Estimating eigenvectors and eigenspaces of covariance matrices: Optimal Bounds and Conditions for Consistency

What the math gives to ML

The transferable asset is an optimal-order analysis of empirical covariance eigenspaces in terms of effective rank, eigengaps, and sample count rather than ambient dimension. This gives a principled way to decide when a low-dimensional activation subspace is statistically reliable, instead of choosing bottleneck rank or calibration-set size heuristically. A practical neural-network use is a sample-aware spectral bottleneck for activation or key/value compression: estimate the covariance eigenspace, retain only directions whose eigengaps exceed the finite-sample uncertainty, and fall back to the uncompressed representation when the estimate is unreliable. The main falsifiable benefit is lower memory or FLOPs at matched accuracy, with failures concentrated exactly where the estimated eigengap is too small.