Learning Coherent Representations: A Topological Approach to Interpretability

Sigurd Gaukstad, Melvin Vaupel, Valdemar Kargård Olsen, Erik Hermansen, Benjamin Adric Dunn
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:34299-34333, 2026.

Abstract

Deep neural networks learn representations where individual features often lack interpretable meaning; a single neuron may activate for scattered, unrelated inputs. We introduce coherence, a geometric property inspired by neural coding in the brain, where neurons like grid cells and head direction cells respond to contiguous regions of state space. A non-negative matrix is coherent if each row (sample) attends to geometrically clustered columns (features) and vice versa, and in addition every sample is well described by some feature and every feature is needed by some sample. We prove that coherent matrices induce a bounded interleaving between the Vietoris-Rips filtrations of samples and features, guaranteeing that both spaces share compatible topological structure. This geometric constraint facilitates interpretability. For example, if data lies on a circle, coherent features must tile that circle into contiguous arcs. We introduce Coh, a differentiable objective function based on Fréchet variance that enforces coherence during training. Unlike sparsity, which bounds how many samples a feature activates on, coherence bounds which samples, requiring geometric connectivity rather than only rarity. This yields not just interpretable features but an interpretable feature space. We validate Coh in an auto-encoder using synthetic and rotated MNIST datasets and in a token embedding of BERT using language data.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-gaukstad26a, title = {Learning Coherent Representations: A Topological Approach to Interpretability}, author = {Gaukstad, Sigurd and Vaupel, Melvin and Karg{\aa}rd Olsen, Valdemar and Hermansen, Erik and Dunn, Benjamin Adric}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {34299--34333}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/gaukstad26a/gaukstad26a.pdf}, url = {https://proceedings.mlr.press/v306/gaukstad26a.html}, abstract = {Deep neural networks learn representations where individual features often lack interpretable meaning; a single neuron may activate for scattered, unrelated inputs. We introduce coherence, a geometric property inspired by neural coding in the brain, where neurons like grid cells and head direction cells respond to contiguous regions of state space. A non-negative matrix is coherent if each row (sample) attends to geometrically clustered columns (features) and vice versa, and in addition every sample is well described by some feature and every feature is needed by some sample. We prove that coherent matrices induce a bounded interleaving between the Vietoris-Rips filtrations of samples and features, guaranteeing that both spaces share compatible topological structure. This geometric constraint facilitates interpretability. For example, if data lies on a circle, coherent features must tile that circle into contiguous arcs. We introduce Coh, a differentiable objective function based on Fréchet variance that enforces coherence during training. Unlike sparsity, which bounds how many samples a feature activates on, coherence bounds which samples, requiring geometric connectivity rather than only rarity. This yields not just interpretable features but an interpretable feature space. We validate Coh in an auto-encoder using synthetic and rotated MNIST datasets and in a token embedding of BERT using language data.} }
Endnote
%0 Conference Paper %T Learning Coherent Representations: A Topological Approach to Interpretability %A Sigurd Gaukstad %A Melvin Vaupel %A Valdemar Kargård Olsen %A Erik Hermansen %A Benjamin Adric Dunn %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-gaukstad26a %I PMLR %P 34299--34333 %U https://proceedings.mlr.press/v306/gaukstad26a.html %V 306 %X Deep neural networks learn representations where individual features often lack interpretable meaning; a single neuron may activate for scattered, unrelated inputs. We introduce coherence, a geometric property inspired by neural coding in the brain, where neurons like grid cells and head direction cells respond to contiguous regions of state space. A non-negative matrix is coherent if each row (sample) attends to geometrically clustered columns (features) and vice versa, and in addition every sample is well described by some feature and every feature is needed by some sample. We prove that coherent matrices induce a bounded interleaving between the Vietoris-Rips filtrations of samples and features, guaranteeing that both spaces share compatible topological structure. This geometric constraint facilitates interpretability. For example, if data lies on a circle, coherent features must tile that circle into contiguous arcs. We introduce Coh, a differentiable objective function based on Fréchet variance that enforces coherence during training. Unlike sparsity, which bounds how many samples a feature activates on, coherence bounds which samples, requiring geometric connectivity rather than only rarity. This yields not just interpretable features but an interpretable feature space. We validate Coh in an auto-encoder using synthetic and rotated MNIST datasets and in a token embedding of BERT using language data.
APA
Gaukstad, S., Vaupel, M., Kargård Olsen, V., Hermansen, E. & Dunn, B.A.. (2026). Learning Coherent Representations: A Topological Approach to Interpretability. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:34299-34333 Available from https://proceedings.mlr.press/v306/gaukstad26a.html.

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