Related papers: Symplectic Bregman divergences
Non-Hermitian random matrices with symplectic symmetry provide examples for Pfaffian point processes in the complex plane. These point processes are characterised by a matrix valued kernel of skew-orthogonal polynomials. We develop their…
We study subsets in possibly degenerate symplectic vector spaces over finite fields, which are stable under a given Coxeter/Weyl reflection group. These symplectic root systems provide crucial combinatorical data to classify…
Discrete diffusion models have recently emerged as a promising alternative to the autoregressive approach for generating discrete sequences. Sample generation via gradual denoising or demasking processes allows them to capture hierarchical…
We adapt Gromov's notion of ideal-valued measures to symplectic topology, and use it for proving new results on symplectic rigidity and symplectic intersections. Furthermore, it enables us to discuss three "big fiber theorems", the…
The bias-variance decomposition is a central result in statistics and machine learning, but is typically presented only for the squared error. We present a generalization of the bias-variance decomposition where the prediction error is a…
A method of constructing a class of bihamiltonian structures is presented. Elements of this class are generalizations of the so-called bihamiltonian structures of general position on odd-dimensional manifolds. The method consists in a…
Starting from a given norm on the vector space of exact 1-forms of a compact symplectic manifold, we produce pseudo-distances on its symplectomorphism group by generalizing an idea due to Banyaga. We prove that in some cases (which include…
Metaplectic Wigner distributions were recently investigated as natural generalizations of the classical Wigner distribution, and provide a wide class of time-frequency representations that exploits the structure of the symplectic group.…
The idea of slicing divergences has been proven to be successful when comparing two probability measures in various machine learning applications including generative modeling, and consists in computing the expected value of a `base…
In knowledge graph embedding, the theoretical relationship between the softmax cross-entropy and negative sampling loss functions has not been investigated. This makes it difficult to fairly compare the results of the two different loss…
We formulate em algorithm in the framework of Bregman divergence, which is a general problem setting of information geometry. That is, we address the minimization problem of the Bregman divergence between an exponential subfamily and a…
We establish Bernstein's inequalities for functions of general (general-state-space and possibly non-reversible) Markov chains. These inequalities achieve sharp variance proxies and encompass the classical Bernstein inequality for…
This paper introduces a new notion of a Fenchel conjugate, which generalizes the classical Fenchel conjugation to functions defined on Riemannian manifolds. We investigate its properties, e.g.,~the Fenchel--Young inequality and the…
We examine the duality theory for a class of non-convex functions obtained by composing a convex function with a continuous one. Using Fenchel duality, we derive a dual problem that satisfies weak duality under general assumptions. To…
In the recent years, considerable attention has been paid to preserving structures and invariants in reduced basis methods, in order to enhance the stability and robustness of the reduced system. In the context of Hamiltonian systems,…
Stochastic gradient methods for minimizing nonconvex composite objective functions typically rely on the Lipschitz smoothness of the differentiable part, but this assumption fails in many important problem classes like quadratic inverse…
We propose a novel framework for the regularised inversion of deep neural networks. The framework is based on the authors' recent work on training feed-forward neural networks without the differentiation of activation functions. The…
We compute the semi-global symplectic invariants near the hyperbolic equilibrium points of the Euler top. The Birkhoff normal form at the hyperbolic point is computed using Lie series. The actions near the hyperbolic point are found using…
The dynamics of symplectic gauge fields provides a consistent framework for fundamental interactions based on spin three gauge fields. One remarkable property is that symplectic gauge fields only have minimal couplings with gravitational…
Skew-symmetric differential forms play an unique role in mathematics and mathematical physics. This relates to the fact that closed exterior skew-symmetric differential forms are invariants. The concept of "Exterior differential forms" was…