# CLEARMIND: Closed-Form Laplacian Embeddings of Arakelov-Green Resistance for Morphological Intrinsic Neuronal Distances

arXiv: https://arxiv.org/abs/2608.04460 · HTML: https://mlsanity.com/paper/2608.04460/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 59 references that are in the map.
- First posted on arXiv 5 Aug 2026.

## Builds on (most cited first, 20 of 20)

- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks](https://mlsanity.com/paper/1810.02244/) (2018)
- [Representation Learning on Graphs with Jumping Knowledge Networks](https://mlsanity.com/paper/1806.03536/) (2018)
- [Benchmarking Graph Neural Networks](https://mlsanity.com/paper/2003.00982/) (2020)
- [Provably Powerful Graph Networks](https://mlsanity.com/paper/1905.11136/) (2019)
- [Improved Semantic Representations From Tree-Structured Long Short-Term Memory Networks](https://mlsanity.com/paper/1503.00075/) (2015)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [Sign and Basis Invariant Networks for Spectral Graph Representation Learning](https://mlsanity.com/paper/2202.13013/) (2022)
- [Nested Graph Neural Networks](https://mlsanity.com/paper/2110.13197/) (2021)
- [Rethinking the Expressive Power of GNNs via Graph Biconnectivity](https://mlsanity.com/paper/2301.09505/) (2023)
- [How Powerful are K-hop Message Passing Graph Neural Networks](https://mlsanity.com/paper/2205.13328/) (2022)
- [Boosting the Cycle Counting Power of Graph Neural Networks with I2-GNNs](https://mlsanity.com/paper/2210.13978/) (2022)
- [An Empirical Study of Realized GNN Expressiveness](https://mlsanity.com/paper/2304.07702/) (2023)
- [Laplacian Canonization: A Minimalist Approach to Sign and Basis Invariant Spectral Embedding](https://mlsanity.com/paper/2310.18716/) (2023)
- [Affinity-Aware Graph Networks](https://mlsanity.com/paper/2206.11941/) (2022)
- [Self-Supervised Graph Representation Learning for Neuronal Morphologies](https://mlsanity.com/paper/2112.12482/) (2021)
- [A polynomial time algorithm for solving the closest vector problem in zonotopal lattices](https://mlsanity.com/paper/2004.07574/) (2020)
- [Metric Graph Kernels via the Tropical Torelli Map](https://mlsanity.com/paper/2505.12129/) (2025)
- [GraPHFormer: A Multimodal Graph Persistent Homology Transformer for the Analysis of Neuroscience Morphologies](https://mlsanity.com/paper/2603.20970/) (2026)

## Abstract (arXiv)

Tropical geometry turns a metric graph into a flat torus, its tropical Jacobian, and measures distances between the images of its points by a closest vector problem on the period lattice. We prove that for points of the graph this problem is solved by geodesics: the squared tropical polarization distance equals the path metric minus the Arakelov-Green (AG) distance, an effective-resistance kernel given in closed form by the Laplacian pseudoinverse. Equivalently, the AG distance and the path metric are the minimal energies of real and of integral unit flows, and the squared tropical distance is exactly their integrality gap. The full tropical distance matrix is thus computable in cubic time. On this identity we build CLEARMIND, a training-free descriptor of 3D neuronal morphology. A reconstruction is reduced to its branching skeleton, neurite tips near the soma are joined to it, short bridges are contracted, and the AG matrix of the result is summarized by its spectrum. Every step has an exact algebraic description; for instance, the period matrix records the shared soma-to-tip path lengths, and the spectrum has exactly one positive eigenvalue, so its absolute values determine it. The signature needs no lattice search, is invariant to vertex order, rigid motions and subdivision, and is Lipschitz in the edge lengths. On ACT-4, JML-4 and BIL-6, appending the 64-dimensional spectrum improves a point-cloud GNN and MorphVAE on every dataset, by up to 20.3 points; a Tree-LSTM with the spectrum outperforms every reimplemented baseline; and an MLP on the spectrum alone is a competitive classifier that surpasses every reimplemented deep baseline on ACT-4. On BREC, a GIN with AG eigenvector encodings distinguishes 70.0% of the pairs beyond the 1-WL limit, including 33% of the CFI pairs, more than PPGN (23%) and I2-GNN (21%), and the spectrum alone separates 217 of 400 pairs without training.