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METHOD:PUBLISH
BEGIN:VEVENT
DTSTAMP:20261023T100534Z
DTSTART:20261028T150000Z
DTEND:20261028T160000Z
SUMMARY:Probability Seminar: Yihan Zhang
UID:{http://www.columbasystems.com/customers/uom/gpp/eventid/}hef-mvkt0bq
 h-fgu4sk
DESCRIPTION:Yihan Zhang (University of Bristol) will speak at the Probabi
 lity seminar.\n\nTitle: Eigenvalue and Eigenvector Approximation for Ran
 dom Matrices Using Low-Degree Polynomials\n\nAbstract: We initiate the s
 tudy of approximating the top eigenvalue and eigenvector of a random sym
 metric matrix $ A \\in \\mathbb{R}^{n\\times n} $ using $ q(A)b $ where 
 $q$ is a degree-$d$ polynomial and $b$ is a standard Gaussian vector ind
 ependent of $A$. For spiked GOE $ Y = \\lambda vv^\\top + X $\, we ident
 ify $ d_\\star = \\frac{\\log(n)}{2\\log(\\lambda)} $ to be the critical
  degree threshold above which accurate approximation of the top eigenval
 ue and eigenvector is possible. This sharpens the common belief that spe
 ctral methods can be implemented by $ O(\\log(n)) $-step power iteration
 s and offers a precise connection between spectral methods and low-degre
 e polynomial algorithms\, a popular proxy for all polynomial-time algori
 thms. For GOE $X$\, we identify $ d_\\star = n^{1/3+o(1)} $ to be the cr
 itical degree threshold for top eigenvector approximation\, whereas cons
 tant degree suffices for top eigenvalue approximation. Moreover\, in the
  limit where $ d/n^{1/3} $ converges to a positive finite constant\, we 
 compute the exact asymptotic eigenvector approximation accuracy in terms
  of the expected squared overlap. These results significantly improve up
 on predictions made in randomized numerical linear algebra for determini
 stic data matrices that the iteration count of power methods with random
  initialization is governed by the inverse spectral gap. Technically\, o
 ur analyses leverage extremal properties of Chebyshev polynomials and dr
 aw upon the rich literature of random matrix theory. Based on https://ar
 xiv.org/abs/2609.28781 [arxiv.org] . 
STATUS:TENTATIVE
TRANSP:TRANSPARENT
CLASS:PUBLIC
LOCATION:Frank Adams 2\, Alan Turing Building\, Manchester
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