Approximating SPR Distance Between Phylogenetic Trees with Graph Neural Networks
Authors: Renata Martins Castanheira, Miguel Bugalho, Cátia Vaz
Organizations: ISEL – Instituto Superior de Engenharia de Lisboa, Instituto Politécnico de Lisboa, Portugal · ALGORITMI Research Center/CCPM, Portugal · INESC-ID Lisboa, Portugal
Abstract
Comparing phylogenetic tree topologies is essential for understanding epidemic dynamics, yet biologically meaningful distances such as the Subtree Prune and Regraft (SPR) distance are NP-hard to compute and intractable on large datasets. We investigate whether a Graph Neural Network (GNN) can approximate SPR distances in near-constant time per comparison after training. Our contributions are fourfold. First, we build and publicly release a dataset of 864 phylogenetic trees inferred with UPGMA and Neighbor-Joining over four bacterial species, spanning up to 9{,}500 isolates, together with 388 labelled tree pairs. Second, we establish a reproducible pre-processing pipeline including midpoint re-rooting, which reduces tree depth and supplies the rooting required for exact distance computation and for the model's root-based features. Third, we validate the supervision target: on small trees, where exact SPR is tractable, the unrooted phangorn::SPR.dist heuristic correlates almost perfectly with the exact rooted distance computed by rspr (Pearson 0.98--0.99), making it an excellent monotonic surrogate. Lastly, we train a Siamese Graph Isomorphism Network (GIN) regressor. In-distribution, i.e., held-out trees from the same species and size range as training, it explains roughly 87--90% of the variance (R2≈0.87 on a held-out split; 0.90±0.19 under stratified cross-validation), with about four times lower error than a mean-predictor baseline, and shows partial transfer to unseen species (R2≈0.37). Its main limitation is extrapolation to trees larger than those seen in training, where accuracy collapses. The released dataset and the validated heuristic versus exact relationship provide a reproducible basis for scaling learned SPR approximation.
In this work, we introduce a Tropical Axial Attention neural reasoning architecture that replaces vanilla softmax dot-product attention with max-plus operators, inducing a piecewise-linear structure aligned with dynamic programming formulations. From multi-species sequence alignments, our model learns all possible pairwise distances and is trained using a combination of ℓ1 and tropical symmetric distance metric losses with an ultrametric violation penalty. We leverage the well known isomorphic relationship between the space of all phylogenetic trees with n species and tropical Grassmannian to show that tropical attention provides a natural geometric framework for phylogenetic inference. On empirical DS1−DS11 alignments, where true trees are unknown, the tropical model produces distance matrices that are substantially closer to their BME-induced tree metrics than the baseline models. These results suggest that tropical attention is a useful geometric inductive bias for neural phylogenetic inference, especially under distribution shift and when tree-metric consistency is important.
Phylogenetics is a branch of computational biology that studies the evolutionary relationships among biological entities. Its long history and numerous applications notwithstanding, inference of phylogenetic trees from sequence data remains challenging: the high complexity of tree space poses a significant obstacle for the current combinatorial and probabilistic techniques. In this paper, we adopt the framework of generative flow networks (GFlowNets) to tackle two core problems in phylogenetics: parsimony-based and Bayesian phylogenetic inference. Because GFlowNets are well-suited for sampling complex combinatorial structures, they are a natural choice for exploring and sampling from the multimodal posterior distribution over tree topologies and evolutionary distances. We demonstrate that our amortized posterior sampler, PhyloGFN, produces diverse and high-quality evolutionary hypotheses on real benchmark datasets. PhyloGFN is competitive with prior works in marginal likelihood estimation and achieves a closer fit to the target distribution than state-of-the-art variational inference methods. Our code is available at https://github.com/zmy1116/phylogfn.
Graphs with a simple spectrum admit cubic-time isomorphism testing, yet we prove that for every natural number k, the k-Weisfeiler-Leman (k-WL) test cannot distinguish all non-isomorphic graphs with a simple spectrum. As the WL hierarchy upper-bounds the distinguishing power of widely-used Graph Neural Networks (GNNs), this incompleteness applies to all such GNNs, ruling out completeness for every k-WL-aligned GNN family. To close this gap, we introduce PRiSM (Partition, Refine, Solve, Match), the first provably complete canonicalization of simple-spectrum eigendecompositions. PRiSM obtains the completeness guarantee that prior canonicalizations provably lack, and resolves the open problem of achieving complete expressivity on simple-spectrum graphs. When composed with DeepSets or a Transformer, PRiSM achieves universal approximation on simple-spectrum graphs, justifying the use of canonicalized Laplacian positional encodings. Empirically, PRiSM performs comparably to or outperforms existing spectral canonicalizations on graph regression, classification, and expressivity