Large language models use high-dimensional latent spaces to encode and process textual information.Much work has investigated how the conceptual content of words translates into geometrical relationships between their vector representations.Fewer studies analyze how the cumulative information of an entire prompt becomes condensed into individual embeddings under the action of transformer layers.We use literary pieces to show that information about intangible, rather than factual, aspects of the prompt are contained in deep representations.We observe that short excerpts (10 -100 tokens) from different novels separate in the latent space independently from what next-token prediction they converge towards.Ensembles from books from the same authors are much more entangled than across authors, suggesting that embeddings encode stylistic features.This geometry of style may have applications for authorship attribution and literary analysis, but most importantly reveals the sophistication of information processing and compression accomplished by language models.

Large language models (LLMs) demonstrate remarkable emergent abilities to perform in-context learning across various tasks, including time series forecasting. This work investigates LLMs' ability to estimate probability density functions (PDFs) from data observed in-context; such density estimation (DE) is a fundamental task underlying many probabilistic modeling problems. We leverage the Intensive Principal Component Analysis (InPCA) to visualize and analyze the in-context learning dynamics of LLaMA-2 models. Our main finding is that these LLMs all follow similar learning trajectories in a low-dimensional InPCA space, which are distinct from those of traditional density estimation methods like histograms and Gaussian kernel density estimation (KDE). We interpret the LLaMA in-context DE process as a KDE with an adaptive kernel width and shape. This custom kernel model captures a significant portion of LLaMA's behavior despite having only two parameters. We further speculate on why LLaMA's kernel width and shape differs from classical algorithms, providing insights into the mechanism of in-context probabilistic reasoning in LLMs. Our codebase, along with a 3D visualization of an LLM's in-context learning trajectory, is publicly available at https://github.com/AntonioLiu97/LLMICL_inPCA.

Large Language Models achieve next-token prediction by transporting a vectorized piece of text (prompt) across an accompanying embedding space under the action of successive transformer layers. The resulting high-dimensional trajectories realize different contextualization, or 'thinking', steps, and fully determine the output probability distribution. We aim to characterize the statistical properties of ensembles of these 'lines of thought.' We observe that independent trajectories cluster along a low-dimensional, non-Euclidean manifold, and that their path can be well approximated by a stochastic equation with few parameters extracted from data. We find it remarkable that the vast complexity of such large models can be reduced to a much simpler form, and we reflect on implications.

We study LLMs' ability to extrapolate the behavior of various dynamical systems, including stochastic, chaotic, continuous, and discrete systems, whose evolution is governed by principles of physical interest.Our results show that LLaMA-2, a language model trained on text, achieves accurate predictions of dynamical system time series without fine-tuning or prompt engineering.Moreover, the accuracy of the learned physical rules increases with the length of the input context window, revealing an in-context version of a neural scaling law.Along the way, we present a flexible and efficient algorithm for extracting probability density functions of multi-digit numbers directly from LLMs. 1

Inferring electromagnetic (EM) propagation characteristics within the marine atmospheric boundary layer (MABL) from data in real time is crucial for modern maritime navigation and communications. The propagation of EM waves is well modeled by a partial differential equation (PDE): a Helmholtz equation. A natural way to solve the MABL characterization inverse problem is to minimize what is observed and what is predicted by the PDE. However, this optimization is difficult because it has many local minima. We propose an alternative solution that relies on the properties of the PDE but does not involve solving the full forward model. Ducted environments result in an EM field which can be decomposed into a few propagating, trapped modes. These modes are a subset of the solutions to a Sturm-Liouville eigenvalue problem. We design a new objective function that measures the distance from the observations to a subspace spanned by these eigenvectors. The resulting optimization problem is much easier than the one that arises in the standard approach, and we show how to solve the associated nonlinear eigenvalue problem efficiently, leading to a real-time method.

Predicting in situ maritime electromagnetic (EM) propagation conditions is of great importance to radar operations within the marine atmospheric boundary layer. To characterize the EM propagation conditions, at a specific location in time and space, the current research constructs (offline) a library of sparsely sampled EM coverage data, expressed as proper orthogonal modes, so as to enable the solution of an inverse problem for the current EM propagation conditions (online). The online inversion is effected within a context that exploits an implied differential geometric structure associated with the manifold containing the proper orthogonal mode library entries.