arXiv:2507. 03065v2 Announce Type: replace Abstract: Why do some macroscopic structures remain identifiable even though their microscopic constituents continually change?
arXiv:2606. 07303v4 Announce Type: replace Abstract: Representation learning is central to modern machine learning, yet most research focuses on optimizing representations after a framework has been selected.
The paper introduces persistent magnitude homology as a functorial invariant for quantitative equational theories, providing a barcode that captures the metric structure of the free algebra generated by a metric space of generators. It shows how this invariant combines graded magnitude homology with persistence, yielding stability estimates and a method to compare barcodes when theories are extended. Four concrete examples illustrate the theory in each homological degree.
The paper investigates how neural representations maintain the structure of input changes, linking representation analysis with internal interventions. It characterises when transformations can be applied through an encoder, providing linear settings where defects depend on discarded information and detailing failure modes for rectifiers and harmonic carriers. Using colour as a case study, the authors show that hue orbits in frozen visual features concentrate most energy in the first two harmonics, that this structure is inherited from input and architecture, and that a compact, fixed‑action interface can read hue zero‑shot with low error on unseen shapes.
arXiv:2506. 20699v2 Announce Type: replace Abstract: Learning in non-stationary and multi-context environments requires more than ordinary within-task generalization.
arXiv:2606. 11911v1 Announce Type: cross Abstract: Persistence diagrams are common representations in topological data analysis, but they do not naturally live in a vector space, and the statistical tools developed for comparing them have largely evolved separately from those used for downstream prediction.
How neural representations preserve the structure of input changes connects representation analysis with internal intervention. We study operable representational content through compatible actions of...
arXiv:2606. 14737v1 Announce Type: cross Abstract: Molecular dynamics (MD) simulations generate trajectories in a high-dimensional configuration space whose analysis critically depends on molecular descriptors, typically handcrafted observables or learned kinetic embeddings.
arXiv:2606. 07303v1 Announce Type: new Abstract: Representation learning is central to modern machine learning, enabling transitions from handcrafted features to learned embeddings, latent spaces, foundation models, world models, and digital twins.
arXiv:2608. 02816v1 Announce Type: new Abstract: We study the topology of learned representations in predictive coding networks (PCNs), a neuro-inspired bidirectional architecture, using a quantitative layer-wise persistent homology analysis.
arXiv:2605. 14998v3 Announce Type: replace Abstract: From subcellular structures to entire organisms, many natural systems generate complex organisation through self-organisation: local interactions that collectively give rise to global structure without any blueprint of the outcome.
arXiv:2512. 18471v2 Announce Type: replace Abstract: Continual learning systems face a fundamental geometric obstacle: as experience accumulates on a fixed-capacity manifold, covering numbers grow linearly with time, eventually forcing representational overlap and catastrophic interference.
arXiv:2512.23348v3 Announce Type: replace-cross Abstract: We introduce a data-analysis framework based on filtrations of finite topological spaces. Starting from a finite metric data set, we construc...