Exact Minimax One-Bit Unbiased Compression: Heavy-Tail Necessity and Finite-Randomness Approximation
A pointwise-unbiased one-bit compressor reconstructs every real input in expectation while transmitting one bit. For a scalar source $P$ with CDF $F$, mean $m$, and $\mathcal J(P)=\int_{\mathbb R}\sqrt{F(r)(1-F(r))}\,dr$, we prove that the infimum of the source-averaged reconstruction second moment over all public-coin one-bit codes unbiased on $\mathbb R$ is $m^2+\mathcal J(P)^2$. For regular full-support sources, a distribution-centered random-threshold code attains this value; a converse over arbitrary randomized binary encoders and an equality analysis characterize every attaining code up to null sets, bit relabeling, and public-seed refinement. For the Gaussian location family $\math...
| Authors | Tao Jiang, Minbo Gao, Shaowei Cai |
|---|---|
| Primary category | cs.LG |
| Published | 20 August 2026 |
| Last updated at source | 20 August 2026 |
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