Robust Recovery of Subspace Structures by Low-Rank Representation

In this work we address the subspace recovery problem. Given a set of data samples (vectors) approximately drawn from a union of multiple subspaces, our goal is to segment the samples into their respective subspaces and correct the possible errors as well. To this end, we propose a novel method termed Low-Rank Representation (LRR), which seeks the lowest-rank representation among all the candidates that can represent the data samples as linear combinations of the bases in a given dictionary. It is shown that LRR well solves the subspace recovery problem: when the data is clean, we prove that LRR exactly captures the true subspace structures; for the data contaminated by outliers, we prove that under certain conditions LRR can exactly recover the row space of the original data and detect the outlier as well; for the data corrupted by arbitrary errors, LRR can also approximately recover the row space with theoretical guarantees. Since the subspace membership is provably determined by the row space, these further imply that LRR can perform robust subspace segmentation and error correction, in an efficient way.

💡 Research Summary

The paper tackles the fundamental problem of subspace recovery: given a collection of high‑dimensional data points that are approximately drawn from a union of several low‑dimensional linear subspaces, the goal is to assign each point to its correct subspace while simultaneously correcting possible corruptions (outliers or dense noise). To achieve this, the authors introduce Low‑Rank Representation (LRR), a novel optimization framework that seeks the lowest‑rank representation of the data over a prescribed dictionary (or “basis”) matrix.

Formally, let (X\in\mathbb{R}^{d\times n}) be the data matrix, and let (A\in\mathbb{R}^{d\times m}) be a dictionary (often chosen as the data itself, i.e., (A=X), or an over‑complete set of atoms). The representation model is
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