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Person# Clément Hongler

Biography

2006: B.Sc. Math, EPFL2008: M.Sc. Math, EPFL2010: Ph.D. Math, Université de Genève2010-2014: Ritt Assistant Professor, Columbia University2014-2018 Tenure-Track Assistant Professor, EPFL2019-present: Associate Professor, EPFL

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Related research domains (7)

Ising model

The Ising model (ˈiːzɪŋ) (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The m

Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many bra

Continuum limit

In mathematical physics and mathematics, the continuum limit or scaling limit of a lattice model refers to its behaviour in the limit as the lattice spacing goes to zero. It is often useful to use la

Courses taught by this person (3)

MATH-200: Analysis III

Apprendre les bases de l'analyse vectorielle et de l'analyse complexe.

MATH-434: Lattice models

Lattice models consist of (typically random) objects living on a periodic graph. We will study some models that are mathematically interesting and representative of physical phenomena seen in the real world.

MATH-642: Artificial Life

We will study the emergence of life-like phenomena, such as locomotion, resilience,, reproduction, and evolution in mathematical models, in particular in discrete and continuous cellular automata, developing techniques to analyze, connect and characterize life-like features in such systems.

Related units (3)

People doing similar research (138)

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Critical statistical mechanics and Conformal Field Theory (CFT) are conjecturally connected since the seminal work of Beliavin et al. (Nucl Phys B 241(2):333-380, 1984). Both exhibit exactly solvable structures in two dimensions. A long-standing question (Itoyama and Thacker in Phys Rev Lett 58:1395-1398, 1987) concerns whether there is a direct link between these structures, that is, whether the Virasoro algebra representations of CFT, the distinctive feature of CFT in two dimensions, can be found within lattice models of statistical mechanics. We give a positive answer to this question for the discrete Gaussian free field and for the Ising model, by connecting the structures of discrete complex analysis in the lattice models with the Virasoro symmetry that is expected to describe their scaling limits. This allows for a tight connection of a number of objects from the lattice model world and the field theory one. In particular, our results link the CFT local fields with lattice local fields introduced in Gheissari et al. (Commun Math Phys 367(3):771-833, 2019) and the probabilistic formulation of the lattice model with the continuum correlation functions. Our construction is a decisive step towards establishing the conjectured correspondence between the correlation functions of the CFT fields and those of the lattice local fields. In particular, together with the upcoming (Chelkak et al. in preparation), our construction will complete the picture initiated in Hongler and Smirnov (Acta Math 211:191-225, 2013), Hongler (Conformal invariance of ising model correlations, 2012) and Chelkak et al. (Annals Math 181(3):1087-1138, 2015), where a number of conjectures relating specific Ising lattice fields and CFT correlations were proven.

Franck Raymond Gabriel, Clément Hongler

The Neural Tangent Kernel is a new way to understand the gradient descent in deep neural networks, connecting them with kernel methods. In this talk, I'll introduce this formalism and give a number of results on the Neural Tangent Kernel and explain how they give us insight into the dynamics of neural networks during training and into their generalization features.

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We study how permutation symmetries in overparameterized multi-layer neural networks generate `symmetry-induced' critical points. Assuming a network with $L$ layers of minimal widths $r_1^*, \ldots, r_{L-1}^*$ reaches a zero-loss minimum at $r_1^*! \cdots r_{L-1}^*!$ isolated points that are permutations of one another, we show that adding one extra neuron to each layer is sufficient to connect all these previously discrete minima into a single manifold. For a two-layer overparameterized network of width $r^*+ h =: m$ we explicitly describe the manifold of global minima: it consists of $T(r^*, m)$ affine subspaces of dimension at least $h$ that are connected to one another. For a network of width $m$, we identify the number $G(r,m)$ of affine subspaces containing only symmetry-induced critical points that are related to the critical points of a smaller network of width r<r^*. Via a combinatorial analysis, we derive closed-form formulas for $T$ and $G$ and show that the number of symmetry-induced critical subspaces dominates the number of affine subspaces forming the global minima manifold in the mildly overparameterized regime (small $h$) and vice versa in the vastly overparameterized regime ($h \gg r^*$). Our results provide new insights into the minimization of the non-convex loss function of overparameterized neural networks.

2021