1D Wasserstein distance (Earth Mover's Distance) · W_1(μ,ν)

Read the equation

This relation defines W_1(μ,ν) in the source section “1D Wasserstein distance (Earth Mover's Distance) · W_1(μ,ν)”. Integrates contributions over the variable and bounds stated in the expression; the measure and domain must come from this source. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or cognitive meaning to them.

Terms & definitions

The recovered passage does not supply a separate symbol table. Read the source context for the notation used in this version; familiar letters may carry different meanings in other builds.

How it works

1. Read W_1(μ,ν) as the quantity being defined or updated by this relation.

2. Integrates contributions over the variable and bounds stated in the expression; the measure and domain must come from this source.

3. Use this version’s variable definitions and update order. The same symbol may be assigned a different meaning in another build.

Assumptions & limits

• This source expression was recovered from the pinned code revision. Its presence does not establish that the associated program was built, run, or scientifically validated.

Original context

// ═══════════════════════════════════════════════════════════════════════════════
// IV. OPTIMAL TRANSPORT
// ═══════════════════════════════════════════════════════════════════════════════

/**
 * @brief 1D Wasserstein distance (Earth Mover's Distance)
 * W_1(μ,ν) = ∫|F_μ^{-1}(t) - F_ν^{-1}(t)| dt
 */
inline double wasserstein_1d(const double* p, const double* q, size_t n) {
    // Compute CDFs
    std::vector<double> cdf_p(n), cdf_q(n);
    cdf_p[0] = p[0];
    cdf_q[0] = q[0];
    for (size_t i = 1; i < n; ++i) {
        cdf_p[i] = cdf_p[i-1] + p[i];
        cdf_q[i] = cdf_q[i-1] + q[i];
    }

W_1(μ,ν) = ∫|F_μ^{-1}(t) - F_ν^{-1}(t)| dt

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