arXiv research

New method finds global Lagrangians for variational systems.

Introduces a restricted Chen-Nagano variational principle for the Einstein-Hilbert functional.

We derive both {\em local} and {\em global} generalized {\em Bianchi identities} for classical Lagrangian field theories on gauge-natural bundles. We show that globally defined generalized Bianchi identities can be found without the {\em a priori} introduction of a connection. The proof is based on a {\em global} decom…

Unified approach classifies stable and minimal elastic curves.

New proof for global rigidity of vertex scaling on polyhedral surfaces.

Federated learning for Bayesian clustering of large datasets.

Representation learning seeks to expose certain aspects of observed data in a learned representation that's amenable to downstream tasks like classification. For instance, a good representation for 2D images might be one that describes only global structure and discards information about detailed texture. In this paper…

Study curve flows with global forcing terms using a distance comparison principle.

Proposes a variational approach to shallow neural networks, bypassing optimization.

This paper investigates the use of methods from partial differential equations and the Calculus of variations to study learning problems that are regularized using graph Laplacians. Graph Laplacians are a powerful, flexible method for capturing local and global geometry in many classes of learning problems, and the tec…

New variational principle found for non-variational differential equations.

Local-to-global principle for Morse actions on symmetric spaces.

Some intrinsic tools from the formal theory of variational equations are being demonstrated at work in application to one concrete example of the third-order evolution equation of free relativistic top in three-dimensional space-time. The main goal is to introduce a combined approach consisting in the simultaneous util…

Inversive distance circle packing metric was introduced by P Bowers and K Stephenson \cite{BS} as a generalization of Thurston's circle packing metric \cite{T1}. They conjectured that the inversive distance circle packings are rigid. For nonnegative inversive distance, Guo \cite{Guo} proved the infinitesimal rigidity a…

Internal Lagrangians derived from variational principles.

Non-trivial obstructions found for topological solitons in Yang-Mills-Chern-Simons theories.

New variational principles found for conformal geodesics.

We give a brief introduction to some of the recent works on finding geometric structures on triangulated surfaces using variational principles.

PA reinterpreted as SB problem, unifying thermodynamics and optimal transport.

New mass inequalities and proofs for causal variational principles.

This paper improves a local-to-global principle for Morse quasigeodesics.

Variational reduction simplifies Lagrangian systems with scaling symmetries.

Autoregressive feedback is considered a necessity for successful unconditional text generation using stochastic sequence models. However, such feedback is known to introduce systematic biases into the training process and it obscures a principle of generation: committing to global information and forgetting local nuanc…

The paper studies the dimension of limit sets using variational principles and stationary measures.

The paper proves rigidity of bordered polyhedral surfaces using variational principles.

We provide a variational description of any Liouville (i.e. volume preserving) autonomous vector fields on a smooth manifold. This is obtained via a ``maximal degree'' variational principle; critical sections for this are integral manifolds for the Liouville vector field. We work in coordinates and provide explicit for…

We discuss a recently proposed variational principle for deriving the variational equations associated to any Lagrangian system. The principle gives simultaneously the Lagrange and the variational equations of the system. We define a new Lagrangian in an extended configuration space ---which we call D'Alambert's--- com…

For any positive integer $n$ and any Lie group $\mathfrak{G}$, given a definite symmetric bilinear form on $\mathbb{R}^n$ and an $\hbox{Ad}$-invariant scalar product on the Lie algebra of $\mathfrak{G}$, we construct a variational problem on fields defined on an arbitrary oriented $(n+\hbox{dim}\mathfrak{G})$-dimension…

Variationality of conformal geodesics fails in higher dimensions.

In this paper, we provide an information-theoretic interpretation of the Vector Quantized-Variational Autoencoder (VQ-VAE). We show that the loss function of the original VQ-VAE can be derived from the variational deterministic information bottleneck (VDIB) principle. On the other hand, the VQ-VAE trained by the Expect…

PRI-VAE learns disentangled representations by optimizing principle-of-relevant-information.

We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded sectional curvature.

A variational principle is proposed for obtaining the Jacobi equations in systems admitting a Lagrangian description. The variational principle gives simultaneously the Lagrange equations of motion and the Jacobi variational equations for the system. The approach can be of help in finding constants of motion in the Jac…

Paper variates Navier-Stokes-Fourier system for thermodynamic consistency.

The paper explores variational principles for equations of maximal symmetry, providing new insights and results.

We prove that under certain assumptions a partial differential equation can be derived from a variational principle. It is well-known from Noether's theorem that symmetries of a variational functional lead to conservation laws of the corresponding Euler-Lagrange equation. We reverse this statement and prove that a diff…

Variational inference methods for latent variable statistical models have gained popularity because they are relatively fast, can handle large data sets, and have deterministic convergence guarantees. However, in practice it is unclear whether the fixed point identified by the variational inference algorithm is a local…

The variational principle and the corresponding differential equation for geodesic circles in two dimensional (pseudo)-Riemannian space are being discovered. The relationship with the physical notion of uniformly accelerated relativistic particle is emphasized. The known form of spin-curvature interaction emerges due t…

Gradients help find global optima in complex functions.

The paper provides guarantees for a tangent transform algorithm in logistic regression models.

The calculus of variations for lagrangians which are not functions on the tangent bundle, but sections certain affine bundles is developed. We follow a general approach to variational principles which admits boundary terms of variations.

Variational autoencoders provide a principled framework for learning deep latent-variable models and corresponding inference models. In this work, we provide an introduction to variational autoencoders and some important extensions.

Proves a principle for one-phase Bernoulli problem minimizers.

In this paper, we develop several related finite dimensional variational principles for discrete optimal transport (DOT), Minkowski type problems for convex polytopes and discrete Monge-Ampere equation (DMAE). A link between the discrete optimal transport, discrete Monge-Ampere equation and the power diagram in computa…

A new method for automatically learning metric scaling in metric-based meta-learning.

A conjecture of Hirschowitz's predicts that a globally generated vector bundle $W$ on a compact complex manifold $A$ satisfies the formal principle, i.e., the formal neighborhood of its zero section determines the germ of neighborhoods in the underlying complex manifold of the vector bundle $W$. By applying Cartan's eq…

New discrete conformal structures on surfaces with boundary, proving global rigidity and constructing hyperbolic metrics.

Alternative proof of weak solutions to mean curvature flow using minimizing movements.