arXiv research

Derives Lagrangian for minimal surfaces, proving tangential variations vanish.

Generalizes Hamiltonian theory for variational problems, applied to first order gravity.

The first Betti number for a lattice in a classifying space for variations of Hodge structures vanishes.

We show that in the first sub-Riemannian Heisenberg group there are intrinsic graphs of smooth functions that are both critical and stable points of the sub-Riemannian perimeter under compactly supported variations of contact diffeomorphisms, despite the fact that they are not area-minimizing surfaces. In particular, w…

We derive the first and second variation formula for the Green's function pole's value of Paneitz operator on the standard three sphere. In particular it is shown that the first variation vanishes and the second variation is nonpositively definite. Moreover, the second variation vanishes only at the direction of confor…

In this paper, we study monotonicity for the first eigenvalue of a class of $(p,q)$-Laplacian. We find the first variation formula for the first eigenvalue of $(p,q)$-Laplacian on a closed Riemannian manifold evolving by the Ricci-harmonic flow and construct various monotic quantities by imposing some conditions on ini…

Unified approach for first-order methods with Markovian noise in stochastic optimization and variational inequalities.

In this note, we establish the first variation formula of the adjusted log entropy functional $\mathcal Y_a$ introduced by Ye in \cite{Y2}. As a direct consequence, we also obtain the monotonicity of $\mathcal Y_a$ along the Ricci flow.

We derive a formula for the first variation of horizontal perimeter measure for $C^2$ hypersurfaces of completely general sub-Riemannian manifolds, allowing for the existence of characteristic points. For $C^2$ hypersurfaces in vertically rigid sub-Riemannian manifolds we also produce a second variation formula for var…

Paper derives formulas for surface variations in shell theory.

Improves bandit convex optimization with gradient variations.

Derive Dirichlet scalar curvature energy functional variation formula

First variation of fractional $k$-dimensional measure for submanifolds

Minimal surfaces in spheres have unique energy properties.

Over the years data has become increasingly higher dimensional, which has prompted an increased need for dimension reduction techniques. This is perhaps especially true for clustering (unsupervised classification) as well as semi-supervised and supervised classification. Although dimension reduction in the area of clus…

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

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…

Geodesics become an essential element of the geometry of a semi-Riemannian manifold. In fact, their differences and similarities with the (positive definite) Riemannian case, constitute the first step to understand semi-Riemannian Geometry. The progress in the last two decades has become impressive, being especially re…

Splitting theorem for non-positively curved Lorentzian spaces.

The article concerns the problem if a~given system of differential equations is identical with the Euler--Lagrange system of an~appropriate variational integral. Elementary approach is applied. The main results involve the determination of the first--order variational integrals related to the second--order Euler--Lagra…

We prove the existence of a continuous $BV$ minimizer with $C^{0}$ 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…

We calculate the first and the second variation formula for the sub-Riemannian area in three dimensional pseudo-hermitian manifolds. We consider general variations that can move the singular set of a C^2 surface and non-singular variation for C_H^2 surfaces. These formulas enable us to construct a stability operator fo…

We prove that, in the first Heisenberg group $\mathbb{H}$, an entire locally Lipschitz intrinsic graph admitting vanishing first variation of its sub-Riemannian area and non-negative second variation must be an intrinsic plane, i.e., a coset of a two dimensional subgroup of $\mathbb{H}$. Moreover two examples are given…

Schwarzian derivative connects to Euler-Lagrange equations in variational calculus.

In this work it is shown that every integral varifold in an open subset of Euclidian space of locally bounded first variation can be covered by a countable collection of submanifolds of class C^2. Moreover, the mean curvature of each member of the collection agrees with the mean curvature of the varifold almost everywh…

We compute first variation formulas for the complex components of the Bakry-Emery-Ricci endomorphism along Kähler structures. Our formulas show that the principal parts of the variations are quite standard complex differential operators with particular symmetry properties on the complex decomposition of the variation o…

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…

Recent progress in variational inference has paid much attention to the flexibility of variational posteriors. One promising direction is to use implicit distributions, i.e., distributions without tractable densities as the variational posterior. However, existing methods on implicit posteriors still face challenges of…

We present a family of complexes playing the same role, for homogeneous variational problems, that the horizontal parts of the variational bicomplex play for variational problems on a fibred manifold. We show that, modulo certain pullbacks, each of these complexes (apart from the first one) is globally exact. All the c…

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.

We study the volume of compact Riemannian manifolds which are Einstein with respect to a metric connection with (parallel) skew-torsion. We provide a result for the sign of the first variation of the volume in terms of the corresponding scalar curvature. This generalizes a result of M. Ville, related with the first var…

The motivations for using variational inference (VI) in neural networks differ significantly from those in latent variable models. This has a counter-intuitive consequence; more expressive variational approximations can provide significantly worse predictions as compared to those with less expressive families. In this …

The concept of a symplectic structure first appeared in the works of Lagrange on the so-called "method of variation of the constants". These works are presented, together with those of Poisson, who first defined the composition law called today the "Poisson bracket". The method of variation of the constants is presente…

Paper derives second variational formula for statistical manifold mappings.

A practical guide to Variational Bayes methods.

Develops a first-order interior-point method for solving constrained variational inequalities.

This note compares two recently published machine learning methods for constructing flexible, but tractable families of variational hidden-variable posteriors. The first method, called "hierarchical variational models" enriches the inference model with an extra variable, while the other, called "auxiliary deep generati…

The paper studies stability of discrete planar curves using variational methods.

After a brief introduction to several variational problems in the study of shapes of thin thickness structures, we deal with variational problems on 2-dimensional surface in 3-dimensional Euclidian space by using exterior differential forms. The morphological problems of lipid bilayers and stabilities of cell membranes…

Paper derives second variation formula for eigenvalue functionals on surfaces.

A new EVI framework improves ParVI methods by maintaining variational structure and reducing KL-divergence.

A new particle algorithm improves mean-field variational inference.

We analyze variational inference for highly symmetric graphical models such as those arising from first-order probabilistic models. We first show that for these graphical models, the tree-reweighted variational objective lends itself to a compact lifted formulation which can be solved much more efficiently than the sta…

Recent advances in stochastic gradient variational inference have made it possible to perform variational Bayesian inference with posterior approximations containing auxiliary random variables. This enables us to explore a new synthesis of variational inference and Monte Carlo methods where we incorporate one or more s…

We study underlying geometric structures for integral variational functionals, depending on submanifolds of a given manifold. Applications include (first order) variational functionals of Finsler and areal geometries with integrand the Hilbert 1-form, and admit immediate extensions to higher-order functionals.

Variational calculus on a vector bundle E equipped with a structure of a general algebroid is developed, together with the corresponding analogs of Euler-Lagrange equations. Constrained systems are introduced in the variational and in the geometrical setting. The constrained Euler-Lagrange equations are derived for ana…

The paper calculates variations of Einstein-Hilbert action on CR manifolds.

We show how the homogeneous variational bicomplex provides a useful formalism for describing a number of properties of single-integral variational problems, and we introduce a subsequence of one of the rows of the bicomplex which is locally exact with respect to the variational derivative. We are therefore able to reco…