Tom Talpir | Searcharxiv

arXiv2026

Sample complexity bounds for the Jensen-Shannon divergence

The Jensen-Shannon divergence (JSD) is a symmetric and bounded measure of the dissimilarity of two probability distributions, which has become a standard tool in statistics, information theory, and machine learning. We complement the understanding of its mathematical properties by presenting an analysis of the amount of data that is needed to distinguish between two distributions, given the value of JSD between them. We find the number of independent and identically distributed samples that suffice for a classifier to determine which of two distributions generated observed data at a desired error rate, for two complementary classifiers: we show that for the log-likelihood-ratio classifier, a sample size that grows as the inverse JSD is sufficient, whereas for a majority-vote classifier assembled from independent single-sample decisions, the sufficient size grows as the squared inverse JSD. These distinct scalings offer operational readings of JSD values and their translation into distinguishability in different contexts.

cs.IT↗

arXiv2026

Identifying structural design principles shaping the computational abilities of recurrent neural networks

Understanding how the architecture of neural networks shapes the computations they carry is a central challenge in neuroscience and machine learning. While specific circuit architectures have been linked to particular network computations and theoretical bounds on expressivity of broad classes of networks have been found, we are still missing general principles connecting the structure of finite networks to their computational capabilities. Here, we characterize the computational abilities of recurrent neural networks as a function of their connectivity by training a large collection of different networks to compute a large set of Boolean functions. For small networks, we constructed the complete ``catalogs'' of network-function performance, which revealed that computational capacity varies widely across architectures and that most networks show poor performance, and most functions are hard to compute. However, we show that having local 2- and 3-cycles in a network strongly enhances its computational ability, and networks with such cycles are often the minimal architectures that can solve particular functions. We further show that a small set of structural statistics accurately predict networks' performance. Extending our analysis to large networks showed that typical networks fail even to approximate a randomly selected function. Surprisingly, adding a small number of sparsely connected biologically-inspired interneurons to the network dramatically increases computational capacity. As in small networks, adding short cycles improved networks' capacity, outperforming acyclic or reachability-matched controls. Thus, our results identify local cycles as design principles linking neural connectivity to computational power, and offer a general framework to explore structure-function relations in computing networks.

q-bio.NC↗