二次优化的秩崩溃原理
The Rank-Collapse Principle for Quadratic Optimization
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中文总结 AI 辅助
本文研究二次优化中与矩阵小秩相关的易处理现象，提出秩崩溃原理，关联多种现有方法并推导相关约束结果，给出实验与证书，构建统一严谨框架。
中文摘要 AI 辅助
二次优化问题在二次型矩阵具有不利曲率，或可行集为离散、组合或非凸时会变得困难。信号处理与优化领域存在一种互补现象：当二次型矩阵的秩较小时，部分看似困难的二次规划问题对固定秩而言存在精确多项式时间算法。我们研究该现象背后共同的半正定几何：若Q=BB^T为半正定矩阵，目标函数仅通过秩空间投影y=B^T x依赖于x；每个最优投影y^*会唯一最大化其自身方向定义的线性泛函，并满足定量二次余量，由此非线性最优性会坍缩为低维、自生成的线性暴露方向，我们将此称为秩崩溃原理。该原理本身不意味着存在有限候选集，高效精确优化还取决于可行族的投影或活动几何。我们通过投影-投影散射和活动-结构坍缩组织该区别，明确关联已有的zonotope、凸组合、边骨架、投影法向锥及固定秩稀疏PCA方法，并推导二元和有限相位向量、基数约束、拟阵基及稀疏PCA的保序结果，还给出方向稳定性和近似低秩证书及可复现实验。本文贡献是提出统一严谨的框架及一组定量结论，而非声称其源自作为研究动机的已知固定秩易处理结果。
英文摘要
Quadratic optimization becomes hard as soon as either the matrix in the quadratic form has an unfavorable curvature or the feasible set is discrete, combinatorial, or otherwise nonconvex. A complementary phenomenon is also well known in the signal-processing and optimization communities: when the matrix in the quadratic form has small rank, some hard-looking quadratic programs admit exact polynomial-time algorithms for fixed rank. We study the common positive-semidefinite geometry behind this phenomenon. If $Q=BB^\top$ is positive semidefinite, the objective depends on $x$ only through the rank-space shadow $y=B^\top x$. Every optimal shadow $y^*$ uniquely maximizes the linear functional defined by its own direction and satisfies a quantitative quadratic margin. Thus nonlinear optimality collapses to a low-dimensional, self-generated linear exposure direction. We call this the rank-collapse principle. The principle alone does not imply a finite candidate set: efficient exact optimization additionally depends on the projected or active geometry of the feasible family. We organize this distinction through projected-shadow scattering and active-structure collapse, relate it explicitly to established zonotope, convex-combinatorial, edge-skeleton, projected-normal-fan, and fixed-rank sparse-PCA methods, and derive tie-safe consequences for binary and finite-phase vectors, cardinality constraints, matroid bases, and sparse PCA. We also give directional-stability and approximately low-rank certificates, together with reproducible experiments. The paper's contribution is a unified, careful framework and a set of quantitative consequences, rather than a claim to originate the known fixed-rank tractability results that motivate it.
