# Nested subspace learning with flags

arXiv: https://arxiv.org/abs/2502.06022 · HTML: https://mlsanity.com/paper/2502.06022/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 138 references that are in the map.
- First posted on arXiv 9 Feb 2025.

## Builds on (most cited first, 24 of 24)

- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [A Brief Review of Domain Adaptation](https://mlsanity.com/paper/2010.03978/) (2020)
- [Manopt, a Matlab toolbox for optimization on manifolds](https://mlsanity.com/paper/1308.5200/) (2013)
- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [Dimensionality Reduction on SPD Manifolds: The Emergence of Geometry-Aware Methods](https://mlsanity.com/paper/1605.06182/) (2016)
- [An Overview of Robust Subspace Recovery](https://mlsanity.com/paper/1803.01013/) (2018)
- [Fast, Robust and Non-convex Subspace Recovery](https://mlsanity.com/paper/1406.6145/) (2014)
- [Dual Principal Component Pursuit](https://mlsanity.com/paper/1510.04390/) (2015)
- [A Well-Tempered Landscape for Non-convex Robust Subspace Recovery](https://mlsanity.com/paper/1706.03896/) (2017)
- [High-Dimensional Bayesian Optimization via Nested Riemannian Manifolds](https://mlsanity.com/paper/2010.10904/) (2020)
- [Convex Sparse Spectral Clustering: Single-view to Multi-view](https://mlsanity.com/paper/1511.06860/) (2015)
- [Nested Hyperbolic Spaces for Dimensionality Reduction and Hyperbolic NN Design](https://mlsanity.com/paper/2112.03402/) (2021)
- [Optimization on flag manifolds](https://mlsanity.com/paper/1907.00949/) (2019)
- [The Flag Median and FlagIRLS](https://mlsanity.com/paper/2203.04437/) (2022)
- [Chordal Averaging on Flag Manifolds and Its Applications](https://mlsanity.com/paper/2303.13501/) (2023)
- [Closed-form geodesics and trust-region method to calculate Riemannian logarithms on Stiefel and its quotient manifolds](https://mlsanity.com/paper/2103.13327/) (2021)
- [Fun with Flags: Robust Principal Directions via Flag Manifolds](https://mlsanity.com/paper/2401.04071/) (2024)
- [Principal nested shape space analysis of molecular dynamics data](https://mlsanity.com/paper/1903.09445/) (2019)
- [The curse of isotropy: from principal components to principal subspaces](https://mlsanity.com/paper/2307.15348/) (2023)
- [Numerical algorithms on the affine Grassmannian](https://mlsanity.com/paper/1607.01833/) (2016)
- [Backward Nested Descriptors Asymptotics with Inference on Stem Cell Differentiation](https://mlsanity.com/paper/1609.00814/) (2016)
- [Trace Ratio Optimization with an Application to Multi-view Learning](https://mlsanity.com/paper/2101.04292/) (2021)
- [A Manifold Approach to Learning Mutually Orthogonal Subspaces](https://mlsanity.com/paper/1703.02992/) (2017)
- [Geometric Optimization of Restricted-Open and Complete Active Space Self-Consistent Field Wavefunctions](https://mlsanity.com/paper/2404.14655/) (2024)

## Built on by (most cited first, 2 of 2)

- [Deep Hierarchical Learning with Nested Subspace Networks for Large Language Models](https://mlsanity.com/paper/2509.17874/) (2025)
- [A Flag Decomposition for Hierarchical Datasets](https://mlsanity.com/paper/2502.07782/) (2025)

## Abstract (arXiv)

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.