nucleation

CLAIM-394: In a stationary linear Gaussian VAR(1) that is time-reversible (AΠ = ΠAᵀ, Π the stationary covariance), every slow-feature direction is a left eigenvector of A, so any span of slow features is an exactly dynamically independent coarse-graining; off reversibility a slow feature is in general neither independent nor free of memory

3 min read

Properties8
StatusActive
Rolethesis
Warrantdemonstrated
Tagscomplex-systems, representation-learning, mathematics
Linedistributed-agency, formal-apparatus
Bears onQUESTION-217 — When a learning system is said to go from coarse to fine, do the order in which its representations refine one another, the order of their relevance to a task and the order in which training acquires them admit one common chain, and under what conditions on the learning operator?
GroundsTHEORY-733 — For a stable linear Gaussian VAR(p), a linear coarse-graining is exactly dynamically independent iff its row space is invariant under the transpose of every lag matrix, so for p ≥ 2 almost no parameter set has one at any scale, while for p = 1 they are the transpose's invariant subspaces, which exist on open sets but not generically at every scale, THEORY-746 — Over a fixed finite set of basis functions, slow feature analysis is a generalized eigenproblem: the outputs that minimize ⟨ẏ²⟩ under zero mean, unit variance and decorrelation are the bottom generalized eigenvectors of the derivative covariance against the covariance, in order of slowness, and a stack of such stages, each of whose function spaces contains its inputs, can only keep or lower each of its first m slowness values on the training signal, and may keep them, LIT-1296 — Slow Feature Analysis as Variational Inference Objective
Cited as “complements” byCLAIM-478 — Scale separation helps make structure and not only marks it: where some degrees of freedom settle fast relative to others, the relation they settle into is a constraint on the system over the window between the two time scales, so what is a realization at one scale works as a constraint at another; and organization can make new separations in turn, as when modes slow near an instability and become order parameters, so time-scale hierarchies and constraints can produce one another

Graph View

The claim

Take X_t = AX_{t−1} + ε_t, stable, with stationary covariance Π and lag-one covariance Cov(X_t, X_{t−1}) = AΠ. Linear slow feature analysis minimizes E(wᵀX_t − wᵀX_{t−1})² at unit variance wᵀΠw = 1, which is 2(1 − wᵀAΠw). Its directions solve the generalized eigenproblem ½(AΠ + ΠAᵀ)w = μΠw, slowest first by largest μ.

Reversible case. If AΠ = ΠAᵀ, the symmetrization is AΠ itself, so AΠw = μΠw. Substituting ΠAᵀ for AΠ gives ΠAᵀw = μΠw, and since Π ≻ 0, Aᵀw = μw: w is a left eigenvector of A. A span L of such directions then satisfies LA = BL with B diagonal, which is THEORY-733’s condition for exact dynamical independence. By CLAIM-389 it then also has no Mori–Zwanzig memory.

Off reversibility. The symmetrization differs from AΠ and the step fails. In a random irreversible 4-dimensional example the largest residual ‖wᵀA − (wᵀAw/wᵀw)wᵀ‖/‖w‖ was 0.71. In a reversible one built as A = Π^{1/2}SΠ^{−1/2} with S symmetric, it was 8 × 10⁻¹⁶, and ‖AΠ − ΠAᵀ‖ was 1 × 10⁻¹⁴. The reader of Schüler and Wiskott found the same contrast (0.26 against 0.0015 in their examples).

The reversible stationary Gaussian VAR(1)s are exactly the noise-free square mixtures of independent AR(1)s: A = Π^{1/2}SΠ^{−1/2} with S symmetric diagonalizes as a mixing of independent AR(1)s with S’s eigenvalues as coefficients. In that class the reader also found that maximum likelihood recovers the slow-feature directions exactly, with no limit taken (NOTE-1015).

What it does not say

  • Not for Markov chains as stated. There a slow eigenfunction closes only in conditional mean; closing the whole law needs eigenvectors constant on the blocks, that is strong lumpability (THEORY-822). Slowness then gives metastability, not exact closure.
  • Not that slowness finds closure in general. In irreversible linear dynamics, in nonlinear dynamics, or at lags above one, a slow feature can be dynamically dependent and carry memory.
  • Not that every closed coarse-graining is slow. Any span of left eigenvectors is closed. Slowness picks one order among them.
  • Not that the features are individually identified. With repeated μ only the eigenspace is fixed.
  • Not a reading of Turner and Sahani (LIT-1287, Unavailable), though its noise-free mixing model is of their kind.

Note of 2026-11-02: a quantum parallel

Yang and Abanin 2025 (NOTE-1534; THEORY-1323) have a generator that is not self-adjoint: a weakly dissipative Lindbladian, whose Hamiltonian part is anti-Hermitian. In it the slowest-decaying operators, small conserved quantities, nearly coincide with the slowest eigenoperators. The leak out of the conserved direction is bounded by ε over a gap, small when ε ≲ γ ≪ 1, an argument they make at first order and check at small sizes. Under a noise channel that is itself not Hermitian, the slow eigenoperators lose their orthogonality. This is a parallel, not a test: the setting is not linear Gaussian, and what they compare is norm decay against eigenoperators, not slow features against closure. It shows the off-reversibility worry can be tamed by a small parameter where one exists.

The claim is unchanged.

Backlinks