2026-09-22
DSA compares systems via Koopman spectra. Mouse compass data matches ring attractors; macaque motor cortex drifts on one reach task over years despite stable kinematics.
Geometric metrics compare the shape of a point cloud. Topological ones compare how a manifold is connected. For brains, RNNs, and climate, the object that actually defines the system is the flow field: how states move through time.
The strongest notion of equivalence is smooth topological conjugacy, a smooth invertible map that turns one vector field into the other. Computing that distance means searching an infinite-dimensional map space, and experimental systems are usually partially observed, noisy, and driven by time-varying inputs. CKA, Procrustes, and RSA need time alignment. MMD and TDA throw the time order away. Neural comparators scale poorly once you need thousands of pairwise distances. A direct metric on dynamics has been missing.
DSA relaxes conjugacy into three computations that run on raw time series.
Lengthening the prediction horizon s lets slow modes dominate fast ones, so the same distance can be sliced by timescale. Entropic regularization turns Wasserstein into a Sinkhorn distance, which is differentiable, so DSA can be a training objective.
On synthetic ring attractors, cut rings, and limit cycles, delay DSA linearly decodes the dynamics class at 100% accuracy with a silhouette score of 0.67. Most geometric and topological baselines sit near silhouette 0. Even with tiny cuts that leave the point cloud indistinguishable, DSA still separates the classes; MMD, Procrustes, and MARBLE do not. MARBLE is about 1000× slower. Sweeping the Lorenz parameter ρ in the chaotic regime, spectral DSA correlates with the difference in largest Lyapunov exponent at R² = 0.859, PAVF at 0.849, MARBLE at 0.61, MMD at 0.036, Procrustes at 0.005.
On mouse anterodorsal thalamic head-direction data, DSA places the recordings with neural-network ring attractors and away from cut rings, limit cycles, and time-shuffled controls (N=12 sessions, Cliff's δ=1.00, zero overlap). Distances from the NN ring to data fall inside the interquartile range of animal-to-animal distances. Rankings hold when observational noise matches data variance and when only 1% of neurons are observed. Short-horizon DSA groups wake with REM; long-horizon DSA groups wake with slow-wave sleep, matching the known split between fast radial relaxation and slower angular sweeps.
In macaque reaching, hand kinematics stay similar across animals and across Center-Out versus Random-Target (cross-task Cliff's δ=0.05, permutation n.s.). Cortical dynamics split by task (p=9.00×10^{-5}, δ=0.36) and more so between preparatory and movement epochs (δ=0.66). Center-Out neural dynamics stay stable across years (R²=0.006). Random-Target dynamics drift (R²=0.384, p=9.995×10^{-4}). CCA on trajectory geometry misses this. PMd and M1 become dynamically distinct around movement onset, as a persistent rotational mode appears in M1.
Used as a repulsive regularizer, DSA finds that 430 integration RNNs trained the usual way collapse near line attractors. Five iterative ∂DSA runs recover a feedforward chain, a planar attractor, an oscillating integrator, and a hybrid that treats positive and negative inputs differently, all with low task error. On 3-bit flip-flop, it needs about one-tenth the iterations of Qian and Pehlevan. The paper reports on the order of 2 million pairwise DSA comparisons.
This is a missing ruler for neuroscience and RNN interpretability: you can ask whether two systems are doing the same thing without aligning trajectories or hand-crafting features. pip install dsa-metric is enough to try it. Model-to-data checks, cross-animal comparisons, and mining unconventional solutions as a training regularizer are the obvious uses. It is a tool upgrade. Geometry still answers shape questions. DSA answers flow-field questions. It does not replace CKA.
Distances have no absolute scale and stretch nonlinearly with hyperparameters, so they have to be read against a within-condition baseline. The default spectral distance is a pseudometric: isospectral operators with different modes (and therefore different non-normal transients) score as similar. Embedding dimension, operator rank, delay length, and sampling rate need AIC and ResDMD; a bad choice can wash out the comparison. The macaque drift result has 70 Center-Out sessions versus 16 Random-Target sessions, so practice volume is a live confound; the paper subsampled CO and still saw a flat slope, but the mechanism is open. DSA is symmetric: it can score coupling strength, not direction.