On Toeplitz determinants with slow Fourier decay

What the math gives to ML

The paper isolates an explicit finite-size quadratic contribution to the log-determinant of Toeplitz matrices whose log-symbol has only 1/|k| Fourier decay, while showing that each fixed higher-order term remains bounded. This suggests a structured neural operator with deliberately slow spectral decay: it can represent long-range interactions without treating all frequencies as independent, while a computable quadratic spectral budget controls the dominant growth of its log-volume or normalization. The most practical transfer is to use Toeplitz or circulant layers parameterized in the Fourier domain, together with a log-determinant regularizer after subtracting the analytically known quadratic term. The proposal is falsifiable through conditioning, long-context accuracy, and cost comparisons against dense attention and ordinary convolution.