Random ε-Cover on Compact Symmetric Space

04/15/2023

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by   Somnath Chakraborty, et al.

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A randomized scheme that succeeds with probability 1-δ (for any δ>0) has been devised to construct (1) an equidistributed ϵ-cover of a compact Riemannian symmetric space 𝕄 of dimension d_𝕄 and antipodal dimension d̅_𝕄, and (2) an approximate (λ_r,2)-design, using n(ϵ,δ)-many Haar-random isometries of 𝕄, where n(ϵ,δ):=O_𝕄(d_𝕄ln(1/ϵ)+log(1/δ)) , and λ_r is the r-th smallest eigenvalue of the Laplace-Beltrami operator on 𝕄. The ϵ-cover so-produced can be used to compute the integral of 1-Lipschitz functions within additive Õ(ϵ)-error, as well as in comparing persistence homology computed from data cloud to that of a hypothetical data cloud sampled from the uniform measure.