• Speaker: Sergey Shadrin, Korteweg-de Vries Instituut
• Title: Non-commucative $M_{0,1+n}$, https://msu.zoom.us/j/94925518997
• Date: 05/14/2020
• Time: 9:30 AM - 10:30 AM
• Place: C204A Wells Hall

There is a huge system of parallels between various geometric and topological properties of the moduli space of curves of genus $0$ with marked points $M_{0,1+n}$ and the so-called brick manifolds $B_n$, defined a few years ago by Laura Escobar. In fact, it appears that all interesting properties relevant for the topological field theory (considered as general phenomena surrounding $M_{0,1+n}$) find their non-commutative analogs in $B_n$: -- stratification, description of cohomology (that implies the WDVV equations), psi-classes, structure of topological operad, cohomological field theories, Givental group action, surrounding topological operads (little disks, framed little discs, gravity), algebraic structures implied by their representations (hypercommutative algebra, also known in the literature as flat F-manifold, BV/Gerstenhaber algebras, etc.), and so on. (To this end, my favourite alternative title for this talk would be "What is a non-commutative hypercommutative algebra?" ) In addition the brick manifolds have very rich geometric structure: they are toric varieties of Loday's realisations of associahedra and have interpretation as wonderful models of De Concini-Procesi. In the talk, we'll try to show some of the properties mentioned above, step-by-step constructing them both on the side of moduli spaces of curves and on the side of brick manifolds. (along the lines of my joint work with Volodya Dotsenko and Bruno Vallette).

• Speaker: Daniel Spielman, Yale
• Title: Balancing covariates in randomized experiments using the Gram–Schmidt walk
• Date: 05/21/2020
• Time: 2:30 PM - 3:30 PM
• Place: 

(Part of the One World MINDS series: https://sites.google.com/view/minds-seminar/home) In randomized experiments, such as a medical trials, we randomly assign the treatment, such as a drug or a placebo, that each experimental subject receives. Randomization can help us accurately estimate the difference in treatment effects with high probability. We also know that we want the two groups to be similar: ideally the two groups would be similar in every statistic we can measure beforehand. Recent advances in algorithmic discrepancy theory allow us to divide subjects into groups with similar statistics. By exploiting the recent Gram-Schmidt Walk algorithm of Bansal, Dadush, Garg, and Lovett, we can obtain random assignments of low discrepancy. These allow us to obtain more accurate estimates of treatment effects when the information we measure about the subjects is predictive, while also bounding the worst-case behavior when it is not. We will explain the experimental design problem we address, the Gram-Schmidt walk algorithm, and the major ideas behind our analyses. This is joint work with Chris Harshaw, Fredrik Sävje, and Peng Zhang. Paper: https://arxiv.org/abs/1911.03071 Code: https://github.com/crharshaw/GSWDesign.jl

• Speaker:  Ronald Coifman, Yale
• Title: The Analytic Geometries of Data; zoom link @ https://sites.google.com/view/minds-seminar/home
• Date: 05/28/2020
• Time: 2:30 PM - 2:30 PM
• Place: 

(Part of the One World MINDS series: https://sites.google.com/view/minds-seminar/home) \[\] We will describe methodologies to build data geometries designed to simultaneously analyze and process data bases. The different geometries or affinity metrics arise naturally as we learn to contextualize and conceptualize. I.e; relate data regions, and data features (which we extend to data tensors). Moreover we generate tensorial multiscale structures. We will indicate connection to analysis by deep nets and describe applications to modeling observations of dynamical systems, from stochastic molecular dynamics to calcium imaging of brain activity.

• Speaker: Ben Adcock, Simon Fraser University
• Title: The troublesome kernel: instabilities in deep learning for inverse problems; zoom link @ https://sites.google.com/view/minds-seminar/home
• Date: 06/04/2020
• Time: 2:30 PM - 3:30 PM
• Place: 

(Part of One World MINDS seminar: https://sites.google.com/view/minds-seminar/home) \[\] Due to their stunning success in traditional machine learning applications such as classification, techniques based on deep learning have recently begun to be actively investigated for problems in computational science and engineering. One of the key areas at the forefront of this trend is inverse problems, and specifically, inverse problems in imaging. The last few years have witnessed the emergence of many neural network-based algorithms for important imaging modalities such as MRI and X-ray CT. These claim to achieve competitive, and sometimes even superior, performance to current state-of-the-art techniques. However, there is a problem. Techniques based on deep learning are typically unstable. For example, small perturbations in the data can lead to a myriad of artefacts in the recovered images. Such artifacts can be hard to dismiss as obviously unphysical, meaning that this phenomenon has potentially serious consequences for the safe deployment of deep learning in practice. In this talk, I will first showcase the instability phenomenon empirically in a range of examples. I will then focus on its mathematical underpinnings, the consequences of these insights when it comes to potential remedies, and the future possibilities for computing genuinely stable neural networks for inverse problems in imaging. This is joint work with Vegard Antun, Nina M. Gottschling, Anders C. Hansen, Clarice Poon, and Francesco Renna Papers: https://www.pnas.org/content/early/2020/05/08/1907377117 https://arxiv.org/abs/2001.01258

• Speaker: Jelani Nelson, UC Berkeley
• Title: Optimal terminal dimensionality reduction in Euclidean space; zoom link @ https://sites.google.com/view/minds-seminar/home
• Date: 06/11/2020
• Time: 2:30 PM - 3:30 PM
• Place: 

(Part of One World MINDS seminar: https://sites.google.com/view/minds-seminar/home) \[ \] The Johnson-Lindenstrauss lemma states that for any X a subset of R^d with |X| = n and for any epsilon, there exists a map f:X-->R^m for m = O(log n / epsilon^2) such that: for all x in X, for all y in X, (1-epsilon)|x - y|_2 <= |f(x) - f(y)|_2 <= (1+epsilon)|x - y|_2. We show that this statement can be strengthened. In particular, the above claim holds true even if "for all y in X" is replaced with "for all y in R^d". Joint work with Shyam Narayanan.

• Speaker: Gitta Kutyniok, TU Berlin
• Title: Understanding Deep Neural Networks: From Generalization to Interpretability; zoom link @ https://sites.google.com/view/minds-seminar/home
• Date: 06/18/2020
• Time: 2:30 PM - 3:30 PM
• Place: 

(Part of One World MINDS seminar: https://sites.google.com/view/minds-seminar/home) \[ \] Deep neural networks have recently seen an impressive comeback with applications both in the public sector and the sciences. However, despite their outstanding success, a comprehensive theoretical foundation of deep neural networks is still missing. For deriving a theoretical understanding of deep neural networks, one main goal is to analyze their generalization ability, i.e. their performance on unseen data sets. In case of graph convolutional neural networks, which are today heavily used, for instance, for recommender systems, already the generalization capability to signals on graphs unseen in the training set, typically coined transferability, was not rigorously analyzed. In this talk, we will prove that spectral graph convolutional neural networks are indeed transferable, thereby also debunking a common misconception about this type of graph convolutional neural networks. If such theoretical approaches fail or if one is just given a trained neural network without knowledge of how it was trained, interpretability approaches become necessary. Those aim to "break open the black box" in the sense of identifying those features from the input, which are most relevant for the observed output. Aiming to derive a theoretically founded approach to this problem, we introduced a novel approach based on rate-distortion theory coined Rate-Distortion Explanation (RDE), which not only provides state-of-the-art explanations, but in addition allows first theoretical insights into the complexity of such problems. In this talk we will discuss this approach and show that it also gives a precise mathematical meaning to the previously vague term of relevant parts of the input.

• Speaker: Daping Weng, MSU
• Title: Augmentations, Fillings, and Clusters of Positive Braid Closures, https://msu.zoom.us/j/94925518997
• Date: 07/02/2020
• Time: 9:30 AM - 10:30 AM
• Place: 

Consider the standard contact structure on $\mathbb{R}^3_{xyz}$ with contact 1-form $\alpha=dz-ydx$. A Legendrian link $\Lambda$ is a link in $\mathbb{R}^3$ along which $\alpha$ vanishes. Chekanov associated a dga (also known as the Chekanov-Eliashberg dga) to every Legendrian link $\Lambda$ and proved that the stable-tame equivalence class of such dga is invariant under Legendrian isotopy of Legendrian links. The augmentation variety of $\Lambda$ is defined to be the moduli space of augmentations of its CE dga. Let $\mathbb{R}^4_{xyzt}$ be the symplectization of the standard contact $\mathbb{R}^3_{xyz}$ and consider the symplectic field theory of exact Lagrangian cobordisms between Legendrian links placed at different constant $t$ slices. Ekholm, Honda, Kalman constructed a contravariant functor that maps an exact Lagrangian cobordism to a dga homomorphism between the corresponding CE dga’s associated to the Legendrian links at the two ends. This induces a covariant functor that maps exact Lagrangian cobordisms to morphisms between augmentation varieties. In the special case of an exact Lagrangian filling of a Legendrian link $\Lambda$, i.e., an exact Lagrangian cobordism from the empty link to $\Lambda$, such covariant functor defines an algebraic torus inside the augmentation variety of $\Lambda$. In this talk, I will focus on the cases of rainbow closures for positive braids, which are naturally Legendrian links. By using an enhanced version of the CE dga, my collaborators and I construct a cluster $K_2$ structure on the augmentation variety of the rainbow closure $\Lambda_\beta$ for any positive braid $\beta$, and prove that the algebraic tori arisen from exact Lagrangian fillings constructed by pinching crossings are cluster charts in this cluster structure. We also relate the Kalman automorphism to the cluster Donaldson-Thomas transformation on these augmentation varieties. As an application of this new cluster structure, we give a sufficient condition on which $\Lambda_\beta$ admits infinitely many non-Hamiltonian-isotopic exact Lagrangian fillings based on the order of DT, solving a conjecture on the existence of Legendrian links that admit infinitely many fillings. This is joint work in progress with Honghao Gao and Linhui Shen.

• Speaker: Holger Rauhut, RWTH Aachen University
• Title: Convergence of gradient flows for learning deep linear neural networks; zoom link @ https://sites.google.com/view/minds-seminar/home
• Date: 07/09/2020
• Time: 2:30 PM - 3:30 PM
• Place: 

(Part of One World MINDS seminar: https://sites.google.com/view/minds-seminar/home) \[ \] Learning neural networks amounts to minimizing a loss function over given training data. Often gradient descent algorithms are used for this task, but their convergence properties are not yet well-understood. In order to make progress we consider the simplified setting of linear networks optimized via gradient flows. We show that such gradient flow defined with respect to the layers (factors) can be reinterpreted as a Riemannian gradient flow on the manifold of rank-r matrices in certain cases. The gradient flow always converges to a critical point of the underlying loss functional and, for almost all initializations, it converges to a global minimum on the manifold of rank-k matrices for some k.

• Speaker: Tamara Kolda, Sandia National Laboratories
• Title: Practical Leverage-Based Sampling for Low-Rank Tensor Decomposition; zoom link @ https://sites.google.com/view/minds-seminar/home
• Date: 07/23/2020
• Time: 2:30 PM - 3:30 PM
• Place: 

(Part of One World MINDS seminar: https://sites.google.com/view/minds-seminar/home) \[ \] Conventional algorithms for finding low-rank canonical polyadic (CP) tensor decompositions are unwieldy for large sparse tensors. The CP decomposition can be computed by solving a sequence of overdetermined least problems with special Khatri-Rao structure. In this work, we present an application of randomized algorithms to fitting the CP decomposition of sparse tensors, solving a significantly smaller sampled least squares problem at each iteration with probabilistic guarantees on the approximation errors. Prior work has shown that sketching is effective in the dense case, but the prior approach cannot be applied to the sparse case because a fast Johnson-Lindenstrauss transform (e.g., using a fast Fourier transform) must be applied in each mode, causing the sparse tensor to become dense. Instead, we perform sketching through leverage score sampling, crucially relying on the fact that the structure of the Khatri-Rao product allows sampling from overestimates of the leverage scores without forming the full product or the corresponding probabilities. Naive application of leverage score sampling is ineffective because we often have cases where a few scores are quite large, so we propose a novel hybrid of deterministic and random leverage-score sampling which consistently yields improved fits. Numerical results on real-world large-scale tensors show the method is significantly faster than competing methods without sacrificing accuracy. This is joint work with Brett Larsen, Stanford University.

• Speaker:  Tselil Schramm, MIT
• Title: Reconciling Statistical Queries and the Low Degree Likelihood Ratio; zoom link @ https://sites.google.com/view/minds-seminar/home
• Date: 08/06/2020
• Time: 2:30 PM - 3:30 PM
• Place: 

(Part of One World MINDS seminar: https://sites.google.com/view/minds-seminar/home) \[ \] In many high-dimensional statistics problems, we observe information-computation tradeoffs: given access to more data, statistical estimation and inference tasks require fewer computational resources. Though this phenomenon is ubiquitous, we lack rigorous evidence that it is inherent. In the current day, to prove that a statistical estimation task is computationally intractable, researchers must prove lower bounds against each type of algorithm, one by one, resulting in a "proliferation of lower bounds". We scientists dream of a more general theory which unifies these lower bounds and explains computational intractability in an algorithm-independent way. In this talk, I will make one small step towards realizing this dream. I will demonstrate general conditions under which two popular frameworks yield the same information-computation tradeoffs for high-dimensional hypothesis testing: the first being statistical queries in the "SDA" framework, and the second being hypothesis testing with low-degree hypothesis tests, also known as the low-degree-likelihood ratio. Our equivalence theorems capture numerous well-studied high-dimensional learning problems: sparse PCA, tensor PCA, community detection, planted clique, and more. Based on joint work with Matthew Brennan, Guy Bresler, Samuel B. Hopkins and Jerry Li.