT, which minimizes the Wasserstein distance that measures the distance between the input tensor and that of the reconstruction. In particular, we define the N-th order tensor Wasserstein loss for the widely used tensor CP factorization and derive the optimization algorithm that minimizes it. By leveraging sparsity structure and different equivalent formulations for optimizing computational efficiency, SWIFT is as scalable as other well-known CP algorithms. Using the factor matrices as features, SWIFT achieves up to 9.65% and 11.31% relative improvement over baselines for downstream prediction tasks. Under the noisy conditions, SWIFT achieves up to 15% and 17% relative improvements over the best competitors for the prediction tasks.Comment: Accepted by AAAI-2