Two derivations of PCA for distributional data.
problem PCA for datasets of distributions.
method Two derivations: variance maximization and reconstruction error minimization.
result Closed-form solution for distributional PCA.
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48 results for “distributional derivative”
Study h-principles for non-integrable distributions on manifolds.
problem Existence and classification of maximally non-integrable distributions of derived length one.
method Introduced formal structures and used h-principles to discuss existence and classification.
result Discussed existence and classification of maximally non-integrable distributions of derived length one.
A new gradient estimator for categorical distributions reduces bias and variance.
problem Intractability of gradients for categorical distributions in discrete latent variable models.
method CatLog-Derivative trick and IndeCateR gradient estimator.
result IndeCateR reduces bias and variance of gradients for categorical distributions.
Researchers derived formulas for joint moments of elliptical distributions.
problem Calculating joint moments of elliptical distributions.
method Used Stein's lemma and two different methods to derive expressions.
result New formulae for expectations of product of normally distributed random variables and simplified expressions for other distributions.
New method uses DistRL to estimate entire payoff distribution for financial derivatives.
problem Traditional methods focus on expected option value; this tackles risk-aware pricing.
method Reinterprets and proposes a framework using Distributional Reinforcement Learning (DistRL).
result Demonstrates enhanced risk-aware pricing and uncertainty quantification on Asian options.
A strictification result is proved for isotropic distributions on derived schemes equipped with negatively shifted homotopically closed -forms. It is shown that any derived scheme over equipped with a -shifted symplectic structure, and having a Hausdorff space of classical points, admits a globally …
The paper derives risk measures for metalog distributions.
problem Deriving risk measures for metalog distributions.
method Closed-form expressions for Conditional Value at Risk and first-order partial moments.
result First-order partial moments are convex with respect to metalog parameters.
The Poisson distribution has been widely studied and used for modeling univariate count-valued data. Multivariate generalizations of the Poisson distribution that permit dependencies, however, have been far less popular. Yet, real-world high-dimensional count-valued data found in word counts, genomics, and crime statis…
A new operator based on t-distributions improves NN classifiers' robustness to out-of-distribution samples.
problem NN classifiers assign extreme probabilities to out-of-distribution samples, leading to unreliable predictions.
method Derive a novel operator using t-distributions to model uncertainty more accurately.
result Classifiers using the new operator are more robust to out-of-distribution samples.
Constructs covariant derivatives for Ehresmann connections.
problem Developing a method for covariant derivatives in fibre bundles.
method Introducing a vertical endomorphism to construct covariant derivatives on vertical and horizontal distributions.
result Covariant derivatives can be constructed separately on vertical and horizontal distributions and then glued together.
New potentials found for sheaves on Calabi-Yau 4-folds.
problem Understanding sheaves on Calabi-Yau 4-folds.
method Derived Quot-stacks and Lagrangian distributions.
result Globally defined -shifted potentials on sheaves.
We exploit the link between the transport equation and derivatives of expectations to construct efficient pathwise gradient estimators for multivariate distributions. We focus on two main threads. First, we use null solutions of the transport equation to construct adaptive control variates that can be used to construct…
Sampling Distributions of Optimal Portfolio Weights and Characteristics in Low and Large Dimensionsq-fin.PM
Optimal portfolio selection problems are determined by the (unknown) parameters of the data generating process. If an investor wants to realise the position suggested by the optimal portfolios, he/she needs to estimate the unknown parameters and to account for the parameter uncertainty in the decision process. Most oft…
New theory of distributions on spaces with singular submanifolds.
problem Defining distributions on spaces with singular submanifolds.
method Construction of thick distributions, operations, and special distributions.
result Clarified connection between thick and classical distributions.
Paper studies M-estimators with derivatives and residual distribution for robust adaptive tuning.
problem Tackles robustness and adaptive tuning of M-estimators with heavy-tailed noise.
method Provides formulae for derivatives, characterizes residual distribution, proposes adaptive criterion.
result Characterizes distribution of residuals and proposes adaptive criterion as out-of-sample error proxy.
We present the first treatment of the arc length of the Gaussian Process (GP) with more than a single output dimension. GPs are commonly used for tasks such as trajectory modelling, where path length is a crucial quantity of interest. Previously, only paths in one dimension have been considered, with no theoretical con…
Optimizes material distribution on surfaces using topological derivatives.
problem Optimal distribution of two materials on smooth submanifolds in .
method Topological derivative approach for shape optimization constrained by PDEs.
result Numerical solution of topology optimization problem on surfaces.
A method to estimate high order derivatives of data distributions from samples.
problem Estimating high order derivatives of data distributions efficiently and accurately.
method Generalizing denoising score matching via Tweedie's formula to estimate higher order derivatives.
result Models trained with the proposed method can approximate second order derivatives more efficiently and accurately than via automatic differentiation.
Optimal high-dimensional and nonparametric distributed testing under communication constraintsmath.ST
Optimal distributed testing under communication constraints with shared randomness.
problem Signal detection in a distributed system with limited communication.
method Derivation of minimax testing errors, distributed testing algorithms, and theoretical lower bounds.
result Consistent nonparametric distributed testing is possible even with minimal communication.
Extends curve theory to non-smooth data with finite curvature and torsion.
problem Applying classical curve theory to non-smooth data.
method Using distributional derivative measures of functions of bounded variation.
result Essentially unique non-smooth curve solution with finite total curvature and torsion.
In this paper we discuss the asymptotic behaviour of random contractions , where , with distribution function , is a positive random variable independent of . Random contractions appear naturally in insurance and finance. Our principal contribution is the derivation of the tail asymptotics of $X…
We derive and approximate the conjugate prior of Dirichlet and beta distributions.
problem Intractability of conjugate prior for Dirichlet and beta distributions.
method Derive conjugate prior, define closed-form approximation, and provide algorithm.
result Closed-form approximation enables fully tractable Bayesian treatment.
Coupled entropy corrects flaws in Tsallis entropy for complex systems.
problem Misinterpretation of generalized temperature and entropy.
method Derived from generalized Pareto and Student's t distributions.
result Provides balanced measure of uncertainty for complex systems.
Generic Unsupervised Optimization for a Latent Variable Model With Exponential Family Observablescs.LG
New LVMs optimize any exponential family distribution without specific assumptions.
problem Optimizing latent variable models with non-Gaussian observables.
method Generic optimization using EM approach for exponential family distributions.
result Concise parameter update equations applicable to various data types.
Paper develops a new algorithm for distribution regression with optimal learning rates.
problem Distribution regression with limited second-stage samples.
method Multi-penalty regularization in a reproducing kernel Hilbert space.
result Derives optimal learning rates for distribution regression.
Exponential distribution is ubiquitous in the framework of multi-agent systems. An alternative approach with an economic motivation to derive the exponential distribution in the framework of iterations in the space of distributions is disclosed.
Employing profits data of Japanese companies in 2002 and 2003, we identify the non-Gibrat's law which holds in the middle profits region. From the law of detailed balance in all regions, Gibrat's law in the high region and the non-Gibrat's law in the middle region, we kinematically derive the profits distribution funct…
Best- and worst-case Scenarios for GlueVaR distortion risk measure with Incomplete informationq-fin.RM
Paper derives best- and worst-case GlueVaR measures with incomplete data.
problem Risk measurement with limited information and shape constraints.
method Unified framework based on partial distribution information and shape properties.
result Characterization of extremal GlueVaR distributions with convex envelopes.
In this paper, we derive Hybrid, Bayesian and Marginalized Cramér-Rao lower bounds (HCRB, BCRB and MCRB) for the single and multiple measurement vector Sparse Bayesian Learning (SBL) problem of estimating compressible vectors and their prior distribution parameters. We assume the unknown vector to be drawn from a compr…
In this expository paper we illustrate the generality of game theoretic probability protocols of Shafer and Vovk (2001) in finite-horizon discrete games. By restricting ourselves to finite-horizon discrete games, we can explicitly describe how discrete distributions with finite support and the discrete pricing formulas…
We analyse derivative securities whose value is NOT a deterministic function of an underlying which means presence of a basis risk at any time. The key object of our analysis is conditional probability distribution at a given underlying value and moment of time. We consider time evolution of this probability distributi…
The problem of categorical data analysis in high dimensions is considered. A discussion of the fundamental difficulties of probability modeling is provided, and a solution to the derivation of high dimensional probability distributions based on Bayesian learning of clique tree decomposition is presented. The main contr…
Analyzes geodesic lengths in sparse networks, deriving a distribution.
problem Understanding connectivity and robustness in networked systems.
method Analytic derivation of geodesic length distribution in sparse networks.
result Simple closed-form expression for geodesic length distribution.
We develop a comprehensive geometric framework for defining spaces of nonlinear generalized sections of vector bundles containing spaces of distributional sections . Our theory incorporates classical differential geometric operations (like tensor products, covariant deri…
Employing profits data of Japanese firms in 2003--2005, we kinematically exhibit the static log-normal distribution in the middle scale region. In the derivation, a Non-Gibrat's law under the detailed balance is adopted together with following two approximations. Firstly, the probability density function of profits gro…
Asymptotic properties of one-layer artificial neural networks with sparse connectivitycond-mat.dis-nn
Law derived for neural networks with sparse connections.
problem Understanding the behavior of neural networks with sparse connections.
method Law of large numbers for empirical distribution of parameters derived.
result Law for neural networks with sparse connections derived.
Derivatives of Risk Measuresq-fin.RM
Derives derivatives of risk measures for various types of portfolio losses.
problem Calculating precise risk measures for portfolio losses.
method Analyzes first and second order derivatives of risk measures for both continuous and discrete portfolio loss scenarios.
result Provides asymptotic results for conditional moments of heavy-tailed portfolio losses.
The paper explores arbitrage opportunities in derivative markets under specific conditions.
problem Arbitrage opportunities in derivative markets under different conditions.
method Analyzes the relationship between pricing kernel monotonicity and stochastic arbitrage opportunities.
result Pricing kernel nonmonotonicity is equivalent to stochastic arbitrage opportunities under adequacy.
Moate Simulation improves accuracy and speed of financial derivative pricing.
problem Efficiently pricing financial derivatives with high accuracy.
method Discrete time simulation of probability distributions using Moate Simulation.
result Moate Simulation provides highly accurate distributions for financial derivatives pricing.
Forecast stock return distributions using neural networks.
problem Accurately modeling non-Gaussian stock return features.
method Two-stage quantile neural network with spline interpolation.
result Improved mean and variance forecasts compared to standard models.
New method for robust learning from batches, even adversarial ones.
problem Learning from batches that may be corrupt or adversarial.
method General framework for robust learning, derived from optimal robust algorithms.
result First robust agnostic learning algorithms for various distributions.
We develop a new model for VIX derivatives with closed-form solutions.
problem VIX derivatives pricing and risk management.
method Data-driven Legendre polynomial model for VIX volatility, deriving analytical series solutions.
result Equal or superior accuracy compared to existing models, offering an efficient alternative.
The paper extends Bochner's technique to singular distributions on manifolds.
problem Analyzing the curvature and null space of Hodge Laplacian on singular distributions.
method Defining modified statistical connection, exterior derivative, and Weitzenbock type curvature operator.
result Derivation of Bochner-Weitzenbock type formula leading to vanishing theorems.
The paper identifies generators of linear SDEs with noise types.
problem Identifying the generator of linear SDEs from their solution distribution.
method Deriving sufficient and necessary conditions for additive noise, and sufficient conditions for multiplicative noise.
result Generic conditions for identifying the generator of linear SDEs with both types of noise.
Nonholonomic Controlled Hamiltonian System: Symmetric Reduction and Hamilton-Jacobi Equationsmath.SG
This paper studies nonholonomic constraints in Hamiltonian systems, deriving equations and theorems.
problem Analyzing nonholonomic constraints in Hamiltonian systems.
method Deriving distributional RCH systems, geometric constraint conditions, and Hamilton-Jacobi theorems.
result Derives precise geometric constraint conditions and Hamilton-Jacobi theorems for nonholonomic systems.
In this paper we propose a generalized numerical scheme for backward stochastic differential equations(BSDEs). The scheme is based on approximation of derivatives via Lagrange interpolation. By changing the distribution of sample points used for interpolation, one can get various numerical schemes with different stabil…
This paper proposes to model asset price dynamics with a mixture of diffusion processes where the instantaneous volatility of the underlying diffusion process contains a random vector. The marginal probability distributions of the proposed process can match exactly the risk-neutral distributions implied by both spot va…
We study distributed estimation methods under communication constraints in a distributed version of the nonparametric random design regression model. We derive minimax lower bounds and exhibit methods that attain those bounds. Moreover, we show that adaptive estimation is possible in this setting.