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.

翻译：许多机器学习方法致力于寻找数据的低维表示。通常通过先选择维度$q$，然后在$q$维子空间（格拉斯曼流形）上优化特定目标函数来估计潜在子空间。尝试不同的$q$值通常会产生非嵌套的子空间，这引发了数据表示一致性的重要问题。本文提出一种简单且易于实现的原理，用于在子空间学习方法中强制实现嵌套性。该方法通过嵌套投影算子将格拉斯曼流形优化准则提升至旗流形（维度递增的嵌套子空间空间）。我们将旗流形技巧应用于多种经典机器学习方法，并证明其能有效解决嵌套性问题。