Correcting Biased Centered Kernel Alignment Measures in Biological and Artificial Neural Networks — part 3/3
Cleaned-text code points [30101, 34286). Bibliography omitted here; source.md includes it.
In order to quantify the effect of CKA in its biased and debiased form on random data for a specific fMRI ROI or MEG time window / sensor group, we sampled from a multivariate standard Normal distribution of the same shape, where each entry is independent of the other. Our later analyses suggested structure of the neural data is in part responsible for biased CKA scores (as seen with shuffled data). We then repeated our random analyses where we estimated each voxel / sensor dimension by taking the mean over responses to the entire image set (and similarly for the standard deviation). We then repeated the random analyses by sampling from Normal distributions that were more similar to the structure of the neural data and confirmed that our results did not differ from sampling from the original case when using Standard Normal distribution.
### A.2 MEG Preprocessing & Sensor Selection
We used the raw version of MEG data in the THINGS database and wrote custom code to apply preprocessing on each session. Specifically, we band-passed filtered the data between [0.1, 40] Hz using a firwin filter and extracted a window of [- 0.1, 1.5] second around each stimulus event, baseline-corrected each epoch using the 100 ms pre-stimulus interval by subtracting the mean and downsampled the data to 200 Hz. We added a mapping to a predefined train, validation and test split designed in a separate analysis and concatenated MEG responses associated with the data set reported in Section 3.1 . We did not perform analyses to detect bad segments or channels, with the exception of dropping MRO11 from all participants as Hebart et al. (2023) advised this due to issues observed during the original data acquisition session that resulted in this channel being excessively noisy. We extracted all MEG channels that were labelled as O[ccipital] in the original data structure for our analyses, resulting a feature dimension of N = 39 N=39 for each time step of MEG data.
### A.3 Code Implementation
Our code was written in Python / PyTorch and we used the implementation given in Kornblith et al. (2019) to calculate biased & debiased CKA. The code used to run the analysis and generate the figures in this work are available at the following link: https://github.com/Alxmrphi/correcting_CKA_alignment .
### A.4 Per-Participant Functional MRI ROI Sizes
### A.5 Unbiased HSIC Kernel-Centering Formula
Székely & Rizzo (2014) introduced the following formula to compute the unbiased HSIC estimator for an n × n n\times n matrix, A A . The only requirements are: (i) the main diagonal contains 0 0 s and (ii) n > 2 n>2 . The unbiased centered matrix, A ~ \tilde{A} is defined below.
A ~ i , j = { a i , j − 1 n − 2 ∑ ℓ = 1 n a i , ℓ − 1 n − 2 ∑ k = 1 n a k , j + 1 ( n − 1 ) ( n − 2 ) ∑ k , l = 1 n a k , l if i ≠ j 0 if i = j \tilde{A}_{i,j}=\begin{cases}a_{i,j}-\frac{1}{n-2}\sum_{\ell=1}^{n}a_{i,\ell}-\frac{1}{n-2}\sum_{k=1}^{n}a_{k,j}+\frac{1}{(n-1)(n-2)}\sum_{k,l=1}^{n}a_{k,l}\hskip 14.22636pt\text{ if }i\neq j\\ 0\hskip 278.837pt\text{ if }i=j\end{cases} (4)
### A.6 Results for CKA Sensitivity to Shuffled Data (CORnet-S)
### A.7 Results for CKA Sensitivity to Partially Shuffled Data (ResNet18)
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