Optimality of the Johnson-Lindenstrauss Dimensionality Reduction for Practical Measures

It is well known that the Johnson-Lindenstrauss dimensionality reduction method is optimal for worst case distortion. While in practice many other methods and heuristics are used, not much is known in terms of bounds on their performance. The question of whether the JL method is optimal for practical measures of distortion was recently raised in \cite{BFN19} (NeurIPS'19). They provided upper bounds on its quality for a wide range of practical measures and showed that indeed these are best possible in many cases. Yet, some of the most important cases, including the fundamental case of average distortion were left open. In particular, they show that the JL transform has $1+\epsilon$ average distortion for embedding into $k$-dimensional Euclidean space, where $k=O(1/\eps^2)$, and for more general $q$-norms of distortion, $k = O(\max\{1/\eps^2,q/\eps\})$, whereas tight lower bounds were established only for large values of $q$ via reduction to the worst case. In this paper we prove that these bounds are best possible for any dimensionality reduction method, for any $1 \leq q \leq O(\frac{\log (2\eps^2 n)}{\eps})$ and $\epsilon \geq \frac{1}{\sqrt{n}}$, where $n$ is the size of the subset of Euclidean space. Our results imply that the JL method is optimal for various distortion measures commonly used in practice, such as {\it stress, energy} and {\it relative error}. We prove that if any of these measures is bounded by $\eps$ then $k=\Omega(1/\eps^2)$, for any $\epsilon \geq \frac{1}{\sqrt{n}}$, matching the upper bounds of \cite{BFN19} and extending their tightness results for the full range moment analysis. Our results may indicate that the JL dimensionality reduction method should be considered more often in practical applications, and the bounds we provide for its quality should be served as a measure for comparison when evaluating the performance of other methods and heuristics.