With data growing in scale and complexity, traditional linear dimension reduction techniques are becoming inadequate in some settings. Manifold fitting offers an important alternative by capturing low-dimensional latent geometric structures within high-dimensional spaces. This capability allows it to support downstream analysis in complex data settings. In this review, we explore the development and applications of manifold fitting. First, we introduce the basic concepts of manifold fitting and distinguish it from related techniques such as manifold embedding and denoising. We review the development of manifold fitting with three distinct stages: early nonparametric statistical methods, insights from mathematical analysis, and contemporary practical statistical approaches. Furthermore, we present diverse applications of manifold fitting, particularly in neural networks and bioinformatics, which illustrate its utility in complex data scenarios. Despite considerable progress, manifold fitting remains a fertile area for research. Many theoretical and practical questions remain unanswered, and ongoing investigations will further clarify its role in modern data science as a geometric tool for a wide range of data analysis challenges.

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