提出用嵌套投影统一不同维度子空间，解决降维表示不一致问题。
Nested subspace learning with flags
- 将子空间优化从格拉斯曼流形升级到旗流形，通过嵌套投影强制维度间包含关系
- 在经典方法上应用后，不同维度的子空间实现严格嵌套，避免表示不一致
- 适合需要多尺度降维且关注维度间一致性场景，如特征分析与可视化
许多机器学习方法致力于寻找数据的低维表示。通常通过选定维度 $q$，并在 $q$-维子空间（即格拉斯曼流形）上优化目标函数来估计潜在子空间。但尝试不同 $q$ 值时，一般得到非嵌套的子空间，引发表示不一致的问题。本文提出一种简单且易实现的原则，以在子空间学习中强制嵌套性：通过嵌套投影将格拉斯曼优化准则提升至旗流形（即维度递增的嵌套子空间空间）。我们将该方法应用于多种经典机器学习算法，结果表明其能有效解决嵌套性问题。
原文摘要 · Abstract (English)
Many machine learning methods look for low-dimensional representations of the data. The underlying subspace can be estimated by first choosing a dimension $q$ and then optimizing a certain objective function over the space of $q$-dimensional subspaces (the Grassmannian). Trying different $q$ yields in general non-nested subspaces, which raises an important issue of consistency between the data representations. In this paper, we propose a simple and easily implementable principle to enforce nestedness in subspace learning methods. It consists in lifting Grassmannian optimization criteria to flag manifolds (the space of nested subspaces of increasing dimension) via nested projectors. We apply the flag trick to several classical machine learning methods and show that it successfully addresses the nestedness issue.
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