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We thank all reviewers for their careful reading and their valuable comments. As seen in the figure on the right, the performance of Lucic et al. (2016) We now included this baseline in the paper. R1: The dimension of B is stated wrongly [..] Thank you for pointing "In contrast to k-means, we assume that the mean .." is not clear to me. Thank you for raising this issue. Reviewer 3 also pointed this out.

A prominent technique for self-supervised representation learning has been to contrast semantically similar and dissimilar pairs of samples.

First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. This work presents a method to efficiently train object detectors in the presence of geometric transformations that can be represented as vector-matrix multiplications. This has recently been developed for the case of translation transformations ([1,8,14]) but has not been too obvious for other transformations, such as rotations, let alone non-rigid deformations. The authors propose to adapt the Fourier-based method that originally allowed the development of efficient algorithms for the case of translations so that we can now deal with rotations and other `cyclic' signal transformations (e.g. the walking pattern of a pedestrian). The condition for this to hold is that the transformation is norm-preserving, can be represented as a matrix multiplication, x_transformed = Q x, has an inverse Q^{-1} = Q^T and for some s Q^s = I. The starting point for the previous works was the fact that the'data matrix' obtained by'stacking' together all translated versions of a signal is a circulant NxN matrix, where N is the length of the signal, and as such can be diagonalized using the discrete harmonic basis (or, Discrete Fourier Transform-DFT matrix), Eq. 3, and reference [5].

Linear regression is a fundamental problem in machine learning.