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Self-Weighted Robust LDA for Multiclass Classification with Edge Classes
Caixia Yan, Xiaojun Chang, Minnan Luo, Qinghua Zheng, Xiaoqin Zhang, Zhihui Li, Feiping Nie
TL;DR
Edge classes can bias LDA’s total mean and cause extreme pairwise distances to dominate, compromising separation among other classes. SWRLDA uses a robust, self-weighted pairwise criterion; experiments report favorable classification performance and computational efficiency against compared methods on synthetic and real-world datasets.
Problem
Edge classes can bias the total mean used for between-class scatter and cause ℓ2-norm criteria to magnify extreme class-pair distances.
Source: §1 · p. 2Method
SWRLDA avoids optimal-mean calculation and uses an ℓ2,1-norm pairwise distance criterion to learn adaptive class-pair weights without extra parameters.
Source: §1 · p. 2, §3.1 · pp. 4–5Results
Experiments on synthetic and real-world datasets report favorable classification performance and superior computational efficiency against compared methods.
Source: §1 · p. 3Takeaways & Limitations
SWRLDA’s self-adaptive class-pair weights are learned in the desired embedding space rather than requiring additional parameter tuning.
Source: §1 · p. 3
Abstract
from arXiv · showhide
Linear discriminant analysis (LDA) is a popular technique to learn the most discriminative features for multi-class classification. A vast majority of existing LDA algorithms are prone to be dominated by the class with very large deviation from the others, i.e., edge class, which occurs frequently in multi-class classification. First, the existence of edge classes often makes the total mean biased in the calculation of between-class scatter matrix. Second, the exploitation of l2-norm based between-class distance criterion magnifies the extremely large distance corresponding to edge class. In this regard, a novel self-weighted robust LDA with l21-norm based pairwise between-class distance criterion, called SWRLDA, is proposed for multi-class classification especially with edge classes. SWRLDA can automatically avoid the optimal mean calculation and simultaneously learn adaptive weights for each class pair without setting any additional parameter. An efficient re-weighted algorithm is exploited to derive the global optimum of the challenging l21-norm maximization problem. The proposed SWRLDA is easy to implement, and converges fast in practice. Extensive experiments demonstrate that SWRLDA performs favorably against other compared methods on both synthetic and real-world datasets, while presenting superior computational efficiency in comparison with other techniques.
1 Introduction
Edge classes can bias LDA’s total mean and let extreme class-pair distances dominate, leaving other classes overlapping. SWRLDA replaces the mean-based criterion with robust, self-weighted pairwise distances and performs favorably on synthetic and real-world datasets.
1 Introduction
- Edge classes can bias the total mean and make ℓ2-norm distances overemphasize extreme class pairs, leaving non-edge classes overlapping.Figure 1 contrasts four nearby bird categories with a distant Black-footed Albatross class; the desired projection trades off separation of edge and non-edge classes.Source: §1 · p. 2, Figure 1 · p. 3
- Earlier robust-mean and weighted pairwise LDA variants face computational complexity, imperfect weights, slow optimization, or approximate solutions to class-separation problems.The section cites M-estimator, S-estimators, MCD, MVE, approximate pairwise accuracy, KL-based criteria, and max-min distance analysis.Source: §1 · p. 2
- SWRLDA avoids optimal-mean calculation and uses an ℓ2,1-norm pairwise criterion to reduce edge-class influence while learning class-pair weights without extra parameters.Its re-weighted algorithm optimizes projection directions simultaneously.Source: §1 · pp. 2–3
- Experiments on synthetic and real-world datasets report favorable classification performance and computational efficiency for SWRLDA against compared methods.Source: §1 · p. 3
2 Related Work
Traditional and pairwise LDA formulations aim to separate classes, but total-mean dependence, equal weighting, or demanding optimization can limit robustness and scalability. Weighted pairwise methods address class-pair importance with different criteria and assumptions.
2.1 Linear Discriminant Analysis
- Traditional LDA measures class separation from the total mean, while FLDA uses pairwise distances but can lose information across its two optimization stages and overemphasize remote class pairs.Source: §2.1 · p. 3
2.2 Weighted Pairwise Distance Criterion
- Loog et al.’s [28] weights use original-space distances, so they may not be optimal when the projected distribution changes or data contain large noise.Source: §2.2 · pp. 3–4
- GMSS and HMSS use weighted geometric- and harmonic-mean criteria under a homoscedastic Gaussian assumption, but gradient optimization can converge slowly.Source: §2.2 · p. 4
- Max-min and simultaneous pairwise-distance approaches target separation of all class pairs, yet rely on complex iterative optimization that limits scalability for large-scale dimension reduction.Source: §2.2 · p. 4
3 The Proposed Methodology
SWRLDA replaces total-mean-based separation with pairwise class-center distances and an ℓ2,1-norm criterion to address edge-class effects. Its re-weighted procedure learns class-pair weights, has complexity O(cmdt) when d is much larger than c and m, and is shown to converge to its maximum.
3.1 Problem Formulation
- SWRLDA measures between-class variability with pairwise class-center distances, avoiding the edge-sensitive total-mean calculation, and uses an ℓ2,1-norm criterion to reduce domination by large-deviation pairs.The method learns class-pair weights adaptively without additional parameters.Source: §3.1 · pp. 4–5
3.2 Optimization Procedure
- The re-weighted procedure alternates updates to pairwise variables and the projection matrix; inverse projected distances act as adaptive weights, giving greater weight to closer class pairs.The weights are derived without additional parameters.Source: §3.2 · p. 6
3.3 Computational Complexity Reduction Analysis
- When feature dimensionality d is much larger than class count c and reduced dimensionality m, SWRLDA's time complexity simplifies to O(cmdt), where t is the iteration count.The analysis also states that this complexity does not increase in contrast to the cited state-of-the-art LDA methods.Source: §3.3 · pp. 6–7
3.4 Convergence Analysis
- The objective increases monotonically under Algorithm 1 and converges to its maximum because it is upper bounded.The upper-bound result is stated for optimization problem (10).Source: §3.4 · pp. 7–8
4 Experiments
The experiments evaluate SWRLDA on synthetic and real-world datasets against seven state-of-the-art algorithms. The comparison includes total-mean-based LDA and robust or weighted pairwise Fisher-criterion methods.
4 Experiments
- The experiments use synthetic and real-world datasets and run SWRLDA in Matlab on Windows 10 with an Intel Core i5-6200U CPU and 16 GB RAM.Source: §4 · p. 8
- Seven state-of-the-art algorithms are selected for comparison, including traditional LDA and robust LDA RSLDA.Source: §4 · p. 8
- The comparison also covers weighted pairwise methods aPAC, GMSS, HMSS, STRDA, and MMDA, which use different pair-weighting or separation criteria.The passage describes their respective criteria, including Mahalanobis distance, divergence means, trace ratios, and minimal pairwise distance.Source: §4 · p. 8
4.1 Experiments on Synthetic Datasets
On synthetic data, SWRLDA provides a better overall projection trade-off across classes when an edge class is present. The experiments compare projections for three closely located classes with and without a distant fourth class.
4.1.1 Experimental Setup
- The synthetic comparison uses syn1 with three closely located classes and syn2 with a distant fourth, edge class; each class has 200 Gaussian-generated points with the same standard deviation.The class means are [−5, −4], [−3, 1], [−1, 6], and [10, −2] for classes 1–4, respectively.Source: §4.1.1 · p. 8
4.1.2 Performance Comparison
- On syn2, most compared methods overemphasize separating edge class 4, while MMDA underemphasizes it; aPAC and SWRLDA perform better on the dataset with an edge class.The projected histograms show overlap among classes 1–3 for over-sensitive methods and overlap involving class 4 for MMDA.Source: §4.1.2 · pp. 9–10, Figure 3 · p. 9
- With an edge class, SWRLDA provides a better overall projection trade-off than aPAC, whose projection leaves classes 3 and 4 with relatively large overlap.The synthetic data add a distant fourth class to three closely located classes.Source: §4.1.2 · pp. 9–10, Figure 3 · p. 9
4.2 Experiments on Real-World Datasets
The real-world evaluation uses seven datasets and compares SWRLDA across accuracy, class separation, visualization, running time, robustness, and convergence. Across these tests, SWRLDA performs well against comparison methods, including under corruption and on large-scale data.
4.2.1 Experimental Setup
- Experiments on seven real-world datasets use five-fold cross-validation, with k-NN (k = 1) on the learned projected space.The evaluation includes accuracy, minimum pairwise distance, two-dimensional visualization, running time, and robustness.Source: §4.2 · p. 10, §4.2.1 · p. 10
4.2.2 Classification Accuracy
- Accuracy improves by 2.4% on COIL20, 6.7% on COIL100, 2.7% on Isolet, 2.2% on CMU PIE, and 0.1% on YaleB, with SWRLDA highest among comparisons.It performs best in most projected dimensions and is more stable as dimension decreases, except when m is very small, such as m = 1.Source: §4.2.2 · p. 11, Figure 4 · p. 11
4.2.3 Average Minimum Pairwise Distance
- As projected dimension increases, SWRLDA gains more significant average minimum pairwise distance than other methods, while methods are nearly tied at one dimension.A larger minimum pairwise distance indicates better separation of all class pairs in the low-dimensional subspace.Source: §4.2.3 · pp. 11–12, Figure 5 · p. 11
4.2.4 Two-Dimensional Visualization of Wine Dataset
- On Wine, SWRLDA produces a two-dimensional projection with no overlapping samples and the largest minimum between-class distance.The comparison includes methods with serious overlap and methods with slight overlap; the training data contain no edge class.Source: §4.2.4 · p. 12, Figure 6 · p. 12
4.2.5 Average Running Time
- On 20Newsgroups, comparison methods are infeasible at more than 24 hours, while SWRLDA takes several seconds; on smaller datasets, it is 15 times shorter.Table 3 presents average running times across six benchmark datasets.Source: Table 3 · p. 12, §4.2.5 · p. 13
4.2.6 The Evaluation of Robustness
- With increasing numbers of corrupted classes, competitors’ recognition performance degrades dramatically while SWRLDA remains more stable.The experiment corrupts 20%, 40%, 60%, or 80% of samples in each corrupted class on CMU PIE and YaleB; without corrupted classes, SWRLDA is only slightly better.Source: §4.2.6 · p. 13, Figure 7 · p. 13
4.2.7 Convergence Analysis
- The objective converges in four iterations on all employed datasets, and different random initializations reach a stable value.The authors interpret this stability as indicating a global optimum in practice on these datasets.Source: §4.2.7 · p. 13, Figure 8 · p. 14
5 Conclusion
The conclusion presents SWRLDA as robust LDA for multiclass classification with edge classes. It combines pairwise class-center distances, an ℓ2,1-norm criterion, and learned class-pair weights; experiments on synthetic and real-world datasets support its effectiveness.
5 Conclusion
- SWRLDA replaces total-mean-based between-class scatter with distances between every pair of class centers, avoiding optimal mean calculation.The paper identifies edge classes as a reason the total mean can be dominated.Source: §5 · p. 14
- Its ℓ2,1-norm between-class distance criterion is intended to reduce overemphasis on distant class pairs and improve robustness to edge classes.The paper contrasts this with the ℓ2-norm criterion, which can neglect smaller distances.Source: §5 · p. 14
- An effective re-weighted algorithm learns adaptive contributions for each class pair without additional parameter tuning.Experiments on two synthetic and six real-world datasets demonstrate the approach’s effectiveness and superiority.Source: §5 · p. 14