Minimax Rates for Homology Inference
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چکیده
Lower bound Here we describe the densities on the two manifolds M1 and M2. There are two sets of interest to us: W1 = M1 \M2 which corresponds to the two “holes” of radius 4τ in the annulus, and W2 = M2\M1 which corresponds to the d-dimensional piece added to smoothly join the inner pieces of the two annuli in M2. By construction, vol(W1) = 2vd(4τ) d where vd is the volume of the unit d-ball. vol(W2) is somewhat tricky to calculate exactly due to the curvature of W2 but it is easy to see that vol(W2) is also O(τ ) with the constant depending on d.
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