We study the problem of low-rank tensor factorization in the presence of missing data. We ask the following question: how many sampled entries do we need, to efficiently and exactly reconstruct a tensor with a low-rank orthogonal decomposition? We propose a novel alternating minimization based method which iteratively refines estimates of the singular vectors. We show that under certain standard assumptions, our method can recover a three-mode $n\times n\times n$ dimensional rank-$r$ tensor exactly from $O(n^{3/2} r^5 \log^4 n)$ randomly sampled entries. In the process of proving this result, we solve two challenging sub-problems for tensors with missing data. First, in analyzing the initialization step, we prove a generalization of a celebrated result by Szemer\'edie et al. on the spectrum of random graphs. Next, we prove global convergence of alternating minimization with a good initialization. Simulations suggest that the dependence of the sample size on dimensionality $n$ is indeed tight.

Warlop, Romain (Inria Lille) | Lazaric, Alessandro (Inria Lille) | Mary, Jérémie (University of Lille 1, Inria)

Many modern recommender systems rely on matrix factor-ization techniques to produce personalized recommendationson the basis of the feedback that users provided on differ-ent items in the past. The feedback may take different forms,such as the rating of a movie, or the number of times a userlistened to the songs of a given music band. Nonetheless, insome situations, the user can perform several actions on eachitem, and the feedback is multidimensional (e.g., the user ofan e-commerce website can either click on a product, add theproduct to her cart or buy it). In this case, one can no longerview the recommendation problem as a matrix completion,unless the problem is reduced to a series of multiple inde-pendent problems, thus loosing the correlation between thedifferent actions. In this case, the most suitable approach is touse a tensor approach to learn all dimensions of the feedbacksimultaneously. In this paper, we propose a specific instanceof tensor completion and we show how it can be heavily par-allelized over both the dimensions (i.e., items, users, actions)and within each dimension (i.e., each item separately). Wevalidate the proposed method both in terms of prediction ac-curacy and scalability to large datasets.

Bhojanapalli, Srinadh, Sanghavi, Sujay

In this paper we propose new techniques to sample arbitrary third-order tensors, with an objective of speeding up tensor algorithms that have recently gained popularity in machine learning. Our main contribution is a new way to select, in a biased random way, only $O(n^{1.5}/\epsilon^2)$ of the possible $n^3$ elements while still achieving each of the three goals: \\ {\em (a) tensor sparsification}: for a tensor that has to be formed from arbitrary samples, compute very few elements to get a good spectral approximation, and for arbitrary orthogonal tensors {\em (b) tensor completion:} recover an exactly low-rank tensor from a small number of samples via alternating least squares, or {\em (c) tensor factorization:} approximating factors of a low-rank tensor corrupted by noise. \\ Our sampling can be used along with existing tensor-based algorithms to speed them up, removing the computational bottleneck in these methods.

An important reason for such an increase is the effective representation of multiway data using a tensor structure. One example is the recommender system (Bi et al., 2018), which can be naturally described as a three-way tensor of user item context and each entry indicates the user-item interaction. Another example is the DBLP database (Zhe et al., 2016), which is organized into a three-way tensor of author word venue and each entry indicates the co-occurrence of the triplets. Whereas many real-world multiway datasets have continuous-valued entries, there have recently emerged more instances of binary tensors, in which all tensor entries are binary indicators 0/1. Examples include click/no-click action in recommender systems (Sun et al., 2017), multi-relational social networks (Nickel et al., 2011), and brain structural connectivity networks (Wang et al., 2017a).

Nimishakavi, Madhav, Jawanpuria, Pratik Kumar, Mishra, Bamdev

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.