Approximately Hadamard matrices and random frames
Holder： Mark Rudelson (UMich)
Time：2023-12-25 14:00-15:00
Location：WANG Xuan Lecture Theater，Zhi Hua Building-101
Abstract:
We will discuss a problem concerning random frames which arises in signal processing. A frame is an overcomplete set of vectors in the n-dimensional linear space which allows a robust decomposition of any vector in this space as a linear combination of these vectors. Random frames are used in signal processing as a means of encoding since the loss of a fraction of coordinates does not prevent the recovery. We will discuss a question when a random frame contains a copy of a nice (almost orthogonal) basis.
Despite the probabilistic nature of this problem it reduces to a completely deterministic question of existence of approximately Hadamard matrices. An n by n matrix with plus-minus 1 entries is called Hadamard if it acts on the space as a scaled isometry. Such matrices exist in some, but not in all dimensions. Nevertheless, we will construct plus-minus 1 matrices of every size which act as approximate scaled isometries. This construction will bring us back to probability as we will have to combine number-theoretic and probabilistic methods.
Joint work with Xiaoyu Dong.
About the Speaker:
Mark Rudelson is a professor of University of Michigan. He received his PhD from the Hebrew University of Jerusalem, Israel. He works in geometric functional analysis and probability, especially random matrix theory.
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