# Machine Learning Friendly Set Version of Johnson-Lindenstrauss Lemma

## TL;DR

This paper introduces a probabilistic approach to the Johnson-Lindenstrauss Lemma, enabling pre-determined target dimensions for data projection with high confidence, and demonstrates its effectiveness for clustering and k-means algorithms.

## Contribution

It formulates a theorem for selecting target dimensions in JL transformations with probabilistic guarantees, improving applicability in data clustering and k-means analysis.

## Key findings

- Probabilistic bounds for target dimension selection in JL transformations
- Preservation of clustering quality in projected space
- Conditions for clusterability transfer to the projected space

## Abstract

In this paper we make a novel use of the Johnson-Lindenstrauss Lemma. The Lemma has an existential form saying that there exists a JL transformation $f$ of the data points into lower dimensional space such that all of them fall into predefined error range $\delta$. We formulate in this paper a theorem stating that we can choose the target dimensionality in a random projection type JL linear transformation in such a way that with probability $1-\epsilon$ all of them fall into predefined error range $\delta$ for any user-predefined failure probability $\epsilon$. This result is important for applications such a data clustering where we want to have a priori dimensionality reducing transformation instead of trying out a (large) number of them, as with traditional Johnson-Lindenstrauss Lemma. In particular, we take a closer look at the $k$-means algorithm and prove that a good solution in the projected space is also a good solution in the original space. Furthermore, under proper assumptions local optima in the original space are also ones in the projected space. We define also conditions for which clusterability property of the original space is transmitted to the projected space, so that special case algorithms for the original space are also applicable in the projected space.