We study the recovery of Hermitian low rank matrices X ∈ C n×n from undersampled measurements via nuclear norm minimization. We consider the particular scenario where the measurements are Frobenius inner products with random rank-one matrices of the form a j a * j for some measurement vectors a 1 , . . . , am, i.e., the measurements are given by y j = tr(Xa j a * j ). The case where the matrix X = xx * to be recovered is of rank one reduces to the problem of phaseless estimation (from measurements, y j = | x, a j | 2 via the PhaseLift approach, which has been introduced recently. We derive bounds for the number m of measurements that guarantee successful uniform recovery of Hermitian rank r matrices, either for the vectors a j , j = 1, . . . , m, being chosen independently at random according to a standard Gaussian distribution, or a j being sampled independently from an (approximate) complex projective t-design with t = 4. In the Gaussian case, we require m ≥ Crn measurements, while in the case of 4-designs we need m ≥ Crn log(n). Our results are uniform in the sense that one random choice of the measurement vectors a j guarantees recovery of all rank r-matrices simultaneously with high probability. Moreover, we prove robustness of recovery under perturbation of the measurements by noise. The result for approximate 4-designs generalizes and improves a recent bound on phase retrieval due to Gross, Kueng and Krahmer. In addition, it has applications in quantum state tomography. Our proofs employ the so-called bowling scheme which is based on recent ideas by Mendelson and Koltchinskii.Date: October 25, 2014.

The problem of recovering a matrix of low rank from an incomplete and possibly noisy set of linear measurements arises in a number of areas such as quantum state tomography, machine learning and the PhaseLift approach to phaseless reconstruction problems. In order to derive rigorous recovery results, the measurement map is usually modeled probabilistically and convex optimization approaches including nuclear norm minimization are often used as recovery method. In this article, we derive sufficient conditions on the minimal amount of measurements that ensure recovery via convex optimization. We establish our results via certain properties of the null space of the measurement map. In the setting where the measurements are realized as Frobenius inner products with independent standard Gaussian random matrices we show that m > 10r(n 1 + n 2 ) measurements are enough to uniformly and stably recover an n 1 × n 2 matrix of rank at most r. Stability is meant both with respect to passing from exactly rank-r matrices to approximately rank-r matrices and with respect to adding noise on the measurements. We then significantly generalize this result by only requiring independent mean-zero, variance one entries with four finite moments at the cost of replacing 10 by some universal constant. We also study the particular case of recovering Hermitian rank-r matrices from measurement matrices proportional to rank-one projectors. For r = 1, such a problem reduces to the PhaseLift approach to phaseless recovery, while the case of higher rank is relevant for quantum state tomography. For m ≥ Crn rank-one projective measurements onto independent standard Gaussian vectors, we show that nuclear norm minimization uniformly and stably reconstructs Hermitian rank-r matrices with high probability. Subsequently, we partially de-randomize this result by establishing an analogous statement for projectors onto independent elements of a complex projective 4-designs at the cost of a slightly higher sampling rate m ≥ Crn log n. Complex projective t-designs are discrete sets of vectors whose uniform distribution reproduces the first t moments of the uniform distribution on the sphere. Moreover, if the Hermitian matrix to be recovered is known to be positive semidefinite, then we show that the nuclear norm minimization approach may be replaced by the simpler optimization program of minimizing the ℓ 2 -norm of the residual subject to the positive semidefinite constraint. This has the additional advantage that no estimate of the noise level is required a priori. We discuss applications of such a result in quantum physics and the phase retrieval problem. Apart from the case of independent Gaussian measurements, the analysis exploits Mendelson's small ball method.Keywords. low rank matrix recovery, quantum state tomography, phase retrieval, convex optimization, nuclear norm minimization, positive semidefinite least squares problem, complex projective designs, random measurements MSC 2010. 94A20, 94A12, 60B20, 90C25, 81P50 Date: October 16, 2018.

The arithmetic fundamental lemma conjecture of the third author connects the derivative of an orbital integral on a symmetric space with an intersection number on a formal moduli space of p-divisible groups of Picard type. It arises in the relative trace formula approach to the arithmetic Gan-Gross-Prasad conjecture. We prove this conjecture in the minuscule case.

We study the convergence of gradient flows related to learning deep linear neural networks (where the activation function is the identity map) from data. In this case, the composition of the network layers amounts to simply multiplying the weight matrices of all layers together, resulting in an overparameterized problem. The gradient flow with respect to these factors can be re-interpreted as a Riemannian gradient flow on the manifold of rank-$r$ matrices endowed with a suitable Riemannian metric. We show that the flow always converges to a critical point of the underlying functional. Moreover, we establish that, for almost all initializations, the flow converges to a global minimum on the manifold of rank $k$ matrices for some $k\leq r$.

Introduction 1 2. The moduli space 4 3. Vertex lattices 7 4. The pointwise Bruhat-Tits stratification of N 0 red 8 5. The Deligne-Lusztig variety for the symplectic group 10 6. Bruhat-Tits-strata 15 7. Applications to Shimura varieties 18 References 20