Partial Matrix Completion

The matrix completion problem aims to reconstruct a low-rank matrix based on a revealed set of possibly noisy entries. Prior works consider completing the entire matrix with generalization error guarantees. However, the completion accuracy can be drastically different over different entries. This work establishes a new framework of partial matrix completion, where the goal is to identify a large subset of the entries that can be completed with high confidence. We propose an efficient algorithm with the following provable guarantees. Given access to samples from an unknown and arbitrary distribution, it guarantees: (a) high accuracy over completed entries, and (b) high coverage of the underlying distribution. We also consider an online learning variant of this problem, where we propose a low-regret algorithm based on iterative gradient updates. Preliminary empirical evaluations are included.Comment: NeurIPS 202

Partial matrix completion

The matrix completion problem involves reconstructing a low-rank matrix by using a given set of revealed (and potentially noisy) entries. Although existing methods address the completion of the entire matrix, the accuracy of the completed entries can vary significantly across the matrix, due to differences in the sampling distribution. For instance, users may rate movies primarily from their country or favorite genres, leading to inaccurate predictions for the majority of completed entries.We propose a novel formulation of the problem as Partial Matrix Completion, where the objective is to complete a substantial subset of the entries with high confidence. Our algorithm efficiently handles the unknown and arbitrarily complex nature of the sampling distribution, ensuring high accuracy for all completed entries and sufficient coverage across the matrix. Additionally, we introduce an online version of the problem and present a low-regret efficient algorithm based on iterative gradient updates. Finally, we conduct a preliminary empirical evaluation of our methods

Information overload in structured data

Information overload refers to the difficulty of making decisions caused by too much information. In this dissertation, we address information overload problem in two separate structured domains, namely, graphs and text. Graph kernels have been proposed as an efficient and theoretically sound approach to compute graph similarity. They decompose graphs into certain sub-structures, such as subtrees, or subgraphs. However, existing graph kernels suffer from a few drawbacks. First, the dimension of the feature space associated with the kernel often grows exponentially as the complexity of sub-structures increase. One immediate consequence of this behavior is that small, non-informative, sub-structures occur more frequently and cause information overload. Second, as the number of features increase, we encounter sparsity: only a few informative sub-structures will co-occur in multiple graphs. In the first part of this dissertation, we propose to tackle the above problems by exploiting the dependency relationship among sub-structures. First, we propose a novel framework that learns the latent representations of sub-structures by leveraging recent advancements in deep learning. Second, we propose a general smoothing framework that takes structural similarity into account, inspired by state-of-the-art smoothing techniques used in natural language processing. Both the proposed frameworks are applicable to popular graph kernel families, and achieve significant performance improvements over state-of-the-art graph kernels. In the second part of this dissertation, we tackle information overload in text. We first focus on a popular social news aggregation website, Reddit, and design a submodular recommender system that tailors a personalized frontpage for individual users. Second, we propose a novel submodular framework to summarize videos, where both transcript and comments are available. Third, we demonstrate how to apply filtering techniques to select a small subset of informative features from virtual machine logs in order to predict resource usage