• More motivation and derivations of the different Fisher scores U x
(for GANs, VAEs, supervised) would be beneficial for understanding better the paper.
• More discussions on the non-diagonal version of the Fisher information matrix models (see comments below) would be beneficial.
• Discussion on the dependence of the quality of the NFK embedding on the quality of the pre-trained neural network.
• Discussion on the choice of the low-dimensional embedding dimensionality k
General comments:
• Using the diagonal of the Fisher information matrix (FIM) seems desirable from a computational reason, however a natural question is what happens if one tries to use the full matrix. Given the size of the parameters θ
in a neural network, estimating the whole matrix would indeed be extremely computationally expensive, but by discarding them, one loses significant information. Could the authors comment on that? Is the diag of FIM related to other known concepts in statistics? Does using only the diagonal imply that the off-diagonal elements are zero meaning the parameters are orthogonal? How does this affect the results and interpretation?
• Could the authors give the derivation of U x
in eq (1) for GANs (I see part of the derivation is in the paper by Zhai et al, 2019)? For VAEs and supervised case, FIM I
is the inner product using U x
, but for GANs it acts in the output space of the generator. Why is this the case (some explanations are given in the original paper but would be helpful to discuss this a bit more here)? The derivation of eq (4) would also be useful.
• Low-rank structure of NFK and Alg 1: How does one choose the feature dimensionality k
? Many methods that rely on kernels and manifold learning make the assumption of low-dimensionality/low-rankness and show that a small number of eigenfunctions is sufficient to reconstruct the input data. How is this different for NFK? The way I understand “low rank” is that the data has its own rank, which is low, and could potentially be learned. However here the authors input the dimensionality/rank k
which might be or not close to the true rank in real applications.
• How does this work relate to the work of Belkin et al (2018) – “To understand deep learning we need to understand kernel learning”, where the authors look at other kernels (Laplacian and Gaussian)?
• Could the approach be used for neural networks that are not pre-trained, as the neural tangent kernel NTK?
• The experimental results are nice, however the focus on computation is not so relevant given that it only uses the diagonal of the Fisher information matrix. Comparisons using the whole matrix would also be needed. What error is used in Table 3 (MSE, MAE, RMSE)? The goal of the paper is to present a method for supervised and unsupervised settings, however in the results an example on semi-supervised is also presented. I wonder if the examples on semi-supervised and knowledge distillation could leave room to improve the supervised and unsupervised settings discussions, and potentially be moved to the Appendix?
Other comments:
• Please update reference (Jaakkola et al): year, conference, also there should be no “et al” there are only two authors
• Doesn’t, don’t, won’t, it’s , etc -> does not, do not, would not, it is
• Both the concepts of “data representation” and “feature representation” are used. Do they always refer to the same thing? If yes, would be good to specify that.
• Expression of K f i s h e r
=> second U
should be subscript z not x ?
• “FIM defined as the variance of the score …” -> the FIM matrix is defined between all pairs of parameters θ i and θ j
, so it should be a covariance?
• Appendix Fig 3: Not sure I fully understand this example. Could one try the reconstruction of the digits using a simple method, such as PCA using the first 100 principal components as a baseline?
• Not familiar with the “Fisher vector” terminology, except in image classification and the “Adversarial Fisher vector” from Zhai et al, 2019. Are there other references?
• Could the approach be used for neural networks that are not pre-trained, as the neural tangent kernel NTK?