Research map / Learning theory and algorithms / Optimal transport
Comparing noisy neural population dynamics using optimal transport distances
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Proposes Causal OT distance, a causal optimal transport metric between Gaussian processes, to compare noisy dynamic neural trajectories, and applies it to motor-cortex models and latent diffusion models.
Problem
Existing neural similarity metrics (Procrustes, SSD, DSA) assume deterministic or static responses, so they can fail to distinguish noisy dynamic neural systems that differ in across-time statistical structure.
Approach
Derive a metric as a bi-causal / causal synchronous optimal transport distance between Gaussian processes, using full across-time covariance with a temporal-causality constraint on the couplings (adapted Bures distance), which reduces to alpha-SSD when across-time correlations vanish.
Key results
- On the scalar example, causal OT distance converges to sigma as epsilon goes to 0 while SSD and Wasserstein converge to 0 and Procrustes gives 0.
- In the motor-system simulation, Procrustes and SSD distances shrink toward zero as preparatory activity shrinks, while Causal OT and DSA distances plateau and remain distinguishable.
- For latent diffusion trajectories, Causal OT and SSD show a 3x3 prompt-category block structure invariant to model version, while Procrustes and DSA do not.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Compare full noisy trajectories while enforcing temporal causality | Causal OT · Scalar example | distance: d_\text{1-causal}(x,y)=\sqrt{\sigma^2+\epsilon^2} | Helped |
| Use mean trajectories only instead of full noisy trajectories | Procrustes · Scalar example | distance: d_\text{Procrustes}(x,y)=0 | Hurt |
| Compare marginal statistics without across-time dependencies | SSD · Scalar example | distance: d_\text{1-SSD}(x,y)=\epsilon | Hurt |
| Allow acausal couplings that can change temporal ordering | Wasserstein · Scalar example | distance: d_\text{Wasserstein}(x,y)=\epsilon | Hurt |
| Shrink preparatory activity relative to motor activity | Procrustes and SSD · Synthetic motor systems | distance | Hurt |
| Shrink preparatory activity relative to motor activity | Causal OT and DSA · Synthetic motor systems | distance | Helped |
| Account for stochastic correlation structure when comparing trajectories | SSD and Causal OT · Stable Diffusion v1-1 and v1-2 prompt-conditioned trajectories | within-category versus between-category distance structure | Helped |
| Use mean trajectories only | Procrustes · Stable Diffusion v1-1 and v1-2 prompt-conditioned trajectories | prompt-category structure | Hurt |
| Use DSA instead of stochastic distances | DSA · Stable Diffusion v1-1 and v1-2 prompt-conditioned trajectories | prompt-category structure | Hurt |
| Project latent trajectories onto top principal components before computing distances | Causal OT, SSD and Procrustes · Stable Diffusion latent trajectories | normalized distance convergence: k ∈ {2,4,6,8,10,15,20} | Helped |
Surprising findings
- Decoded mean latent trajectories from the diffusion model show few prompt-related visual features, although individual sample trajectories do.
Limitations
- Captures only first- and second-order moments, so may miss differences between non-Gaussian processes.
- Estimating NT x NT covariance requires many sample trajectories when N or T is large.
- DSA depends on hyperparameters (delays, rank) that must be tuned.
Experimental setup
Datasets: synthetic linear and nonlinear dynamical systems latent trajectories sampled from Stable Diffusion v1-1 and v1-2 for 10 prompts
Metrics: Causal OT distance Procrustes shape distance alpha-SSD DSA distance Wasserstein distance pairwise distance matrices MDS embedding
Baselines: Procrustes shape distance Stochastic Shape Distance Dynamical Similarity Analysis Wasserstein distance between Gaussian processes
Builds on: Procrustes shape distance Stochastic Shape Distance Dynamical Similarity Analysis adapted Wasserstein distance Bures metric
Closest papers by meaning
- Geometry-Aware Similarity Metrics for Neural Representations on Riemannian and Statistical Manifolds (NeurIPS 2026 · similarity 0.86)
- Modeling Latent Neural Dynamics with Gaussian Process Switching Linear Dynamical Systems (NeurIPS 2024 · similarity 0.86)
- Optimal Recurrent Network Topologies for Dynamical Systems Reconstruction (ICML 2024 · similarity 0.86)
- Meta-Dynamical State Space Models for Integrative Neural Data Analysis (ICLR 2025 · similarity 0.86)
- Partial observation can induce mechanistic mismatches in data-constrained models of neural dynamics (NeurIPS 2024 · similarity 0.85)
- KoopSTD: Reliable Similarity Analysis between Dynamical Systems via Approximating Koopman Spectrum with Timescale Decoupling (ICML 2025 · similarity 0.85)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.