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FDS Member Colloquium: Van Vu (Department of Mathematics), “New perturbation bounds for matrices with rapidly decaying spectrum”

Wednesday, April 17, 2024    
11:30AM to 1:00PM
Yale Institute for Foundations of Data Science
Kline Tower 13th Floor, Room 1327
New Haven, CT 06511
Van H. Vu
Speaker: Van Vu

Percey F. Smith Professor of Mathematics
Department of Mathematics,
Yale University

Wednesday, April 17, 2024

11:30 am: Lunch (Kitchen)
12:00 pm – 1:00 pm: Talk (Seminar Room #1327)

Location: Yale Institute for Foundations of Data Science, Kline Tower, 13th floor and via Zoom (starts at 12:00 pm): https://yale.zoom.us/s/7859884026 (meeting ID: 7859884026)

Abstract: Matrix perturbation bounds (such as Weyl and Davis-Kahan) form an important part of the mathematical foundation of data science.

In many recent studies, it has been observed/assumed that large data matrices have rapidly decaying spectrum
(low rank is an extreme example).

We are going to discuss new perturbation bounds for these matrices. Our focus will be on the perturbation of eigenvectors and eigensubspaces (Davis-Kahan type bounds). Our new results will improve the original Davis-Kahan bound in several aspects. In particular, we will not require the gap-to-noise ratio to be large. More importantly, the new bounds take into account the interaction between the noise matrix and the eigenvectors of the data matrix.

We will present few applications, including a new error analysis for:

1, PCA of sample covariance matrices.
2, Low rank approximation.

(Partially joined with P. Tran, S. O’rourke, K. Wang)


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