# Rank-One Small-Gain Recurrent Controller

- ID: 2535
- Canonical URL: https://synthcore.org/idea/2535/rank-one-small-gain-recurrent-controller
- API JSON: https://synthcore.org/api/idea/2535.json
- API Markdown: https://synthcore.org/api/idea/2535.md
- Verification status: unverified
- Source: [arXiv:2608.18561](https://arxiv.org/abs/2608.18561)
- Category: dynamics
- Solves: stability, speedup
- ML areas: rnn, ssm, optimizer, architecture
- Math tags: control-theory, dynamical-systems, spectral-theory, linear-algebra
- Ratings: usefulness 6/10; difficulty 5/10; novelty 6/10

## Idea description

When a recurrent or graph coupling matrix is approximately rank one, replace expensive full spectral monitoring with a scalar small-gain controller. Adapt a residual mixing coefficient so that the dominant coupled mode remains below a prescribed contraction threshold.

## Mathematical statement

Approximate the coupling matrix A by u v-transpose, where u and v are vectors. For agent transfer functions h_i, the rank-one feedback loop has scalar gain g equal to v-transpose times diag(h_i) times u. The paper's small-gain mechanism states that stability follows when the infinity norm of g multiplied by the infinity norm of the uncertainty is less than one. For a discrete neural layer, define A_gamma = (1 minus gamma) I plus gamma u v-transpose, where gamma is an adaptive mixing coefficient. If the activation function has Lipschitz constant L_phi, then L_phi times the spectral norm of A_gamma below one gives contraction. A conservative rank-one estimate is q_gamma = L_phi times the absolute value of 1 minus gamma plus gamma times v-transpose u, plus the spectral norm of the residual A minus u v-transpose.

## Key formulas

- A approximately equals u v-transpose
- g(s) = v-transpose diag(h_1(s), ..., h_n(s)) u
- ||g||_infinity ||Delta||_infinity < 1 implies small-gain stability
- x_next = phi(A_gamma x + B u), A_gamma = (1 - gamma) I + gamma u v-transpose

## Implementation notes

Integration point: place the controller in an RNN, state-space layer, graph neural layer, or learned optimizer whose update contains a dense coupling matrix A. Every few optimizer steps, estimate the leading singular vectors using one or two power iterations. Normalize u and v, compute the dominant scalar coefficient c = v-transpose u, and estimate the residual norm with a few power iterations on A minus u v-transpose. Choose gamma by a one-dimensional grid search or analytic clipping so that q_gamma = L_phi times the absolute value of 1 minus gamma plus gamma c, plus the residual norm, remains below a target q_target less than one. Then apply x_next = phi(((1 minus gamma) I plus gamma u v-transpose) x plus B u_input). Use L_phi equal to one for tanh or ReLU as a conservative bound, and use the active-mask estimate for a less conservative ReLU bound. The paper contributes the rank-one reduction and small-gain condition; the rank-one factorization and residual gain are estimated from current weights. First experiment: compare this controller with unconstrained recurrence and standard spectral normalization on sequential MNIST and the adding problem. Sweep q_target over 0.8, 0.9, 0.99, and 1.05. Measure the ratio of perturbation norms at successive steps and compare it with q_target. Prediction: perturbations decay exponentially when the estimated q is below one, with log decay slope close to log(q); long-horizon instability should appear sharply as q crosses one.

## Disclaimer

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