Strict Total Positivity from Spectral Darboux and Toeplitz Smoothing Mechanisms

What the math gives to ML

The paper isolates strict total positivity as a mechanism for turning ordered one-dimensional families into matrices whose every ordered minor is positive, with the modified-Bessel kernel I_s(x) as a concrete infinite-order example. This structure is transferable to sequence models because attention and convolution operators are ordered kernel matrices, and positive minors impose an anti-oscillatory or variation-diminishing bias beyond entrywise positivity. The most direct experiment is a content-light attention layer whose weights are generated by the Bessel kernel on positive ordered scalar coordinates, followed by row normalization. A cheaper variant can regularize only low-order minors of an ordinary learned attention matrix.