Branch-Gated Ambiguity Layer

Integrate this into an inverse model whose output is intrinsically multimodal, such as an intrinsic-image decomposer, inverse renderer, diffusion inverse, or VAE decoder. Start with a baseline encoder $h=E(x)$ and replace its final prediction head by two experts $f_1(h)$ and $f_2(h)$ plus a scalar gate logit $a(h)$; compute $p=\sigma(a)$ and return either the mixture $pf_1+(1-p)f_2$ during ordinary training or both branches at inference. The pseudocode is: (1) sample input $x$ and target $y$; (2) compute $h=E(x)$; (3) compute $y_1=f_1(h)$, $y_2=f_2(h)$, $p=\sigma(a(h))$; (4) use the permutation-invariant reconstruction loss $\min(\ell(y_1,y),\ell(y_2,y))$ when only one target decomposition is available; (5) add a gate entropy term with a small coefficient and a diversity term $-\|y_1-y_2\|^2$ on examples known or estimated to be ambiguous; (6) optimize the weighted output or select the lower-loss branch. Estimate the paper’s diagnostic by finite differences: perturb an ambiguity control $c$ by $\pm\epsilon$, compute $J_f\approx[f(x+\epsilon v)-f(x-\epsilon v)]/(2\epsilon)$, and for an encoder compute $\gamma_-,\gamma_0,\gamma_+$ and finite-difference curvature from their first and second differences. The mathematical prediction is that the single-head baseline’s peak Jacobian scales approximately as $1/\sqrt{\lambda}$ when its smoothness coefficient is varied; estimate the proportionality constant from a log-log fit. Use this diagnostic to route only high-score samples to the branch module, keeping ordinary samples on the cheaper baseline path. The first experiment should use a small U-Net on a synthetic intrinsic-image dataset with two illuminant explanations for the same RGB image, comparing a single-head U-Net, unconditional two-head output, and diagnostic-routed branches. Measure reconstruction error to the correct branch, branch diversity, ambiguity calibration, peak Jacobian, and compute overhead. Success means lower error on ambiguous examples without harming unambiguous examples, plus disappearance or reduction of the baseline’s $1/\sqrt{\lambda}$ Jacobian blow-up at equal parameter count.