使用高斯加權線性變換的可解釋非線性降維

Interpretable non-linear dimensionality reduction using gaussian weighted linear transformation

April 24, 2025

作者: Erik Bergh

cs.AI

摘要

降維技術是分析和可視化高維數據的基礎。現有方法如t-SNE和PCA在表徵能力與可解釋性之間存在權衡。本文提出了一種新穎的方法，通過結合線性方法的可解釋性和非線性變換的表達力來彌合這一差距。所提出的算法通過一系列由高斯函數加權的線性變換，構建了高維與低維空間之間的非線性映射。這種架構在實現複雜非線性變換的同時，保留了線性方法的可解釋性優勢，因為每個變換都可以獨立分析。最終模型既提供了強大的降維能力，又對變換後的空間提供了透明的洞察。本文還介紹了解釋學習到的變換的技術，包括識別被抑制的維度以及空間如何擴展和收縮的方法。這些工具使實踐者能夠理解算法在降維過程中如何保留和修改幾何關係。為了確保該算法的實用性，本文強調了開發用戶友好軟件包的重要性，以促進其在學術界和工業界的應用。

English

Dimensionality reduction techniques are fundamental for analyzing and visualizing high-dimensional data. With established methods like t-SNE and PCA presenting a trade-off between representational power and interpretability. This paper introduces a novel approach that bridges this gap by combining the interpretability of linear methods with the expressiveness of non-linear transformations. The proposed algorithm constructs a non-linear mapping between high-dimensional and low-dimensional spaces through a combination of linear transformations, each weighted by Gaussian functions. This architecture enables complex non-linear transformations while preserving the interpretability advantages of linear methods, as each transformation can be analyzed independently. The resulting model provides both powerful dimensionality reduction and transparent insights into the transformed space. Techniques for interpreting the learned transformations are presented, including methods for identifying suppressed dimensions and how space is expanded and contracted. These tools enable practitioners to understand how the algorithm preserves and modifies geometric relationships during dimensionality reduction. To ensure the practical utility of this algorithm, the creation of user-friendly software packages is emphasized, facilitating its adoption in both academia and industry.