We introduce Pion, a spectrum-preserving optimizer for large language model (LLM) training based on orthogonal equivalence transformation. Unlike additive optimizers such as Adam and Muon, Pion updates each weight matrix through left and right orthogonal transformations, preserving its singular values throughout training. This yields an optimization mechanism that modulates the geometry of weight matrices while keeping their spectral norm fixed. We derive the Pion update rule, systematically examine its design choices, and analyze its convergence behavior along with several key properties. Empirical results show that Pion offers a stable and competitive alternative to standard optimizers for both LLM pretraining and finetuning.

The Procrustes-Wasserstein problem consists in matching two high-dimensional point clouds in an unsupervised setting, and has many applications in natural language processing and computer vision. We consider a planted model with two datasets $X,Y$ that consist of $n$ datapoints in $\mathbb{R}^d$, where $Y$ is a noisy version of $X$, up to an orthogonal transformation and a relabeling of the data points. This setting is related to the graph alignment problem in geometric models.In this work, we focus on the euclidean transport cost between the point clouds as a measure of performance for the alignment. We first establish information-theoretic results, in the high ($d \gg \log n$) and low ($d \ll \log n$) dimensional regimes. We then study computational aspects and propose the'Ping-Pong algorithm', alternatively estimating the orthogonal transformation and the relabeling, initialized via a Franke-Wolfe convex relaxation. We give sufficient conditions for the method to retrieve the planted signal after one single step. We provide experimental results to compare the proposed approach with the state-of-the-art method of Grave et al. (2019).

A longstanding challenge in neuroscience is to understand neural mechanisms underlying the brain's remarkable ability to learn and detect transformations of objects due to motion. Translations and rotations of images can be viewed as orthogonal transformations in the space of pixel intensity vectors. Every orthogonal transformation can be decomposed into rotations within irreducible two-dimensional subspaces (or representations). For sets of commuting transformations, known as toroidal groups, Cohen and Welling proposed a mathematical framework for learning the irreducible representations. We explore the possibility that the brain also learns irreducible representations using a biologically plausible learning mechanism. The first is based on SVD of the anti-symmetrized outer product of the vectors representing consecutive images and is implemented by a single-layer neural network. The second is based on PCA of the difference between consecutive frames and is implemented in a two-layer network but with greater biological plausibility. Both networks learn image rotations (replicating Cohen and Welling's results) as well as translations. It would be interesting to search for the proposed networks in nascent connectomics and physiology datasets.

Another recent contribution is that of Grave et al. [2019], where a method is proposed to

Embedding models trained separately on similar data often produce representations that encode stable information but are not directly interchangeable. This lack of interoperability raises challenges in several practical applications, such as model retraining, partial model upgrades, and multimodal search. Driven by these challenges, we study when two sets of embeddings can be aligned by an orthogonal transformation. We show that if pairwise dot products are approximately preserved, then there exists an isometry that closely aligns the two sets, and we provide a tight bound on the alignment error. This insight yields a simple alignment recipe, Procrustes post-processing, that makes two embedding models interoperable while preserving the geometry of each embedding space. Empirically, we demonstrate its effectiveness in three applications: maintaining compatibility across retrainings, combining different models for text retrieval, and improving mixed-modality search, where it achieves state-of-the-art performance.

Another recent contribution is that of Grave et al. [2019], where a method is proposed to