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04.03.2024 15:00 Amit Einav (Durham University):
About chaos and order in many element systemsMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

Systems that involve many elements, be it a gas of particles or a herd of animals, are ubiquitous in our day to day lives. Their investigation, however, is hindered by the complexity of such systems and the amount of (usually coupled) equations that are needed to be solved. The late 50’s has seen the birth of the so-called mean field limit approach as an attempt to circumvent some of the difficulties arising in treating such systems. Conceived by Kac as a way to give justification to the validity of the Boltzmann equation, the mean field limit approach attempts to find the behaviour of a limiting “average” element in a many element system and relies on two ingredients: an average model of the system (i.e. an evolution equation for the probability density of the ensemble), and an asymptotic correlation relation that expresses the emerging phenomena we expect to get as the number of elements goes to infinity. Mean field limits of average models, originally applied to particle models, have permeated to fields beyond mathematical physics in recent decades. Examples include models that pertain to biological, chemical, and even societal phenomena. However, to date we use only one asymptotic correlation relation – chaos, the idea that the elements become more and more independent. While suitable when considering particles in a certain cavity, this assumption doesn’t seem reasonable in models that pertain to biological and societal phenomena. In our talk we will introduce Kac’s particle model and the notions of chaos and mean field limits. We will discuss the problem of having chaos as the sole asymptotic correlation relation and define a new asymptotic relation of order. We show that this is the right relation for a recent animal based model suggested by Carlen, Degond, and Wennberg, and highlight the importance of appropriate scaling in its investigation.

05.03.2024 16:00 Maximilian Gläser:
Sub-Exponential Lower Bounds for Branch-and-Bound with General Disjunctions via InterpolationMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

We investigate linear programming based branch-and-bound using general disjunctions, also known as stabbing planes and derive the first sub-exponential lower bound (i.e., 2^{L^\omega(1)}) in the encoding length L of the integer program for the size of a general branch-and-bound tree. In fact, this is even the first super-linear bound.

This is achieved by showing that general branch-and-bound admits quasi-feasible monotone real interpolation. Specialized to a certain infeasible integer program whose infeasibility expresses that no graph can have both a k-coloring and a (k+1)-clique, this property states that any branch-and-bound tree for this program can be turned into a monotone real circuit of similar size which distinguishes between graphs with a k-coloring and graphs with a (k+1)-clique. Thus, we can utilize known lower-bounds for such circuits.

Using Hrubeš' and Pudlák's notion of infeasibility certificates, one can show that certain random CNFs are sub-exponentially hard to refute for branch-and-bound with high probability via the same technique.

This is joint work with Marc Pfetsch.

06.03.2024 13:00 Pirmin Schlicke:
My Research At TUM. Mathematical Modeling in Medicine to Enhance Clinical Decision-MakingMI 03.04.011 (Boltzmannstr. 3, 85748 Garching)

During my six-year journey as a doctoral and postdoctoral researcher, I have explored how mathematical modeling can help clinicians to understand and fight diseases, particularly cancer. In this talk, I share the main projects I have worked on. The central work of my research was to formalize how cancer can metastasize. By creating simple mathematical models that are calibrated with routine patient data from early clinical care phases, we have identified computational biomarkers for timely and accurate assessment of metastatic progression. In addition, I have worked on predictive tools for renal diseases to forecast time points of disease stages as well as to predict immunotherapy outcomes in lung cancer patients. As I am getting ready to move to the University of Texas MD Anderson Cancer Center as a postdoctoral research fellow, I will briefly conclude the talk with a preview of what I will be working on in the future.

13.03.2024 12:30 Bryon Aragam (University of Chicago):
Optimal structure learning in structural equation modelsBC1 2.01.10 / 8101.02.110 (Parkring 11, 85748 Garching)

We study the optimal sample complexity of structure learning in Gaussian structural equation models. In the first part of the talk, we compare the complexity of structure learning via the PC algorithm and distribution learning via the Chow-Liu algorithm in directed polytrees. We will show how both algorithms are optimal under different assumptions, and lead to different statistical complexities. Moving beyond polytrees, we then investigate the problem of neighbourhood selection, which is an important primitive when learning the overall structure of a graphical model. We will introduce a new estimator, called klBSS, and compare its performance to best subset selection (BSS). We show by example that—even when the structure is unknown—the existence of underlying structure can reduce the sample complexity of neighbourhood selection compared to classical methods such as BSS and the Lasso.

19.03.2024 14:00 Mikael Vejdemo-Johansson (CUNY College of Staten Island):
Highly Symmetric Point CloudsMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

Point clouds with high degrees of symmetry have additional structure that may allow us to significantly speed up topological computations. In this talk I will report on work in progress on studying the persistent (co)homology pipeline on point clouds with a known (and large) group acting on them. I will describe the setup, the amount of information about the group action needed for improvements, a (partially) failed conjecture, and a 10x speedup in generating and traversing 60k simplices in the Vietoris-Rips complex of the 4-dimensional hypercube.

20.03.2024 12:15 Yichen Zhu (Università Bocconi, Milano):
Posterior Contraction Rates for Vecchia Approximations of Gaussian ProcessesBC1 2.01.10 / 8101.02.110 (Parkring 11, 85748 Garching)

Gaussian Processes (GP) are widely used to model spatial dependency in geostatistical data, yet the exact Bayesian inference has an intractable time complexity of $O(n^3)$. Vecchia approximation has become a popular solution to this computational issue, where spatial dependency is characterized by a sparse directed acyclic graph (DAG) that allows scalable Bayesian inference. Despite the popularity in practice, little is understood about its theoretical properties. In this paper, we systematically study the posterior contraction rates of Vecchia approximations of GP. Under minimal regularity conditions, we prove that by appropriate selection of the underlying DAG, the Vecchia approximated GP possess the same posterior contraction rates as the mother GP. Therefore, by optimal choices of the tunning hyper-parameters, the Vecchia approximation can achieve the minimax contraction rate, providing strong frequentist guarantees to the procedure. Our theoretical findings are demonstrated numerically as well using synthetic and real world data sets.

25.03.2024 15:00 Gautham Govind:
When can we Approximate Wide Contrastive Models with Neural Tangent Kernels and Principal Component Analysis?MI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

Contrastive learning is a paradigm for learning representations from unlabelled data that has been highly successful for image and text data. Several recent works have examined contrastive losses to claim that contrastive models effectively learn spectral embeddings, while few works show relations between (wide) contrastive models and kernel principal component analysis (PCA). However, it is not known if trained contrastive models indeed correspond to kernel methods or PCA. In this work, we analyze the training dynamics of two-layer contrastive models, with non-linear activation, and answer when these models are close to PCA or kernel methods. It is well known in the supervised setting that neural networks are equivalent to neural tangent kernel (NTK) machines, and that the NTK of infinitely wide networks remains constant during training. We provide the first convergence results of NTK for contrastive losses, and present a nuanced picture: NTK of wide networks remains almost constant for cosine similarity based contrastive losses, but not for losses based on dot product similarity. We further study the training dynamics of contrastive models with orthogonality constraints on the output layer, which is implicitly assumed in works relating contrastive learning to spectral embedding. Our deviation bounds suggest that representations learned by contrastive models are close to the principal components of a certain matrix computed from random features. We empirically show that our theoretical results possibly hold beyond two-layer networks.

26.03.2024 13:00 Tobias Boege (KTH Royal Institute of Technology, Stockholm):
Colored Gaussian DAG modelsBC1 2.01.10 / 8101.02.110 (Parkring 11, 85748 Garching)

Colored Gaussian DAG models generalize linear structural equation models by allowing additional equalities to be specified among the error variances and regression coefficients. We show that these models are smooth manifolds and give a characterization of their vanishing ideals up to a saturation. We also initiate the study of faithfulness and structural identifiability. Our results are facilitated by an in-depth analysis of parameter identification maps for ordinary Gaussian DAG models and our techniques carry over easily to other classes of rationally identifiable statistical models. This is joint work with Kaie Kubjas, Pratik Misra and Liam Solus.