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UID:news1931@dmi.unibas.ch
DTSTAMP;TZID=Europe/Zurich:20251030T111715
DTSTART;TZID=Europe/Zurich:20251106T121500
SUMMARY:Bernoullis Tafelrunde: Roberto Colombo (EPFL)
DESCRIPTION:Gradient flows in the Wasserstein space of probability measures
  have proven very useful for providing new interpretations of many parabol
 ic partial differential equations in relation to Optimal Transport theory.
  Recently\, it has been observed that this formalism can also be used to d
 escribe the learning dynamics of certain (continuous) two-layer neural net
 works. In a model case\, this evolution corresponds to the Wasserstein gra
 dient flow of the energy given by a negative Sobolev distance to a fixed t
 arget measure. The resulting active-scalar continuity equation shares seve
 ral analogies with the vorticity formulation of the Euler equations for 2D
  fluids\, while exhibiting markedly different qualitative long time behavi
 or. A natural question is to identify conditions under which the system c
 onverges to the target\, possibly with an explicit rate of convergence. In
  joint work with Lénaïc Chizat\, Maria Colombo\, and Xavier Fernández-R
 eal\, we address this question from a PDE perspective\, obtaining precise 
 exponential or polynomial convergence rates under suitable smoothness assu
 mptions. In this seminar\, we will introduce Wasserstein gradient flows o
 f negative Sobolev discrepancies and describe the main ideas behind the af
 orementioned quantitative convergence results.\\r\\npdf_version [t3://file
 ?uid=4063]
X-ALT-DESC:<p>Gradient flows in the Wasserstein space of probability measur
 es have proven very useful for providing new interpretations of many parab
 olic partial differential equations in relation to Optimal Transport theor
 y. Recently\, it has been observed that this formalism can also be used to
  describe the learning dynamics of certain (continuous) two-layer neural n
 etworks. In a model case\, this evolution corresponds to the Wasserstein g
 radient flow of the energy given by a negative Sobolev distance to a fixed
  target measure. The resulting active-scalar continuity equation shares se
 veral analogies with the vorticity formulation of the Euler equations for 
 2D fluids\, while exhibiting markedly different qualitative long time beha
 vior.&nbsp\;A natural question is to identify conditions under which the s
 ystem converges to the target\, possibly with an explicit rate of converge
 nce. In joint work with Lénaïc Chizat\, Maria Colombo\, and Xavier Fern
 ández-Real\, we address this question from a PDE perspective\, obtaining 
 precise exponential or polynomial convergence rates under suitable smoothn
 ess assumptions.&nbsp\;In this seminar\, we will introduce Wasserstein gra
 dient flows of negative Sobolev discrepancies and describe the main ideas 
 behind the aforementioned quantitative convergence results.</p>\n<p><a hre
 f="t3://file?uid=4063">pdf_version</a></p>
DTEND;TZID=Europe/Zurich:20251106T130000
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