  We propose and analyze two new MCMC sampling algorithms, the Vaidya walk and
the John walk, for generating samples from the uniform distribution over a
polytope. Both random walks are sampling algorithms derived from interior point
methods. The former is based on volumetric-logarithmic barrier introduced by
Vaidya whereas the latter uses John's ellipsoids. We show that the Vaidya walk
mixes in significantly fewer steps than the logarithmic-barrier based Dikin
walk studied in past work. For a polytope in $\mathbb{R}^d$ defined by $n >d$
linear constraints, we show that the mixing time from a warm start is bounded
as $\mathcal{O}(n^{0.5}d^{1.5})$, compared to the $\mathcal{O}(nd)$ mixing time
bound for the Dikin walk. The cost of each step of the Vaidya walk is of the
same order as the Dikin walk, and at most twice as large in terms of constant
pre-factors. For the John walk, we prove an
$\mathcal{O}(d^{2.5}\cdot\log^4(n/d))$ bound on its mixing time and conjecture
that an improved variant of it could achieve a mixing time of
$\mathcal{O}(d^2\cdot\text{polylog}(n/d))$. Additionally, we propose variants
of the Vaidya and John walks that mix in polynomial time from a deterministic
starting point. The speed-up of the Vaidya walk over the Dikin walk are
illustrated in numerical examples.
