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# System prepended metadata

title: Response^2

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# Response^2

Consider the following example: there are 4 frequencies $f_1, f_2, f_3, f_4$. Now, let there be two disjoint sets of frequencies : $\{f_1, f_2\}, \{f_3,f_4\}$. Consider arbitrary perturbations for the two sets : $\delta_1 = \{f_1 : 2, f_2 : 3\}; \delta_2 = \{f_3 : 6, f_4 : 7\}$. Now, the total $\delta = \{f_1 : 2, f_2 : 3, f_3 : 6, f_4 : 7\}$. Now, by the definition of L2 norm, $||\delta||_2^2$ is equal to the sum of squares of the magnitudes of each component, i.e., $||\delta||^2_2 = 2^2 + 3^2 + 6^2 + 7^2$. Also by definition of L2 norm, $||\delta_i||_2^2$ is equal to the sum of squares of magnitudes of frequency components in the respective subset, i.e., $||\delta_1||^2_2 = 2^2 + 3^2$ and $||\delta_2||^2_2 = 6^2 + 7^2$. Since the frequency subsets are disjoint, and their union is equal to the set of all frequencies, this example thus shows that $||\delta||_2^2 = ||\delta_1||^2_2 + ||\delta_2||^2_2$. We are unclear why this needs to be verified via an experiment. Also, we hope that the other concerns have been addressed and we are happy to offer any further clarifications if needed.




Consider the following example: there are 4 frequencies $f_1, f_2, f_3, f_4$. Now, let there be two disjoint sets of frequencies : $\{f_1, f_2\}, \{f_3,f_4\}$. Consider arbitrary perturbations for the two sets : $\delta_1 = \{f_1 : 2, f_2 : 3\}; \delta_2 = \{f_3 : 6, f_4 : 7\}$. Now, the total $\delta = \{f_1 : 2, f_2 : 3, f_3 : 6, f_4 : 7\}$. Now, by the definition of L2 norm, $||\delta||^2_2 = 2^2 + 3^2 + 6^2 + 7^2$. And $||\delta_1||^2 = 2^2 + 3^2$, $||\delta_2||^2 = 6^2 + 7^2$. This example shows that $||\delta||_2^2 = ||\delta_1||^2_2 + ||\delta_2||^2_2$ which can also be inferred from the definition of L2 norm. Specifically, by the definition of L2 norm, $||\delta||_2^2$ is equal to the sum of squares of the magnitudes of each component. Using this, $||\delta_i||_2^2$ is equal to the sum of squares of magnitudes of frequency components in the respective subset. Now, the subsets are disjoint, and their union is equal to the set of all frequencies. Since, $||\delta||_2^2$ is the sum of squares of all the frequency components, $||\delta||_2^2 = \sum_i ||\delta_i||_2^2$. We are unclear why this needs to be verified via an experiment. Also, we hope that the other concerns have been addressed and we are happy to offer any further clarifications if needed.

We don't agree with the reviewer's hypothesis. By the definition of L2 norm, $||.||_2^2$ is equal to the sum of squares of the magnitudes of each component. Using this, $||\delta_i||_2^2$ is equal to the sum of squares of magnitudes of frequency components in the respective subset. Now, the subsets are disjoint, and their union is equal to the set of all frequencies. Since, $||\delta||_2^2$ is the sum of squares of all the frequency components, $||\delta||_2^2 = \sum_i ||\delta_i||_2^2$. We are unclear why this needs to be verified via an experiment. Also, we hope that the other concerns have been addressed and we are happy to offer any further clarifications if needed.