Obtain a closed-form transition point for heterogeneous atom exponents

Derive a closed-form expression for the value of the false-negative threshold at which the confidence transition occurs for a set of atoms having arbitrary, nonidentical exponents a(φ_i), rather than equal exponents.

Background

The supplementary analysis derives the confidence transition for a set of atoms with equal exponent a and obtains the practical condition Z/(a·tildeΔ)=1. The authors then consider the more general case in which the atoms have different exponents a(φ_i), associated with different maturity and training histories.

For heterogeneous exponents, the confidence is expressed through a Lagrange-interpolation formula. The paper states that the exact equation defining the 0.5 confidence transition cannot be resolved in closed form, and it therefore derives an approximate transition criterion instead. A closed-form solution for this general case remains unresolved in the text.

References

The equation $P(\omega(\phi_1,....\phi_Z) \geq \Delta) = 0.5$ yields the location at which the confidence transitions but it cannot be resolved.

— Algebraic Machine Learning: Learning as computing an algebraic decomposition of a task  (2502.19944 - Martin-Maroto et al., 27 Feb 2025) in Supplementary Section “General case for the transition point,” Section 3.3