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SUMMARY:Fundamental limits in structured PCA\, and how to reach them - Dr 
 Jean Barbier\, International Center for Theoretical Physics (ICTP)
DTSTART:20230315T140000Z
DTEND:20230315T150000Z
UID:TALK194131@talks.cam.ac.uk
CONTACT:Prof. Ramji Venkataramanan
DESCRIPTION:I will discuss the paradigmatic spiked matrix model of princip
 al components analysis\, where a rank-one signal is corrupted by additive 
 noise. While the noise is typically taken from a Wigner matrix with indepe
 ndent entries\, here the potential acting on the eigenvalues has a quadrat
 ic plus a quartic component. The quartic term induces strong correlations 
 between the matrix elements\, which makes the setting relevant for applica
 tions but analytically challenging. Our work provides the first characteri
 zation of the Bayes-optimal limits for inference in this model with struct
 ured noise. If the signal prior is rotational-invariant\, then we show tha
 t a spectral estimator is optimal. In contrast\, for more general priors\,
  the existing approximate message passing algorithm (AMP) falls short of a
 chieving the information-theoretic limits\, and we provide a justification
  for this sub-optimality. Finally\, by generalizing the theory of Thouless
 -Anderson-Palmer equations\, we cure the issue by proposing a novel AMP wh
 ich matches the theoretical limits. Our information-theoretic analysis is 
 based on the replica method\, a powerful heuristic from statistical mechan
 ics\; instead\, the novel AMP comes with a rigorous state evolution analys
 is tracking its performance in the high-dimensional limit. \n
LOCATION:B1.19 (MR19)\, CMS Pavilion B 
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