Optimal linear encoding rate for lossy compression of Bernoulli sources
Ancheta's answer to Massey's question about linear encoders for lossy compression is extended from the fair-coin case to every source bias below one half.
- Lab
- Independent
- Model
- GPT-5.6 Sol
- Field
- Computer science
- Date
- 2026-08-24
- Human collaborators
- Yihong Wu
- Problem posed
- 1978 · open 48 yrs
Sources
Original work
Independent commentary
What was found
Linear encoders achieve the entropy for lossless compression of a Bernoulli source, but for lossy compression linearity is known to be strictly suboptimal against the rate-distortion function. Massey asked in 1978 whether the optimal rate for linear encoding is achieved simply by compressing a fraction of the bits linearly and losslessly and estimating the rest by zero. Ancheta answered yes for bias one half in the same year, and this note extends that to every bias below one half. The key step bounds the entropy of the posterior distribution conditioned on an affine subspace in terms of its marginals.
Novelty check
The question and its partial answer are both named, dated and cited: Massey posed it in 1978 and Ancheta settled the p = 1/2 case that same year, so what remained was every other bias. The note also records, as a correction to its own novelty, that a literature search aided by Codex found several proof ingredients already present in or deducible from prior work, and devotes a section to that discussion. That is a partial-novelty flag surfaced by the author rather than by a later reader.
Caveats and known objections
A short preprint, unrefereed, not formalized, with no independent check on record. The novelty is qualified by the author's own hindsight statement that several ingredients of the proof have appeared in, or can be deduced from, the prior literature, discussed in the note's Section 4.1; what is claimed as new is the extension itself and a simplified self-contained presentation. Autonomy is graded ai-led on the declaration of AI use, which credits discovery to the model and simplification to the author: the first version of the proof was discovered by GPT-5.6 Sol in an interactive process guided by the author, who subsequently simplified the program and developed the self-contained proof presented in the note.
Also recorded at
vibemathedentropy-of-bernoulli-measures-conditioned-on-affine-subspaces-and-a-problem-of-a
Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.
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Entry history (1 event)
- AddedEntered the registry graded Author verified and AI-led.
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Cite this entry
whataifound.org. (2026). Optimal linear encoding rate for lossy compression of Bernoulli sources. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-24-ancheta-massey-linear-codingBibTeX
@misc{whataifound-independent-2026-coding,
title = {Optimal linear encoding rate for lossy compression of Bernoulli sources},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Author verified. Autonomy: AI-led.},
url = {https://whataifound.org/finding/2026-08-24-ancheta-massey-linear-coding}
}