Prof. Dr. Matus Telgarsky (NYU, US)
Feature learning, lower-homogeneity, and normalization layers abstract
Abstract:
The first half of this talk will describe the feature learning problem in deep learning optimization, its statistical consequences, and an approach to proving general theorems with a heavy reliance on normalization layers, which are common to all modern architectures but typically treated as an analytic nuisance. Theorems will cover two settings: concrete results for shallow networks, and abstract template theorems for general architectures. The shallow network results allow for globally maximal margins at the cost of large width and no further assumptions, while the general architecture theorems give convergence rates to KKT points for a new general class of architectures satisfying “partial lower-homogeneity”.The second half will be technical, demonstrating two core proof techniques. The first ingredient, essential to the shallow analysis, is a new mirror descent lemma, strengthening a beautiful idea discovered by Chizat and Bach. The second ingredient is the concept of “partial lower-homogeneity” and its consequences.Joint work with Danny Son; not currently on arXiv, but “coming soon”.
10:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Ruiwen Dong (Saarbrucken)
The Identity Problem in subsemigroups of metabelian groups abstract
Abstract:
Algorithmic problems in metabelian groups have been studied as early as the 1950s since the work of Hall. In the 1970s Romanovskii proved decidability of the Group Membership problem (given the generators of a subgroup and a target element, decide whether the target element is in the subgroup) in metabelian groups. However, Semigroup Membership (same as Group Membership, but with sub-semigroups) has been shown to be undecidable in several instances of metabelian groups. In this talk we consider two "intermediate" decision problems: the Identity Problem (deciding if a sub-semigroup contains the neutral element) and the Group Problem (deciding if a sub-semigroup is a group). We reduce them to solving linear equations over the polynomial semiring N[X] and show decidability using a Positivstellensatz-type local-global principle.
10:30 • Université de Genève, Conseil Général 7-9, Room 1-05
Andrey Pilipenko (Kyviv)
On a selection problem for small noise perturbation of unstable dynamical systems abstract
Abstract:
The well-known Peano existence theorem states that an ordinary differential equation (ODE) with continuous coefficients has a local solution, which, however, may not be unique. On the contrary, an addition of a non-degenerate noise term usually yields the existence of a unique solution to the corresponding stochastic differential equation, even if the drift term is discontinuous. We study the limit of the stochastic equations as the noise intensity converges to zero. This limit can be interpreted as a natural selection of a solution for the initial ODE.The identification of the limit is closely related to a study of the exact growth rate of certain stochastic equations, as well as to an averaging principle.
14:00 • Université de Genève, Conseil Général 7-9, Room 1-05
Christopher Criscitiello (EPFL, CH)
The sensor network localization problem has benign landscape under mild rank relaxation abstract
Abstract:
We consider the sensor network localization problem (also called metric multidimensional scaling): we observe some pairwise distances between n ground-truth points in R^d, and our goal is to recover this cloud of ground-truth points (up to translation and rotation). The corresponding optimization problem is nonconvex, and we show that it can have spurious local minima. However, inspired by numerical experiments, we argue that if one relaxes the problem by optimizing over clouds of n points in dimension k greater than d, then all second-order critical points of the problem are global minima. Specifically, we show this for two settings: (1) for arbitrary ground-truth points, when all pairwise distances are known, and k = O(sqrt{n d}), and: (2) for isotropic random ground-truth points, when most (but not necessarily all) pairwise distances are known, and k = O(d log(n)). To the best of our knowledge, these are the first landscape results for this nonconvex version of sensor network localization.
14:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.1
Alexander Thomas (Heidelberg University)
Fock bundles and the Hitchin component abstract
Abstract:
The study of representations of a fundamental group of a surface into a Lie group is described by the character variety. I will present a new approach, via so-called Fock bundles, to study character varieties. Despite similarities with nonabelian Hodge theory, the crucial difference is that no complex structure is fixed on the surface. Fock bundles are closely related to higher complex structures and Hitchin components. Joint work with Georgios Kydonakis and Charlie Reid.
15:00 • Université de Genève, Conseil Général 7-9, Room 1-07
Prof. Dr. Melanie Rupflin (University of Oxford)
Sharp quantitative results for maps from $S^2$ to $S^2$ of general degree abstract
Abstract:
As the energy of any map $v$ from $S^2$ to $S^2$ is at least $4\\pi vert deg(v)\\vert$ with equality if and only if $v$ is a rational map it is natural to ask whether maps with small energy defect $\\de_v=E(v)-4\\pi \\abs{\\deg(v)}$ are necessarily close to a rational map.While such a rigidity statement turns out to be false for maps of general degree, we will prove that any map $v$ with small energy defect is essentially given by a collection of rational maps that describe the behaviour of $v$ at very different scales and that the corresponding distance is controlled by a quantitative estimate of the form $\\text{dist}^2\\leq C \\delta_v(1+\\abs{\\log\\delta_v})$ which is indeed sharp.
15:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Ana Marija Vego (ETHZ)
What is... an Iwasawa algebra? abstract
Abstract:
\'\'The Iwasawa algebra is a key object in the study of p-adic L-functions, which are a central topic in number theory. The Iwasawa algebra arises naturally in this context as a tool for understanding the behavior of certain arithmetic invariants, such as Selmer groups and class groups, in towers of number fields. It provides a framework for studying these invariants in a unified way over all the levels of the tower. This allows us to investigate the arithmetic properties of number fields and their associated objects, particularly in the context of p-adic L-functions and Galois representations. It has applications in various areas of number theory, including the study of special values of L-functions, the Birch and Swinnerton-Dyer conjecture, and the structure of class groups of number fields. In this talk we will introduce Iwasawa algebras and give some basic properties. We\'ll then explore how these algebras are used in constructing Euler systems and obtaining p-adic L-functions. If time allows, we\'ll also touch on the main conjecture of Iwasawa theory.
16:30 • UZH Zentrum, Building KO2, Room F 150
Prof. Dr. Bruno Colbois (Uni Neuchâtel)
Isoperimetric inequalities and spectrum of the Laplacian abstract
Abstract:
In this talk, I will relate some fairly classical results concerning isoperimetric inequalities to results concerning the spectrum of the Laplacian in Euclidean domains. In the last part of the talk, I will present some new results. Throughout the talk, I will also present some open questions that are fairly simple to formulate (but perhaps not to solve).
17:15 • Université de Fribourg, room Phys 2.52