信息诱导的训练几何：精确约化、典范完备化与结构化表达性
Information-Induced Training Geometry: Exact Reduction, Canonical Completion, and Structured Expressivity
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中文总结 AI 辅助
本研究在仿射不变黎曼几何下，证明信息诱导的压缩映射存在唯一完备化，实现精确变分约化，并刻画了正定锥上的分层结构与结构化表达性。
中文摘要 AI 辅助
训练数据通过声明信息通道中可见的余向量约束优化器的几何结构。我们研究这种部分信息如何相对于参考确定一个完整的正度量，以及哪些自由度仍未确定。我们的核心结果解决了仿射不变黎曼几何下的满列秩正定压缩问题。压缩映射是一个分裂-哈达玛度量子映射，并具有显式的唯一完备化，该完备化是仿射不变最近的全几何，实现可见目标，并产生精确的全到可见变分约化。当通道移动时，完备化形成正定锥的规范不变秩分层。其闭式拉回对度量通过参考失配权重将可见度量运动与子空间旋转分离，产生显式的半正定多方向格拉姆矩阵，并揭示参考值模式的精确奇异性。该机制由一个度量定理解释，该定理将球子映射、达到的纤维距离和每个单调径向可见决策问题的无损约化等同起来。一个光滑分裂-哈达玛定理提供相干信息片、近端交换和逐解梯度流提升。正定实现也给出闭式先验数据收缩。对角和块优化器族约化为具有有效面证书的相对内部锥图像测试，而确定性和有限样本界量化可见几何及其子空间的恢复。这些结果共同刻画了所述有限维仿射不变模型的精确约化、参考依赖完备化和结构化表达性。
英文摘要
Training data constrains optimizer geometry through the covectors visible to a declared information channel. We study how such partial information determines a full positive cometric relative to a reference and which degrees of freedom remain unidentified. Our central result resolves full-column-rank positive-definite compression under affine-invariant Riemannian geometry. The compression map is a split-Hadamard metric submetry and admits an explicit unique completion that is the affine-invariant nearest full geometry realizing a visible target and yields exact full-to-visible variational reduction. When the channel moves, the completions form a gauge-invariant rank stratification of the positive-definite cone. Its closed-form pullback pair metric separates visible-metric motion from subspace rotation through a reference-mismatch weight, yields an explicit positive-semidefinite multi-direction Gram matrix, and exposes the precise singularity of reference-valued modes. The mechanism is explained by a metric theorem equating ball submetry, attained fiber distance, and lossless reduction of every monotone radial visible decision problem. A smooth split-Hadamard theorem supplies coherent information sheets, proximal commutation, and solution-wise gradient-flow lifting. The positive-definite realization also gives closed-form prior-data shrinkage. Diagonal and block optimizer families reduce to relative-interior conic image tests with valid facial certificates, while deterministic and finite-sample bounds quantify recovery of the visible geometry and its subspace. Together these results characterize exact reduction, reference-dependent completion, and structured expressivity for the stated finite-dimensional affine-invariant model.
发表机构
- Xidian University（西安电子科技大学）
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