By starting a Markov chain from the empirical distribution of samples, mixing time depends on a higher-order spectral gap, and sample complexity grows only linearly in the number of modes.
Hence 1 2 ‖∇V (Eπx)‖ ≤ Eπ[‖∇V ‖] + β √ CP( √ d + ln 6) ≤ √ βd + β √ CP( √ d + ln 6) where the last inequality follows from Eπ[‖∇V ‖] ≤ Eπ[‖∇V ‖2]1/2 ≤ √βd by Lemma 58
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Efficiently learning and sampling multimodal distributions with data-based initialization
By starting a Markov chain from the empirical distribution of samples, mixing time depends on a higher-order spectral gap, and sample complexity grows only linearly in the number of modes.