A dual framework for low-rank tensor completion

One of the popular approaches for low-rank tensor completion is to use the latent trace norm regularization. However, most existing works in this direction learn a sparse combination of tensors. In this work, we fill this gap by proposing a variant of the latent trace norm that helps in learning a non-sparse combination of tensors. We develop a dual framework for solving the low-rank tensor completion problem. We first show a novel characterization of the dual solution space with an interesting factorization of the optimal solution. Overall, the optimal solution is shown to lie on a Cartesian product of Riemannian manifolds. Furthermore, we exploit the versatile Riemannian optimization framework for proposing computationally efficient trust region algorithm. The experiments illustrate the efficacy of the proposed algorithm on several real-world datasets across applications.Comment: Aceepted to appear in Advances of Nueral Information Processing Systems (NIPS), 2018. A shorter version appeared in the NIPS workshop on Synergies in Geometric Data Analysis 201

Greedy Algorithms for Cone Constrained Optimization with Convergence Guarantees

Greedy optimization methods such as Matching Pursuit (MP) and Frank-Wolfe (FW) algorithms regained popularity in recent years due to their simplicity, effectiveness and theoretical guarantees. MP and FW address optimization over the linear span and the convex hull of a set of atoms, respectively. In this paper, we consider the intermediate case of optimization over the convex cone, parametrized as the conic hull of a generic atom set, leading to the first principled definitions of non-negative MP algorithms for which we give explicit convergence rates and demonstrate excellent empirical performance. In particular, we derive sublinear ($\mathcal{O}(1/t)$) convergence on general smooth and convex objectives, and linear convergence ($\mathcal{O}(e^{-t})$) on strongly convex objectives, in both cases for general sets of atoms. Furthermore, we establish a clear correspondence of our algorithms to known algorithms from the MP and FW literature. Our novel algorithms and analyses target general atom sets and general objective functions, and hence are directly applicable to a large variety of learning settings.Comment: NIPS 201

Structured Tensor Recovery and Decomposition

Tensors, a.k.a. multi-dimensional arrays, arise naturally when modeling higher-order objects and relations. Among ubiquitous applications including image processing, collaborative filtering, demand forecasting and higher-order statistics, there are two recurring themes in general: tensor recovery and tensor decomposition. The first one aims to recover the underlying tensor from incomplete information; the second one is to study a variety of tensor decompositions to represent the array more concisely and moreover to capture the salient characteristics of the underlying data. Both topics are respectively addressed in this thesis. Chapter 2 and Chapter 3 focus on low-rank tensor recovery (LRTR) from both theoretical and algorithmic perspectives. In Chapter 2, we first provide a negative result to the sum of nuclear norms (SNN) model---an existing convex model widely used for LRTR; then we propose a novel convex model and prove this new model is better than the SNN model in terms of the number of measurements required to recover the underlying low-rank tensor. In Chapter 3, we first build up the connection between robust low-rank tensor recovery and the compressive principle component pursuit (CPCP), a convex model for robust low-rank matrix recovery. Then we focus on developing convergent and scalable optimization methods to solve the CPCP problem. In specific, our convergent method, proposed by combining classical ideas from Frank-Wolfe and proximal methods, achieves scalability with linear per-iteration cost. Chapter 4 generalizes the successive rank-one approximation (SROA) scheme for matrix eigen-decomposition to a special class of tensors called symmetric and orthogonally decomposable (SOD) tensor. We prove that the SROA scheme can robustly recover the symmetric canonical decomposition of the underlying SOD tensor even in the presence of noise. Perturbation bounds, which can be regarded as a higher-order generalization of the Davis-Kahan theorem, are provided in terms of the noise magnitude