Related papers: Error Bounds for Compositions of Piecewise Affine …
This paper demonstrates that the space of piecewise smooth functions can be well approximated by the space of functions defined by a set of simple (non-linear) operations on smooth uniform splines. The examples include bivariate functions…
The problem of piecewise affine (PWA) regression and planning is of foundational importance to the study of online learning, control, and robotics, where it provides a theoretically and empirically tractable setting to study systems…
The analysis of complex nonlinear systems is often carried out using simpler piecewise linear representations of them. A principled and practical technique is proposed to linearize and evaluate arbitrary continuous nonlinear functions using…
In this paper, we investigate a class of non-convex sum-of-ratios programs relevant to decision-making in key areas such as product assortment and pricing, and facility location and cost planning. These optimization problems, characterized…
Given a function dictionary $\cal D$ and an approximation budget $N\in\mathbb{N}^+$, nonlinear approximation seeks the linear combination of the best $N$ terms $\{T_n\}_{1\le n\le N}\subseteq{\cal D}$ to approximate a given function $f$…
We propose two optimization-based heuristics for structure selection and identification of PieceWise Affine (PWA) models with exogenous inputs. The first method determines the number of affine sub-models assuming known model order of the…
In this paper, we propose a simple but effective semantic part-based weighting aggregation (PWA) for image retrieval. The proposed PWA utilizes the discriminative filters of deep convolutional layers as part detectors. Moreover, we propose…
In the context of functional data analysis, probability density functions as non-negative functions are characterized by specific properties of scale invariance and relative scale which enable to represent them with the unit integral…
This paper is concerned with incremental stability properties of nonlinear systems. We propose conditions to compute an upper bound on the incremental L2-gain and to assess incremental asymptotic stability of piecewise-affine (PWA) systems.…
Recent advancements in visualizing deep neural networks provide insights into their structures and mesh extraction from Continuous Piecewise Affine (CPWA) functions. Meanwhile, developments in neural surface representation learning…
We present a dimension-incremental algorithm for the nonlinear approximation of high-dimensional functions in an arbitrary bounded orthonormal product basis. Our goal is to detect a suitable truncation of the basis expansion of the…
Piecewise affine functions are widely used to approximate nonlinear and discontinuous functions. However, most, if not all existing models only deal with fitting continuous functions. In this paper, we investigate the problem of fitting a…
Most of machine learning approaches have stemmed from the application of minimizing the mean squared distance principle, based on the computationally efficient quadratic optimization methods. However, when faced with high-dimensional and…
As demonstrated in many areas of real-life applications, neural networks have the capability of dealing with high dimensional data. In the fields of optimal control and dynamical systems, the same capability was studied and verified in many…
In this paper, we introduce a method known as polynomial frame approximation for approximating smooth, multivariate functions defined on irregular domains in $d$ dimensions, where $d$ can be arbitrary. This method is simple, and relies only…
In this paper, we focus on approximating a natural class of functions that are compositions of smooth functions. Unlike the low-dimensional support assumption on the covariate, we demonstrate that composition functions have an intrinsic…
We study the basic computational problem of detecting approximate stationary points for continuous piecewise affine (PA) functions. Our contributions span multiple aspects, including complexity, regularity, and algorithms. Specifically, we…
Functional principal component analysis (FPCA) is an important technique for dimension reduction in functional data analysis (FDA). Classical FPCA method is based on the Karhunen-Lo\`{e}ve expansion, which assumes a linear structure of the…
Symbolic models have been recently used as a sound mathematical formalism for the formal verification and control design of purely continuous and hybrid systems. In this paper we propose a sequence of symbolic models that approximates a…
We study mixed-integer programming formulations for the piecewise linear lower and upper bounds (in other words, piecewise linear relaxations) of nonlinear functions that can be modeled by a new class of combinatorial disjunctive…