Lipschitz Observable Factor Bottleneck

Integrate this at the encoder output of an autoencoder, contrastive model, vision backbone, or transformer embedding layer. Let \(x\) be an input, \(z=\pi_\theta(x)\) its latent representation, and let \(f_j(x)\) be scalar probe targets. Use two probe families: fixed random 1-Lipschitz projections of the input or intermediate features, and trainable adversarial probes that maximize factorization error. Enforce approximate 1-Lipschitzness with spectral normalization on every linear layer of \(\pi_\theta\) and \(g_j\). Estimate each network's global Lipschitz upper bound by multiplying its per-layer spectral norms, and penalize values above one. For each minibatch, compute \(z_b=\pi_\theta(x_b)\), center each probe over the batch, and evaluate \(g_j(z_b)\). Update the probes for two to five ascent steps to maximize the factor loss under their Lipschitz constraints; then update the encoder and task model using task loss plus \(\alpha\mathcal L_{\mathrm{factor}}\). Use \(\ell_\tau(u)=\tau^{-1}\log(1+\exp(\tau u))\) as a smooth approximation to the positive part. The paper supplies the 1-Lipschitz factor and \(\varepsilon\)-tail target; probe count, \(\tau\), \(\varepsilon\), and \(\alpha\) are empirical. First test CIFAR-10 with ResNet-18 and a 4x-dimensional latent bottleneck against weight decay and a standard contractive penalty. Measure clean accuracy, corruption accuracy, latent dimension at fixed accuracy, and perturbation sensitivity. Success means lower latent dimension or better corruption accuracy at equal clean accuracy, with the empirical fraction of probe errors exceeding \(\varepsilon\) approaching the target \(\varepsilon\).