Algebraic Manifold Coding: Near-Optimal Lossy Compression via Implicit Quadratic Constraints

We present Algebraic Manifold Coding (AMC), a generalized framework for lossy-to-lossless compressionof structured data governed by explicit physical or biological laws. Unlike data-driven manifoldlearning, AMC analytically exploits deterministic redundancies—manifested as implicit algebraicconstraints (e.g., quadratic invariants)—to confine data to low-dimensional submanifolds. Using thepervasive constraint b2 = ac + 1 as a canonical model, we prove that AMC encodes only independentcoordinates and an orthogonal residual, achieving a theoretical compression ratio of (n − m)/n forexact-manifold sources. For near-manifold sources, we derive the asymptotic rate-distortion function anddemonstrate a 1.76 dB coding gain over optimal vector quantization. The framework is generalized toarbitrary algebraic varieties with adaptive Jacobian-based bit allocation. Experiments on synthetic data,tuning-fork parameters, and monozygotic-twin physiological signals confirm that AMC consistently outperformsstate-of-the-art floating-point compressors and PCA-based methods. We also discuss extensionsto microwave networks, stereo vision, and financial time series, where such constraints naturally arise.