Showing papers by "Joel A. Tropp published in 2018"
TL;DR: It is proved that there is a phase transition in the success probability of the dimension reduction map as the embedding dimension increases, and each map has the same stability properties, as quantified through the restricted minimum singular value.
Abstract: Dimension reduction is the process of embedding high-dimensional data into a lower dimensional space to facilitate its analysis. In the Euclidean setting, one fundamental technique for dimension reduction is to apply a random linear map to the data. This dimension reduction procedure succeeds when it preserves certain geometric features of the set.
The question is how large the embedding dimension must be to ensure that randomized dimension reduction succeeds with high probability.
This paper studies a natural family of randomized dimension reduction maps and a large class of data sets. It proves that there is a phase transition in the success probability of the dimension reduction map as the embedding dimension increases. For a given data set, the location of the phase transition is the same for all maps in this family. Furthermore, each map has the same stability properties, as quantified through the restricted minimum singular value. These results can be viewed as new universality laws in high-dimensional stochastic geometry.
Universality laws for randomized dimension reduction have many applications in applied mathematics, signal processing, and statistics. They yield design principles for numerical linear algebra algorithms, for compressed sensing measurement ensembles, and for random linear codes. Furthermore, these results have implications for the performance of statistical estimation methods under a large class of random experimental designs.
67 citations
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TL;DR: In this article, a non-asymptotic confidence region is derived for the projection of the least squares estimator as a sum of random matrices and applying a matrix-valued concentration inequality.
Abstract: Projected least squares (PLS) is an intuitive and numerically cheap technique for quantum state tomography. The method first computes the least-squares estimator (or a linear inversion estimator) and then projects the initial estimate onto the space of states. The main result of this paper equips this point estimator with a rigorous, non-asymptotic confidence region expressed in terms of the trace distance. The analysis holds for a variety of measurements, including 2-designs and Pauli measurements. The sample complexity of the estimator is comparable to the strongest convergence guarantees available in the literature and---in the case of measuring the uniform POVM---saturates fundamental lower bounds.The results are derived by reinterpreting the least-squares estimator as a sum of random matrices and applying a matrix-valued concentration inequality. The theory is supported by numerical simulations for mutually unbiased bases, Pauli observables, and Pauli basis measurements.
18 citations
TL;DR: In this article, the facial geometry of the set of n×n correlation matrices was studied and it was shown that almost every set of r vertices generates a simplicial face, provided that r ≤ √cn, where c is an absolute constant.
Abstract: This paper concerns the facial geometry of the set of n×n correlation matrices. The main result states that almost every set of r vertices generates a simplicial face, provided that r ≤ √cn, where c is an absolute constant. This bound is qualitatively sharp because the set of correlation matrices has no simplicial face generated by more than √2n vertices.
15 citations
TL;DR: In this paper, the authors identify one of the sources of the dimensional term and exploit this insight to develop sharper matrix concentration inequalities, which use information beyond the matrix variance to improve the dimensional dependence.
14 citations
TL;DR: It is stated that almost every set of r vertices generates a simplicial face, provided that r≤cn is an absolute constant, and this bound is qualitatively sharp.
Abstract: This paper concerns the facial geometry of the set of $n \times n$ correlation matrices. The main result states that almost every set of $r$ vertices generates a simplicial face, provided that $r \leq \sqrt{\mathrm{c} n}$, where $\mathrm{c}$ is an absolute constant. This bound is qualitatively sharp because the set of correlation matrices has no simplicial face generated by more than $\sqrt{2n}$ vertices.
2 citations