A full understanding of the behavior of a large language model (LLM) requires our grasp of its input token space. If this space differs from our assumptions, our comprehension of and conclusions about the LLM will likely be flawed. We elucidate the structure of the token embeddings both empirically and theoretically. We present a novel statistical test assuming that the neighborhood around each token has a relatively flat and smooth structure as the null hypothesis. Failing to reject the null is uninformative, but rejecting it at a specific token $\psi$ implies an irregularity in the token subspace in a $\psi$-neighborhood, $B(\psi)$. The structure assumed in the null is a generalization of a manifold with boundary called a \emph{smooth fiber bundle} (which can be split into two spatial regimes -- small and large radius), so we denote our new hypothesis test as the ``fiber bundle hypothesis.'' By running our test over several open-source LLMs, each with unique token embeddings, we find that the null is frequently rejected, and so the evidence suggests that the token subspace is not a fiber bundle and hence also not a manifold. As a consequence of our findings, when an LLM is presented with two semantically equivalent prompts, if one prompt contains a token implicated by our test, the response to that prompt will likely exhibit less stability than the other.

A recent trend in explainable AI research has focused on surrogate modeling, where neural networks are approximated as simpler ML algorithms such as kernel machines. A second trend has been to utilize kernel functions in various explain-by-example or data attribution tasks. In this work, we combine these two trends to analyze approximate empirical neural tangent kernels (eNTK) for data attribution. Approximation is critical for eNTK analysis due to the high computational cost to compute the eNTK. We define new approximate eNTK and perform novel analysis on how well the resulting kernel machine surrogate models correlate with the underlying neural network. We introduce two new random projection variants of approximate eNTK which allow users to tune the time and memory complexity of their calculation. We conclude that kernel machines using approximate neural tangent kernel as the kernel function are effective surrogate models, with the introduced trace NTK the most consistent performer.

We investigate the spectral properties of linear-width feed-forward neural networks, where the sample size is asymptotically proportional to network width. Empirically, we show that the spectra of weight in this high dimensional regime are invariant when trained by gradient descent for small constant learning rates; we provide a theoretical justification for this observation and prove the invariance of the bulk spectra for both conjugate and neural tangent kernels. We demonstrate similar characteristics when training with stochastic gradient descent with small learning rates. When the learning rate is large, we exhibit the emergence of an outlier whose corresponding eigenvector is aligned with the training data structure. We also show that after adaptive gradient training, where a lower test error and feature learning emerge, both weight and kernel matrices exhibit heavy tail behavior. Simple examples are provided to explain when heavy tails can have better generalizations. We exhibit different spectral properties such as invariant bulk, spike, and heavy-tailed distribution from a two-layer neural network using different training strategies, and then correlate them to the feature learning. Analogous phenomena also appear when we train conventional neural networks with real-world data. We conclude that monitoring the evolution of the spectra during training is an essential step toward understanding the training dynamics and feature learning.

MOTIVATION: The IntAct repository is one of the largest and most widely used databases for the curation and storage of molecular interaction data. These datasets need to be analyzed by computational methods. Software packages in the statistical environment R provide powerful tools for conducting such analyses. RESULTS: We introduce Rintact, a Bioconductor package that allows users to transform PSI-MI XML2.5 interaction data files from IntAct into R graph objects. On these, they can use methods from R and Bioconductor for a variety of tasks: determining cohesive subgraphs, computing summary statistics, fitting mathematical models to the data or rendering graphical layouts. Rintact provides a programmatic interface to the IntAct repository and allows the use of the analytic methods provided by R and Bioconductor. AVAILABILITY: Rintact is freely available at http://bioconductor.org

MOTIVATION: Proteins work together to drive biological processes in cellular machines. Summarizing global and local properties of the set of protein interactions, the interactome, is necessary for describing cellular systems. We consider a relatively simple per-protein feature of the interactome: the number of interaction partners for a protein, which in graph terminology is the degree of the protein. RESULTS: Using data subject to both stochastic and systematic sources of false positive and false negative observations, we develop an explicit probability model and resultant likelihood method to estimate node degree on portions of the interactome assayed by bait-prey technologies. This approach yields substantial improvement in degree estimation over the current practice that naively sums observed edges. Accurate modeling of observed data in relation to true but unknown parameters of interest gives a formal point of reference from which to draw conclusions about the system under study. AVAILABILITY: All analyses discussed in this text can be performed using the ppiStats and ppiData packages available through the Bioconductor project (http://www.bioconductor.org).