This tutorial explains Linear Discriminant Analysis (LDA) and Quadratic Discriminant Analysis (QDA) as two fundamental classification methods in statistical and probabilistic learning. We start with the optimization of decision boundary on which the posteriors are equal. Then, LDA and QDA are derived for binary and mul…
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48 results for “Mahalanobis discriminant analysis”
Survey of spectral, probabilistic, and deep metric learning methods.
problem Developing effective distance metrics for various machine learning tasks.
method Divided into spectral, probabilistic, and deep approaches, covering various techniques and their applications.
result Comprehensive overview of metric learning methods, including new developments and applications.
We consider an enlarged dimension reduction space in functional inverse regression. Our operator and functional analysis based approach facilitates a compact and rigorous formulation of the functional inverse regression problem. It also enables us to expand the possible space where the dimension reduction functions bel…
We derive a new radial link for binary classification under shared elliptical distributions.
problem Binary classification under shared-generator elliptical class-conditional distributions.
method We derive the Bayes radial-link family from the within-class radius law and estimate it by a finite fractional-power stochastic-polynomial projection.
result The derived link is asymptotically Bayes-optimal and significantly better than QDA on various benchmarks.
Flexible classifier using Mahalanobis distances for non-elliptical distributions.
problem Classifying non-elliptical and multimodal distributions.
method Semiparametric classifier based on Mahalanobis distances and generalized additive models.
result The proposed classifiers outperform traditional methods in high-dimensional, low-sample-size scenarios.
Robust GQDA improves classification accuracy in non-Normal data.
problem Non-robustness of GQDA under data contamination.
method Introduced robust estimators for mean vector and dispersion matrix.
result Robust GQDA classifiers perform significantly better in real data applications.
Feature selection in omics prediction problems using cat scores and false nondiscovery rate controlstat.AP
We revisit the problem of feature selection in linear discriminant analysis (LDA), that is, when features are correlated. First, we introduce a pooled centroids formulation of the multiclass LDA predictor function, in which the relative weights of Mahalanobis-transformed predictors are given by correlation-adjusted …
This paper presents a general notion of Mahalanobis distance for functional data that extends the classical multivariate concept to situations where the observed data are points belonging to curves generated by a stochastic process. More precisely, a new semi-distance for functional observations that generalize the usu…
Enhances forecasting of complex systems using FKMD.
problem Forecasting high-dimensional dynamical systems with unknown features.
method Featurized Koopman Mode Decomposition (FKMD) using delay embedding and learned Mahalanobis distance.
result Improves prediction accuracy for various complex systems.
A new method detects anomalies in multivariate streams without unit dependence.
problem Detect anomalies in multivariate streams without unit dependence.
method Proposes SigMahaKNN combining variance norm and path signature.
result SigMahaKNN detects anomalies better than existing methods.
Extends Mahalanobis distance to Banach spaces for anomaly detection.
problem Anomaly detection in infinite-dimensional spaces.
method Generalizes Mahalanobis distance to Banach spaces via Cameron-Martin norm and variance norm.
result Kernelized nearest-neighbour Mahalanobis distance outperforms traditional methods for time series novelty detection.
Mahalanobis distance detects anomalies well, but not for classification.
problem Detecting anomalies in neural classifier outputs.
method Analyzes Mahalanobis distance-based anomaly detection method, revealing its reliance on information not useful for classification.
result Combining Mahalanobis and ODIN methods improves anomaly detection performance and robustness.
A method uses Shapley values and Mahalanobis distances to explain multivariate outliers.
problem Explaining multivariate outlyingness in data.
method Decomposing squared Mahalanobis distance using Shapley values.
result Shapley values provide variable contributions to outlying observations.
The paper connects neural networks to Mahalanobis distance for interpretability.
problem Lack of interpretability in neural networks.
method Establishes a connection between neural network linear layers and Mahalanobis distance.
result Provides a foundation for more interpretable neural network models.
Compressing data helps learn Mahalanobis metrics effectively.
problem Learning Mahalanobis metrics in high-dimensional spaces.
method Randomly compress data to train a full-rank metric in a reduced feature space.
result Theoretical guarantees on error for Mahalanobis metric learning, independent of ambient dimension.
Proposes a text perturbation method using a Mahalanobis metric to balance privacy and utility.
problem Low utility of text analysis when using spherical noise for privacy-preserving text embedding.
method Regularized Mahalanobis metric to add elliptical noise, accounting for embedding space density.
result Improves privacy statistics while maintaining utility, outperforming Laplace mechanism.
A Randomized Permutation Whole-Model Test Heuristic for Self-Validated Ensemble Models (SVEM)stat.ME
A new test validates ensemble models against the null hypothesis.
problem Validating ensemble models against the null hypothesis of a constant response.
method Randomized permutation test on SVEM model predictions.
result The test maintains Type I error rate even with more parameters than observations.
Informative and discriminative feature descriptors play a fundamental role in deformable shape analysis. For example, they have been successfully employed in correspondence, registration, and retrieval tasks. In the recent years, significant attention has been devoted to descriptors obtained from the spectral decomposi…
Bayesian method selects subsets for LMMs with structured dependence.
problem Subset selection challenge in LMMs with structured dependence.
method Bayesian decision analysis with Mahalanobis loss function.
result Optimal linear coefficients for subsets and cardinality constraints.
Some Targets Are Harder to Identify than Others: Quantifying the Target-dependent Membership Leakagecs.LG
This paper quantifies how hard it is to identify specific data points in machine learning models.
problem Quantifying the difficulty of identifying specific data points in machine learning models.
method Characterizing optimal attacks and privacy defences, deriving impacts of noise and misspecification, and proposing a new covariance attack.
result The Mahalanobis distance explains the hardness of fixed-target membership inference attacks.
Learning Mahalanobis metric spaces is an important problem that has found numerous applications. Several algorithms have been designed for this problem, including Information Theoretic Metric Learning (ITML) [Davis et al. 2007] and Large Margin Nearest Neighbor (LMNN) classification [Weinberger and Saul 2009]. We study…
Connects robust optimization to conformal prediction for uncertainty sets.
problem Decision-making under uncertainty in sensitive data.
method Defines Mahalanobis distance as a conformity score and generates conformal uncertainty sets.
result Conformal uncertainty sets provide valid and conservative ellipsoidal regions.
We propose a low-rank approach to learning a Mahalanobis metric from data. Inspired by the recent geometric mean metric learning (GMML) algorithm, we propose a low-rank variant of the algorithm. This allows to jointly learn a low-dimensional subspace where the data reside and the Mahalanobis metric that appropriately f…
Kernel discriminant analysis uses nonlinear embeddings to improve classification.
problem Limited effectiveness of linear discriminant analysis in capturing nonlinear features.
method Study of nonlinear embeddings in kernel discriminant analysis using polynomial and Gaussian kernels, solving generalized eigenvalue problems.
result Polynomial and Gaussian discriminants capture class differences through population moments and randomized projections.
Recent work in metric learning has significantly improved the state-of-the-art in k-nearest neighbor classification. Support vector machines (SVM), particularly with RBF kernels, are amongst the most popular classification algorithms that uses distance metrics to compare examples. This paper provides an empirical analy…
Paper proposes distributed sparse multicategory discriminant analysis for classification.
problem Sparse multicategory classification with distributed data.
method Convex formulation, distributed setting, invariant discriminant subspace recovery.
result Distributed sparse multicategory linear discriminant analysis performs as good as centralized version after a few rounds of communications.
Detecting test samples drawn sufficiently far away from the training distribution statistically or adversarially is a fundamental requirement for deploying a good classifier in many real-world machine learning applications. However, deep neural networks with the softmax classifier are known to produce highly overconfid…
A Bayesian Nonparametric Perspective on Mahalanobis Distance for Out of Distribution Detectionstat.ML
Bayesian nonparametric models improve OOD detection, especially with complex covariance structures.
problem Improving out-of-distribution detection methods, especially in complex scenarios.
method Proposes Bayesian nonparametric mixture models with hierarchical priors that generalize the Mahalanobis distance score.
result Bayesian nonparametric methods outperform existing OOD methods, especially in complex scenarios.
Paper proposes faster incremental subclass discriminant analysis.
problem Efficiently classify subclasses in incremental data.
method Exact and approximate linear and kernelized solutions.
result Superior training time and accuracy compared to existing methods.
Extends metrics for SPD matrices to infinite dimensions.
problem Lack of generalized forms for Riemannian metrics.
method Unitized Hilbert-Schmidt operators and extended Mahalanobis norm.
result Improved performance in high-dimensional comparisons.
Spectrally-Corrected and Regularized Linear Discriminant Analysis for Spiked Covariance Modelstat.ML
Improved LDA method for better classification and dimensionality reduction.
problem Improving linear discriminant analysis for better classification performance.
method Integrates spectrally-corrected covariance matrix and regularized discriminant analysis.
result SRLDA has a linear classification global optimal solution under spiked model assumption.
PANDA improves linear discriminant analysis in high dimensions with minimal tuning.
problem Linear discriminant analysis in high-dimensional settings.
method PANDA: a tuning-insensitive method for linear discriminant analysis.
result PANDA achieves optimal convergence rates in estimation error and misclassification rate.
New robust discriminant analysis for non-Gaussian data.
problem Classical discriminant analysis struggles with non-Gaussian distributions and contaminated datasets.
method Each data point follows its own ES distribution with arbitrary scale, leading to robust classification.
result Maximum-likelihood estimation and classification are simple, fast, and robust.
For many tasks and data types, there are natural transformations to which the data should be invariant or insensitive. For instance, in visual recognition, natural images should be insensitive to rotation and translation. This requirement and its implications have been important in many machine learning applications, a…
A fundamental question in data analysis, machine learning and signal processing is how to compare between data points. The choice of the distance metric is specifically challenging for high-dimensional data sets, where the problem of meaningfulness is more prominent (e.g. the Euclidean distance between images). In this…
Proposes KMvDA for object recognition from multi-view data.
problem Recognizing objects from different views, even when views are heterogeneous.
method Introduces kernel multi-view discriminant analysis (KMvDA) and uses random Fourier features (RFF) for large-scale learning.
result KMvDA and RFF approximation improve object recognition from multi-view data.
Distance metric learning is a successful way to enhance the performance of the nearest neighbor classifier. In most cases, however, the distribution of data does not obey a regular form and may change in different parts of the feature space. Regarding that, this paper proposes a novel local distance metric learning met…
This paper analyzes implicit bias in Deep Linear Discriminant Analysis.
problem The implicit bias of Deep Linear Discriminant Analysis.
method Analyzing gradient flow on a L-layer diagonal linear network.
result Under balanced initialization, the network transforms additive updates into multiplicative updates, conserving the (2/L) quasi-norm.
This paper learns user preferences from comparisons using Mahalanobis metrics.
problem Learning user preferences from paired comparisons with unknown metric.
method Estimates user ideal point and Mahalanobis metric from comparisons.
result Effectiveness of the algorithm demonstrated on synthetic and real-world data.
DFSOS improves sparse discriminant analysis for high-dimensional data.
problem Sparse discriminant analysis in high-dimensional settings with feature selection.
method Deflation-Free Sparse Optimal Scoring (DFSOS) using Bregman iteration and orthogonality-constrained optimization.
result DFSOS achieves comparable or better classification accuracy than deflation-based methods.
Discriminative Dictionary Learning (DL) methods have been widely advocated for image classification problems. To further sharpen their discriminative capabilities, most state-of-the-art DL methods have additional constraints included in the learning stages. These various constraints, however, lead to additional computa…
Proposes ridge regression on Riemannian manifolds for time-series prediction.
problem Time-series prediction on Riemannian manifolds.
method Combines Riemannian least-squares fitting via Bézier curves, empirical covariance on manifolds, and Mahalanobis distance regularization.
result Significant error reduction in synthetic spherical experiments and hurricane forecasting.
Proposes improved classification via transfer learning with regularized linear discriminant analysis.
problem High dimensionality and small sample sizes lead to poor classification performance.
method Regularized random-effects linear discriminant analysis, combining ridge estimates from target and source models.
result Explicit derivation of asymptotic weights and classification error rates in high-dimensional settings.
New method for tensor classification with missing data.
problem Handling incomplete tensor data in high-dimensional classification.
method High-dimensional tensor linear discriminant analysis with TGMM and Tensor LDA-MD.
result Established convergence rates and minimax optimal bounds for misclassification rate.
New framework improves classification accuracy using Pillai's trace and ULDA.
problem Traditional LDA's limitations in noise sensitivity and non-invertible matrices.
method Integrates Pillai's trace with ULDA for a unified classifier.
result Effective control of Type I error rates and improved classification accuracy.
New method compresses large sample data for faster discriminant analysis.
problem Large sample sizes in discriminant analysis increase computational burden.
method Proposes a new compression approach for reducing training samples.
result Significant computational gains and superior predictive ability compared to random sub-sampling.
This paper proposes a new subspace learning method, named Quantized Fisher Discriminant Analysis (QFDA), which makes use of both machine learning and information theory. There is a lack of literature for combination of machine learning and information theory and this paper tries to tackle this gap. QFDA finds a subspac…
We introduce a new discriminant analysis method (Empirical Discriminant Analysis or EDA) for binary classification in machine learning. Given a dataset of feature vectors, this method defines an empirical feature map transforming the training and test data into new data with components having Gaussian empirical distrib…