arXiv research

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

Maximum likelihood learning with exponential families leads to moment-matching of the sufficient statistics, a classic result. This can be generalized to conditional exponential families and/or when there are hidden data. This document gives a first-principles explanation of these generalized moment-matching conditions…

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…

AdamNX improves Adam's stability by adjusting its learning rate.

In this paper we study the exponential functionals of the processes $X$ with independent increments , namely $I_t= \int _0^t\exp(-X_s)ds, _,\,\, t\geq 0,$ and also $I_{\infty}= \int _0^{\infty}\exp(-X_s)ds.$ When $X$ is a semi-martingale with absolutely continuous characteristics, we derive recurrent integral equat…

We provide a surprising new application of classical approximation theory to a fundamental asset-pricing model of mathematical finance. Specifically, we calculate an analytic value for the correlation coefficient between exponential Brownian motion and its time average, and we find the use of divided differences greatl…

The hidden tail of empirical distributions is analyzed using extreme value theory.

A tractable pseudo-metric for non-parametric distributions via SPD geometry.

Study differentially private linear regression with heavy-tailed data.

New method tightens sub-Gaussian concentration inequalities.

Study on U-statistics with heavy-tailed samples, providing tail bounds and LDP.

Study well-posedness of SPDE on Riemannian manifolds with rough initial conditions.

The paper uses moment matching method for pricing spread options under Lévy models.

Adaptive t-distribution estimates nonstationary time series using moving moments.

We study concentration phenomena of eigenfunctions of the Laplacian on closed Riemannian manifolds. We prove that the volume measure of a closed manifold concentrates around nodal sets of eigenfunctions exponentially. Applying the method of Colding and Minicozzi we also prove restricted exponential concentration inequa…

Barren plateaus are not an average-case phenomenon, but a highly non-unique problem.

Study shows exponential growth of Laplacian determinant on random hyperbolic surfaces.

Improved Sobolev mappings in Carnot groups with weaker assumptions.

The study of random walks on hyperbolic spaces and Teichmüller spaces, proving central limit theorems and geodesic tracking.

Time homogeneous polynomial processes are Markov processes whose moments can be calculated easily through matrix exponentials. In this work, we develop a notion of time inhomogeneous polynomial processes where the coeffiecients of the process may depend on time. A full characterization of this model class is given by m…

We analyze exponential integrability properties of the Cox-Ingersoll-Ross (CIR) process and its Euler discretizations with various types of truncation and reflection at 0. These properties play a key role in establishing the finiteness of moments and the strong convergence of numerical approximations for a class of sto…

Exponential family distributions are highly useful in machine learning since their calculation can be performed efficiently through natural parameters. The exponential family has recently been extended to the t-exponential family, which contains Student-t distributions as family members and thus allows us to handle noi…

Efficient method for learning continuous exponential families beyond Gaussian.

The asymptotic behavior of the implied volatility associated with a general call pricing function has been extensively studied in the last decade. The main topics discussed in this paper are Lee's moment formulas for the implied volatility, and Piterbarg's conjecture, describing how the implied volatility behaves in th…

MGD combines maximum entropy and diffusion methods for efficient sampling.

We introduce a novel approach, requiring only mild assumptions, for the characterization of deep neural networks at initialization. Our approach applies both to fully-connected and convolutional networks and easily incorporates batch normalization and skip-connections. Our key insight is to consider the evolution with …

The paper improves PAC-Bayes bounds for losses with finite moments.

The paper analyzes stability of random matrix products with Markovian noise.

In the setting of polynomial jump-diffusion dynamics, we provide an explicit formula for computing correlators, namely, cross-moments of the process at different time points along its path. The formula appears as a linear combination of exponentials of the generator matrix, extending the well-known moment formula for p…

DGMM improves Gaussian mixture modeling efficiency and stability.

The paper introduces a new method for tail bounds of random vectors and matrices.

Nonlinear SGD achieves high-probability rates in non-convex optimization with heavy-tailed noise.

A new method for generating samples without training, using smoothed score matching.

We consider the problem of predicting as well as the best linear combination of d given functions in least squares regression, and variants of this problem including constraints on the parameters of the linear combination. When the input distribution is known, there already exists an algorithm having an expected excess…

We consider the problem of learning a mixture of linear regressions (MLRs). An MLR is specified by $k$ nonnegative mixing weights $p_1, \ldots, p_k$ summing to $1$, and $k$ unknown regressors $w_1,...,w_k\in\mathbb{R}^d$. A sample from the MLR is drawn by sampling $i$ with probability $p_i$, then outputting $(x, y)$ wh…

A new filter reduces density fitting to a linear solve, improving performance on nonlinear systems.

Paper presents robust confidence sequences for means with known moment bounds and arbitrary corruption.

Paper proposes ClipSMT algorithm for better ATE estimation.

New algorithms achieve high-probability parameter-free regret in online convex optimization with heavy-tailed data.

This paper provides an insight to the time-varying dynamics of the shape of the distribution of financial return series by proposing an exponential weighted moving average model that jointly estimates volatility, skewness and kurtosis over time using a modified form of the Gram-Charlier density in which skewness and ku…

We introduce a new set of consistent measures of risks, in terms of the semi-invariants of pdf's, such that the centered moments and the cumulants of the portfolio distribution of returns that put more emphasis on the tail the distributions. We derive generalized efficient frontiers, based on these novel measures of ri…

Improved concentration inequalities for sub-Weibull variables enhance statistical and machine learning applications.

Factorial moments are convenient tools in particle physics to characterize the multiplicity distributions when phase-space resolution ($Δ$) becomes small. They include all correlations within the system of particles and represent integral characteristics of any correlation between these particles. In this letter, we sh…

In this paper we propose a closed-form approximation for the price of basket options under a multivariate Black-Scholes model, based on Taylor expansions and the calculation of mixed exponential-power moments of a Gaussian distribution. Our numerical results show that a second order expansion provides accurate prices o…

Bayesian framework uses AI-generated data to improve parameter estimation.

The well known maximum-entropy principle due to Jaynes, which states that given mean parameters, the maximum entropy distribution matching them is in an exponential family, has been very popular in machine learning due to its "Occam's razor" interpretation. Unfortunately, calculating the potentials in the maximum-entro…

Intertemporal decision making involves choices among options whose effects occur at different moments. These choices are influenced not only by the effect of rewards value perception at different moments, but also by the time perception effect. One of the main difficulties that affect standard experiments involving int…

This paper compares VaR estimation methods under tail misspecification, finding importance sampling underestimates VaR.