Deterministic Causal Quantization with Bounded Error: Adaptive Reconstruction and Entropy bounds
We study compression of real-valued stationary sources under a deterministic error bound: every sample must be reconstructed within a prescribed tolerance $\varepsilon$ on every input sequence. Reconstructions are causal, depending only on present and past inputs. Rates are measured by the entropy of the quantizer output, which block entropy coding can approach, and compared with the minimum rate of unrestricted block quantization under the same error constraint. A uniform scalar quantizer with cells of width $2\varepsilon$ is within one bit per sample of this minimum whenever finite-entropy scalar quantization is possible, without density, independence, or mixing assumptions. Under entropy and Sobolev regularity conditions, the gap is at most $2\mathcal{J}\varepsilon^2/(π^2\ln 2)$, where $\mathcal{J}$ is the asymptotic Fisher information per sample. Thus the cost of causal reconstruction vanishes at a quadratic rate as the tolerance decreases. A uniform grid, however, cannot adapt to the source. We construct an adaptive quantizer whose encoder and decoder maintain a common finite set of reconstruction values, updated from the transmitted symbols alone; suitable initialization reproduces any uniform grid. When its initial values are at least $\varepsilon$ apart, its rate is within two bits per sample of the minimum for every such source. For sources uniform within separated intervals of width $2\varepsilon$, storing the interval centers attains the minimum and its Shannon lower bound. Such configurations can outperform every uniform grid of the same cell width: over sources with well-separated intervals, the best uniform grid's worst-case excess rate is $1/(2\ln 2)\approx 0.72$ bit per sample.