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.
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. The paper calibrates a model to market quotes efficiently and arbitrage-free.
problem Calibrating a model to market option quotes efficiently and without arbitrage.
method Piecewise-linear local variance function for efficient calibration.
result Arbitrage-free interpolation of class C2 achieved under one millisecond. We consider the quasiconformal dilatation of projective transformations of the real projective plane. For non-affine transformations, the contour lines of dilatation form a hyperbolic pencil of circles, and these are the only circles that are mapped to circles. We apply this result to analyze the dilatation of the circ…
This paper is about interpolating minimal surfaces between two real analytic curves, a and b, each of which are simple real analytic curves, using the Björling-Schwarz formula in the domain where it is valid, changing the normal distributions on inital curves. We insert curves l1,...lL at specific locations and cla…
Piecewise polynomial interpolation-based gradient descent reduces oracle complexity for smooth loss functions.
problem Optimizing empirical risk minimization loss functions
method Piecewise polynomial interpolation-based gradient descent
result Oracle complexity is reduced for smooth loss functions
This note is the updated outline of the article "Interpolational properties of planar spiral curves", Fund. and Applied Math., 2001, Vol.7, N.2, 441-463, published in Russian. The main result establishes boundary regions for spiral and piecewise spiral splines, matching given data. The width of such region can serve as…
Deep ReLU networks generalize via piecewise linear interpolation.
problem Understanding how deep neural networks generalize well on real datasets despite memorizing training samples.
method Demonstrated that deep ReLU networks generalize via piecewise linear interpolation, providing a quantified analysis of generalization intervals.
result Generalization intervals of ReLU networks behave similarly along pairwise directions between samples of the same label in both real and random cases, suggesting the same interpolation mechanism.
SURF simplifies distribution estimation with simple, robust, and fast algorithms.
problem Efficient and accurate distribution estimation in statistics and machine learning.
method Piecewise polynomial approximation using empirical probability interpolation and divide-and-conquer merging.
result Surpassing state-of-the-art algorithms in efficiency and accuracy, SURF estimates distributions robustly and quickly.
NODEs with explicit time dependence can interpolate and generalize like piecewise-constant estimators.
problem Learning from finite datasets with neural ODEs.
method Control-theoretic perspective applied to semi-autonomous NODEs.
result SA-NODEs can interpolate and satisfy SCC, leading to generalization rates similar to histogram and nearest-neighbor estimators.
Simpler algorithms for morphing planar and toroidal graphs.
problem Constructing smooth transitions between isomorphic drawings of planar and toroidal graphs.
method Barycentric interpolation and scaling strategy.
result Simplified and more natural morphs with improved computational efficiency.
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 method efficiently interpolates nonparametric density estimators.
problem Efficient evaluation of nonparametric density estimators.
method Piecewise multivariate polynomial interpolation scheme.
result New estimator with low space requirements and efficient querying.
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. …
PARC uses piecewise linear predictors for regression and classification.
problem Multivariate regression and classification problems.
method Alternates between ridge and softmax regression, and cluster assignment based on accuracy and separability.
result Converges to a local minimum in a finite number of steps.
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.
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.
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.
We prove that every piecewise linear manifold of dimension up to four on which a finite group acts by piecewise linear homeomorphisms admits a compatible smooth structure with respect to which the group acts smoothly. This solves a challenge posed by Thurston in dimension three and confirms a conjecture by Kwasik and L…
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.
Signature uniquely identifies piecewise linear surfaces up to thin homotopy.
problem Characterizing piecewise linear surfaces up to equivalence.
method Crossed module of piecewise linear surfaces and signature homomorphism.
result Signature uniquely characterizes surfaces up to translation and thin homotopy.
The paper characterizes functions of shallow ReLU NN denoisers under minimal norm constraints.
problem Understanding the theoretical success of neural network denoisers.
method Characterization of functions realized by shallow ReLU NN denoisers under minimal norm constraints.
result The functions realized by shallow ReLU NN denoisers are contractive toward clean data points and generalize better than the empirical MMSE estimator at low noise levels.
In this paper, we introduce a bordism category CdPL whose objects are bundles of closed (d−1)-dimensional piecewise linear manifolds and whose morphisms are bundles of d-dimensional piecewise linear cobordisms. In the main theorem of this article, we show that the classifying space $B\mathcal{C}_d^{…
First explicit isometric immersion of a flat Klein bottle in 3D space.
problem Finding an isometric embedding of a Klein bottle in 3D.
method Piecewise-linear map from a Klein bottle to Euclidean 3-space.
result Explicit numerical data for a flat Klein bottle isometrically immersed in 3D.
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.
Paper proposes variational inference for piecewise-linear systems.
problem Intractability of switching dynamical systems.
method Variational approximation and expectation-maximization algorithms.
result Parameters can be estimated off-line, including the number of linear modes.
Nonlinearity is crucial to the performance of a deep (neural) network (DN). To date there has been little progress understanding the menagerie of available nonlinearities, but recently progress has been made on understanding the rôle played by piecewise affine and convex nonlinearities like the ReLU and absolute value …
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.
Paper investigates optimal interpolation methods in linear regression.
problem Understanding when interpolating methods generalize well in linear regression.
method Investigates optimal response-linear interpolators using functions linear in the response variable.
result Provides a closed-form expression for the optimal interpolator and shows it can be derived as the limit of gradient descent.
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.
Equivariant neural networks use symmetry to interpret complex data.
problem Interpreting and understanding the behavior of equivariant neural networks.
method Decompose layers into simple representations and analyze nonlinear activation functions.
result Equivariant neural networks can be interpreted using a filtration generalizing Fourier series.
A new complexity measure for neural networks improves upon classical methods.
problem Lack of a refined complexity measure for comparing different neural network architectures, especially permutation-invariant ones.
method Introduced an equivalence relation among linear functions and counted them relative to this relation.
result The new complexity measure clearly distinguishes between different models and increases exponentially with depth.
We present a detailed analysis of the class of regression decision tree algorithms which employ a regulized piecewise-linear node-splitting criterion and have regularized linear models at the leaves. From a theoretic standpoint, based on Rademacher complexity framework, we present new high-probability upper bounds for …
Paper presents ABGD for efficient piecewise linear regression in high dimensions.
problem Efficiently solving piecewise linear regression in high-dimensional spaces.
method Parametrizes piecewise linear functions as difference of max-affine functions, using ABGD algorithm.
result ABGD converges linearly to an ε-accurate estimate with optimal sample complexity.
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.
The paper analyzes discrete approximations to minimize curve length in Euclidean space.
problem Minimizing the length of curves between two sets in Euclidean space.
method Finite differences and numerical integration for discrete approximations.
result The squared length of the reconstructed curve converges to the squared minimal length with rate O(N−1/2). We consider smooth isotropic immersions from the 2-dimensional torus into R2n, for n≥2. When n=2 the image of such map is an immersed Lagrangian torus of R4. We prove that such isotropic immersions can be approximated by arbitrarily C0-close piecewise linear isotropic maps. If n≥3 the piece…
New cones in 4D space found with minimal mass.
problem Finding mass-minimizing piecewise linear cones in 4D space.
method Mass minimization via Lipschitz maps, classification of candidates.
result No additional mass-minimizing cones found outside five known cases.
We study algebraic varieties of ReLU networks to understand their representable functions.
problem Understanding the functions that ReLU neural networks can represent.
method We introduce algebraic varieties associated with ReLU networks and derive polynomial equations to characterize representable functions.
result Conditions under which ReLU networks attain their expected dimension, providing insight into their structural properties.
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.
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.
The paper tackles prescribing discrete Gaussian curvature on polyhedral surfaces.
problem Prescribing discrete Gaussian curvature on polyhedral surfaces.
method Discrete conformal theory and variational principles with constraints.
result Proves Kazdan-Warner type theorems for polyhedral surfaces.
A hybrid model combines piecewise linear and neural components for interpretable predictions.
problem Post-hoc interpretable methods lead to contradictory explanations and lower prediction accuracy.
method Hybrid model with piecewise linear and neural components.
result The model achieves good interpretability and state-of-the-art accuracy.
The paper studies optimal transport in linear quadratic systems and derives interpolation inequalities.
problem Optimal transport problem in Linear Quadratic optimal control systems.
method Well-posedness of the Monge problem, regularity of optimal transport map, displacement interpolation of measures.
result Derivation of general interpolation inequalities for entropy functionals.
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…
Deep linear networks can closely approximate interpolants without improving risk.
problem Understanding the risk bounds of deep linear networks compared to minimum ℓ2-norm solutions. method Bounding excess risk of interpolating deep linear networks trained using gradient flow.
result Deep linear networks can closely approximate or match minimum ℓ2-norm solutions in terms of risk. XGBoost is often presented as the algorithm that wins every ML competition. Surprisingly, this is true even though predictions are piecewise constant. This might be justified in high dimensional input spaces, but when the number of features is low, a piecewise linear model is likely to perform better. XGBoost was exten…
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…