We consider the question whether algorithmic choices in over-parameterized linear matrix factorization introduce implicit low-rank regularization.We focus on the noiseless matrix sensing scenario over low-rank positive semi-definite (PSD) matrices over the reals, with a sensing mechanism that satisfies restricted isometry properties.Surprisingly, it was recently argued that for recovery of PSD matrices, gradient descent over a squared, \textit{full-rank} factorized space introduces implicit low-rank regularization.Thus, a clever choice of the recovery algorithm avoids the need for explicit low-rank regularization. In this contribution, we prove that in fact, under certain conditions, the PSD constraint by itself is sufficient to lead to a unique low-rank matrix recovery, without explicit or implicit regularization.Therefore, under these conditions, the set of PSD matrices that are consistent with the observed data, is a singleton, regardless of the algorithm used. Our numerical study indicates that this result is general and extends to cases beyond the those covered by the proof.

We give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speed-ups. We consider SDP instances with m constraint matrices, each of dimension n, rank at most r, and sparsity s. The first algorithm assumes an input model where one is given access to an oracle to the entries of the matrices at unit cost. We show that it has run time O~(s^2 (sqrt{m} epsilon^{-10} + sqrt{n} epsilon^{-12})), with epsilon the error of the solution. This gives an optimal dependence in terms of m, n and quadratic improvement over previous quantum algorithms (when m ~~ n). The second algorithm assumes a fully quantum input model in which the input matrices are given as quantum states. We show that its run time is O~(sqrt{m}+poly(r))*poly(log m,log n,B,epsilon^{-1}), with B an upper bound on the trace-norm of all input matrices. In particular the complexity depends only polylogarithmically in n and polynomially in r. We apply the second SDP solver to learn a good description of a quantum state with respect to a set of measurements: Given m measurements and a supply of copies of an unknown state rho with rank at most r, we show we can find in time sqrt{m}*poly(log m,log n,r,epsilon^{-1}) a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as rho on the m measurements, up to error epsilon. The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes' principle from statistical mechanics. As in previous work, we obtain our algorithm by "quantizing" classical SDP solvers based on the matrix multiplicative weight update method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians, given quantum states encoding these Hamiltonians, with a poly-logarithmic dependence on its dimension, which is based on ideas developed in quantum principal component analysis. We also develop a "fast" quantum OR lemma with a quadratic improvement in gate complexity over the construction of Harrow et al. [Harrow et al., 2017]. We believe both techniques might be of independent interest.