Wenqi Wang

Wenqi Wang, Vaneet Aggarwal, Shuchin Aeron

Tensor train is a hierarchical tensor network structure that helps alleviate the curse of dimensionality by parameterizing large-scale multidimensional data via a set of network of low-rank tensors. Associated with such a construction is a notion of Tensor Train subspace and in this paper we propose a TT-PCA algorithm for estimating this structured subspace from the given data. By maintaining low rank tensor structure, TT-PCA is more robust to noise comparing with PCA or Tucker-PCA. This is borne out numerically by testing the proposed approach on the Extended YaleFace Dataset B.

Wenqi Wang, Yifan Sun, Brian Eriksson et al.

Deep neural networks have demonstrated state-of-the-art performance in a variety of real-world applications. In order to obtain performance gains, these networks have grown larger and deeper, containing millions or even billions of parameters and over a thousand layers. The trade-off is that these large architectures require an enormous amount of memory, storage, and computation, thus limiting their usability. Inspired by the recent tensor ring factorization, we introduce Tensor Ring Networks (TR-Nets), which significantly compress both the fully connected layers and the convolutional layers of deep neural networks. Our results show that our TR-Nets approach {is able to compress LeNet-5 by $11\times$ without losing accuracy}, and can compress the state-of-the-art Wide ResNet by $243\times$ with only 2.3\% degradation in {Cifar10 image classification}. Overall, this compression scheme shows promise in scientific computing and deep learning, especially for emerging resource-constrained devices such as smartphones, wearables, and IoT devices.

Wenqi Wang, Vaneet Aggarwal, Shuchin Aeron

In this paper, we propose a Tensor Train Neighborhood Preserving Embedding (TTNPE) to embed multi-dimensional tensor data into low dimensional tensor subspace. Novel approaches to solve the optimization problem in TTNPE are proposed. For this embedding, we evaluate novel trade-off gain among classification, computation, and dimensionality reduction (storage) for supervised learning. It is shown that compared to the state-of-the-arts tensor embedding methods, TTNPE achieves superior trade-off in classification, computation, and dimensionality reduction in MNIST handwritten digits and Weizmann face datasets.

Wenqi Wang, Vaneet Aggarwal, Shuchin Aeron

In this paper, we consider clustering data that is assumed to come from one of finitely many pointed convex polyhedral cones. This model is referred to as the Union of Polyhedral Cones (UOPC) model. Similar to the Union of Subspaces (UOS) model where each data from each subspace is generated from a (unknown) basis, in the UOPC model each data from each cone is assumed to be generated from a finite number of (unknown) \emph{extreme rays}.To cluster data under this model, we consider several algorithms - (a) Sparse Subspace Clustering by Non-negative constraints Lasso (NCL), (b) Least squares approximation (LSA), and (c) K-nearest neighbor (KNN) algorithm to arrive at affinity between data points. Spectral Clustering (SC) is then applied on the resulting affinity matrix to cluster data into different polyhedral cones. We show that on an average KNN outperforms both NCL and LSA and for this algorithm we provide the deterministic conditions for correct clustering. For an affinity measure between the cones it is shown that as long as the cones are not very coherent and as long as the density of data within each cone exceeds a threshold, KNN leads to accurate clustering. Finally, simulation results on real datasets (MNIST and YaleFace datasets) depict that the proposed algorithm works well on real data indicating the utility of the UOPC model and the proposed algorithm.

Wenqi Wang, Shuchin Aeron, Vaneet Aggarwal

In this paper we present deterministic analysis of sufficient conditions for sparse subspace clustering under missing data, when data is assumed to come from a Union of Subspaces (UoS) model. In this context we consider two cases, namely Case I when all the points are sampled at the same co-ordinates, and Case II when points are sampled at different locations. We show that results for Case I directly follow from several existing results in the literature, while results for Case II are not as straightforward and we provide a set of dual conditions under which, perfect clustering holds true. We provide extensive set of simulation results for clustering as well as completion of data under missing entries, under the UoS model. Our experimental results indicate that in contrast to the full data case, accurate clustering does not imply accurate subspace identification and completion, indicating the natural order of relative hardness of these problems.