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Regularized Linear Discriminant Analysis Using a Nonlinear Covariance Matrix Estimator

Maaz Mahadi, Tarig Ballal, Muhammad Moinuddin, Tareq Y. Al-Naffouri, Ubaid M. Al-Saggaf

TL;DR

The paper tackles the degradation of Linear Discriminant Analysis (LDA) in high-dimensional settings where the covariance matrix is ill-conditioned. It introduces NL-RLDA, a classifier that uses a nonlinear inverse-covariance estimator derived from reformulating the Bayes score, yielding a tractable and robust discriminant rule. The authors establish double-asymptotic misclassification rate results and provide a consistent estimator for this rate, enabling a grid-search strategy to select the regularization parameter $\gamma$. Empirical results on synthetic and real datasets show NL-RLDA to outperform state-of-the-art RLDA methods and be competitive with non-LDA classifiers, particularly in low-sample or highly correlated scenarios. This work advances high-dimensional LDA by integrating nonlinear covariance filtering with principled performance guarantees and practical parameter-tuning tools.

Abstract

Linear discriminant analysis (LDA) is a widely used technique for data classification. The method offers adequate performance in many classification problems, but it becomes inefficient when the data covariance matrix is ill-conditioned. This often occurs when the feature space's dimensionality is higher than or comparable to the training data size. Regularized LDA (RLDA) methods based on regularized linear estimators of the data covariance matrix have been proposed to cope with such a situation. The performance of RLDA methods is well studied, with optimal regularization schemes already proposed. In this paper, we investigate the capability of a positive semidefinite ridge-type estimator of the inverse covariance matrix that coincides with a nonlinear (NL) covariance matrix estimator. The estimator is derived by reformulating the score function of the optimal classifier utilizing linear estimation methods, which eventually results in the proposed NL-RLDA classifier. We derive asymptotic and consistent estimators of the proposed technique's misclassification rate under the assumptions of a double-asymptotic regime and multivariate Gaussian model for the classes. The consistent estimator, coupled with a one-dimensional grid search, is used to set the value of the regularization parameter required for the proposed NL-RLDA classifier. Performance evaluations based on both synthetic and real data demonstrate the effectiveness of the proposed classifier. The proposed technique outperforms state-of-art methods over multiple datasets. When compared to state-of-the-art methods across various datasets, the proposed technique exhibits superior performance.

Regularized Linear Discriminant Analysis Using a Nonlinear Covariance Matrix Estimator

TL;DR

The paper tackles the degradation of Linear Discriminant Analysis (LDA) in high-dimensional settings where the covariance matrix is ill-conditioned. It introduces NL-RLDA, a classifier that uses a nonlinear inverse-covariance estimator derived from reformulating the Bayes score, yielding a tractable and robust discriminant rule. The authors establish double-asymptotic misclassification rate results and provide a consistent estimator for this rate, enabling a grid-search strategy to select the regularization parameter . Empirical results on synthetic and real datasets show NL-RLDA to outperform state-of-the-art RLDA methods and be competitive with non-LDA classifiers, particularly in low-sample or highly correlated scenarios. This work advances high-dimensional LDA by integrating nonlinear covariance filtering with principled performance guarantees and practical parameter-tuning tools.

Abstract

Linear discriminant analysis (LDA) is a widely used technique for data classification. The method offers adequate performance in many classification problems, but it becomes inefficient when the data covariance matrix is ill-conditioned. This often occurs when the feature space's dimensionality is higher than or comparable to the training data size. Regularized LDA (RLDA) methods based on regularized linear estimators of the data covariance matrix have been proposed to cope with such a situation. The performance of RLDA methods is well studied, with optimal regularization schemes already proposed. In this paper, we investigate the capability of a positive semidefinite ridge-type estimator of the inverse covariance matrix that coincides with a nonlinear (NL) covariance matrix estimator. The estimator is derived by reformulating the score function of the optimal classifier utilizing linear estimation methods, which eventually results in the proposed NL-RLDA classifier. We derive asymptotic and consistent estimators of the proposed technique's misclassification rate under the assumptions of a double-asymptotic regime and multivariate Gaussian model for the classes. The consistent estimator, coupled with a one-dimensional grid search, is used to set the value of the regularization parameter required for the proposed NL-RLDA classifier. Performance evaluations based on both synthetic and real data demonstrate the effectiveness of the proposed classifier. The proposed technique outperforms state-of-art methods over multiple datasets. When compared to state-of-the-art methods across various datasets, the proposed technique exhibits superior performance.
Paper Structure (18 sections, 3 theorems, 72 equations, 4 figures, 3 tables)

This paper contains 18 sections, 3 theorems, 72 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Notations
  3. Background
  4. Linear Discriminant Analysis (LDA)
  5. Regularized Linear Discriminant Analysis (RLDA)
  6. The proposed Regularized LDA Approach
  7. Interpretation of the proposed (inverse) covariance matrix estimator
  8. Definitions
  9. Asymptotic Performance
  10. The Consistent Estimator
  11. Setting the regularization parameter value
  12. Summary of the proposed classifier
  13. Computational Complexity
  14. Performance Evaluation
  15. Synthetic Data
  16. ...and 3 more sections

Key Result

Theorem 1

Under Assumptions (1)--(4), and ${\bf S}({\bf m}_0 - {\bf m}_1) \neq \boldsymbol{0}$, the following relations hold true for the quantities defined in eqn:G and eqn:D: where and with ${ \eta}_{\boldsymbol{\mu \mu}^T}$ and $\eta_{_\text{\tiny \boldmath$\Sigma$}}$ obtained from (eqn:asym) using some of the definitions in Table tab:def.

Figures (4)

  • Figure 1: RLDA classification error versus the regularization parameter ($\gamma$) for Gaussian data: A comparison of the proposed nonlinear covariance matrix estimator and commonly used linear estimators for different Mahalanobis distance ($\nu$) and training data size ($n$) values and a fixed $p=100$.
  • Figure 2: Histograms of the regularization parameter ($\gamma$) values together with RLDA classification error versus $\gamma$ curves for Gaussian data: A comparison of the proposed nonlinear covariance matrix estimator and commonly used linear estimators for different Mahalanobis distance ($\nu$) and training data size ($n$) values and a fixed $p=100$.
  • Figure 3: Model 1: The diagonal elements of the covariance are equal to 1, and the off-diagonal elements are 0.1.
  • Figure 6: The average computational time (in seconds) of all methods using Matlab R2019a running on a 64-bit, Core(TM) i7-2600K 3.40 GHz Windows PC, and R version 4.0.1. For all methods that require cross-validation or searching, we consider a grid of 21 values.

Theorems & Definitions (6)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Theorem 2
  • Remark 3
  • Lemma 1