Schatten spectral transport — interactive experiment

Numerical experiment · transport between point clouds

Change the spectral gauge, change the transport

Two irregular, multimodal point clouds, with equal mass at every point. Change the exponent to see how the coupling balances its directional costs, then move time to follow the transported particles.

γ_p(S) = (λ₁^p + λ₂^p)^1/pγ_∞(S) = max(λ₁, λ₂)

Schatten exponent p 2.00 ½ favors low rank · 1 is quadratic OT Points per cloud n 48 Each point has mass 1/n; sampling is reproducible. Interpolation time t 0.50 Move along the computed transport segments. Use p = ∞: balance the worst direction

Coupling and displacement interpolation

SourceTargetAt time t

Edge opacity and thickness indicate mass; green particle area is proportional to mass.

Displacement second moment S

Ellipse semiaxes: √λ₁ and √λ₂. This is the uncentered second moment of displacement, not a centered covariance. Its scale stays fixed when p changes; the two plots use independent spatial scales.

Transport length √γ_p(S_π): —
Objective γ_p(S_π): —
Eigenvalues λ₁, λ₂: —
Nonzero transport edges: —

Computing the transport coupling…

No entropic regularization is added. For p ≥ 1, the displayed gap bounds the remaining error in the objective, not in the coupling; these bounds are evaluated in floating-point arithmetic. For ½ ≤ p < 1, the displayed value is only a numerical upper bound on the minimum: local search is not a global certificate. The time slider only moves particles; it does not solve a new transport problem.