[Paper Review] Tight Lower Bounds for Homology Inference

Sivaraman Balakrishnan, Alessandro Rinaldo+2 more|arXiv (Cornell University)|Jul 29, 2013

Topological and Geometric Data Analysis4 references3 citations

TL;DR

This paper establishes tight minimax lower bounds for homology inference from noiseless samples on a d-dimensional manifold embedded in ℝᴰ. Using a likelihood ratio test on a constructed testing problem with m d-spheres, it proves the minimax risk is Ω(1/τᵈ exp(−nτᵈ)), matching the known upper bound and establishing rate optimality in the noiseless setting.

ABSTRACT

The homology groups of a manifold are important topological invariants that provide an algebraic summary of the manifold. These groups contain rich topological information, for instance, about the connected components, holes, tunnels and sometimes the dimension of the manifold. In earlier work, we have considered the statistical problem of estimating the homology of a manifold from noiseless samples and from noisy samples under several different noise models. We derived upper and lower bounds on the minimax risk for this problem. In this note we revisit the noiseless case. In previous work we used Le Cam's lemma to establish a lower bound that differed from the upper bound of Niyogi, Smale and Weinberger by a polynomial factor in the condition number. In this note we use a different construction based on the direct analysis of the likelihood ratio test to show that the upper bound of Niyogi, Smale and Weinberger is in fact tight, thus establishing rate optimal asymptotic minimax bounds for the problem. The techniques we use here extend in a straightforward way to the noisy settings considered in our earlier work.

Motivation & Objective

• To close the gap between existing upper and lower bounds for minimax risk in homology inference under noiseless sampling.
• To establish rate-optimality of homology estimation by proving a tight lower bound matching the known upper bound.
• To develop a novel construction based on manifold separation via sphere removal to analyze the fundamental limits of homology estimation.
• To extend the analysis to noisy settings, though the focus here is on the noiseless case with asymptotic bounds.
• To provide a foundation for lower bounds in related problems such as manifold estimation in Hausdorff distance and persistence diagram estimation.

Proposed method

• Construct a null manifold M₀ composed of m d-spheres of radius τ, spaced 4τ apart, embedded in ℝᴰ with D ≥ d+1.
• Define alternate manifolds M₁ᵢ by removing one sphere from M₀, creating m distinct alternatives.
• Use uniform distributions P₀ on M₀ and P₁ᵢ on M₁ᵢ, ensuring bounded density from below.
• Frame the problem as a hypothesis testing task: H₀ (M₀) vs. H₁ (M₁ᵢ with i uniform over {1,…,m}).
• Apply the likelihood ratio test T(X) = 1 if L(X) > 1, where L(X) = L₁(X)/L₀(X) is the ratio of likelihoods under H₁ and H₀.
• Leverage the coupon collector problem to bound the probability of not observing any sample from a given sphere, which implies rejection of H₀.
• Derive asymptotic lower bound on Type I error via limit behavior of the coupon collector distribution.

Experimental results

Research questions

• RQ1 What is the fundamental minimax rate for estimating the homology of a d-dimensional manifold from noiseless i.i.d. samples?
• RQ2 Can the previously known lower bound via Le Cam’s lemma be improved to match the known upper bound?
• RQ3 Does the likelihood ratio test achieve the optimal error rate in the homology inference problem?
• RQ4 Can the construction of m disjoint d-spheres be used to derive tight non-asymptotic and asymptotic lower bounds?
• RQ5 To what extent can this method be extended to noisy sampling models and other topological inference problems?

Key findings

• The minimax risk for homology inference is Ω(1/τᵈ exp(−nτᵈ)) as n → ∞, matching the known upper bound and establishing rate optimality.
• The lower bound is derived via a likelihood ratio test on a testing problem involving m = Θ(1/(4τ)ᵈ) d-spheres.
• When n = m log m + m log(1/δ), the probability of not observing a sample from each sphere is at least cδ for some constant c, leading to a Type I error rate of at least cδ.
• This implies Rₙ ≥ c′δ for some c′ > 0, showing the risk cannot decay faster than Ω(1/τᵈ exp(−nτᵈ)).
• The construction and analysis extend naturally to noisy settings, suggesting similar lower bounds can be derived in those cases.
• Finite-sample lower bounds can be obtained by replacing the asymptotic coupon collector limit with finite-sample estimates.