Error Bounds for Compositions of Piecewise Affine Approximations

TL;DR

This work tackles efficient approximation of nonlinear functions via piecewise affine (PWA) representations by leveraging functional decomposition to express multivariate functions as compositions of unary/binary components. It advances two breakpoint-placement strategies: (i) an optimization-free bisection method (Method 1) and (ii) an algebraic, curvature-based method (Method 2) that uses a closed-form bound to adapt breakpoint locations while meeting a specified tolerance. The authors then derive rigorous error propagation bounds for compositions of PWA approximations, including a principal bound $\varepsilon_{f\circ g} = \tau_f + d_{\bar{f}}^\top \varepsilon_g$ and extensions to PWA and unary cases, enabling P1/P2 formulations that trade off tolerance against complexity. Collectively, these results enable scalable construction of PWA approximations with guaranteed accuracy, offering practical impact for nonlinear analysis and control, and enabling applications in neural networks, reachability, and set-valued estimation.

Abstract

Nonlinear expressions are often approximated by piecewise affine (PWA) functions to simplify analysis or reduce computational costs. To reduce computational complexity, multivariate functions can be represented as compositions of functions with one or two inputs, which can be approximated individually. This paper provides efficient methods to generate PWA approximations of nonlinear functions via functional decomposition. The key contributions focus on intelligent placement of breakpoints for PWA approximations without requiring optimization, and on bounding the error of PWA compositions as a function of the error tolerance for each component of that composition. The proposed methods are used to systematically construct a PWA approximation for a complicated function, either to within a desired error tolerance or to a given level of complexity.

Figures (6)

• Figure 1: Construction of PWA approximation of $y=\sin (x)$ using Method 1. (a) Bisection to determine $x_{2}$. (b) Bisection to determine $x_3$. (c) The algorithm concludes with a breakpoint at the upper limit of the domain $\Bar{x}=2\pi$, for which the the error tolerance is satisfied without requiring iteration.
• Figure 2: Comparison of Method 1 and Method 2 for functions and domains given in Table \ref{['tab:M1vM2_SOS_FuncAndDomain']}. Figure \ref{['fig:M1vM2_SOS_TimeAndBreak']}(a) and Figure \ref{['fig:M1vM2_SOS_TimeAndBreak']}(b) show the number of breakpoints and computation time as a function of tolerance, respectively. Method 1 is significantly slower to compute than Method 2, but constructs an approximation to a specified error tolerance using fewer breakpoints.
• Figure 3: Number of PWA approximation breakpoints required to meet a specified tolerance for the function $\sin(x),\ x\in[0,2\pi]$ using Method 1. Dashed lines indicate where the number of breakpoints "jumps".
• Figure 4: Graphs of the decomposed functions and their PWA approximation error bounds from Example \ref{['ex:ErrProp_affOnly']}.
• Figure 5: Using Corollary \ref{['cor:MVT_err_bound_PWA']} yields a more conservative (less tight) error bound than Corollary \ref{['cor:PWA_err_bound']}, but with the benefit of only depending on the original functions.