Related papers: Generalized Reduction Procedure: Symplectic and Po…
Recent theoretical work has identified random projection as a promising dimensionality reduction technique for learning mixtures of Gausians. Here we summarize these results and illustrate them by a wide variety of experiments on synthetic…
In this paper, we show that contraction operations preserve the homology of $n$D generalized maps, under some conditions. Removal and contraction operations are used to propose an efficient algorithm that compute homology generators of $n$D…
While the existence of low-dimensional embedding manifolds has been shown in patterns of collective motion, the current battery of nonlinear dimensionality reduction methods are not amenable to the analysis of such manifolds. This is mainly…
A class of two dimensional field theories, based on (generically degenerate) Poisson structures and generalizing gravity-Yang-Mills systems, is presented. Locally, the solutions of the classical equations of motion are given. A general…
Fully non-linear equations of motion for dissipative general relativistic multi-fluids can be obtained from an action principle involving the explicit use of lower dimensional matter spaces. More traditional strategies for incorporating…
We propose a produre of reduction a locally conformal symplectic structure. This procedure of reduction can be applied to wide class of submanifolds. There are no local obstructions for this procedure. But there are global obstructions. We…
For a system of partial differential equations admitting point, contact, or higher symmetries, the framework of invariant reduction systematically computes how invariant geometric structures, such as conservation laws, presymplectic…
The version of Marsden-Ratiu reduction theorem for Nambu-Poisson manifolds by a regular distribution has been studied by Ib$\acute{\text{a}}\tilde{\text{n}}$ez et al. In this paper we show that the reduction is always ensured unless the…
Probabilistic Manifold Decomposition (PMD)\cite{doi:10.1137/25M1738863}, developed in our earlier work, provides a nonlinear model reduction by embedding high-dimensional dynamics onto low-dimensional probabilistic manifolds. The PMD has…
Generalized planning is concerned with the computation of plans that solve not one but multiple instances of a planning domain. Recently, it has been shown that generalized plans can be expressed as mappings of feature values into actions,…
The goal of this paper is to study approaches to bridge the gap between first-order and second-order type methods for composite convex programs. Our key observations are: i) Many well-known operator splitting methods, such as…
This work proposes a model-reduction approach for the material point method on nonlinear manifolds. Our technique approximates the $\textit{kinematics}$ by approximating the deformation map using an implicit neural representation that…
This note gives an overview on the construction of symplectic groupoids as reduced phase spaces of Poisson sigma models and its generalization in the infinite dimensional setting (before reduction).
We give a general approach to infinite dimensional non-Gaussian Analysis for measures which need not have a logarithmic derivative. This framework also includes the possibility to handle measures of Poisson type.
Retraction maps have been generalized to discretization maps in (Barbero Li\~n\'an and and Mart\'{\i}n de Diego, 2022). Discretization maps are used to systematically derive numerical integrators that preserve the symplectic structure, as…
This paper presents a regularized Newton method (RNM) with generalized regularization terms for unconstrained convex optimization problems. The generalized regularization includes quadratic, cubic, and elastic net regularizations as special…
The concept of Gaussian process tomography along with nonnegative constraints is applied in the context of high-resolution image reconstruction using segmented planar detectors with few readout channels.Expanding on the concept of 2-D…
This work introduces a Hamiltonian approach to regularization and linearization of central-force particle dynamics through a new canonical extension of the so-called "projective decomposition". The regularization scheme is formulated within…
We give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry (classical and non-classical) reductions of partial differential equations. Using the introduced definition of reduction of…
I present in this paper some tools in Symplectic and Poisson Geometry in view of their applications in Geometric mechanics and Mathematical Physics. After a short discussion of the Lagrangian and Hamiltonian formalisms, including the use of…