The standard n-simplex <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Delta ^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="normal">Δ</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> serves as a natural geometric space for representing discrete probability distributions and compositional data. Quantifying dissimilarity between points <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$P, Q \in \Delta ^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mo>,</mo> <mi>Q</mi> <mo>∈</mo> <msup> <mi mathvariant="normal">Δ</mi> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> is essential, with metrics respecting ordinal structures being particularly valuable. The 1-Wasserstein distance <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$W_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>W</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> excels in this regard but lacks a direct geometric interpretation as a path length on the simplex. This paper addresses this gap by defining a canonical path from P to Q in an “outside-in” recursive manner using dimension-reducing projections. We prove that the length of this path, under a weighted Euclidean metric with weights <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textbf{a} = (a_1, \dots , a_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">a</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>a</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> encoding ordinal spacings, equals the weighted 1-Wasserstein distance <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$W_{1,\textbf{a}}(P, Q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi mathvariant="bold">a</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, we situate this result within Finsler geometry, demonstrating that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$W_{1,\textbf{a}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi mathvariant="bold">a</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> is the geodesic distance induced by a continuous Finsler metric on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Delta ^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="normal">Δ</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>. The canonical path thus emerges as a non-smooth length-minimizing geodesic.

A dyadic path construction as a geometric realization of the 1-Wasserstein distance on the simplex

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2026年5月20日

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Beniamino Cappelletti-Montano,
Monica Musio

摘要

The standard n-simplex $\Delta ^n$ $Δ^{n}$ serves as a natural geometric space for representing discrete probability distributions and compositional data. Quantifying dissimilarity between points $P, Q \in \Delta ^n$ $P, Q \in Δ^{n}$ is essential, with metrics respecting ordinal structures being particularly valuable. The 1-Wasserstein distance $W_1$ $W_{1}$ excels in this regard but lacks a direct geometric interpretation as a path length on the simplex. This paper addresses this gap by defining a canonical path from P to Q in an “outside-in” recursive manner using dimension-reducing projections. We prove that the length of this path, under a weighted Euclidean metric with weights $\textbf{a} = (a_1, \dots , a_n)$ $a = (a_{1}, \dots, a_{n})$ encoding ordinal spacings, equals the weighted 1-Wasserstein distance $W_{1,\textbf{a}}(P, Q)$ $W_{1, a} (P, Q)$ . Furthermore, we situate this result within Finsler geometry, demonstrating that $W_{1,\textbf{a}}$ $W_{1, a}$ is the geodesic distance induced by a continuous Finsler metric on $\Delta ^n$ $Δ^{n}$ . The canonical path thus emerges as a non-smooth length-minimizing geodesic.