BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Gabriel Peyré (CNRS\, Ecole Normale Supérieure) DTSTART:20200420T120000Z DTEND:20200420T124500Z DTSTAMP:20260422T225755Z UID:OWMADS/1 DESCRIPTION:Title: S caling Optimal Transport for High dimensional Learning\nby Gabriel Pey ré (CNRS\, Ecole Normale Supérieure) as part of One World seminar: Mathe matical Methods for Arbitrary Data Sources (MADS)\n\n\nAbstract\nOptimal t ransport (OT) has recently gained lot of interest in machine learning. It is a natural tool to compare in a geometrically faithful way probability d istributions. It finds applications in both supervised learning (using geo metric loss functions) and unsupervised learning (to perform generative mo del fitting). OT is however plagued by the curse of dimensionality\, since it might require a number of samples which grows exponentially with the d imension. In this talk\, I will review entropic regularization methods whi ch define geometric loss functions approximating OT with a better sample c omplexity. More information and references can be found on the website of our book Computational Optimal Transport.\n LOCATION:https://researchseminars.org/talk/OWMADS/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Marie-Therese Wolfram (Warwick University\, UK) DTSTART:20200420T130000Z DTEND:20200420T134500Z DTSTAMP:20260422T225755Z UID:OWMADS/2 DESCRIPTION:Title: I nverse Optimal Transport\nby Marie-Therese Wolfram (Warwick University \, UK) as part of One World seminar: Mathematical Methods for Arbitrary Da ta Sources (MADS)\n\n\nAbstract\nDiscrete optimal transportation problems arise in various contexts in engineering\, the sciences and the social sci ences. Examples include the marriage market in economics or international migration flows in demographics. Often the underlying cost criterion is un known\, or only partly known\, and the observed optimal solutions are corr upted by noise. In this talk we discuss a systematic approach to infer unk nown costs from noisy observations of optimal transportation plans. The pr oposed methodologies are developed within the Bayesian framework for inver se problems and require only the ability to solve the forward optimal tran sport problem\, which is a linear program\, and to generate random numbers . We illustrate our approach using the example of international migration flows. Here reported migration flow data captures (noisily) the number of individuals moving from one country to another in a given period of time. It can be interpreted as a noisy observation of an optimal transportation map\, with costs related to the geographical position of countries. We use a graph-based formulation of the problem\, with countries at the nodes of graphs and non-zero weighted adjacencies only on edges between countries which share a border. We use the proposed algorithm to estimate the weight s\, which represent cost of transition\, and to quantify uncertainty in th ese weights.\n LOCATION:https://researchseminars.org/talk/OWMADS/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Lorenzo Rosasco (Universitá di Genova\, IT - MIT\, US) DTSTART:20200504T120000Z DTEND:20200504T124500Z DTSTAMP:20260422T225755Z UID:OWMADS/3 DESCRIPTION:Title: E fficient kernel-PCA by Nyström sampling\nby Lorenzo Rosasco (Universi tá di Genova\, IT - MIT\, US) as part of One World seminar: Mathematical Methods for Arbitrary Data Sources (MADS)\n\n\nAbstract\nIn this talk\, we discuss and study a Nyström based approach to efficient large scale kern el principal component analysis (PCA). The latter is a natural nonlinear e xtension of classical PCA based on considering a nonlinear feature map or the corresponding kernel. Like other kernel approaches\, kernel PCA enjoys good mathematical and statistical properties but\, numerically\, it scale s poorly with the sample size. Our analysis shows that Nyström sampling g reatly improves computational efficiency without incurring any loss of sta tistical accuracy. While similar effects have been observed in supervised learning\, this is the first such result for PCA. Our theoretical findings \, which are also illustrated by numerical results\, are based on a combin ation of analytic and concentration of measure techniques. Our study is mo re broadly motivated by the question of understanding the interplay betwee n statistical and computational requirements for learning.\n LOCATION:https://researchseminars.org/talk/OWMADS/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Lars Ruthotto (Emory University\, US) DTSTART:20200518T120000Z DTEND:20200518T124500Z DTSTAMP:20260422T225755Z UID:OWMADS/4 DESCRIPTION:Title: M achine learning meets optimal transport: old solutions for new problems an d vice versa\nby Lars Ruthotto (Emory University\, US) as part of One World seminar: Mathematical Methods for Arbitrary Data Sources (MADS)\n\n\ nAbstract\nThis talk presents new connections between optimal transport (O T)\, which has been a critical problem in applied mathematics for centurie s\, and machine learning (ML)\, which has been receiving enormous attentio n in the past decades. In recent years\, OT and ML have become increasingl y intertwined. This talk contributes to this booming intersection by provi ding efficient and scalable computational methods for OT and ML.\nThe firs t part of the talk shows how neural networks can be used to efficiently ap proximate the optimal transport map between two densities in high dimensio ns. To avoid the curse-of-dimensionality\, we combine Lagrangian and Euler ian viewpoints and employ neural networks to solve the underlying Hamilton -Jacobi-Bellman equation. Our approach avoids any space discretization and can be implemented in existing machine learning frameworks. We present nu merical results for OT in up to 100 dimensions and validate our solver in a two-dimensional setting. \nThe second part of the talk shows how optimal transport theory can improve the efficiency of training generative models and density estimators\, which are critical in machine learning. We consi der continuous normalizing flows (CNF) that have emerged as one of the mos t promising approaches for variational inference in the ML community. Our numerical implementation is a discretize-optimize method whose forward pro blem relies on manually derived gradients and Laplacian of the neural netw ork and uses automatic differentiation in the optimization. In common benc hmark challenges\, our method outperforms state-of-the-art CNF approaches by reducing the network size by 8x\, accelerate the training by 10x- 40x a nd allow 30x-50x faster inference.\n LOCATION:https://researchseminars.org/talk/OWMADS/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Vincent Duval (Inria\, FR) DTSTART:20200608T130000Z DTEND:20200608T134500Z DTSTAMP:20260422T225755Z UID:OWMADS/5 DESCRIPTION:Title: R epresenting the solutions of total variation regularized problems\nby Vincent Duval (Inria\, FR) as part of One World seminar: Mathematical Meth ods for Arbitrary Data Sources (MADS)\n\n\nAbstract\nRepresenting the solu tions of total variation regularized problems\n\nThe total (gradient) vari ation is a regularizer which has been widely used in inverse problems aris ing in image processing\, following the pioneering work of Rudin\, Osher a nd Fatemi. In this talk\, I will describe the structure the solutions to t he total variation regularized variational problems when one has a finite number of measurements.\nFirst\, I will present a general representation p rinciple for the solutions of convex problems\, then I will apply it to th e total variation by describing the faces of its unit ball.\n\nIt is a joi nt work with Claire Boyer\, Antonin Chambolle\, Yohann De Castro\, Frédé ric de Gournay and Pierre Weiss.\n LOCATION:https://researchseminars.org/talk/OWMADS/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael Unser (École polytechnique fédérale de Lausanne\, CH) DTSTART:20200608T120000Z DTEND:20200608T124500Z DTSTAMP:20260422T225755Z UID:OWMADS/6 DESCRIPTION:Title: R epresenter theorems for machine learning and inverse problems\nby Mich ael Unser (École polytechnique fédérale de Lausanne\, CH) as part of On e World seminar: Mathematical Methods for Arbitrary Data Sources (MADS)\n\ n\nAbstract\nRegularization addresses the ill-posedness of the training pr oblem in machine learning or the reconstruction of a signal from a limited number of measurements. The standard strategy consists in augmenting the original cost functional by an energy that penalizes solutions with undesi rable behaviour. In this presentation\, I will present a general represent er theorem that characterizes the solutions of a remarkably broad class of optimization problems in Banach spaces and helps us understand the effect of regularization. I will then use the theorem to retrieve some classical characterizations such as the celebrated representer theorem of machine l eaning for RKHS\, Tikhonov regularization\, representer theorems for spars ity promoting functionals\, as well as a few new ones\, including a result for deep neural networks.\n LOCATION:https://researchseminars.org/talk/OWMADS/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Nicolás García Trillos (University of Wisconsin-Madison\, US) DTSTART:20200615T130000Z DTEND:20200615T134500Z DTSTAMP:20260422T225755Z UID:OWMADS/7 DESCRIPTION:Title: R egularity theory and uniform convergence in the large data limit of graph Laplacian eigenvectors on random data clouds.\nby Nicolás García Tri llos (University of Wisconsin-Madison\, US) as part of One World seminar: Mathematical Methods for Arbitrary Data Sources (MADS)\n\n\nAbstract\nGrap h Laplacians are omnipresent objects in machine learning that have been us ed in supervised\, unsupervised and semi supervised settings due to their versatility in extracting local and global geometric information from data clouds. In this talk I will present an overview of how the mathematical t heory built around them has gotten deeper and deeper\, layer by layer\, si nce the appearance of the first results on pointwise consistency in the 20 00’s\, until the most recent developments\; this line of research has fo und strong connections between PDEs built on proximity graphs on data clou ds and PDEs on manifolds\, and has given a more precise mathematical meani ng to the task of “manifold learning”. In the first part of the talk I will highlight how ideas from optimal transport made some of the initial steps\, which provided L2 type error estimates between the spectra of gra ph Laplacians and Laplace-Beltrami operators\, possible. In the second par t of the talk\, which is based on recent work with Jeff Calder and Marta L ewicka\, I will present a newly developed regularity theory for graph Lapl acians which among other things allow us to bootstrap the L2 error estimat es developed through optimal transport and upgrade them to uniform converg ence and almost C^{0\,1} convergence rates. The talk can be seen as a tale of how a flow of ideas from optimal transport\, PDEs\, and in general\, a nalysis\, has made possible a finer understanding of concrete objects popu lar in data analysis and machine learning.\n LOCATION:https://researchseminars.org/talk/OWMADS/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Michaël Fanuel (KU Leuven\, BE) DTSTART:20200504T130000Z DTEND:20200504T134500Z DTSTAMP:20260422T225755Z UID:OWMADS/8 DESCRIPTION:Title: D iversity sampling in kernel method\nby Michaël Fanuel (KU Leuven\, BE ) as part of One World seminar: Mathematical Methods for Arbitrary Data So urces (MADS)\n\n\nAbstract\nA well-known technique for large scale kernel methods is the Nyström approximation. Based on a subset of landmarks\, it gives a low rank approximation of the kernel matrix\, and is known to pro vide a form of implicit regularization. We will discuss the impact of samp ling diverse landmarks for constructing the Nyström approximation in supe rvised and unsupervised problems. In particular\, three methods will be co nsidered: uniform sampling\, leverage score sampling and Determinantal Poi nt Processes (DPP). The implicit regularization due the diversity of the l andmarks will be made explicit by numerical simulations and analysed furth er in the case of DPP sampling by some theoretical results.\n LOCATION:https://researchseminars.org/talk/OWMADS/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Francis Bach (Inria\, FR) DTSTART:20200518T130000Z DTEND:20200518T134500Z DTSTAMP:20260422T225755Z UID:OWMADS/9 DESCRIPTION:Title: O n the convergence of gradient descent for wide two-layer neural networks\nby Francis Bach (Inria\, FR) as part of One World seminar: Mathematica l Methods for Arbitrary Data Sources (MADS)\n\n\nAbstract\nMany supervised learning methods are naturally cast as optimization problems. For predict ion models which are linear in their parameters\, this often leads to conv ex problems for which many guarantees exist. Models which are non-linear i n their parameters such as neural networks lead to non-convex optimization problems for which guarantees are harder to obtain. In this talk\, I will consider two-layer neural networks with homogeneous activation functions where the number of hidden neurons tends to infinity\, and show how qualit ative convergence guarantees may be derived. I will also highlight open pr oblems related to the quantitative behavior of gradient descent for such m odels. (Based on joint work with Lénaïc Chizat\, https://arxiv.org/abs/1 805.09545\, https://arxiv.org/abs/2002.04486)\n\nPlease note that this is a joint talk with the One World Optimization Seminar.\n LOCATION:https://researchseminars.org/talk/OWMADS/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Andrea Braides (University of Rome Tor Vergata) DTSTART:20200615T120000Z DTEND:20200615T124500Z DTSTAMP:20260422T225755Z UID:OWMADS/10 DESCRIPTION:Title: Continuum limits of interfacial energies on (sparse and) dense graphs\ nby Andrea Braides (University of Rome Tor Vergata) as part of One World s eminar: Mathematical Methods for Arbitrary Data Sources (MADS)\n\n\nAbstra ct\nI review some results on the convergence of energies defined on graphs . My interest in such energies comes from models in Solid Mechanics (where the bonds in the graph represent the relevant atomistic interactions) or Statistical Physics (Ising systems)\, but the nodes of the graph can also be thought as a collection of data on which the bonds describe some relati on between the data.\nThe typical objective is an approximate (simplified) continuum description of problems of minimal cut as the number N of the n odes of the graphs diverges.\nIf the graphs are sparse (i.e. the number of bonds is much less than the total number of pairs of nodes as N goes to i nfinity)\, often (more precisely when we have some control on the range or on the decay of the interactions) such minimal-cut problems translate int o minimal-perimeter problems for sets or partitions on the continuum. This description is easily understood for periodic lattice systems\, but carri es on also for random distributions of nodes. In the case of a (locally) u niform Poisson distribution\, actually the limit minimal-cut problems are described by more regular energies than in the periodic-lattice case. \nWh en we relax the hypothesis on the range of interactions\, the description of the limit of sparse graphs becomes more complex\, as it depends subtly on geometric characteristics of the graph\, and is partially understood. S ome easy examples show that\, even though for the continuum limit we still remain in a similar analytical environment\, the description as (sharp) i nterfacial energies can be lost in this case\, and more “diffuse” inte rfaces must be taken into account.\nIf instead we consider dense sequences of graphs (i.e.\, the number of bonds is of the same order as the total n umber of pairs as N goes to infinity) then a completely different limit en vironment must be used\, that of graphons (which are abstract limits of gr aphs)\, for which sophisticated combinatoric results can be used. We can r e-read the existing notion of convergence of graphs to graphons as a conve rgence of the related cut functionals to non-local energies on a simple re ference parameter set. This convergence provides an approximate descriptio n of the corresponding minimal-cup problems.\nWorks in collaboration with Alicandro\, Cicalese\, Piatnitski and Solci (sparse graphs) and Cermelli a nd Dovetta (dense graphs).\n LOCATION:https://researchseminars.org/talk/OWMADS/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Jana de Wiljes (Universität Potsdam\, DE) DTSTART:20200629T120000Z DTEND:20200629T124500Z DTSTAMP:20260422T225755Z UID:OWMADS/11 DESCRIPTION:Title: Sequential learning for decision support under uncertainty\nby Jana de Wiljes (Universität Potsdam\, DE) as part of One World seminar: Mathemat ical Methods for Arbitrary Data Sources (MADS)\n\n\nAbstract\nIn many appl icational areas there is a need to determine a control variable that optim izes a pre-specified objective. This problem is particularly challenging w hen knowledge on the underlying dynamics is subject to various sources of uncertainty. A scenario such as that arises for instance in the context of therapy individualization to improve the efficacy and safety of medical treatment. Mathematical models describing the pharmacokinetics and pharma codynamics of a drug together with data on associated biomarkers can be le veraged to support decision-making by predicting therapy outcomes. We pres ent a continuous learning strategy which follows a novel sequential Monte Carlo tree search approach and explore how the underlying uncertainties re flect in the approximated control variable.\n LOCATION:https://researchseminars.org/talk/OWMADS/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Björn Sprungk (TU Freiberg\, DE) DTSTART:20200629T130000Z DTEND:20200629T134500Z DTSTAMP:20260422T225755Z UID:OWMADS/12 DESCRIPTION:Title: Noise-level robust Monte Carlo methods for Bayesian inference with infomat ive data\nby Björn Sprungk (TU Freiberg\, DE) as part of One World se minar: Mathematical Methods for Arbitrary Data Sources (MADS)\n\n\nAbstrac t\nThe Bayesian approach to inverse problems provides a rigorous framework for the incorporation and quantification of uncertainties in measurements \, parameters and models. However\, sampling from or integrating w.r.t. th e resultung posterior measure can become computationally challenging. In r ecent years\, a lot of effort has been spent on deriving dimension-indepen dent methods and to combine efficient sampling strategies with multilevel or surrogate methods in order to reduce the computational burden of Bayesi an inverse problems.\nIn this talk\, we are interested in designing numeri cal methods which are robust w.r.t. the size of the observational noise\, i.e.\, methods which behave well in case of concentrated posterior measure s. The concentration of the posterior is a highly desirable situation in p ractice\, since it relates to informative or large data. However\, it can pose as well a significant computational challenge for numerical methods b ased on the prior or reference measure. We propose to employ the Laplace a pproximation of the posterior as the base measure for numerical integratio n in this context. The Laplace approximation is a Gaussian measure centere d at the maximum a-posteriori estimate (MAPE) and with covariance matrix d epending on the Hessian of the log posterior density at the MAPE. We discu ss convergence results of the Laplace approximation in terms of the Hellin ger distance and analyze the efficiency of Monte Carlo methods based on it . In particular\, we show that Laplace-based importance sampling and quasi -Monte-Carlo as well as Laplace-based Metropolis-Hastings algorithms are r obust w.r.t. the concentration of the posterior for large classes of poste rior distributions and integrands whereas prior-based Monte Carlo sampling methods are not.\n LOCATION:https://researchseminars.org/talk/OWMADS/12/ END:VEVENT END:VCALENDAR