Testing the manifold hypothesis

Source abstract

The hypothesis that high dimensional data tend to lie in the vicinity of a low dimensional manifold is the basis of manifold learning. The goal of this paper is to develop an algorithm (with accompanying complexity guarantees) for testing the existence of a manifold that fits a probability distribution supported in a separable Hilbert space, only using i.i.d. samples from that distribution. More precisely, our setting is the following. Suppose that data are drawn independently at random from a probability distribution P \mathcal {P} supported on the unit ball of a separable Hilbert space H \mathcal {H} . Let G ( d , V , τ ) \mathcal {G}(d, V, \tau ) be the set of submanifolds of the unit ball of H \mathcal {H} whose volume is at most V V and reach (which is the supremum of all r r such that any point at a distance less than r r has a unique nearest point on the manifold) is at least τ \tau . Let L ( M , P ) \mathcal {L}(\mathcal {M}, \mathcal {P}) denote the mean-squared distance of a random point from the probability distribution P \mathcal {P} to M \mathcal {M} . We obtain an algorithm that tests the manifold hypothesis in the following sense. The algorithm takes i.i.d. random samples from P \mathcal {P} as input and determines which of the following two is true (at least one must be): There exists M ∈ G ( d , C V , τ C ) \mathcal {M} \in \mathcal {G}(d, CV, \frac {\tau }{C}) such that L ( M , P ) ≤ C ϵ . \mathcal {L}(\mathcal {M}, \mathcal {P}) \leq C {\epsilon }. There exists no M ∈ G ( d , V / C , C τ ) \mathcal {M} \in \mathcal {G}(d, V/C, C\tau ) such that L ( M , P ) ≤ ϵ C . \mathcal {L}(\mathcal {M}, \mathcal {P}) \leq \frac {\epsilon }{C}. The answer is correct with probability at least 1 − δ 1-\delta .