Sequence distortion for metric spaces

What the math gives to ML

The paper introduces a coarse metric notion for the distance profile of an indexed sequence: pairwise distances between positions separated by N should grow like a prescribed rate f(N), up to multiplicative and additive distortion. This gives a concrete way to regularize hidden-state trajectories so that nearby positions remain locally similar while distant positions do not collapse or become arbitrarily separated. The Euclidean classification is especially relevant: an N^alpha profile requires alpha>1/k in k-dimensional Euclidean state space, suggesting a dimension-aware choice of trajectory regularizer and a diagnostic for insufficient representation width. The transfer is best tested as an auxiliary loss on recurrent or Transformer hidden states, rather than as a claim that every learned representation should satisfy a fixed geometry.