Chi-Square-Calibrated Covariance Matching

(1) Add the mechanism to any training loss that compares covariance or correlation matrices between two augmentations, domains, modalities, teacher-student branches, or time windows. Compute the quotient distance \(d_Q\) using the inner diagonal-scale optimization described in the first idea, or use the faster correlation-normalized affine-invariant approximation. (2) For each minibatch pair, compute \(p\) from the covariance dimension, \(d=p(p-1)/2\), and \(n_{\mathrm{eff}}=m_am_b/(m_a+m_b)\), where \(m_a,m_b\) are the two sample counts. Set \(T=(n_{\mathrm{eff}}/4)d_Q^2\). Obtain \(q_{d,1-\alpha}\) from a standard chi-squared quantile function, with \(\alpha=0.05\) initially. Use \(\lambda_t=\lambda_0\max(0,(T-q_{d,1-\alpha})/q_{d,1-\alpha})\), optionally clipping \(\lambda_t\) to \([0,\lambda_{\max}]\), and add \(\lambda_t d_Q^2\) to the task objective. Stop gradients through \(\lambda_t\) so the gate does not create unstable second-order effects. (3) The factor 4, degrees of freedom \(d\), effective sample size, and threshold rule are derived from the paper; the Gaussian null approximation is not exact for deep features and should be empirically checked. Estimate a correction factor by measuring the null distribution of \(T\) between independent augmentations of the same examples. (4) First test a SimCLR-style ResNet-18 on CIFAR-10 with covariance alignment between augmented views. Compare fixed-weight alignment, no alignment, and the calibrated gate at equal compute. Track representation linear-probe accuracy, covariance loss, fraction of batches with nonzero gate, gradient norm variance, and performance when batch size changes from 64 to 512. The expected signal is reduced sensitivity to batch size and lower gradient variance, while preserving or improving downstream accuracy.