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Geometry and topology of spaces of structured matrices

By Tom Needham

Appears in collection : Geometric Sciences in Action: from geometric statistics to shape analysis / Les sciences géometriques en action: des statistiques géometriques à l'analyse de forme

A finite unit norm tight frame (FUNTF) is a spanning set of unit vectors in a finite-dimensional Hilbert space such that the spectrum of singular values of an associated operator is constant. In signal processing applications, it is desirable to use FUNTFs to encode signals, as such representations are proven to be optimally robust to noise. This naturally gives rise to questions about the geometry and topology of the space of FUNTFs. For example, the conjecture that every space of FUNTFs is connected was open for 15 years, and slight variants of this problem still remain open. I will discuss recent work with Clayton Shonkwiler, where we answer several questions about random matrix theory and optimization in spaces of structured matrices, using tools from symplectic geometry and geometric invariant theory.

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Citation data

  • DOI 10.24350/CIRM.V.20185303
  • Cite this video Needham, Tom (30/05/2024). Geometry and topology of spaces of structured matrices. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20185303
  • URL https://dx.doi.org/10.24350/CIRM.V.20185303


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