Measuring overlap, or measuring displacement – KL and Wasserstein are not competitors.
Paper 137 wrote that the KL divergence is not a distance. This paper sets its counterpart, the Wasserstein distance, beside it, and writes that the two are not competitors but answer different questions. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED. Monge, Kantorovich, Otto’s gradient flow and the Fournier-Guillin rates are all standard. A word on the numerics: in one dimension the transport cost is computed exactly by sorting, and in two dimensions and above by log-domain Sinkhorn with entropic regularisation, whose implementation was validated against the exact one-dimensional answer (relative error 0.1 to 1.1 per cent). The sliced Wasserstein distance was deliberately not used, since it averages one-dimensional projections, its rate is dimension-free, and it would have hidden the very effect under test.
First, the Wasserstein distance is a metric. Over three thousand triples of samples drawn from Gaussians on the line, symmetry and the triangle inequality were tested, and there were zero asymmetries and zero failures of the triangle inequality. The KL divergence of Paper 137 failed the same test 25.4 per cent of the time. Wasserstein therefore satisfies the metric axioms and gives a consistent ordering of which is further from which, exactly what KL could not give.
Second, and this is the core. What separates the two is disjoint support. Shifting a Gaussian of fixed width by a distance, KL grows as half the square of that distance and the second Wasserstein distance grows as the distance itself, and the numerics here agree exactly with the closed forms (12.500000 against 5 at a separation of five). While the supports overlap, in other words, both see the separation. But for two point masses a given distance apart, KL is infinite whether the distance is one half, one, five or a hundred, and cannot tell them apart, whereas the second Wasserstein distance returns the distance itself. So the question of how different two distributions are is in fact two questions: do they overlap, and how far apart are they. KL answers the first and Wasserstein the second, and the single word apartness has been lumping them together (Paper 17).
Third, Wasserstein is nevertheless not the better object; it asks for more. Its definition contains the distance on the underlying space, and the definition of KL does not. For distributions over three unordered categories, KL is well defined at 0.223805, while Wasserstein cannot be formed until someone says how far red is from green. It requires more input, and where that input does not exist it cannot be formed. This is not a ranking but two tools with two prerequisites (Paper 95).
Fourth, the heat equation can be written twice. The heat kernel started from a point is a Gaussian of zero mean whose variance is twice the elapsed time, so its second Wasserstein distance from the point mass is the square root of twice the time. At times of 0.25, 1 and 4 this paper obtains 0.708606, 1.414860 and 2.829112, against theoretical values of 0.707107, 1.414214 and 2.828427. This square-root behaviour means that time has the dimension of length squared, which is exactly the parabolic scaling Paper 119 wrote for its fourth clock. And by Otto’s theorem the heat equation is the gradient flow of entropy in this metric: one equation written twice, as a partial differential equation and as steepest descent, the second reading having meaning only once this metric is in place. The corpus contains the same shape, since Paper 110 wrote that Ricci flow is a renormalisation-group flow; one equation with two descriptions happened there too, at a different address, there the string worldsheet and here the space of probability measures.
Fifth, the rate of convergence changes character in three dimensions. Measuring the Wasserstein distance between two samples of the same standard normal, the true value is zero, so what is measured is pure estimation error. A fourfold sample improves the estimate by a factor of 2.18 in one dimension, 2.00 in two, 1.59 in three, 1.39 in five and 1.22 in eight, against predictions of 2.00, 2.00, 1.59, 1.32 and 1.19. What matters here is that the rate is not uniformly the reciprocal of the dimension. It is the reciprocal of the square root up to two dimensions, the central-limit rate, and the reciprocal of the dimension from three, the covering rate. So the curse of dimensionality begins at three, where the reciprocal of the dimension drops below one half, and the two rates change places there. Paper 114 separated facts that select a dimension from facts that do not, and this three is on the selecting side, though what is selected is not a geometry but which of two rates wins. It shares its number with the three of James-Stein in Paper 133 while having a different root, there admissibility and here covering numbers. And satisfying the metric axioms does not make a quantity easy to estimate: the shape seen in Paper 136 returns, and being right and being measurable are different.
IMPORTANT (scope): a census of established standard material (Layer 2). Every work cited is standard. What is not claimed: any new mathematical theorem (there is none) / any new physical law (there is none) / that Wasserstein is better than KL (it is not called better; the prerequisites differ) / that KL is in error (it is not; it answers a different question). What is added: counting the metric properties beside KL’s, writing disjoint support as the separator, writing the asymmetry of prerequisites, and checking the two regimes of the convergence rate numerically. What remains: KL and Wasserstein are not in competition. One measures overlap and the other measures displacement, and which to use is decided by which of the two questions one is holding.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). Measuring Overlap, or Measuring Displacement [Paper 139]. Zenodo.
DOI: 10.5281/zenodo.22081801
ORCID: 0009-0006-8587-3450