Probability Seminar: Difference between revisions

From UW-Math Wiki
Jump to navigation Jump to search
VisualWikitext
No edit summary
 
(236 intermediate revisions by 10 users not shown)
Line 1: Line 1:
__NOTOC__
__NOTOC__
[[Probability | Back to Probability Group]]


= Fall 2020 =
* '''When''': Thursdays at 2:30 pm
* '''Where''': 901 Van Vleck Hall
* '''Organizers''': Hanbaek Lyu, Tatyana Shcherbyna, David Clancy
* '''To join the probability seminar mailing list:''' email probsem+subscribe@g-groups.wisc.edu.
* '''To subscribe seminar lunch announcements:''' email lunchwithprobsemspeaker+subscribe@g-groups.wisc.edu


<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted.
[[Past Seminars]]
<b>We  usually end for questions at 3:20 PM.</b>
 
<b> IMPORTANT: </b> In Fall 2020 the seminar is being run online. [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK]


If you would like to sign up for the email list to receive seminar announcements then please join [https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem our group].
== September 17, 2020, [https://www.math.tamu.edu/~bhanin/ Boris Hanin] (Princeton and Texas A&M) ==


'''Pre-Talk: (1:00pm)'''


'''Neural Networks for Probabilists''' 
= Spring 2025 =
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b>


Deep neural networks are a centerpiece in modern machine learning. They are also fascinating probabilistic models, about which much remains unclear. In this pre-talk I will define neural networks, explain how they are used in practice, and give a survey of the big theoretical questions they have raised. If time permits, I will also explain how neural networks are related to a variety of classical areas in probability and mathematical physics, including random matrix theory, optimal transport, and combinatorics of hyperplane arrangements.
We usually end for questions at 3:20 PM.


'''Talk: (2:30pm)'''
== January 23, 2025: ==
No seminar 


'''Effective Theory of Deep Neural Networks'''  
== January 30, 2025: Promit Ghosal (UChicago) ==
'''Bridging Theory and Practice in Stein Variational Gradient Descent: Gaussian Approximations, Finite-Particle Rates, and Beyond'''


Deep neural networks are often considered to be complicated "black boxes," for which a full systematic analysis is not only out of reach but also impossible. In this talk, which is based on ongoing joint work with Sho Yaida and Daniel Adam Roberts, I will make the opposite claim. Namely, that deep neural networks with random weights and biases are exactly solvable models. Our approach applies to networks at finite width n and large depth L, the regime in which they are used in practice. A key point will be the emergence of a notion of "criticality," which involves a finetuning of model parameters (weight and bias variances). At criticality, neural networks are particularly well-behaved but still exhibit a tension between large values for n and L, with large values of n tending to make neural networks more like Gaussian processes and large values of L amplifying higher cumulants. Our analysis at initialization has many consequences also for networks during after training, which I will discuss if time permits.
Stein Variational Gradient Descent (SVGD) has emerged as a powerful interacting particle-based algorithm for nonparametric sampling, yet its theoretical properties remain challenging to unravel. This talk delves into two complementary perspectives about SVGD. First, we explore Gaussian-SVGD, a framework that projects SVGD onto the family of Gaussian distributions via a bilinear kernel. We establish rigorous convergence results for both mean-field dynamics and finite-particle systems, demonstrating linear convergence to equilibrium in strongly log-concave settings and unifying recent algorithms for Gaussian variational inference (GVI) under a single framework. Second, we analyze the finite-particle convergence rates of SVGD in Kernelized Stein Discrepancy (KSD) and Wasserstein-2 metrics. Leveraging a novel decomposition of the relative entropy time derivative, we achieve near-optimal rates with polynomial dimensional dependence and extend these results to bilinear-enhanced kernels.  


== September 24, 2020, [https://people.ucd.ie/neil.oconnell Neil O'Connell] (Dublin) ==
== February 6, 2025: Subhabrata Sen (Harvard) ==
'''Community detection on multi-view networks''' 


'''Some new perspectives on moments of random matrices'''
The community detection problem seeks to recover a latent clustering of vertices from an observed random graph. This problem has attracted significant attention across probability, statistics and computer science, and the fundamental thresholds for community recovery have been characterized in the last decade. Modern applications typically collect more fine-grained information on the units under study. For example, one might measure relations of multiple types among the units, or observe an evolving network over time. In this talk, we will discuss the community detection problem on such ‘multi-view’ networks. We will present some new results on the fundamental thresholds for community detection in these models. Finally, we will introduce algorithms for community detection based on Approximate Message Passing. 


The study of `moments' of random matrices (expectations of traces of powers of the matrix) is a rich and interesting subject, with fascinating connections to enumerative geometry, as discovered by Harer and Zagier in the 1980’s. I will give some background on this and then describe some recent work which offers some new perspectives (and new results). This talk is based on joint work with Fabio Deelan Cunden, Francesco Mezzadri and Nick Simm.
This is based on joint work with Xiaodong Yang and Buyu Lin (Harvard University).


== October 1, 2020, [https://marcusmichelen.org/ Marcus Michelen] (UIC) ==
== February 13, 2025: Hanbaek Lyu (UW-Madison) ==
'''Large random matrices with given margins''' 


'''Roots of random polynomials near the unit circle'''
We study large random matrices with i.i.d. entries conditioned to have prescribed row and column sums (margin). This problem has rich connections to relative entropy minimization,  Schr\"{o}dinger bridge, the enumeration of contingency tables, and random graphs with given degree sequences. We show that such a margin-constrained random matrix is sharply concentrated around a certain deterministic matrix, which we call the ''typical table''. Typical tables have dual characterizations: (1) the expectation of the random matrix ensemble with minimum relative entropy from the base model constrained to have the expected target margin, and (2) the expectation of the maximum likelihood model obtained by rank-one exponential tilting of the base model. The structure of the typical table is dictated by two potential functions, which give the maximum likelihood estimates of the tilting parameters. Based on these results, for a sequence of "tame" margins that converges in $L^{1}$ to a limiting continuum margin as the size of the matrix diverges, we show that the sequence of margin-constrained random matrices converges in cut norm to a limiting kernel, which is the $L^{2}$-limit of the corresponding rescaled typical tables. The rate of convergence is controlled by how fast the margins converge in $L^{1}$.  We also propose a generalized Sinkhorn algorithm for computing typical tables and establish its linear convergence. We derive several new results for random contingency tables from our general framework. 


It is a well-known (but perhaps surprising) fact that a polynomial with independent random coefficients has most of its roots very close to the unit circle.  Using a probabilistic perspective, we understand the behavior of roots of random polynomials exceptionally close to the unit circle and prove several limit theorems; these results resolve several conjectures of Shepp and Vanderbei.  We will also discuss how our techniques provide a heuristic, probabilistic explanation for why random polynomials tend to have most roots near the unit circle.  Based on joint work with Julian Sahasrabudhe.
Based on a joint work with Sumit Mukherjee (Columbia) 


== October 8, 2020, [http://sites.harvard.edu/~sus977/index.html Subhabrata Sen] (Harvard) ==
== February 20, 2025: Mustafa Alper Gunes (Princeton) ==
'''Characteristic Polynomials of Random Matrices, Exchangeable Arrays & Painlevé Equations''' 


'''Large deviations for dense random graphs: beyond mean-field'''
Joint moments of characteristic polynomials of unitary random matrices and their derivatives have gained attention over the last 25 years, partly due to their conjectured relation to the Riemann zeta function. In this talk, we will consider the asymptotics of these moments in the most general setting allowing for derivatives of arbitrary order, generalising previous work that considered only the first derivative. Along the way, we will examine how exchangeable arrays and integrable systems play a crucial role in understanding the statistics of a class of infinite Hermitian random matrices. Based on joint work with Assiotis, Keating and Wei.


In a seminal paper, Chatterjee and Varadhan derived an Erdős-Rényi random graph, viewed as a random graphon. This directly provides LDPs for continuous functionals such as subgraph counts, spectral norms, etc. In contrast, very little is understood about this problem if the underlying random graph is inhomogeneous or constrained.
== February 27, 2025: Souvik Dhara (Purdue) ==
 
In this talk, we will explore large deviations for dense random graphs, beyond the “mean-field” setting. In particular, we will study large deviations for uniform random graphs with given degrees, and a family of dense block model
'''Propagation of Shocks on Networks: Can Local Information Predict Survival?'''  
random graphs. We will establish the LDP in each case, and identify the rate function. In the block model setting, we will use this LDP to study the upper tail problem for homomorphism densities of regular sub-graphs. Our results establish that this problem exhibits a symmetry/symmetry-breaking transition, similar to one observed for Erdős-Rényi random graphs.
 
Based on joint works with Christian Borgs, Jennifer Chayes, Souvik Dhara, Julia Gaudio and Samantha Petti.
 
== October 15, 2020, [https://math.cornell.edu/philippe-sosoe Philippe Sosoe] (Cornell) ==
 
Title: '''Concentration in integrable polymer models'''
 
I will discuss a general method, applicable to all known integrable stationary polymer models, to obtain nearly optimal bounds on the
central moments of the partition function and the occupation lengths for each level of the polymer system. The method was developed
for the O'Connell-Yor polymer, but was subsequently extended to discrete integrable polymers. As an application, we obtain
localization of the OY polymer paths along a straight line on the scale O(n^{2/3+o(1)}).
 
Joint work with Christian Noack.
 
==October 22, 2020, [http://www.math.toronto.edu/balint/ Balint Virag] (Toronto) ==
 
Title: '''The heat and the landscape'''
 
Abstract: The directed landscape is the conjectured universal scaling limit of the
most common random planar metrics. Examples are planar first passage
percolation, directed last passage percolation, distances in percolation
clusters, random polymer models, and exclusion processes. The limit laws of distances of objects are given by the KPZ fixed point.
 
We show that the KPZ fixed point is characterized by the Baik Ben-Arous
Peche statistics well-known from random matrix theory.
 
This provides a general and elementary method for showing convergence to
the KPZ fixed point. We apply this method to two models related to
random heat flow: the O'Connell-Yor polymer and the KPZ equation.
 
Note: there will be a follow-up talk with details about the proofs at 11am, Friday, October 23.


==October 29, 2020, [https://www.math.wisc.edu/node/80 Yun Li] (UW-Madison) ==
Abstract: Complex systems are often fragile, where minor disruptions can cascade into dramatic collapses. Epidemics serve as a prime example of this phenomenon, while the 2008 financial crisis highlights how a domino effect, originating from the small subprime mortgage sector, can trigger global repercussions. The mathematical theory underlying these phenomena is both elegant and foundational, profoundly shaping the field of Network Science since its inception. In this talk, I will present a unifying mathematical model for network fragility and cascading dynamics, and explore its deep connections to the theory of local-weak convergence, pioneered by Benjamini-Schramm and Aldous-Steele.


Title: '''Operator level hard-to-soft transition for β-ensembles'''
== March 6, 2025: Alexander Meehan (UW-Madison, Department of Philosophy) ==
'''What conditional probability could (probably) be'''  


Abstract: It was shown that the soft and hard edge scaling limits of β-ensembles can be characterized as the spectra of certain random Sturm-Liouville operators. By tuning the parameter of the hard edge process one can obtain the soft edge process as a scaling limit. In this talk, I will present the corresponding limit on the level of the operators. This talk is based on joint work with Laure Dumaz and Benedek Valkó.
According to orthodox probability theory, when B has probability zero, the conditional probability of A given B can depend on the partition or sub-sigma-field that B is relativized to. This relativization to sub-sigma-fields, a hallmark of Kolmogorov's theory of conditional expectation, is traditionally seen as appropriate in a treatment of conditioning with continuous variables, and it is what allows the theory to preserve Total Disintegrability, a generalization of the Law of Total Probability to uncountable partitions. In this talk, I will argue that although the relativization of conditional probability to sub-sigma-fields has advantages, it also has an underrecognized cost: it leads to puzzles for the treatment of ''iterated conditioning''. I will discuss these puzzles and some possible implications for the foundations of conditional probability.


== November 5, 2020, [http://sayan.web.unc.edu/ Sayan Banerjee] (UNC at Chapel Hill) ==
This talk is based on joint work with Snow Zhang (UC Berkeley).


Title: '''Persistence and root detection algorithms in growing networks'''
== March 13, 2025: Klara Courteaut (Courant) ==
TBD 


Abstract: Motivated by questions in Network Archaeology, we investigate statistics of dynamic networks
== March 20, 2025: Ewain Gwynne (UChicago) ==
that are ''persistent'', that is, they fixate almost surely after some random time as the network grows. We
TBD 
consider ''generalized attachment models'' of network growth where at each time $n$, an incoming vertex
attaches itself to the network through $m_n$ edges attached one-by-one to existing vertices with probability
proportional to an ''arbitrary function'' $f$ of their degree. We identify the class of attachment functions $f$ for
which the ''maximal degree vertex'' persists and obtain asymptotics for its index when it does not. We also
show that for tree networks, the ''centroid'' of the tree persists and use it to device polynomial time root
finding algorithms and quantify their efficacy. Our methods rely on an interplay between dynamic
random networks and their continuous time embeddings.


This is joint work with Shankar Bhamidi.
== March 27, 2025: SPRING BREAK ==
No seminar 


== November 12, 2020, [https://cims.nyu.edu/~ajd594/ Alexander Dunlap] (NYU Courant Institute) ==
== April 3, 2025: Jimme He (OSU) ==
TBD 


Title: '''A forward-backward SDE from the 2D nonlinear stochastic heat equation'''
== April 10, 2025: Evan Sorensen (Columbia) ==
TBD 


Abstract: I will discuss a two-dimensional stochastic heat equation in the weak noise regime with a nonlinear noise strength. I will explain how pointwise statistics of solutions to this equation, as the correlation length of the noise is taken to 0 but the noise is attenuated by a logarithmic factor, can be related to a forward-backward stochastic differential equation (FBSDE) depending on the nonlinearity. In the linear case, the FBSDE can be explicitly solved and we recover results of Caravenna, Sun, and Zygouras. Joint work with Yu Gu (CMU).
== April 17, 2025: ==
TBD 


== November 19, 2020, [https://statistics.wharton.upenn.edu/profile/dingjian/ Jian Ding] (University of Pennsylvania) ==
== April 24, 2025: William Leep (University of Minnesota, Twin Cities) ==
TBD 


Title: '''Correlation length of two-dimensional random field Ising model via greedy lattice animal'''
== May 1, 2025: ==
 
No seminar
Abstract: In this talk, I will discuss two-dimensional random field Ising model where the disorder is given by i.i.d. mean zero Gaussian variables with small variance. In particular, I will present a recent joint work with Mateo Wirth on (one notion of) the correlation length, which is the critical size of the box at which the influences to spin magnetization from the boundary conditions and from the random field are comparable. Our work draws a connection to the greedy lattice animal normalized by the boundary size.
 
== December 3, 2020, [https://www.math.wisc.edu/people/faculty-directory Tatyana Shcherbina] (UW-Madison) ==
 
Title: '''SUSY transfer matrix approach for the real symmetric 1d random band matrices '''
 
Abstract: Random band matrices (RBM) are natural intermediate models to study
eigenvalue statistics and quantum propagation in disordered systems,
since they interpolate between mean-field type Wigner matrices and
random Schrodinger operators. In particular, RBM can be used to model the
Anderson metal-insulator phase transition. The conjecture states that the eigenvectors
of $N\times N$ RBM are completely delocalized and the local spectral statistics governed 
by  the Wigner-Dyson statistics for large bandwidth $W$ (i.e. the local behavior is
the same as for Wigner matrices), and by Poisson statistics for a small $W$
(with exponentially localized eigenvectors). The transition is conjectured to
be sharp and for RBM in  one  spatial  dimension  occurs around  the  critical 
value $W=\sqrt{N}$. Recently, we proved the universality of the correlation
functions for the whole delocalized region $W\gg \sqrt{N}$ for a certain type
of Hermitian Gaussian RBM. This result was obtained by
application of the supersymmetric method (SUSY) combined with the transfer matrix approach.
In this talk I am going to discuss how this techniques can be adapted to the
real symmetric case.
 
== December 10, 2020, [https://www.ewbates.com/ Erik Bates] (UW-Madison) ==
 
Title: '''TBA'''
 
Abstract: TBA
 
 
[[Past Seminars]]

Latest revision as of 19:17, 5 February 2025

Back to Probability Group

  • When: Thursdays at 2:30 pm
  • Where: 901 Van Vleck Hall
  • Organizers: Hanbaek Lyu, Tatyana Shcherbyna, David Clancy
  • To join the probability seminar mailing list: email probsem+subscribe@g-groups.wisc.edu.
  • To subscribe seminar lunch announcements: email lunchwithprobsemspeaker+subscribe@g-groups.wisc.edu

Past Seminars


Spring 2025

Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom

We usually end for questions at 3:20 PM.

January 23, 2025:

No seminar

January 30, 2025: Promit Ghosal (UChicago)

Bridging Theory and Practice in Stein Variational Gradient Descent: Gaussian Approximations, Finite-Particle Rates, and Beyond

Stein Variational Gradient Descent (SVGD) has emerged as a powerful interacting particle-based algorithm for nonparametric sampling, yet its theoretical properties remain challenging to unravel. This talk delves into two complementary perspectives about SVGD. First, we explore Gaussian-SVGD, a framework that projects SVGD onto the family of Gaussian distributions via a bilinear kernel. We establish rigorous convergence results for both mean-field dynamics and finite-particle systems, demonstrating linear convergence to equilibrium in strongly log-concave settings and unifying recent algorithms for Gaussian variational inference (GVI) under a single framework. Second, we analyze the finite-particle convergence rates of SVGD in Kernelized Stein Discrepancy (KSD) and Wasserstein-2 metrics. Leveraging a novel decomposition of the relative entropy time derivative, we achieve near-optimal rates with polynomial dimensional dependence and extend these results to bilinear-enhanced kernels.

February 6, 2025: Subhabrata Sen (Harvard)

Community detection on multi-view networks

The community detection problem seeks to recover a latent clustering of vertices from an observed random graph. This problem has attracted significant attention across probability, statistics and computer science, and the fundamental thresholds for community recovery have been characterized in the last decade. Modern applications typically collect more fine-grained information on the units under study. For example, one might measure relations of multiple types among the units, or observe an evolving network over time. In this talk, we will discuss the community detection problem on such ‘multi-view’ networks. We will present some new results on the fundamental thresholds for community detection in these models. Finally, we will introduce algorithms for community detection based on Approximate Message Passing.

This is based on joint work with Xiaodong Yang and Buyu Lin (Harvard University).

February 13, 2025: Hanbaek Lyu (UW-Madison)

Large random matrices with given margins

We study large random matrices with i.i.d. entries conditioned to have prescribed row and column sums (margin). This problem has rich connections to relative entropy minimization,  Schr\"{o}dinger bridge, the enumeration of contingency tables, and random graphs with given degree sequences. We show that such a margin-constrained random matrix is sharply concentrated around a certain deterministic matrix, which we call the typical table. Typical tables have dual characterizations: (1) the expectation of the random matrix ensemble with minimum relative entropy from the base model constrained to have the expected target margin, and (2) the expectation of the maximum likelihood model obtained by rank-one exponential tilting of the base model. The structure of the typical table is dictated by two potential functions, which give the maximum likelihood estimates of the tilting parameters. Based on these results, for a sequence of "tame" margins that converges in $L^{1}$ to a limiting continuum margin as the size of the matrix diverges, we show that the sequence of margin-constrained random matrices converges in cut norm to a limiting kernel, which is the $L^{2}$-limit of the corresponding rescaled typical tables. The rate of convergence is controlled by how fast the margins converge in $L^{1}$.  We also propose a generalized Sinkhorn algorithm for computing typical tables and establish its linear convergence. We derive several new results for random contingency tables from our general framework.

Based on a joint work with Sumit Mukherjee (Columbia)

February 20, 2025: Mustafa Alper Gunes (Princeton)

Characteristic Polynomials of Random Matrices, Exchangeable Arrays & Painlevé Equations

Joint moments of characteristic polynomials of unitary random matrices and their derivatives have gained attention over the last 25 years, partly due to their conjectured relation to the Riemann zeta function. In this talk, we will consider the asymptotics of these moments in the most general setting allowing for derivatives of arbitrary order, generalising previous work that considered only the first derivative. Along the way, we will examine how exchangeable arrays and integrable systems play a crucial role in understanding the statistics of a class of infinite Hermitian random matrices. Based on joint work with Assiotis, Keating and Wei.

February 27, 2025: Souvik Dhara (Purdue)

Propagation of Shocks on Networks: Can Local Information Predict Survival?

Abstract: Complex systems are often fragile, where minor disruptions can cascade into dramatic collapses. Epidemics serve as a prime example of this phenomenon, while the 2008 financial crisis highlights how a domino effect, originating from the small subprime mortgage sector, can trigger global repercussions. The mathematical theory underlying these phenomena is both elegant and foundational, profoundly shaping the field of Network Science since its inception. In this talk, I will present a unifying mathematical model for network fragility and cascading dynamics, and explore its deep connections to the theory of local-weak convergence, pioneered by Benjamini-Schramm and Aldous-Steele.

March 6, 2025: Alexander Meehan (UW-Madison, Department of Philosophy)

What conditional probability could (probably) be

According to orthodox probability theory, when B has probability zero, the conditional probability of A given B can depend on the partition or sub-sigma-field that B is relativized to. This relativization to sub-sigma-fields, a hallmark of Kolmogorov's theory of conditional expectation, is traditionally seen as appropriate in a treatment of conditioning with continuous variables, and it is what allows the theory to preserve Total Disintegrability, a generalization of the Law of Total Probability to uncountable partitions. In this talk, I will argue that although the relativization of conditional probability to sub-sigma-fields has advantages, it also has an underrecognized cost: it leads to puzzles for the treatment of iterated conditioning. I will discuss these puzzles and some possible implications for the foundations of conditional probability.

This talk is based on joint work with Snow Zhang (UC Berkeley).

March 13, 2025: Klara Courteaut (Courant)

TBD

March 20, 2025: Ewain Gwynne (UChicago)

TBD

March 27, 2025: SPRING BREAK

No seminar

April 3, 2025: Jimme He (OSU)

TBD

April 10, 2025: Evan Sorensen (Columbia)

TBD

April 17, 2025:

TBD

April 24, 2025: William Leep (University of Minnesota, Twin Cities)

TBD

May 1, 2025:

No seminar

Retrieved from "https://wiki.math.wisc.edu/index.php?title=Probability_Seminar&oldid=28061"