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Abstract:
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In this work we study the framework of topological inference through the lens of classical parametric inference in statistics. Given ??n={X1,X2,…,Xn} observed iid from a distribution ??, the Betti numbers associated with the ?ech complex of ??n encode both the topological and parametric information of the distribution of points. When two distinct distributions admit the same thermodynamic limit for the random Betti numbers, the distributions are said to be ?-equivalent. By studying families of distributions which admit ?-equivalence, we investigate conditions under which topological inference is possible in this parametric setup.
We characterize necessary and sufficient conditions for ?-equivalence by imposing an algebraic structure on the distributions and examining their maximal invariants. We further relax the algebraic structure to provide sufficient conditions for ?-equivalence. To this end, we first present a result for ?-equivalence when the underlying space admits a fiber bundle structure. Next, we present a result which guarantees ?-equivalence when the density and score function satisfy an orthogonality condition. We illustrate all results through supporting examples.
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