Riccardo Ali

Riccardo Ali, Paulina Kulytė, Haitz Sáez de Ocáriz Borde et al.

Clifford Group Equivariant Neural Networks (CGENNs) leverage Clifford algebras and multivectors as an alternative approach to incorporating group equivariance to ensure symmetry constraints in neural representations. In principle, this formulation generalizes to orthogonal groups and preserves equivariance regardless of the metric signature. However, previous works have restricted internal network representations to Euclidean or Minkowski (pseudo-)metrics, handpicked depending on the problem at hand. In this work, we propose an alternative method that enables the metric to be learned in a data-driven fashion, allowing the CGENN network to learn more flexible representations. Specifically, we populate metric matrices fully, ensuring they are symmetric by construction, and leverage eigenvalue decomposition to integrate this additional learnable component into the original CGENN formulation in a principled manner. Additionally, we motivate our method using insights from category theory, which enables us to explain Clifford algebras as a categorical construction and guarantee the mathematical soundness of our approach. We validate our method in various tasks and showcase the advantages of learning more flexible latent metric representations. The code and data are available at https://github.com/rick-ali/Metric-Learning-for-CGENNs

Riccardo Ali, Alessio Borgi, Christopher Irwin et al.

Rotary Positional Encoding (RoPE) is widely used in modern large language models. However, when sequences are extended beyond the range seen during training, rotary phases can enter out-of-distribution regimes, leading to spurious long-range alignments, diffuse attention, and degraded retrieval. Existing remedies only partially address these failures, as they often trade local positional resolution for long-context stability. We propose GAPE (Gated Adaptive Positional Encoding), a drop-in augmentation for positional encodings that introduces a content-aware bias directly into the attention logits while preserving the rotary geometry. GAPE decouples distance-based suppression from token importance through a query-dependent gate that contracts irrelevant context and a key-dependent gate that preserves salient distant tokens. We prove that protected tokens remain accessible, while the attention mass assigned to unprotected distant tokens decays as a function of the query gate. We further show that GAPE can be implemented within standard scaled dot-product attention. We validate these properties empirically, finding that GAPE consistently yields sharper attention and improved long-context robustness over rotary baselines across both synthetic retrieval and long-context benchmarks.

Riccardo Ali, Francesco Caso, Christopher Irwin et al.

Transformer models map input token sequences to output token distributions, layer by layer. While most interpretability work focuses on internal latent representations, we study the evolution of these token-level distributions directly in vocabulary space. However, such distributions are high-dimensional and defined on an unordered support, making common descriptors like moments or cumulants ill-suited. We address this by computing the Shannon entropy of each intermediate predicted distribution, yielding one interpretable scalar per layer. The resulting sequence, the entropy profile, serves as a compact, information-theoretic signature of the model's computation. We introduce Entropy-Lens, a model-agnostic framework that extracts entropy profiles from frozen, off-the-shelf transformers. We show that these profiles (i) reveal family-specific computation patterns invariant under depth rescaling, (ii) are predictive of prompt type and task format, and (iii) correlate with output correctness. We further show that Rényi entropies yield similar results within a broad range of $α$ values, justifying the use of Shannon entropy as a stable and principled summary. Our results hold across different transformers, without requiring gradients, fine-tuning, or access to model internals.

Riccardo Ali, Pietro Liò, Jamie Vicary

Equivariant neural networks incorporate symmetries through group actions, embedding them as an inductive bias to improve performance on a wide variety of tasks. However, existing equivariant methods can be computationally intensive, with high parameter counts, and are often tied to a specific architecture. We propose a simple zero-parameter approach that imposes approximate equivariance for a finite group in the latent representation, as an additional term in the loss function. We conduct experiments which allow the network to learn a group representation on the latent space, and show in every case it prefers to learn the regular representation. Fixing this action on the latent space, this yields a simple method to impose approximate equivariance as an additional loss penalty. We benchmark our approach on three datasets and compare it against several existing equivariant methods, showing that in many cases it achieves similar or better performance for a fraction of the parameters.