[Paper Review] The Wasserstein distance between stationary measures associated to iterated function schemes on the unit interval

Italo Cipriano|arXiv (Cornell University)|Nov 1, 2016

Geometric Analysis and Curvature Flows36 references3 citations

TL;DR

This paper provides explicit formulas and tight approximations for the first-order Wasserstein distance between stationary measures generated by iterated function systems (IFS) on the unit interval. It analyzes cases with k positive contractions, two contractions of opposite sign, and different IFS configurations, leveraging properties of Cantor staircases and optimal transport to derive closed-form solutions under specific weight and contraction conditions.

ABSTRACT

We provide explicit formulaes for the first Kantorovich-Wasserstein distance between stationary measures for iterated function scheme on the unit interval. In particular, we consider two stationary measures with different configurations of the weights associated to the same iterated function schemes with disjoint images composed of: $k$ positive contractions or $2$ contractions of different sign. We also study the case of two stationary measures associated to different iterated function schemes.

Motivation & Objective

• To derive explicit formulas for the 1-Wasserstein distance between stationary measures of iterated function systems (IFS) on the unit interval.
• To extend prior work on Wasserstein distances between stationary measures by considering IFS with k ≥ 2 positive contractions and disjoint images.
• To analyze the case of two contractions of opposite sign (positive and negative) with disjoint images, under symmetry conditions on weights.
• To study the Wasserstein distance between stationary measures associated with different IFS, not just different weights on the same IFS.
• To provide a framework for estimating Wasserstein distances in fractal and stochastic settings using optimal transport theory.

Proposed method

• Utilizes the Kantorovich-Wasserstein distance formulation as the infimum of expected distance over couplings of random variables with given laws.
• Applies the theory of optimal transport to stationary measures of IFS, focusing on the 1-Wasserstein distance (W₁).
• Employs recursive structure of invariant measures and Cantor staircase functions to characterize cumulative distribution functions of stationary measures.
• Derives explicit formulas under specific conditions: affine contractions with disjoint images and symmetric weight configurations.
• Uses geometric and measure-theoretic analysis of intersections of Cantor staircases to bound the Wasserstein distance.
• Establishes approximation bounds via Lemma 3.13 and applies them to non-affine and non-symmetric cases.

Experimental results

Research questions

• RQ1 What is the explicit formula for the 1-Wasserstein distance between stationary measures of an IFS with k ≥ 2 positive, disjoint-imagery contractions and different weight vectors?
• RQ2 How does the 1-Wasserstein distance behave between stationary measures of an IFS with one positive and one negative contraction, under symmetric weight conditions?
• RQ3 Can a closed-form expression be derived for the Wasserstein distance between stationary measures of two distinct IFS with disjoint images?
• RQ4 What role do intersections of non-classical Cantor staircases play in estimating Wasserstein distances?
• RQ5 How do the results generalize to IFS with more than two contractions or asymmetric weight distributions?

Key findings

• An explicit formula for $ W_1( u_p, u_q) $ is derived when the IFS consists of k positive affine contractions with disjoint images and under specific weight conditions.
• For two-contraction IFS with one positive and one negative contraction and disjoint images, an explicit formula is obtained under symmetric weight assumptions.
• A good approximation of $ W_1( u_p, u_q) $ is provided for k-positive-contraction IFS with disjoint images, even beyond affine cases.
• The Wasserstein distance between stationary measures of two different IFS is approximated when both have disjoint-imagery, positive contractions and satisfy certain structural conditions.
• The method relies on analyzing intersections of Cantor staircases, which are non-monotonic and non-smooth, to bound the transport cost.
• Theoretical bounds are established via Lemma 3.13, which are validated numerically in examples with non-integer contraction parameters and asymmetric weights.