In this paper we generalize the 1-bit matrix completion problem to higher order tensors. We prove that when r = O(1) a bounded rank-r, order-d tensor T in R N × R N × · · · × R N can be estimated efficiently by only m = O(N d) binary measurements by regularizing its max-qnorm and M-norm as surrogates for its rank. We prove that similar to the matrix case, i.e., when d = 2, the sample complexity of recovering a low-rank tensor from 1-bit measurements of a subset of its entries is the same as recovering it from unquantized measurements. Moreover, we show the advantage of using 1-bit tensor completion over matricization both theoretically and numerically. Specifically, we show how the 1-bit measurement model can be used for context-aware recommender systems.

We investigate whether quantum annealers with select chip layouts can outperform classical computers in reinforcement learning tasks. We associate a transverse field Ising spin Hamiltonian with a layout of qubits similar to that of a deep Boltzmann machine (DBM) and use simulated quantum annealing (SQA) to numerically simulate quantum sampling from this system. We design a reinforcement learning algorithm in which the set of visible nodes representing the states and actions of an optimal policy are the first and last layers of the deep network. In absence of a transverse field, our simulations show that DBMs are trained more effectively than restricted Boltzmann machines (RBM) with the same number of nodes. We then develop a framework for training the network as a quantum Boltzmann machine (QBM) in the presence of a significant transverse field for reinforcement learning. This method also outperforms the reinforcement learning method that uses RBMs.

We analyze low rank tensor completion (TC) using noisy measurements of a subset of the tensor. Assuming a rank-r, order-d, N × N × · · · × N tensor where r = O(1), the best sampling complexity that was achieved is O(N d 2 ), which is obtained by solving a tensor nuclear-norm minimization problem. However, this bound is significantly larger than the number of free variables in a low rank tensor which is O(dN ). In this paper, we show that by using an atomic-norm whose atoms are rank-1 sign tensors, one can obtain a sample complexity of O(dN ). Moreover, we generalize the matrix max-norm definition to tensors, which results in a maxquasi-norm (max-qnorm) whose unit ball has small Rademacher complexity. We prove that solving a constrained least squares estimation using either the convex atomic-norm or the nonconvex max-qnorm results in optimal sample complexity for the problem of low-rank tensor completion. Furthermore, we show that these bounds are nearly minimax rate-optimal. We also provide promising numerical results for max-qnorm constrained tensor completion, showing improved recovery results compared to matricization and alternating least squares.

We investigate whether quantum annealers with select chip layouts can outperform classical computers in reinforcement learning tasks. We associate a transverse field Ising spin Hamiltonian with a layout of qubits similar to that of a deep Boltzmann machine (DBM) and use simulated quantum annealing (SQA) to numerically simulate quantum sampling from this system. We design a reinforcement learning algorithm in which the set of visible nodes representing the states and actions of an optimal policy are the first and last layers of the deep network. In absence of a transverse field, our simulations show that DBMs are trained more effectively than restricted Boltzmann machines (RBM) with the same number of nodes. We then develop a framework for training the network as a quantum Boltzmann machine (QBM) in the presence of a significant transverse field for reinforcement learning. This method also outperforms the reinforcement learning method that uses RBMs.

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