arXiv research

We construct a tangent bundle exponential map and locally autoparallel coordinates for geometries based on a general connection on the tangent bundle of a manifold. As concrete application we use these new coordinates for Finslerian geometries and obtain Finslerian geodesic coordinates. They generalise normal coordinat…

We represent the coordinate ring of algebraic hulls (which are generalizations of the Malcev completions of nilpotent groups for solvable groups) of solvmanifolds $G/Γ$ by using Miller's exponential iterated integrals (which are extensions of Chen's iterated integrals) of invariant differential forms.

We prove an exponential estimate for the asymptotics of Bergman kernels of a positive line bundle under hypotheses of bounded geometry. We give further Bergman kernel proofs of complex geometry results, such as separation of points, existence of local coordinates and holomorphic convexity by sections of positive line b…

In every point of a Kähler manifold there exist special holomorphic coordinates well adapted to the underlying geometry. Comparing these Kähler normal coordinates with the Riemannian normal coordinates defined via the exponential map we prove that their difference is a universal power series in the curvature tensor and…

We design a randomised parallel version of Adaboost based on previous studies on parallel coordinate descent. The algorithm uses the fact that the logarithm of the exponential loss is a function with coordinate-wise Lipschitz continuous gradient, in order to define the step lengths. We provide the proof of convergence …

This paper summarizes closed-form relations for SE(3) maps and their derivatives.

Normal and almost normal surfaces are essential tools for algorithmic 3-manifold topology, but to use them requires exponentially slow enumeration algorithms in a high-dimensional vector space. The quadrilateral coordinates of Tollefson alleviate this problem considerably for normal surfaces, by reducing the dimension …

We develop a class of integrals on a manifold M called exponential iterated integrals, an extension of K. T. Chen's iterated integrals. It is shown that the matrix entries of any upper triangular representation of the fundamental group of M can be expressed via these new integrals. The ring of exponential iterated inte…

Improved Sobolev mappings in Carnot groups with weaker assumptions.

Chekhov, Fock and Kashaev introduced a quantization of the Teichmüller space $\mathcal{T}^q(S)$ of a punctured surface $S$, and an exponential version of this construction was developed by Bonahon and Liu. The construction of the quantum Teichmüller space crucially depends on certain coordinate change isomorphisms betw…

The paper tackles sampling from Gibbs measures with constrained support, providing a sampling guarantee.

Kahn and Markovic \cite{KahnMark} proved that the fundamental group of each closed hyperbolic three manifold contains a closed surface subgroup. One of the main ingredients in their proof is a theorem which states that an assignment of nearly real, complex Fenchel-Nielsen coordinates to the cuffs of a pants decompositi…

In this paper, we examine the dependence of standard gluing process for pseudoholomorphic curves under the change of the length $T$ of the neck-region with respect to the cylindrical metrics associated to the given analytic coordinates near the punctures in the setting of bordered open Riemann surface with boundary pun…

In the field of statistics, many kind of divergence functions have been studied as an amount which measures the discrepancy between two probability distributions. In the differential geometrical approach in statistics (information geometry), dually flat spaces play a key role. In a dually flat space, there exist dual a…

In the quantum Teichmuller theory, based on Penner coordinates, the mapping class groups of punctured surfaces are represented projectively. The case of a genus three surface with one puncture is worked out explicitly. The projective factor is calculated. It is given by the exponential of the Liouville central charge.

Constructs hyperkähler metrics on Higgs bundle moduli spaces using Gaiotto coordinates.

We develop randomized (block) coordinate descent (CD) methods for linearly constrained convex optimization. Unlike most CD methods, we do not assume the constraints to be separable, but let them be coupled linearly. To our knowledge, ours is the first CD method that allows linear coupling constraints, without making th…

The paper proves the stability of a flow in Schwarzschild space.

CAVI converges for log-concave measures via optimal transport.

In multi-agent reinforcement learning, discovering successful collective behaviors is challenging as it requires exploring a joint action space that grows exponentially with the number of agents. While the tractability of independent agent-wise exploration is appealing, this approach fails on tasks that require elabora…

We propose a new stochastic dual coordinate ascent technique that can be applied to a wide range of regularized learning problems. Our method is based on Alternating Direction Multiplier Method (ADMM) to deal with complex regularization functions such as structured regularizations. Although the original ADMM is a batch…

For a fixed marked surface $S$, we show that the problem of deciding whether or not a mapping class is reducible lies in $\textbf{NP}$. As usual this immediately gives an exponential time algorithm to decide whether or not a mapping class is reducible. To do this we use an (ideal) triangulation to obtain a coordinate s…

CAVI converges globally or locally exponentially for two-block models.

We derive scaling laws for optimizing neural networks in hardware.

A simple log-transform fixes heavy-tailed data for generative models.

We explore the geometry that underlies the osculating nilpotent group structures of the Heisenberg calculus. For a smooth manifold $M$ with a distribution $H\subseteq TM$ analysts use explicit (and rather complicated) coordinate formulas to define the nilpotent groups that are central to the calculus. Our aim in this p…

Identifying the unknown underlying trend of a given noisy signal is extremely useful for a wide range of applications. The number of potential trends might be exponential, which can be computationally exhaustive even for short signals. Another challenge, is the presence of abrupt changes and outliers at unknown times w…

BCD algorithm finds global minima in neural networks.

CAVI converges exponentially fast for Bayesian PCA models.

Coordinate ascent variational inference is an important algorithm for inference in probabilistic models, but it is slow because it updates only a single variable at a time. Block coordinate methods perform inference faster by updating blocks of variables in parallel. However, the speed and stability of these algorithms…

New tensor framework connects Fisher information, hypergraphs, and multi-observable correlations.

We define and study a family of distributions with domain complete Riemannian manifold. They are obtained by projection onto a fixed tangent space via the inverse exponential map. This construction is a popular choice in the literature for it makes it easy to generalize well known multivariate Euclidean distributions. …

This paper develops a path-first theory using signatures and jump lifts for self-exiting processes.

Holomorphic actions on complex spaces for nilpotent groups.

The paper studies the stability of volume and area preserving mean curvature flows in Schwarzschild and asymptotic Schwarzschild spaces.

Inhomogeneous plasmas filaments instabilities are investigated by using the techniques of classical differential geometry of curves where Frenet torsion and curvature describe completely the motion of curves. In our case the Frenet frame changes in time and also depends upon the other coordinates taking into account th…

The paper analyzes and validates two step size schedules for SGD: exponential and cosine, proving their adaptivity and performance.

Enhances LMC for log-concave sampling, reducing computational cost.

These notes on Riemannian geometry use the bases bundle and frame bundle, as in Geometry of Manifolds, to express the geometric structures. It has more problems and omits the background material. It starts with the definition of Riemannian and semi-Riemannian structures on manifolds. Affine connections, geodesics, tors…

We study the performance of a family of randomized parallel coordinate descent methods for minimizing the sum of a nonsmooth and separable convex functions. The problem class includes as a special case L1-regularized L1 regression and the minimization of the exponential loss ("AdaBoost problem"). We assume the input da…

In intractable, undirected graphical models, an intuitive way of creating structured mean field approximations is to select an acyclic tractable subgraph. We show that the hardness of computing the objective function and gradient of the mean field objective qualitatively depends on a simple graph property. If the tract…

Data structure for Heegaard splittings reduces complexity.

We provide upper bounds of the expected Wasserstein distance between a probability measure and its empirical version, generalizing recent results for finite dimensional Euclidean spaces and bounded functional spaces. Such a generalization can cover Euclidean spaces with large dimensionality, with the optimal dependence…

Develops Riemannian geometry for optimization on manifolds with detailed derivations.

New framework improves EM algorithm convergence under log-Sobolev inequality.

This paper contains the technical foundations from stochastic differential geometry for the construction of geometrically intrinsic nonlinear recursive filters. A diffusion X on a manifold N is run for a time interval T, with a random initial condition. There is a single observation consisting of a nonlinear function o…

We study the asymptotics of the natural $L^2$ metric on the Hitchin moduli space with group $G = \mathrm{SU}(2)$. Our main result, which addresses a detailed conjectural picture made by Gaiotto, Neitzke and Moore \cite{gmn13}, is that on the regular part of the Hitchin system, this metric is well-approximated by the se…

This paper is a sequel of arxiv:1709.09045 and deals with privileged coordinates and nilpotent approximation of Carnot manifolds. By a Carnot manifold it is meant a manifold equipped with a filtration by subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. In this paper, we single…