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Bounds on the Number of Pieces in Continuous Piecewise Affine Functions

Leo Zanotti

TL;DR

This work analyzes the relationship between the number of pieces $p$ and the number of affine components $n$ in continuous piecewise affine functions on $\mathbb{R}^d$. It achieves an explicit upper bound $p = O(n^{d+1})$ through a hyperplane-arrangement framework in $\mathbb{R}^{d+1}$ and a projection argument, showing the piece-count growth is polynomial in $n$ with exponent $d+1$. Complementarily, it proves a near-tight lower bound $p = \Omega(n^{d+1-\frac{c}{\sqrt{\log_2(n)}}})$ by leveraging Balogh's monotone-path results in the plane and applying a dimensional lifting via the $m$-sawtooth construction. Collectively, the results nearly close the gap between upper and lower bounds, providing insights into the combinatorial complexity of CPA functions and informing their potential neural-network representations with ReLU activations.

Abstract

The complexity of continuous piecewise affine (CPA) functions can be measured by the number of pieces $p$ or the number of distinct affine functions $n$. For CPA functions on $\mathbb{R}^d$, this paper shows an upper bound of $p=O(n^{d+1})$ and constructs a family of functions achieving a lower bound of $p=Ω(n^{d+1-\frac{c}{\sqrt{\log_2(n)}}})$.

Bounds on the Number of Pieces in Continuous Piecewise Affine Functions

TL;DR

This work analyzes the relationship between the number of pieces and the number of affine components in continuous piecewise affine functions on . It achieves an explicit upper bound through a hyperplane-arrangement framework in and a projection argument, showing the piece-count growth is polynomial in with exponent . Complementarily, it proves a near-tight lower bound by leveraging Balogh's monotone-path results in the plane and applying a dimensional lifting via the -sawtooth construction. Collectively, the results nearly close the gap between upper and lower bounds, providing insights into the combinatorial complexity of CPA functions and informing their potential neural-network representations with ReLU activations.

Abstract

The complexity of continuous piecewise affine (CPA) functions can be measured by the number of pieces or the number of distinct affine functions . For CPA functions on , this paper shows an upper bound of and constructs a family of functions achieving a lower bound of .

Paper Structure

This paper contains 4 sections, 9 theorems, 9 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definition of CPA Functions
  3. Upper Bound
  4. Lower Bound

Key Result

Lemma 2.1

Let $f:\mathds{R}^d\to\mathds{R}$ be a CPA function with $n$ distinct affine components. Let $\lambda$ be the minimum number of convex sets needed to satisfy the conditions of def:CPA. Then,

Figures (3)

  • Figure 1: $f(x,y):=\min(y,\min(\max(-x,-1),\max(3-2x,-x)))$. Left: Piecewise definition of $f$. Right: Slice of $f$ at $y=0$. If the pieces are only required to be closed connected sets, the dashed line $l=\lbrace(x,y):y=-x\rbrace$ can belong to both the red and grey pieces, as $f(x,y)=y=-x$ along this line. Thus, defining $f$ with closed connected sets would require only four pieces, whereas under \ref{['def:CPA']}, at least five pieces are necessary.
  • Figure 2: $3$-sawtooth function.
  • Figure 3: The middle figure shows the pieces of $h(x,t)=\min(f(x),s(t))$, where $f(x)$ and $s(t)$ are plotted along the left and bottom axes, respectively. $h(x,t)=s(t)$ in horizontally patterned pieces and $h(x,t)=f(x)$ in vertically patterned pieces. For the dashed segment $L$, we have $h|_{L}=h|_{P_{k1}}=h|_{P_{l1}}=0.4$. However, no $(x_0,t_0)\in L$ has a neighbourhood $U$ where $h|_U=0.4$, implying that $(x_0,t_0)$ is not contained in any piece with affine component $0.4$.

Theorems & Definitions (21)

  • Definition 1.1
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3
  • Theorem 3.4: balogh2003
  • ...and 11 more