Cris Moore
Tensor networks, tensor cumulants, and tensor factorization abstract
Abstract:
Statistical physics and statistical inference have a deep connection, and ideas continue to flow between the two fields. Consider the following problem in high-dimensional statistics: there is a hidden vector v, and we observe v’s p-th outer product with itself, giving a tensor of arity p, with Gaussian noise in every entry. Given this observed tensor, our goal is to reconstruct v, as well as to reject the null hypothesis that the tensor consists only of noise. To a physicist, this is a planted p-spin model, where p is the arity of the tensor, and where the coupling matrix is correlated with the hidden vector. To a computer scientist, it is a tensor version of PCA (principal component analysis). But we lack a theory of linear algebra for tensors, so we can’t simply look at the dominant eigenvector and hope it is correlated with v. What should we do instead? What is the “spectrum” of a tensor anyway? Many algorithms approach problems like this by “flattening” the tensor into a matrix, and then looking at the spectrum of the resulting matrix. But a more natural approach is to form a tensor network with copies of the observed tensor, and contract it to obtain a scalar or vector. I’ll describe how this picture helps us understand a conjectured phase transition where this problem becomes exponentially hard, and provides algorithms that work all the way up to that transition. This is joint work with Tim Kunisky (Yale) and Alex Wein (UC Davis).
10:15 • EPF Lausanne, CM 1 120
David Beers
Title T.B.A.
10:15 • EPF Lausanne, MA B1 524
Patrick Ghanaat (UniFr)
Almost flat manifolds abstract
Abstract:
This expository talk will be centered on one of the classical “pinching”theorems of Riemannian geometry, due to Gromov in its original form:Compact connected manifolds admitting a Riemannian metric with suitably small curvature are diffeomorphic to infra-nilmanifolds.Flat Riemannian manifolds are quotients Λ\\Rn of euclidean Rn bya discrete group Λ of isometries. More generally, infra-nilmanifolds arequotients Λ\\N of a nilpotent Lie group by a discrete group Λ of isome-tries of a left invariant Riemannian metric on N .We will begin with background on nilpotent Lie groups, their metrics,isometries and quotients, then explain the history and versions of thetheorem, outline a proof following ideas of Auslander, and discuss someclosely related questions.
10:20 • Université de Fribourg, room Phys 2.52
Hajime Yoshino
Spatially heterogeneous learning in a deep neural network abstract
Abstract:
In this talk I discuss statistical mechanical properties of typical machines in an ensemble of multi-layer perceptrons of width N and depth L which are performing exactly the same learning task independently from each other [1] following the line of research started by E. Gardner in 1980s. The natural order parameters associated with the ensemble are the overlaps that measure similarity between the configurations of the machines in the hidden layers. By theoretical and numerical analysis we found that the order parameters evolve in space, along the axis perpendicular to the layers. Typically we find that the order parameters become smaller in the center of the network suggesting that the system is more constrained close to the input/output boundaries while less constrained in the center. The situation is reminiscent of the wetting transitions switched on by \'walls\' found in physical systems [2] and the spatially coupled inference problems [3]. On the theoretical side, we developed a replica method to analyze the two canonical learning scenariosused often in statistical mechanics of machine learning, namely random scenario and Bayes-optimal teacher-student scenario. We found not only the amplitude of the order parameters but also the hierarchy of replica symmetry breaking (when it happens) evolves in space. On the numerical side, we performed simulations analyzing the two learning scenarios as in the theory and also analyzed realistic learning tasks on MNIST data. Numerical simulations suggest that the dynamics (MC or SGD) eventually bring the system to thermal equilibrium characterized by space-dependent overlaps.
11:15 • EPF Lausanne, CM 1 120
Aryan Mokhtari (UT Austin)
Online Learning Guided Quasi-Newton Methods: Improved Global Non-asymptotic Guarantees abstract
Abstract:
Quasi-Newton (QN) methods are popular iterative algorithms known for their superior practical performance compared to Gradient Descent (GD)-type methods. However, the existing theoretical results for this class of algorithms do not sufficiently justify their advantage over GD-type methods. In this talk, we discuss our recent efforts to address this issue. Specifically, in the strongly convex setting, we propose the first “globally” convergent QN method that achieves an explicit “non-asymptotic superlinear” rate. We show that the rate presented for our method is provably faster than GD after at most $O(d)$ iterations, where $d$ is the problem dimension. Additionally, in the convex setting, we present an accelerated variant of our proposed method that provably outperforms the accelerated gradient method and converges at a rate of $O(\\min\\{1/k^2, \\sqrt{d \\log k }/ k^{2.5}\\})$, where $k$ is the number of iterations. To attain these results, we diverge from conventional approaches and construct our QN methods based on the Hybrid Proximal Extragradient (HPE) framework and its accelerated variants. Furthermore, a pivotal algorithmic concept underpinning our methodologies is an online learning framework for updating the Hessian approximation matrices. Specifically, we relate our method\'s convergence rate to the regret of a specific online convex optimization problem in the matrix space and choose the sequence of Hessian approximation matrices to minimize its overall regret. Bio: Aryan Mokhtari is an Assistant Professor in the Electrical and Computer Engineering (ECE) Department at the University of Texas at Austin (UT Austin), where he holds the title of Fellow of Texas Instruments/Kilby. Prior to joining UT Austin, he was a Postdoctoral Associate in the Laboratory for Information and Decision Systems (LIDS) at MIT. Before that, he worked as a Research Fellow at the Simons Institute for the program on “Bridging Continuous and Discrete Optimization.” He earned his Ph.D. in electrical and systems engineering from the University of Pennsylvania (Penn). Aryan has received the NSF CAREER Award, the Army Research Office (ARO) Early Career Program Award, Google Research Scholar Award, UT Austin ECE Department’s Junior Faculty Excellence in Teaching Award, the Simons-Berkeley Research Fellowship, and Penn’s Joseph and Rosaline Wolf Award for Best Doctoral Dissertation.
11:30 • ETH Zentrum, Building CAB, Room H 53
Mirko Mauri (Ecole Polytechnique)
An algebro-geometric version of the Poincaré conjecture abstract
Abstract:
Dual complexes are CW-complexes, encoding the combinatorial data of how the irreducible components of a simple normal crossing pair intersect. The algebraic geometry of the divisor is reflected in the topology of the dual complex. One of the most tantalizing conjectures in the field is the expectation that the dual complex of an anticanonical divisor is a sphere or a finite quotient of a sphere. Equivalently, a combinatorial Calabi-Yau variety should resemble a sphere. I will provide an overview of the current state of the conjecture, and report on joint work with Joaquin Moraga. We introduce a numerical invariant called birational complexity that, among other properties, measures the degree to which the dual complex of an anticanonical divisor is close to be a sphere.
13:15 • EPF Lausanne, CM 1 517
Dr. Homin Lee (Northwestern University)
Absolute continuity of stationary measure abstract
Abstract:
In this talk, we discuss about the smooth random dynamical systems on surfaces. Based on the measure rigidity work by Aaron Brown and Federico Rodriguez Hertz, we know that a stationary measure has SRB property unless there is a certain obstruction. Here, SRB property implies that, morally, the measure is 'absolutely continuous' along 1 dimensional piece. In this talk, we consider about a different mechanism to get \'measure rigidity\' which promote SRB to absolute continuity using \'transversality' motivated by Tsujii\'s works on partially hyperbolic endomorphisms and Bernoulli convolution. This is a work in progress with Aaron Brown, Davi Obata, and Yuping Ruan.
13:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.1
Prof. Dr. Francois Greer (Michigan State University)
Quasi-modular special cycles abstract
Abstract:
A classical theorem of Kudla and Millson states that the cohomology classes of Noether-Lefschetz loci in a moduli space of K3-type Hodge structures form the coefficients of a modular form. We investigate how well this theorem survives upon passing to a smooth compactification. As an application, we sketch a counterexample to the Severi Problem for rational surfaces.
13:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Riccardo Tione (MPI Leipzig)
Abstract:
This talk concerns critical points $u$ of polyconvex energies of the form $f(X) = g(det(X))$, where $g$ is (uniformly) convex. It is not hard to see that, if $u$ is smooth, then $\\det(Du)$ is constant. I will show that the same result holds for Lipschitz critical points $u$ in the plane. I will also discuss how to obtain rigidity for approximate solutions. This is a joint work with A. Guerra.
14:15 • Universität Basel, Spiegelgasse 5, Seminarraum 05.002
Min Jun Jo (Duke University)
Abstract:
We prove the instantaneous cusp formation from a single corner of the vortex patch solutions. This positively settles the conjecture given by Cohen-Danchin in Multiscale approximation of vortex patches, SIAM J. Appl. Math. 60 (2000), no. 2, 477-502.
15:15 • Universität Basel, Spiegelgasse 5, Seminarraum 05.002
Alexandra Kjuchukova (University of Notre Dame)
Slice and ribbon obstructions from irregular branched covers of knots abstract
Abstract:
I will give an introduction on how to obtain invariants of a knot $K$ from irregular covers of $S^3$ branched along $K$. In an extended example, I will show how examining a particular unknot in a Seifert surface for a twist knot $K_n$ can in one infinitude of cases detect that $K_n$ is not ribbon and, in another, that it is not slice. The talk will draw on old, recent and ongoing works with Cahn, Orr and Shaneson.
15:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Gilles Vilmart (Uni Geneva)
Langevin dynamics with fast perturbation driven by Stratonovich noise for enhancing probability measure sampling abstract
Abstract:
We propose a perturbation of Langevin dynamics for enhancing the sampling from probability measures in, possibly, high dimensional spaces. Precisely, by perturbing Langevin dynamics by a suitable Stratonovich noise that preserves the invariant measure of the original system, we show that accelerated convergence to equilibrium and reduced asymptotic variance can be achieved.This talk is based on joint work with A. Abdulle† and G. A. Pavliotis (Imperial College London).
16:00 • EPF Lausanne, Bernoulli Center
Stephan Schnez
Fundamental Challenges and Opportunities for the ICT-Industry abstract
Abstract:
The ICT (Information and Communication Technologies) industry faces some fundamental challenges because certain engineering guardrails which have been guiding the industry for the last decades approach physical and/or economic limits. In my presentation, I will outline these and explain why these not only lead to performance challenges but also raise concerns in terms of the sustainability of the ICT industry and its services. I will then present some approaches of how to potentially overcome these challenges and what opportunities these present - for both researchers in academia and in the industry, as well as start-ups and established companies. Stephan Schnez joined Huawei\'s Central Research Institute in 2019 where he is responsible for technology strategy and planning, based in Zurich. His core interests are related to quantum technologies and optics/photonics for novel approaches in communication and computing technologies. Before that, he spent more than seven years as a researcher and a project manager at ABB with a focus on energy storage and renewable energy systems. Stephan Schnez studied physics at the University of Heidelberg (Germany) and at the University of Manchester (UK) and received his Ph.D. in experimental physics from ETH Zurich (Switzerland) with a thesis on quantum electronics and graphene nanostructures. He has been a lecturer on "Energy and Digitalization" at the University of Freiburg(Germany) since 2017.
16:15 • Uni St. Gallen, 58-022
Dr. Leonardo Zepeda-Nunez (Google Research, USA)
Recent Advances in Probabilistic Scientific Machine learning abstract
Abstract:
The advent of generative AI has turbocharged the development of a myriad of commercial applications, and it has slowly started to permeate to scientific computing. In this talk we discussed how recasting the formulation of old and new problems within a probabilistic approach opens the door to leverage and tailor state-of-the-art generative AI tools. As such, we review recent advancements in Probabilistic SciML – including computational fluid dynamics, inverse problems, and particularly climate sciences, with an emphasis on statistical downscaling.Statistical downscaling is a crucial tool for analyzing the regional effects of climate change under different climate models: it seeks to transform low-resolution data from a (potentially biased) coarse-grained numerical scheme (which is computationally inexpensive) into high-resolution data consistent with high-fidelity models.We recast this problem in a two-stage probabilistic framework using unpaired data by combining two transformations: a debiasing step performed by an optimal transport map, followed by an upsampling step achieved through a probabilistic conditional diffusion model. Our approach characterizes conditional distribution without requiring paired data and faithfully recovers relevant physical statistics, even from biased samples.We will show that our method generates statistically correct high-resolution outputs from low-resolution ones, for different chaotic systems, including well known climate models and weather data. We show that the framework is able to upsample resolutions by 8x and 16x while accurately matching the statistics of physical quantities – even when the low-frequency content of the inputs and outputs differs. This is a crucial yet challenging requirement that existing state-of-the-art methods usually struggle with.
16:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room E 1.2
Dr. Geronimo Uribe Bravo (Universidad Nacional Autónoma de México)
A pathwise approach to time change abstract
Abstract:
Time-change equations are a generalization of ordinary differentialequations which are driven by the random, irregular, and possibly denselydiscontinuous sample paths of the typical stochastic process. They can be thought of as a multiparameter version of the methodof time-change and can be given a pathwise theory. Time-change equations can lead to deep results on weak existence anduniqueness of stochastic differential equations and posses a robuststrong approximation theory. However, time-change equations are notrestricted to Markovian or semimartingale settings. In this talk, we will go through some examples of time-changeequations which can be succesfully analyzed (such as (multidimensional) affine processes, sticky Lévy processes orDoeblin´s mostly unknown proposal for diffusion processes)as well as some open problems they suggest.
17:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43