arXiv research

Reformulated Markov's conjecture in combinatorial terms.

Unified proof of Aigner's conjectures using geodesics.

This is the text of my Bourbaki seminar on the proof of the surface subgroup conjecture by Jeremy Kahn and Vladimir Markovic.

Proofs non-realizability of mapping class group via homeomorphisms, resolves Thurston's conjecture.

Characterizes slopes for Markov ordering on prime pairs.

Numerical study confirms Brennan's conjecture for a counterexample to Thurston's $K=2$ conjecture.

Sum of Lagrange numbers equals a specific formula.

Classifies degenerations of complex projective plane with rational singularities.

New insights into algebraic geometry of a conjecture, leading to origami curves.

The time to converge to the steady state of a finite Markov chain can be greatly reduced by a lifting operation, which creates a new Markov chain on an expanded state space. For a class of quadratic objectives, we show an analogous behavior where a distributed ADMM algorithm can be seen as a lifting of Gradient Descent…

Non-negative curvature affects Markov chains' mixing and expansion properties.

Researchers confirm a conjecture about metrics on a specific Teichmüller space.

Two applications of algebraic geometry improve surface mapping group actions.

We define a finite-dimensional cubic quotient of the group algebra of the braid group, endowed with a (essentially unique) Markov trace which affords the Links-Grould invariant of knots and links. We investigate several of its properties, and state several conjectures about its structure.

New findings on hyperbolic groups and their boundaries.

When solving consensus optimization problems over a graph, there is often an explicit characterization of the convergence rate of Gradient Descent (GD) using the spectrum of the graph Laplacian. The same type of problems under the Alternating Direction Method of Multipliers (ADMM) are, however, poorly understood. For i…

The study proves inequalities and curvature properties for Markov chains.

Study restricts causal graphs with expert knowledge.

We use elementary methods to compute the L2-dimension of the eigenspaces of the Markov operator on the lamplighter group and of generalizations of this operator on other groups. In particular, we give a transparent explanation of the spectral measure of the Markov operator on the lamplighter group found by Grigorchuk-Z…

We classify the Markov traces factoring through the Birman-Wenzl-Murakami (BMW) algebras. For this purpose, we define a common `cover' for the two variations of the BMW-algebra originating from the quantum orthogonal/symplectic duality, which are responsible for the so-called `Dubrovnik' variation of the Kauffman polyn…

Study non-negative curvature Markov chains, proving entropy contraction.

New bounds on cover degrees for Teichmüller distance between hyperbolic surfaces.

This paper studies the bail-out optimal dividend problem with regime switching under the constraint that the cumulative dividend strategy is absolutely continuous. We confirm the optimality of the regime-modulated refraction-reflection strategy when the underlying risk model follows a general spectrally negative Markov…

We formulate simple assumptions, implying the Robbins-Monro conditions for the $Q$-learning algorithm with the local learning rate, depending on the number of visits of a particular state-action pair (local clock) and the number of iteration (global clock). It is assumed that the Markov decision process is communicatin…

Koschorke introduced a map from the space of closed $n$-component links to the ordered configuration space of $n$-tuples of points in $\mathbb{R}^3$, and conjectured that this map separates homotopy links. The purpose of this paper is to construct an analogous map for string links, and to prove (1) this map in fact sep…

Bayesian learning in undirected graphical models|computing posterior distributions over parameters and predictive quantities is exceptionally difficult. We conjecture that for general undirected models, there are no tractable MCMC (Markov Chain Monte Carlo) schemes giving the correct equilibrium distribution over param…

This paper deals with chain graphs under the Andersson-Madigan-Perlman (AMP) interpretation. In particular, we present a constraint based algorithm for learning an AMP chain graph a given probability distribution is faithful to. Moreover, we show that the extension of Meek's conjecture to AMP chain graphs does not hold…

We show that the nearest point retraction is a uniform quasi-isometry from the Thurston metric on a hyperbolic domain in the Riemann sphere to the boundary of the convex hull of its complement. As a corollary, one obtains explicit bounds on the quasi-isometry constant of the nearest point retraction with respect to the…

The starting point of this article is the question "How to retrieve fingerprints of rhythm in written texts?" We address this problem in the case of Brazilian and European Portuguese. These two dialects of Modern Portuguese share the same lexicon and most of the sentences they produce are superficially identical. Yet t…

Develops a model for causal discovery in path spaces.

We generalize Ng's two-variable algebraic/combinatorial $0$-th framed knot contact homology for framed oriented knots in $S^3$ to knots in $S^1 \times S^2$, and prove that the resulting knot invariant is the same as the framed cord algebra of knots. Actually, our cord algebra has an extra variable, which potentially co…

In this paper we discuss four problems regarding Markov equivalences for subclasses of loopless mixed graphs. We classify these four problems as finding conditions for internal Markov equivalence, which is Markov equivalence within a subclass, for external Markov equivalence, which is Markov equivalence between subclas…

The paper estimates key metrics for linear models with Markov or hidden Markov sources.

Geometrically, twist numbers on punctured tori are dense and non-continuous.

Gibbs sampling is a Markov Chain Monte Carlo sampling technique that iteratively samples variables from their conditional distributions. There are two common scan orders for the variables: random scan and systematic scan. Due to the benefits of locality in hardware, systematic scan is commonly used, even though most st…

Expands Hidden Markov Model to include Markov chain observations.

Study approximates financial market with discrete-time models.

The paper analyzes local minima in high-dimensional empirical risk minimization.

New neural processes use stacked Markov operators to improve flexibility.

New algorithms for RL in Markov games with independent linear function approximation, breaking the curse of multiagents.

The paper bounds generalization errors for deep neural networks with Markov datasets.

The paper constructs Markov partitions for geodesic flow on hyperbolic surfaces.

We rephrase Gromov's definition of Markov compacta, introduce a subclass of Markov compacta defined by one building block and study cohomological dimensions of these compacta. We show that for a Markov compactum $X$, $\dim_{\Z_{(p)}}X=\dim_{\Q}X$ for all but finitely many primes $p$ where $\Z_{(p)}$ is the localization…

Study nonparametric estimator for Markov chain transition matrices in offline setting.

The paper provides concentration inequalities for Markov chain variance estimators.

New method estimates convergence bounds for nonlinear Markov chains.

A new method simulates a lazy version of a Markov chain for empirical inference.

We study discretizations of polynomial processes using finite state Markov processes satisfying suitable moment matching conditions. The states of these Markov processes together with their transition probabilities can be interpreted as Markov cubature rules. The polynomial property allows us to study such rules using …