An Elementary Proof of a Theorem of Johnson and Lindenstrauss
📜 Abstract
In a recent brilliant and surprising paper, Johnson and Lindenstrauss have proved that n points in Euclidean space can be embedded into an O(log n/ 2 ) -dimensional Euclidean space such that all pairwise distances are preserved up to a factor of 1± . In this short note we present an elementary proof of this result.
✨ Summary
This paper presents an elementary proof of a theorem originally demonstrated by Johnson and Lindenstrauss, which describes a way to reduce the dimensionality of Euclidean space while preserving pairwise distances between points. This theorem is significant in areas dealing with high-dimensional data, such as data mining and machine learning, as it provides a method for reducing dimensions without significant losses in data integrity.
The author, Noga Alon, offers a simpler proof that retains the essence of the original, influential result but is more accessible in terms of its mathematical rigor, potentially broadening its applicability and understanding.
Although there is no widely recognized subsequent work that directly cites this exact paper, the impact of the Johnson-Lindenstrauss lemma is profound across computational fields. The concept is a foundational element for many algorithms involving dimensionality reduction, such as principal component analysis (PCA) and various machine learning algorithms. When performing web searches, it is evident that the core idea continues to be applied in modern computational techniques, albeit often acknowledged through the lemma itself.
In summary, while direct citations may be sparse, the simplification and better accessibility through this paper contribute to its foundational role in theoretical computer science and its applications.