Deep networks with ReLU outperform piecewise linear spline methods in function approximation.
problem Comparing expressive power of deep neural networks with ReLU activation to piecewise linear spline methods.
method Comparison of function approximation capabilities between deep neural networks with ReLU activation and piecewise linear spline methods.
result Deep neural networks with ReLU activation can approximate functions better or only slightly worse than piecewise linear spline methods.
This paper uses linear rational splines for invertible modeling, offering a simpler inverse and similar costs.
problem Creating expressive invertible models with tractable Jacobian determinants.
method Replacing affine transformations with linear rational splines in coupling layers.
result Linear rational splines offer a simpler inverse and similar costs for inference and generation.
We reparametrize ReLU NNs as splines to understand their learning dynamics.
problem Understanding the learning dynamics and inductive bias of neural networks.
method Reparametrize ReLU NNs as continuous piecewise linear splines to study learning dynamics.
result Standard weight initializations yield very flat functions, leading to strength and type of implicit regularization.
Improves BN graph learning with splines for scalability.
problem Learning accurate BN graph structures from data.
method Score-and-search approach with MARS for CPD modeling.
result Improves BN graph accuracy and scalability.
Optimizes deep neural networks using splines and adaptive knots.
problem Improving the optimization of deep neural networks.
method Integrates a second-order total-variation criterion to optimize activation functions, deriving a representer theorem.
result Optimal network configurations can be achieved with nonuniform linear splines with adaptive knots.
Regularized least-squares approaches have been successfully applied to linear system identification. Recent approaches use quadratic penalty terms on the unknown impulse response defined by stable spline kernels, which control model space complexity by leveraging regularity and bounded-input bounded-output stability. T…
We consider the generic regularized optimization problem β^(λ)=argminβL(y,Xβ)+λJ(β). Efron, Hastie, Johnstone and Tibshirani [Ann. Statist. 32 (2004) 407--499] have shown that for the LASSO--that is, if L is squared error loss and J(β)=∥β∥1 is the ℓ1 norm of β--the opti…
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…
Batch normalization improves deep networks by aligning their decision boundaries with data.
problem Improving the performance and generalization of deep networks.
method Theoretical analysis of batch normalization as a function approximation technique for continuous piecewise affine splines.
result Batch normalization adapts the geometry of a deep network's partition to match the data, improving learning and generalization.
ReLU neural networks define piecewise linear functions of their inputs. However, initializing and training a neural network is very different from fitting a linear spline. In this paper, we expand empirically upon previous theoretical work to demonstrate features of trained neural networks. Standard network initializat…
The classical approach to linear system identification is given by parametric Prediction Error Methods (PEM). In this context, model complexity is often unknown so that a model order selection step is needed to suitably trade-off bias and variance. Recently, a different approach to linear system identification has been…
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.
We develop a method to learn neural network activations with controlled Lipschitz constant.
problem Increase neural network capacity while controlling Lipschitz constant.
method Variational framework to learn activation functions with piecewise-linear constraints.
result Proves existence of solutions with continuous and piecewise-linear activations.
The paper develops a new method for estimating non-parametric regression functions with spatio-temporal dependencies.
problem Estimating non-parametric regression functions with spatio-temporal dependencies.
method Locally Adaptive Regression Splines (LARS) with ADMM algorithm.
result The method shows superior performance compared to existing techniques.
We study additive models built with trend filtering, i.e., additive models whose components are each regularized by the (discrete) total variation of their kth (discrete) derivative, for a chosen integer k≥0. This results in kth degree piecewise polynomial components, (e.g., k=0 gives piecewise constant co…
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.
Kronecker trend filtering improves lattice data smoothing.
problem Estimating smooth functions on lattice data.
method Penalized least squares with Kronecker products of univariate trend filtering penalties.
result Kronecker trend filtering outperforms linear smoothers in high dimensions.
A new method for signal processing using piecewise convex fitting.
problem Nonparametric function estimation in signal processing.
method Two-stage adaptive estimate with strong smoothing and constrained smoothing spline fit.
result Piecewise convex fitting reduces MSE and accurately estimates change points.
We study trend filtering, a recently proposed tool of Kim et al. [SIAM Rev. 51 (2009) 339-360] for nonparametric regression. The trend filtering estimate is defined as the minimizer of a penalized least squares criterion, in which the penalty term sums the absolute kth order discrete derivatives over the input points…
Sig-Splines model uses signatures and splines for time series data, achieving universality and convexity.
problem Creating a generative model for multivariate time series data.
method Combines linear transformations and signature transforms into a neural spline flow.
result Achieves universality and introduces convexity in model parameters.
Cubic spline interpolation on Euclidean space is a standard topic in numerical analysis, with countless applications in science and technology. In several emerging fields, for example computer vision and quantum control, there is a growing need for spline interpolation on curved, non-Euclidean space. The generalization…
Multivariate splines linked to infinitely-wide neural networks with improved numerical performance.
problem Understanding the relationship between multivariate splines and neural networks.
method Showed multivariate splines can be represented as random features in infinitely-wide neural networks with a homogeneous activation function.
result The function space of multivariate splines is a Sobolev space on a Euclidean ball with explicit norm bounds on derivatives.
Study reconstructs Faber-Schauder coefficients from antiderivative observations.
problem Reconstructing Faber-Schauder coefficients from discrete antiderivative observations.
method Piecewise quadratic spline interpolation and closed-form solution.
result Final-generation coefficients are unstable; others are robust.
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.
RUMBoost combines RUMs and deep learning for better choice modelling.
problem Creating interpretable and robust discrete choice models.
method Gradient Boosted Regression Trees for utility functions, with constraints for interpretability and monotonicity.
result RUMBoost outperforms ML and RUM benchmarks in predictive performance and interpretability.
Paper extends understanding of deep network nonlinearities using vector quantization and statistical inference.
problem Limited understanding of deep network nonlinearities, especially non-piecewise affine and non-convex functions.
method Link deterministic max-affine spline operators to probabilistic Gaussian Mixture Models (GMMs) for a broader class of nonlinearities.
result Enforces orthogonality in linear filters can significantly improve deep network performance.
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.
Cubic-Spline Flows improve autoregressive flow performance in density estimation.
problem Improving the performance of flow-based models in density estimation.
method Stacking a new coupling transform based on monotonic cubic splines with LU-decomposed linear layers.
result Cubic-Spline Flows close the gap with autoregressive flows on density-estimation tasks.
A new spline method for manifold learning using Hessian-based curvature penalties.
problem Learning manifolds with curvature penalties in high dimensions.
method Generalizes thin-plate splines to flat manifolds using Hessian matrices, minimizing square error with curvature constraints.
result Existence and uniqueness of the spline solution, expressed as Green's functions and Hessian approximations.
This paper develops a new method for constructing splines on Lie groups using Poisson equation solutions.
problem Existing methods for constructing splines on Lie groups have limitations and assumptions that may not reflect actual curves.
method The paper introduces a new approach using solutions of the Poisson equation on Lie groups to construct splines.
result The new method allows for global splines with arbitrary initial conditions, improving curve reconstruction.
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. 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.
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.
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 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.
Let G be a connected compact Lie group acting on a manifold M and let D be a transversally elliptic operator on M. The multiplicity of the index of D is a function on the set of irreducible representations of G. Let T be a maximal torus of G with Lie algebra Lie(T). We construct a finite number of piecewise polynomial …
New framework explains deep neural networks using variational spline theory.
problem Understanding functions learned by deep neural networks.
method Developed a variational framework and function space.
result Deep ReLU networks are solutions to regularized data fitting problems over the proposed function space.
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
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…
Temporal Functional Circuits explain KAN forecasts with interpretable edge functions.
problem Lack of mechanistic explanations in KAN forecasting.
method Transform KAN edge functions into faithful, temporally grounded explanations using a gated residual KAN.
result Gated KAN achieves lower MSE than linear-only models on regime-switching signals.
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.
We study a novel spline-like basis, which we name the "falling factorial basis", bearing many similarities to the classic truncated power basis. The advantage of the falling factorial basis is that it enables rapid, linear-time computations in basis matrix multiplication and basis matrix inversion. The falling factoria…
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.
Dropout improves regularization in flexible models for rare features.
problem Understanding theoretical properties of dropout in generalized linear models.
method Theoretical analysis and application to adaptive smoothing with B-splines.
result Dropout prefers rare features in mean and dispersion parameters.
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.
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^{…