arXiv research

Study improves variance calculation for random zero sets on complex manifolds.

The paper connects bundle curvature to random zero currents.

Study on zeros of Gaussian sections on semipositive line bundles on punctured Riemann surfaces.

In this work we prove an universality result regarding the equidistribution of zeros of random holomorphic sections associated to a sequence of singular Hermitian holomorphic line bundles on a compact Kähler complex space $X$. Namely, under mild moment assumptions, we show that the asymptotic distribution of zeros of r…

Study Bergman kernels and zero distributions of random sections on Kähler manifolds.

The paper studies Fubini-Study metrics and zero distributions on CR manifolds.

This thesis predicts the distribution of smoothed zeros of random sections on line bundles.

We study a tower of normal coverings over a compact Kähler manifold with holomorphic line bundles. When the line bundle is sufficiently positive, we obtain an effective estimate, which implies the Bergman stability. As a consequence, we deduce the equidistribution for zero currents of random holomorphic sections. Furth…

We establish the equidistribution of zeros of random holomorphic sections of powers of a semipositive singular Hermitian line bundle, with an estimate of the convergence speed.

The paper equidistributes zeros of random polynomials and sections on manifolds.

The study proves a central limit theorem for Gaussian holomorphic sections on Kähler manifolds.

The paper studies random systems of holomorphic sections on compact Kähler manifolds and proves equidistribution results.

Surveying random sections on Kähler manifolds, leading to metrics.

A new gradient estimator for online optimization with two function evaluations.

We estimate generic statistical properties of a structural credit risk model by considering an ensemble of correlation matrices. This ensemble is set up by Random Matrix Theory. We demonstrate analytically that the presence of correlations severely limits the effect of diversification in a credit portfolio if the corre…

Paper introduces S-SSE for stable sparse subspace embedding.

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…

Boosted CVaR Classification improves tail performance in classification tasks.

Rough volatility models are becoming increasingly popular in quantitative finance. In this framework, one considers that the behavior of the log-volatility process of a financial asset is close to that of a fractional Brownian motion with Hurst parameter around 0.1. Motivated by this, we wish to define a natural and re…

Randomly sampled interpolators achieve zero generalization error with enough data.

Consider a random smooth Gaussian field $G(x):F\to\mathbb{R}$, where $F$ is a compact in $\mathbb{R}^d$. We derive a formula for average area of a surface generated by the equation $G(x)=0$ and give some applications. As an auxiliary result we obtain an integral expression for area of a surface induced by zeros of a \e…

New approach to handle ranking function variation in zero-shot NAS.

We prove that the Euler form of a metric connection on real oriented vector bundle $E$ over a compact oriented manifold $M$ can be identified, as a current, with the expectation of the random current defined by the zero-locus of a certain random section of the bundle. We also explain how to reconstruct probabilisticall…

Tian's theorem connects Chern classes of bundles to random section zeros and degeneracy sets.

KSG mutual information estimator, which is based on the distances of each sample to its k-th nearest neighbor, is widely used to estimate mutual information between two continuous random variables. Existing work has analyzed the convergence rate of this estimator for random variables whose densities are bounded away fr…

Study optimizes zero-order strongly convex function minimization with higher order smoothness.

In a previous work, the first and third authors studied a random knot model for all two-bridge knots using billiard table diagrams. Here we present a closed formula for the distribution of the crossing numbers of such random knots. We also show that the probability of any given knot appearing in this model decays to ze…

New algorithm optimizes positions of CountSketch non-zero entries for better data compression.

Robust Lasso-Zero handles missing covariates and sparse corruptions.

Study of random sections on complex spaces converging to equilibrium metrics.

We show that the exterior powers of the matrix valued random walk invariant of string links, introduced by Lin, Tian, and Wang, are isomorphic to the graded components of the tangle functor associated to the Alexander Polynomial by Ohtsuki divided by the zero graded invariant of the functor. Several resulting propertie…

Study uses zero-shot models to forecast mortality rates globally.

We study the asymptotics of Fubini-Study currents and zeros of random holomorphic sections associated to a sequence of singular Hermitian line bundles on a compact normal Kaehler complex space.

Consider a $d\times d$ matrix $M$ whose rows are independent centered non-degenerate Gaussian vectors $ξ_1,...,ξ_d$ with covariance matrices $Σ_1,...,Σ_d$. Denote by $\mathcal{E}_i$ the location-dispersion ellipsoid of $ξ_i:\mathcal{E}_i={\mathbf{x}\in\mathbb{R}^d : \mathbf{x}^\topΣ_i^{-1} \mathbf{x}\leqslant1}$. We sh…

Study shows zero probability of cut locus for Fréchet mean on Riemannian manifolds.

We prove that a Gaussian ensemble of smooth random sections of a real vector bundle over compact manifold canonically defines a metric on the bundle together with a connection compatible with it. Additionally, we prove a refined Gauss-Bonnet-Chern theorem stating that if the bundle and the manifold are oriented, then t…

A new definition of events of game-theoretic probability zero in continuous time is proposed and used to prove results suggesting that trading in financial markets results in the emergence of properties usually associated with randomness. This paper concentrates on "qualitative" results, stated in terms of order (or or…

We present the results of computer experiments suggesting that the probability that a random multiword in a free group is virtually geometric decays to zero exponentially quickly in the length of the multiword. We then prove this fact.

Analyzes bias-variance in overparameterized linear models using random features.

This study examines a single attention layer's capabilities using random features.

The issue of giving an explicit description of the flow of information concerning the time of bankruptcy of a company (or a state) arriving on the market is tackled by defining a bridge process starting from zero and conditioned to be equal to zero when the default occurs. This enables to catch some empirical facts on …

Study shows normal distribution in divisor counts of random sections on complex manifolds.

Gradient descent learns ReLU functions with non-zero bias efficiently.

In principle, zero-shot learning makes it possible to train a recognition model simply by specifying the category's attributes. For example, with classifiers for generic attributes like \emph{striped} and \emph{four-legged}, one can construct a classifier for the zebra category by enumerating which properties it posses…

Study on geodesics and eigenvalues on random hyperbolic surfaces with cusps.

New method uncovers zero entropy in dependent observations after finite samples.

This paper proposes a parsimoniously time varying parameter vector autoregressive model (with exogenous variables, VARX) and studies the properties of the Lasso and adaptive Lasso as estimators of this model. The parameters of the model are assumed to follow parsimonious random walks, where parsimony stems from the ass…

The paper estimates variance of random sections on complex manifolds.