Graded Projection Recursion (GPR): A Framework for Controlling Bit-Complexity of Algebraic Packing
Jeffrey Uhlmann
TL;DR
The paper introduces Graded Projection Recursion (GPR), a framework to convert blocked recursive algorithms into model-honest, near-quadratic procedures by combining a three-band packing with a two-round extractor and a per-node centering plus dyadic L2 normalization. It proves an exact integral kernel in Part I with a recurrence $T(n)=4T(n/2)+O(n^2)$ and a soft-quadratic bit/word cost under staged precision; Part II extends to float-GPR with a drift-ledger model ensuring commensurate accuracy; Part III demonstrates reusable templates and numerous case studies in linear algebra and semiring problems; Part IV abstracts admissible packings via a master condition and BWBM, enabling multilinear extensions and extractor abstractions. Collectively, GPR provides a robust pathway to model-honest complexity bounds, controlling intermediate growth and enabling efficient reductions of core tasks like LUP, Solve, Det, Inverse, LDL$^{\top}$, QR, and SDP/LP kernels. The approach offers practical implications for numerical linear algebra and combinatorial computations by delivering near-quadratic complexity while preserving accuracy guarantees under fixed budgets. The framework also outlines general extensions to multilinear and semiring domains, with explicit templates for constructing new GPR reductions and evaluating their complexity and correctness in practice.
Abstract
We present Graded Projection Recursion (GPR), a framework for converting certain blocked recursive algorithms into model-honest (i.e., reflects full bit complexity) near-quadratic procedures under bounded intermediate budgets. Part I gives a proof-complete {\em integral specification} of a three-band packing identity and a two-round middle-band extractor, and shows how per-node centering and sqrt-free dyadic l2 normalization (recursive invariant amplification) gives a sufficient packing base that grows linearly with the scaling depth (i.e., logarithmic bit-growth) in exact arithmetic. On fixed-width hardware, exact evaluation of the packed recursion generally requires either an extended-precision path or digit-band (slice) staging} that emulates b(n) bits of mantissa precision using w-bit words, incurring only polylogarithmic overhead; This leads to a soft-quadratic bit cost O(n^2) when b(n)=Θ(\log n) in the basic 2x2 recursion). Part II introduces execution-format comparators (e.g., IEEE-754), a drift ledger, and a decision-invariance theorem that supports commensurate-accuracy claims in floating arithmetic (and that cleanly accounts for any staged/truncated auxiliary drift). Part III provides case-study reductions (LUP/solve/det/inv, LDL^T, blocked QR, SOI/SPD functions, GSEVP, dense LP/SDP IPM kernels, Gaussian process regression, and representative semiring problems) showing how to export the kernel advantage without reintroducing uncontrolled intermediate growth. Part IV abstracts admissible packings and extractors via a master condition and an easily checkable BWBM sufficient condition, and sketches extensions to multilinear/multigraded kernels and non-rounding extractors (e.g., CRT and semiring bucket projections).