Ball-Box Theorem proven for non-differentiable subbundles.
problem Proving a theorem for non-differentiable subbundles.
method Analogue of Ball-Box Theorem for step 2, non-integrable bundles.
result The Ball-Box Theorem holds for a specific class of non-differentiable subbundles.
Anosov diffeomorphisms with integrable subbundles have coherent dynamics and spectral rigidity.
problem Characterizing Anosov diffeomorphisms with integrable subbundles.
method Joint integrability of strong stable and unstable subbundles leads to coherent dynamics and spectral rigidity.
result Anosov diffeomorphisms with integrable subbundles are dynamically coherent and have spectral rigidity.
Given a Dirac subbundle and an isotropic subbundle of a Courant algebroid, we provide a canonical method to obtain a new Dirac subbundle. When the original Dirac subbundle is involutive (i.e., a Dirac structure) this construction has interesting applications, for instance to Dirac's theory of constraints and to the Mar…
Extends involutivity to non-Lipschitz subbundles and proves the Frobenius Theorem.
problem Defining involutivity for non-Lipschitz subbundles and proving the Frobenius Theorem.
method Using generalized functions, the Frobenius Theorem is extended to log-Lipschitz subbundles with sharp regularity estimates.
result For log-Lipschitz involutive subbundles, there exists a homeomorphism with specific regularity properties.
The paper explores invariant subbundles in nonholonomic mechanics.
problem Determining invariant affine subbundles in nonholonomic and constrained variational mechanics.
method Using Spencer cohomology and iterative formulae, the paper formalizes the integrability of linear partial differential equations and determines the largest invariant affine subbundle.
result Iterative formulae for determining the largest invariant affine subbundle are provided.
Study real line subbundles on curves, extending classical work.
problem Understanding real line subbundles in real bundles on curves.
method Application of Atiyah's techniques and work of Lange-Narasimhan.
result Describes the Galois action on the set of lines through a real point in the moduli space of such bundles.
Study on embedding sphere subbundles with prescribed mean curvature in Riemannian vector bundles.
problem Embedding sphere subbundles with prescribed mean curvature in Riemannian vector bundles.
method Analyzes embeddings of sphere subbundles into Riemannian vector bundles with a prescribed mean curvature.
result Conditions for the existence of such embeddings are derived.
Abstract reviews distributions and subbundles in differential geometry.
problem Understanding distributions and subbundles in differential geometry.
method Systematic review of distributions and subbundles, including sheaves and differentiability cases.
result Detailed consideration of Orbit Theorem and its applications.
The study finds conditions for embedding sphere subbundles with specific mean curvatures.
problem Embedding sphere subbundles with prescribed mean curvatures in Riemannian vector bundles.
method Analyzes embeddings of sphere subbundles into Riemannian vector bundles with prescribed mean curvatures.
result Conditions for embedding sphere subbundles with specific mean curvatures are identified.
Formulas for spectra of higher spin operators on sphere subbundles.
problem Finding spectra of higher spin operators on specific subbundles of spinor-valued tensors.
method Explicit formulas derived for spectra in both even and odd dimensions.
result Spectra formulas for higher spin operators and their squares.
Paper proves autodiff systems are correct for non-differentiable functions.
problem Correctness of autodiff systems for non-differentiable functions in deep learning.
method Investigation of PAP functions and introduction of intensional derivatives.
result Intensional derivatives always exist and coincide with standard derivatives for almost all inputs.
New algorithms improve inference in non-differentiable models.
problem Inference and learning in latent variable models with non-differentiable densities.
method Proximal interacting particle Langevin algorithms (PIPLA).
result Nonasymptotic bounds and effectiveness demonstrated in various models.
New algorithm for variational inference on non-differentiable models.
problem Challenges in stochastic variational inference for non-differentiable models.
method Generalizes reparameterization trick for non-differentiable models, splitting latent variables into differentiable and non-differentiable regions.
result Our algorithm reduces variance and remains unbiased for non-differentiable models.
Study particle dynamics in non-differentiable fractal spaces.
problem Understanding motion in non-smooth, probabilistic geometries.
method Use fiber bundle theory to characterize multivalued geodesic trajectories.
result Developed a hybrid theory combining surface and stochastic process theories.
The paper constructs L∞-algebras from contact Courant algebroids and isotropic subbundles.
problem Understanding the structure of contact Courant algebroids and their associated L∞-algebras. method The construction of L∞-algebras from L-Courant algebroids and isotropic subbundles. result A relationship between constructed L∞-algebras is established by a morphism. We generalize stochastic smoothing for gradient estimation of non-differentiable functions.
problem Gradient estimation for non-differentiable functions.
method Developed a general framework for relaxation and gradient estimation of non-differentiable black-box functions using stochastic smoothing with reduced assumptions.
result Empirically validated the effectiveness of variance reduction strategies for various non-differentiable tasks.
The study connects Hilbert entropy to non-differentiability points of limit sets in flag spaces.
problem Understanding non-differentiability points in limit sets of convex projective structures.
method Introduces hyperplane conicality for θ-Anosov representations and uses it to prove properties of boundary maps. result Hilbert entropy is linked to the Hausdorff dimension of non-differentiability points in flag spaces.
The paper studies deformations of calibrated subbundles in special holonomy manifolds.
problem Deforming calibrated subbundles in noncompact manifolds of special holonomy.
method Twisting calibrated subbundles by special sections and deriving conditions for deformations to remain calibrated.
result Twisting conormal bundles of Lagrangian submanifolds in T∗Sn by 1-forms does not provide new examples. The paper introduces a method for dimension reduction using sub-Riemannian geometry.
problem Dimension reduction for manifold learning and surface reconstruction.
method Combining local linear approximations of a point cloud to obtain lower dimensional bundles.
result Sub-Riemannian geodesics can successfully be applied to problems like constructing an approximating submanifold and computing distances.
ES for non-differentiable parameters scales to large models.
problem Learning non-differentiable parameters in large models.
method Hybrid approach combining ES for non-differentiable and gradient-based methods for differentiable parameters.
result Hybrid approach is competitive and allows training sparse models from the start.
New method solves non-convex constrained optimization problems with non-differentiable constraints.
problem Training non-convex models with non-differentiable constraints.
method Proxy-Lagrangian formulation and semi-coarse correlated equilibrium.
result Solves non-convex constrained optimization problems with theoretical guarantees.
Lecture notes introduce differential geometry using sheaves and differential operators.
problem Exploring differential geometry concepts.
method Using sheaves, differential operators, and horizontal subbundles.
result Presented an approach to fundamental differential geometry structures.
Algorithm finds optimal investment strategies for non-differentiable preferences.
problem Optimal investment strategies under non-differentiable preferences.
method Reduces problem to a discrete grid, uses efficient method to find strategies.
result Optimal strategies lie on a discrete grid, allowing efficient computation.
New examples of real hypersurfaces found in complex hyperbolic quadrics.
problem Existence of specific types of real hypersurfaces in complex hyperbolic quadrics.
method Construction of a one-parameter family of homogeneous Hopf hypersurfaces.
result First known examples of real hypersurfaces with integrable maximal complex subbundle in irreducible Kahler manifolds.
We extend the "bundle constructions" of calibrated submanifolds, due to Harvey--Lawson in the special Lagrangian case, and to Ionel--Karigiannis--Min-Oo in the cases of exceptional calibrations, by "twisting" the bundles by a special (harmonic, holomorphic, parallel) section of a complementary bundle. The existence of …
The existence of a recurrent spinor field on a pseudo-Riemannian spin manifold (M,g) is closely related to the existence of a parallel 1-dimensional complex subbundle of the spinor bundle of (M,g). We characterize the following simply connected pseudo-Riemannian manifolds admitting such subbundles in terms of their…
A Dirac structure is a Lagrangian subbundle of a Courant algebroid, L⊂E, which is involutive with respect to the Courant bracket. In particular, L inherits the structure of a Lie algebroid. In this paper, we introduce the more general notion of a pseudo-Dirac structure: an arbitrary subbundle, $W\sub…
We define integrable, big-isotropic structures on a manifold M as subbundles E⊆TM⊕T∗M that are isotropic with respect to the natural, neutral metric (pairing) g of TM⊕T∗M and are closed by Courant brackets (this also implies that [E,E⊥g]⊆E⊥g). We give the interp…
We prove that the universal covering of a complete locally symmetric normal metric contact pair manifold is a Calabi-Eckmann manifold. Moreover we show that a complete, simply connected, normal metric contact pair manifold such that the foliation induced by the vertical subbundle is regular and reflections in the integ…
Study shows AD for neural nets with machine-representable numbers can be incorrect.
problem Correctness of AD for neural nets with machine-representable numbers.
method Analyzed two sets of parameters: incorrect and non-differentiable. Proved bounds and conditions for AD correctness.
result AD can be incorrect for machine-representable numbers, but provides a Clarke subderivative on non-differentiable set.
Unified approach for sampling non-differentiable and heavy-tailed targets.
problem Sampling non-differentiable and heavy-tailed distributions using Langevin algorithms.
method Anchored Langevin dynamics, which modifies the Langevin diffusion with a smooth reference potential and multiplicative scaling.
result Non-asymptotic guarantees in the 2-Wasserstein distance to the target distribution.
Study improves understanding of non-differentiable penalties in high-dimensional settings.
problem Theoretical understanding of non-differentiable penalties like generalized LASSO and nuclear norm in high-dimensional settings.
method Proportional high-dimensional regime analysis with finite sample upper bounds on expected squared error.
result LO provides accurate estimation of out-of-sample risk in high-dimensional settings.
Proves lower bounds for Lyapunov exponents of flat bundles on curves.
problem Proving lower bounds for Lyapunov exponents of flat bundles on curves.
method Generalized from Teichmueller curves to any local system over a curve with non-expanding cusp monodromies.
result Obtained large genus limits of individual Lyapunov exponents in hyperelliptic strata of Abelian differentials.
The authors study a generalized notion of null geodesic defined by the Legendrian dynamics of a regular conical subbundle of the tangent bundle on a manifold. A natural extension of the Weyl tensor is shown to exist, and to depend only on this conical subbundle. Given a suitable defining function of the conical bundle,…
Suppose N is an affine SL(2,R)-invariant submanfold of the moduli space of pairs (M,w) where M is a curve, and w is a holomorphic 1-form on M. We show that the Forni bundle of N (i.e. the maximal SL(2,R)-invariant isometric subbundle of the Hodge bundle of N) is always flat and is always orthogonal to the tangent space…
We establish a version of the complex Frobenius theorem in the context of a complex subbundle S of the complexified tangent bundle of a manifold, having minimal regularity. If the subbundle S defines the structure of a Levi-flat CR-manifold, it suffices that S be Lipschitz for our results to apply. A principal tool in …
Study identifies conditions for proxy adjustment in confounded binary treatment outcomes.
problem Average causal effect estimation with a non-differentially mismeasured binary confounder.
method Identifies conditions for proxy adjustment in the presence of a non-differentially mismeasured binary confounder.
result Adjusting for a non-differentially mismeasured binary proxy can improve estimation of the average causal effect.
Develops LF-PPL for non-differentiable models with automatic boundary checks.
problem Handling non-differentiable models in probabilistic programming.
method Introduces LF-PPL with automatic boundary checks and a formalism ensuring measure zero discontinuities.
result Demonstrates efficient inference for non-differentiable models using DHMC.
AVO optimizes simulators without likelihoods, combining GANs and variational methods.
problem Inference in non-differentiable simulators is difficult.
method Adversarial Variational Optimization (AVO) using GANs and variational techniques.
result AVO minimizes JS divergence between synthetic and empirical data distributions.
Differentiable pipeline replaces non-differentiable CAE components for shape optimization.
problem Gradient-based optimization is limited by non-differentiable components in CAE workflows.
method Surrogate models replace non-differentiable pipeline components, enabling gradient-based optimization.
result Gradient-based shape optimization possible without differentiable solvers.
Study on spectrum of Dirac operator on pseudo-Riemannian manifolds.
problem Spectrum of Dirac operator on pseudo-Riemannian spin manifolds.
method Analyzing the spectrum of the Dirac operator D in Lξ2(S), considering maximal time-like subbundles. result Spectra of D induced by two maximal time-like subbundles are equal if the base manifold is compact. This paper is an attempt at understanding the quantum-like dynamics of financial markets in terms of non-differentiable price-time continuum having fractal properties. The main steps of this development are the statistical scaling, the non-differentiability hypothesis, and the equations of motion entailed by this hypot…
Smooth Contextual Bandits bridge two previously studied extremes of non-differentiable and parametric-response bandits.
problem Nonparametric contextual bandits with Hölder smoothness.
method Developed a novel algorithm that optimally balances between non-differentiable and parametric-response bandits.
result Proved the algorithm achieves rate-optimal regret for all smoothness settings.
SoDeep learns approximations of ranking metrics for deep learning tasks.
problem Non-differentiable metrics in machine learning tasks.
method Sorting deep (SoDeep) net trained to approximate sorting of scores.
result Competitive results on Cross-modal text-image retrieval, multi-label image classification, and visual memorability ranking tasks.
Hyperbolic geometry explained without calculus.
problem Understanding hyperbolic geometry without differential calculus.
method Soft presentation of hyperbolic spaces, avoiding differential apparatus.
result Spheres, hyperbolic, and Euclidean spaces are the only three-point homogeneous locally compact geodesic metric spaces.
Paper generalizes Hardy-Rogers maps for market equilibrium analysis in duopoly markets.
problem Existence and uniqueness of market equilibrium in duopoly markets with non-differentiable, nonlinear response functions.
method Coupled fixed points approach for generalized Hardy-Rogers maps.
result Enriched understanding of market equilibrium in duopoly markets with non-differentiable response functions.
Proposes a method for inference in high-dimensional classification with non-differentiable surrogate losses.
problem Lack of inference procedures for identifying driving factors in high-dimensional classification with non-differentiable surrogate losses.
method Kernel-smoothed decorrelated score and cross-fitted version for hypothesis tests and interval estimators.
result Valid and superior inference methods for high-dimensional classification with non-differentiable surrogate losses.
Paper proposes a method to minimize non-differentiable loss functions.
problem Minimizing non-differentiable and non-decomposable loss functions.
method Learn smooth relaxations of true losses through surrogate neural networks, then optimize jointly with the prediction model.
result Empirical results show the efficiency of learning surrogate losses.