29 research outputs found
On the stability of the polygonal isoperimetric inequality
We obtain a sharp lower bound on the isoperimetric deficit of a general
polygon in terms of the variance of its side lengths, the variance of its
radii, and its deviation from being convex. Our technique involves a functional
minimization problem on a suitably constructed compact manifold and is based on
the spectral theory for circulant matrices
The variational structure and time-periodic solutions for mean-field games systems
Here, we observe that mean-field game (MFG) systems admit a two-player
infinite-dimensional general-sum differential game formulation. We show that
particular regimes of this game reduce to previously known variational
principles. Furthermore, based on the game-perspective we derive new
variational formulations for first-order MFG systems with congestion. Finally,
we use these findings to prove the existence of time-periodic solutions for
viscous MFG systems with a coupling that is not a non-decreasing function of
density.Comment: 31 page
Applications of No-Collision Transportation Maps in Manifold Learning
In this work, we investigate applications of no-collision transportation maps
introduced in [Nurbekyan et. al., 2020] in manifold learning for image data.
Recently, there has been a surge in applying transportation-based distances and
features for data representing motion-like or deformation-like phenomena.
Indeed, comparing intensities at fixed locations often does not reveal the data
structure. No-collision maps and distances developed in [Nurbekyan et. al.,
2020] are sensitive to geometric features similar to optimal transportation
(OT) maps but much cheaper to compute due to the absence of optimization. In
this work, we prove that no-collision distances provide an isometry between
translations (respectively dilations) of a single probability measure and the
translation (respectively dilation) vectors equipped with a Euclidean distance.
Furthermore, we prove that no-collision transportation maps, as well as OT and
linearized OT maps, do not in general provide an isometry for rotations. The
numerical experiments confirm our theoretical findings and show that
no-collision distances achieve similar or better performance on several
manifold learning tasks compared to other OT and Euclidean-based methods at a
fraction of a computational cost
A mean-field game economic growth model
Here, we examine a mean-field game (MFG) that models the economic growth of a
population of non-cooperative rational agents. In this MFG, agents are
described by two state variables - the capital and consumer goods they own.
Each agent seeks to maximize their utility by taking into account statistical
data of the total population. The individual actions drive the evolution of the
players, and a market-clearing condition determines the relative price of
capital and consumer goods. We study the existence and uniqueness of optimal
strategies of the agents and develop numerical methods to compute these
strategies and the equilibrium price