# High-dimensional Linear Discriminant Analysis: Optimality, Adaptive Algorithm, and Missing Data

arXiv: https://arxiv.org/abs/1804.03018 · HTML: https://mlsanity.com/paper/1804.03018/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Apr 2018.

## Built on by (most cited first, 17 of 17)

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- [Deep learning with missing data](https://mlsanity.com/paper/2504.15388/) (2025)
- [Optimal Discriminant Analysis in High-Dimensional Latent Factor Models](https://mlsanity.com/paper/2210.12862/) (2022)
- [Sharp-SSL: Selective high-dimensional axis-aligned random projections for semi-supervised learning](https://mlsanity.com/paper/2304.09154/) (2023)
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- [High-Dimensional Gaussian Mean Estimation under Realizable Contamination](https://mlsanity.com/paper/2603.16798/) (2026)
- [Classification of high-dimensional data with spiked covariance matrix structure](https://mlsanity.com/paper/2110.01950/) (2021)
- [Varying Coefficient Linear Discriminant Analysis for Dynamic Data](https://mlsanity.com/paper/2203.06371/) (2022)
- [Interpolating Discriminant Functions in High-Dimensional Gaussian Latent Mixtures](https://mlsanity.com/paper/2210.14347/) (2022)
- [Pivotal Estimation of Linear Discriminant Analysis in High Dimensions](https://mlsanity.com/paper/2309.09831/) (2023)
- [GradMix: Gradient-based Selective Mixup for Robust Data Augmentation in Class-Incremental Learning](https://mlsanity.com/paper/2505.08528/) (2025)

## Abstract (arXiv)

This paper aims to develop an optimality theory for linear discriminant analysis in the high-dimensional setting. A data-driven and tuning free classification rule, which is based on an adaptive constrained ℓ1 minimization approach, is proposed and analyzed. Minimax lower bounds are obtained and this classification rule is shown to be simultaneously rate optimal over a collection of parameter spaces. In addition, we consider classification with incomplete data under the missing completely at random (MCR) model. An adaptive classifier with theoretical guarantees is introduced and optimal rate of convergence for high-dimensional linear discriminant analysis under the MCR model is established. The technical analysis for the case of missing data is much more challenging than that for the complete data. We establish a large deviation result for the generalized sample covariance matrix, which serves as a key technical tool and can be of independent interest. An application to lung cancer and leukemia studies is also discussed.