On multistochastic Monge-Kantorovich problem, bitwise operations, and fractals

TL;DR

This paper extends the Monge--Kantorovich problem to multistochastic cases, analyzing well-posedness and providing explicit solutions involving bitwise operations and fractal structures like the Sierpiński tetrahedron.

Contribution

It introduces a novel multistochastic Monge--Kantorovich framework and explicitly solves a key model case using bitwise XOR operations and fractal geometry.

Findings

Optimal transport map involves bitwise XOR operation.

Support of the optimal measure is the Sierpiński tetrahedron.

Dual problem solution is explicitly described.

Abstract

The multistochastic $ (n,k)$-Monge--Kantorovich problem on a product space $\prod_{i=1}^n X_i$ is an extension of the classical Monge--Kantorovich problem. This problem is considered on the space of measures with fixed projections onto $X_{i_1} \times \ldots \times X_{i_k}$ for all $k$-tuples $\{i_1, \ldots, i_k\} \subset \{1, \ldots, n\}$ for a given $1 \le k < n$. In our paper we study well-posedness of the primal and the corresponding dual problem. Our central result describes a solution $\pi$ to the following important model case: $n=3, k=2, X_i = [0,1]$, the cost function $c(x,y,z) = xyz$, and the corresponding two--dimensional projections are Lebesgue measures on $[0,1]^2$. We prove, in particular, that the mapping $(x,y) \to x \oplus y$, where $\oplus$ is the bitwise addition (xor- or Nim-addition) on $[0,1] \cong \mathbb{Z}_2^{\infty}$, is the corresponding optimal transportation. In particular, the support of $\pi$ is the Sierpi\'nski tetrahedron. In addition, we describe a solution to the corresponding dual problem.