Uniform sine-kernel determinant asymptotics, tail-side quantiles, and prolate eigenvalue bounds

What the math gives to ML

The paper gives explicit, uniform estimates for how many eigenmodes of a time-frequency concentration operator remain above a prescribed energy threshold, including the logarithmic-width transition region around the Shannon number. This is directly transferable to deterministic sequence and feature compression: a neural layer can project signals onto discrete prolate spheroidal sequences (DPSS), while the theorem supplies a principled rank selected from bandwidth, window length, and tolerated discarded energy rather than an arbitrary truncation. The most promising use is a trainable or frozen spectral bottleneck for long-context models, convolutional feature streams, and multidimensional grid data. The main risk is that the guarantee assumes approximately bandlimited inputs, so experiments should compare against learned low-rank projections and Fourier truncation on both smooth and non-bandlimited data.