06.02.2025 16:30 Shahar Mendelson (Australian National University): Structure recovery from a geometric and probabilistic perspective
Structure recovery is at the heart of modern Data Science. Roughly put, the goal in structure recovery problems is to identify (or at least approximate) an unknown object using limited, random information – e.g., a random sample of the unknown object.
As it happens, key questions on recovery are fundamental (open) problems in Asymptotic Geometric Analysis and High Dimensional Probability. In this talk I will give one example (out of many) that exhibits the rather surprising ties between those seemingly unrelated areas.
I will explain why noise-free recovery is dictated by the geometry of natural random sets: for a class of functions 𝐹 and n i.i.d random variables 𝜎 = (𝑋_1,…,𝑋_n), the random sets are 𝑃_𝜎 (𝐹) = { (𝑓(𝑋_1),….,𝑓(𝑋_n)) : 𝑓 ∈ 𝐹 }.
I will outline a (sharp) estimate on the structure of a typical 𝑃_𝜎 (𝐹) that leads to the solution of the noise-free recovery problem under minimal assumptions. I will explain why the same estimate resolves various questions in high dimensional probability (e.g., the smallest singular values of certain random matrices) and high dimensional geometry (e.g., the Gelfand width of a convex body).
The optimality of the solution is implied by a exposing a “hidden extremal structure” contained in 𝑃_𝜎 (𝐹), which in turn is based on a complete answer to Talagrand’s celebrated entropy problem.
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Invited by Prof. Holger Rauhut.
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10.02.2025 15:00 Theresa Lange: On convex integration solutions to the surface quasi-geostrophic equation with generic additive noise
This talk shall be concerned with the surface quasi-geostrophic equation driven by a generic additive noise process W. By means of convex integration techniques, we establish existence of weak solutions whenever the stochastic convolution z associated with W is well defined and fulfils certain regularity constraints. Quintessentially, we show that the so constructed solutions to the non-linear equation are controlled by z in a linear fashion. This allows us to deduce further properties of the so constructed solutions, without relying on structural probabilistic properties such as Gaussianity, Markovianity or a martingale property of the underlying noise W. This is joint work with Florian Bechtold (University of Bielefeld) and Jörn Wichmann (Monash University) (cf. [1]).
This activity receives partial funding from the European Research Council (ERC) under the EU-HORIZON EUROPE ERC-2021-ADG research and innovation programme (project „Noise in Fluids“, grant agreement no. 101053472).
[1] F. Bechtold, T. Lange, J. Wichmann, "On convex integration solutions to the surface quasi-geostrophic equation driven by generic additive noise", Electronic Journal of Probability 29 (2024): 1-38
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10.02.2025 16:30 Daniel Sharon (Technion, Haifa, Israel): The Cluster Cluster Model
We consider a stochastic process on the graph $\mathds{Z}^d$.
Each $x\in \mathds{Z}^d$ starts with a cluster of size 1 with probability $p \in (0,1]$ independently.
Each cluster $C$ of performs a continuous time SRW with rate $\abs{C}^{-\alpha}$.
If it attempts to move to a vertex occupied by another cluster, it does not move, and instead the two clusters connect via a new edge.
Focusing on dimension $d=1$, we show that for $\alpha>-2$, at time $t$, the cluster size is of order $t^\frac{1}{\alpha + 2}$, and for $\alpha \le -2$ we get an infinite component.
Additionally, for $\alpha = 0$ we show convergence in distribution of the scaling limit.
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17.02.2025 15:00 Dennis Chemnitz: Dynamic Stability in Stochastic Gradient Descent
Most modern machine learning applications are based on overparameterized neural networks trained by variants of stochastic gradient descent. To explain the performance of these networks from a theoretical perspective (in particular the so-called "implicit bias"), it is necessary to understand the random dynamics of the optimization algorithms. Mathematically this amounts to the study of random dynamical systems with manifolds of equilibria. In this talk, I will give a brief introduction to machine learning theory and explain how the Lyapunov exponents of random matrix products can be used to characterize the set of possible limit points for stochastic gradient descent. The talk is based on joint work with Maxmilian Engel.
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19.02.2025 12:15 Jane Coons (Max Planck Institute of Molecular Cell Biology and Genetics, Dresden): Iterative Proportional Scaling and Log-Linear Models with Rational Maximum Likelihood Estimator
In the field of algebraic statistics, we view statistical models as part of an algebraic variety and use tools from algebra, geometry, and combinatorics to learn statistically relevant information about these models. In this talk, we discuss the algebraic interpretation of likelihood inference for discrete statistical models. We present recent work on the iterative proportional scaling (IPS) algorithm, which is used to compute the maximum likelihood estimate (MLE), and give algebraic conditions under which this algorithm outputs the exact MLE in one cycle. Next, we introduce quasi-independence models, which describe the joint distribution of two random variables where some combinations of their states cannot co-occur, but they are otherwise independent. We combinatorially classify the quasi-independence models whose MLEs are rational functions of the data. We show that each of these has a parametrization which satisfies the conditions that guarantee one-cycle convergence of the IPS algorithm.
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