Unified deep metric learning approach using neural networks.
problem Learning embeddings of data and extending Euclidean distances.
method Deep Bregman divergences based on neural networks.
result Superior performance on benchmark datasets compared to existing methods.
A cylindrical stretch line is a stretch line, in the sense of Thurston, whose horocyclic lamination is a weighted multicurve. In this paper, we show that two correctly parameterized cylindrical lines are parallel if and only if these lines converge towards the same point in Thurston's boundary of Teichmüller space.
We consider on-line density estimation with a parameterized density from the exponential family. The on-line algorithm receives one example at a time and maintains a parameter that is essentially an average of the past examples. After receiving an example the algorithm incurs a loss which is the negative log-likelihood…
Study random walks on groups with superlinear divergent geodesics.
problem Existence of superlinear divergent geodesics in groups.
method Developed theory of superlinear divergence and applied Gouëzel's pivoting technique.
result Established a central limit theorem for random walks on groups with superlinear divergent geodesics.
On a compact Kahler manifold, one can define global invariants by integrating local invariants of the metric. Assume that a global invariant thus obtained depends only on the Kahler class. Then we show that the integrand can be decomposed into a Chern polynomial (the integrand of a Chern number) and divergences of one …
New optimal transport divergences derived from scoring functions.
problem Developing new divergences for optimal transport.
method Using scoring functions as cost functions in optimal transport.
result Comonotonic coupling is optimal for many new divergences.
Computes circular area and spherical volume invariants via integrals.
problem Computing circular area and spherical volume invariants for curves and surfaces.
method Using the Divergence Theorem, express area and volume integrals as line and surface integrals against kernels, then compute analytically on triangulated meshes.
result Simple algorithm for computing spherical volume invariant for triangulated surfaces without discretizing ambient space.
Robust Bayesian changepoint detection with β-divergences reduces false discovery rates.
problem Detecting changepoints in non-stationary streaming data with high accuracy.
method Doubly robust Bayesian Online Changepoint Detection (BOCD) using β-divergences. result False discovery rates of changepoints reduced from over 90% to 0%.
The transformation formula of the Berezin integral holds, in the non-compact case, only up to boundary integrals, which have recently been quantified by Alldridge-Hilgert-Palzer. We establish divergence theorems in semi-Riemannian supergeometry by means of the flow of vector fields and these boundary integrals, and sho…
Constructs minimal graphs over curved surfaces and proves harmonic diffeomorphisms.
problem Constructing minimal graphs over curved surfaces and proving harmonic mappings.
method Uses divergence lines to construct minimal graphs and analyzes curvature conditions.
result Proves the existence of harmonic diffeomorphisms under specific curvature conditions.
Geodesics in curved spaces spread evenly over time.
problem Equidistribution of geodesics in negatively curved spaces.
method Proving equidistribution of geodesic flow orbits towards measures of maximal entropy and Bowen-Margulis measure.
result Equidistribution of divergent geodesics in negative curvature as their complexity increases.
Study shows how a strip's twisting increases at infinity, affecting its spectrum.
problem Understanding the spectrum of a strip with diverging twisting.
method Analyzing the Dirichlet Laplacian in a two-dimensional strip with segments rotating at increasing velocity.
result Essential spectrum forms a three-dimensional tube at infinity, with discrete eigenvalues possible if the tube's cross-section is a disk.
The study characterizes straight-line flows in dynamic measure transport.
problem Tackles the challenge of designing flows that are easy to integrate.
method Characterizes straight-line flows using a PDE and Reynolds tensor.
result Characterizes affine-in-time interpolants and necessary conditions for flow geometry.
New deep clustering network uses divergence measures for unlabeled data.
problem Discovering cluster structure in unlabeled data without supervision.
method Discriminative loss function incorporating geometric regularization.
result Competitive performance on synthetic and real datasets.
EM algorithm converges in KL divergence for exponential families via mirror descent.
problem Lack of understanding of EM's non-asymptotic convergence properties.
method Viewing EM as a mirror descent algorithm, showing convergence rates in KL divergence.
result KL divergence rates for EM in exponential families, invariant to parametrization.
A solution of a problem by V.I.Arnol'd about higher analog of the asymptotic Hopf invariant of divergence-free vector fields is presented. A higher invariant of magnetic fields, which is not expressed from the asymptotic linking numbers of magnetic lines is constructed and examples of an asymptotic invariants is constr…
The paper studies the behavior of Möbius-invariant Willmore flow in 3-sphere, proving convergence to Clifford torus.
problem Investigating the behavior of Möbius-invariant Willmore flow in 3-sphere.
method Analyzing flow lines of the Möbius-invariant Willmore flow in 3-sphere, constructing divergent and convergent flow lines, and identifying limit surfaces.
result The flow lines of the Möbius-invariant Willmore flow in 3-sphere converge to parametrizations of the Clifford torus, up to Möbius transformations.
Study scaling of optimal solutions for reliability constraints in resource provisioning.
problem Achieving high reliability in resource provisioning under stringent requirements.
method Chance-constrained optimization, distributionally robust optimization, f-divergence balls, line search.
result Correct scaling properties of optimal decisions are preserved by using appropriate f-divergence balls, leading to conservative yet near-optimal solutions.
Unified framework for various probability distribution distances.
problem Handling diverse probability distribution distances in statistics.
method General framework covering density-based and distribution-function-based divergences.
result Unified approach to classical and modern statistical procedures.
On-line estimation plays an important role in process control and monitoring. Obtaining a theoretical solution to the simultaneous state-parameter estimation problem for non-linear stochastic systems involves solving complex multi-dimensional integrals that are not amenable to analytical solution. While basic sequentia…
Unified view linking differential privacy and adversarial robustness.
problem Combining differential privacy and adversarial robustness in machine learning.
method Abstracting definitions and using probabilistic mappings and Renyi divergence.
result Results from one domain can be transferred to the other.
Solves Jenkins-Serrin problem in 3-manifolds with Killing vector fields.
problem Minimal surface equation in 3-manifolds with Killing vector fields.
method Riemannian submersion, divergence lines, maximum principles.
result Existence and uniqueness of minimal surfaces under specific conditions.
This paper generalizes Sinkhorn algorithm for unbalanced optimal transport.
problem Handling distributions with different total mass and robustness to outliers.
method Alternates between standard Sinkhorn updates and pointwise application of a contractive function.
result Defines Sinkhorn divergences that are differentiable, positive, definite, convex, and robust.
A graph theory approach defines curl and decomposes vector fields.
problem Defining curl for vector fields on graphs and decomposing them.
method Definition of curl as orthogonal complement of circulation-free fields, proving analogues of vector field theorems.
result Helmholtz-Hodge decomposition on graphs: gradient, curl, and harmonic fields.
In this paper, we shall study the Dirichlet problem for the minimal surfaces equation. We prove some results about the boundary behaviour of a solution of this problem. We describe the behaviour of a non-converging sequence of solutions in term of lines of divergence in the domain. Using this second result, we build so…
Faster algorithm for sampling logconcave densities in high dimensions.
problem Cubic barrier in sampling logconcave densities from a cold start.
method Two key ingredients: weaker distance sampling and refined log-Sobolev inequality.
result First sub-cubic sampling algorithms for isotropic position.
The extragradient method accelerates convergence in complex game dynamics.
problem Complex interactions in game dynamics cause simple methods to diverge, necessitating more sophisticated approaches.
method A polynomial-based analysis to identify three scenarios for accelerated convergence of the momentum extragradient method.
result The momentum extragradient method achieves faster convergence under specific eigenvalue conditions.
We show that Caratheodory's conjecture, on umbilical points of closed convex surfaces, may be reformulated in terms of the existence of at least one umbilic in the graphs of functions f: R^2-->R whose gradient decays uniformly faster than 1/r. The divergence theorem then yields a pair of integral equations for the norm…
GenDICE estimates stationary values from offline data.
problem Estimating stationary values from limited offline data.
method Ratio correction based on stationary distribution properties, variational divergence minimization.
result Consistent estimation of stationary values possible in offline settings.
We study the Asymptotic Cone of Teichmüller space equipped with the Weil-Petersson metric. In particular, we provide a characterization of the canonical finest pieces in the tree-graded structure of the asymptotic cone of Teichmüller space along the same lines as a similar characterization for right angled Artin groups…
This paper unifies GANs and VAEs through a new formulation.
problem The distinction between GANs and VAEs in generative model learning.
method Interpreting GANs as posterior inference and showing KL divergence minimization similarities.
result Unified view of GANs and VAEs provides a powerful tool for analysis and technique transfer.
The paper constructs non-convergent solutions to Vafa-Witten equations with specific harmonic 2-form limits.
problem Constructing solutions to Vafa-Witten equations with non-zero mass term.
method Constructs divergent sequences of solutions, renormalizes them, and defines harmonic 2-form data sets.
result Defines an 'interesting' harmonic 2-form data set with specific properties.
New method estimates density functionals using polynomial basis without full distribution knowledge.
problem Estimating quantities like information divergence functions requires complete distribution knowledge and integration.
method Introduces data-driven basis functions and develops methods for basis expansions of functionals of two distributions.
result Approximates functions of distributions as closely as desired using the new basis set.
This thesis bridges Lie theory and sketch theory using tangent categories.
problem Two diverging lines of research in Lie theory.
method Developing involution algebroids and using tangent categories to connect Lie algebroids and Weil algebras.
result The category of Lie algebroids is a functor category, and the Lie functor is a composition with a tangent categorical functor.
We consider the limit set in Thurston's compactification PMF of Teichmueller space of some Teichmueller geodesics defined by quadratic differentials with minimal but not uniquely ergodic vertical foliations. We show that a) there are quadratic differentials so that the limit set of the geodesic is a unique point, b) th…
New research shows DDPM can adapt to data's intrinsic low dimensionality efficiently.
problem Theoretical inefficiency of DDPM in high-dimensional data.
method Investigates how DDPM can exploit intrinsic low dimensionality of data.
result Proves DDPM's iteration complexity scales nearly linearly with intrinsic dimension k. The study analyzes transfer learning using information theory.
problem Transfer learning in different distributions.
method Information-theoretic analysis, focusing on KL divergence.
result Upper bounds for general transfer learning algorithms and specific ERM.
Study geometric quantization and measured Gromov-Hausdorff convergence of frame bundle metrics.
problem Understanding the convergence of frame bundle metrics and their relation to Gromov-Hausdorff convergence.
method Analyzes the one-parameter family of Riemannian metrics on the frame bundle of a prequantum line bundle, considering the convergence to metric measure spaces with S1-actions. result The frame bundle metrics converge to metric measure spaces with S1-actions, which depend on the choice of base point in Bohr-Sommerfeld fibers. New variational inference approach using Hilbert space for robotic state estimation.
problem Robotic state estimation with high-dimensional data.
method Variational inference reformulated in a Bayesian Hilbert space, using iterative projection.
result Variational inference can be seen as iterative projection in Euclidean space.
The paper studies curvature surfaces in conformally flat hypersurfaces and their extensions and approximations.
problem Analyzing curvature surfaces in conformally flat hypersurfaces and their properties.
method Using the Poincaré metric to determine curvature surfaces and extending them analytically.
result Curvature surfaces extend to certain sets in \(\mathbb{R}^2\) and have specific properties like parallel small circles at limits.
New divergences extend Bregman and skew Jensen, including f-divergences.
problem Developing new divergences to include f-divergences.
method Introducing g-Bregman and skew g-Jensen divergences, showing they include f-divergences.
result g-divergences generalize existing divergences and inequalities.
In the 3-dimensional Riemannian geometry, contact structures equipped with an adapted Riemannian metric are divergence-free, nondegenerate eigenforms of the Laplace-Beltrami operator. We trace out a 2-d analogue of this fact: there is a close relationship between the topology of the contact structure on a convex surfac…
The paper models asset and bond prices with equations leading to periodic bubbles that collapse into crashes.
problem Modeling the formation and collapse of asset price bubbles.
method Developed a reduced model of asset and bond prices using coupled continuous time equations.
result The model predicts the emergence of periodically collapsing bubbles, which are analogous to phase transitions.
Paper shows how to break down a specific type of divergence into simpler parts.
problem Understanding and simplifying divergence functions.
method Decomposes the symmetric Bregman divergence into two types of Jensen divergences and a Bregman divergence, and extends this to include f-divergences.
result Sum decomposition of divergence into simpler parts is possible.
DORO improves DRO's performance and stability in tasks with subpopulation shift.
problem DRO's poor performance and instability in tasks with subpopulation shift.
method DORO, a refined risk function that prevents overfitting to outliers.
result DORO improves DRO's performance and stability on large modern datasets.
Recent work by Jaffe and Scardicchio has expressed the optical approximation to the Casimir effect as a sum over geometric quantities. The first two authors have developed a technique which uses the complex geometry of the space of oriented affine lines in R3 to describe reflection of rays off a surface. Thi…
EGAB algorithms improve online portfolio selection.
problem Online portfolio selection problem.
method Generalized exponentiated gradient (EG) updates with Alpha-Beta divergence regularization.
result EGAB algorithms enhance portfolio performance, especially with transaction costs.
Study explores relationship between Hölder and FDPD divergences.
problem Understanding the relationship between Hölder and FDPD divergences.
method Intersection and generalization of divergence families, proving nonnegativity, deriving inequalities.
result Established ξ-Hölder divergence and derived inequalities.