Learning Tensor Latent Features

We study the problem of learning latent feature models (LFMs) for tensor data commonly observed in science and engineering such as hyperspectral imagery. However, the problem is challenging not only due to the non-convex formulation, the combinatorial nature of the constraints in LFMs, but also the high-order correlations in the data. In this work, we formulate a tensor latent feature learning problem by representing the data as a mixture of high-order latent features and binary codes, which are memory efficient and easy to interpret. To make the learning tractable, we propose a novel optimization procedure, Binary matching pursuit (BMP), that iteratively searches for binary bases via a MAXCUT-like boolean quadratic solver. Such a procedure is guaranteed to achieve an? suboptimal solution in O($1/\epsilon$) greedy steps, resulting in a trade-off between accuracy and sparsity. When evaluated on both synthetic and real datasets, our experiments show superior performance over baseline methods.Comment: 10 pages, 3 figure

Higher-Order Low-Rank Regression

This paper proposes an efficient algorithm (HOLRR) to handle regression tasks where the outputs have a tensor structure. We formulate the regression problem as the minimization of a least square criterion under a multilinear rank constraint, a difficult non convex problem. HOLRR computes efficiently an approximate solution of this problem, with solid theoretical guarantees. A kernel extension is also presented. Experiments on synthetic and real data show that HOLRR outperforms multivariate and multilinear regression methods and is considerably faster than existing tensor methods.Comment: submitted to ICML 201

Tensor Regression Networks with various Low-Rank Tensor Approximations

Tensor regression networks achieve high compression rate of neural networks while having slight impact on performances. They do so by imposing low tensor rank structure on the weight matrices of fully connected layers. In recent years, tensor regression networks have been investigated from the perspective of their compressive power, however, the regularization effect of enforcing low-rank tensor structure has not been investigated enough. We study tensor regression networks using various low-rank tensor approximations, aiming to compare the compressive and regularization power of different low-rank constraints. We evaluate the compressive and regularization performances of the proposed model with both deep and shallow convolutional neural networks. The outcome of our experiment suggests the superiority of Global Average Pooling Layer over Tensor Regression Layer when applied to deep convolutional neural network with CIFAR-10 dataset. On the contrary, shallow convolutional neural networks with tensor regression layer and dropout achieved lower test error than both Global Average Pooling and fully-connected layer with dropout function when trained with a small number of samples

Learning Depthwise Separable Graph Convolution from Data Manifold

Convolution Neural Network (CNN) has gained tremendous success in computer vision tasks with its outstanding ability to capture the local latent features. Recently, there has been an increasing interest in extending convolution operations to the non-Euclidean geometry. Although various types of convolution operations have been proposed for graphs or manifolds, their connections with traditional convolution over grid-structured data are not well-understood. In this paper, we show that depthwise separable convolution can be successfully generalized for the unification of both graph-based and grid-based convolution methods. Based on this insight we propose a novel Depthwise Separable Graph Convolution (DSGC) approach which is compatible with the tradition convolution network and subsumes existing convolution methods as special cases. It is equipped with the combined strengths in model expressiveness, compatibility (relatively small number of parameters), modularity and computational efficiency in training. Extensive experiments show the outstanding performance of DSGC in comparison with strong baselines on multi-domain benchmark datasets

Tensor Regression Meets Gaussian Processes

Low-rank tensor regression, a new model class that learns high-order correlation from data, has recently received considerable attention. At the same time, Gaussian processes (GP) are well-studied machine learning models for structure learning. In this paper, we demonstrate interesting connections between the two, especially for multi-way data analysis. We show that low-rank tensor regression is essentially learning a multi-linear kernel in Gaussian processes, and the low-rank assumption translates to the constrained Bayesian inference problem. We prove the oracle inequality and derive the average case learning curve for the equivalent GP model. Our finding implies that low-rank tensor regression, though empirically successful, is highly dependent on the eigenvalues of covariance functions as well as variable correlations.Comment: 17 page

FasTer: Fast Tensor Completion with Nonconvex Regularization

Low-rank tensor completion problem aims to recover a tensor from limited observations, which has many real-world applications. Due to the easy optimization, the convex overlapping nuclear norm has been popularly used for tensor completion. However, it over-penalizes top singular values and lead to biased estimations. In this paper, we propose to use the nonconvex regularizer, which can less penalize large singular values, instead of the convex one for tensor completion. However, as the new regularizer is nonconvex and overlapped with each other, existing algorithms are either too slow or suffer from the huge memory cost. To address these issues, we develop an efficient and scalable algorithm, which is based on the proximal average (PA) algorithm, for real-world problems. Compared with the direct usage of PA algorithm, the proposed algorithm runs orders faster and needs orders less space. We further speed up the proposed algorithm with the acceleration technique, and show the convergence to critical points is still guaranteed. Experimental comparisons of the proposed approach are made with various other tensor completion approaches. Empirical results show that the proposed algorithm is very fast and can produce much better recovery performance

Time-varying Autoregression with Low Rank Tensors

We present a windowed technique to learn parsimonious time-varying autoregressive models from multivariate timeseries. This unsupervised method uncovers interpretable spatiotemporal structure in data via non-smooth and non-convex optimization. In each time window, we assume the data follow a linear model parameterized by a system matrix, and we model this stack of potentially different system matrices as a low rank tensor. Because of its structure, the model is scalable to high-dimensional data and can easily incorporate priors such as smoothness over time. We find the components of the tensor using alternating minimization and prove that any stationary point of this algorithm is a local minimum. We demonstrate on a synthetic example that our method identifies the true rank of a switching linear system in the presence of noise. We illustrate our model's utility and superior scalability over extant methods when applied to several synthetic and real-world example: two types of time-varying linear systems, worm behavior, sea surface temperature, and monkey brain datasets

Multiresolution Tensor Learning for Efficient and Interpretable Spatial Analysis

Efficient and interpretable spatial analysis is crucial in many fields such as geology, sports, and climate science. Large-scale spatial data often contains complex higher-order correlations across features and locations. While tensor latent factor models can describe higher-order correlations, they are inherently computationally expensive to train. Furthermore, for spatial analysis, these models should not only be predictive but also be spatially coherent. However, latent factor models are sensitive to initialization and can yield inexplicable results. We develop a novel Multi-resolution Tensor Learning (MRTL) algorithm for efficiently learning interpretable spatial patterns. MRTL initializes the latent factors from an approximate full-rank tensor model for improved interpretability and progressively learns from a coarse resolution to the fine resolution for an enormous computation speedup. We also prove the theoretical convergence and computational complexity of MRTL. When applied to two real-world datasets, MRTL demonstrates 4 ~ 5 times speedup compared to a fixed resolution while yielding accurate and interpretable models

Theoretical and Experimental Analyses of Tensor-Based Regression and Classification

We theoretically and experimentally investigate tensor-based regression and classification. Our focus is regularization with various tensor norms, including the overlapped trace norm, the latent trace norm, and the scaled latent trace norm. We first give dual optimization methods using the alternating direction method of multipliers, which is computationally efficient when the number of training samples is moderate. We then theoretically derive an excess risk bound for each tensor norm and clarify their behavior. Finally, we perform extensive experiments using simulated and real data and demonstrate the superiority of tensor-based learning methods over vector- and matrix-based learning methods

Long-Short Term Spatiotemporal Tensor Prediction for Passenger Flow Profile

Spatiotemporal data is very common in many applications, such as manufacturing systems and transportation systems. It is typically difficult to be accurately predicted given intrinsic complex spatial and temporal correlations. Most of the existing methods based on various statistical models and regularization terms, fail to preserve innate features in data alongside their complex correlations. In this paper, we focus on a tensor-based prediction and propose several practical techniques to improve prediction. For long-term prediction specifically, we propose the "Tensor Decomposition + 2-Dimensional Auto-Regressive Moving Average (2D-ARMA)" model, and an effective way to update prediction real-time; For short-term prediction, we propose to conduct tensor completion based on tensor clustering to avoid oversimplifying and ensure accuracy. A case study based on the metro passenger flow data is conducted to demonstrate the improved performance