arXiv research

Study uses orthogonal polynomials to solve option pricing equations.

A method for interpreting SVMs using polynomial kernels, revealing model complexity.

We study the Chern-Simons partition function of orthogonal quantum group invariants, and propose a new orthogonal Labastida-Mariño-Ooguri-Vafa conjecture as well as degree conjecture for free energy associated to the orthogonal Chern-Simons partition function. We prove the degree conjecture and some interesting cases o…

Study shows RFRR's effectiveness with nearly orthogonal data in overparameterized settings.

We provide new examples of diffusion operators in dimension 2 and 3 which have orthogonal polynomials as eigenvectors. Their construction rely on the finite subgroups of O(3) and their invariant polynomials.

In this paper we derive a series expansion for the price of a continuously sampled arithmetic Asian option in the Black-Scholes setting. The expansion is based on polynomials that are orthogonal with respect to the log-normal distribution. All terms in the series are fully explicit and no numerical integration nor any …

The paper connects Riemannian Gaussian distributions to random matrix theory and diffusion kernels.

A new method tests Expected Shortfall by analyzing both duration and severity of VaR violations.

New theory shows EDMD works well in chaotic systems.

The noncommutative A-ideal of a knot is a generalization of the A-polynomial, defined using Kauffman bracket skein modules. In this paper we show that any knot that has the same noncommutative A-ideal as the (2,2p+1)-torus knot has the same colored Jones polynomials. This is a consequence of the orthogonality relation,…

OPAA estimates probability densities using functional analysis.

Universal preconditioning reduces sequential prediction regret.

Studying the isotropy orbits of compact symmetric spaces Reiswich introduced a family of explicit polynomials in one variable in order to describe the unique minimal isotropy orbit of compact symmetric spaces with Dynkin diagram of type $D_m$. Based on this geometric interpretation he conjectured that these polynomials…

We determine the expected curvature polynomial of random real projective varieties given as the zero set of independent random polynomials with Gaussian distribution, whose distribution is invariant under the action of the orthogonal group. In particular, the expected Euler characteristic of such random real projective…

This work improves SGD minibatch sampling using determinantal point processes based on orthogonal polynomials.

The study finds resonance points in polarised curves with polynomial conserved quantities.

For Poincare series of binary polyhedral groups and Coxeter polynomials there are obtained statements close to the Euclid algorithm and orthogonal polynomials theory: generalized Ebeling formula, decompositions into ramified continued fractions, Christoffel-Darboux identity, combinatorial formula. Known results about t…

We derive analytic series representations for European option prices in polynomial stochastic volatility models. This includes the Jacobi, Heston, Stein-Stein, and Hull-White models, for which we provide numerical case studies. We find that our polynomial option price series expansion performs as efficiently and accura…

The study characterizes and constructs polynomial harmonic morphisms on spheres.

Unified approach to tensor PCA and related problems using tensor cumulants.

The radar experiment connects the geometry of spacetime with an observers measurement of spatial length. We investigate the radar experiment on Finsler spacetimes which leads to a general definition of radar orthogonality and radar length. The directions radar orthogonal to an observer form the spatial equal time surfa…

This work proves the asymptotic freeness of layerwise Jacobians in MLPs with Haar orthogonal matrices.

A novel online GP model captures long-term memory in sequential data.

This paper simplifies conditional Sobol' indices calculation using PCE bases.

We present a riemannian structure on the disk that has a remarkably rich structure. Geodesics are hypocycloids and the (negative of the) laplacian has integer spectrum with multiplicity the Dirichlet divisor function. Eigenfunctions of the laplacian are orthogonal polynomials naturally suited to the analysis of acousti…

We obtain the first polynomial-time algorithm for exact tensor completion that improves over the bound implied by reduction to matrix completion. The algorithm recovers an unknown 3-tensor with $r$ incoherent, orthogonal components in $\mathbb R^n$ from $r\cdot \tilde O(n^{1.5})$ randomly observed entries of the tensor…

The paper computes characteristic classes for Lie group representations.

The paper proves deep neural networks with analytic activation can approximate any function.

The study examines how gamma positivity and PL homeomorphism types affect simplicial spheres.

Study reveals an equivalence principle for the spectrum of random inner-product kernel matrices in polynomial scaling.

Study the spectrum of Poincaré operator in triaxial ellipsoids.

A new neural network model uses polynomial chaos theory to improve neural signal processing.

The paper analyzes tensor recovery from symmetric rank-one measurements using information theory.

New algorithm learns ReLU networks efficiently using Schur polynomials.

Muon with Newton-Schulz converges to the same stationary point as SVD-polar, up to a constant factor.

Racah matrices and higher $j$-symbols are used in description of braiding properties of conformal blocks and in construction of knot polynomials. However, in complicated cases the logic is actually inverted: they are much better deduced from these applications than from the basic representation theory. Following the re…

New methods improve neural connectivity analysis at submillisecond timescales.

Non-Gaussian component analysis (NGCA) is a problem in multidimensional data analysis which, since its formulation in 2006, has attracted considerable attention in statistics and machine learning. In this problem, we have a random variable $X$ in $n$-dimensional Euclidean space. There is an unknown subspace $Γ$ of the …

PolyNSD improves Neural Sheaf Diffusion with polynomial operators and spectral rescaling.

This paper presents a novel one-factor stochastic volatility model where the instantaneous volatility of the asset log-return is a diffusion with a quadratic drift and a linear dispersion function. The instantaneous volatility mean reverts around a constant level, with a speed of mean reversion that is affine in the in…

Two criteria for a closed connected definite 4-manifold with infinite cyclic fundamental group to be TOP-split are given. One criterion extends a sufficient condition made in a previous paper. The result is equivalent to a purely algebraic result on the question asking when a positive definite Hermitian form over the r…

The ``Links-Gould invariant'' is a two-variable Laurent polynomial invariant of oriented (1,1) tangles, which is derived from the representation of the braid generator associated with the one-parameter family of four dimensional representations with highest weights (0,0|a) of the quantum superalgebra U_q[gl(2|1)]. We u…

New polynomial-time solutions found for training ReLU networks, mirroring Max-Cut complexity.

This paper provides mathematical foundations for regression methods used in forward initial margin approximation.

New method uses almost orthonormal bases to prove low-degree lower bounds in complex statistical models.

The paper explores statistical limits for detecting correlation in tree structures.

We analyze stochastic gradient descent for optimizing non-convex functions. In many cases for non-convex functions the goal is to find a reasonable local minimum, and the main concern is that gradient updates are trapped in saddle points. In this paper we identify strict saddle property for non-convex problem that allo…

This is mainly a survey, explaining how the probabilistic (statistical mechanical) construction of Kahler-Einstein metrics on compact complex manifolds, introduced in a series of works by the author, naturally arises from classical approximation and interpolation problems in complex n-space. A fair amount of background…