Robust Supervised Spline Embedding.

High-dimensional data present significant challenges such as inadequate sample size, abundance of noise, and the curse of dimensionality, which make many traditional classification algorithms inapplicable. To provide valid inference for such data, it requires finding a noise-free low-dimensional representation that preserves both the underlying manifold structure and discriminative information. However, the existing methods often fail to take full consideration of these requirements. In this article, we introduce a robust supervised spline embedding (RS2E) algorithm for high-dimensional classification. The proposed approach is highlighted in four aspects: 1) it preserves the class-aware submanifold structure in the thin plate spline embedding space; 2) it eliminates noise and outliers to recover the clean manifold by exploiting its intrinsic low complexity; 3) it separates the class-aware submanifolds by maximizing the distance between each data point and the marginal data points of other class-aware submanifolds; and 4) it applies the alternating direction method of multipliers with generalized power iteration to solve the objective function. Promising experimental results on the real-world, generative adversarial network (GAN)-generated and artificially corrupted datasets demonstrate that RS2E outperforms other supervised dimensionality reduction algorithms in terms of classification accuracy.