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UID:20250714T203922EDT-3344BIiGUM@132.216.98.100
DTSTAMP:20250715T003922Z
DESCRIPTION:Title: An Approximation Theory for Metric Space-Valued Function
 s With A View Towards Deep Learning\n\nAbstract :\n\nWe build universal ap
 proximators of continuous maps between arbitrary Polish metric spaces X an
 d Y using universal approximators between Euclidean spaces as building blo
 cks. Earlier results assume that the output space Y is a topological vecto
 r space. We overcome this limitation by 'randomization': our approximators
  output discrete probability measures over Y. When X and Y are Polish with
 out additional structure\, we prove very general qualitative guarantees\; 
 when they have suitable combinatorial structure\, we prove quantitative gu
 arantees for Holder-like maps\, including maps between finite graphs\, sol
 ution operators to rough differential equations between certain Carnot gro
 ups\, and continuous non-linear operators between Banach spaces arising in
  inverse problems. In particular\, we show that the required number of Dir
 ac measures is determined by the combinatorial structure of X and Y. For b
 arycentric Y\, including Banach spaces\, R-trees\, Hadamard manifolds\, or
  Wasserstein spaces on Polish metric spaces\, our approximators reduce to 
 Y-valued functions. When the Euclidean approximators are neural networks\,
  our constructions generalize transformer networks\, providing a new proba
 bilistic viewpoint of geometric deep learning.\n
DTSTART:20240311T200000Z
DTEND:20240311T210000Z
LOCATION:Room 1104\, Burnside Hall\, CA\, QC\, Montreal\, H3A 0B9\, 805 rue
  Sherbrooke Ouest
SUMMARY:Anastasis Kratsios (McMaster University)
URL:/mathstat/channels/event/anastasis-kratsios-mcmast
 er-university-355908
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