KUKULKAN: A STATE SPACE MODEL FOR TEMPORAL POINT PROCESSES WITH INTERPRETABILITY

We write the temporal point process as a state space model and use the parallel scan of Mamba to model the excitation kernel explicitly: the Hawkes excitation is the state of a diagonal linear system, its compensator has a closed form and its branching ratio is the steady-state gain. Through structured state space duality the same computation is an attention over past events, which places the classical kernel, the attention models and the state space models in one form. The Mamba Hawkes process and S2P2 are built on that form but give the branching ratio up in exchange for expressivity. Kukulkan keeps it and relaxes it from a number to an interval. The context selects every channel’s decay rate and the background rate freely, and the output map only through a gate between zero and one, so the likelihood stays exact and the offspring of every event lies below a number fixed at its birth. The branching ratio is partially identified, with a bound, a conditional estimate and a simulated estimate. Under an evaluation protocol that removes the quadrature bias of the standard held-out likelihood, Kukulkan matches the unconstrained neural output map on seven of nine public datasets and improves on the model with an exact branching ratio on all. On exchange order flow it improves on the unconstrained output map while bounding the offspring of every order, and its estimate of the market stays sub-critical.