The paper extends a theorem about momentum maps to singular symplectic spaces.
problem Extending a theorem about momentum maps to singular symplectic spaces.
method Using integral affine stratification and equivariant locally trivial fibrations, the paper extends the linear variation theorem to singular values of the momentum map.
result Cohomology classes of symplectic forms on reduced spaces vary linearly within strata.
A scalable method for accurate inference of low-dimensional parameters in high-dimensional linear regression.
problem Statistical inference for low-dimensional parameters in high-dimensional linear regression models.
method Mean-field variational Bayes approach, focusing on nuisance parameters and conditional distributions.
result Competitive numerical performance and theoretical guarantees for estimation and uncertainty quantification.
The paper tackles prescribing discrete Gaussian curvature on polyhedral surfaces.
problem Prescribing discrete Gaussian curvature on polyhedral surfaces.
method Discrete conformal theory and variational principles with constraints.
result Proves Kazdan-Warner type theorems for polyhedral surfaces.
A key challenge for modern Bayesian statistics is how to perform scalable inference of posterior distributions. To address this challenge, variational Bayes (VB) methods have emerged as a popular alternative to the classical Markov chain Monte Carlo (MCMC) methods. VB methods tend to be faster while achieving comparabl…
New insights into cascade feedback linearization of control systems.
problem Obtaining a cascade feedback linearization for invariant control systems.
method Introducing truncated versions of operators from the calculus of variations to prove new theorems.
result Established new geometry and foundational theorems for future work.
The paper explores variational principles for equations of maximal symmetry, providing new insights and results.
problem Exploring variational principles for equations of maximal symmetry.
method Study of variational and divergence symmetries for linear and nonlinear equations of maximal symmetry, providing first integrals in explicit form.
result Significantly different results and more general variational symmetry algebra for linear and nonlinear equations compared to previous studies.
In this paper we prove a variation of the theorem in title, for equations with periodic coefficients, in Frechet spaces. The main result gives equivalent conditions ensuring the reduction of such an equation to one with constant coefficient. In the particular case of C∞, we obtain the exact analogue of the cl…
Study introduces a new Allen-Cahn energy on hypersurfaces and analyzes its properties.
problem Analyzing geometric variations of the Allen-Cahn energy on hypersurfaces.
method Establishes Γ-convergence, computes variations, and analyzes the linearized equation.
result Shows that the index and nullity of the energy are related to the Allen-Cahn index and nullity.
Noether theorem applied to variational problems on hyperbolic surfaces.
problem Variational problems on hyperbolic surfaces.
method Noether's theorem on symmetry and conservation laws.
result Application to geometric problems on hyperbolic surfaces.
We prove sectional and Ricci-type comparison theorems for the existence of conjugate points along sub-Riemannian geodesics. In order to do that, we regard sub-Riemannian structures as a special kind of variational problems. In this setting, we identify a class of models, namely linear quadratic optimal control systems,…
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.
We formulate stochastic partial differential equations on Riemannian manifolds, moving surfaces, general evolving Riemannian manifolds (with appropriate assumptions) and Riemannian manifolds with random metrics, in the variational setting of the analysis to stochastic partial differential equations. Considering mainly …
New method proves Alexandrov theorem for curved spacetimes.
problem Proving Alexandrov theorem for curved surfaces with conical singularities.
method Adapting Volkov's variational method to handle Lorentzian angles.
result Existence of a locally Minkowski 3-manifold with conical singularities isometric to a given surface.
The paper proves signatures of non-geometric rough paths can approximate functionals uniformly.
problem Approximating functionals of non-geometric rough paths.
method Extending rough paths with time and quadratic variation terms, proving uniform approximation.
result Linear functionals of extended signatures uniformly approximate continuous functionals.
Represents neural networks as solutions to inverse problems in Banach spaces.
problem Understanding the function learned by neural networks.
method Variational framework, representer theorem, polynomial ridge splines.
result Neural networks are solutions to inverse problems in Banach spaces.
Variational inference is a powerful tool for approximate inference, and it has been recently applied for representation learning with deep generative models. We develop the variational Gaussian process (VGP), a Bayesian nonparametric variational family, which adapts its shape to match complex posterior distributions. T…
We will study a linear first order system, a connection $\db$ problem, on a vector bundle equipped with a connection, over a Riemann surface. We show optimal conditions on the connection forms which allow one to find a holomorphic frame, or in other words to prove the optimal regularity of our solution. The underlying …
This paper considers a new family of variational distributions motivated by Sklar's theorem. This family is based on new copula-like densities on the hypercube with non-uniform marginals which can be sampled efficiently, i.e. with a complexity linear in the dimension of state space. Then, the proposed variational densi…
In this work we consider a question in the calculus of variations motivated by riemannian geometry, the isoperimetric problem. We show that solutions to the isoperimetric problem, close in the flat norm to a smooth submanifold, are themselves smooth and C2,α-close to the given sub manifold. We show also a version …
Paper introduces vector-valued variation spaces for multi-output neural networks.
problem Understanding and optimizing multi-output neural networks.
method Development of vector-valued variation spaces and representer theorem.
result Novel bounds for layer widths in deep networks and a convex optimization method for compression.
This paper studies arbitrage pricing theory in financial markets with implicit transaction costs. We extend the existing theory to include the more realistic possibility that the price at which the investors trade is dependent on the traded volume. The investors in the market always buy at the ask and sell at the bid p…
Paper develops formulas and theorems in Hermitian geometry.
problem None explicitly stated in the abstract.
method Develops second variational formulas and index forms in Hermitian geometry.
result Establishes results analogous to classical theorems in Riemannian geometry.
Proof of Tait-Kneser theorem and related variations using Lorentzian geometry.
problem Proving variations of the Tait-Kneser theorem for different conics.
method Using Lorentzian geometry to prove the theorem and its variations.
result Proof of the theorem and its variations concerning different conics.
We provide a new extension of Breiman's Theorem on computing tail probabilities of a product of random variables to a multivariate setting. In particular, we give a complete characterization of regular variation on cones in [0,∞)d under random linear transformations. This allows us to compute probabilities of a…
First, we review the Dirac operator folklore about basic analytic and geometrical properties of operators of Dirac type on compact manifolds with smooth boundary and on closed partitioned manifolds and show how these properties depend on the construction of a canonical invertible double and are related to the concept o…
Noether's First Theorem yields conservation laws for Lagrangians with a variational symmetry group. The explicit formulae for the laws are well known and the symmetry group is known to act on the linear space generated by the conservation laws. In recent work the authors showed the mathematical structure behind both th…
Extends curve theory to non-smooth data with finite curvature and torsion.
problem Applying classical curve theory to non-smooth data.
method Using distributional derivative measures of functions of bounded variation.
result Essentially unique non-smooth curve solution with finite total curvature and torsion.
This paper belongs to the realm of conformal geometry and deals with Euclidean submanifolds that admit smooth variations that are infinitesimally conformal. Conformal variations of Euclidean submanifolds is a classical subject in differential geometry. In fact, already in 1917 Cartan classified parametrically the Eucli…
Splitting theorem for non-positively curved Lorentzian spaces.
problem Understanding curvature in Lorentzian spaces.
method Proving a splitting theorem with global non-positive timelike curvature and extending first variation formula.
result Splitting theorem for Lorentzian pre-length spaces with global non-positive timelike curvature.
Paper proves Toponogov's theorem in Alexandrov geometry.
problem Proving Toponogov's theorem in Alexandrov geometry with lower curvature bound.
method Inspired by Riemannian geometry, uses second variation formula.
result Elementary proof of Toponogov's theorem in Alexandrov geometry.
We propose to optimize the activation functions of a deep neural network by adding a corresponding functional regularization to the cost function. We justify the use of a second-order total-variation criterion. This allows us to derive a general representer theorem for deep neural networks that makes a direct connectio…
The necessary and sufficient conditions for existence of a generalized representer theorem are presented for learning Hilbert space-valued functions. Representer theorems involving explicit basis functions and Reproducing Kernels are a common occurrence in various machine learning algorithms like generalized least squa…
We address the integrability conditions of the inverse problem of the calculus of variations for time-dependent SODE using the Spencer version of the Cartan-Kähler theorem. We consider a linear partial differential operator P given by the two Helmholtz conditions expressed in terms of semi-basic 1-forms and study its…
Study joint invariants on symplectic spaces, extending group and space variations.
problem Computing joint invariants on linear symplectic spaces.
method Review and extend previous work on group and space variations, relate to differential invariants.
result New computations and extensions of joint invariants.
We develop a variational theory of geodesics for the canonical variation of the metric of a totally geodesic foliation. As a consequence, we obtain comparison theorems for the horizontal and vertical Laplacians. In the case of Sasakian foliations, we show that sharp horizontal and vertical comparison theorems for the s…
The paper studies stability of mean-field variational inference for log-concave distributions.
problem Stability of mean-field variational inference for log-concave distributions.
method Novel approach via linearized optimal transport, lifting non-convex problem to convex optimization over transport maps.
result Dimension-free Lipschitz continuity of the MFVI optimizer with respect to the target distribution, measured in 2-Wasserstein distance.
New mass inequalities and proofs for causal variational principles.
problem Proving new mass inequalities for causal variational principles.
method Proved a new inequality for minimizers of causal variational principles and applied it to prove the positive mass theorem.
result Introduced a positive quasilocal mass and proved new mass inequalities.
Study high-dimensional Bayesian linear regression using variational inference.
problem High-dimensional Bayesian linear regression with product priors.
method Non-linear large deviations theory and variational inference.
result Unique optimizer in variational problem governs posterior distribution under separation condition.
Study proves Liouville theorem for specific curvature equations with boundary conditions.
problem Proving Liouville theorem for σk-curvature equations in half spaces with nonlinear boundary conditions. method Established using positive constant curvature equations and variational functional approach.
result Proved Liouville theorem for positive constant σk-curvature equations in R+n and boundary conditions. This work interprets SFA through variational inference, relaxing linearity constraints.
problem Recover non-linear SFA from variational inference.
method Probabilistic interpretation of SFA through variational inference, relaxing linearity constraints.
result Reinterprets SFA as a variational framework, allowing slowness as a regularizer to reconstruction loss.
We provide a constructive, variational proof of Rivin's realization theorem for ideal hyperbolic polyhedra with prescribed intrinsic metric, which is equivalent to a discrete uniformization theorem for spheres. The same variational method is also used to prove a discrete uniformization theorem of Gu et al. and a corres…
New integral theorems improve density function estimations.
problem Improving density function estimations.
method Integrals based on cyclic functions and Riemann sums, Fourier integral theorem, Monte Carlo methods, variational approach, Cauchy residue theorem.
result Optimal cyclic functions minimize square integrals, improving density estimations.
Internal Lagrangians derived from variational principles.
problem Reproducing the principle of stationary action in variational geometry.
method Introducing stationary points of internal Lagrangians, establishing connections with symmetries and conservation laws, and investigating relations between non-degenerate and internal Lagrangians.
result Noether's theorem reformulated in terms of internal Lagrangians.
In the classical Lagrangian approach to conservation laws of gauge-natural field theories a suitable (vector) density is known to generate the so--called {\em conserved Noether currents}. It turns out that along any section of the relevant gauge--natural bundle this density is the divergence of a skew--symmetric (tenso…
Continuous family of elliptic operators' projections maintain Cauchy data spaces.
problem Maintaining Cauchy data spaces for a continuous family of elliptic operators.
method Elementary tools and classical results applied to operator graphs, Sobolev spaces, and Green's formula.
result Orthogonalized Calderón projections form a continuous family of projections.
The paper proves index theorems for graph-based optimal control problems.
problem Optimal control problems on graphs with constraints.
method Proves Morse index theorems for a broad class of variational problems on graphs.
result Formulas compute the difference of Hessians related to different graphs or boundary conditions.
MCMC complexity matches optimization for large n and d.
problem Lack of theoretical understanding of MCMC complexity for large n and d. method Comparison of MCMC, LA, and VI complexities for linear, logistic, and Poisson regression.
result MCMC complexity matches optimization complexity for n≳d. We consider an arbitrary linear elliptic first--order differential operator A with smooth coefficients acting between sections of complex vector bundles E,F over a compact smooth manifold M with smooth boundary N. We describe the analytic and topological properties of A in a collar neighborhood U of N and analyze vario…