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This work proves nearly quadratic lower bounds for randomized algorithms for linear optimization and uniform sampling over convex bodies in the membership oracle model and implies the same lower bound for volume estimation.
We show that logconcave probability measures along the Gaussian cooling path have thin-shell stability, generalizing the thin-shell theorem. This result leads to improved complexity for the fundamental problem of sampling an arbitrary logconcave distribution from a cold start. For (near-)isotropic logconcave distributi...
For any convex body $\mathcal{K}\subset\mathbb{R}^{n}$ containing a unit ball, the spectral gap of Hit-and-Run is $\Omega(1/(n^2 C_{\mathsf{PI}}))$, where $C_{\mathsf{PI}}$ is the Poincar\'e constant of the uniform distribution $\pi$ over $\mathcal{K}$. This implies that Hit-and-Run converges to a distribution within $...
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