Boundary-normal trust-region flow

1. Integration point: apply this to any neural module whose parameters are represented in a Lie-algebra chart, such as rotation-valued layers, equivariant networks, neural ODE states constrained to a ball, or a low-dimensional latent code with a required trust region. Compute the ordinary minibatch gradient velocity \(v=-g\), then modify it before the optimizer update. Start with the cheap projection rule; add the barrier as a differentiable alternative. For a discrete step, use \(x_{k+1}=x_k+\eta v_{\mathrm{proj}}(x_k)\), or retract a Lie-group parameter with \(R_{k+1}=R_k\exp(\eta\widehat v_{\mathrm{proj}})\). 2. Pseudocode: compute loss and gradient \(g\); set \(v=-g\); if \(\|x\|\geq\rho-\epsilon\), compute \(a=x^\top v/\|x\|^2\) and replace \(v\leftarrow v-\max(0,a)x\); update using the exponential-map retraction. For the barrier version, use \(v\leftarrow-g-\lambda_B 2x/[\rho^2(1-\|x\|^2/\rho^2)]\). 3. Computed versus estimated: \(x\), its norm, and the boundary projection are computed exactly per batch; the paper's curvature certificate is not needed by the implementation, while the actual outward-rate distribution is estimated by logging \(x^\top v/\|x\|\) on near-boundary iterates. 4. First experiment: train a small rotation-equivariant MLP or SO(3)-parameterized linear layer on synthetic rotated-MNIST, comparing Adam, unconstrained SGD, projected flow, and barrier flow with \(\rho=0.3\). Initialize near the paper's quadratic example and use the same \(H\) as a controlled test. The falsifiable signature is that the unconstrained method exhibits positive boundary-normal velocity near the boundary, with a controlled reproduction near \(0.042\), and exits the chart; projection should make every logged boundary-normal velocity nonpositive up to numerical tolerance and produce zero chart exits. In discrete time, measure the largest safe learning rate and test whether it scales approximately as the distance-to-boundary divided by the maximum outward velocity.