A New Covariance Estimator for Sufficient Dimension Reduction in High-Dimensional and Undersized Sample Problems
This addresses a known bottleneck in sufficient dimension reduction for high-dimensional data where p >> n, though it appears incremental as an enhancement to existing SIR/SAVE methods.
The paper tackles the problem of covariance matrix singularity in high-dimensional, undersized sample settings for sufficient dimension reduction methods. They propose a Maximum Entropy Covariance (MEC) estimator that combines sliced covariance matrices using maximum entropy principles, demonstrating its efficiency on real-world leukemia cancer and electricity load data.
The application of standard sufficient dimension reduction methods for reducing the dimension space of predictors without losing regression information requires inverting the covariance matrix of the predictors. This has posed a number of challenges especially when analyzing high-dimensional data sets in which the number of predictors $\mathit{p}$ is much larger than number of samples $n,~(n\ll p)$. A new covariance estimator, called the \textit{Maximum Entropy Covariance} (MEC) that addresses loss of covariance information when similar covariance matrices are linearly combined using \textit{Maximum Entropy} (ME) principle is proposed in this work. By benefitting naturally from slicing or discretizing range of the response variable, y into \textit{H} non-overlapping categories, $\mathit{h_{1},\ldots ,h_{H}}$, MEC first combines covariance matrices arising from samples in each y slice $\mathit{h\in H}$ and then select the one that maximizes entropy under the principle of maximum uncertainty. The MEC estimator is then formed from convex mixture of such entropy-maximizing sample covariance $S_{\mbox{mec}}$ estimate and pooled sample covariance $\mathbf{S}_{\mathit{p}}$ estimate across the $\mathit{H}$ slices without requiring time-consuming covariance optimization procedures. MEC deals directly with singularity and instability of sample group covariance estimate in both regression and classification problems. The efficiency of the MEC estimator is studied with the existing sufficient dimension reduction methods such as \textit{Sliced Inverse Regression} (SIR) and \textit{Sliced Average Variance Estimator} (SAVE) as demonstrated on both classification and regression problems using real life Leukemia cancer data and customers' electricity load profiles from smart meter data sets respectively.