SOS programming verifies MTW tensor non-negativity for optimal transport maps.
problem Verifying MTW tensor non-negativity for general cost functions is difficult.
method Sum-of-Squares (SOS) programming for verifying and approximating MTW non-negativity.
result SOS programming provides certificates and approximations of MTW non-negativity.
Study optimal transport costs with zero MTW tensor, finding new families of costs and divergence functions.
problem Characterize optimal transport costs with zero MTW tensor.
method Optimal transport theory, information geometry, solving nonlinear ODEs.
result Found new families of costs and divergence functions.
This paper connects optimal transport and information geometry using pseudo-Riemannian geometry.
problem Understanding the geometric structures of probability distributions.
method Introducing a new differential geometric connection between optimal transport and information geometry.
result A new information-geometric interpretation of the MTW tensor.
We define a new type of metric comparison similar to the comparison of Alexandrov. We show that it has strong connections to continuity of optimal transport between regular measures on a Riemannian manifold, in particular to the so called MTW condition introduced by Xi-Nan Ma, Neil Trudinger and Xu-Jia Wang.
The Markov Theorem Without Stabilization (MTWS) established the existence of a calculus of braid isotopies that can be used to move between closed braid representatives of a given oriented link type without having to increase the braid index by stabilization. Although the calculus is extensive there are three key isoto…
Kähler-Ricci flow preserves negative anti-bisectional curvature.
problem Preserving curvature under Kähler-Ricci flow.
method Study of Kähler-Ricci flow behavior on anti-bisectional curvature.
result Non-positive anti-bisectional curvature is preserved under Kähler-Ricci flow.
The Markov Theorem Without Stabilization (MTWS) (see math.GT/0310279) established the existence of a calculus of braid isotopies that can be used to move between closed braid representatives of a given oriented link type without having to increase the braid index by stabilization. Although the calculus is extensive the…
We study a parabolic equation for finding solutions to the optimal transport problem on compact Riemannian manifolds with general cost functions. We show that if the cost satisfies the strong MTW condition and the stay-away singularity property, then the solution to the parabolic flow with any appropriate initial condi…
We show that if a Riemannian manifold satisfies (3,3)-bipolar comparisons and has an open flat subset then it is flat. The same holds for a version of MTW where the perpendicularity is dropped. In particular we get that the (3,3)-bipolar comparison is strictly stronger than the Alexandrov comparison.
Withdrawn and replaced by two related manuscripts: (1) "Stabilization in the braid groups I:MTWS", published in Geometry and Topology Volume 10 (2006), 413-540, arXiv:math.GT/0310279, and (2) "Stabilization in the braid groups II: Transversal simplicity of knots", Geometry and Topology Volume 10 (2006), to appear, arXi…
Study C2 estimates for p-Hessian equations on closed manifolds.
problem Estimating solutions to p-Hessian equations on closed Riemannian manifolds. method Introducing pseudo-solutions to generalize C-subsolution and proving C1 and C2 estimates. result Proves C2 estimates for general p-Hessian equations on closed manifolds under sharp conditions. Let X and Y be domains of Rn equipped with respective probability measures μ and ν. We consider the problem of optimal transport from μ to ν with respect to a cost function c:X×Y→R. To ensure that the solution to this problem is smooth, it is necessary to make several ass…
We introduce a new braid-theoretic framework with which to understand the Legendrian and transversal classification of knots, namely a Legendrian Markov Theorem without Stabilization which induces an associated transversal Markov Theorem without Stabilization. We establish the existence of a nontrivial knot-type specif…
Choose any oriented link type X and closed braid representatives X[+], X[-] of X, where X[-] has minimal braid index among all closed braid representatives of X. The main result of this paper is a `Markov theorem without stabilization'. It asserts that there is a complexity function and a finite set of `templates' such…
Global geometric expressions derived for manifold embeddings.
problem Expressing geometric quantities globally on manifolds.
method Global formulas using operator-valued expressions and affine projection.
result Explicit cross-curvature results for specific metrics.
Stochastic optimization improves semi-discrete OT map estimation with a minimax rate.
problem Empirical success of SGD in semi-discrete OT, but lack of theoretical guarantees.
method Averaged projected SGD with a minimax convergence rate of O(1/√n).
result SGD methods can estimate the OT map with a minimax convergence rate of O(1/√n).
Generates samples conditioned on labels using optimal transport.
problem Estimating conditional distributions for specific labels.
method Wasserstein geodesic generator based on optimal transport theory.
result Learned conditional distributions and optimal transport maps.
The paper classifies Finsler surfaces satisfying the T-condition or σT-condition.
problem Characterizing Finsler surfaces based on specific tensor conditions.
method Analyzing Finsler surfaces in dimensions n≥3, proving conditions equivalence, and solving PDEs.
result All Finsler surfaces satisfying the T-condition or σT-condition are classified.
The paper develops a new approach to conditional risk measures using modular convex analysis.
problem Developing a new method for conditional risk measures.
method Random modular approach to conditional certainty equivalents and niveloids in the conditional L∞-space. result Retrieves a conditional variational formula for optimized certainty equivalents and applies it to the conditional entropic risk measure.
Paper constructs solutions to Bogomolny equations with specific boundary and asymptotic conditions.
problem Constructing solutions to Bogomolny equations with given boundary and asymptotic conditions.
method Using generalized Nahm pole boundary condition and real symmetry breaking condition.
result Solutions analogous to instanton solutions, satisfying different asymptotic conditions.
We extend probabilistic programming to handle conditioning on marginal distributions.
problem Conditioning probabilistic programs on marginal distributions of observable variables.
method We define and implement stochastic conditioning, allowing inference in probabilistic programs conditioned on marginal distributions.
result We demonstrate the effectiveness of stochastic conditioning in various real-life scenarios.
New tests for conditional copulas based on decision trees.
problem Testing constancy of conditional dependence structure given conditioning events.
method Data-driven decision trees to maximize differences in conditional Kendall's tau.
result Asymptotic distributions of test statistics under the null hypothesis.
Paper finds necessary condition for logarithmic Minkowski problem in higher dimensions.
problem Logarithmic Minkowski problem in higher dimensions.
method Established a necessary condition through generalization and refinement of previous work.
result Generalizes and refines necessary condition for logarithmic Minkowski problem.
This paper introduces a neural operator for probabilistic conditioning.
problem Probabilistic conditioning of random variables X given Y. method Develops a single operator that maps any joint density to its conditional, approximated by neural operators.
result Neural operators can approximate the conditioning operator to arbitrary accuracy.
CSI method learns conditional distributions by estimating flow equations.
problem Learning conditional distributions in generative models.
method Estimates probability flow equations to transport reference to target distribution.
result Derives explicit expressions for conditional drift and score functions.
New conditional risk measures called conditional generalized quantiles defined and characterized.
problem Developing new risk measures for dynamic risk assessment.
method Propose and characterize conditional generalized quantiles using expected utility model and equivalent conditions.
result Characterized conditional generalized quantiles as well-defined and equivalent to a conditional first order condition.
A new method for learning conditional distributions using ODEs and neural networks.
problem Learning conditional distributions efficiently and accurately.
method Conditional Föllmer Flow, discretized with Euler's method, using nonparametric velocity estimation.
result Effective approximation of target conditional distributions, with convergence results for Wasserstein-2 distance.
Sharp statistical theory for conditional diffusion models.
problem Lack of theoretical foundation for conditional diffusion models.
method Sharp statistical theory with approximation of conditional score function.
result Sample complexity bound that adapts to data distribution smoothness.
An analysis is made of reality conditions within the context of noncommutative geometry. We show that if a covariant derivative satisfies a given left Leibniz rule then a right Leibniz rule is equivalent to the reality condition. We show also that the matrix which determines the reality condition must satisfy the Yang-…
New conditions prevent gaps in optimal control problems.
problem Preventing gaps in optimal control problems with state constraints.
method Developed new sufficient conditions not relying on convexity.
result Derived bounds for the size of the relaxation gap.
We identify conditional parity as a general notion of non-discrimination in machine learning. In fact, several recently proposed notions of non-discrimination, including a few counterfactual notions, are instances of conditional parity. We show that conditional parity is amenable to statistical analysis by studying ran…
We consider families of strongly consistent multivariate conditional risk measures. We show that under strong consistency these families admit a decomposition into a conditional aggregation function and a univariate conditional risk measure as introduced Hoffmann et al. (2016). Further, in analogy to the univariate cas…
Develops a rigorous theory for conditional mean embeddings.
problem Efficient conditioning of probability distributions in RKHSs.
method Mathematical theory for both centred and uncentred covariance operators.
result Significantly weakens conditions for applicability of CMEs.
Proposes a new method for interpreting feature importance and effects in dependent feature models.
problem Challenges in interpreting feature importance when features are dependent and interactions are present.
method Conditional Subgroup Approach
result Conditional PFI and PDP estimates based on this approach often outperform existing methods.
This paper investigates how policy conditioning affects reinforcement learning stability.
problem Improving stability and generalization of reinforcement learning agents.
method The authors study Jacobian conditioning behavior during policy optimization and propose a conditioning regularization algorithm.
result The proposed conditioning regularization algorithm enhances reinforcement learning agent generalization.
New boundary conditions solve Cauchy problem for Dirac operators on spacetimes.
problem Understanding non-local boundary conditions for Dirac operators on spacetimes.
method Define and analyze a class of Lorentzian boundary conditions that are local in time and non-local in spatial directions.
result Well-posed Cauchy problem for the Dirac operator is established under these conditions.
We extend CS divergence to conditional distributions and show its advantages in time series data and sequential decision making.
problem Quantifying the closeness between conditional distributions.
method Developed and estimated a conditional Cauchy-Schwarz divergence using kernel density estimation.
result Conditional CS divergence outperforms previous methods in time series clustering and sequential decision making.
The Samuelson condition is not satisfied by tangent lines of quadratic curves.
problem Area condition for Lagrangian 2-web
method Show that the Samuelson condition is not satisfied
result The Samuelson condition is not satisfied by tangent lines of quadratic curves.
We describe a Groebner basis of relations among conditional probabilities in a discrete probability space, with any set of conditioned-upon events. They may be specialized to the partially-observed random variable case, the purely conditional case, and other special cases. We also investigate the connection to generali…
A new method tests conditional independence by transforming it into an unconditional problem using transport maps.
problem Testing conditional independence between two random vectors given a third.
method Constructing transport maps to transform conditional independence into unconditional independence, estimating these maps from data using conditional continuous normalizing flow models.
result The proposed method is validated through simulations and real-data analysis, demonstrating practical effectiveness.
Generalizes moment-matching for exponential families with conditioning or hidden data.
problem Generalizing moment-matching conditions for exponential families with conditioning or hidden data.
method First-principles explanation and self-contained derivation of generalized moment-matching conditions.
result Derives generalized moment-matching conditions for conditional exponential families and hidden data.
DG algorithms often fail to generalize well in limited domains, highlighting necessary vs. sufficient conditions.
problem DG algorithms fail to consistently outperform ERM in limited domains.
method Examined necessary and sufficient conditions for DG, proposing a subspace alignment method.
result DG methods focus on sufficient conditions, often neglecting necessary conditions, leading to generalization failures.
Conditional probabilities modeled using Riemann-Theta Boltzmann Machines.
problem Modeling conditional probabilities in Boltzmann machines.
method Deriving conditional density functions from Riemann-Theta Boltzmann machines.
result Conditional densities can be directly inferred from Riemann-Theta Boltzmann machines.
In this paper, we examine higher order difference problems. Using the "squeezing" argument, we derive both Euler's condition and the transversality condition. In order to derive the two conditions, two needed assumptions are identified. A counterexample, in which the transversality condition is not satisfied without th…
Study on estimating conditional risk in machine learning.
problem Estimating expected loss of prediction models given input features.
method Analyzed in classification and regression settings, showing equivalence to standard regression. Developed theoretical insights and empirical validation.
result Conditional risk calibration is distinct from existing uncertainty quantification problems.
The paper extends static Systemic Risk Measures to a conditional setting.
problem Investigating how static Systemic Risk Measures can be adapted to a conditional framework.
method Providing a general dual representation result, analyzing Conditional Shortfall Systemic Risk Measures, and providing explicit formulas for exponential preferences.
result Explicit formulas for Conditional Shortfall Systemic Risk Measures and a time consistency property.
This paper simplifies conditional Sobol' indices calculation using PCE bases.
problem Computational inefficiency and lack of consistency in evaluating conditional Sobol' indices.
method Analytical extraction of conditional Sobol' indices via basis decomposition of PCE expansions.
result Derives closed-form expressions for conditional Sobol' indices.
A new method uses Schrödinger bridges for deep conditional generative learning.
problem Learning conditional distributions with additional information.
method Schrödinger bridge approach with discretized SDE and deep neural network.
result Generated samples have higher quality and can estimate conditional density.