Conformal symmetries, i.e.\ coordinate transformations that preserve angles, play a key role in many fields, including physics, mathematics, computer vision and (geometric) machine learning. Here we build a neural network that is equivariant under general conformal transformations. To achieve this, we lift data from flat Euclidean space to Anti de Sitter (AdS) space. This allows us to exploit a known correspondence between conformal transformations of flat space and isometric transformations on the AdS space. We then build upon the fact that such isometric transformations have been extensively studied on general geometries in the geometric deep learning literature. We employ message-passing layers conditioned on the proper distance, yielding a computationally efficient framework. We validate our model on tasks from computer vision and statistical physics, demonstrating strong performance, improved generalization capacities, and the ability to extract conformal data such as scaling dimensions from the trained network.

Field-based reactive control provides a minimalist, decentralized route to guiding robots that lack onboard computation. Such schemes are well suited to resource-limited machines like microrobots, yet implementation artifacts, limited behaviors, and the frequent lack of formal guarantees blunt adoption. Here, we address these challenges with a new geometric approach called artificial spacetimes. We show that reactive robots navigating control fields obey the same dynamics as light rays in general relativity. This surprising connection allows us to adopt techniques from relativity and optics for constructing and analyzing control fields. When implemented, artificial spacetimes guide robots around structured environments, simultaneously avoiding boundaries and executing tasks like rallying or sorting, even when the field itself is static. We augment these capabilities with formal tools for analyzing what robots will do and provide experimental validation with silicon-based microrobots. Combined, this work provides a new framework for generating composed robot behaviors with minimal overhead.

Transformers with linear attention offer significant computational advantages over softmax-based transformers but often suffer from degraded performance. The symmetric power (sympow) transformer, a particular type of linear transformer, addresses some of this performance gap by leveraging symmetric tensor embeddings, achieving comparable performance to softmax transformers. However, the finite capacity of the recurrent state in sympow transformers limits their ability to retain information, leading to performance degradation when scaling the training or evaluation context length. To address this issue, we propose the conformal-sympow transformer, which dynamically frees up capacity using data-dependent multiplicative gating and adaptively stores information using data-dependent rotary embeddings. Preliminary experiments on the LongCrawl64 dataset demonstrate that conformal-sympow overcomes the limitations of sympow transformers, achieving robust performance across scaled training and evaluation contexts.

In this article we investigate the effects of conformal transformations on kernel functions used in Support Vector Machines. Our focus lies in the task of text document categorization, which involves assigning each document to a particular category. We introduce a new Gaussian Cosine kernel alongside two conformal transformations. Building upon previous studies that demonstrated the efficacy of conformal transformations in increasing class separability on synthetic and low-dimensional datasets, we extend this analysis to the high-dimensional domain of text data. Our experiments, conducted on the Reuters dataset on two types of binary classification tasks, compare the performance of Linear, Gaussian, and Gaussian Cosine kernels against their conformally transformed counterparts. The findings indicate that conformal transformations can significantly improve kernel performance, particularly for sub-optimal kernels. Specifically, improvements were observed in 60% of the tested scenarios for the Linear kernel, 84% for the Gaussian kernel, and 80% for the Gaussian Cosine kernel. In light of these findings, it becomes clear that conformal transformations play a pivotal role in enhancing kernel performance, offering substantial benefits.

We propose conformal generative modeling, a framework for generative modeling on 2D surfaces approximated by discrete triangle meshes. Our approach leverages advances in discrete conformal geometry to develop a map from a source triangle mesh to a target triangle mesh of a simple manifold such as a sphere. After accounting for errors due to the mesh discretization, we can use any generative modeling approach developed for simple manifolds as a plug-and-play subroutine. We demonstrate our framework on multiple complicated manifolds and multiple generative modeling subroutines, where we show that our approach can learn good estimates of distributions on meshes from samples, and can also learn simultaneously from multiple distinct meshes of the same underlying manifold.