Global approximation for piecewise linear paths via signatures.
problem Global approximation theorems for piecewise linear paths.
method Using signatures of piecewise linear paths and their density in Lp-norms. result Linear functionals of signatures are dense in Lp-norms under an integrability condition. Binary BPS improves sampling for easy mixtures.
problem Sampling from binary distributions efficiently.
method Generalized Bouncy Particle Sampler for binary variables.
result Binary BPS outperforms binary HMC for easy mixtures.
Paper introduces differentiable sorting and ranking with O(nlogn) time complexity.
problem Non-differentiability of sorting and ranking operations in machine learning.
method Differentiable proxies constructed as projections onto the permutahedron and reduction to isotonic optimization.
result First differentiable sorting and ranking operators with O(nlogn) time and O(n) space complexity. Generalizes Thurston's jiggling lemma for piecewise smooth solutions.
problem Creating piecewise smooth solutions of differential relations without homotopical assumptions.
method Jiggling arbitrary sections of E to construct solutions of R. result Generalization of Thurston's lemma for piecewise smooth solutions of differential relations.
Injectivity of geodesic ray transform for piecewise constants on compact manifolds.
problem Injectivity of geodesic ray transform for piecewise constant functions.
method Injectivity of geodesic ray transform on piecewise constant functions weighted by a continuous matrix weight.
result Injectivity of the geodesic X-ray transform on piecewise constant functions.
A fundamental object in a hyperbolic 3-manifold M is its convex core C(M), defined as the smallest closed non-empty convex subset of M. We investigate the way the geometry of the boundary S of C(M) varies as we vary the hyperbolic metric of M. Thurston observed that the intrinsic metric of S is hyperbolic, and that its…
Constructs finite element spaces for (p,q)-forms, excluding one subspace.
problem Constructing finite element spaces for (p,q)-forms. method Piecewise polynomial finite element spaces for all natural subspaces of (p,q)-forms, excluding one subspace. result Recovers known finite element spaces and introduces new ones.
Unified formula for optimal portfolio under piecewise hyperbolic risk aversion.
problem Optimizing portfolios with piecewise hyperbolic risk aversion utilities.
method Derive a unified closed-form formula for the optimal portfolio.
result Unified formula reflects risk aversion behaviors and risk-taking behaviors.
New method samples from piecewise smooth distributions using Hamiltonian Monte Carlo.
problem Sampling from distributions with discontinuous gradients.
method Generalized Randomized Hamiltonian Monte Carlo (GRHMC) for piecewise smooth targets.
result GRHMC processes sample from piecewise smooth target distributions with the desired distribution as the invariant distribution.
We recast basic topological concepts underlying differential geometry using the language and tools of noncommutative geometry. This way we characterize principal (free and proper) actions by a density condition in (multiplier) C*-algebras. We introduce the concept of piecewise triviality to adapt the standard notion of…
The paper approximates Levi-Civita connection and curvature on 2D manifolds using finite elements.
problem Approximating Levi-Civita connection and curvature on 2D manifolds with finite elements.
method Using Regge finite elements, piecewise polynomial symmetric (0,2)-tensor fields, and distributional sense for non-regular tensors.
result Distributional quantities converge to their smooth counterparts under refinement of triangulation.
Hilbert initiated the standpoint in foundations of mathematics. From this standpoint, we allow only a finite number of repetitions of elementary operations when we construct objects and morphisms. When we start from a subset of a Euclidean space. Then we assume that any element of the line has only a finite number of c…
Optimizes shapes in uncertain Navier-Stokes flow problems.
problem Optimizing shapes with geometric constraints and physical uncertainty.
method Multi-shape calculus and stochastic augmented Lagrangian method.
result Successfully optimized shapes in uncertain Navier-Stokes flow.
Given a piecewise linear (PL) function p defined on an open subset of Rn, one may construct by elementary means a unique polyhedron with multiplicities $\D(p)$ in the cotangent bundle Rn×Rn∗ representing the graph of the differential of p. Restricting to dimension 2, we show that any smooth functi…
This paper tackles discontinuous neural networks for better approximation of piecewise continuous functions.
problem Limitation of neural networks in approximating piecewise continuous functions due to discontinuities.
method Proposes a decoupled two-step procedure to train a discontinuous deep neural network model.
result Provides approximation guarantees for the proposed model in piecewise continuous function spaces.
Discrete vector bundles are important in Physics and recently found remarkable applications in Computer Graphics. This article approaches discrete bundles from the viewpoint of Discrete Differential Geometry, including a complete classification of discrete vector bundles over finite simplicial complexes. In particular,…
In this paper, we study eigenvalues of the closed eigenvalue problem of the differential operator L, which is introduced by Colding and Minicozzi in [4], on an n-dimensional compact self-shrinker in Rn+p. Estimates for eigenvalues of the differential operator L are obtained. Our estimates for eigenvalues…
Unified method for estimating properties of large domain distributions efficiently.
problem Estimating properties of distributions over large domains efficiently.
method Piecewise-polynomial approximation technique for constructing sample- and time-efficient estimators.
result Near-linear-time computable estimators with optimal and highly-concentrated approximation values.
Improved Gaussian Process model for predicting trajectories without independence assumption errors.
problem Incorrect independence assumption in previous work on Gaussian Process uncertainty propagation.
method Proposed a novel piecewise linear approximation to correct the independence assumption in continuous models.
result Corrected the independence assumption in Gaussian Process models for predicting trajectories.
Extends Local Variance Gamma model with geometric Brownian motion and piecewise linear local variance.
problem Modeling volatility dynamics in financial markets.
method Develops a geometric version of the Local Variance Gamma model with drift and piecewise linear local variance functions.
result Derives an ordinary differential equation for option prices and solves it in closed form.
A new shape space allows optimization of non-smooth shapes in fluid mechanics.
problem Optimizing non-smooth shapes in fluid mechanics.
method Constructing a product manifold to include piecewise-smooth shapes.
result Numerical results show applicability in minimizing viscous energy dissipation.
Constructs commensurating actions for groups of piecewise transformations.
problem Classifying and understanding actions of groups of piecewise transformations.
method Partial actions and commensurating actions to model geometric structures.
result Conjugacy results for subgroups with specific properties.
We introduce a nonparametric approach for estimating drift and diffusion functions in systems of stochastic differential equations from observations of the state vector. Gaussian processes are used as flexible models for these functions and estimates are calculated directly from dense data sets using Gaussian process r…
The study connects cubic differentials to convex RP^2-structures and their ends.
problem Understanding the relationship between cubic differentials and convex RP^2-structures.
method Affine sphere construction and analysis of poles of cubic differentials.
result Poles of cubic differentials correspond to ends of convex RP^2-structures.
This thesis is divided into three parts. In the first part, we give an introduction to J. Harrison's theory of differential chains. In the second part, we apply these tools to generalize the Cauchy theorems in complex analysis. Instead of requiring a piecewise smooth path over which to integrate, we can now do so over …
Investigates stability of piecewise flat Ricci flow using analysis and simulations.
problem Stability of piecewise flat Ricci flow.
method Linear stability analysis and numerical simulations.
result Adaptations avoided numerical instability and led to convergence to smooth solutions.
Extends gradient-based optimization to spline functions.
problem Limitations of standard differentiable programming methods.
method Derives Jacobian of spline functions and uses it in predictive models.
result Improved performance in various applications.
A new method solves complex financial equations efficiently.
problem Solving worst-case and best-case prices for two-factor uncertain volatility models.
method Decompose and integrate, then optimize; piecewise constant control; closed-form Green's functions; 2D convolution integrals; monotone numerical integration; Fast Fourier Transforms.
result The method efficiently computes the value function and optimal control, converging to the viscosity solution of the HJB equation.
Arguments on PL,(=piecewise linear) topology work over any ordered field in the same way as over the real field, and those on differential topology do over a real closed field R in an o-minimal structure that expands (R,<,0,1,+,cdot). One of the most fundamental properties of definable sets is that a compact definable …
LinXGBoost extends XGBoost for better regression of piecewise linear functions.
problem Regression of functions with jumps or discontinuities is challenging.
method LinXGBoost stores linear models at each leaf, equivalent to piecewise regularized least-squares.
result LinXGBoost outperforms vanilla XGBoost and Random Forest in experiments.
Neural networks can represent complex piecewise functions efficiently.
problem Representing continuous piecewise affine functions with neural networks.
method Two hidden layers with ReLU activation, O(p) neurons for p pieces. result CPA functions can be represented by a neural network with linear size.
Theorem proves integrability for piecewise-smooth distributions.
problem Integrability of piecewise-smooth distributions.
method Generalizations of Frobenius integrability theorem.
result Sufficient criteria for complete integrability with bi-Lipschitz coordinates.
The paper develops methods to approximate quantities of interest in insurance models using deterministic integration.
problem Computing quantities of interest in insurance models, such as the probability of ruin and insurance company value.
method Adapting the problem to allow for deterministic numerical integration algorithms, including quasi-Monte Carlo rules and smoothing techniques.
result Convergence result justifying phase-type approximations on the process level.
We propose a numerical recipe for risk evaluation defined by a backward stochastic differential equation. Using dual representation of the risk measure, we convert the risk valuation to a stochastic control problem where the control is a certain Radon-Nikodym derivative process. By exploring the maximum principle, we s…
Piecewise flat approximations for curvature in Euclidean and non-Euclidean spaces.
problem Approximating local extrinsic curvature on discrete manifolds.
method Constructing discrete curvature forms on piecewise flat manifolds, using weighted sums of hinge angles.
result Converges to smooth curvature values as mesh refinement occurs, favorably comparing with other discrete approaches.
We study the differential geometric consequences of our previous result on the existence of fat triangulations, in conjunction with a result of Cheeger, Müller and Schrader, regarding the convergence of Lipschitz-Killing curvatures of piecewise-flat approximations of smooth Riemannian manifolds. A further application t…
New method characterizes surface quadrilateral layouts as special immersions.
problem Characterize surface quadrilateral layouts mathematically.
method Characterizes quadrilateral layouts as special immersions of a cut representation of the surface into the Euclidean plane.
result Mathematically describes and generalizes integer grid maps.
New MFG model for MV portfolio management with peer-based risk aversion.
problem Time-inconsistent mean-variance portfolio management with peer-based risk aversion.
method Mean-field game, smooth regularization, fixed-point arguments, convergence analysis.
result Existence of mean-field equilibrium in time-inconsistent MFG.
Study groups of piecewise isometries in tessellations of Euclidean space.
problem Understanding the structure of groups formed by cutting and gluing tessellations.
method Proving structure results about groups of piecewise isometries of tessellations, including elementary amenability.
result Groups of piecewise isometries of tessellations are elementary amenable.
Study geometrically characterizes piecewise circular curves with decreasing curvature.
problem Characterizing piecewise circular curves with decreasing curvature.
method Introducing moduli spaces and relating them to Legendrian polygons.
result Proves the moduli space contains a connected component homeomorphic to the Fock-Goncharov space of positive flags.
The center of a quotient group of piecewise linear homeomorphisms is trivial.
problem Understanding the structure of a specific group of homeomorphisms.
method Analyzing a quotient of piecewise linear homeomorphisms of the real line.
result The center of the quotient group is trivial.
New algorithms cluster non-stationary time series data.
problem Clustering time series generated by piecewise stationary processes.
method Proposed a natural formulation and introduced a notion of consistency for clustering.
result Simple, efficient algorithms that work without additional assumptions.
GraN-GAN normalizes gradients for better GAN performance.
problem Improving image generation in GANs with piecewise linear discriminators.
method Piecewise Gradient Normalization (GraN) for input-dependent normalization.
result Significant performance gains in image generation across various datasets.
Expanded Local Variance Gamma model adds drift and simplifies calibration.
problem Calibration of complex local volatility surfaces.
method Adding drift to the underlying process, deriving an ODE, piecewise linear and constant local variance, closed-form solution using hypergeometric functions.
result Calibration to market smiles can be done term-by-term and is fast.
The paper extends a variance gamma model to quadratic functions, reducing arbitrage and computational costs.
problem Creating an arbitrage-free interpolation for option pricing models.
method Generalizing the local variance gamma model to a piecewise quadratic local variance function.
result The quadratic model results in an arbitrage-free interpolation of class C3, reducing knots and computational cost.
New algorithm reveals piecewise affine structure of neural networks.
problem Lack of strong guarantees on deep neural networks' behavior in safety-critical applications.
method Developed a novel algorithm to compute the piecewise affine form of neural networks.
result Computed piecewise affine representations of neural networks with rectified linear unit activations.
Formalizes quantum path integrals using groupoids and differential forms.
problem Formalizing Feynman's path integral in quantum mechanics.
method Shifted focus to pair groupoid, using van Est map and piecewise linear structures.
result Developed a coordinate-free approach to integration of differential forms.
This article provides an attempt to extend concepts from the theory of Riemannian manifolds to piecewise linear spaces. In particular we propose an analogue of the Ricci tensor, which we give the name of an Einstein vector field. On a given set of piecewise linear spaces we define and discuss (normalized) Ricci flows. …