Recursive algorithms that use algebraic packing may appear to achieve reduced computational complexity that does not reflect the dominating bit-complexity, i.e., the implemented performance would not exhibit the claimed scaling. This paper introduces Graded Projection Recursion (GPR), a formal framework designed to address this problem by defining a rigorous mechanism that provably controls bit growth if specific conditions are satisfied. We use GPR to develop a matrix multiplication algorithm that is model-honest and achieves a true near-quadratic bit-complexity. This framework also provides a general GPR Substitution Principle recipe for accelerating a significant class of related algorithms in a provably rigorous way.

翻译：使用代数打包的递归算法可能表面上实现了降低的计算复杂度，但这并未反映主导的位复杂度，即实际实现的性能不会表现出所声称的缩放特性。本文引入了分级投影递归（GPR），这是一个旨在通过定义一种严格的机制来解决此问题的正式框架，该机制在满足特定条件时可证明地控制位增长。我们利用GPR开发了一种模型诚实且实现真正近二次位复杂度的矩阵乘法算法。该框架还提供了一种通用的GPR替代原则方法，用于以可证明严格的方式加速一类重要的相关算法。