Elliptical processes generalize Gaussian and Student-t models with fat tails and computational efficiency.
problem Need for models with fat tails and computational tractability.
method Represent elliptical distributions as continuous mixtures of Gaussian distributions, derive closed-form expressions for marginal and conditional distributions.
result Elliptical processes offer advantages in robust regression compared to Gaussian processes.
Elliptical processes extend Gaussian models with heavier tails.
problem Regression and classification with non-Gaussian likelihoods or heavy tails.
method Spline normalizing flow for variational inference of elliptical distributions.
result Elliptical processes outperform Gaussian processes in non-Gaussian settings.
Introduces spectral-domain Wasserstein distance and Gelbrich bound for elliptical processes.
problem Estimating distances and bounds for elliptical stochastic processes.
method Defines spectral-domain W2 Wasserstein distance and Gelbrich bound. result Develops new spectral-domain bounds for non-elliptical processes.
The Heston stochastic volatility process is a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this process with killing, called the elliptic Heston operator, is a second-order, degenerat…
The Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance, is a paradigm for a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this p…
Paper explores Elliptical Wishart distributions in signal processing and machine learning.
problem Estimating parameters of Elliptical Wishart distributions.
method Proposes fixed point and Riemannian optimization algorithms for maximum likelihood estimation.
result Characterizes existence, uniqueness, and convergence of the MLE.
The Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance, is a paradigm for a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this p…
Elliptical slice sampling converges geometrically, providing reliable sampling for Bayesian learning.
problem Sampling from posterior distributions in Bayesian learning.
method Elliptical slice sampling, geometric ergodicity.
result Elliptical slice sampling yields geometric convergence guarantees under weak regularity assumptions.
We establish Schauder a priori estimates and regularity for solutions to a class of boundary-degenerate elliptic linear second-order partial differential equations. Furthermore, given a smooth source function, we prove regularity of solutions up to the portion of the boundary where the operator is degenerate. Degenerat…
We show that an Osserman-type inequality holds for spacelike surfaces of constant mean curvature (CMC) 1 with singularities and with elliptic ends in de Sitter 3-space. An immersed end of a CMC 1 surface is an ``elliptic end'' if the monodromy representation at the end is diagonalizable with eigenvalues in the unit cir…
Let x denote a diffusion process defined on a closed compact manifold. In an earlier article, the author introduced a new approach to constructing admissible vector fields on the associated space of paths, under the assumption of ellipticity of x. In this article, this method is extended to yield similar results fo…
New algorithm learns value and advantage functions for continuous-time Markov processes without structural assumptions.
problem Learning value and advantage functions for continuous-time Markov processes without structural assumptions.
method Proposes Sobolev-prox fitted q-learning algorithm based on Hilbert-space positive definiteness and boundedness properties of Bellman operators. result Identifies ellipticity as a key structural property enabling reinforcement learning for Markov diffusions.
We prove Feynman-Kac formulas for solutions to elliptic and parabolic boundary value and obstacle problems associated with a general Markov diffusion process. Our diffusion model covers several popular stochastic volatility models, such as the Heston model, the CEV model and the SABR model, which are widely used as ass…
A justification of the Basel liquidity formula for risk capital in the trading book is given under the assumption that market risk-factor changes form a Gaussian white noise process over 10-day time steps and changes to P&L are linear in the risk-factor changes. A generalization of the formula is derived under the more…
Consistent estimator for mixtures of nonparametric elliptical distributions helps cluster analysis.
problem Consistency of maximum likelihood estimator for mixtures of nonparametric elliptical distributions.
method Maximum likelihood estimation for mixtures of elliptically-symmetric distributions under nonparametric P. result Components of the estimator correspond to well-separated components of the underlying distribution P. Correlation mixtures of elliptical copulas arise when the correlation parameter is driven itself by a latent random process. For such copulas, both penultimate and asymptotic tail dependence are much larger than for ordinary elliptical copulas with the same unconditional correlation. Furthermore, for Gaussian and Stude…
Researchers find a method to construct projective structures on a specific surface.
problem Constructing projective structures with given holonomy and tameness conditions.
method Grafting circular triangles determined by a natural framing of the representation.
result All structures satisfying the conditions can be obtained through this method.
We describe two topologies on the space of unbounded Fredholm operators and we explain their K-theoretic relevance. In the process we also prove a very general result concerning the continuity of families of first order, elliptic boundary value problems.
Local well-posedness proved for Bartnik static extension near Schwarzschild spheres.
problem Proving well-posedness for the Bartnik static extension problem near Schwarzschild spheres.
method Introduced a geodesic gauge to formulate governing equations as coupled elliptic and transport equations; used Bochner-measurable functions for transport equations.
result Established local well-posedness for arbitrary Bartnik data near Schwarzschild spheres, including those with small mean curvature.
The study finds the best elliptical trajectory for planets using a variation of the hodograph theorem.
problem Finding the best elliptical trajectory for planets.
method Using a variation of the circular hodograph theorem, the study finds the best fitting ellipse for planetary trajectories by minimizing the sum of square distances from the points to the plane.
result The study finds that the best fitting ellipse for planetary trajectories minimizes the sum of square distances from the points to the plane.
We construct non-symmetric diffusion processes associated with Dirichlet forms consisting of uniformly elliptic forms and derivation operators with killing terms on RCD spaces by aid of non-smooth differential structures introduced by Gigli '16. After constructing diffusions, we investigate conservativeness and the wea…
Constructs Lie groupoid integrating elliptic tangent bundles and Poisson structures.
problem Existence and integration of elliptic tangent bundles and Poisson structures.
method Explicit construction of Lie groupoids and local models for symplectic integration.
result Necessary and sufficient topological condition for integration of elliptic tangent bundles.
The paper (in French) exemplifies graphically a solution of the heat equation which is a 1-dimensional unfolding of an elliptic umbilic catastrophe. The example is due to James Damon and adapts Thom-Mather's singularity theory to multiscale models of scale-space analysis in image processing.
Study finds negatively curved spheres in elliptic surfaces and their modifications.
problem Finding negatively curved spheres in elliptic surfaces.
method Using elliptic fibrations with specific singular fibers.
result Identifies spheres with very negative self-intersections in elliptic surfaces and their modifications.
In this paper, we construct for the first time the projective elliptic genera for a compact oriented manifold equipped with a projective complex vector bundle. Such projective elliptic genera are rational q-series that have topological definition and also have analytic interpretation via the fractional index theorem in…
New method for analyzing elliptic and parabolic equations.
problem Analyzing elliptic and parabolic equations.
method Level set version of partial uniform ellipticity.
result Effective approach to investigate equations.
We prove a uniform Sobolev inequality along the Sasaki-Ricci flow. In the process, we develop the theory of basic Lebesgue and Sobolev function spaces, and prove some general results about the decomposition of the heat kernel for a class of elliptic operators on a Sasaki manifold.
We give a systematic method to calculate some homological data from the global monodromy of a topological elliptic surface. We apply this method to the cases 1) the transcendental lattice of an extremal elliptic K3 surface, 2) the torsion part of Mordell-Weil group of a general elliptic surface, and 3) the Mordell-Weil…
New topological obstructions found for elliptic and quasiregularly elliptic manifolds.
problem Connection between ellipticity and quasiregular ellipticity.
method Proved new topological obstructions.
result Closed manifolds of both types must have virtually abelian fundamental group.
This study explores the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators using KK-theory.
problem Analyzing the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators.
method Applying Kasparov's methodology and examining specific conditions using Fourier transform of the nilpotent group C∗-algebra. result Demonstrated enhanced methods for analyzing hypoellipticity and defined transversal Heisenberg ellipticity in a KK-theoretic context. Elliptic bouquets defined for spin manifolds with circular actions.
problem Defining integration in elliptic cohomology.
method Introducing elliptic bouquets of germs of holomorphic equivariant cohomology classes, integrating them as in K-theory.
result Witten's rigidity theorem follows from integration of elliptic bouquets.
Paper classifies minimal graph transformations into new families of surfaces.
problem Classifying minimal graph transformations into new families of surfaces.
method Formulated and solved a coupled system of partial differential equations, reduced to solving an ordinary differential equation.
result Established rigorous equivalence to a modified problem for a harmonic function, yielding new families of minimal surfaces.
We give a parameterization of Alfred Gray's Elliptical Catenoid and Elliptical Hellicoid using Jacobi's elliptic functions. This parameterization avoids some problems present in the original depiction of these surfaces.
Study the relationship between canonical polynomials and elliptic sequences for elliptic singularities.
problem Understanding the relationship between canonical polynomials and elliptic sequences for elliptic singularities.
method An inductive setup of elliptic germs and comparison of their canonical polynomials.
result The exponents of the canonical polynomial determine the elliptic sequence and vice versa under certain conditions.
Elliptic Chern characters and Atiyah-Witten formula generalized to double loop spaces.
problem Generalizing classical Atiyah-Witten formula to double loop spaces.
method Constructing elliptic Chern and Bismut-Chern characters, defining elliptic holonomy, and using equivariant twisted parallel transport.
result Established elliptic Atiyah-Witten formula on double loop space.
Mathai, Melrose, and Singer introduced the notion of projective elliptic operators on manifolds equipped with an Azumaya bundle. In this note we compute the equivariant index of transversally elliptic operators that are the pullback of projective elliptic operators on the trivialization of the Azumaya bundle. It encomp…
Paper proposes a new algorithm for graph learning with covariance constraints.
problem Graphical models and factor analysis not jointly leveraged in graph learning processes.
method Penalized maximum likelihood estimation of an elliptical distribution with Riemannian optimization.
result Effectiveness of the proposed approach demonstrated on real-world data sets.
New deficit functions link elliptic and parabolic inequalities, proving log Sobolev.
problem Proving log Sobolev inequality using deficit functions.
method Introducing two deficit functions, one elliptic and one parabolic, and showing their pointwise convergence and equations.
result Elliptic deficit converges to parabolic deficit, leading to an elliptic proof of log Sobolev inequality.
New proof for curved 3-cohom manifold rational ellipticity.
problem Rational ellipticity of curved manifolds with specific cohomogeneity.
method Proved rationally elliptic for cohomogeneity-three manifolds with positive curvature and no boundary quotient.
result Closed, simply connected, positively curved, cohomogeneity-three manifolds without boundary are rationally elliptic.
Characterizes totally elliptic surface group representations into Lie groups.
problem Understanding totally elliptic surface group representations into Lie groups.
method Characterization of representations into PSL2R and PSL2C by their mapping properties. result They are either into a compact subgroup or Deroin--Tholozan representations.
Study homogenizes equations on parallelizable manifolds using tensor localization and periodicity.
problem Homogenizing oscillating linear elliptic equations on parallelizable manifolds.
method Two-scale convergence through localization and periodicity induced by geometry.
result Explicit cell formulae for the homogenization limit and a theory of two-scale convergence of tensors.
A guide for solving first-order elliptic boundary value problems.
problem Solving first-order elliptic boundary value problems on manifolds.
method Operator methods and general elliptic boundary conditions.
result Characterization of a new subclass of elliptic boundary conditions.
New Witten rigidity theorems for elliptic genus in various dimensions.
problem Proving rigidity theorems for elliptic genus in different dimensions.
method Combining Liu's and Han-Yu's methods to prove Witten rigidity theorems for elliptic genus in even and odd dimensions.
result Several new Witten rigidity theorems for elliptic genus in even and odd dimensions have been established.
Researchers compute the cohomology of an elliptic tangent bundle.
problem Computing the cohomology of a specific Lie algebroid.
method Direct computation of cohomology.
result The cohomology of the elliptic tangent bundle is computed.
Paper defines invariants for elliptic Weyl groups and connects them to Frobenius structures.
problem Defining invariants for elliptic Weyl groups.
method Defines a set of good basic invariants and shows their connection to Frobenius structures.
result Good basic invariants give flat invariants and structure constants of Frobenius structures.
This note shows how independent elliptical distributions minimize the Wasserstein distance.
problem Minimizing the Wasserstein distance between elliptical distributions.
method Analyzing the Wasserstein distance between independent elliptical distributions with the same density generators.
result Independent elliptical distributions minimize their Wasserstein distance from other elliptical distributions with the same density generators.
Study of 17 surface behaviors and singularities for elliptic Weingarten equations.
problem Characterizing and understanding elliptic Weingarten surfaces and their singularities.
method Phase space analysis and classification of surface behaviors.
result Classification of 17 possible qualitative behaviors for rotational surfaces.
New Kelvin transform for anisotropic elliptic problems.
problem Semilinear and quasilinear anisotropic elliptic problems.
method Introducing a new Kelvin-type transform in the anisotropic setting.
result New insights into anisotropic elliptic problems.