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.
We show a very simple and general total second variation formula for Perelman's W-functional at arbitrary points in the space of Riemannian metrics. Moreover we perform a study of the properties of the variations of Kähler structures. We deduce a quite simple and general total second variation formula for P…
The paper studies variational functionals for submanifolds using the Lepage form.
problem Variational functionals for submanifolds in Grassmann fibrations.
method Introduces the fundamental Lepage form and uses it to study variations of submanifolds.
result Proves the first infinitesimal variation formula and Euler-Lagrange equations.
Geometric framework analyzes bias in variational inference for posterior functionals.
problem Analyzing the bias of posterior functionals under variational approximations.
method Developed a geometric framework to evaluate the bias of posterior functionals using the variational tangent space.
result The leading-order bias of a posterior functional is determined by its component orthogonal to the variational tangent space.
Study on new Monge-Ampère functionals and their variational problems.
problem Existence and uniqueness of solutions for nonlinear eigenvalue problems.
method Introduction of a family of real Monge-Ampère functionals and proving Sobolev type inequalities.
result Existence of solutions for a nonlinear eigenvalue problem.
Formulas for mean curvature in various spaces.
problem Understanding mean curvature in different geometric settings.
method First and second variation formulas for arbitrary variations in Riemannian and sub-Riemannian manifolds.
result Formulas derived for mean curvature in Heisenberg group via approximation.
Derive Dirichlet scalar curvature energy functional variation formula
problem Dirichlet scalar curvature energy functional
method First variation formula
result Introduce Dirichlet-Einstein metrics
New method quantifies and bounds changes in non-stationary optimization problems.
problem Optimizing under changing cost functions with limited variation.
method Proposes Lp,q-variation functional for quantifying changes and derives regret bounds. result Upper and lower bounds for smooth and strongly convex functions, matching optimal rates.
This tutorial derives the VAE loss function under Gaussian assumptions.
problem Computational intractability of posterior distributions in Bayesian machine learning.
method Derives the variational lower bound loss function of a standard VAE.
result The Kullback-Leibler divergence has a closed form solution under Gaussian assumptions.
Derives a new variational approach to information bottleneck.
problem Information theoretic inference and predictive modeling.
method Variational lower bound of predictive information bottleneck.
result Generalizes modern inference procedures and suggests new ones.
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.
Extends calculus of variations to generalized functions.
problem Extending calculus of variations to generalized functions.
method Category of generalized smooth functions, proving connections and conditions.
result Full connections between extremals and Euler-Lagrange equations, necessary conditions for minimizers.
Variational neural networks optimize activation functions using gradient descent.
problem Lack of guiding principles for choosing activation functions in neural networks.
method Variational neural networks use a linear combination of candidate functions, optimizing via gradient descent.
result Optimal activation functions can be found using gradient descent.
New approach to handle ranking function variation in zero-shot NAS.
problem Variation in ranking function outputs due to randomness.
method Viewing ranking function output as a random variable and constructing a stochastic ordering.
result Stochastic ordering boosts performance in neural architecture search.
The paper studies variations of σu-curvature for submanifolds in Riemannian manifolds.
problem Understanding the behavior of σu-curvature under variations of submanifolds. method Analyzes the functional of σu-curvature for submanifolds of arbitrary codimension in Riemannian manifolds. result Provides insights into the variational properties of σu-curvature. New method finds global Lagrangians for variational systems.
problem Constructing global variational principles for variational systems.
method Analyzing Lepage 2-forms and finding global Lagrangians for systems defined by homogeneous functions of degree \(c
eq 0, 1\).
result Locally variational systems defined by homogeneous functions of degree \(c
eq 0, 1\) are globally variational.
The paper studies stability of discrete planar curves using variational methods.
problem Stability of discrete planar curves under area constraints.
method Unified interpretation of discrete curvatures, determination of equilibrium curves, stability analysis.
result Equilibrium curves for the length functional under area-constraint conditions are determined and their stability is studied.
GWI combines deep neural networks with Gaussian processes for better predictive performance and uncertainty quantification.
problem Combining deep learning with Gaussian process uncertainty quantification.
method Gaussian Wasserstein inference (GWI) using Wasserstein distance between Gaussian measures.
result GWI achieves state-of-the-art performance on benchmark datasets.
FTIP uses normalizing flows to improve posterior inference in function space.
problem Challenges in posterior inference with implicit-process priors.
method FTIP uses normalizing flows to define a richer variational distribution over combination weights.
result FTIP captures asymmetric and multimodal posterior structure better than Gaussian coefficient approximations.
In Heisenberg group, certain graphs are stable under contact variations but not area-minimizing.
problem Characterizing stable sub-Riemannian graphs in the Heisenberg group.
method Analyzing intrinsic graphs of smooth functions in the first Heisenberg group under contact variations.
result Intrinsic graphs of smooth functions are stable points of sub-Riemannian perimeter under contact variations, but not area-minimizing.
Proposes a new DQN framework using variational inference.
problem Efficient exploration and large-scale MDPs.
method Variational inference for value function parameters.
result Achieves efficient exploration on large-scale MDPs.
EigenVI uses orthogonal function expansions for efficient variational inference.
problem Efficiently approximate complex distributions in variational inference.
method EigenVI constructs variational approximations using orthogonal function expansions, minimizing Fisher divergence.
result EigenVI provides more accurate approximations than existing methods for Gaussian BBVI.
New method approximates diffusion process posteriors using moment functions.
problem Approximating posteriors of stochastic differential equations.
method Constructs variational process as controlled prior, approximates posterior with moment functions, uses natural gradient descent.
result Richer variational approximations for state-dependent diffusion terms.
The paper studies curves in Riemannian manifolds using total variation flow.
problem Analyzing the evolution of curves in Riemannian manifolds using total variation.
method Defining and proving the existence of strong solutions to the flow equations, showing variational equality, and proving convergence.
result Strong solutions converge to a constant map in finite time for non-positive sectional curvature.
fBNNs use stochastic processes for variational inference in neural networks.
problem Difficulties in specifying priors and posteriors in high-dimensional weight spaces.
method Maximize Evidence Lower Bound (ELBO) on stochastic processes, using spectral Stein gradient estimator.
result fBNNs provide reliable uncertainty estimates and extrapolate well with structured priors.
Semi-Implicit Variational Inference broadens variational distributions.
problem Expanding the variational family to include implicit distributions.
method Mixing variational parameters with a flexible distribution.
result SIVI provides an asymptotically exact surrogate ELBO for optimization.
New variational principle found for PDEs with symmetries and conservation laws.
problem Finding variational principles for PDEs with symmetries and conservation laws.
method Proving existence of a variational principle for PDEs with symmetries and conservation laws.
result A differential equation with sufficient symmetries and conservation laws leads to a variational functional.
Study introduces indecomposability for varifolds, leading to geometric consequences.
problem Understanding the structure of varifolds and their connectedness properties.
method Introducing indecomposability and related concepts for varifolds.
result Substantial geometric consequences derived from the connectedness properties of varifolds.
Variational method for eigenvalues on manifolds.
problem Optimizing functionals involving eigenvalues of Riemannian manifolds.
method New Palais-Smale sequences and min-max methods for locally-Lipschitz functionals.
result Convergence of Palais-Smale sequences in Laplace and Steklov eigenvalues.
The Riemann-Theta Boltzmann machine's visible sector is sampled using a discrete multi-variate Gaussian.
problem Sampling the visible sector of the Riemann-Theta Boltzmann machine.
method Discrete multi-variate Gaussian over the hidden state space.
result The visible sector probability density function is an infinite mixture of multi-variate Gaussians.
A new algorithm for optimizing probability distributions converges linearly.
problem Optimizing functionals over families of probability distributions.
method Variational transport: particle-based algorithm approximating Wasserstein gradient descent.
result Variational transport converges linearly to the global minimum of the objective functional.
The article describe the model, derivation, and implementation of variational Bayesian inference for linear and logistic regression, both with and without automatic relevance determination. It has the dual function of acting as a tutorial for the derivation of variational Bayesian inference for simple models, as well a…
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.
Extends Einstein-Hilbert functional definition for stable manifolds.
problem Stability of Einstein manifolds on Riemannian manifolds.
method Second variation of generalized Einstein-Hilbert functional.
result Properties of stable Einstein manifolds presented.
A new method solves variational inequality problems with multiple constraints without needing optimal Lagrange multipliers.
problem Solving variational inequality problems with multiple functional constraints efficiently.
method Constrained Gradient Method (CGM) for Minty variational inequality problems.
result The Constrained Gradient Method achieves complexity similar to projection-based methods but with cheaper oracles.
New method improves variational bounds on partition function.
problem Computing the partition function of discrete graphical models is intractable.
method Combines gauge transformations with weighted mini-bucket elimination (WMBE).
result WMBE-G strictly improves earlier WMBE approximation for symmetric models.
Proposes a new method for continual learning in neural networks.
problem Challenges in applying sequential Bayesian inference to neural networks.
method Sequential function-space variational inference.
result Neural networks trained with the proposed method achieve better predictive accuracy.
Elvet solves differential equations and variational problems with neural networks.
problem Solving complex differential and variational equations with arbitrary conditions.
method Machine learning, specifically neural networks, to represent and solve equations.
result Elvet can solve a wide range of differential and variational problems.
Unified approach for predicting missing segments in partially observed functions.
problem Predicting missing segments in partially observed functions with complex dependence and irregular noise.
method Unified registration and prediction approach under the conformal prediction framework, integrating amplitude and phase components.
result Effective prediction bands with finite-sample marginal coverage guarantees under weak assumptions.
New method for constrained sampling using gradient flows.
problem Sampling from constrained domains.
method Introducing a boundary condition for gradient flow to confine particles within the domain.
result Provable continuous-time convergence in total variation for constrained sampling.
VANO uses neural operators for unsupervised learning of functional data.
problem Learning operators between infinite dimensional spaces for functional data.
method Variational Autoencoding Neural Operators (VANO) approach.
result VANO can learn and reconstruct functional data without supervision.
Paper develops a classification method using matrix-variate t-distributions.
problem Classifying matrix-valued observations with dependence structure.
method Develops an Expectation-Maximization algorithm for discriminant analysis.
result Method shows promise on various datasets.
Investigates invariant hulls of functionals on manifolds.
problem Understanding meaningful functionals on manifolds through reparameterizations.
method Uses inner-variations to transform arbitrary functionals into invariant realizations.
result Explicit computations for volume functional in N-dimensional manifolds, especially in N=2. New minimax rates for total variation denoising in higher dimensions, showing limitations of linear smoothers.
problem Estimating functions with bounded total variation over high-dimensional grids.
method Minimax analysis, focusing on linear and non-linear estimators.
result Linear estimators like Laplacian smoothing and eigenmaps are suboptimal for total variation denoising in higher dimensions.
Study variational problem for time-like curves in Einstein universe.
problem Variational problem for time-like curves in Einstein universe.
method Conformally invariant variational problem, analysis of stationary curves, integration by quadratures.
result Stationary curves are trapped into Einsetin universes of dimension 2, 3, or 4.
Study on a new Yang-Mills type functional and its critical points.
problem Deriving a new functional from Yang-Mills and Born-Infeld theories.
method First and second variation formulas established.
result Existence of critical points proved.
We prove the existence of a continuous BV minimizer with C0 boundary value for the p-area (pseudohermitian or horizontal area) in a parabolically convex bounded domain. We extend the domain of the area functional from BV functions to vector-valued measures. Our main purpose is to study the first and second v…
Paper proposes a new method to minimize submodular functions with fewer calls to simpler oracles.
problem Minimizing the sum of submodular set functions with limited information.
method Introduces a modified convex problem requiring constrained total variation oracles that can be solved with fewer calls to minimization oracles.
result Shows significant reduction in the number of calls to minimization oracles.