Conjugacy-Regularized Latent Dynamics

(1) Integration point: apply this to the latent transition of two RNN, SSM, or world-model instances that should encode the same process. Let encoder A produce z_t from observations and encoder B produce w_t; use F_theta and G_phi for their latent one-step transitions. Insert H_psi between the two latent spaces only inside the conjugacy loss, not in the observation decoder. Start with H_psi(z)=Az+b, parameterize A as a full nonsingular matrix initialized near identity, or use a RealNVP coupling flow for a more expressive bijection.

(2) Pseudocode:

for batch of trajectories x[0:T]:
    z = encoder_A(x); w = encoder_B(x)
    loss = task_loss_A + task_loss_B
    for t in 0..T-1:
        z_next = F_theta(z[t])
        loss += lambda1 * mse(H_psi(z_next), G_phi(H_psi(z[t])))
    for t in sampled_times and k in 2..K:
        loss += lambda_roll * mse(H_psi(iterate(F_theta,z[t],k)), iterate(G_phi,H_psi(z[t]),k))
    loss += lambda_inv * mse(H_inv(H(z)), z)
    update theta, phi, psi with AdamW

The one-step residual is the adapted form of h\circ f=g\circ h, and the multi-step term tests trajectory consistency rather than only local consistency.

(3) Computed from the mathematics: the commuting-map residual, inverse consistency, and optional observable residual. Estimated empirically: latent states, transition maps, rollout horizon K, and loss weights. Track the singular values and condition number of A, or the Jacobian determinants of the flow; reject runs where H becomes numerically singular. Compare against orthogonal Procrustes alignment and an unconstrained MLP alignment.

(4) First experiment: train two 2-layer GRU predictors on Lorenz-63 or damped-pendulum trajectories, with model B observations generated by a fixed shear and scaling of model A's state. Use 16- or 32-dimensional latents and evaluate one- to 50-step rollout error after mapping B into A coordinates. The conjugacy model should achieve low transition-commutation error under the deliberately non-orthogonal transform, while orthogonal alignment should show a residual floor. A second test should use independently initialized GRUs on identical sequences and measure whether conjugacy regularization improves long-horizon prediction at equal parameter count.