Advances variational Bayesian neural networks using singular learning theory.
problem Discrepancies between predictive performance and variational objective in BNNs.
method Corrected asymptotic form of singular posterior distributions to inform variational family design.
result Improvements in variational free energy and generalization error with proposed normalizing flow.
New variational principle found for non-variational differential equations.
problem Non-variational differential equations without variational multipliers.
method Connecting functional forms with antiexact differential forms to identify obstructions.
result Formulation of variational problem for non-variational equations.
Geometrically describes Jacobi equations for field theories with dissipation.
problem Describing field theories with dissipation geometrically.
method Prolongation of the Lagrangian on a k-cosymplectic formulation to describe Jacobi equations and a modified Lagrangian for variational formulation.
result Variational formulation of field theories with dissipation.
Study variation spaces for neural networks, linking them to approximation theory.
problem Understanding the variation spaces of shallow neural networks.
method Examined variation spaces defined by convex hulls and integral representations for a dictionary of functions.
result Found that Barron space, spectral Barron space, and Radon BV space are variation spaces for certain neural networks.
Variational inference has become one of the most widely used methods in latent variable modeling. In its basic form, variational inference employs a fully factorized variational distribution and minimizes its KL divergence to the posterior. As the minimization can only be carried out approximately, this approximation i…
Paper derives formulas for surface variations in shell theory.
problem Deriving first variation formulas for surfaces in thin shell theory.
method Using strain-displacement relations from thin shell theory.
result Provides formulas for linear Weingarten surfaces as stationary points.
Generalizes Hamiltonian theory for variational problems, applied to first order gravity.
problem Formulating Hamiltonian field theory for variational problems of general nature.
method Introduces a generalized Hamiltonian formalism without requiring a Hamiltonian section.
result Develops a novel multisymplectic Hamiltonian field theory for first order gravity.
Not only the Dirac operator, but also the spinor bundle of a pseudo-Riemannian manifold depends on the underlying metric. This leads to technical difficulties in the study of problems where many metrics are involved, for instance in variational theory. We construct a natural finite dimensional bundle, from which all th…
A setting for global variational geometry on Grassmann fibrations is presented. The integral variational functionals for finite dimensional immersed submanifolds are studied by means of the fundamental Lepage equivalent of a homogeneous Lagrangian, which can be regarded as a generalization of the well-known Hilbert for…
We present a Donaldson-Witten type field theory in eight dimensions on manifolds with Spin(7) holonomy. We prove that the stress tensor is BRST exact for metric variations preserving the holonomy and we give the invariants for this class of variations. In six and seven dimensions we propose similar theories on Calabi…
New framework explains deep neural networks using variational spline theory.
problem Understanding functions learned by deep neural networks.
method Developed a variational framework and function space.
result Deep ReLU networks are solutions to regularized data fitting problems over the proposed function space.
Derives stress-energy identities in Liouville theory on compact surfaces.
problem Stress-energy tensor correlation functions on compact Riemann surfaces.
method Varying correlation functions with respect to background metric, treating different types of variations separately.
result Stress-energy correlation functions expressed as differential operators acting on primary field correlation functions.
Systems of ordinary differential equations (or dynamical forms in Lagrangian mechanics), induced by embeddings of smooth fibered manifolds over one-dimensional basis, are considered in the class of variational equations. For a given non-variational system, conditions assuring variationality (the Helmholtz conditions) o…
Study on geometric variational problems for existence, regularity, and uniqueness of solutions.
problem Geometric variational problems, focusing on existence, regularity, and uniqueness of solutions.
method Formulated in Federer and Fleming's theory of currents, discussed the existence theory, and presented core ideas of the (interior) regularity theory for area-minimizing currents and optimal transport paths. Two original results on generic uniqueness of solutions were presented.
result Generic uniqueness of solutions for both Plateau's problem and optimal branched transport problem.
This paper presents a geometric-variational approach to continuous and discrete {\it second-order} field theories following the methodology of \cite{MPS}. Staying entirely in the Lagrangian framework and letting Y denote the configuration fiber bundle, we show that both the multisymplectic structure on J3Y as well…
Develops Palatini formalism in generalized geometry for string theory.
problem Formulating Palatini variation in generalized geometry.
method Palatini formalism within generalized Riemannian geometry of Courant algebroids.
result Natural emergence of generalized Levi-Civita connection and string effective actions.
Discrete Lagrange problems solved with Lie group constraints.
problem Solving discrete Lagrange problems with Lie group constraints.
method Proving critical sections are solutions of unconstrained variational problems, applying Noether theory and multisymplectic forms.
result Critical sections of discrete Lagrange problems are solutions of unconstrained variational problems.
An intrinsic description of the Hamilton-Cartan formalism for first-order Berezinian variational problems determined by a submersion of supermanifolds is given. This is achieved by studying the associated higher-order graded variational problem through the Poincaré-Cartan form. Noether theorem and examples from superfi…
Paper extends variational formula for Bismut-Cheeger eta form, proving key theorems in K-theory.
problem Extending variational formula for Bismut-Cheeger eta form without kernel bundle assumption.
method Twisting spinc Dirac operators by isomorphic vector bundles, proving Z2-graded additivity. result Analytic index in differential K-theory is a well-defined group homomorphism, and Riemann-Roch-Grothendieck theorem in R/Z K-theory. We study the obstruction to the exactness of the variational complex for a field theory on an affine bundle.
Finite energy pluripotential theory accommodates the variational theory of equations of complex Monge-Ampère type arising in Kähler geometry. Recently it has been discovered that many of the potential spaces involved have a rich metric geometry, effectively turning the variational problems in question into problems of …
The variational formalism for classical field theories is extended to the setting of Lie algebroids. Given a Lagrangian function we study the problem of finding critical points of the action functional when we restrict the fields to be morphisms of Lie algebroids. In addition to the standard case, our formalism include…
This work provides statistical guarantees for VAEs using PAC-Bayesian theory.
problem Theoretical properties of VAEs remain open questions.
method PAC-Bayesian theory to derive statistical guarantees.
result Upper bounds on Wasserstein distance between input and generative model.
Unified theory linking Bayesian and ensemble methods in deep learning.
problem Uncertainty quantification in deep learning.
method Reformulating optimisation as convex optimisation in probability measures, studying Wasserstein gradient flows.
result Unified theory explaining success of deep ensembles over variational inference.
The geometric Lagrangian theory (of arbitrary order) is based on the analysis of some basic mathematical objects such as: the contact ideal, the (exact) variational sequence, the existence of Euler-Lagrange and Helmholtz-Sonin forms, etc. In this paper we give new and much simpler proofs for the whole theory using Fock…
Develops methods for structured variational inference with star-structured models.
problem Inference in models with interdependent variables.
method Star-structured variational inference, existence, uniqueness, self-consistency proofs, approximation error bounds, gradient-based algorithm.
result First results for existence, uniqueness, and self-consistency of variational approximations in star-structured models.
Non-trivial obstructions found for topological solitons in Yang-Mills-Chern-Simons theories.
problem Existence of topological solitons in Yang-Mills-Chern-Simons theories on compact manifolds.
method Cohomological formulations of the calculus of variations, focusing on Yang-Mills-Chern-Simons theories on compact manifolds in odd dimensions.
result Non-trivial obstructions leading to a strong non-existence theorem for topological solitons.
DisCoPyro combines category theory with machine learning for program learning.
problem Applying category theory to machine learning tasks.
method Introducing DisCoPyro, a framework combining categorical structures with amortized variational inference.
result DisCoPyro can be applied in program learning for variational autoencoders and potentially contributes to AGI.
The abstract discusses extending learning objectives to measure theory for better generalization.
problem Improving out-of-distribution generalization and weakly-supervised learning.
method Extending variational learning objectives to measures.
result New objectives on measures may lead to practical algorithms.
Unified theory for semi-implicit variational inference, bridging approximation and optimization.
problem Developing a statistical theory for semi-implicit variational inference.
method Unified theory combining approximation and optimization analyses.
result Unified theory characterizes SIVI's ability to recover target distributions and governs asymptotic behavior.
We characterize the Lie derivative of spinor fields from a variational point of view by resorting to the theory of the Lie derivative of sections of gauge-natural bundles. Noether identities from the gauge-natural invariance of the first variational derivative of the Einstein(--Cartan)--Dirac Lagrangian provide restric…
New theory for area of Legendrian surfaces, proving smoothness and variational results.
problem Understanding the area of Legendrian surfaces under constraints.
method Introducing PHSLVs, proving sequential compactness, regularity, and variational results.
result Generalized regularity theory for Legendrian surfaces, achieving variational minima.
Develops variational Bayesian neural network for complex biomedical applications.
problem High computational cost of Markov Chain Monte Carlo in BNN.
method Variational Bayes inference for posterior consistency and classification accuracy.
result Developed statistical theory for posterior consistency and prediction accuracy.
A method to derive Lagrangians from field equations in metric-affine theories of gravity.
problem Deriving Lagrangians from field equations in metric-affine theories of gravity.
method Variational completion method to transform field equations into Euler-Lagrange equations and find a Lagrangian.
result Starting from metric equations, full metric equations and Lagrangian can be derived up to metric-independent terms.
The paper studies the smoothness of critical points of variational integrals on Hessian spaces.
problem The study focuses on the regularity of critical points of variational integrals defined on Hessian spaces.
method The approach involves solving a fourth order nonlinear equation and analyzing the Hessian of the critical points.
result Smooth critical points with bounded Hessian are shown to be smooth provided their Hessian has small BMO.
Generalizes Carathéodory form for higher-order field theories.
problem Extending the Carathéodory form to second and higher-order Lagrangians.
method Geometric operations applied to the Poincaré--Cartan form and Lepage forms.
result Generalized Carathéodory form for second and higher-order Lagrangians.
Extends Campanato theory to multi-valued functions for geometric variational problems.
problem Regularity of multi-valued functions in geometric variational problems.
method Adapting Campanato's ideas to multi-valued functions, proving regularity theorems.
result Established regularity for multi-valued harmonic functions and stationary integral varifolds.
A new framework for Einstein-Hilbert action with topological variations.
problem Understanding critical points and dimensionality in Einstein-Hilbert action.
method Localized Einstein-Hilbert variational principle, topology on Sobolev configurations, topological variations.
result No critical points in dimension 4, higher dimensions free of this problem.
New definition of disentanglement for non-independent factors of variation.
problem Current disentanglement definitions assume independent factors of variation, limiting their applicability.
method Definition based on information theory, related to Information Bottleneck Method, proposed measurement method.
result Proposed method correctly measures disentanglement with non-independent factors of variation.
We introduce a new geometric evolution equation for hypersurfaces in asymptotically flat spacetime initial data sets, that unites the theory of marginally outer trapped surfaces (MOTS) with the study of inverse mean curvature flow in asymptotically flat Riemannian manifolds. A theory of weak solutions is developed usin…
Lecture notes from the Concentrated Graduate Course preceding the Workshop on Hodge Theory in String Theory at the Fields Institute in Toronto, November 11--15, 2013.
This short note contains an explicit proof of the Jacobi identity for variational Schouten bracket in Z2-graded commutative setup. For the reasoning to be rigorous, it refers to the product bundle geometry of iterated variations (see arXiv:1312.1262 [math-ph]); no ad hoc regularizations occur anywhere in this theory…
This paper introduces Wasserstein variational inference, a new form of approximate Bayesian inference based on optimal transport theory. Wasserstein variational inference uses a new family of divergences that includes both f-divergences and the Wasserstein distance as special cases. The gradients of the Wasserstein var…
New insights into variational inference using Monte Carlo estimates.
problem Improving variational bounds in latent variable models.
method Analyzing properties of Monte Carlo estimates and their impact on variational gaps.
result Negative correlation reduces variational gaps, contrary to intuition.
We study a functional that derives from the classical Yang-Mills functional and Born-Infeld theory. We establish its first variation formula and prove the existence of critical points. We also obtain the second variation formula.
By resorting to Noether's Second Theorem, we relate the generalized Bianchi identities for Lagrangian field theories on gauge-natural bundles with the kernel of the associated gauge-natural Jacobi morphism. A suitable definition of the curvature of gauge-natural variational principles can be consequently formulated in …
Proposes a new variational principle for Einstein gravity.
problem Formulating Einstein gravity as a gauge theory for the conformal group.
method First order formulation of conformal tractor geometry, variational principle based on abstract principal bundle.
result Provides first order field equations without requiring supplementary constraints.
Lecture notes on crystallography and discrete surfaces.
problem Mathematical modeling of crystal structures.
method Variational principle and discrete surface theory.
result Most symmetric crystal structures identified.