流形學習是一類用於識別高維度資料的技術。其基本假設是資料來源於高維度的空間低維度子流形上或是在其附近。流形學習的目的在於探索該種低維度結構並使數據用相對簡單、更具解釋性的形式來表達。常見的的流形學習包含主成分分析、多維尺度法、ISOMAP、t-隨機鄰近嵌入法等等。這些方法主要的差異為對於數據的假設、適用的數據種類與將數據表示為低維度的計算複雜度。

在本門課程中，我們將介紹許多流形學習技術中的數學概念、演算法以及這些方法的基本假設和局限性。在學期結束時，學生需要完成一個應用流形學習技術的期末專案。該專案應該包含一種或多種流形學習方法處理並應用於真實世界數據的分析上，並將結果清晰易懂的方式呈現出來。

Manifold learning is a class of techniques used to identify patterns in high-dimensional data. The underlying assumption is that the data lies on or near a lower-dimensional manifold embedded in the high-dimensional space. The goal of manifold learning is to discover this lower-dimensional structure and represent the data in a simpler, more interpretable form. Common manifold learning techniques include principal component analysis (PCA), multidimensional scaling (MDS), ISOMAP, t-SNE, and many others. These methods differ in their assumptions, the types of data they are suitable for, and the complexity of the resulting representation.

In this course, we will introduce the mathematical concepts and algorithms used in many manifold learning techniques, along with their underlying assumptions and limitations. At the end of this semester, students are required to complete a final project that utilizes manifold learning techniques. The project should involve preparing and analyzing real-world data using one or more manifold learning methods, and, presenting the results in a clear and informative manner.