Papers for
data visualization specialists
Papers whose findings have a practical use for this group, as judged from the abstract. Open a paper to read what it means in practice.
Language model weekday concepts behave as points not smooth circles
Spokes, not a Ring: Revisiting the Weekday Manifold in Language Models
Abstract: Recent work has hypothesised that large language models may represent some concepts as multi-dimensional manifolds instead of single directions. Using these manifolds to interpret or steer representations often assumes that moving along them corresponds to a continuous change in the represented concept. In this work, we test this continuous manifold assumption on the prototypical weekday manifold. Despite its striking circular geometry, we find that the weekday representations behave more like seven discrete states arranged in a circle than a continuous ring. First, we show that local sensitivity has seven minima aligned with the weekdays, producing a step-like mapping between representation and behaviour under on-manifold steering. Second, we test whether any concepts are represented along the manifold outside the seven discrete states, and find neither time of day, uncertainty, nor natural text populate the intermediate space. Together, these results challenge the continuous interpretation of the weekday manifold and motivate testing whether movement along other representation manifolds correspond to meaningful changes in model representations and behaviour.
Deep learning reveals complex geometric structures in data spaces
The Geometry of Manifold Learning: Singular foliations via Deep Learning models
Abstract: Deep learning algorithms have become widely used and successful across a wide range of tasks. A key challenge is determining whether these algorithms truly capture the intrinsic geometry of their training data in order to better understand what they learn from data. In this theoretical work, we explore a new geometric structure defined on the data space by a trained neural network through the Data Information Matrix (DIM), a variation of the Fisher Information Matrix. This new geometric structure is a foliation and the main focus of this work is to explore the non-smooth case of ReLU activation functions, where traditional geometric tools often fail. The foliation is defined through a distribution spanned by the columns of the DIM. We first prove that this distribution is non-smooth in the sense of distributions, rendering the existence of an associated foliation much more complex. To circumvent this issue, we introduce a smoothed version of this distribution that differs from the original only on a set of measure zero. Finally, we prove that this smoothed distribution induces a singular foliation on the data space, which is based on the data the network has been trained with. Our theoretical contributions provide a geometric framework to better understand how neural networks organize their input space in non-smooth settings, bridging differential geometry and deep learning theory.