[tech_ken]

But this argument seems analogous to someone saying that average height is less “correct” than the full original data set because every individual’s height is different. In one sense it’s not wrong, but it kind of misses the point of averaging. The full local metric tensor defined on the manifold is going to have the same “complexity” as the manifold itself; it’s a bad way of summarizing a model because it’s not any simpler than the model. Their approach is to average that metric tensor over the region of the manifold swept out by the training data, and they show that this average empirically reflects something meaningful about the underlying response manifold in problems that we’re interested in. Whether or not this average quantity can entirely reproduce that original manifold is kind of irrelevant (and indeed undesirable), the point is that it (a) represents something meaningful about the model and (b) it’s low dimensional enough for a human to reason about. Although globally it will not be accurate to distances along the surface, presumably it is “good enough” to at least first order for much of the support of the training data.