Fast dynamical similarity analysis

TL;DR

The paper tackles the computational bottleneck of dynamical similarity analysis by introducing fastDSA, which combines automatic rank selection via SVHT with three efficient vector-field alignment optimizers. It preserves the core invariances and sensitivities of the original DSA while delivering substantial speedups, enabling scalable comparisons across neural networks, circuits, and data-model pairs. The authors validate fastDSA on synthetic and controlled dynamical scenarios, showing robustness to geometric deformations, effective detection of dynamical shifts, and superior sensitivity to subtle changes compared to KWDSA, with practical implications for large-scale neuroscience and AI research. Limitations include embedding-parameter sensitivity and SVHT caveats at extreme SNRs, motivating future work on adaptive parameter selection and integration with noisy-DMD variants.

Abstract

To understand how neural systems process information, it is often essential to compare one circuit with another, one brain with another, or data with a model. Traditional similarity measures ignore the dynamical processes underlying neural representations. Dynamical similarity methods offer a framework to compare the temporal structure of dynamical systems by embedding their (possibly) nonlinear dynamics into a globally linear space and there computing conjugacy metrics. However, identifying the best embedding and computing these metrics can be computationally slow. Here we introduce fast Dynamical Similarity Analysis (fastDSA), which is computationally far more efficient than previous methods while maintaining their accuracy and robustness. FastDSA introduces two key components that boost efficiency: (1) automatic selection of the effective model order of the Hankel (delay) embedding from the data via a data-driven singular-value threshold that identifies the informative subspace and discards noise to lower computational cost without sacrificing signal, and (2) a novel optimization procedure and objective, which replaces the slow exact orthogonality constraint in finding a minimal distance between dynamics matrices with a lightweight process to keep the search close to the space of orthogonal transformations. We demonstrate that fastDSA is at least an order of magnitude faster than the previous methods. Furthermore, we demonstrate that fastDSA has the properties of its ancestor, including its invariances and sensitivities to system dynamics. FastDSA, therefore, provides a computationally efficient and accurate method for dynamical similarity analysis.

Figures (7)

• Figure 1: Benchmark of delay–embedding selectors on synthetic Lorenz data. Columns: method = parameter–selection procedure; channel = state variable used ($x,y,z$); $N$ = number of time samples; time_sec = wall-clock runtime per run (seconds); mem_MB = peak resident memory (MB); $(\tau,\mu)$ = selected delay interval and number of delays; quality_NRMSE = normalized RMSE of DMD reconstruction on the holdout set (lower is better).
• Figure 2: Algorithmic representations of DSA, fastDSA variants, and kwDSA:(a) DSA. (b) Regularized optimization (RO-fastDSA). (c) On-manifold Riemannian method (Rim-fastDSA). (d) Landing algorithm (Land-fastDSA). (e) kwDSA.
• Figure 3: Schematic overview of methods for estimating dynamic (dis)similarity: DSA, family of fastDSA methods, and kwDSA: (Top) Depict the overall pipeline common to all methods. In particular, the similarity computation is decomposed into three consecutive steps: lifting to a higher-dimensional representation (purple), $(x_1, x_2, \ldots, x_n) \;\longmapsto\; (\tilde{x}_1, \tilde{x}_2, \ldots, \tilde{x}_m)$, where $m > n$, linear approximation of the dynamics based on the Koopman theory (green), and optimization of the similarity transform (brown). (Bottom) The chart summarizes how each method (DSA, the three fastDSA variants, and kwDSA) instantiates these three stages. In summary, all three fastDSA variants employ automatic rank detection in the linear approximation step. For kwDSA there is an implicit lifting to higher dimension through a kernel function bernhard2004bernhard2001. Furthermore, for kwDSA, the automatic rank selection can also be applied. At the optimization stage, however, the methods differ substantially, including the three fastDSA variants, which leads to distinct computational and accuracy profiles across algorithms (see section \ref{['sec:fastDSA-effic-assess']} for more details).
• Figure 4: Demonstration of automatic rank reduction with SVHT:(a) (top-left) Synthetic Lorenz attractor. (top-right) Depiction of the projected Lorenz attractor timeseries into a higher-dimensional (128-dimensional) observation space kapoor2024latent. X-axis of the heatmap indicates time steps, and the y-axis indicates the dimensions. (bottom-left) Comparison of optimal rank detection by SVHT (brown broken line) against traditional model selection criteria (AIC, red; AICc, purple; BIC, green). Due to the overlap, they are not all fully visible. SVHT accurately identifies the rank of underlying latent dynamics (rank 3) that precisely overlap with the ground truth (GT, vertical continuous blue line). AIC, AICc (corrected AIC for small samples), and BIC after the knee point on $r = 3$, exhibit a monotonic decrease in their values with increasing rank $r$. (bottom-right) Stability of SVHT-detected optimal rank under varying levels of additive noise. (b) Schematics of the process of injecting noise to synthetic Lorenz attractor data with isotropic noise (left) or non-linear noise (right), both were generated with variance $\sigma = 5$, and (bottom) DMD reconstructed data using the optimal rank detected by SVHT. (c-d) Mean Squared Error (MSE), on left, and AIC values, on right, for DMD-based reconstruction across varying levels of (c) isotropic and (d) nonlinear noise. (e) (top) MSE and (bottom) AIC of DMD-based reconstruction under nonlinear noise, and the SVHT-selected optimal rank (brown broken vertical line) for different values of SNR. Darker colors correspond to higher SNR values (lower noise levels), and brighter colors to lower SNR values (higher noise levels). SNR values are noted on top of individual columns.
• Figure 5: Speed–accuracy trade-offs between DSA and the fastDSA variants under matched experimental conditions.(a) Runtime (in seconds) as a function of matrix size for DSA (blue) and the three fastDSA variants (red, RO-fastDSA; green, Rim-fastDSA; purple, Land-fastDSA). (b) Frobenius alignment error across matrix sizes for all methods. (c) Frequency with which each method reports a (numerically) zero Euclidean alignment score (dissimilarity $< 10^{-3}$) as a function of matrix size, estimated over 10 independent runs per method and size. (d) Angular alignment error across matrix sizes for all methods. (e) Frequency with which each method reports an angular score of zero (dissimilarity $< 10^{-3}$), indicating almost perfect angular alignment. All were estimated over 10 independent runs per method and size. In all line plots, shaded regions indicate $\pm 1$ standard error of the mean across runs.