ESyMath
Spotlights
Mathematics of machine learning
Conservative SPDEs as fluctuating mean field limits of stochastic gradient descent
(arXiv-preprint by Benjamin Gess, Rishabh S. Gvalani and Vitalii Konarovskyi)
The surprising, empirically observed convergence of stochastic optimization in supervised machine learning is among the many unsolved mysteries of the field. The hunt for hidden, underlying mathematical structures is essential for the development of a systematic understanding. In this work, conservative stochastic partial differential equations are identified as such a universal structure that encodes essential information on the dynamics of the learning process and its fluctuations.
Stochastic analysis meets control theory
On a Class of Infinite-Dimensional Singular Stochastic Control Problems
(paper by Salvatore Federico, Giorgio Ferrari, Frank Riedel, and Michael Röckner published in SIAM Journal on Control and Optimization)
In this work, the authors study a class of infinite-dimensional singular stochastic control problems that might find applications in economic theory and finance. The control process linearly affects an abstract evolution equation on a suitable partially ordered infinite-dimensional space X, it takes values in the positive cone of X, and it has right-continuous and nondecreasing paths. The main contribution of the paper is to provide a rigorous formulation of the problem by properly defining the controlled dynamics and integrals with respect to the control process, and then to derive necessary and sufficient first-order conditions for optimality. The latter are finally exploited in a specification of the model where an optimal control is determined. The techniques used are those of semigroup theory, vector-valued integration, convex analysis, and general theory of stochastic processes.
Spectral methods in algebraic geometry
The Balmer spectrum of certain Deligne-Mumford stacks
(paper by Eike Lau published in Compositio Mathematica)
This preprint considers a Deligne-Mumford stack X which is the quotient of an affine scheme SpecA by the action of a finite group G. It is shown that the Balmer spectrum of the tensor triangulated category of perfect complexes on X is homeomorphic to the space of homogeneous prime ideals in the group cohomology ring H*(G,A).