Study improves probabilistic circuits using transformations for better predictions.
problem Predictive limitations of probabilistic circuits in robotic scenarios.
method Integrates transformations into joint probability trees, extending their capabilities.
result Achieves higher likelihoods with fewer parameters on various data sets.
A new test evaluates risk estimation accuracy using probability integral transform.
problem Measuring the accuracy of financial market risk estimations.
method Probability Integral Transform (PIT) of ex post realized returns against ex ante probability distributions.
result The new test shows the importance of capturing the dynamic of financial markets.
New method calibrates photometric redshift PDFs more accurately.
problem Inaccurate photometric redshift uncertainties lead to systematic errors.
method Local re-calibration using feature-space regression of Probability Integral Transform (PIT) distributions.
result Calibrated PDFs are more accurate at all locations in feature space.
New methods for Z-transform inversion and Wiener-Hopf factorization.
problem Efficient numerical inversion of Z-transforms and factorization of functions. method Sinh-deformations of contours, variable changes, and simplified trapezoid rule.
result High precision and speed in evaluating moments and constructing filters.
This work integrates differentiation and integration in Physics-Informed Neural Networks.
problem Solving integro-differential equations and computing integral transforms.
method Augmenting Physics-Informed Neural Networks with automatic integration.
result Solving complex integral transforms and integro-differential equations.
The paper provides statistical guarantees for generative models using dimension reduction.
problem Improving the quality of generative models without increasing dimensionality.
method Modeling generative devices as smooth transformations of a lower-dimensional space and using integral probability metrics.
result Established a risk bound showing the impact of dimension reduction on generative model error.
Squared families are a new model class derived from linear transformations, offering convenient properties and universal approximation.
problem Developing a new class of probability models that are easier to handle and have useful properties.
method Introducing squared families as families of probability densities obtained by squaring a linear transformation of a statistic, and showing their properties and applications.
result Squared families have convenient properties and can approximate target densities well.
We present a new distributed fuzzy partitioning method to reduce the complexity of multi-way fuzzy decision trees in Big Data classification problems. The proposed algorithm builds a fixed number of fuzzy sets for all variables and adjusts their shape and position to the real distribution of training data. A two-step p…
Paper integrates real data into probabilistic models using Fourier transform.
problem Learning from constrained data sets in high dimensions.
method Functional approach based on weak formulation of Fourier transform of probability measures.
result Estimation of posterior probability measures for QoI and QoI with control parameter.
Econometric framework integrates heavy-tailed distributions with behavioral probability weighting for better asset pricing.
problem Underestimation of Value-at-Risk by traditional models in asset pricing.
method Developed an econometric framework combining heavy-tailed Student's t distributions with behavioral probability weighting. result Student's t specifications outperform Gaussian models in 88.4% of cases, reducing underestimation of Value-at-Risk by 16.5 percentage points. Connections between integration along hypersufaces, Radon transforms, and neural networks are exploited to highlight an integral geometric mathematical interpretation of neural networks. By analyzing the properties of neural networks as operators on probability distributions for observed data, we show that the distribu…
In this paper we revisit the integral functional of geometric Brownian motion It=∫0te−(μs+σWs)ds, where μ∈R, σ>0, and (Ws)s>0 is a standard Brownian motion. Specifically, we calculate the Laplace transform in t of the cumulative distribution function and of the probability density …
PITMonitor monitors model calibration over time with formal error guarantees.
problem Fixed-sample tests applied to models over time can lead to false alarms.
method PITMonitor uses mixture e-processes to detect distributional shifts in probability integral transforms.
result PITMonitor achieves competitive detection rates on river's FriedmanDrift benchmark.
Introduces a new geometric framework for probability distributions.
problem Developing a geometric framework for probability distributions.
method Introduces ℓp-information geometry and defines the ℓ2-probability simplex via the q-root transform. result Defines a noncanonical differentiable structure and q-root map as an isometry. Method orders Pareto solutions using transformed objective scores.
problem Lack of orderability in multi-objective optimization solutions.
method Probability integral transform to map objectives to scores.
result Pareto efficient solutions can be ordered in score space.
Derives integral representations for a Lévy process and its extremum, hitting time, with fast evaluation.
problem Efficiently evaluating the joint probability density function of a Lévy process, its supremum, and hitting time.
method Integral representations, Laplace-Fourier transforms, summation by parts, conformal deformation, trapezoid rules, Gaver-Wynn-Rho algorithm.
result Explicit calculations and fast evaluation of the joint cpdf for Lévy processes.
We propose an analytical approach to the computation of tail probabilities of compound distributions whose individual components have heavy tails. Our approach is based on the contour integration method, and gives rise to a representation of the tail probability of a compound distribution in the form of a rapidly conve…
Transformers recall from long distributions with statistical guarantees.
problem Designing Transformers that can recall from arbitrarily long, distributional contexts.
method Recast associative memory as probability measures, decomposing the task into recall and prediction.
result A shallow measure-theoretic Transformer learns the recall-and-predict map under spectral assumptions.
This paper introduces a novel recalibration method for multivariate forecasts.
problem Multivariate calibration for potentially misspecified models.
method Local mappings between marginal probability integral transform values and observed space, using K-nearest neighbors or normalizing flows.
result Demonstrated effectiveness on currency exchange rate and childhood malnutrition data.
Flexible copula model using implicit generative neural networks.
problem Limited flexibility of parametric copulas and curse of dimensionality in non-parametric methods.
method Implicit generative neural networks to model high-dimensional copula distributions with unspecified marginals.
result Demonstrated flexibility and performance on various datasets.
Introduces new geometric framework for probability densities on manifolds.
problem Developing a new geometric framework for probability densities on manifolds.
method Introduces ℓp-information geometry and defines ℓ2-probability simplex with q-root transform. result Explicit solution of gradient flow and geodesic completeness of e-connection. Geometric approach uses Bäcklund transformations to create integrable discrete analogs of surface nets.
problem Creating integrable discrete analogs of surface nets and conjugate nets.
method Interpreting classical differential geometry results through Bäcklund transformations and applying permutability properties.
result Integrable discrete analogs of asymptotic and conjugate nets are constructed.
It is known since 40 years old paper by M. Keane that minimality is a generic (i.e. holding with probability one) property of an irreducible interval exchange transformation. If one puts some integral linear restrictions on the parameters of the interval exchange transformation, then minimality may become an "exotic" p…
Legendre transformations link related integrable hierarchies.
problem Understanding relationships between integrable hierarchies.
method Legendre-type transformations of generalized Frobenius manifolds.
result Linear reciprocal transformations link related hierarchies.
Proposes a method to create shorter, more accurate prediction intervals.
problem Challenges in achieving both conditional validity and interval efficiency in complex settings.
method Uses a conformal-style calibration method for neural network responses, adjusting to empirical PIT distribution.
result Demonstrates better conditional calibration and shorter intervals than existing methods.
The paper explores geometric aspects of Miura transformations in integrable systems.
problem Relating different integrable equations and classifying bi-Hamiltonian structures.
method Construction of generalized Miura transformations under algebraic and geometric settings.
result Miura transformations relate integrable curve flows in different geometries and induce moving frame transitions.
Convolutional Bayesian filtering generalizes state estimation by incorporating inequality conditions.
problem Standard Bayesian filtering assumes exact conditional probabilities, limiting its applicability.
method Introducing inequality conditions transforms conditional probabilities into convolutional forms, expanding the filtering framework.
result Convolutional Bayesian filtering encompasses standard Bayesian filtering and allows for more nuanced model consideration.
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.
We prove that second-order hyperbolic Monge-Ampere equations for one function of two variables are connected to the wave equation by a Backlund transformation if and only if they are integrable by the method of Darboux at second order. One direction of proof, proving Darboux integrability, follows the implications of t…
Transformer with denoising diffusion improves probabilistic density estimation.
problem Estimating non-Gaussian and multimodal probability distributions for regression problems.
method Training a denoising diffusion head on top of a Transformer model.
result The model provides reasonable probability density estimation for high-dimensional inputs.
Proposes a new feature preprocessing method using kernel density integral transformation.
problem Feature preprocessing for tabular data in machine learning and statistics.
method Kernel density integral transformation as a drop-in replacement or improved alternative to min-max scaling and quantile transformation.
result Frequently outperforms min-max scaling and quantile transformation with hyperparameter tuning.
Develops support theorem for analytic transforms in tomography.
problem Analytic wave front set resolution for integral transforms.
method Microlocal analysis, double fibration framework, wave packet transforms.
result Uniqueness and support theorems for analytic transforms.
A framework uses free probability to analyze Transformer models.
problem Understanding the dynamics and complexity of Transformer-based language models.
method Formal operator-theoretic analysis using free probability theory.
result Entropy-based generalization bounds derived under freeness assumptions.
UT module refines VAE latent space, improving disentanglement and interpretability.
problem Irregular latent distributions cause posterior collapse and misalignment in VAEs.
method UT module uses G-KDE clustering, GM modeling, and PIT to transform latent space into uniform distribution.
result UT module enhances disentanglement and interpretability of latent representations.
We give a new mechanism for constructing Backlund transformations by using symmetry reduction of differential systems. We then characterize a family of Backlund transformations between Darboux integrable systems where the Backlund transformation can be constructed by the proposed symmetry reduction method.
Deep neural networks forecast financial return distributions accurately.
problem Forecasting probability distributions of financial returns.
method Used 1D CNN and LSTM architectures with custom loss functions to optimize distribution parameters.
result LSTM with skewed Student's t distribution outperformed classical models in multiple evaluation metrics.
Computation of moments of transformed random variables is a problem appearing in many engineering applications. The current methods for moment transformation are mostly based on the classical quadrature rules which cannot account for the approximation errors. Our aim is to design a method for moment transformation for …
New integral theorems improve density function estimations.
problem Improving density function estimations.
method Integrals based on cyclic functions and Riemann sums, Fourier integral theorem, Monte Carlo methods, variational approach, Cauchy residue theorem.
result Optimal cyclic functions minimize square integrals, improving density estimations.
The continuous-time random walk (CTRW) is a pure-jump stochastic process with several applications in physics, but also in insurance, finance and economics. A definition is given for a class of stochastic integrals driven by a CTRW, that includes the Ito and Stratonovich cases. An uncoupled CTRW with zero-mean jumps is…
We introduce the Koenigs lattice, which is a new integrable reduction of the quadrilateral lattice (discrete conjugate net) and provides natural integrable discrete analogue of the Koenigs net. We construct the Darboux-type transformations of the Koenigs lattice and we show permutability of superpositions of such trans…
Transformers improve with Fourier integral attentions.
problem Inefficiency of dot-product attention in capturing feature dependencies.
method Interpreted attention as kernel regression, proposed FourierFormer with generalized Fourier integral kernels.
result FourierFormer achieves better accuracy and reduces redundancy.
Characteristic functions of several popular classes of distributions and processes admit analytic continuation into unions of strips and open coni around R⊂C. The Fourier transform techniques reduces calculation of probability distributions and option prices to evaluation of integrals whose i…
We review the main "omnibus procedures" for goodness-of-fit testing for copulas: tests based on the empirical copula process, on probability integral transformations, on Kendall's dependence function, etc, and some corresponding reductions of dimension techniques. The problems of finding asymptotic distribution-free te…
New integral transforms solve multilayer heat equations.
problem Solving multilayer heat equations with moving boundaries.
method Expanding Dirac delta function in eigenfunctions, constructing oscillating integral transforms.
result Semi-analytical solutions for various problems.
New framework transforms labeled datasets for various machine learning tasks.
problem Lack of principled methods to transform labeled datasets.
method Wasserstein gradient flows in probability space for optimization of data-generating distributions.
result Framework can impose constraints, adapt for transfer learning, or re-purpose models.
The paper discusses a new method for constructing two-step Darboux transforms of isothermic surfaces.
problem Constructing two-step Darboux transforms of isothermic surfaces.
method Sym-type construction using parallel sections of the associated family.
result All two-step Darboux transforms of an isothermic surface are given without further integration.
New method for curve comparison using iterated integrals and moving frames.
problem Comparing curves robustly to noise and transformations.
method Moving frame method paired with log-signature transform.
result Algorithmic construction of invariants for curve equivalence under rigid motions.
The study evaluates financial risk using copulas and statistical tests.
problem Validating bivariate forecasts in risk evaluation.
method Using copulas to characterize dependencies, applying statistical tests to validate forecasts, removing heteroskedasticity.
result A Student copula accurately describes financial time series dependencies.