8th International Conference on
Computational Harmonic Analysis

September 12-16, 2022

Ingolstadt, Germany

"Lipschitz Embeddings and Riemannian Properties of Spaces of Low-Rank Symmetric Matrices"

Balan, Radu

For the quotient space of rectangular matrices identified modulo right-side multiplicative action by unitary matrices, we introduce a new metric and compare it to existing metrics (the natural metric a.k.a. the Bures-Wasserstein distance). We show how these metric spaces can be Lipschitz embedded in Euclidean spaces. Next we apply these results to the generalized phase retrieval problem. The classical phase retrieval problem arises in contexts ranging from speech recognition to x-ray crystallography and quantum state tomography. The generalization to $U(r)$ phase retrieval of matrix frames is natural in the sense that it corresponds to quantum tomography of impure states. We provide computable global stability bounds for understanding related problems in terms of the differential geometry of key spaces. In particular, we manifest a Whitney stratification of the positive semidefinite matrices of low rank which allows us to ``stratify'' the computation of the global stability bound. We show that for the impure state case no such global stability bounds can be obtained for the non-linear analysis map $\alpha$ with respect to certain natural distance metrics. Finally, our computation of the global lower Lipschitz constant for the $\beta$ analysis map provides novel conditions for a matrix frame to be generalized phase retrievable when $r>1$. These results are contained in the arXiv preprint [quant-ph] 2109.14522.

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