A convex optimization approach to high-dimensional sparse quadratic discriminant analysis
A convex optimization approach to high-dimensional sparse quadratic discriminant analysis
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DOI：
10.1214/20-aos2012
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发表时间：
2019-12
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通讯作者：
T. Cai;Linjun Zhang
中科院分区：
文献类型：
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作者：
T. Cai;Linjun Zhang
本文研究了高维稀疏二次判别分析（QDA），旨在建立分类误差的最优收敛率。建立了极大极小下界，证明了判别方向和微分图上的稀疏性条件等结构假设对于可能构造一致高维QDA规则的必要性。然后，在稀疏性假设下，我们提出了一种基于约束凸优化的分类算法SDAR。得到了极大极小上界和下界，并证明该分类规则在参数空间集合上同时是速率最优的，可达对数因子。仿真研究表明，SDAR具有良好的数值性能。该算法还通过对前列腺癌数据和结肠组织数据的分析来说明。在此基础上，将高斯环境下高维两组QDA方法和理论扩展到多组分类和高斯copula模型下的分类。
In this paper, we study high-dimensional sparse Quadratic Discriminant Analysis (QDA) and aim to establish the optimal convergence rates for the classification error. Minimax lower bounds are established to demonstrate the necessity of structural assumptions such as sparsity conditions on the discriminating direction and differential graph for the possible construction of consistent high-dimensional QDA rules. We then propose a classification algorithm called SDAR using constrained convex optimization under the sparsity assumptions. Both minimax upper and lower bounds are obtained and this classification rule is shown to be simultaneously rate optimal over a collection of parameter spaces, up to a logarithmic factor. Simulation studies demonstrate that SDAR performs well numerically. The algorithm is also illustrated through an analysis of prostate cancer data and colon tissue data. The methodology and theory developed for high-dimensional QDA for two groups in the Gaussian setting are also extended to multi-group classification and to classification under the Gaussian copula model.