t-SNE能量的临界点丰度
On the Abundance of Critical Points of the t-SNE Energy
- North Carolina State University（北卡罗来纳州立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对含t-SNE算法的通用能量族，利用特征空间与目标嵌入空间的离散对称对，构造无穷多临界点，解释了t-SNE能量景观中局部极小值不尊重数据拓扑的现象。
AI中文摘要：
本文研究t-SNE算法的能量景观。尽管该算法已被广泛采用，但相关能量的非凸性使得难以严格理解该算法在多种场景下所捕捉的内容。特别是，许多著名的数值示例（本文复现了其中几个）表明，存在复杂的能量景观，包含大量局部极小值，这些极小值不尊重底层数据的拓扑结构或聚类结构。本研究旨在为这些现象提供严格解释的初步步骤。具体而言，对于包含原始t-SNE算法及近期确定的大数据极限的通用能量族，以及特征空间中服从连续对称性的密度，我们构造了无穷多组不同的临界点。这些临界点基于识别离散对称对，一个在原始特征空间，另一个在目标嵌入空间，二者在梯度动力学下保持不变。这些临界构型展现出许多经验中常观察到的特征，如拓扑破坏和虚假聚类。最后，全文给出了数值和分析示例以说明该方法。
英文摘要：
This paper considers the energy landscape of the t-SNE algorithm. While this algorithm has enjoyed broad adoption, the non-convexity of the associated energy has made it difficult to rigorously understand what the algorithm captures in many settings. In particular, a number of well-known numerical examples, several of which are reproduced in this article, suggest a complicated energy landscape with many local minimizers that do not respect the topology or clustering structure of the underlying data. This work seeks to provide first steps towards a rigorous explanation of these phenomena. Specifically, for a general family of energies, which include both the original t-SNE algorithm and recently identified large data limits, and for densities in feature space which obey a continuous symmetry, we construct infinite families of distinct critical points. These critical points are based upon identifying pairs of discrete symmetries, one in the original feature space and the other in the target embedding space, which are preserved under gradient dynamics. These critical configurations exhibit many characteristics, such as topology breaking and spurious clustering, which are often observed empirically. Finally, numerical and analytical examples are given throughout as a means of illustrating the approach.
