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.
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.
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.
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.
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.
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.
Optimizes piecewise local-linear approximations for global model understanding.
problem Global model behavior interpretation for black-box models.
method Dynamic programming framework for piecewise local-linear approximations with fidelity guarantees.
result Polynomial time algorithm for optimal clustering.
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.
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.
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.
This paper proposes a distributed Bayesian method for piecewise sparse linear models.
problem High computational cost in simultaneous model selection for piecewise linear models.
method Distributed factorized asymptotic Bayesian (FAB) inference on distributed memory architectures.
result Achieves high prediction accuracy and performance scalability.
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. …
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. 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.
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.
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.
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.
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…
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.
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.
This paper concerns a method of selecting a subset of features for a sequential logit model. Tanaka and Nakagawa (2014) proposed a mixed integer quadratic optimization formulation for solving the problem based on a quadratic approximation of the logistic loss function. However, since there is a significant gap between …
Simplifies PLNNs to interpretable models for better explainability.
problem Challenges in interpretability of PLNNs for high-stakes applications.
method Trained deep network simplification and algorithm for reducing flat networks.
result Improved interpretability of PLNNs without sacrificing performance.
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.
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. 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…
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.
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.
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.
BART and MOTR-BART improve tree-based predictions with local linear models.
problem Non-linearity and high-order interactions in data.
method Bayesian Additive Regression Trees (BART) and Model Trees BART (MOTR-BART) using piecewise linear functions.
result MOTR-BART achieves equal or better performance with fewer trees than BART.
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.
Proposes new models to solve portfolio selection with cardinality constraints using factor models.
problem Solving portfolio selection with cardinality constraints using factor models.
method Developed 0-1 linear models and a minimum edge-weighted clique problem to solve the cardinality constrained portfolio problem.
result Piecewise linear approximation reduces computation time for solving the quadratic problem.
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.
This paper tightens bounds on linear regions in piecewise linear neural networks.
problem Understanding the expressiveness of piecewise linear neural networks.
method Analyzes the maximal number of linear regions in single and multi-layer PLNNs, and rectifier networks.
result Derives tighter bounds on the maximal number of linear regions in PLNNs and rectifier networks.
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.
A new tree-based model for multivariate responses interprets piecewise linear regimes.
problem Recovering piecewise multivariate linear regimes in complex data.
method Twoblock clustering trees with coskewness-based dimension reduction.
result Recovery of piecewise linear regimes in data.
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.
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.
Open and discrete maps with specific branch set images are equivalent to PL branched covers.
problem Understanding the equivalence of open and discrete maps and PL branched covers.
method Demonstrated that an open and discrete map f:SnoSn with a specific branch set image is equivalent to a PL branched cover up to homeomorphism. result Open and discrete maps with a specific branch set image are equivalent to PL branched covers.
New algorithms improve neural network verification by exploiting piecewise linear structure.
problem Efficiently verify the correctness of neural networks, especially those with high-dimensional inputs.
method Branch-and-Bound (BaB) framework applied to Mixed Integer Linear Programming (MIP) formulation.
result Significant performance improvements and new branching strategies for neural networks.
Algorithm identifies best arm in piecewise stationary linear bandits with minimal samples.
problem Identifying the best arm in a piecewise stationary linear bandit model with unknown contexts and changepoints.
method Design of PSεBAI+ algorithm, consisting of PSεBAI and NεBAI subroutines. result PSεBAI+ achieves optimal sample complexity up to a logarithmic factor. Paper proposes LANN to measure model complexity of neural networks with curve activation functions.
problem Measuring model complexity of neural networks with curve activation functions.
method Proposes LANN, a piecewise linear framework to approximate curve activation functions, and derives complexity measure based on the number of linear regions.
result Demonstrates positive correlation between overfitting and model complexity during training.
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…
Unified piecewise-linear Ricci flows improve community detection.
problem Improving community detection in graph neural networks.
method Proposed piecewise-linear Ricci curvature flows with surgeries.
result Flow consistently outperforms baseline models on real-world datasets.