Scale-Quotient Covariance Alignment

(1) Integrate this into a covariance-alignment loss. Choose a layer producing centered activations \(H_x\in\mathbb{R}^{m\times p}\), with batch size \(m\), channel dimension \(p\), and two groups or views \(a,b\). Form \(\widehat\Sigma_x=(H_x^{\mathsf T}H_x)/(m-1)+\varepsilon I\), where \(\varepsilon>0\) ensures positive definiteness. Add \(\lambda\mathcal{L}_{\mathrm{quot}}\) to the task loss. (2) Compute the loss with a small number of Newton or gradient iterations over unconstrained variables \(u\in\mathbb{R}^p\), setting \(D=\operatorname{diag}(\exp u)\). Initialize \(u=0\); at each inner iteration compute \(M=\widehat\Sigma_a^{-1/2}D\widehat\Sigma_bD\widehat\Sigma_a^{-1/2}\), eigendecompose \(M=V\operatorname{diag}(s)V^{\mathsf T}\), evaluate \(\|\log s\|_2^2\), backpropagate through the eigendecomposition, update \(u\leftarrow u-\eta_u\nabla_u\|\log s\|_2^2\), and detach or stop after 3--5 iterations. Use the resulting value as the outer loss. (3) The quotient action and affine-invariant metric come from the paper; the minimizing diagonal \(D\), covariance estimates, and matrix eigendecomposition are computed numerically. For a cheaper approximation, replace the inner minimization by correlation normalization \(R_{ij}=\Sigma_{ij}/\sqrt{\Sigma_{ii}\Sigma_{jj}}\), then use the affine-invariant distance between \(R_a+\varepsilon I\) and \(R_b+\varepsilon I\). (4) First test on CIFAR-10 or Office-31 with a small ResNet-18, using supervised classification plus covariance alignment between two augmented views or source and target domains. Compare against no alignment, Frobenius covariance loss, and standard correlation-matrix loss. Measure target accuracy, robustness under random per-channel activation rescaling, training variance across seeds, and wall-clock overhead. Success is higher accuracy or domain-transfer performance under scale perturbations at comparable task loss, without the covariance term destabilizing optimization.