Projected Bures Covariance Pooling

(1) Integration point: add this as a differentiable covariance-aggregation module in a CNN, graph neural network, or set encoder. For each sample, form SPD feature covariances \(A_i=H_i^\top H_i/n+\epsilon I\), where \(H_i\) is a patch, token-neighborhood, or augmentation feature matrix. Replace arithmetic pooling with \(K\) projected BW barycenter iterations. (2) Pseudocode: initialize \(S\leftarrow\sum_iw_iA_i\). For \(t=1,\ldots,K\), compute \(R_i=(S^{1/2}A_iS^{1/2})^{1/2}\), form the unit-step proposal \(X\leftarrow(\sum_iw_iR_i)^2\), eigendecompose \(X=U\operatorname{diag}(\lambda)U^\top\), and update \(S\leftarrow U\operatorname{diag}(\operatorname{clip}(\lambda,\alpha,\beta))U^\top\). Feed \(S\), or \(\log S\), to the next layer and backpropagate through the matrix operations with stabilized eigendecomposition gradients. (3) Compute the BW update and clipping exactly; choose \(\alpha\) and \(\beta\) from running lower and upper eigenvalue quantiles of the inputs, with gradients stopped through these bounds. Estimate the empirical contraction by measuring successive \(d_{\mathrm{BW}}(S_t,S_{t-1})\); the theorem supplies the non-expansiveness and claimed rate, while task improvement is empirical. (4) First experiment: compare arithmetic pooling, log-Euclidean pooling, unprojected BW iterations, and projected BW iterations on CIFAR-100 using a small ResNet and covariance features, with \(K\in\{3,5,10\}\). Measure validation accuracy, barycenter objective, condition numbers, NaN or gradient-failure frequency, and loss versus eigendecomposition FLOPs. Success is lower instability and faster objective decrease at equal compute, with no extra eigendecomposition relative to the unprojected BW baseline.