We develop a new statistical test for comparing variables with varying scales.
problem Comparing variables with different scales in multidimensional spaces.
method Order based on expectations of random variables, generalized stochastic dominance (GSD) order, regularized statistical test, linear optimization, imprecise probability models.
result Validated through multidimensional data from various fields.
New method for learning multidimensional CDFs using Archimedean copulas.
problem Learning multidimensional CDFs in high dimensions.
method Generative modeling technique using Archimedean copulas as mixture models with latent variables from neural networks.
result Efficacy and computational efficiency compared to existing methods.
Develops a cumulant-based algorithm for optimizing investment portfolios.
problem Optimizing investment portfolios with low variability in non-Gaussian data.
method Alternating Least Square method applied to 2nd-6th cumulants of multidimensional random variables.
result The algorithm outperforms benchmarks and other methods during recent crashes.
RS-HDMR-GPR simplifies complex functions with machine-learned lower-dimensional terms.
problem Representing and understanding complex multidimensional functions with sparse data.
method Random Sampling High Dimensional Model Representation Gaussian Process Regression (RS-HDMR-GPR).
result Facilitates recovery of functional dependence and adds insight into input variable importance.
The paper analyzes the asymptotic sequential Rademacher complexity for finite function classes.
problem Understanding the complexity of finite function classes in asymptotic settings.
method Using viscosity solutions of a G-heat equation and sublinear expectation theory, the paper derives the asymptotic sequential Rademacher complexity.
result The asymptotic sequential Rademacher complexity is expressed in terms of the viscosity solution of a G-heat equation and the expected value of the largest order statistics of a multidimensional G-normal random variable.
Study improves sample complexity for distinguishing continuous distributions and causal relationships.
problem Distinguishing continuous distributions and causal relationships in the presence of unobserved confounding.
method Proposed an estimator of KL divergence based on von Mises expansion for closeness testing.
result Established sample complexity guarantees for causal discovery in non-linear models with continuous variables and unobserved confounding.
To any positive number ε and any nonnegative even Schwartz function w:R→R we associate the random function uε on the m-torus Tεm:=Rm/(ε−1Z)m defined as the real part of the random Fourier series $$ \sum_{ν\in\mathbb{Z}^m} X_…
MDFS selects important variables considering variable interactions, improving over simple filtering.
problem Discarding variable interactions leads to loss of relevant variables.
method MultiDimensional Feature Selection (MDFS) using information theory and CUDA C.
result Multidimensional analysis provides more reliable rankings of variable importance.
We propose a simulation method for multidimensional Hawkes processes with differing decays.
problem Simulating and calibrating Hawkes processes with various decay rates.
method Superposition theory of point processes, decomposition of inter-arrival times, auxiliary variables, Gibbs samplers, adaptive rejection sampling.
result Significant improvement in algorithm speed and accurate simulation of Hawkes processes.
We introduce a general setting for multidimensional dispersionless integrable hierarchy in terms of differential m-form Ωm with the coefficients satisfying the Plücker relations, which is gauge-invariantly closed and its gauge-invariant coordinates (ratios of coefficients) are (locally) holomorphic with respect to…
Geometrically convex return risk measures on AM-algebras
problem Quantifying risk in time series analysis
method Extending return risk measures to general ordered vector spaces
result Establishing results on finiteness, continuity, separability, and dual and aggregation-based representations
Efficiently analyzes multidimensional functional data using separable basis functions.
problem Curse of dimensionality in traditional functional data analysis.
method Marginal product basis systems for multidimensional data, tensor decomposition, differential operator-based penalties.
result Efficient estimation of multidimensional functional data representations.
Scalable model detects multidimensional changes in data.
problem Detecting and characterizing smooth multidimensional changepoints.
method Random Kitchen Sink features and spectral mixture kernels for flexible and expressive modeling, with additive non-separable kernels for scalability.
result Model identifies previously unknown heterogeneous changes in space and time.
A method for multidimensional probabilistic electricity market forecasting is proposed.
problem Uncertainty in simultaneous multivariate predictions of electricity markets.
method Repeated resampling to estimate uncertainty of simultaneous multivariate predictions.
result The method provides highly accurate predictions and gains are largest when considering functions of variables.
New analysis shows ESNs can handle multidimensional inputs without scaling network size.
problem Understanding the memory capacity of ESNs for multidimensional inputs.
method Advanced random matrix theory applied to ESNs with structured inputs.
result Linear scaling of network size with information rate and poly-logarithmic scaling with input dimension.
Extends Bayesian theory to handle complex interdependencies in multidimensional event spaces.
problem Complex interdependencies between events and hypotheses sets in real-world systems.
method Developed a mathematical formalism for modeling complex relationships through rigorous derivation and validated using analytical proofs, simulations, and case studies.
result MDSE theory improves prediction accuracy by 15-20% compared to standard Bayesian methods in high interdimensionality datasets.
Two new methods for analyzing repeated measures data using embeddings into Reproducing Kernel Hilbert Spaces.
problem Analyzing complex data structures with multiple features over time.
method Two generalizations of canonical correlation analysis for repeated measures data using embeddings into Reproducing Kernel Hilbert Spaces.
result Consistency rates for transformation and correlation estimators, relaxing common assumptions.
Study calculates Poisson cohomology for scalar multi-D brackets, finding non-trivial deformation theory.
problem Analyzing non-trivial deformation theory in multi-dimensional scalar Poisson brackets.
method Computed Poisson cohomology groups for scalar Poisson brackets with D independent variables.
result Second and third cohomology groups are non-vanishing in D>1, indicating non-trivial deformation theory.
sWk-means clusters multidimensional financial time series into distinct market regimes.
problem Classifying distinct market regimes in multidimensional financial time series.
method Approximated multidimensional Wasserstein distance as sliced Wasserstein distance for clustering.
result sWk-means successfully identifies distinct market regimes in real financial data.
A monopolist sells goods with possibly a characteristic consumers dislike (for instance, he sells random goods to risk averse agents), which does not affect the production costs. We investigate the question whether using undesirable goods is profitable to the seller. We prove that in general this may be the case, depen…
Deep learning detects arrhythmias from ECGs using multidimensional representations.
problem Detecting arrhythmias from ECGs using traditional methods.
method Convert 1-D ECG data into 2-D images, then use deep learning for classification.
result Deep learning outperforms existing methods in arrhythmia detection.
This paper addresses the identification of insurance models with multidimensional screening where insurees have private information about their risk and risk aversion. The model includes a random damage and the possibility of several claims. Screening of insurees relies on their certainty equivalence. The paper then in…
The paper introduces new KMEs to capture stochastic process filtrations.
problem Missing filtration information in stochastic processes.
method Higher order kernel mean embeddings (KMEs) conditioned on filtrations.
result Consistent estimators and tests for filtration-sensitive information.
The CHAMPION study clusters multi-dimensional accelerometer data to understand health links.
problem Clustering multi-dimensional data from pediatric longitudinal studies.
method Developed a finite mixture of multidimensional arrays model for clustering 4-dimensional accelerometer data.
result Demonstrated the feasibility and utility of clustering higher order data.
Tensor completion improves EEG-based BCI performance with missing data.
problem Improving classification accuracy in BCI systems with noisy EEG data.
method Tensor decomposition models to infer missing entries in multidimensional EEG datasets.
result Tensor completion algorithms enhance BCI classification accuracy with missing data.
New change surfaces for multidimensional changes and counterfactuals.
problem Limited expressiveness of standard changepoint models in multidimensional settings.
method Model-agnostic formalization of change surfaces, using Gaussian Process Change Surfaces (GPCS).
result Discovery of complex, heterogeneous changes in measles incidence and lead testing kit requests.
Study shows how certain stochastic models reach a steady state over time.
problem Understanding long-term behavior of stochastic volatility models.
method Novel coupling technique for Markov chains, applicable to random environments.
result Convergence to an invariant measure for multidimensional fractional models.
NGCA identifies non-Gaussian components in multidimensional data.
problem Identifying non-Gaussian components in multidimensional data.
method Uses relative entropy to approximate the non-Gaussian subspace.
result Algorithm approximates non-Gaussian subspace with polynomial time complexity.
We investigate the class of σ-stable Poisson-Kingman random probability measures (RPMs) in the context of Bayesian nonparametric mixture modeling. This is a large class of discrete RPMs which encompasses most of the the popular discrete RPMs used in Bayesian nonparametrics, such as the Dirichlet process, Pitman-Yor p…
We derive error estimates for multinomial approximations of American options in a multidimensional jump--diffusion Merton's model. We assume that the payoffs are Markovian and satisfy Lipschitz type conditions. Error estimates for such type of approximations were not obtained before. Our main tool is the strong approxi…
The paper studies deformations of Poisson brackets in two dimensions, finding non-trivial cohomology groups.
problem Deformations of multidimensional Poisson brackets of hydrodynamic type.
method Cohomology computation of PVAs associated with Poisson brackets at third differential degree.
result Non-trivial third cohomology group indicates non-equivalent infinitesimal deformations.
Building on the work of Schweizer (1995) and Cern and Kallseny (2007), we present discrete time formulas minimizing the mean square hedging error for multidimensional assets. In particular, we give explicit formulas when a regime-switching random walk or a GARCH-type process is utilized to model the returns. Monte Carl…
This paper deals with multidimensional dynamic risk measures induced by conditional g-expectations. A notion of multidimensional g-expectation is proposed to provide a multidimensional version of nonlinear expectations. By a technical result on explicit expressions for the comparison theorem, uniqueness theorem and…
We investigate the relative information content of six measures of dependence between two random variables X and Y for large or extreme events for several models of interest for financial time series. The six measures of dependence are respectively the linear correlation ρv+ and Spearman's rho ρs(v) conditio…
In the last chapter of his book "The Algebraic Theory of Modular Systems " published in 1916, F. S. Macaulay developped specific techniques for dealing with " unmixed polynomial ideals " by introducing what he called " inverse systems ". The purpose of this paper is to extend such a point of view to differential module…
This paper improves conditional multidimensional scaling for incomplete data.
problem Handling missing data in known features for multidimensional scaling.
method Proposes a method to learn low-dimensional configurations with missing known feature values.
result Can learn low-dimensional configurations and impute missing values.
Modified multidimensional scaling improves clustering in noisy high-dimensional data.
problem Improving clustering accuracy in noisy high-dimensional data.
method Unified framework of multidimensional scaling, modified with nonlinear transformation.
result Modified multidimensional scaling achieves exact recovery of cluster labels with high probability.
Paper solves complex game theory problems with new equations.
problem Zero-sum stochastic games with non-Markovian switching.
method New multidimensional SRE and BSDE solutions.
result Existence and uniqueness of SRE solutions.
This chapter covers methods for identifying and inferring graph topologies.
problem Identifying and inferring graph topologies from multidimensional relational data.
method Overview of methods including correlation metrics, covariance selection, kernels, structural equations, and vector autoregressions.
result Supports both batch and online learning with convergence guarantees and leverages high-order statistical information.
The paper explores multidimensional critic output in GANs, improving convergence and diversity.
problem Underexplored in GANs literature, multidimensional critic output.
method Generalized Wasserstein GAN framework, SRVT block, maximal p-centrality discrepancy.
result High-dimensional critic output improves GAN performance in convergence and diversity.
Improved algorithm for multidimensional scaling reduces stress.
problem Stress in multidimensional scaling.
method Proposed modifications of the smacof algorithm.
result Convergent majorization algorithm for Kruskal's stress formula two.
The paper tackles estimating vectors from binary comparisons, providing bounds and adaptive strategies.
problem Estimating a vector from binary comparisons of preference.
method Theoretical bounds and adaptive strategies for estimating vectors from noisy and randomized comparisons.
result Stable embedding of the space of target vectors and significant gains from adaptive distribution changes.
A method to visualize multidimensional local subspaces using implicit differentiation.
problem Understanding the effect of multidimensional projection on local subspaces.
method Implicit function differentiation to analyze local subspaces shaped by multidimensional ellipses.
result Visualization of local subspaces provides insights into the global structure of data.
Timer-XL predicts multidimensional time series using a unified Transformer approach.
problem Unified time series forecasting across various tasks and contexts.
method Decoder-only Transformers with a universal TimeAttention mechanism and deft position embedding.
result State-of-the-art performance across multiple forecasting benchmarks.
The paper introduces CoCoCat bonds for multi-region natural catastrophes, accounting for complex dependencies.
problem Valuation of multi-region contingent convertible bonds under complex dependencies.
method Developed a model accounting for inter-regional dependencies using change-of-measure techniques.
result Significant impact of inter-regional dependencies on CoCoCat bond pricing.
Paper analyzes classical multidimensional scaling for cluster recovery.
problem Cluster recovery from noisy data.
method Classical multidimensional scaling followed by distance-based clustering.
result Scaling conditions for high probability cluster recovery.
Bayesian Complementary Kernelized Learning models complex spatiotemporal data.
problem Modeling complex, nonstationary, and nonseparable spatiotemporal data.
method Integrates kernelized low-rank tensor factorization and short-range spatiotemporal Gaussian Processes.
result BCKL offers superior performance in providing accurate posterior mean and high-quality uncertainty estimates.
3D RadViz improves 3D data visualization of multidimensional datasets.
problem Tackles the challenge of visualizing multidimensional datasets in 3D.
method Develops RadViz3D, a 3D radial visualization tool with uniform anchor points.
result Improves the display of multidimensional datasets, especially for uncorrelated variables.