Exactly Unbiased Single-Stage Randomized Hadamard Companders: Uniform High-Rate Limits, Minimax Reconstruction, and Fixed-Query Tradeoffs

Abstract

For every continuous, strictly positive probability density $f$ and every $B = 2^b \ge 4$, we construct a $B$-ary companded scalar quantizer with a calibrated reconstruction map whose box average in the quantile domain equals the inverse compander. Uniform dithering therefore yields exact pointwise unbiasedness for every deterministic scalar input. Combined with one randomized Hadamard transform and affine pairwise-independent coordinate dithers, the construction gives exact finite-rate identities that reduce the vector-reconstruction MSE and the MSE for a single fixed inner-product query to, respectively, an unweighted Rademacher risk and a shared-sign Rademacher risk with a correlated quadratic weight. For each fixed Hadamard-admissible density, a central-tail argument establishes the corresponding high-rate limits uniformly over all finite dimensions and all signal and query directions. These limits reduce point-density design to two scalar functionals. Over the Hadamard-admissible class, central-limit sequences of flat Rademacher sums and a sharp Hölder argument show that the minimax reconstruction constant is $\pi\sqrt{3}/2$, uniquely attained by the Gaussian density with variance three. The fixed-query problem behaves differently. Within the Gaussian family, the exact constants are $C_{\mathrm{rec}}(v) = (\pi/6)v^{3/2}(v-2)^{-1/2}$ and $C_{\mathrm{IP}}(v) = (\pi/6)v^{5/2}(v-2)^{-3/2}$, for $v>2$, so the query optimum is $v=5$ and the Gaussian-family Pareto interval is $[3,5]$. Over the full Hadamard-admissible class, the exact fixed-query infimum and whether it is attained remain unresolved. A dense-aligned adversary and a continuum of two-dimensional cancellation adversaries yield an optimized lower bound, while an explicit admissible two-cosh density yields an upper bound through an exact one-dimensional risk formula. High-precision numerical evaluation places the two endpoints at approximately 4.619123 and 4.846854, an upper-to-lower relative gap of about 4.93%. Exact unbiasedness holds at finite rate, whereas the minimax statements concern fixed-density high-rate functionals for the declared single-stage one-RHT scalar-compander architecture, not arbitrary quantizers or rate-dependent densities.