Identifiability of latent dynamics

Identifiability asks whether the thing a model recovers is determined by the data or is one of many equally good alternatives. For latent dynamical models the answer is usually the latter: they are generically identifiable only up to transformations. A linear-Gaussian state-space model is fixed only up to an invertible linear change of the latent basis (with rotations and scalings), and a nonlinear latent model is, without extra structure, identifiable only up to smooth invertible reparameterizations of the latent space — so the axes, the geometry of the inferred manifold, and the specific form of the vector field are not by themselves meaningful.

This has hard consequences for interpretation. Two networks trained on the same data can find latent dynamics that look different yet are the same up to a warp; conversely, matching latent trajectories across animals or models requires methods invariant to these transformations. Progress comes from adding identifying structure: known inputs or interventions, auxiliary variables (as in identifiable-VAE / iVAE results that recover latents up to permutation and scaling given side information), sparsity, or physical constraints. The honest reporting standard is to state the invariances explicitly and to base claims on quantities that are invariant to them (topology of fixed points, spectra, decoding performance) rather than on raw latent coordinates.