# C1 Homogeneous Lyapunov Critic

- ID: 1004
- Canonical URL: https://synthcore.org/idea/1004/c1-homogeneous-lyapunov-critic
- API JSON: https://synthcore.org/api/idea/1004.json
- API Markdown: https://synthcore.org/api/idea/1004.md
- Verification status: unverified
- Source: [arXiv:2607.16171](https://arxiv.org/abs/2607.16171)
- Category: dynamics
- Solves: stability, generalization
- ML areas: training-dynamics, rnn, ssm, regularization
- Math tags: dynamical-systems, control-theory, optimization
- Ratings: usefulness 6/10; difficulty 6/10; novelty 8/10

## Idea description

For a neural ODE or recurrent state update, learn a positive-definite degree-two homogeneous Lyapunov function that is only C1, rather than restricting the certificate to polynomials or analytic neural networks. Parameterize its angular dependence with a positive spline or softplus mixture, and train it to decrease along the learned vector field; this can certify stable dynamics that polynomial Lyapunov searches systematically miss.

## Mathematical statement

The paper considers a homogeneous vector field f satisfying f(lambda z)=lambda^d f(z), where d>0 is the vector-field degree, and a homogeneous Lyapunov function V satisfying V(lambda z)=lambda^m V(z), where m>0 is its degree. Its Lie derivative is L_f V(z)=nabla V(z)^T f(z). The counterexample has d=3 and a positive-definite, radially unbounded, degree-two Lyapunov function that is globally continuously differentiable but not polynomial or real analytic. Homogeneity implies L_f V(lambda z)=lambda^(m+d-1)L_f V(z), because the gradient of V has degree m-1. If L_f V(z)<=-c||z||^(m+d-1) on the unit sphere for c>0, norm equivalence gives dot V<=-alpha V^((m+d-1)/m), where alpha>0. The resulting decay is V(t)=O(t^(-m/(d-1))); for the cubic field and quadratic certificate, V(t)=O(t^(-1)).

## Key formulas

- $$f(\lambda z)=\lambda^d f(z),\qquad V(\lambda z)=\lambda^mV(z),\qquad \mathcal L_fV(z)=\nabla V(z)^\top f(z).$$
- $$V(z)=\|z\|^m q\!\left(\frac{z}{\|z\|}\right),\qquad q(u)>0\ \text{for }\|u\|=1.$$
- $$\mathcal L_fV(z)\le-\alpha V(z)^{(m+d-1)/m}\ \Longrightarrow\ V(t)\le\left[V(0)^{-(d-1)/m}+\frac{d-1}{m}\alpha t\right]^{-m/(d-1)}.$$
- $$\mathcal L_fV(z)\le0\ \forall z\ne0\quad\text{while no positive-definite polynomial }P\text{ satisfies }\mathcal L_fP\le0.$$

## Implementation notes

1. Integration point: use this as a Lyapunov critic for a neural ODE dot z=f_theta(z), an SSM transition, or the continuous interpolation of an RNN update. Restrict the critic to degree two, V_phi(z)=||z||^2 q_phi(u), with u=z/(||z||+epsilon). Represent q_phi on the circle or sphere using periodic cubic B-splines, Fourier features followed by softplus, or a small angular MLP; enforce q_phi(u)>=q_min>0 with q_phi=softplus(h_phi)+q_min. 2. Pseudocode: sample states z_i from training trajectories and random perturbations; compute u_i=z_i/(||z_i||+epsilon), V_i=||z_i||^2 q_phi(u_i), and D_i=gradient_z V_phi(z_i)^T f_theta(z_i) by automatic differentiation. Minimize task loss plus lambda times softplus(D_i+alpha V_i^((d+1)/2)), together with penalties for q_phi(u_i)<q_min and angular derivative discontinuities. For the cubic case set d=3, so the target is D_i<=-alpha V_i^2. 3. Computed from the paper: homogeneity, the Lie-derivative certificate, and the algebraic decay exponent. Estimated empirically: alpha, the minimum sphere margin, and C1 behavior near the origin. 4. First cheap experiment: train a two-dimensional neural ODE on a stable synthetic trajectory-reconstruction task, comparing a polynomial quadratic or quartic Lyapunov critic with the angular C1 critic. Roll out from radii 10^-2 through 10^2. The quantitative prediction is that successful cubic dynamics satisfy a log-log slope d log V/d log t approximately -1 for the quadratic certificate, while polynomial-certificate training can fail despite bounded trajectories; the measured slope should approach -1 over at least one decade and the estimated sphere margin -D/V^2 should remain positive.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.