We thank the Reviewer for their time, consideration, and invaluable feedback.
## Weaknesses:
1. See Question 1 below.
2. We explicitate the mentioned phrase in the Appendix A.4 "LCOT Barycenter". We thank the reviewer for this comment since now our definition is more concise.
Precisely, the *LCOT barycenter* is defined as follows:
Given $N$ target measures $\nu_1,\dots,\nu_N\in\mathcal{P}(\mathbb{S}^1)$, as $LCOT_2(\cdot,\cdot)$ is a distance, their *LCOT barycenter* is defined by the measure $\nu_b$ such that
$$
\nu_b=argmin_{\nu\in\mathcal{P}(\mathbb{S}^1)} \frac{1}{N}\sum_{j=1}^N LCOT_2(\nu,\nu_j)= argmin_{\nu\in\mathcal{P}(\mathbb{S}^1)} \frac{1}{N}\sum_{j=1}^N ||\widehat{\nu}-\widehat{\nu_j}||^2_{L^2(\mathbb{S}^1)}.
$$
In the embedding space, it can be shown that the minimizer of
$
argmin_{\widehat{\nu}} \frac{1}{N}\sum_{j=1}^N ||\widehat{\nu}-\widehat{\nu_j}||^2_{L^2(\mathbb{S}^1)}
$
is given by the circular mean
$$\overline{\nu}(x):=\text{circle mean}(\{\widehat{\nu_1}(x),\ldots \widehat{\nu_N}(x)\}):=
\frac{1}{2\pi}\arg\tan\left({\frac{\sum_{i=1}^N\sin(2\pi\widehat{\nu_i}(x))}{\sum_{i=1}^N\cos(2\pi\widehat{\nu_i}(x))}}\right).
$$
For each $x\in[0,1)$, the last value is the average of the angles ${2\pi \widehat{\nu_1}(x), \ldots, 2\pi\widehat{\nu_N}(x)}$, which is then normalized to fall within the range $[-0.5, 0.5]$.
By using the closed formula for the inverse of the LCOT Embedding provided by Proposition 3.5, we can go back to the measure space obtaining the *LCOT barycenter* between $\nu_1,\dots,\nu_N$ as
$\nu_b=(\overline{\nu}+\mathrm{id})$#$\mu$.
In our experiments, we use the last expression.
3. We included the units (*seconds*) in the caption of Figure 4.
## Questions:
1. Consider two circular measures $\nu_1$ and $\nu_2$ and the uniform circular measure as the reference $\mu$. Calculating COT from $\nu_i$ to $\mu$ can be intuitively thought as finding an optimal cut on the circle for $\nu_i$, cutting the circle into the line $[0,1)$ and then solving a one-dimensional OT problem in this flat space, which has a closed-form solution. This process allows us to find a Monge map (and the corresponding optimal displacement) between $\nu_i$ and $\mu$. Now, given that this mapping is invertible (i.e., for each push forward we also have a pull back), we can first push $\nu_1$ to the reference, $\mu$, and then pull back $\mu$ to $\nu_2$. This process will result in a transport map between $\nu_1$ and $\nu_2$, which is unique, but is not necessarily the optimal transport map between $\nu_1$ and $\nu_2$. Below, we describe this process more formally.
The LCOT-Embedding encodes the optimal circular displacement given in Equation (15). That is, a reference measure $\mu$ is fixed and for each target measure $\nu$, the LCOT-Embedding $\widehat{\nu}^{\mu,h}(x)$ is defined as $M_\mu^\nu(x)-x$, where $M_\mu^\nu(x)$ is the optimal location to place $x$.
For simplicity in the notation, let us denote $\widehat{\nu}^{\mu,h}$ by $\widehat{\nu}$.
Since we have an optimal displacement for each $x$, we have that $\widehat{\nu}$ is a function. To measure the disimilarity between to target measures $\nu_1$ and $\nu_2$ in the embedding space, we use a definition that allows us to use the standard $p$-norms when $h(x)=|x|^p$. The particularity here is that these distances are for functions supported on the circle which makes the formulas more convoluted.
The take away from Def 3.2 is that the LCOT distance between $\nu_1$ and $\nu_2$ is essentially a $p$-norm ($||\widehat{\nu}_1-\widehat{\nu}_2||_p$).
2. In general, COT and LCOT are not equivalent distances. In Remark 3.4 we pointed out that the COT cost between $\mu$ and $\nu$ can be recovered by using the LCOT distance when $\mu$ is the reference. For other pairs of measures, LCOT is still a 'transport based' distance, just not the same as COT. As a consequence, COT and LCOT barycenter are not equivalent for arbitrary sets of measures.