Deep neural nets converge exponentially fast for analytic functions.
problem Analytic function approximation in low dimensions.
method Exponential convergence rate of deep neural networks for analytic functions.
result Deep neural nets achieve exponential convergence for analytic functions.
Proves new concentration inequalities for sub-gaussian and sub-exponential variables.
problem Understanding functions of independent random variables better.
method Sub-gaussian and sub-exponential conditions, Rademacher complexities, Lipschitz function classes.
result Extension of Rademacher complexities to unbounded sub-exponential distributions.
Researchers prove constant solutions for a specific Finslerian equation.
problem Investigating exponentially harmonic functions on Finslerian spaces.
method Analyzing the exponential energy functional and using nonnegative Ricci curvature conditions.
result Any bounded solution to the Finslerian equation is constant.
Covering spaces with exponential growth have non-constant positive harmonic functions.
problem Existence of non-constant positive harmonic functions on covering spaces with exponential growth.
method Normal Riemannian covering, exponential volume growth, Lyons and Sullivan conjecture.
result Existence of non-constant positive harmonic functions on M. Paper evaluates squared-exponential covariance function for Gaussian processes with integral observations.
problem Evaluating double line integrals of the squared exponential covariance function in Gaussian processes.
method Proposes a new approach to reduce double integrals to a single integral using the error function and efficiently computed with numerical techniques.
result Shows superior numerical robustness and accuracy compared to existing methods.
Study portfolio optimization with an exponential utility function and illiquid asset.
problem Optimizing a portfolio with a risk-free, liquid, and illiquid risky asset.
method Analytical substitution, Lie algebraic reduction, solving PDEs.
result Different optimization results for exponential utility function compared to HARA.
The purpose of this note is to describe, in terms of a power series, the distribution function of the exponential functional, taken at some independent exponential time, of a spectrally negative Lévy process ξwith unbounded variation. We also derive a Geman-Yor type formula for Asian options prices in a financial marke…
Proposes a new classification model using extended exponential functions.
problem Improving classification accuracy in binary linear classification problems.
method Developed a Bregman-Tweedie classification model based on extended exponential functions.
result The H-Bregman and L-Bregman sub-models outperform traditional methods in ranking and classification accuracy.
Kernel estimator optimally recovers function from noisy exponential Radon transform.
problem Inverting noisy exponential Radon transform of a function.
method Proposed a kernel estimator to estimate the true function.
result The estimator converges to the true function at minimax optimal rate.
A new method for exponentially weighted moving models using approximations.
problem Efficiently updating moving averages for time series data.
method Approximates EWMM using a fixed window and quadratic term, solving non-growing problems.
result Approximation produces estimates similar to exact EWMM.
A new machine learning model uses matrix exponentials for universal approximation.
problem Developing a robust and efficient machine learning model.
method Introduces a novel architecture using matrix exponentials as the only nonlinearity.
result The model achieves universal approximation properties and outperforms other models on benchmark tasks.
New energy functional and fields for Yang-Mills theory, proving monotonicity and vanishing theorems.
problem Developing new mathematical tools for Yang-Mills theory.
method Introducing normalized exponential Yang-Mills energy functional, deriving monotonicity formula and vanishing theorem.
result Monotonicity and vanishing theorems for exponential Yang-Mills fields.
Exponential functionals of Brownian motion have been extensively studied in financial and insurance mathematics due to their broad applications, for example, in the pricing of Asian options. The Black-Scholes model is appealing because of mathematical tractability, yet empirical evidence shows that geometric Brownian m…
Three-hidden-layer neural networks can approximate Hölder continuous functions uniformly with exponential rate.
problem Approximating Hölder continuous functions with neural networks.
method Introduced Floor-Exponential-Step (FLES) networks with three hidden layers.
result Uniform approximation of Hölder continuous functions with an exponential rate.
Paper solves best approximation by exponential functions for economic data.
problem Best approximation of data by exponential functions.
method TAC algorithm for fitting data by exponential patterns.
result TAC algorithm successfully applied to demand curves and nonlinear time series.
The paper classifies statistical Einstein manifolds in exponential families.
problem Classifying statistical Einstein manifolds in exponential families.
method Deriving partial differential equations for potential functions, obtaining special and group-invariant solutions.
result Special and group-invariant solutions of the equations for potential functions of exponential families.
This paper studies a class of exponential family models whose canonical parameters are specified as linear functionals of an unknown infinite-dimensional slope function. The optimal minimax rates of convergence for slope function estimation are established. The estimators that achieve the optimal rates are constructed …
We produce examples of groups of type F_3 with 2-dimensional Dehn functions of the form exp^n(x) (a tower of exponentials of height n), where n is any natural number.
In the paper, the author studies properties of three functions relating to the exponential function and the existence of partitions of unity, including accurate and explicit computation of their derivatives, analyticity, complete monotonicity, logarithmically complete monotonicity, absolute monotonicity, and the like.
A new model relaxes constraints on exponential dispersion models.
problem Tight conditions on cumulant function limit the class of exponential dispersion models.
method Introduces K-LED model with Legendre cumulant function and Bregman divergence guidance.
result The model allows for easier computation of mean parameter and includes various distributions.
Correspondence found between exponential families and affine Grassmannians.
problem Understanding the relationship between exponential families and geometric structures.
method Established a one-to-one correspondence between exponential families and affine Grassmannians.
result Found a correspondence between minimal exponential families and affine Grassmannians.
We prove that the number of combinatorially distinct causal 3-dimensional triangulations homeomorphic to the 3-dimensional sphere is bounded by an exponential function of the number of tetrahedra. It is also proven that the number of combinatorially distinct causal 4-dimensional triangulations homeomorphic to the 4-sph…
A function is exponentially concave if its exponential is concave. We consider exponentially concave functions on the unit simplex. In a previous paper we showed that gradient maps of exponentially concave functions provide solutions to a Monge-Kantorovich optimal transport problem and give a better gradient approximat…
The exponential mechanism is extended to infinite dimensional outputs, leading to a Central Limit Theorem and a non-negligible privacy noise.
problem Extending the exponential mechanism to infinite dimensional outputs for privacy and statistical estimation.
method Designing the exponential mechanism with respect to a specific base measure over the output space, and proving a Central Limit Theorem.
result The magnitude of privacy noise is asymptotically non-negligible relative to the statistical estimation error.
The study shows exponential distortion in virtually special groups containing free subgroups.
problem Understanding distortion in virtually special groups containing free subgroups.
method Constructing examples of virtually special groups with finite rank free subgroups.
result Distortion functions grow like exp^k(x^m) and can be superexponential.
Paper proposes a new e-exponentiated transformation to make convex loss functions more robust to outliers.
problem Making convex loss functions robust to outliers in the presence of label noise.
method Introduces a novel e-exponentiated transformation for loss functions and proves its effectiveness through theoretical and empirical analysis. result The transformed loss function achieves tighter generalization error bounds and higher accuracy in noisy datasets.
The logistic regression model is known to converge to a Poisson point process model if the binary response tends to infinitely imbalanced. In this paper, it is shown that this phenomenon is universal in a wide class of link functions on binomial regression. The proof relies on the extreme value theory. For the logit, p…
We analyze the data on personal income distribution from the Australian Bureau of Statistics. We compare fits of the data to the exponential, log-normal, and gamma distributions. The exponential function gives a good (albeit not perfect) description of 98% of the population in the lower part of the distribution. The lo…
Paper introduces a method for supervised hierarchical clustering with Exponential Linkage.
problem Discrepancy between training and clustering objectives in supervised clustering.
method Tightly couples supervised training of dissimilarity function with hierarchical clustering, using Exponential Linkage.
result Joint training procedure consistently matches or outperforms other methods, improving dendrogram purity by up to 8 points.
Paper explores connections between loss functions and consistency in binary classification and regression.
problem Consistency in binary classification and regression applications.
method Characterization of conformable loss functions and derivation of a new Huber-type loss function.
result Margin-based loss functions are equivalent to loss functions of squared standardized logistic regression residuals.
Exponential Lasso improves Lasso's robustness to outliers and heavy-tailed noise.
problem Lasso's sensitivity to outliers and heavy-tailed noise in high-dimensional statistics.
method Integrates an exponential-type loss function into the Lasso framework.
result Achieves strong statistical convergence rates robust to heavy-tailed contamination.
First order methods can take extremely long to find global minima of non-convex functions.
problem Finding global minimizers of non-convex functions.
method Designing a family of non-convex functions and using statistical lower bounds for parameter estimation.
result First order methods can take exponential time to converge to a global minimizer.
Unified method for deriving formulas on almost complex manifolds.
problem Deriving formulas for derivations conjugated by exponential functions.
method Unified method using exponential functions.
result Corrected mistakes in previous versions.
Incorporates matrix exponential into generative flows for improved performance.
problem Improving generative flow models for better density estimation.
method Integrates matrix exponential into generative flows, proposing new layers and modifying network architecture.
result The proposed model achieves great performance on density estimation.
New RL approach handles non-exponential discounting for sequential decisions.
problem Modeling human discounting in sequential decision-making tasks.
method Generalized model-based reinforcement learning with arbitrary discount functions, using Hamilton-Jacobi-Bellman equation and collocation method.
result Validated approach on simulated problems, showing applicability to human discounting.
Introduces a new theoretical framework for exponential smoothing.
problem Theoretical foundation and robustness of simple exponential smoothing.
method Stochastic gradient ascent to optimize Gaussian log-likelihood functions.
result Simple exponential smoothing converges to the trend of a trend-stationary process.
Optimal insurance policy for exponential utility maximization with convex premium calculation.
problem Maximizing terminal wealth utility with exponential utility function and convex premium formula.
method Necessary condition for optimal indemnity, numerical algorithm to compute it, convergence proof.
result Numerical algorithm converges to unique optimal indemnity.
This paper studies stability of the exponential utility maximization when there are small variations on agent's utility function. Two settings are considered. First, in a general semimartingale model where random endowments are present, a sequence of utilities defined on R converges to the exponential utility. Under a …
Study on harmonic functions in spaces with collapsing behaviors.
problem Harmonic functions on spaces with inhomogeneous collapsing behaviors at infinity.
method Analysis of complete and incomplete spaces with nonnegative Ricci curvature.
result Any nonconstant harmonic function yields a definite exponential growth rate.
New algorithm trains deep neural networks without global optimization.
problem Training deep neural networks efficiently and without global optimization.
method Uses random complex exponential activation functions and Markov Chain Monte Carlo sampling.
result Consistently attains theoretical approximation rate for residual networks.
New approach to concentration inequalities for unbounded state space dynamical systems.
problem Concentration inequalities for unbounded state space dynamical systems.
method Functional analytic framework, transport-entropy inequality.
result Exponential concentration inequalities for sampling from stationary distribution.
Signals are generally modeled as a superposition of exponential functions in spectroscopy of chemistry, biology and medical imaging. For fast data acquisition or other inevitable reasons, however, only a small amount of samples may be acquired and thus how to recover the full signal becomes an active research topic. Bu…
Adapts to estimate functions from noisy ERT data.
problem Estimating functions from noisy Exponential Radon Transform data.
method Locally adaptive kernel type estimator for functions of varying smoothness.
result Achieves minimax optimal rate up to a log(n) factor for Sobolev functions.
New activation function SERLU improves neural network performance.
problem Improving neural network performance and avoiding overfitting.
method Introducing a new activation function (SERLU) that breaks monotonicity while preserving self-normalizing property and developing shift-dropout for regularization.
result SERLU-based neural networks provide consistently promising results compared to other activation functions.
Unified framework for analyzing gradient flows of measures with exponential decay of entropy.
problem Analyzing exponential decay of entropy functionals in gradient flows of measures.
method Characterization of global exponential decay behaviors using Hellinger-Kantorovich geometry, shape-mass decomposition, and Polyak-Łojasiewicz-type inequalities.
result Unified theoretical framework for gradient flows with complete analysis of exponential decay behaviors.
The paper explores the limits of deep neural networks in approximating various function classes.
problem Characterizing the limits of deep neural networks in function approximation.
method Develops a theory relating function complexity and network complexity, using Kolmogorov complexity.
result Deep networks are optimal approximants for various function classes and provide exponential approximation accuracy.
Study dynamics of Lp-multipliers on harmonic manifolds with exponential volume growth.
problem Characterize the behavior of Lp-multipliers on harmonic manifolds of purely exponential volume growth. method Analyzing the dynamics of Lp-multipliers on non-compact harmonic manifolds, using Fourier transformation and properties of radial functions. result Show that Lp-multipliers acting nicely on smooth functions with compact support for p≤2 cannot be chaotic. Exponential rate of convergence for harmonic heat flow maps.
problem Analyzing the convergence rate of harmonic heat flow maps.
method Proving exponential convergence rate for harmonic heat flow maps.
result Exponential convergence rate of the harmonic heat flow.