Spectral analysis of representational similarity with limited neurons

Understanding how neural representations align between biological and artificial systems has emerged as a central challenge in computational neuroscience. While deep neural networks now reliably predict neural responses across multiple brain areas, their utility for understanding biological computation remains limited by our ability to accurately measure representational similarities. This limitation becomes particularly acute when working with sparse neural recordings, where traditional similarity metrics may fail to capture true representational relationships. Recent spectral analyses of similarity measures provided careful decomposition of neural similarities in the regime of small sample sizes. Here, we consider Centered Kernel Alignment (CKA) as a similarity measure and, using techniques from random matrix theory, identify what spectral aspects affect the representational similarity in the limited neuron regime. We show that the true CKA is underestimated when a small population of neurons is randomly sampled and compared with deterministic neural network representations. We find that increasing sample size may cause underestimating the true CKA. When the number of neurons is small, we demonstrate that only information up to a certain eigenvector threshold can be resolved. We develop a systematic method to denoise the CKA and demonstrate a similarity measure that is robust against changes in population size.