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Fast and Guaranteed Tensor Decomposition via Sketching

Neural Information Processing Systems

Tensor CANDECOMP/PARAFAC (CP) decomposition has wide applications in statistical learning of latent variable models and in data mining. In this paper, we propose fast and randomized tensor CP decomposition algorithms based on sketching. We build on the idea of count sketches, but introduce many novel ideas which are unique to tensors. We develop novel methods for randomized com- putation of tensor contractions via FFTs, without explicitly forming the tensors. Such tensor contractions are encountered in decomposition methods such as ten- sor power iterations and alternating least squares. We also design novel colliding hashes for symmetric tensors to further save time in computing the sketches. We then combine these sketching ideas with existing whitening and tensor power iter- ative techniques to obtain the fastest algorithm on both sparse and dense tensors. The quality of approximation under our method does not depend on properties such as sparsity, uniformity of elements, etc. We apply the method for topic mod- eling and obtain competitive results.


Low-Rank Tucker Decomposition of Large Tensors Using TensorSketch

Neural Information Processing Systems

We propose two randomized algorithms for low-rank Tucker decomposition of tensors. The algorithms, which incorporate sketching, only require a single pass of the input tensor and can handle tensors whose elements are streamed in any order. To the best of our knowledge, ours are the only algorithms which can do this. We test our algorithms on sparse synthetic data and compare them to multiple other methods. We also apply one of our algorithms to a real dense 38 GB tensor representing a video and use the resulting decomposition to correctly classify frames containing disturbances.


Low-Rank Tucker Decomposition of Large Tensors Using TensorSketch

Neural Information Processing Systems

We propose two randomized algorithms for low-rank Tucker decomposition of tensors. The algorithms, which incorporate sketching, only require a single pass of the input tensor and can handle tensors whose elements are streamed in any order. To the best of our knowledge, ours are the only algorithms which can do this. We test our algorithms on sparse synthetic data and compare them to multiple other methods. We also apply one of our algorithms to a real dense 38 GB tensor representing a video and use the resulting decomposition to correctly classify frames containing disturbances.


SPALS: Fast Alternating Least Squares via Implicit Leverage Scores Sampling

Neural Information Processing Systems

Tensor CANDECOMP/PARAFAC (CP) decomposition is a powerful but computationally challengingtool in modern data analytics. In this paper, we show ways of sampling intermediate steps of alternating minimization algorithms for computing low rank tensor CP decompositions, leading to the sparse alternating least squares (SPALS) method. Specifically, we sample the Khatri-Rao product, which arises as an intermediate object during the iterations of alternating least squares. This product captures the interactions between different tensor modes, and form the main computational bottleneck for solving many tensor related tasks. By exploiting the spectral structures of the matrix Khatri-Rao product, we provide efficient access to its statistical leverage scores. When applied to the tensor CP decomposition, our method leads to the first algorithm that runs in sublinear time per-iteration and approximates the output of deterministic alternating least squares algorithms.


Stochastically Rank-Regularized Tensor Regression Networks

arXiv.org Machine Learning

Over-parametrization of deep neural networks has recently been shown to be key to their successful training. However, it also renders them prone to overfitting and makes them expensive to store and train. Tensor regression networks significantly reduce the number of effective parameters in deep neural networks while retaining accuracy and the ease of training. They replace the flattening and fully-connected layers with a tensor regression layer, where the regression weights are expressed through the factors of a low-rank tensor decomposition. In this paper, to further improve tensor regression networks, we propose a novel stochastic rank-regularization. It consists of a novel randomized tensor sketching method to approximate the weights of tensor regression layers. We theoretically and empirically establish the link between our proposed stochastic rank-regularization and the dropout on low-rank tensor regression. Extensive experimental results with both synthetic data and real world datasets (i.e., CIFAR-100 and the UK Biobank brain MRI dataset) support that the proposed approach i) improves performance in both classification and regression tasks, ii) decreases overfitting, iii) leads to more stable training and iv) improves robustness to adversarial attacks and random noise.