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Abstract Can one recover a matrix efficiently from only matrix‐vector products? If so, how many are needed? This article describes algorithms to recover matrices with known structures, such as tridiagonal, Toeplitz, Toeplitz‐like, and hierarchical low‐rank, from matrix‐vector products. In particular, we derive a randomized algorithm for recovering an unknown hierarchical low‐rank matrix from only matrix‐vector products with high probability, where is the rank of the off‐diagonal blocks, and is a small oversampling parameter. We do this by carefully constructing randomized input vectors for our matrix‐vector products that exploit the hierarchical structure of the matrix. While existing algorithms for hierarchical matrix recovery use a recursive “peeling” procedure based on elimination, our approach uses a recursive projection procedure.
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A Jensen inequality for partial traces and applications to partially semiclassical limits
Abstract We prove a matrix inequality for convex functions of a Hermitian matrix on a bipartite space. As an application, we reprove and extend some theorems about eigenvalue asymptotics of Schrödinger operators with homogeneous potentials. The case of main interest is where the Weyl expression is infinite and a partially semiclassical limit occurs.
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- Award ID(s):
- 1954995
- PAR ID:
- 10591012
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Letters in Mathematical Physics
- Volume:
- 115
- Issue:
- 3
- ISSN:
- 1573-0530
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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