Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: 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,  have considered the statistical problem of estimating the homology of a manifold from noiseless samples and from noisy samples under several different noise models.  derived upper and lor bounds on the minimax risk for this problem. In this note  revisit the noiseless case. In previous work  used Le Cam's lemma to establish a lor bound that differed from the upper bound of Niyogi, Smale and inberger by a polynomial factor in the condition number. #R##N#In this note  use a different construction based on the direct analysis of the likelihood ratio test to show that the upper bound of Niyogi, Smale and inberger is in fact tight, thus establishing rate optimal asymptotic minimax bounds for the problem. The techniques  use here extend in a straightforward way to the noisy settings considered in our earlier work.
['Sivaraman Balakrishnan', 'Alessandro Rinaldo', 'Aarti Singh', 'Larry Wasserman']
I am writing a research paper on a idea Tight Lower Bounds for Homology Inference
[{"from": "human", "value": "I am writing a research paper on a idea Tight Lower Bounds for Homology Inference"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: 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,  have considered the statistical problem of estimating the homology of a manifold from noiseless samples and from noisy samples under several different noise models.  derived upper and lor bounds on the minimax risk for this problem. In this note  revisit the noiseless case. In previous work  used Le Cam's lemma to establish a lor bound that differed from the upper bound of Niyogi, Smale and inberger by a polynomial factor in the condition number. #R##N#In this note  use a different construction based on the direct analysis of the likelihood ratio test to show that the upper bound of Niyogi, Smale and inberger is in fact tight, thus establishing rate optimal asymptotic minimax bounds for the problem. The techniques  use here extend in a straightforward way to the noisy settings considered in our earlier work."}]