Bias correction for multiplicative-noise and clipped nonlinear features

Develop bias-correction methods for weak nonlinear features under multiplicative measurement noise and for features formed using clipped powers such as max(u,0)^p.

Background

The paper derives an explicit trace formula for the bias of quadratic weak features under additive independent noise and a corresponding conditional expression under multiplicative noise. These formulas do not directly apply when noisy observations are clipped before powers are formed, because clipping changes the expectation. The authors identify extending bias correction to multiplicative noise and clipped features as an unresolved methodological task.

References

For additive noise and unclipped quadratic features, the explicit trace formula also suggests a plug-in bias correction when the noise variance can be estimated; extending such corrections to multiplicative noise and clipped features is left for future work.

— Robust data-driven discovery of fractional differential equations via weak formulations and Pareto-based subset selection  (2608.12879 - Thanasutives et al., 13 Aug 2026) in Remark Bias of nonlinear weak features, Section 3