Positive Bidiagonal Factorizations for Banded Markov Processes

What the math gives to ML

The paper provides an ordered factorization of a nonnegative banded transition matrix into positive bidiagonal factors, with the factor order retaining structure that is hidden in the multiplied matrix. This gives a cheap, positivity-preserving and row-stochastic parameterization of local neural propagation operators, while allowing arbitrary bandwidth through repeated one-step moves. The most direct transfer is a learnable diffusion or state-space layer whose transition kernel is a product of elementary lower- and upper-neighbor updates, rather than a dense unconstrained matrix. Total nonnegativity and stochasticity provide stability and probabilistic interpretability, while the factorization reduces parameter count and computation.