Displacement interpolation and Wasserstein geodesics

Let μ₀ and μ₁ belong to P₂ of Euclidean space, and choose a coupling π that minimizes squared distance. For each t between zero and one, map a pair (x, y) to (1 − t)x + ty and let ρₜ be its distribution under π. At the endpoints these pushforwards are μ₀ and μ₁. Finite second moments persist along the segment by the squared triangle bound, so the full path remains in P₂.

No density assumption is needed for this plan-based construction. If the optimal coupling is induced by a map T, the expression reduces to pushing μ₀ forward through (1 − t)id + tT. When a source atom splits, however, the joint pair variable is essential: different fractions of that atom follow different segments. The interpolation contains information about which destinations are matched to which origins; knowing just the two endpoint marginals does not specify an arbitrary intermediate distribution.

Using the same paired particles couples ρₛ with ρₜ. Their displacement is (t − s)(y − x), giving W₂(ρₛ, ρₜ) at most |t − s| times the endpoint distance. This is initially only an upper bound: a coupling of the intermediate distributions need not look optimal merely because it came from a convenient parameterization. The optimality conclusion comes from comparing all pieces with the original endpoints.

For 0 ≤ s ≤ t ≤ 1, apply the metric triangle inequality along μ₀, ρₛ, ρₜ, μ₁. The three upper bounds sum to exactly W₂(μ₀, μ₁). Any strict inequality in one piece would make the total smaller than the endpoint distance, which is impossible. Hence every piece attains its bound. This proves that the constructed curve is a constant-speed geodesic; straight segments with an arbitrary nonoptimal endpoint coupling do not receive this certificate.

The ordinary mixture (1 − t)μ₀ + tμ₁ keeps the endpoint locations and transfers their weights. It is often appropriate for mixture models, but usually differs from displacement interpolation. Between two distinct atoms, displacement interpolation is one moving atom, whereas the mixture contains two stationary atoms with changing masses. Intermediate locations and spreads therefore distinguish the constructions even when their means happen to coincide.

For one-dimensional Gaussians, the increasing affine optimal map interpolates the mean and the standard deviation linearly. Variance is the square of that interpolated standard deviation, not the linear average of endpoint variances. In higher-dimensional Gaussian examples the covariance formula depends on the optimal matrix map and cannot generally be obtained by interpolating matrix entries arbitrarily. This is a useful warning for geometric visualization: a smooth-looking deformation is not automatically a shortest distributional path.