Splitting-Conjugate Latent Dynamics
Implementation & benchmark of arXiv:2608.05875 — Linearisation, splitting property and homotopy algebras
Source paper: Linearisation, splitting property and homotopy algebras arXiv:2608.05875 ⓘ · analyzed Aug 31, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
Equip a latent transition model with a near-identity polynomial coordinate transform that conjugates the nonlinear transition to a linear latent operator, at least locally around a reference state. Train the transform jointly with the dynamics using both the usual prediction loss and the paper's splitting/intertwining residual, so that multi-step prediction is performed partly in approximately linearised coordinates.
Formulas
Mathematical statement
The paper's coalgebra criterion states that, for a graded vector space V and a coalgebra morphism F:SV->SV with linear part f_1:V->V, the following are equivalent: there is a coalgebra isomorphism Phi:SV->SV with linear part phi_1=id_V and Phi f_1=F Phi; equivalently, there is a degree-preserving linear map tau:V->coDer(SV) such that tau(v)·1=v and tau(f_1(v))·F(mathbf{v})=F(tau(v)·mathbf{v}) for every v in V and every coalgebra element mathbf{v} in SV. Here SV is the symmetric coalgebra over V, coDer(SV) is the space of coderivations, 1 is the coalgebra unit, and a dot denotes coderivation action. The proof constructs tau from the conjugacy by tau(v)·mathbf{v}=Phi(v odot Phi^{-1}(mathbf{v})), where odot is the symmetric product. For ML, use a finite truncation S^{<=K}V of monomials of degree at most K, let A=f_1 be the Jacobian or linear latent transition at the reference point, let F_theta be the truncated polynomial jet of a nonlinear latent transition, and let Phi_eta be a polynomial near-identity map. The directly implementable conjugacy residual is Phi_eta(Az)-F_theta(Phi_eta(z)); the splitting residual is the finite-dimensional analogue tau_eta(A v)F_theta(m)-F_theta(tau_eta(v)m), with v a sampled latent direction and m a sampled monomial feature. The theorem motivates these residuals as certificates for local linearisability, although finite truncation and optimization provide only an approximation.
Implementation notes
Integrate this into a deterministic latent world model or state-space model. Let an encoder produce z_t in R^d, let F_theta be the one-step latent transition, and choose a reference latent equilibrium z=0 by subtracting the batch mean or by training on deviations from a fixed point. Parameterize a near-identity polynomial transform Phi_eta(z)=z+sum_{r=2}^K P_r[z^{\otimes r}], where each P_r is a learned symmetric tensor; initialize all P_r to zero. Parameterize A as a d-by-d matrix, or set A to the Jacobian of F_theta at zero using automatic differentiation. Use this pseudocode: sample sequences; compute z_t=encoder(x_t); compute z_next=F_theta(z_t); compute u_t=Phi_eta(z_t), u_next=Phi_eta(z_next); compute the ordinary prediction loss between decoder(z_next) and x_{t+1}; compute r_conj=Phi_eta(A z_t)-F_theta(Phi_eta(z_t)); add the mean squared r_conj over sampled states. For an optional splitting loss, represent monomials up to degree K as a feature vector m(z), let tau_eta(v) be the directional polynomial insertion operator induced by Phi_eta(v odot Phi_eta^{-1}(m)), and penalize r_split=tau_eta(A v)F_theta(m(z))-F_theta(tau_eta(v)m(z)) for random basis vectors v and minibatch states z. Estimate Phi^{-1} with three to five Newton or fixed-point iterations, or omit the explicit splitting term initially and use the conjugacy residual only. Apply a penalty max(0,rho(A)-1)^2 when stable rollouts are desired. First test on a 2D damped nonlinear oscillator and a small Lorenz or CartPole forecasting task, comparing an MLP transition with and without Phi_eta at equal parameter count. Measure one-step error, 20- and 100-step rollout error, Jacobian spectral radius, and the fraction of trajectories for which Phi remains locally invertible. Success is lower long-horizon error or slower error growth at equal one-step accuracy; failure is instability, non-invertible Phi, or no rollout benefit over the baseline.
Verification
This idea has not been verified yet.
Verification happens in two stages: Stage 1 — a mechanism check on a toy system confirms the claimed mathematical phenomenon reproduces; Stage 2 — a benchmark implements the idea on a real (small) neural network task and compares it against a tuned baseline over 8 paired seeds with a permutation test.
Artifacts
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