The Momentum Sampler

Sampling from discrete distributions usually means random walks: propose a move, accept or reject, repeat. The walk explores the space but does so drunkenly — it doubles back, revisits old states, wastes most of its steps on redundant exploration. The fundamental limitation is Markovian: each step knows nothing about the previous one.

Temporal point processes offer an alternative. By constructing a multivariate point process whose event-count vector in a fixed sliding window converges to the target distribution, the authors create a sampler with a discrete form of momentum. The process is structured as coupled infinite-server queues with deterministic service times — each event persists for a fixed duration, and the current state is the count of active events per component.

The momentum arises because events don’t appear and vanish instantaneously. They persist. A newly generated event influences the state for its entire service time, creating temporal correlation that suppresses random-walk behavior. The sampler remembers where it’s been — not through explicit memory but through the physical persistence of its components.

In simulations across 63 target distributions, the process always outperforms birth-death samplers and frequently outperforms Zanella processes in multivariate effective sample size. Adding auxiliary randomness reduces it to a birth-death process, establishing the latter as a degenerate limit — momentum set to zero.

The structural insight: birth-death processes are samplers without momentum. Adding temporal persistence to events — making each sample “last” for a deterministic duration — introduces the discrete analogue of Hamiltonian Monte Carlo’s momentum term. The connection between sampling efficiency and temporal persistence is the claim: memory, even mechanical memory, beats memorylessness.