The paper proves gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.
problem Proving gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.
method Using Souplet-Zhang type estimates and properties of Bakry-Emery Ricci tensor and weighted mean curvature.
result Gradient estimates for nonlinear parabolic equations on smooth metric measure spaces with Dirichlet boundary condition.
Private method measures nonlinear correlations between data hosted across two entities.
problem Measuring nonlinear correlations between sensitive data hosted across multiple parties while preserving privacy.
method Differentially private estimator of distance correlation.
result First private estimator of nonlinear correlations in a multi-party setup.
The paper solves curvature measure problem in hyperbolic space.
problem Prescribed curvature measure problem in hyperbolic space.
method Establishing C^2 regularity estimates for solutions to fully nonlinear PDE.
result Existence of star-shaped k-convex bodies with prescribed curvature measures.
The paper provides gradient estimates for nonlinear heat-type equations on smooth metric measure spaces.
problem Proving gradient estimates for nonlinear heat-type equations on smooth metric measure spaces.
method Using Hamilton type and Li-Yau type estimates, the paper proves gradient estimates on positive solutions to generalized nonlinear parabolic equations on smooth metric measure spaces with compact boundary.
result Gradient estimates for nonlinear heat-type equations on smooth metric measure spaces.
Study on conditioning Gaussian measures on nonlinear observations, including representer theorem and mode estimation.
problem Conditioning Gaussian measures on nonlinear observations in Bayesian inference and machine learning.
method Representer theorem, novel mode definition, maximum a posteriori estimation, Laplace approximation.
result Identification of infinite-dimensional Gaussian and finite-dimensional non-Gaussian components in conditioned measures.
Adaptive algorithm improves nonlinear data assimilation for non-Gaussian systems.
problem Challenges of non-Gaussian statistics in data assimilation.
method Triangular measure transport with P-spline basis functions and an information criterion.
result Automatic selection of parsimonious parametrization for efficient adaptation.
Measurement noise limits the advantage of nonlinear models over linear models in biomedical prediction
problem Nonlinear models vs. linear models in biomedical prediction
method Measurement reliability
result Measurement noise blurs the population-optimal predictor
The paper provides new gradient estimates for solutions to a nonlinear elliptic equation on smooth metric measure spaces.
problem Gradient estimates for solutions to a specific nonlinear elliptic equation on smooth metric measure spaces.
method Nash-Moser iteration technique to obtain local gradient estimates.
result New local gradient estimates for positive solutions to the equation.
Method improves SINDy for noisy nonlinear systems.
problem Recover nonlinear dynamical systems from noisy data.
method Reweighted ℓ1-regularized least squares. result Improved accuracy and robustness in noisy conditions.
Proposes a new derivative concept for nonlinear DRO problems.
problem Optimizing nonlinear functions in probability space with distributionally robust optimization.
method Introduces Gateaux derivative for smoothness and proposes a Frank-Wolfe algorithm.
result Validates theoretical results on portfolio selection problems with numerical validation.
New techniques improve the accuracy of identifying nonlinear systems from noisy data.
problem Identifying nonlinear dynamical systems from noisy state measurements.
method Comparative study of local and global smoothing techniques to denoise state measurements and improve sparse regression methods.
result Global smoothing methods outperform local methods in improving the accuracy of governing equation recovery.
MASF improves score-based filters for high-dimensional nonlinear systems with spatially sparse measurements.
problem Challenges in data assimilation for nonlinear, high-dimensional systems with spatially sparse measurements.
method Developed a forward process tailored for filtering that transforms the system state toward the measurement space, enabling a theoretically sound formulation of the likelihood score.
result MASF shows improved performance over existing score-based filters and ensemble-type Kalman filters, achieving up to a 28.2× wall-clock speedup.
Bayesian framework improves robustness in nonlinear regression models.
problem Measurement error, model misspecification, and distributional misspecification in regression analyses.
method Joint Dirichlet process prior on latent covariate-response distribution, updating with posterior pseudo-samples.
result Improved stability and consistency in estimators under increasing measurement error.
The paper introduces a new method for risk measurement using weak optimal transport.
problem Risk measurement in insurance and financial contexts.
method Convex risk measures with weak optimal transport penalties, explicit representation via nonlinear transform, computational aspects, and approximations using neural networks.
result Explicit representation and computational methods for risk measures.
New algorithm learns nonlinear phenomena from noisy local measurements without data exchange.
problem Learning nonlinear phenomena from noisy local measurements in a decentralized network.
method Non-parametric learning algorithm that spreads information only between neighboring nodes.
result Non-asymptotic estimation error bounds for the proposed method.
The paper studies gradient estimates and Liouville theorems for a nonlinear elliptic equation on metric measure spaces.
problem Gradient estimates and Liouville theorems for positive solutions to a specific nonlinear elliptic equation.
method Analyzes the nonlinear elliptic equation \( \Delta_{V}u^{m} + \mu(x)u + p(x)u^{\alpha} = 0 \) on smooth metric measure spaces with bounded Bakry-Émery curvature.
result Establishes gradient estimates and related Liouville theorems and Harnack inequalities.
We establish local elliptic and parabolic gradient estimates for positive smooth solutions to a nonlinear parabolic equation on a smooth metric measure space. As applications, we determine various conditions on the equation's coefficients and the growth of solutions that guarantee the nonexistence of nontrivial positiv…
PGD algorithms solve nonlinear inverse problems with generative priors using noisy measurements.
problem Signal estimation from noisy nonlinear measurements with generative priors.
method Projected gradient descent algorithms for two cases: unknown and known nonlinearity.
result PGD algorithms converge linearly to optimal statistical rates using arbitrary initialization.
Most previous contributions to BSDEs, and the related theories of nonlinear expectation and dynamic risk measures, have been in the framework of continuous time diffusions or jump diffusions. Using solutions of BSDEs on spaces related to finite state, continuous time Markov chains, we develop a theory of nonlinear expe…
New calculus on spacetimes for nonlinear differential equations.
problem Nonlinear differential equations on metric measure spacetimes.
method Introduces maximal weak subslope and variational calculus.
result Establishes a comparison theorem for nonlinear p-d'Alembertian. The paper introduces a framework to assess nonlinear causality in financial markets.
problem Identifying and quantifying co-dependence between financial instruments.
method Transfer entropy and convergent cross-mapping methods to assess linear and nonlinear causality.
result Stock indices exhibit significant nonlinear causality, and correlation underestimates causality.
We propose a method for maximizing a partial area under a receiver operating characteristic (ROC) curve (pAUC) for binary classification tasks. In binary classification tasks, accuracy is the most commonly used as a measure of classifier performance. In some applications such as anomaly detection and diagnostic testing…
We consider dynamic sublinear expectations (i.e., time-consistent coherent risk measures) whose scenario sets consist of singular measures corresponding to a general form of volatility uncertainty. We derive a càdlàg nonlinear martingale which is also the value process of a superhedging problem. The superhedging strate…
Proposes a new method for nonlinear Bayesian updates using ensemble kernel regression.
problem Nonlinear and non-Gaussian Bayesian updates for complex systems.
method Combines Kalman filtering for observed components and kernel density estimation for unobserved components, with subsampling and clustering.
result Reduces estimation errors in highly nonlinear scenarios compared to standard linear updates.
Independent Component Analysis (ICA) aims to find a coordinate system in which the components of the data are independent. In this paper we construct a new nonlinear ICA model, called WICA, which obtains better and more stable results than other algorithms. A crucial tool is given by a new efficient method of verifying…
We consider the problem of reconstructing signals and images from periodic nonlinearities. For such problems, we design a measurement scheme that supports efficient reconstruction; moreover, our method can be adapted to extend to compressive sensing-based signal and image acquisition systems. Our techniques can be pote…
We introduce a data-based approach to estimating key quantities which arise in the study of nonlinear control systems and random nonlinear dynamical systems. Our approach hinges on the observation that much of the existing linear theory may be readily extended to nonlinear systems - with a reasonable expectation of suc…
In this paper, we study elliptic gradient estimates for a nonlinear f-heat equation, which is related to the gradient Ricci soliton and the weighted log-Sobolev constant of smooth metric measure spaces. Precisely, we obtain Hamilton's and Souplet-Zhang's gradient estimates for positive solutions to the nonlinear f-…
We consider the problem of recovering a common latent source with independent components from multiple views. This applies to settings in which a variable is measured with multiple experimental modalities, and where the goal is to synthesize the disparate measurements into a single unified representation. We consider t…
Our objective is to estimate the unknown compositional input from its output response through an unknown system after estimating the inverse of the original system with a training set. The proposed methods using artificial neural networks (ANNs) can compete with the optimal bounds for linear systems, where convex optim…
The paper studies solutions to a nonlinear equation on Finsler manifolds with gradient estimates and Harnack inequalities.
problem Exploring positive solutions to a nonlinear parabolic equation on Finsler manifolds.
method Developed new comparison theorems and used Li-Yau estimates.
result Established gradient estimates and Harnack inequalities for solutions.
A joint conditional autoregressive expectile and Expected Shortfall framework is proposed. The framework is extended through incorporating a measurement equation which models the contemporaneous dependence between the realized measures and the latent conditional expectile. Nonlinear threshold specification is further i…
In stochastic decision problems, one often wants to estimate the underlying probability measure statistically, and then to use this estimate as a basis for decisions. We shall consider how the uncertainty in this estimation can be explicitly and consistently incorporated in the valuation of decisions, using the theory …
In this paper we introduce a simple continuous-time asset pricing framework, based on general multi-dimensional diffusion processes, that combines semi-analytic pricing with a nonlinear specification for the market price of risk. Our framework guarantees existence of weak solutions of the nonlinear SDEs under the physi…
New method identifies key genes affecting phenotypes in biological systems.
problem Identifying genes that drive specific phenotypes in complex biological systems.
method Data-driven observability decomposition using Koopman operators.
result Koopman operator representation identifies genes that drive phenotypes.
The paper derives gradient estimates for solutions of certain equations on metric measure spaces.
problem Gradient estimates for solutions of specific nonlinear and elliptic equations on metric measure spaces.
method Derives Li-Yau and Hamilton's type gradient estimates for positive solutions.
result Gradient estimates for positive solutions of the equations on complete noncompact metric measure spaces.
Unified framework for uniform signal recovery in nonlinear GCS with 1-bit/quantized measurements.
problem Uniform recovery guarantees for nonlinear generative compressed sensing.
method Unified framework using generalized Lasso and Lipschitz approximation.
result Uniform recovery of all signals in the ball up to an error of ε using approximately O(k/ε^2) samples.
Wassmap reduces image complexity while preserving key features.
problem Global nonlinear dimensionality reduction in imaging.
method Wassmap uses Wasserstein space and pairwise distances to create isometric embeddings.
result Wassmap can recover parameters of image manifolds like translations and dilations.
Develops a new duality between entropy martingale optimal transport and nonlinear pricing-hedging.
problem Entropy Martingale Optimal Transport problem and its associated optimization problem.
method Combines Entropy Optimal Transport and Martingale Optimal Transport theories, with novel penalization terms and constraints.
result Establishes a nonlinear robust pricing-hedging duality, covering various known robust results.
This paper introduces a fast, general method for dictionary-free parameter estimation in quantitative magnetic resonance imaging (QMRI) via regression with kernels (PERK). PERK first uses prior distributions and the nonlinear MR signal model to simulate many parameter-measurement pairs. Inspired by machine learning, PE…
Method learns dynamics from noisy partial observations.
problem Reconstructing stochastic dynamical systems from indirect noisy data.
method Amortized path generation method for nonlinear stochastic filtering.
result Learned conditional path generator quantifies uncertainty.
In many compressive sensing problems today, the relationship between the measurements and the unknowns could be nonlinear. Traditional treatment of such nonlinear relationships have been to approximate the nonlinearity via a linear model and the subsequent un-modeled dynamics as noise. The ability to more accurately ch…
This paper presents a novel clustering concept that is based on jointly learned nonlinear transforms (NTs) with priors on the information loss and the discrimination. We introduce a clustering principle that is based on evaluation of a parametric min-max measure for the discriminative prior. The decomposition of the pr…
The paper analyzes equity market dynamics and optimal portfolios using time-varying optimization.
problem Analyzing the time-varying structure of equity markets, particularly market capitalization inequality and concentration.
method The study employs mathematical functionals of time-varying portfolios and a Sharpe optimization procedure.
result Optimal portfolios exhibit varying market capitalization exposure over time.
Kernel measures similarity of nonlinear causal structures in heterogeneous populations.
problem Learning causal structure in populations with diverse underlying structures.
method Distance covariance-based kernel for measuring similarity of causal structures.
result Kernel enables clustering of homogeneous subpopulations for causal structure learning.
New approach uses secants to improve sensor placement and feature selection for nonlinear systems.
problem Inadequacy of linear methods for minimal sensor placement and feature selection in nonlinear systems.
method Data-driven approach using secant vectors to develop greedy algorithms for robust, near-minimal reconstruction guarantees.
result Demonstrated on two problems where linear techniques fail, secant-based approach provides robust solutions.
Bell's theorem shows quantum correlations can't be explained by classical causal models, even with some measurement dependence.
problem Quantum correlations violate classical causal models.
method Using causal networks, the study bounds the level of measurement dependence and derives nonlinear Bell inequalities.
result Quantum correlations can't be explained by classical causal models even with some measurement dependence.
We use statistical learning methods to construct an adaptive state estimator for nonlinear stochastic systems. Optimal state estimation, in the form of a Kalman filter, requires knowledge of the system's process and measurement uncertainty. We propose that these uncertainties can be estimated from (conditioned on) past…