Heteroscedastic Condition-Adversarial Representation

1. Exact integration point: choose the penultimate feature tensor of a small MLP, 1-D CNN, Transformer encoder, or state-space model. Add two linear heads on pooled features: one outputs $\mu$ and one outputs $s$, with variance $\sigma^2=\operatorname{softplus}(s)+10^{-6}$. Add a gradient-reversal layer between the feature and condition heads. Train the fault head with cross-entropy and the condition head with the Gaussian negative log-likelihood. 2. Pseudocode: for each minibatch $(x,y,c)$, compute $h=F_\theta(x)$, $p=G_\phi(h)$, $(\mu,s)=D_\psi(\operatorname{GRL}_\lambda(h))$, $v=\operatorname{softplus}(s)+\epsilon$, then $L=-\operatorname{mean}\log p_y+\lambda\operatorname{mean}[0.5\log v+0.5(c-\mu)^2/v]$ and backpropagate once. Warm up with $\lambda=0$ for 5 epochs, then ramp to a target value. 3. Computed from the paper: the Gaussian condition objective, adversarial sign, and mean/variance parameterization. Estimated empirically: the best $\lambda$, feature location, and condition normalization. 4. First cheap experiment: train a 2-layer 1-D CNN on a gearbox or synthetic rotating-signal dataset with fault labels and a continuous speed/load variable. Compare ERM, discrete domain-adversarial training, and this method for $\lambda\in\{0,0.01,0.03,0.1,0.3,1\}$. Hold out operating-condition intervals rather than random samples. The predicted signature is that condition-probe $R^2$ decreases with $\lambda$, while held-out-condition fault accuracy first improves and then collapses beyond an over-adaptation region. Report the crossover and test whether the best $\lambda$ occurs before validation fault loss increases by 10%.