Paper: Faster Sampling from Log-Concave Densities over Polytopes via Efficient Linear Solvers (ICLR-2024)

Reviewer comment (Writing):
The authors should try to make the paper more accessible to people who might not have read [1]. This relates to clarity, which was noted as an issue before, though the reviewer is unsure how exactly to improve it.

Author response:
Thank you for your suggestion. We will add a section in the appendix providing additional background on the soft-threshold Dikin walk Markov chain of [1].

Reviewer comment (Writing):
There are a number of typos throughout the paper. For example, in Theorem 2.1, 3) convex function $f:K \mapsto \mathbb{R}^d$. Here $f$ should be $K \mapsto \mathbb{R}$. This mistake is repeated in several places in the paper and also in [1]. The reviewer considered whether the codomain of a convex function could be $\mathbb{R}^d$ instead of $\mathbb{R}$, but concluded that in that case $e^{-f}$ would not be real, which would be problematic since it must be real for a probability density function.

Author response:
Sorry for the typo. It should read $f :K \rightarrow \mathbb{R}$, where $K\subset \mathbb{R}^d$ is a convex polytope.

Reviewer comment (Writing):
On page 5, line 3: "ball or radius" should be "ball of radius". This line also repeats the $f: K \mapsto \mathbb{R}^d$ issue.

Author response:
Thank you for pointing out these typos. We will correct them in the final version.