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SUMMARY:Structured solutions to nonlinear systems of equations - Justin Ro
 mberg (Georgia Institute of Technology)
DTSTART:20171030T172000Z
DTEND:20171030T181000Z
UID:TALK94027@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:We consider the question of estimating a solution to a system 
 of equations that involve convex nonlinearities\, a problem that is common
  in machine learning and signal processing. Because of these nonlinearitie
 s\, conventional estimators based on empirical risk minimization generally
  involve solving a non-convex optimization program. We propose a method (c
 alled "anchored regression&rdquo\;) that is based on convex programming an
 d amounts to maximizing a linear functional (perhaps augmented by a regula
 rizer) over a convex set.   The proposed convex program is formulated in t
 he natural space of the problem\, and avoids the introduction of auxiliary
  variables\, making it computationally favorable. Working in the native sp
 ace also provides us with the flexibility to incorporate structural priors
  (e.g.\, sparsity) on the solution.  For our analysis\, we model the equat
 ions as being drawn from a fixed set according to a probability law.  Our 
 main results provide guarantees on the accuracy of the estimator in terms 
 of the number of equations weare solving\, the amount of noise present\, a
  measure of statistical complexity of the random equations\, and thegeomet
 ry of the regularizer at the true solution. We also provide recipes for co
 nstructing the anchor vector (that determines the linear functional to max
 imize) directly from the observed data.  We will discuss applications of t
 his technique to nonlinear problems including phase retrieval\, blind deco
 nvolution\, and inverting the action of a neural network.  This is joint w
 ork with Sohail Bahmani.
LOCATION:Seminar Room 1\, Newton Institute
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