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    Online Matrix Completion with Side Information

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    We give an online algorithm and prove novel mistake and regret bounds for online binary matrix completion with side information. The mistake bounds we prove are of the form O~(D/Îł2)\tilde{O}(D/\gamma^2). The term 1/Îł21/\gamma^2 is analogous to the usual margin term in SVM (perceptron) bounds. More specifically, if we assume that there is some factorization of the underlying m×nm \times n matrix into PQâŠșP Q^\intercal where the rows of PP are interpreted as "classifiers" in Rd\mathcal{R}^d and the rows of QQ as "instances" in Rd\mathcal{R}^d, then Îł\gamma is the maximum (normalized) margin over all factorizations PQâŠșP Q^\intercal consistent with the observed matrix. The quasi-dimension term DD measures the quality of side information. In the presence of vacuous side information, D=m+nD= m+n. However, if the side information is predictive of the underlying factorization of the matrix, then in an ideal case, D∈O(k+ℓ)D \in O(k + \ell) where kk is the number of distinct row factors and ℓ\ell is the number of distinct column factors. We additionally provide a generalization of our algorithm to the inductive setting. In this setting, we provide an example where the side information is not directly specified in advance. For this example, the quasi-dimension DD is now bounded by O(k2+ℓ2)O(k^2 + \ell^2)

    Random matrix techniques in quantum information theory

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    The purpose of this review article is to present some of the latest developments using random techniques, and in particular, random matrix techniques in quantum information theory. Our review is a blend of a rather exhaustive review, combined with more detailed examples -- coming from research projects in which the authors were involved. We focus on two main topics, random quantum states and random quantum channels. We present results related to entropic quantities, entanglement of typical states, entanglement thresholds, the output set of quantum channels, and violations of the minimum output entropy of random channels
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