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.
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48 results for “piecewise regular functions”
For a bounded domain equipped with a piecewise Lipschitz continuous Riemannian metric g, we consider harmonic map from to a compact Riemannian manifold without boundary. We generalize the notion of stationary harmonic map and prove the partial regularity. We also discuss the global Li…
New definition of regular points for PL functions on manifolds.
problem Defining regular points for PL functions on combinatorial manifolds.
method Definition based on link of the point, stratification of Jacobi set, Stein factorization of Reeb space.
result Our definition of regularity is distinct from existing definitions.
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.
Identifiability of Potentially Degenerate Gaussian Mixture Models With Piecewise Affine Mixingstat.ML
Method identifies latent variables from high-dimensional data with piecewise affine mixing.
problem Identifying latent variables from high-dimensional observations with dependencies and piecewise affine transformations.
method Proposes a two-stage method with sparsity and Gaussianity regularization.
result Effectively recovers ground-truth latent variables from synthetic and image data.
Piecewise constant denoising can be solved either by deterministic optimization approaches, based on the Potts model, or by stochastic Bayesian procedures. The former lead to low computational time but require the selection of a regularization parameter, whose value significantly impacts the achieved solution, and whos…
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.
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…
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.
We solve ElasticNet regularization tuning across multiple instances with provable guarantees.
problem Tuning ElasticNet regularization coefficients across multiple problem instances.
method Characterized ElasticNet loss as a piecewise-rational function, derived structural complexity bounds, and showed generalization and online learning guarantees.
result First general learning-theoretic guarantees for ElasticNet tuning without strong data distribution assumptions.
We study the Ollivier-Ricci curvature of graphs as a function of the chosen idleness. We show that this idleness function is concave and piecewise linear with at most linear parts, with at most linear parts in the case of a regular graph. We then apply our result to show that the idleness function of the Cartes…
This paper uses the technology of weighted and regular triangulations to study discrete versions of the Laplacian on piecewise Euclidean manifolds. Regular triangulations are studied in some detail, including flip algorithms. The Laplacian is then studied as an operator on functions of the vertices as a generalized wei…
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 …
We introduce a new multi-dimensional nonlinear embedding -- Piecewise Flat Embedding (PFE) -- for image segmentation. Based on the theory of sparse signal recovery, piecewise flat embedding with diverse channels attempts to recover a piecewise constant image representation with sparse region boundaries and sparse clust…
We consider the generic regularized optimization problem . Efron, Hastie, Johnstone and Tibshirani [Ann. Statist. 32 (2004) 407--499] have shown that for the LASSO--that is, if is squared error loss and is the norm of --the opti…
We study additive models built with trend filtering, i.e., additive models whose components are each regularized by the (discrete) total variation of their th (discrete) derivative, for a chosen integer . This results in th degree piecewise polynomial components, (e.g., gives piecewise constant co…
Proposed by Donoho (1997), Dyadic CART is a nonparametric regression method which computes a globally optimal dyadic decision tree and fits piecewise constant functions in two dimensions. In this article we define and study Dyadic CART and a closely related estimator, namely Optimal Regression Tree (ORT), in the contex…
Hi-fi priors enhance BNNs by learning flexible activations.
problem Challenging to impose function-space priors on BNNs.
method Optimization techniques to learn flexible activations.
result BNNs with flexible activations can achieve desired priors.
Neural networks can represent complex piecewise functions efficiently.
problem Representing continuous piecewise affine functions with neural networks.
method Two hidden layers with ReLU activation, neurons for pieces.
result CPA functions can be represented by a neural network with linear size.
We introduce a family of variational functionals for spinor fields on a compact Riemann surface that can be used to find close-to-conformal immersions of into in a prescribed regular homotopy class. Numerical experiments indicate that, by taking suitable limits, minimization of these functionals …
Functional data analysis involves data described by regular functions rather than by a finite number of real valued variables. While some robust data analysis methods can be applied directly to the very high dimensional vectors obtained from a fine grid sampling of functional data, all methods benefit from a prior simp…
We study ray transforms on spherically symmetric manifolds with a piecewise metric. Assuming the Herglotz condition, the X-ray transform is injective on the space of functions on such manifolds. We also prove injectivity results for broken ray transforms (with and without periodicity) on such manifolds …
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.
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 achieved under one millisecond.
We study the necessary and sufficient complexity of ReLU neural networks---in terms of depth and number of weights---which is required for approximating classifier functions in . As a model class, we consider the set of possibly discontinuous piecewise functions $f : [-1/2, 1/2]^…
Introduces HTV to measure function complexity in learning schemes.
problem Assessing the complexity of supervised-learning schemes.
method Defines Hessian-Schatten total variation (HTV) as a seminorm to quantify function complexity.
result HTV is invariant to rotations, scalings, and translations, and its minimum value is achieved for linear mappings.
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.
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 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.
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…
In the paper, we consider the rigidity problem of the infinite hexagonal triangulation of the plane under the piecewise linear conformal changes introduced by Luo in [5]. Our result shows that if a geometric hexagonal triangulation of the plane is PL conformal to the regular hexagonal triangulation and all inner angles…
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 -norms.
result Linear functionals of signatures are dense in -norms under an integrability condition.
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.
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.
Numerical Generalized Randomized Hamiltonian Monte Carlo for piecewise smooth target densitiesstat.CO
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.
A piecewise flat manifold is a triangulated manifold given a geometry by specifying edge lengths (lengths of 1-simplices) and specifying that all simplices are Euclidean. We consider the variation of angles of piecewise flat manifolds as the geometry varies in a particular way, which we call a conformal variation. This…
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.
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.
In this paper, we consider a generic probabilistic discriminative learner from the functional viewpoint and argue that, to make it learn well, it is necessary to constrain its hypothesis space to a set of non-trivial piecewise constant functions. To achieve this goal, we present a scalable unsupervised regularization f…
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.
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.
Study identifies change points in piecewise constant reward functions with fixed exploration budget.
problem Locating abrupt changes in piecewise constant reward functions under bandit feedback.
method Fixed exploration budget, piecewise constant bandit problem, lower bounds, near optimal algorithms.
result Established lower bounds and near matching upper bounds for both small and large budgets.
A new algorithm for faster model selection in twin multi-class SVM.
problem Challenges in effective solution of multi-classification and fast model selection in twin multi-class SVM.
method Sample data set partition strategy, Lagrangian multipliers, piecewise linear update, initialization algorithm, and event-based iteration.
result Comparable classification performance achieved without solving quadratic programming problems.
New insights into Deep Autoencoders for better data approximation and generalization.
problem Understanding and improving generalization of deep learning models with more parameters than data.
method Interpreting Deep Autoencoders' structure and using Lie group theory for regularization.
result Regularizations enable Deep Autoencoders to better approximate data manifolds and generalize.
The paper analyzes a simple neural network model with algebraic methods.
problem Finding minima of a ridge-regularized mean squared error for ReLU perceptrons.
method Developed a Divide-Enumerate-Merge strategy using computational algebra.
result Identifies both isolated and connected minima of the RR-MSE.
The double tetrahedron is the triangulation of the three-sphere gotten by gluing together two congruent tetrahedra along their boundaries. As a piecewise flat manifold, its geometry is determined by its six edge lengths, giving a notion of a metric on the double tetrahedron. We study notions of Einstein metrics, consta…