arXiv research

Characterizes Lie-Backlund vector fields in infinite jet bundles that can be exponentiated.

We construct 2-dimensional CAT(-1) groups which contain free subgroups with arbitrary iterated exponential distortion, and with distortion higher than any iterated exponential.

We produce examples of groups of type F_3 with 2-dimensional Dehn functions of the form exp^n(x) (a tower of exponentials of height n), where n is any natural number.

Study optimal stopping times for multi-dimensional processes with non-exponential discounting.

We prove that the number of combinatorially distinct causal 3-dimensional triangulations homeomorphic to the 3-dimensional sphere is bounded by an exponential function of the number of tetrahedra. It is also proven that the number of combinatorially distinct causal 4-dimensional triangulations homeomorphic to the 4-sph…

Thompson Sampling has been demonstrated in many complex bandit models, however the theoretical guarantees available for the parametric multi-armed bandit are still limited to the Bernoulli case. Here we extend them by proving asymptotic optimality of the algorithm using the Jeffreys prior for 1-dimensional exponential …

Signals are generally modeled as a superposition of exponential functions in spectroscopy of chemistry, biology and medical imaging. For fast data acquisition or other inevitable reasons, however, only a small amount of samples may be acquired and thus how to recover the full signal becomes an active research topic. Bu…

Tensor networks improve integration accuracy for high-dimensional problems.

This paper considers multi-dimensional affine processes with continuous sample paths. By analyzing the Riccati system, which is associated with affine processes via the transform formula, we fully characterize the regions of exponents in which exponential moments of a given process do not explode at any time or explode…

High-dimensional neural network manifolds misalign with human perception, causing adversarial examples.

Study the exponential map on surfaces using fluid dynamics.

Paper shows existence of vortex solutions with specific decay properties.

Moment polytope of toric exponential families is a projection of a simplex.

Paper develops a new algorithm to find shortest paths on surfaces.

Proves properties of sub-Riemannian exponential map, showing it's not injective.

Necessary and sufficient conditions for the exponentiation of finite-dimensional real Lie algebras of linear operators on complete Hausdorff locally convex spaces are obtained, focused on the equicontinuous case - in particular, necessary conditions for exponentiation to compact Lie groups are established. Applications…

Most exact methods for k-nearest neighbour search suffer from the curse of dimensionality; that is, their query times exhibit exponential dependence on either the ambient or the intrinsic dimensionality. Dynamic Continuous Indexing (DCI) offers a promising way of circumventing the curse and successfully reduces the dep…

We construct an infinite-dimensional information manifold based on exponential Orlicz spaces without using the notion of exponential convergence. We then show that convex mixtures of probability densities lie on the same connected component of this manifold, and characterize the class of densities for which this mixtur…

We give a complete classification of 1-dimensional exponential families $\mathcal{E}$ defined over a finite space $Ω=\{x_{0}, ...,x_{n}\}$ whose Hessian scalar curvature is constant. We observe an interesting phenomenon: if $\mathcal{E}$ has constant Hessian scalar curvature, say $λ$, then $λ=\tfrac{2}{k}$ for some pos…

We prove existence and uniqueness of optimal maps on $RCD^*(K,N)$ spaces under the assumption that the starting measure is absolutely continuous. We also discuss how this result naturally leads to the notion of exponentiation.

Novel Bayesian method for high-dimensional count data prediction.

We prove that the Riemannian exponential map of the right-invariant $L^2$ metric on the group of volume-preserving diffeomorphisms of a two-dimensional manifold with a nonempty boundary is a nonlinear Fredholm map of index zero.

Paper efficiently infers differential parameters in time-varying models using time score matching.

We study online learning under logarithmic loss with regular parametric models. Hedayati and Bartlett (2012b) showed that a Bayesian prediction strategy with Jeffreys prior and sequential normalized maximum likelihood (SNML) coincide and are optimal if and only if the latter is exchangeable, and if and only if the opti…

Ancient solutions on a strip are constant if polynomial, and have finite-dimensional space for slower growth.

Constructing exponential families from statistical manifolds.

Method extends option valuation for 2D Lévy models.

This paper studies a class of exponential family models whose canonical parameters are specified as linear functionals of an unknown infinite-dimensional slope function. The optimal minimax rates of convergence for slope function estimation are established. The estimators that achieve the optimal rates are constructed …

The left-invariant sub-Riemannian problem on the Engel group is considered. The problem gives the nilpotent approximation to generic nonholonomic systems in four-dimensional space with two-dimensional control, for instance to a system which describes motion of mobile robot with a trailer. The global optimality of extre…

We provide a complete characterization of the class of one-dimensional time-homogeneous diffusions consistent with a given law at an exponentially distributed time using classical results in diffusion theory. To illustrate we characterize the class of diffusions with the same distribution as Brownian motion at an expon…

We study an optimization problem for a portfolio with a risk-free, a liquid, and an illiquid risky asset. The illiquid risky asset is sold in an exogenous random moment with a prescribed liquidation time distribution. The investor prefers a negative or a positive exponential utility function. We prove that both cases a…

We explore the perspective of a bug living on the two-dimensional surface of a polyhedron. Images of various kinds of effects like lensing and cloaking are shown via color pictures of three viewpoints: the first person perspective of the bug, a map of the bug's viewpoint, and a look at the bug on the embedded polyhedro…

Using symplectic techniques and spectral analysis of smooth paths of self-adjoint operators, we characterize the set of conjugate instants along a geodesic in an infinite dimensional Riemannian Hilbert manifold.

In this paper, we present a simple analysis of {\bf fast rates} with {\it high probability} of {\bf empirical minimization} for {\it stochastic composite optimization} over a finite-dimensional bounded convex set with exponential concave loss functions and an arbitrary convex regularization. To the best of our knowledg…

New bounds for score matching in polynomial exponential families.

High-dimensional unimodal distributions can cause MCMC methods to fail.

New method uses neural exponential families for likelihood-free inference.

The paper develops algorithms to restore monotonicity in non-monotone functions.

We introduce a covariance matrix estimator that both takes into account the heteroskedasticity of financial returns (by using an exponentially weighted moving average) and reduces the effective dimensionality of the estimation (and hence measurement noise) via techniques borrowed from random matrix theory. We calculate…

Global stability proved for Navier-Stokes equations on hyperbolic space.

Closed-form optimal portfolios for exponential utility in small/large markets.

New Thompson sampling algorithm reduces regret for exponential family bandits.

We study the sub-Riemannian exponential for contact distributions on manifolds of dimension greater or equal to 5. We compute an approximation of the sub-Riemannian Hamiltonian flow and show that the conjugate time can have multiplicity 2 in this case. We obtain an approximation of the first conjugate locus for small r…

Gradient descent can take exponentially long to escape saddle points in 2D.

It is shown that every abelian regular Lie group is a quotient of its Lie algebra via the exponential mapping.

There has been an explosion of interest in statistical models for analyzing network data, and considerable interest in the class of exponential random graph (ERG) models, especially in connection with difficulties in computing maximum likelihood estimates. The issues associated with these difficulties relate to the bro…

Classifies toric dually flat manifolds into complex space forms.

The success of machine learning methods heavily relies on having an appropriate representation for data at hand. Traditionally, machine learning approaches relied on user-defined heuristics to extract features encoding structural information about data. However, recently there has been a surge in approaches that learn …