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SUMMARY:A gradient flow on control space with rough initial condition
DTSTART:20240517T140000Z
DTEND:20240517T150000Z
DTSTAMP:20240723T083000Z
UID:indico-event-11755@indico.math.cnrs.fr
DESCRIPTION:Speakers: Florin Suciu (CEREMADE\, Université Paris-Dauphine)
\n\nWe consider the (sub-Riemannian type) control problem of finding a pat
h going from an initial point x to a target point y\, by only moving in ce
rtain admissible directions. We assume that the corresponding vector field
s satisfy the Hörmander condition\, so that the classical Chow-Rashevskii
theorem guarantees the existence of such a path. One natural way to try t
o solve this problem is via a gradient flow on control space. However\, si
nce the corresponding dynamics may have saddle points\, any convergence re
sult must rely on suitable (e.g. random) initialization. We consider the c
ase when this initialization is irregular\, which is conveniently formulat
ed via Lyons' rough path theory. In some simple cases\, we manage to prove
that the gradient flow converges to a solution\, if the initial condition
is the path of a Brownian motion (or rougher). The proof is based on comb
ining ideas from Malliavin calculus with Łojasiewicz inequalities. A poss
ible motivation for our study comes from the training of deep Residual Neu
ral Nets\, in the regime when the number of trainable parameters per layer
is smaller than the dimension of the data vector.\n\nhttps://indico.math.
cnrs.fr/event/11755/
LOCATION:Salle 01 (Institut Henri Poincaré)
URL:https://indico.math.cnrs.fr/event/11755/
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