arXiv research

Research on mixed polynomials, extending non-degeneracy concepts to complex variables.

Study resolves polynomial germs, proving no mixed critical points and strict transform properties.

Milnor fibrations were extended by Mutsuo Oka for certain mixed polynomial. In this paper, we study singular points of differentiable maps into the 2-dimensional torus, called Milnor fibration product maps, obtained by several Milnor fibrations for mixed polynomial. We give a characterization of singular points of such…

Study bi-Lipschitz equivalence of mixed polynomials under specific conditions.

In the present paper, we deform isolated singularities of a certain class of polar weighted homogeneous mixed polynomials, and show that there exists a deformation which has only definite fold singularities and mixed Morse singularities.

Using the Fourier expansion of Markov traces for Ariki-Koike algebras over $\mathbb{Q}(q,u_{1},...,u_{e})$, we give a direct definition of the Alexander polynomials for mixed links. We observe that under the corresponding specialization of a Markov parameter, the Fourier coefficients of Markov traces take quite simple …

New method detects essential tori in mixed singularity links.

Study on singularities of specific polynomial functions.

Paper describes links of mixed polynomials with specific properties.

We present a reduced Burau-like representation for the mixed braid group on one strand representing links in lens spaces and show how to calculate the Alexander polynomial of a link directly from the mixed braid.

The paper extends knot polynomials to annular and toroidal pseudo links.

This work improves mixing rates for Bayesian CART, a key component of BART.

The paper explores knotoids, pseudo knotoids, braidoids, and pseudo braidoids on the torus.

Temporal Difference Learning analysis under non-i.i.d. data and nonlinear approximation.

Character expansion expresses extended HOMFLY polynomials through traces of products of finite dimensional R- and Racah mixing matrices. We conjecture that the mixing matrices are expressed entirely in terms of the eigenvalues of the corresponding R-matrices. Even a weaker (and, perhaps, more reliable) version of this …

Algorithm distinguishes Gaussian mixtures from pure Gaussians in quasi-polynomial time.

Study on moduli spaces of Higgs bundles over Abelian varieties, focusing on their topology and polynomials.

New mixed singularities help classify real algebraic links.

Study on gradient descent in Hilbert spaces with Markov chains, focusing on mixing coefficients.

At the heart of convex geometry lies the observation that the volume of convex bodies behaves as a polynomial. Many geometric inequalities may be expressed in terms of the coefficients of this polynomial, called mixed volumes. Among the deepest results of this theory is the Alexandrov-Fenchel inequality, which subsumes…

Study online learning in RKHS with dependent processes, focusing on \(β\)- and \(φ\)-mixing.

Rapid mixing of Langevin dynamics on Riemannian manifolds

MixCIT tests conditional independence for mixed data types efficiently and reliably.

In this survey we summarize results regarding the Kauffman bracket, HOMFLYPT, Kauffman 2-variable and Dubrovnik skein modules, and the Alexander polynomial of links in lens spaces, which we represent as mixed link diagrams. These invariants generalize the corresponding knot polynomials in the classical case. We compare…

Algorithm recovers graph from Glauber dynamics trajectory without mixing.

Polynomial mixing times for simulated tempering in mixture sampling problems.

Lower bounds on MALA and HMC for well-conditioned distributions.

We present an affine-invariant random walk for drawing uniform random samples from a convex body $\mathcal{K} \subset \mathbb{R}^n$ that uses maximum volume inscribed ellipsoids, known as John's ellipsoids, for the proposal distribution. Our algorithm makes steps using uniform sampling from the John's ellipsoid of the …

We study probability measures induced by set functions with constraints. Such measures arise in a variety of real-world settings, where prior knowledge, resource limitations, or other pragmatic considerations impose constraints. We consider the task of rapidly sampling from such constrained measures, and develop fast M…

The paper proves positivity of characteristic forms for certain vector bundles.

We analyze mixing times of three DA algorithms for regression models.

Improved lower bound for parallel tempering's mixing time.

In this paper we use Bernstein and Chebyshev polynomials to approximate the price of some basket options under a bivariate Black-Scholes model. The method consists in expanding the price of a univariate related contract after conditioning on the remaining underlying assets and calculating the mixed exponential-power mo…

We consider the problem of solving mixed random linear equations with $k$ components. This is the noiseless setting of mixed linear regression. The goal is to estimate multiple linear models from mixed samples in the case where the labels (which sample corresponds to which model) are not observed. We give a tractable a…

New insights into statistical and computational limits for mixed sparse linear regression.

Research connects geometric structures to knot theory and algebraic combinatorics.

Algorithm samples from Bingham distribution efficiently.

Study nearest-neighbor radii under dependent sampling, finding they remain informative.

It is known that evaluating a certain approximation to the Jones polynomial for the plat closure of a braid is a BQP-complete problem. That is, this problem exactly captures the power of the quantum circuit model. The one clean qubit model is a model of quantum computation in which all but one qubit starts in the maxim…

By now it is well established that the quantum dimensions of descendants of the adjoint representation can be described in a universal form, independent of a particular family of simple Lie algebras. The Rosso-Jones formula then implies a universal description of the adjoint knot polynomials for torus knots, which in p…

New bounds for SMC show its advantage over MCMC in multimodal distributions.

We study the problem of sampling from the power posterior distribution in Bayesian Gaussian mixture models, a robust version of the classical posterior. This power posterior is known to be non-log-concave and multi-modal, which leads to exponential mixing times for some standard MCMC algorithms. We introduce and study …

We connect two important conjectures in the theory of knot polynomials. The first one is the property Al_R(q) = Al_{[1]}(q^{|R|}) for all single hook Young diagrams R, which is known to hold for all knots. The second conjecture claims that all the mixing matrices U_{i} in the relation {\cal R}_i = U_i{\cal R}_1U_i^{-1}…

We propose and analyze two new MCMC sampling algorithms, the Vaidya walk and the John walk, for generating samples from the uniform distribution over a polytope. Both random walks are sampling algorithms derived from interior point methods. The former is based on volumetric-logarithmic barrier introduced by Vaidya wher…

The paper generalizes polynomial functions on Lie groups and their properties.

While all kinds of mixed data -from personal data, over panel and scientific data, to public and commercial data- are collected and stored, building probabilistic graphical models for these hybrid domains becomes more difficult. Users spend significant amounts of time in identifying the parametric form of the random va…

Transformers improve solving mixed-integer programs, especially CLSP.

Study confirms nullity of biharmonic maps family, linking spectral and arithmetic geometry.