Extracting meaningful information from high-dimensional data poses a
formidable modeling challenge, particularly when the data is obscured by noise
or represented through different modalities. In this research, we propose a
novel non-parametric modeling approach, leveraging the Gaussian Process (GP),
to characterize high-dimensional data by mapping it to a latent low-dimensional
manifold. This model, named the Latent Discriminative Generative Decoder
(LDGD), utilizes both the data (or its features) and associated labels (such as
category or stimulus) in the manifold discovery process. To infer the latent
variables, we derive a Bayesian solution, allowing LDGD to effectively capture
inherent uncertainties in the data while enhancing the model's predictive
accuracy and robustness. We demonstrate the application of LDGD on both
synthetic and benchmark datasets. Not only does LDGD infer the manifold
accurately, but its prediction accuracy in anticipating labels surpasses
state-of-the-art approaches. We have introduced inducing points to reduce the
computational complexity of Gaussian Processes (GPs) for large datasets. This
enhancement facilitates batch training, allowing for more efficient processing
and scalability in handling extensive data collections. Additionally, we
illustrate that LDGD achieves higher accuracy in predicting labels and operates
effectively with a limited training dataset, underscoring its efficiency and
effectiveness in scenarios where data availability is constrained. These
attributes set the stage for the development of non-parametric modeling
approaches in the analysis of high-dimensional data; especially in fields where
data are both high-dimensional and complex.

我们提出了一种新的非参数建模方法 LDGD，利用高斯过程（GP）将高维数据映射到潜在的低维流形，通过推断潜变量来有效捕捉数据中的不确定性，提高了模型的预测准确性和鲁棒性。LDGD 不仅能准确地推断流形，而且在预测标签方面的准确性也超过了现有的方法。我们还通过引入诱导点来降低大型数据集中高斯过程（GP）的计算复杂度，实现了批处理训练，提高了处理大规模数据集的效率和可扩展性。此外，我们证明了 LDGD 在预测有限的训练数据标签方面的较高准确性，突显了它在数据可用性受限的场景中的效率和有效性。这些特点为高维数据分析中非参数建模方法的发展奠定了基础，特别是对于高维且复杂的数据领域。