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QUESTION-293: Is the distortion of a composite of approximate transports bounded by the sum of its parts' distortions, and does that additive bound survive when the distortion lacks a triangle inequality or a later kernel stretches distances?

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Properties5
StatusAnswered
Tagsmathematics, philosophy-of-language
Linepragmatic-transport
RefinesQUESTION-017 — Where the additive bound fails, what bound on accumulated drift holds when distortion is directed and need not satisfy the triangle inequality?
Cited as “answers” byCLAIM-260 — Approximate transports compose into a category only if the distortion obeys a triangle inequality and later kernels are nonexpansive, so the semigroup claim's "category or semigroup" holds for exact kernels, and for approximate ones only under those conditions

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The question

A50 §2 bounded drift by adding the steps’ distortions, assuming a metric. QUESTION-017 asks what bound holds when that assumption is dropped. This question asks the first part of that: whether the additive bound itself holds, and under which of the metric’s properties.

The answer

CLAIM-260, Active and demonstrated, settles it:

  • With a triangle inequality the composite’s distortion is at most Lε₁ + ε₂, where L is how far the later kernel stretches distances. It is additive when L ≤ 1. In total variation both conditions hold, since it is a metric and every Markov kernel is nonexpansive in it.
  • Without a triangle inequality the additive bound can fail. Under the squared difference of Bernoulli parameters, or squared Hellinger distance, two steps of 1/4, or about 0.29, compose to a distortion of 1.
  • With an expansive kernel it fails too. In Wasserstein-1 on the line, x ↦ 2x makes the composite only a (2ε₁ + ε₂)-transport.

What weaker bound holds where the additive one fails, and what holds for directed distortion, remain QUESTION-017’s.

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