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### Learning from Binary Multiway Data: Probabilistic Tensor Decomposition and its Statistical Optimality

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).

### Robust Tensor Decomposition with Gross Corruption

In this paper, we study the statistical performance of robust tensor decomposition with gross corruption. The observations are noisy realization of the superposition of a low-rank tensor $\mathcal{W}^*$ and an entrywise sparse corruption tensor $\mathcal{V}^*$. Unlike conventional noise with bounded variance in previous convex tensor decomposition analysis, the magnitude of the gross corruption can be arbitrary large. We show that under certain conditions, the true low-rank tensor as well as the sparse corruption tensor can be recovered simultaneously. Our theory yields nonasymptotic Frobenius-norm estimation error bounds for each tensor separately. We show through numerical experiments that our theory can precisely predict the scaling behavior in practice.

### Legendre Decomposition for Tensors

We present a novel nonnegative tensor decomposition method, called Legendre decomposition, which factorizes an input tensor into a multiplicative combination of parameters. Thanks to the well-developed theory of information geometry, the reconstructed tensor is unique and always minimizes the KL divergence from an input tensor. We empirically show that Legendre decomposition can more accurately reconstruct tensors than other nonnegative tensor decomposition methods. Papers published at the Neural Information Processing Systems Conference.

### Tensor Completion Algorithms in Big Data Analytics

Tensor completion is a problem of filling the missing or unobserved entries of partially observed tensors. Due to the multidimensional character of tensors in describing complex datasets, tensor completion algorithms and their applications have received wide attention and achievement in data mining, computer vision, signal processing, and neuroscience, etc. In this survey, we provide a modern overview of recent advances in tensor completion algorithms from the perspective of big data analytics characterized by diverse variety, large volume, and high velocity. Towards a better comprehension and comparison of vast existing advances, we summarize and categorize them into four groups including general tensor completion algorithms, tensor completion with auxiliary information (variety), scalable tensor completion algorithms (volume) and dynamic tensor completion algorithms (velocity). Besides, we introduce their applications on real-world data-driven problems and present an open-source package covering several widely used tensor decomposition and completion algorithms. Our goal is to summarize these popular methods and introduce them to researchers for promoting the research process in this field and give an available repository for practitioners. In the end, we also discuss some challenges and promising research directions in this community for future explorations.

### Statistical Performance of Convex Tensor Decomposition

We analyze the statistical performance of a recently proposed convex tensor decomposition algorithm. Conventionally tensor decomposition has been formulated as non-convex optimization problems, which hindered the analysis of their performance. We show under some conditions that the mean squared error of the convex method scales linearly with the quantity we call the normalized rank of the true tensor. The current analysis naturally extends the analysis of convex low-rank matrix estimation to tensors. Furthermore, we show through numerical experiments that our theory can precisely predict the scaling behaviour in practice.