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.
New GP model estimates piecewise continuous functions.
problem Piecewise continuous regression functions in scientific and engineering applications.
method Local Gaussian process model with partitioned local data and joint estimation of boundaries.
result Superior performance over conventional GP models in estimating piecewise regression functions.
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.
For a bounded domain equipped with a piecewise Lipschitz continuous Riemannian metric g, we consider harmonic map from (Ω,g) to a compact Riemannian manifold (N,h)⊂Rk without boundary. We generalize the notion of stationary harmonic map and prove the partial regularity. We also discuss the global Li…
New algorithm reduces regret in online learning for piecewise continuous functions.
problem Exponential loss in efficiency when moving from classical to adversarial learning.
method Introduces generalized bracketing numbers and Follow-the-Perturbed-Leader algorithm.
result Optimal scaling of optimization oracle calls with average regret.
Deep Jump Gaussian Processes model high-dimensional piecewise functions.
problem Modeling high-dimensional piecewise continuous functions with limited accuracy.
method Integrates region-specific locally linear projections with Jump Gaussian Processes (JGP) to capture local low-dimensional subspace structures.
result DJGP achieves superior predictive accuracy and more reliable uncertainty quantification compared to existing methods.
We study online optimization of smoothed piecewise constant functions over the domain [0, 1). This is motivated by the problem of adaptively picking parameters of learning algorithms as in the recently introduced framework by Gupta and Roughgarden (2016). Majority of the machine learning literature has focused on Lipsc…
We show injectivity of the geodesic X-ray transform on piecewise constant functions when the transform is weighted by a continuous matrix weight. The manifold is assumed to be compact and nontrapping of any dimension, and in dimension three and higher we assume a foliation condition. We make no assumption regarding con…
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.
Piecewise linear activations create many spurious local minima in neural networks.
problem Understanding the loss surface of neural networks with piecewise linear activations.
method Proved the existence of infinite spurious local minima and partitioned the loss surface into smooth cells.
result Piecewise linear activations create many spurious local minima that are invariant under a continuous path.
Smoothing graphons improve link prediction in Bayesian SBM without increasing computational complexity.
problem Accurate modeling of exchangeable relational data with flexible and computationally efficient graphons.
method Introducing smoothing procedures to piecewise-constant graphons to create smoothing graphons, which allow continuous intensity values for relations.
result Smoothing graphons improve AUC and precision for link prediction in real-world data sets.
Given iid observations from an unknown absolute continuous distribution defined on some domain Ω, we propose a nonparametric method to learn a piecewise constant function to approximate the underlying probability density function. Our density estimate is a piecewise constant function defined on a binary partition o…
BN refines local partition geometry in piecewise-affine networks during training.
problem Understanding the effect of BN on the function realized during training in piecewise-affine networks.
method Analyzing the geometry of switching hyperplanes and affine-region partition conditioned on a mini-batch.
result BN increases expected local partition refinement in ReLU and piecewise-affine networks.
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…
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.
GroupSort neural networks can approximate Lipschitz continuous functions.
problem Understanding and improving the expressive power of neural networks with Lipschitz constraints.
method Introduced and studied GroupSort neural networks with constraints on weights, proving their ability to approximate Lipschitz continuous functions.
result GroupSort networks can represent any Lipschitz continuous piecewise linear functions and are well-suited for approximating general Lipschitz continuous functions.
The paper designs neural networks with assurance for controlling nonlinear systems.
problem Designing neural networks with assurance for nonlinear system control.
method Bounding the number of affine functions needed for a CPWA function, connecting it to a TLL NN architecture.
result The TLL NN architecture is parameterized by the number of affine functions in the CPWA function it realizes.
Quantum Monte Carlo speeds up option pricing for complex payoff functions.
problem Efficiently pricing options with complex payoff functions using quantum computing.
method Developed a quantum Monte Carlo algorithm for multidimensional Black-Scholes PDEs.
result Proved polynomial computational complexity and speed-up over classical methods.
We study the isoperimetric problem in Euclidean space endowed with a density. We first consider piecewise constant densities and examine particular cases related to the characteristic functions of half-planes, strips and balls. We also consider continuous modification of Gauss density in R2. Finally, we give a list…
A new invariant captures geometric features of circle embeddings.
problem Capturing geometric features of circle embeddings invariantly.
method Chordal distance transform and persistent homology.
result Persistent homology of chordal distance transform is invariant.
We use partial actions, as formalized by Exel, to construct various commensurating actions. We use this in the context of groups piecewise preserving a geometric structure, and we interpret the transfixing property of these commensurating actions as the existence of a model for which the group acts preserving the geome…
The aim of this work consists in the study of the optimal investment strategy for a behavioural investor, whose preference towards risk is described by both a probability distortion and an S-shaped utility function. Within a continuous-time financial market framework and assuming that asset prices are modelled by semim…
A triangulated piecewise-linear minimal surface in Euclidean 3-space defined using a variational characterization is critical for area amongst all continuous piecewise-linear variations with compact support that preserve the simplicial structure. We explicitly construct examples of such surfaces that are embedded and a…
In this contribution we derive an explicit formula for the boundary non-crossing probabilities for Slepian processes associated with the piecewise linear boundary function. This formula is used to develop an approximation formula to the boundary non-crossing probabilities for general continuous boundaries. The formulas…
A celebrated theorem of Hadwiger states that the Euler-Poincaré characteristic is the the unique invariant and continuous valuation on the distributive lattice of compact polyhedra in R^n that assigns value one to each convex non-empty such polyhedron. This paper provides an analogue of Hadwiger's result for finitely p…
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.
The square root velocity function (SRVF), introduced by Srivastava et al, has proved to be an effective way to compare absolutely continuous curves in RN modulo reparametrization. Several computational papers have been published based on this method. In this paper, we carefully establish the theoretical foundations …
Recently there have been exciting developments in Monte Carlo methods, with the development of new MCMC and sequential Monte Carlo (SMC) algorithms which are based on continuous-time, rather than discrete-time, Markov processes. This has led to some fundamentally new Monte Carlo algorithms which can be used to sample f…
Modified ReLU networks improve regression estimation rates.
problem Regression estimation with smooth functions.
method Using modified ReLU neural networks with specific weight modifications.
result Empirical risk minimizers achieve minimax rate of prediction.
New method for robust learning from batches, even adversarial ones.
problem Learning from batches that may be corrupt or adversarial.
method General framework for robust learning, derived from optimal robust algorithms.
result First robust agnostic learning algorithms for various distributions.
Neural network models improve survival analysis with reduced computation time.
problem Limited expressiveness of standard survival models.
method Piecewise neural network models of hazard and density functions.
result Models outperform state-of-the-art models with less computation time.
New method uses DC functions for piecewise linear regression.
problem Regression with piecewise linear constraints.
method Estimates piecewise linear convex functions using a difference of convex functions.
result Method achieves close to minimax statistical risk and comparable performance to existing methods.
Proposes adaptive ridge regression for functional linear models with piecewise shapes.
problem Functional linear regression with unknown coefficient function.
method Adaptive piecewise function template with L2 penalization. result Improves predictive power and interpretability compared to standard methods.
Deep neural networks with piecewise-polynomial activations can approximate smooth functions and their derivatives.
problem Approximating smooth functions and their derivatives with neural networks.
method Derives the depth, width, and sparsity required for approximation in Hölder norms.
result Deep neural networks with bounded weights can approximate Hölder smooth functions and their derivatives.
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.
The Bouncy Particle Sampler is a novel rejection-free non-reversible sampler for differentiable probability distributions over continuous variables. We generalize the algorithm to piecewise differentiable distributions and apply it to generic binary distributions using a piecewise differentiable augmentation. We illust…
Study approximates nonlinear functionals using deep ReLU networks.
problem Approximating nonlinear continuous functionals with neural networks.
method Constructs continuous piecewise linear interpolation under simple triangulation, analyzes rates of approximation.
result Established rates of approximation for functional deep ReLU networks.
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. Investment strategy optimization from discrete to continuous models.
problem Optimizing investment strategies and stopping times in both continuous and discrete settings.
method Characterized value functions via quadratic reflected BSDEs for continuous case, discretized BSDEs for discrete case, and derived uniform convergence rates.
result Uniform convergence and rate from discrete to continuous quadratic reflected BSDEs.
New proof for global rigidity of vertex scaling on polyhedral surfaces.
problem Global rigidity of vertex scaling on polyhedral surfaces.
method Elementary variational proof based on continuity of eigenvalues and extension of convex functions.
result Global rigidity of vertex scaling proved without involving 3D hyperbolic geometry.
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.
A map f:X→Y between topological spaces is defined to be {\em scatteredly continuous} if for each subspace A⊂X the restriction f∣A has a point of continuity. We show that for a function f:X→Y from a perfectly paracompact hereditarily Baire Preiss-Simon space X into a regular space Y the scattere…
New algorithm predicts piecewise regular functions online.
problem Online prediction of piecewise regular functions.
method Modified sleeping experts aggregation algorithm.
result Oracle risk bounds for all local regions.
PAR provides a flexible framework for quantization in optimization problems.
problem Challenges in optimization problems over discrete or quantized variables.
method Piecewise-affine regularization (PAR) for modeling and computational optimization.
result PAR-regularized loss functions exhibit high quantization at critical points in the overparameterized regime.
New proof shows efficient ReLU networks for piecewise linear functions.
problem Existence of efficient ReLU neural networks for piecewise linear functions.
method Degree 1 triangulations of the relative homology class bounded by polyhedra.
result Existence of efficient ReLU neural networks for functions with compact support.
Anosov maps study with new Banach space and foliation method.
problem Understanding statistical properties of Anosov maps.
method Constructing a new Banach space and using a new foliation method.
result New Banach space provides insights into foliation absolute continuity.
SyMPLER improves time series forecasting in nonstationary environments with explainable models.
problem Nonstationary time series forecasting with limited interpretability.
method Dynamic piecewise-linear approximations based on Statistical Learning Theory generalization bounds.
result SyMPLER achieves comparable performance to black-box and explainable models while maintaining interpretability.
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.