The work, led by Chris Teska, Kurt Pasque and Ruriko Yoshida of the Naval Postgraduate School together with Baran Hashemi of the Max Planck Institute for Mathematics in the Sciences, targets a specific bottleneck in evolutionary tree reconstruction. Standard distance-based methods, such as Neighbor Joining and Balanced Minimum Evolution, first estimate all pairwise evolutionary distances between species from a multiple sequence alignment, then fit a tree to that distance matrix. The second step is notoriously difficult: finding the optimal tree under the Balanced Minimum Evolution criterion is NP-hard, and the number of possible tree topologies grows super-exponentially with the number of species. Even for modest datasets, exhaustive search is hopeless.

Recent neural approaches have tried to shortcut this search. Phyloformer and its successor Phyloformer 2 use transformer models that ingest sequence alignments and output distance matrices or distributions over trees, shifting the computational burden from inference time to training time. But these models operate in ordinary Euclidean space, and that, the authors argue, is precisely the problem. The set of distance matrices that actually correspond to valid phylogenetic trees is not a smooth Euclidean region. It is a polyhedral cone structure, carved out by combinatorial constraints such as the four-point condition, which states that for any four species, the largest of three sums of pairwise distances must be achieved at least twice. Euclidean regression and softmax attention, which blur comparisons into diffuse probability distributions, are poorly matched to that geometry.

The key insight comes from a branch of mathematics called tropical algebra, in which ordinary addition is replaced by taking a maximum and ordinary multiplication is replaced by addition. Under this max-plus arithmetic, polynomials become piecewise-linear functions and geometric objects become polyhedral complexes rather than smooth manifolds. In a landmark result, Speyer and Sturmfels showed in 2004 that the space of phylogenetic trees with a fixed number of leaves is isomorphic to the tropical Grassmannian, an algebraic variety defined entirely by max-plus relations. In other words, the geometry of evolutionary trees is, at its mathematical core, tropical. Tree metrics are characterized by maximum operations, not linear combinations.

The new model, called the Tropical Axial Transformer, builds this algebra directly into the attention mechanism. Instead of computing similarity between query and key vectors with a Euclidean dot product followed by softmax normalization, the model maps inputs into tropical projective space through a learnable valuation map, forms queries, keys and values via tropical matrix multiplication, and scores query-key agreement using the Hilbert projective metric, also known as the tropical metric. Attention is then aggregated by tropical matrix multiplication, which effectively selects the value that aligns best with the query rather than averaging all values with exponentially reweighted weights. The result is a piecewise-linear, scale-invariant computation whose structure mirrors the dynamic programming and combinatorial optimization that underlie tree reconstruction.

The architecture itself follows the Phyloformer blueprint: sequence alignments are one-hot encoded and embedded, with each token representing an unordered pair of species at a particular alignment position. Six axial attention blocks with four heads each then alternate attention along two axes, one across sequence positions and one across species pairs, allowing the model to integrate mutation evidence along alignments while computing the pairwise relationships that the tree constraints actually govern. A convolutional projection and a softplus activation produce non-negative distance predictions for every pair of leaves.

Training is equally geometry-aware. Rather than relying on a single regression loss, the objective combines an L1 term for coordinate accuracy, a tropical symmetric distance loss that penalizes distortions relative to the tropical projective torus, and an ultrametric violation penalty that pushes predictions toward the space of rooted trees with equal root-to-leaf distances. The ultrametric penalty activates only after twenty epochs of training, once the model has learned a reasonable representation of pairwise relationships. An ablation study confirmed that the L1 term is essential for raw accuracy, the tropical loss acts as a geometry-aware regularizer, and the full objective provides the best balance across distance and branch-length-sensitive metrics.

The empirical results are striking. On eleven real-world datasets, labeled DS1 through DS11 and spanning ribosomal RNA, mitochondrial DNA and nuclear protein-coding genes from 27 to 71 species, the tropical model achieved the lowest mean absolute error to its FastME-reconstructed tree metric on every single dataset, with error reductions averaging 81.5 percent relative to Phyloformer and 98.4 percent relative to Phyloformer 2. Because the true trees for these empirical alignments are unknown, the result does not directly measure reconstruction accuracy, but it demonstrates something arguably more important: the tropical model produces distance matrices that are far more internally consistent with valid tree structure, even when the input data lies well outside the training distribution.

On simulated data generated under the same birth-death process as the training set, the tropical model landed between the two baselines, with Phyloformer 2 showing superior topological accuracy and Phyloformer performing better on branch-length-aware metrics. But when tested on data from a different simulation regime, the PhyML benchmark generated under a General Time Reversible model with gamma-distributed rates, the tropical model matched Phyloformer 2 topologically while outperforming both competitors on branch-length-sensitive measures. The authors interpret this as evidence that the model learns genuine tree-metric structure rather than merely fitting the quirks of its training simulator, a form of distributional robustness that classical methods have always held over neural ones.

The work is part of a broader movement in neural algorithmic reasoning, in which the inductive bias of a network is deliberately aligned with the mathematical structure of the problem it is meant to solve. The same tropical attention mechanism was introduced by Hashemi and colleagues at NeurIPS 2025 for combinatorial algorithms, where softmax attention was shown to fail under out-of-distribution shifts because the gaps between top logits shrink and attention becomes diffuse. Phylogenetic inference, with its polyhedral tree space and extremal four-point constraints, turns out to be a natural fit for the same machinery. The authors acknowledge a practical limitation: the current implementation relies on PyTorch broadcast operations that create large intermediate tensors and inflate GPU memory during training, though specialized tropical matrix multiplication routines could alleviate this. If those engineering hurdles fall, the message of the paper is clear. When the geometry of a problem is fundamentally non-Euclidean, the smartest thing a neural network can do is stop pretending otherwise.

Subject of Research: Mathematics

Article Title: Phylogenetic tree inference with tropical axial attention

Article References: Teska, C., Pasque, K., Yoshida, R., & Hashemi, B. (2026). Phylogenetic tree inference with tropical axial attention. Frontiers in Systems Biology, 6, Article 1886918. https://doi.org/10.3389/fsysb.2026.1886918

Image Credits: AI Generated

DOI: 10.3389/fsysb.2026.1886918

Keywords: tropical geometry, phylogenetics, deep learning, transformers, tree inference, max-plus algebra, neural networks, algebraic geometry, Phylogenetic, tree, inference, tropical