Least Variability in a Polynomial-Square Class of Rational Kernels
arXiv:2608.29143
2026
Architecture
1 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper gives a constructive way to build nonnegative rational temporal kernels that approximate a deterministic delay with substantially lower variance than Erlang/gamma cascades at the same polynomial order. The transferable asset is the polynomial-square form, which guarantees positivity while remaining implementable as a finite-dimensional repeated-pole state-space filter, together with the Laguerre-zero characterization that selects the least-variable kernel. This suggests replacing Erlang delay lines or positive exponential smoothing cascades in state-space and recurrent modules with a spectrally designed polynomial-square kernel. The first test should compare equal-state causal temporal filters on long-range sequence prediction, measuring delay fidelity, validation loss, and stability under recurrent rollout.
Ideas from this paper
Unverified
2026
Replace an Erlang delay or exponential smoothing cascade in a recurrent or state-space layer by a positive rational kernel of the form \(\kappa(u)=C e^{-a u}p(u)^2\). Choose the degree-\(m\) polynomial by deleting the adjacent pair of Laguerre zeros with smallest relative gap from \(L_{m+2}\), then rescale the resulting density to unit mean. This preserves a nonnegative impulse response while reducing temporal jitter relative to Erlang filters.
Useful6/10
Difficulty6/10
Novelty7/10