Section 6.4, “Connection to optimisation (informal)”Spectral bias suggests an analogy for parity learning; a hypothesis requiring separate measurements
Informal interpretation Motivation, interpretation or scope; not a formal claim.
1 Paper text
Connection to optimisation (informal). Fourier degree also aligns with a common empirical observation in deep learning: gradient-based training tends to fit low-frequency components before high-frequency ones (“spectral bias” / “frequency principle”) (Rahaman et al., 2019; Xu et al., 2019). This suggests a possible analogy for parity learning, but continuous-domain spectral-bias observations do not prove how SGD learns Boolean parity or restoration under our nonuniform data distribution. We treat that connection as a hypothesis requiring separate measurements.
Generated from paper_jmlr.tex, lines 983–985.
2 Explain this
Gradient-trained networks are often observed to fit low-frequency components first ("spectral bias", "frequency principle"). That suggests a possible analogy with parity learning. The paper is explicit that these continuous-domain observations do not prove how SGD learns Boolean parity or restoration under its data distribution; it is a hypothesis for separate measurement.
3 How this is formalised
An informal hypothesis citing empirical literature; nothing is formalised or claimed.
| Paper anchor | par-connection-to-optimisation-informal-p1 (website anchor; the manuscript has no label here) |
|---|---|
| Displayed as | Section 6.4, “Connection to optimisation (informal)” |
| Claim | Spectral bias suggests an analogy for parity learning; a hypothesis requiring separate measurements |
| Coverage category | Informal interpretation |
| Statement source | paper_jmlr.tex lines 983–985; SHA-256 of the normalised block source 9f45932aaafe7ffd… |
| Explanations | explain/c-spectral-bias, formal/c-spectral-bias |
| Build evidence | fresh isolated rebuild, audit report |
4 Lean statement and source
No Lean statement corresponds to this claim. This is interpretation or motivation, not a formal mathematical claim.
Build evidence: fresh isolated rebuild of all 38 bundled modules with Lean (version 4.26.0, started 2026-10-10T15:21:56Z, finished 2026-10-10T15:27:00Z: 38/38 modules compiled, 1588 axiom reports, no errors. See Build for the commands and the independent audit of 2026-10-10 (dated evidence).