Explicit formula derived for Slepian process boundary non-crossing probabilities.
problem Calculating boundary non-crossing probabilities for Slepian processes.
method Derived explicit formula and approximation formula for general continuous boundaries.
result Easy to implement formulas for boundary non-crossing probabilities.
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.
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…
Polytopes in high dimensions have at least 2n+4 normals.
problem Understanding normals to convex polytopes in high dimensions.
method Proved for generic simple polytopes in R^n, n>3.
result Each polytope contains a point with at least 2n+4 normals.
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.
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…
Researchers can reverse-engineer deep ReLU networks from outputs.
problem Recovering a deep ReLU network from its outputs.
method Dissecting region boundaries to identify neuron states and weights.
result Weights and neuron arrangement of a deep ReLU network can be recovered.
Improves SVM speed by 2 orders of magnitude for 12 out of 17 datasets.
problem Slowness of kernel classifiers like SVM for large problems.
method Piecewise linear classifier trained from kernel-based classifier.
result Improves classification speed by up to 2 orders of magnitude.
The number of linear regions in neural networks is far below exponential, even after training.
problem Understanding the expressivity of neural networks, especially those with ReLU activation.
method Mathematical framework to count linear regions and measure their boundaries, analyzing networks at initialization and after training.
result The average number of regions along any one-dimensional subspace grows linearly in the total number of neurons, not exponentially.
Geodesic tomography identifies piecewise constants on convex manifolds.
problem Determining piecewise constant functions on nontrapping manifolds.
method Iterating local uniqueness results based on geodesic integrals.
result Piecewise constant functions are uniquely determined by their geodesic integrals.
This paper considers affine analogues of the isoperimetric inequality in the sense of piecewise linear topology. Given a closed polygon P embedded in R^d having n edges, we give upper and lower bounds for the minimal number of triangles needed to forma triangulated embedded orientable surface in R^d having P as its geo…
Generalizes Poincaré-Hopf Theorem for piecewise smooth boundaries.
problem Conservation law for vector fields on surfaces with piecewise smooth boundaries.
method Generalization of the Poincaré-Hopf Theorem for real-analytic vector fields on surfaces with piecewise smooth boundaries.
result Conservation law for vector fields on surfaces with piecewise smooth boundaries.
The paper extends Riemannian concepts to piecewise linear spaces, defining Einstein metrics and flows.
problem Extending Riemannian concepts to piecewise linear spaces.
method Defining an Einstein vector field and Ricci flows on piecewise linear spaces.
result Criteria for flows to approach Einstein metrics are formulated.
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.
Moduli spaces of hyperbolic surfaces with geodesic boundary components of fixed lengths may be endowed with a symplectic structure via the Weil-Petersson form. We show that, as the boundary lengths are sent to infinity, the Weil-Petersson form converges to a piecewise linear form first defined by Kontsevich. The proof …
Let K be a closed polygonal curve in $\RR^3$ consisting of n line segments. Assume that K is unknotted, so that it is the boundary of an embedded disk in $\RR^3$. This paper considers the question: How many triangles are needed to triangulate a Piecewise-Linear (PL) spanning disk of K? The main result exhibits …
The Immersed Boundary (IB) method is a widely-used numerical methodology for the simulation of fluid-structure interaction problems. The IB method utilizes an Eulerian discretization for the fluid equations of motion while maintaining a Lagrangian representation of structural objects. Operators are defined for transmit…
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.
Smooth structures on manifolds with group actions.
problem Smooth structures on manifolds with group actions.
method Piecewise linear manifolds with group actions, equivariant smoothing.
result Smooth structures compatible with group actions exist.
Reconstructs piecewise constant functions from geodesic integrals.
problem Recovering piecewise constant functions from X-ray data.
method Injectivity proof using variations through geodesics, improved for simple manifolds.
result Explicit formulas for function values near the boundary and stability analysis.
We construct invariants of four-dimensional piecewise-linear manifolds, represented as simplicial complexes, with respect to rebuildings that transform a cluster of three 4-simplices having a common two-dimensional face in a different cluster of the same type and having the same boundary. Our construction is based on t…
Develops piecewise visual, linearly connected metrics on group boundaries.
problem Creating metrics on group boundaries with cut points.
method Graph of groups decompositions and piecewise visual metrics.
result Linearly connected metrics on boundaries with cut points.
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.
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 approximates smooth isotropic surfaces with piecewise linear ones.
problem Approximating smooth isotropic surfaces with piecewise linear ones.
method Using analogies with infinite dimensional moment map geometry, the authors prove the approximation of smooth isotropic immersions by piecewise linear ones.
result Smooth isotropic immersions can be approximated by piecewise linear isotropic maps.
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.
Piecewise-linear regression trees improve tree-based regression with theoretical and practical benefits.
problem Improving tree-based regression models with theoretical guarantees and practical tractability.
method Regularized piecewise-linear node-splitting criterion, LASSO-type and ℓ2 regularization, variable selection procedure. result New high-probability generalization error bounds for piecewise-linear regression trees.
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.
Piecewise-linear approximation improves feature selection in logit models.
problem Improving feature subset selection in sequential logit models.
method Applied piecewise-linear approximation to logistic loss function to frame feature selection as a mixed integer linear optimization problem.
result Piecewise-linear approximation found a better subset of features than quadratic approximation.
Given a Riemann surface with boundary S, the lengths of a maximal system of disjoint simple geodesic arcs on S that start and end at the boundary of S perpendicularly are coordinates on the Teichmueller space T(S). We compute the Weil-Petersson Poisson structure on T(S) in this system of coordinates and we prove that i…
This paper studies PL cobordism categories and their homotopy types.
problem Understanding the homotopy types of PL cobordism categories.
method Introducing a bordism category and showing weak homotopy equivalence to an infinite loop space.
result The classifying space BCdPL is weak homotopy equivalent to an infinite loop space. 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 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.
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.
New approach treats neural networks with piecewise linear activations using tropical geometry.
problem Upper bounds on linear regions of neural networks with ReLU or leaky ReLU activations.
method Treat neural network layers with piecewise linear activations as tropical polynomials, refining upper bounds using tropical geometry.
result Upper bounds on linear regions improved to $\min\left\{ 2^m, \sum_{j=0}^n \binom{m}{j}
ight\}$, where n,m are the number of inputs and outputs, respectively. The theory of geometric structures on a surface with nonempty boundary can be developed by using a decomposition of such a surface into hexagons, in the same way as the theory of geometric structures on a surface without boundary is developed using the decomposition of such a surface into pairs of pants. The basic elem…
Efficiently checks local robustness in neural networks using geometric projections.
problem Ensuring robustness of neural networks against adversarial inputs.
method Systematic search for decision boundaries in convex polyhedral regions using geometric projections.
result Shows geometric projections can efficiently check robustness in neural networks.
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…
Corners can be identified by a drum's sound spectrum.
problem Determining the presence of corners in a drum's shape from its sound.
method Proving spectral invariance of corners in domains with Lipschitz, piecewise smooth boundaries.
result Corners are uniquely determined by a drum's spectrum among domains with fixed genus.
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.
Smooths local volatility model calibration with piecewise linear variance.
problem Calibrating local volatility models to option prices.
method Extends LiptonSepp2011 approach with piecewise linear variance and non-zero interest rates/dividends.
result Analytical tractability with Kummer's hypergeometric functions.
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.
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.
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.
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.
Embeddings in 5D can be isotoped to closed braids.
problem Embeddings of manifolds in high dimensions.
method Piecewise linear co-dimension two embeddings, isotopy to closed braids.
result Embeddings in 5D can be isotoped to closed braids.
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.