Maps of q-deformed fractional order: From circle to cardioid via crescent
arXiv:2607.15833
2026
Dynamics
2 ideas extracted · analyzed Aug 30, 2026
What the math gives to ML
The paper offers a constructive deformation of fractional discrete dynamics in which Gaussian q-binomial memory kernels interpolate between memoryless maps and classical long-memory fractional maps. Its transferable asset is the combination of an explicitly parameterized memory kernel, a characteristic equation, and a computable complex-plane stability boundary whose geometry changes with the deformation parameter. In neural networks, this can produce recurrent or state-space layers with tunable memory localization: q close to 1 gives fractional power-law memory, while q<1 gives exponentially localized memory. The most useful experiments test whether the predicted characteristic-root boundary matches the empirical exploding-gradient boundary.
Ideas from this paper
△ Mechanism confirmed, baseline not beaten
2026
Replace the uniform or power-law convolution in a recurrent or state-space layer by a Gaussian q-binomial fractional kernel with learnable order alpha and deformation q. The parameter q controls a concrete memory-localization transition: q close to 1 gives classical fractional power-law memory, whereas q<1 produces exponentially localized memory and should reduce long-horizon gradient interference and truncation cost.
Useful8/10
Difficulty6/10
Novelty7/10
✗ Failed on benchmark
2026
Use the q-fractional characteristic equation as an online trust-region controller for recurrent gain or residual-memory strength. Instead of allowing the recurrent Jacobian to cross the unit-circle boundary, estimate the dominant characteristic root and rescale the feedback gain whenever it approaches modulus one.
Useful7/10
Difficulty5/10
Novelty6/10