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Robust and Efficient Subspace Segmentation via Least Squares Regression
Can-Yi Lu, Hai Min, Zhong-Qiu Zhao, Lin Zhu, De-Shuang Huang, Shuicheng Yan
TL;DR
The paper addresses how to segment data drawn from unions of linear subspaces while obtaining useful block-diagonal affinities. It characterizes objective-function conditions for block diagonality and proposes Least Squares Regression (LSR), which exploits data correlation. Experiments report that LSR outperforms SSC and LRR on two databases and is more efficient.
Problem
Subspace segmentation requires clustering data from a union of subspaces, but prior methods differ in objective functions and can have limitations such as excessive sparsity or unclear low-rank justification.
Method
The paper derives Enforced Block Diagonal conditions and presents LSR, a correlation-aware least-squares model with a grouping effect.
Results
LSR outperforms SSC and LRR on the Hopkins 155 database and Extended Yale Database B, while experiments also show greater efficiency.
Takeaways & Limitations
The analysis identifies a common block-diagonality framework for existing criteria, while LSR provides a simpler and more efficient segmentation approach that models data correlation.
Takeaways & Limitations
Original LRR does not guarantee a block-diagonal solution even when subspaces are orthogonal, so its representation may fail to produce the true segmentation.
Abstract
from arXiv · showhide
This paper studies the subspace segmentation problem which aims to segment data drawn from a union of multiple linear subspaces. Recent works by using sparse representation, low rank representation and their extensions attract much attention. If the subspaces from which the data drawn are independent or orthogonal, they are able to obtain a block diagonal affinity matrix, which usually leads to a correct segmentation. The main differences among them are their objective functions. We theoretically show that if the objective function satisfies some conditions, and the data are sufficiently drawn from independent subspaces, the obtained affinity matrix is always block diagonal. Furthermore, the data sampling can be insufficient if the subspaces are orthogonal. Some existing methods are all special cases. Then we present the Least Squares Regression (LSR) method for subspace segmentation. It takes advantage of data correlation, which is common in real data. LSR encourages a grouping effect which tends to group highly correlated data together. Experimental results on the Hopkins 155 database and Extended Yale Database B show that our method significantly outperforms state-of-the-art methods. Beyond segmentation accuracy, all experiments demonstrate that LSR is much more efficient.
1 Introduction
Subspace segmentation groups data drawn from multiple subspaces, typically through an affinity matrix whose block structure supports clustering. Prior methods differ mainly in their representation objectives, while this paper analyzes when block diagonality is guaranteed and proposes LSR as a simpler, correlation-aware alternative.
- Problem: Subspace segmentation clusters data drawn from a union of subspaces according to the underlying subspace generating each point.It appears in applications including image representation, clustering, and motion segmentation.
- Affinity construction: Spectral clustering depends on an affinity matrix with zero between-cluster affinities, ideally yielding a block-diagonal graph.SSC and LRR measure similarity through representational coefficients but impose different regularizations on the coefficient matrix.
- Prior methods: SSC can become too sparse, separating highly correlated within-subspace points, and its repeated ℓ1 minimizations are computationally inefficient.Sparse representation may select one point at random from a group with high pairwise correlations.
- Prior methods: Original LRR does not guarantee a block-diagonal solution even for orthogonal subspaces, so its resulting representation may fail to recover the true segmentation.The example also questions whether low rank itself explains LRR’s effectiveness for segmentation.
- Research questions: The paper asks which objective-function conditions and data assumptions guarantee block-diagonal solutions, and which criterion provides a better choice.Existing methods obtain block-diagonal solutions under independent-subspace assumptions, while SSQP requires orthogonality.
- Contributions: The paper analyzes Enforced Block Diagonal conditions and introduces LSR, which uses data correlation and groups highly correlated points together.It further reports robustness to bounded disturbance, a closed-form solution, and stronger effectiveness and efficiency than SSC and LRR on two databases.
2 Theoretical Analysis
Theoretical analysis establishes when optimization criteria yield block-diagonal affinity solutions for subspace segmentation, under independent or orthogonal subspaces and appropriate sampling. It also clarifies how existing criteria fit these conditions and what their within-cluster affinity assumptions imply.
- Independent Subspaces: Independence guarantees that each data point has a unique representation using only the basis of its own subspace.The proof uses the direct-sum property of independent subspaces to force coefficients for other subspaces to zero.
- Enforced Block Diagonal Conditions: A matrix function satisfying the EBD conditions yields a block-diagonal optimal solution when sampling is sufficient and subspaces are independent.The EBD conditions enforce invariance to permutation, penalize cross-block components, and support blockwise decomposition.
- Block-Diagonal Optimality: Under the EBD conditions, any optimal solution can be replaced by its block-diagonal component without worsening the objective, with equality forcing the cross-block component to vanish.This establishes the block-diagonal structure through feasibility and the strict equality condition.
- Existing Criteria: SSC, LRR, MSR, and SSQP are special cases of the general EBD framework, while the additional decomposition condition makes each within-subspace block independently optimal.The framework separates the shared structural guarantee from differences in the criteria used within clusters.
- Within-Cluster Affinities: The analysis distinguishes within-cluster behaviors: SSC can become too sparse, whereas LRR promotes low nuclear norm, whose physical meaning for graph structure is unclear.The paper notes that nuclear norm has an independent modeling role for subspace segmentation rather than merely approximating rank.
- Insufficient Sampling: For insufficient data, orthogonal subspaces allow a block-diagonal solution despite representational error caused by limited data rather than noise.The resulting guarantee provides a theoretical lower bound for correct segmentation when sampling is insufficient, although orthogonality may not hold in real data.
3 Subspace Segmentation via LSR
LSR combines theoretical block-diagonal guarantees with a least-squares representation that exploits correlation among data and supports efficient spectral clustering.
- Motivation: Sparse representation can miss within-cluster correlations, motivating a criterion that preserves grouping among correlated data.The paper argues that sparsity may separate highly correlated points, whereas most real data exhibit strong correlations.
- Theoretical guarantees: Under sufficient sampling and independent subspaces, the optimal solution is block diagonal.
- LSR formulation: LSR penalizes reconstruction error with a Frobenius-norm term and has an analytical solution.The constraint diag(Z) = 0 can be removed to obtain an alternative formulation based on ridge regression.
- Grouping effect: LSR’s grouping effect makes coefficients for highly correlated data approximately equal, encouraging those points to share a cluster.The paper states that this effectiveness comes from grouping rather than from the low rank of the solution.
- Algorithm: The algorithm symmetrizes the learned representation into an affinity matrix and applies Normalized Cuts for segmentation.It solves either LSR formulation before constructing (|Z*| + |(Z*)T|)/2.
4 Experimental Verification
Experiments evaluate LSR against SSC and LRR on motion and face datasets, reporting segmentation performance and affinity-matrix computation time. The authors report that LSR performs better and more efficiently, attributing its advantage partly to grouping correlated data.
- Experimental setup: LSR is evaluated against SSC and LRR on the Hopkins 155 motion database and Extended Yale Database B.The experiments use two LSR variants and report parameters, segmentation performance, and running time.
- Motion segmentation: 156 Hopkins 155 sequences provide two- or three-motion segmentation tasks after PCA projection to 12 dimensions.Each sequence is treated as a separate task, and reported running time measures affinity-matrix computation.
- Face clustering: The Extended Yale experiment uses the first 5 and 10 classes, with images projected by PCA before clustering.Each class contains 64 resized 32 × 32 images.
- Results: LSR outperforms SSC and LRR on both the Hopkins 155 database and Extended Yale Database B.The paper attributes the advantage mainly to LSR’s grouping effect for modeling data correlation.
- Results: LSR is reported to be robust to bounded matrix disturbances measured with the Frobenius norm.
- Results: LSR is more efficient than SSC and LRR, whose optimization procedures are described as computationally heavier.The comparison cites nonsmooth ℓ1 minimization for SSC and repeated singular value decompositions for LRR.
- Discussion: The authors characterize LSR as simpler and better than prior SSC, LRR, MSR, and SSQP models.
5 Conclusions
The paper develops general block-diagonal conditions and proposes LSR, which uses correlation-driven grouping to achieve effective and efficient subspace segmentation.
- Conclusions: Theoretical analysis identifies general conditions yielding block-diagonal solutions for independent subspaces and permits insufficient sampling for orthogonal subspaces.
- Conclusions: LSR exploits data correlation, groups highly correlated points, and is theoretically shown to be robust to noise.
- Conclusions: Experiments on real data report that LSR is efficient and effective compared with SSC and LRR.
Proof of Theorem 6
The proof derives each LSR coefficient column from a leave-one-out regression and then uses a precomputed matrix inverse to avoid repeated inversions.
- Leave-one-out solution: Removing the i-th data column forms Yi, and the corresponding optimal coefficient column is expressed through (Yi^T Yi + λI)^-1.
- Efficient computation: The data are permuted as XP = [Yi xi] so the leave-one-out computation can be related to XT X + λI.
- Efficient computation: The Woodbury formula is used to compute the required inverse from the precomputable matrix D = (XT X + λI)^-1.
- Efficient computation: Permutation-matrix properties recover the desired coefficient column as [Z*]i = -bi.
Proof of Theorem 7
The proof references the optimal solution z∗ to problem (19) and invokes equations (23) and (24), including a quantity involving r = xT_i xj.
- The proof uses that z∗ is the optimal solution to problem (19).
- Equations (23) and (24) are invoked in the proof.
- The proof introduces r = xT_i xj in an expression involving 2(1 − r).