\n

High-dimensional gener alized linear models are basic building blocks of current data analysis to ols including multilayers neural networks. They arise in signal processing \, statistical inference\, machine learning\, communication theory\, and o ther fields. I will explain how to establish rigorously the intrinsic info rmation-theoretic limitations of inference and learning for a class of ran domly generated instances of generalized linear models\, thus closing seve ral old conjectures. Examples will be shown where one can delimit regions of parameters for which the optimal error rates are efficiently achievable with currently known algorithms. I will discuss how the proof technique\, based on the recently developed adaptive interpolation method\, is able t o deal with the output nonlinearity and also to some extent with non-separ able input distributions. |

https://probability.dmi.unibas.ch/seminar.html

We consider a spin system cont aining pure two spin Sherrington-Kirkpatrick Hamiltonian with Curie-Weiss interaction. The model where the spins are spherically symmetric was consi dered by Baik and Lee and Baik et al. which shows a two dimensional phase transition with respect to temperature and the coupling constant. In this paper we prove a result analogous to Baik and Lee in the “paramagnetic r egime” when the spins are i.i.d. Rademacher. We prove the free energy in this case is asymptotically Gaussian and can be approximated by a suitabl e linear spectral statistics. Unlike the spherical symmetric case the free energy here can not be written as a function of the eigenvalues of the co rresponding interaction matrix. The method in this paper relies on a dense sub-graph conditioning technique introduced by Banerjee . The proof of th e approximation by the linear spectral statistics part is close to Banerje e and Ma. | |

https://probability.dmi.unibas.ch/seminar. html

END:VEVENT BEGIN:VEVENT UID:news988@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20191209T133116 DTSTART;TZID=Europe/Zurich:20191211T110000 SUMMARY:Seminar in probability theory: Daniele Tantari (Scuola Normale Supe riore) DESCRIPTION:Mean-field methods fail to reconstruct the parameters of the mo del when the dataset is clusterized. This situation is found at low temper atures because of the emergence of multiple thermodynamic states. The para digmatic Hopfield model is considered in a teacher-student scenario as a p roblem of unsupervised learning with Restricted Boltzmann Machines (RBM). For different choices of the priors on units and weights\, the replica sym metric phase diagram of random RBM’s is analyzed and in particular the p aramagnetic phase boundary is presented as directly related to the optimal size of the training set necessary for a good generalization. The connect ion between the direct and inverse problem is pointed out by showing that inference can be efficiently performed by suitably adapting both standard learning techniques and standard approaches to the direct problem.\\r\\n\\ r\\nhttps://probability.dmi.unibas.ch/seminar.html [https://probability.dm i.unibas.ch/seminar.html] X-ALT-DESC:Mean-field methods fail to rec onstruct the parameters of the model when the dataset is clusterized. This situation is found at low temperatures because of the emergence of multip le thermodynamic states. The paradigmatic Hopfield model is considered in a teacher-student scenario as a problem of unsupervised learning with Rest ricted Boltzmann Machines (RBM). For different choices of the priors on un its and weights\, the replica symmetric phase diagram of random RBM’s is analyzed and in particular the paramagnetic phase boundary is presented a s directly related to the optimal size of the training set necessary for a good generalization. The connection between the direct and inverse proble m is pointed out by showing that inference can be efficiently performed by suitably adapting both standard learning techniques and standard approach es to the direct problem. | |

https://probability.dmi. unibas.ch/seminar.html

END:VEVENT BEGIN:VEVENT UID:news982@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20191201T140857 DTSTART;TZID=Europe/Zurich:20191106T110000 SUMMARY:Seminar in probability theory: Razvan Gurau (CNRS) DESCRIPTION:I will give an introduction to random tensors and their applica tions. In particular I will describe an universality result for invariant probability measures for tensors: under generic scaling assumptions\, for large tensor sizes any invariant tensor measure approaches a Gaussian. I w ill then discuss the implications of this result\, as well as ways to avoi d it.\\r\\n\\r\\nhttps://probability.dmi.unibas.ch/seminar.html X-ALT-DESC:I will give an introduction to random tensors and their applicat ions. In particular I will describe an universality result for invariant p robability measures for tensors: under generic scaling assumptions\, for l arge tensor sizes any invariant tensor measure approaches a Gaussian. I wi ll then discuss the implications of this result\, as well as ways to avoid it.\n

https://probability.dmi.unibas.ch/s
eminar.html

In this talk we will discuss s ome recent large deviation asymptotics concerning the local behavior of ra ndom interlacements on Z^d\, d≥3. In particular\, we will describe the l ink with previous results concerning macroscopic holes left inside a large box\, by the the adequately thickened connected component of the boundary of the box in the vacant sets of random interlacements. | |

https://probability.dmi.unibas.ch/seminar.html

In this talk we will study t he asymptotic behavior of a random walk that evolves on top of a simple sy mmetric exclusion process. This nice example of a random walk on a dynamic al random environment presents its own challenges due to the slow mixing p roperties of the underlying medium. We will discuss a law of large numbers that has been proved recently for this random walk. Interestingly\, we ca n only prove this law of large numbers for all but two exceptional densiti es of the exclusion process. The main technique that we have employed is a multi-scale renormalization that has been derived from works in percolati on theory. | |

END:VEVENT BEGIN:VEVENT UID:news873@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190503T143804 DTSTART;TZID=Europe/Zurich:20190508T110000 SUMMARY:Seminar in probability theory: Roland Bauerschmidt (University of C ambridge) DESCRIPTION:The classical random walk isomorphism theorems relate the local time of a random walk to the square of a Gaussian free field. I will pres ent non-Gaussian versions of these theorems\, relating hyperbolic and hemi spherical sigma models (and their supersymmetric versions) to non-Markovia n random walks interacting through their local time. Applications include a short proof of the Sabot-Tarres limiting formula for the vertex-reinforc ed jump process (VRJP) and a Mermin-Wagner theorem for hyperbolic sigma mo dels and the VRJP. This is joint work with Tyler Helmuth and Andrew Swan. X-ALT-DESC:

The classical random walk isom orphism theorems relate the local time of a random walk to the square of a Gaussian free field. I will present non-Gaussian versions of these theore ms\, relating hyperbolic and hemispherical sigma models (and their supersy mmetric versions) to non-Markovian random walks interacting through their local time. Applications include a short proof of the Sabot-Tarres limitin g formula for the vertex-reinforced jump process (VRJP) and a Mermin-Wagne r theorem for hyperbolic sigma models and the VRJP. This is joint work wit h Tyler Helmuth and Andrew Swan. | |

END:VEVENT BEGIN:VEVENT UID:news843@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190319T145022 DTSTART;TZID=Europe/Zurich:20190415T130000 SUMMARY:Seminar in probability theory: Marilou Gabrié (ENS) DESCRIPTION:The complexity of deep neural networks remains an obstacle to t he understanding of their great efficiency. Their generalisation ability\, a priori counter intuitive\, is not yet fully accounted for. Recently an information theoretic approach was proposed to investigate this question.R elying on the heuristic replica method from statistical physics we present an estimator for entropies and mutual informations in models of deep mode l networks. Using this new tool\, we test numerically the relation between generalisation and information. X-ALT-DESC:The complexity of deep neural networks remains an obstacle to th e understanding of their great efficiency. Their generalisation ability\, a priori counter intuitive\, is not yet fully accounted for. Recently an i nformation theoretic approach was proposed to investigate this question.

We will start by revi ewing some of the recent literature on the geometry of the loss function\, and how SGD navigates the landscape in the OP regime. Then we will see ho w to define OP by finding a sharp transition described by the models fitti ng abilities to its training set. Finally\, we will discuss how this criti cal threshold is connected to the generalization properties of the model\, and argue that life beyond this threshold is (more or less) as good as it gets. END:VEVENT BEGIN:VEVENT UID:news839@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190319T144242 DTSTART;TZID=Europe/Zurich:20190320T110000 SUMMARY:Seminar in probability theory: David Belius (Universität Basel) DESCRIPTION:This is the first talk in a five part series of talks on deep l earning from a theoretical point of view\, held jointly between the probab ility theory and machine learning groups of the Department of Mathematics and Computer Science. The four invited speakers that follow after this tal k are young researchers who are contributing in different ways to what wil l hopefully eventually be a comprehensive theory of deep neural networks.I n this first talk I will introduce the main theoretical questions about de ep neural networks:1. Representation - what can deep neural networks repr esent?2. Optimization - why and under what circumstances can we successfu lly train neural networks?3. Generalization - why do deep neural networks often generalize well\, despite huge capacity?As a preface I will review the basic models and algorithms (Neural Networks\, (stochastic) gradient d escent\, ...) and some important concepts from machine learning (capacity\ , overfitting/underfitting\, generalization\, ...). X-ALT-DESC:This is the first talk in a five part series of talks on deep le arning from a theoretical point of view\, held jointly between the probabi lity theory and machine learning groups of the Department of Mathematics a nd Computer Science. The four invited speakers that follow after this talk are young researchers who are contributing in different ways to what will hopefully eventually be a comprehensive theory of deep neural networks.

In this first talk I will introduce the main theoretical questio ns about deep neural networks:

1. Representation \;- what can dee p neural networks represent?

2. Optimization \;- why and under wh at circumstances can we successfully train neural networks?

3. Genera lization \;- why do deep neural networks often generalize well\, despi te huge capacity?

As a preface I will review the basic models a nd algorithms (Neural Networks\, (stochastic) gradient descent\, ...) and some important concepts from machine learning (capacity\, overfitting/unde rfitting\, generalization\, ...). END:VEVENT BEGIN:VEVENT UID:news355@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181030T094303 DTSTART;TZID=Europe/Zurich:20181212T110000 SUMMARY:Seminar in probability theory: Ioan Manolescu (Fribourg) DESCRIPTION:TBA X-ALT-DESC:TBA END:VEVENT BEGIN:VEVENT UID:news354@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181030T094136 DTSTART;TZID=Europe/Zurich:20181128T110000 SUMMARY:Seminar in probability theory: Gaultier Lambert (University of Zuri ch) DESCRIPTION:TBA X-ALT-DESC:TBA END:VEVENT BEGIN:VEVENT UID:news326@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181103T093011 DTSTART;TZID=Europe/Zurich:20181121T110000 SUMMARY:Seminar in probability theory: Antti Knowles (Geneva) DESCRIPTION:We consider the adjacency matrix of the Erdos-Renyi graph G(N\, p) in the supercritical regime pN > C log N for some universal constant C . We show that the eigenvalue density is with high probability well appr oximated by the semicircle law on all spectral scales larger than the typ ical eigenvalue spacing. We also show that all eigenvectors are completel y delocalized with high probability. Both results are optimal in the sens e that they are known to be false for pN < log N. A key ingredient of the proof is a new family of large deviation estimates for multilinear forms of sparse vectors. \\r\\nJoint work with Yukun He and Matteo Marcozzi. X-ALT-DESC: We consider the adjacency matrix of the Erdos-Renyi graph G(N\, p) in the supercritical regime pN >\; C log N for some universal consta nt C. We show that the eigenvalue density is with high probability well approximated by the semicircle law on all spectral scales larger than the typical eigenvalue spacing. We also show that all eigenvectors are compl etely delocalized with high probability. Both results are optimal in the sense that they are known to be false for pN <\; log N. A key ingredien t of the proof is a new family of large deviation estimates for multiline ar forms of sparse vectors. \nJoint work with Yukun He and Matteo Marcozz i. END:VEVENT BEGIN:VEVENT UID:news353@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181103T092837 DTSTART;TZID=Europe/Zurich:20181114T110000 SUMMARY:Seminar in probability theory: Marius Schmidt (Basel) DESCRIPTION:Consider the hypercube as a graph with vertex set {0\,1}^N and edges between two vertices if they are only one coordinate flip apart. C hoosing independent standard exponentially distributed lengths for all ed ges and asking how long the shortest directed paths from (0\,..\,0) to (1 \,..\,1) is defines oriented first passage percolation on the hypercube. We will discuss the conceptual steps needed to answer this question to th e precision of extremal process following the two paper series "Oriented first passage percolation in the mean field limit" by Nicola Kistler\, Ad rien Schertzer and Marius A. Schmidt: arXiv:1804.03117 and arXiv:1808.0459 8. X-ALT-DESC: Consider the hypercube as a graph with vertex set {0\,1}^N and edges between two vertices if they are only one coordinate flip apart. C hoosing independent standard exponentially distributed lengths for all ed ges and asking how long the shortest directed paths from (0\,..\,0) to (1 \,..\,1) is defines oriented first passage percolation on the hypercube. We will discuss the conceptual steps needed to answer this question to th e precision of extremal process following the two paper series "\;Ori ented first passage percolation in the mean field limit"\; by Nicola Kistler\, Adrien Schertzer and Marius A. Schmidt: arXiv:1804.03117 and arX iv:1808.04598. END:VEVENT BEGIN:VEVENT UID:news352@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181103T092627 DTSTART;TZID=Europe/Zurich:20181107T110000 SUMMARY:Seminar in probability theory: Dominik Schröder (IST Austria) DESCRIPTION:For Wigner-type matrices\, i.e. Hermitian random matrices with independent\, not necessarily identically distributed entries above the diagonal\, we show that at any cusp singularity of the limiting eigenvalu e distribution the local eigenvalue statistics are universal and form a P earcey process. Since the density of states typically exhibits only squar e root or cubic root cusp singularities\, our work complements previous r esults on the bulk and edge universality and it thus completes the resolu tion of the Wigner-Dyson-Mehta universality conjecture for the last remai ning universality type. X-ALT-DESC: For Wigner-type matrices\, i.e. Hermitian random matrices with independent\, not necessarily identically distributed entries above the diagonal\, we show that at any cusp singularity of the limiting eigenvalu e distribution the local eigenvalue statistics are universal and form a P earcey process. Since the density of states typically exhibits only squar e root or cubic root cusp singularities\, our work complements previous r esults on the bulk and edge universality and it thus completes the resolu tion of the Wigner-Dyson-Mehta universality conjecture for the last remai ning universality type. END:VEVENT BEGIN:VEVENT UID:news327@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20181030T093900 DTSTART;TZID=Europe/Zurich:20181031T110000 SUMMARY:Seminar in probability theory: Anton Klimovsky (Duisburg-Essen) DESCRIPTION:Finding the (space-height) distribution of the (local) extrema of high-dimensional strongly correlated random fields is a notorious hard problem with many applications. Following Fyodorov & Sommers (2007)\, we f ocus on the Gaussian fields with isotropic increments and take the viewpoi nt of statistical physics. By exploiting various probabilistic symmetries\ , we rigorously derive the Fyodorov-Sommers formula for the log-partition function in the high-dimensional limit. The formula suggests a rich pictur e for the distribution of the local extrema akin to the celebrated spheric al Sherrington-Kirkpatrick model with mixed p-spin interactions. X-ALT-DESC: Finding the (space-height) distribution of the (local) extrema of high-dimensional strongly correlated random fields is a notorious hard problem with many applications. Following Fyodorov &\; Sommers (2007)\, we focus on the Gaussian fields with isotropic increments and take the vi ewpoint of statistical physics. By exploiting various probabilistic symmet ries\, we rigorously derive the Fyodorov-Sommers formula for the log-parti tion function in the high-dimensional limit. The formula suggests a rich p icture for the distribution of the local extrema akin to the celebrated sp herical Sherrington-Kirkpatrick model with mixed p-spin interactions. END:VEVENT BEGIN:VEVENT UID:news325@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20180925T212513 DTSTART;TZID=Europe/Zurich:20180906T110000 SUMMARY:Seminar in probability theory: Lisa Hartung (New York University) DESCRIPTION:It was proven by Rider and Virag that the logarithm of the char acteristic polynomial of the Ginibre ensemble converges to a logarithmical ly correlated random field. In this talk we will see how this connection c an be established on the level if powers of the characteristic polynomial by proving convergence to Gaussian multiplicative chaos. We consider the r ange of powers in the L^2 phase. \\r\\n(Joint work in progress with Paul B ourgade and Guillaume Dubach). X-ALT-DESC: It was proven by Rider and Virag that the logarithm of the char acteristic polynomial of the Ginibre ensemble converges to a logarithmical ly correlated random field. In this talk we will see how this connection c an be established on the level if powers of the characteristic polynomial by proving convergence to Gaussian multiplicative chaos. We consider the r ange of powers in the L^2 phase. \n(Joint work in progress with Paul Bourg ade and Guillaume Dubach). DTEND;TZID=Europe/Zurich:20180906T120000 END:VEVENT BEGIN:VEVENT UID:news306@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20180927T122021 DTSTART;TZID=Europe/Zurich:20180822T110000 SUMMARY:Seminar in probability theory: Alexander Drewitz (Köln) DESCRIPTION:We consider two fundamental percolation models with long-range correlations: The Gaussian free field and (the vacant set) of Random Inter lacements. Both models have been the subject of intensive research during the last years and decades\, on Zd as well as on some more general graphs. We investigate some structural percolative properties around their critic al parameters\, in particular the ubiquity of the infinite components of c omplementary phases. \\r\\nThis talk is based on joint works with A. Prév ost (Köln) and P.-F. Rodriguez (Bures-sur-Yvette). X-ALT-DESC: We consider two fundamental percolation models with long-range correlations: The Gaussian free field and (the vacant set) of Random Inter lacements. Both models have been the subject of intensive research during the last years and decades\, on Zd as well as on some more general graphs. We investigate some structural percolative properties around their critic al parameters\, in particular the ubiquity of the infinite components of c omplementary phases. \nThis talk is based on joint works with A. Prévost (Köln) and P.-F. Rodriguez (Bures-sur-Yvette). DTEND;TZID=Europe/Zurich:20180822T120000 END:VEVENT END:VCALENDAR