We thank Reviewer TcMZ for the prompt response. We clarify our problem statements and theorem statements in the following points, and we will add these discussions to the camera-ready version of our paper for better readability.
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### Problem Statements
Our paper consists of two parts, each addressing different problems. We describe the problems in these sections in detail as follows:
1. **Section 3** is focused on the theoretical problem of finding a principled way (i.e., canonicalization) to characterize the complexity of frames $\mathcal{F}(X)$ and frame averaging, a general class of invariant and equivariant learning methods. We have included the definitions of invariant and equivariant learning in Section 2, and the definitions of frames frame averaging in Section 2.1. We will state the main problem more clear in the beginning of Section 3 for better readability.
2. Guided by the theoretical insights in Section 3, **Section 4** is to design better or optimal canonicalization algorithms for a widely appeared class of problems, the sign and basis invariance of eigenvectors. Specifically, we aim to design a canonicalization algorithm $\mathcal{C}$ operating on eigenvectors $\mathbf{U}\in\mathbb{R}^{n\times d}$, that is **invariant** to sign/basis transformations, **equivariant** to permutation transformations, and outputs a set of eigenvectors $\mathbf{U}^*\in\mathbb{R}^{n\times d}$ in the **same eigenspace** as $\mathbf{U}$. We consider two settings of sign and basis invariance: without (Section 4.1) and with (Section 4.2) permutation equivariance, corresponding to different problem scenarios. We will make it more clear in the beginning too.
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### Theorem Statements
We understand that we define some notations out of the theorems, making them less self-contained and easy-to-understand. To ease your concerns, we will define the notations more clearly, point out the key messages of the main theorem, and add references to previous definitions. Below we give two examples of the modified theorem statements:
- **Theorem 4.1.** Given a set of eigenvectors $\mathbf{U}\in\mathbb{R}^{n\times d}$, let $\mathscr{P}=\mathbf{UU}^\mathrm{T}$ denote the projection matrix of the eigenspace. Let $\mathbf e_1,\dots,\mathbf e_n$ denote the standard basis vectors. Then, there exists indices $1\leq i_1<\cdots<i_d\leq n$, such that for all $1\leq j\leq d$, we have $\lVert\mathscr P\mathbf e_{i_j}\rVert>0$, and the vectors $\mathscr P\mathbf e_{i_1},\dots,\mathscr P\mathbf e_{i_d}$ are linearly independent.
- **Theorem 4.3.** Let $\alpha_i\ (i=1,\dots,n)$ be the outputs of the hash function in the OAP algorithm defined in Equation (4), and let $i_j\ (j=1,\dots,d)$ be the indices found in Algorithm 3. Then, the MAP algorithm is equivalent to the OAP algorithm by taking $\alpha_i=\lVert\mathscr P_i\rVert$ for all $1\leq i\leq n$ and $i_j=j$ for all $1\leq j\leq d$. The FA-lap algorithm is equivalent to the OAP algorithm by taking $\alpha_i=\mathscr P_{ii}$ for all $1\leq i\leq n$.
We will modify all the theorems in our paper similarly to enhance readability and self-containedness. These changes will be reflected in the camera-ready revision of our paper. We hope these changes address your concerns. If you have further concerns or suggestions, please feel free to reach out.