Thanks to Jasmina Urdshals, Xavier Poncini, and Justis Mills for comments.

Introduction

At Simplex our mission is to develop a principled science of the representations and emergent behaviors of AI systems. Our initial work showed that transformers linearly represent belief state geometries in their residual streams. We think of that work as providing the first steps into an understanding of what fundamentally we are training AI systems to do, and what representations we are training them to have.

Since that time, we have used that framework to make progress in a number of directions, which we will present in the sections below. The projects ask, and provide answers to, the following questions:

How, mechanistically, do transformers use attention to geometrically arrange their activations according to the belief geometry?
What is the nature of in-distribution in-context learning (ICL), and how does it relate to structure in the training data?
What is the fundamental nature of computation in neural networks? What model of computation should we be thinking about when trying to make sense of these systems?

In answering these questions we find a number of surprising results, described in the rest of this post, and new ways of thinking about both specific parts of the transformer architecture, for instance what exactly the attention heads are doing, and more generally computation in neural networks overall that we believe have important implications for much of interpretability. We are currently expanding on these findings, both on the theoretical side as well as the applied side, pushing towards applications to LLMs at scale.

In addition to the work presented here, we've mentored fellows at MATS and PIBBSS, ran a hackathon with PIBBSS and Apart, that had a number of interesting results^[1], and are currently scaling up our team to build on these insights, with support and funding from the Astera Institute. We will finish by mentioning some directions we are exploring and excited to work on next.

For context, we will begin with a short summary of our foundational work. We will then give paragraph-length summaries of the answers to the 3 questions above, and then we will give longer summaries of each project.

The foundational theory

In our original work, expanded on in our NeurIPS 2024 publication, we developed a general theory connecting next-token pred...

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