Unified approach to totally ramified values in various surface theories.
problem Totally ramified values in value distribution theory, normal family theory, and Gauss maps of surfaces.
method Bloch--Ros principle applied to various surface theories.
result Unified approach to phenomena concerning totally ramified values.
New method uses extreme value theory to estimate neural network errors.
problem Quantifying the error of neural networks, especially for large values.
method Applying extreme value theory to approximate the distribution of error.
result Developed a new estimator for the shape parameter of the Pareto distribution.
A new framework assigns values to data points considering their distribution.
problem Limited applicability of data Shapley to points outside the fixed data set.
method Proposes distributional Shapley, defining point value in context of data distribution.
result Distributional Shapley values are stable under data point and distribution perturbations.
The hidden tail of empirical distributions is analyzed using extreme value theory.
problem Understanding the bias between in-sample mean and true statistical mean for large n. method Extreme value theory applied to empirical distributions and their moments.
result The hidden moment of order 0 for power law distributions follows an exponential distribution with expectation 1/n. We introduce equivariant Liouville forms and Duistermaat-Heckman distributions for Hamiltonian group actions with group valued moment maps. The theory is illustrated by applications to moduli spaces of flat connections on 2-manifolds.
The paper tackles catastrophic risk in reinforcement learning using extreme value theory.
problem Mitigating catastrophic risk in sequential decision making with limited observations.
method Developed POTPG, a policy gradient algorithm based on extreme value theory.
result POTPG outperforms common benchmarks in numerical experiments.
This paper extends Nevanlinna's unicity theorems to complete Kahler manifolds.
problem Generalizing Nevanlinna's unicity theorems to non-compact Kahler manifolds.
method Applying the theorems to complete Kahler manifolds with specific curvature conditions.
result Generalized Nevanlinna's unicity theorems for specific types of Kahler manifolds.
In this paper, we explore various statistical techniques for anomaly detection in conjunction with the popular Long Short-Term Memory (LSTM) deep learning model for transportation networks. We obtain the prediction errors from an LSTM model, and then apply three statistical models based on (i) the Gaussian distribution…
Paper introduces SPADE method to protect classifiers from OOD and adversarial samples.
problem Protecting classifiers from out-of-distribution and adversarial samples.
method SPADE method based on GEV model in latent space.
result Provable protection against OOD and adversarial samples.
Paper analyzes singular subspace estimation in noisy matrix models.
problem Estimating low-rank signals in noisy matrix data.
method Asymptotic distributional theory, extreme value theory, saddle point approximation, random matrix theory.
result Plug-in test statistic based on two-to-infinity norm has higher power for detecting structured alternatives.
In this article we show the relationship between the Pareto distribution and the gamma distribution. This shows that the second one, appropriately extended, explains some anomalies that arise in the practical use of extreme value theory. The results are useful to certain phenomena that are fitted by the Pareto distribu…
Extreme value theory enhances statistical learning extrapolation for rare events.
problem Challenges in traditional machine learning methods for extreme data.
method Asymptotic theory and statistical tools for tail behavior.
result Effective extrapolation methods for extreme quantiles and anomalies.
Several well-established benchmark predictors exist for Value-at-Risk (VaR), a major instrument for financial risk management. Hybrid methods combining AR-GARCH filtering with skewed-t residuals and the extreme value theory-based approach are particularly recommended. This study introduces yet another VaR predictor, …
New neural network models extreme value distributions with preserved shape constraints.
problem Modeling multivariate extreme value distributions with preserved shape constraints.
method d-max-decreasing neural network architecture for non-parametric calibration and generation of MEVs.
result The proposed architecture approximates the dependence structure of MEVs at parametric rate and preserves essential shape constraints.
Proposes a method to model financial returns with extreme shocks using flexible tail transformations.
problem Capturing extreme shocks in financial return data.
method Introduces a transformation layer in normalizing flows to model heavy-tailed distributions.
result Trained models can generate synthetic sets of extreme returns.
Paper improves risk estimation for rare events in sequential decisions.
problem Estimating tail risk in high confidence levels with limited data.
method Combines extreme value theory and automated threshold selection.
result Improves performance in estimating tail risk for some distributions.
We develop a theory to represent dislocated single crystals at the mesoscopic scale by considering concentrated effects, governed by the distribution theory combined with multiple-valued kinematic fields. Our approach gives a new understanding of the continuum theory of defects as developed by Kroener (1980) and other …
For spherically symmetric distributions, efficient quantisation can be achieved with moderate sample sizes.
problem Optimal quantisation in high dimensions requires large sample sizes, making it impractical.
method Uniformly distributed random quantisers on a sphere of suitable radius achieve exceptional performance.
result For moderate sample sizes, quantisation error can be efficiently computed and approximated.
We present in this article a survey of recent results in value distribution theory for the Gauss maps of several classes of immersed surfaces in space forms, for example, minimal surfaces in Euclidean n-space (n=3 or 4), improper affine spheres in the affine 3-space and flat surfaces in hyperbolic 3-space. In parti…
This work analyzes self-attention matrices using random matrix theory.
problem Understanding the theoretical behavior of self-attention layers in neural networks.
method Asymptotic spectral analysis of the attention matrix, Gaussian equivalence, and linearization.
result The singular value distribution of the attention matrix is asymptotically characterized by a linear model.
The distribution of price returns for a class of uncorrelated diffusive dynamics is considered. The basic assumptions are (1) that there is a "consensus" value associated with a stock, and (2) that the rate of diffusion depends on the deviation of the stock price from the consensus value. We find an analytical expressi…
Efficient methods estimate bid and value distributions in auctions.
problem Estimating bid and value distributions in auctions with limited information.
method Non-parametric estimation algorithms for first- and second-price auctions.
result Uniform estimation bounds for bid and value distributions, independent of distributions being estimated.
Paper generalizes Bloch-Ros principle to various surface classes.
problem Understanding the relationship between normal family theory, value distribution theory, and surface theory.
method Formulation and generalization of Bloch-Ros principle to different surface classes.
result Effective criterion for determining Gaussian curvature estimates for various surface classes.
Study shows one-dimensional location-scale-shape models are flat in Wasserstein geometry.
problem Investigating curvature in location-scale-shape models under Wasserstein metric.
method Introduced location-scale-shape model and investigated its geometry.
result Location-scale-shape model is intrinsically flat but extrinsically curved in Wasserstein geometry.
The paper proves a distribution claim for neural network Jacobians.
problem Distribution of singular values in deep neural networks.
method Free probability and random matrix theory techniques.
result Singular value distribution matches for specific cases.
The argument that the alarming level of Gini coefficient is 0.4 is very popular, especially in the media industry, all around the world for a long time. Although the 0.4 standard is widely accepted, the derivation of the value lacks rigid theoretical foundations. In fact, to the best of our knowledge, it is not based o…
Develops a dynamic mean field theory for reinforcement learning.
problem Finite state and action Bayesian reinforcement learning in large state spaces.
method Analogies with statistical physics, interpreting probabilities as couplings and values as spins, solving mean field equations.
result State-action values are statistically independent in the asymptotic state space limit, with exact or approximate equations for computation.
This paper is devoted to study the optimal portfolio problem. Harry Markowitz's Ph.D. thesis prepared the ground for the mathematical theory of finance. In modern portfolio theory, we typically find asset returns that are modeled by a random variable with an elliptical distribution and the notion of portfolio risk is d…
New method attributes feature uncertainty in ML models using cooperative game theory.
problem Lack of feature-level uncertainty attribution in explainable AI.
method Proposes a novel, model-agnostic uncertainty attribution method using cooperative game theory and conformal prediction.
result Demonstrates improved runtime efficiency and practical utility in real-world applications.
Estimation of tail quantities, such as expected shortfall or Value at Risk, is a difficult problem. We show how the theory of nonlinear expectations, in particular the Data-robust expectation introduced in [5], can assist in the quantification of statistical uncertainty for these problems. However, when we are in a hea…
It is well known that the initialization of weights in deep neural networks can have a dramatic impact on learning speed. For example, ensuring the mean squared singular value of a network's input-output Jacobian is O(1) is essential for avoiding the exponential vanishing or explosion of gradients. The stronger condi…
Random matrix analysis reveals that neural network weights are mostly random, with some indicating learned information.
problem Understanding how neural networks store information needed for tasks.
method Random matrix theory (RMT) applied to weight matrices of trained deep neural networks.
result Most singular values and eigenvectors of trained neural networks follow universal RMT predictions, suggesting they are random and do not contain system-specific information.
The thesis evaluates and compares extreme mixture models in finance and insurance.
problem Estimating tail risk measures in finance and insurance.
method Extreme mixture models and methods, including kernel density estimation and GARCH preprocessing.
result Kernel density estimation-based models do not outperform others in tail risk estimation.
Simpler one-step distributional RL framework for control.
problem Lack of a unified theory for DistrRL in control.
method One-step distributional reinforcement learning (OS-DistrRL) framework.
result Unified theory for policy evaluation and control.
Unified framework for feature-based explanations using ANOVA and game theory.
problem Differences between feature-based explanations methods limit their applicability.
method Introduces a unified framework combining fANOVA and cooperative game theory.
result Uncovered similarities and differences between various explanation techniques.
"How much is my data worth?" is an increasingly common question posed by organizations and individuals alike. An answer to this question could allow, for instance, fairly distributing profits among multiple data contributors and determining prospective compensation when data breaches happen. In this paper, we study the…
We present a comprehensive theory of homogeneous volatility (and variance) estimators of arbitrary stochastic processes that fully exploit the OHLC (open, high, low, close) prices. For this, we develop the theory of most efficient point-wise homogeneous OHLC volatility estimators, valid for any price processes. We intr…
We present a method for constructing the log-optimal portfolio using the well-calibrated forecasts of market values. Dawid's notion of calibration and the Blackwell approachability theorem are used for computing well-calibrated forecasts. We select a portfolio using this "artificial" probability distribution of market …
Commentary on Teichmüller's 1938 paper on conformal and quasiconformal mappings.
problem Investigations into conformal and quasiconformal mappings and their applications.
method Detailed development of conformal invariants and applications in value distribution theory.
result Insures the almost circularity of certain loci and the circularity near infinity of quasiconformal maps.
Identifies root causes of outliers using causal DAGs.
problem Detecting and understanding the causes of anomalies in interrelated variables.
method Formal approach using causal directed acyclic graphs (DAGs), outlier scores, and Shapley values.
result Quantifies the extent of outlier scores attributed to ancestors.
Study a market with uncertain informed traders, finding price impact depends on both asset value and informed trader count distribution.
problem Uncertain participation of informed traders in a market with limit orders.
method Characterized equilibrium by a fixed point integral equation, analyzed large order asymptotics, solved numerically.
result Equilibrium price impact depends on both asset value and distribution of informed traders, not just expected number of informed traders.
New theory of sensitivity for unbiased estimators using Wasserstein geometry.
problem Estimating the instability of estimators under small perturbations.
method Developed a new theory based on Wasserstein geometry, analogous to classical Cramér-Rao theory.
result Wasserstein-Cramér-Rao lower bound for sensitivity of unbiased estimators.
The article models financial asset returns using Gaussian mixtures and EVT-based copulas to price equity options.
problem Modeling financial asset returns and pricing equity options considering extreme values.
method Modeling marginal distributions with Gaussian mixtures and joint dependence structure with EVT-based copulas.
result The approach accurately prices various equity options on Atos and Dassault Systems actions.
This paper develops DRO estimators for EVT statistics using point processes.
problem Scarcity of extreme data leads to model misspecification error in EVT.
method Developed DRO estimators informed by semi-parametric max-stable constraints in the space of point processes.
result Proposed DRO estimators improve out-of-sample performance and are validated on synthetic and real data.
This paper uses ML and EVT to analyze tree ring data, improving accuracy of predictions.
problem Analyzing tree ring data for climate modeling and historical studies.
method Combines machine learning algorithms with extreme value theory for data analysis.
result Random Forest method yields the most accurate results for tree ring data analysis.
This paper deals with optimally-robust parameter estimation in generalized Pareto distributions (GPDs). These arise naturally in many situations where one is interested in the behavior of extreme events as motivated by the Pickands-Balkema-de Haan extreme value theorem (PBHT). The application we have in mind is calcula…
Establishes Poincaré's lemma for formal manifolds.
problem Developing smooth relative Lie algebra homologies and cohomologies.
method Theory of formal manifolds and formal Lie groups.
result Poincaré's lemma for de Rham complexes with formal functions and generalized functions.
The geometry of graded principal bundles is discussed in the framework of graded manifold theory of Kostant-Berezin-Leites. In particular, we prove that a graded principal bundle is globally trivial if and only if it admits a global graded section and, further, that the sheaf of vertical derivations on such a bundle co…