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.

翻译：众所周知, Johnson- Lindenstraus 维度的减少方法是用于最坏情况扭曲的最佳方法 。 虽然在实践中使用了许多其他的方法和超偏度, 但通常在性能的界限上并不为人所知。 最近在\ cite{Bfur19} (NeurIPS'19) 中提出了JL 方法是否对扭曲的实用度度量最理想的问题。 它们为一系列广泛的实际措施提供了质量的上限, 并表明, 在许多情形下, 这些都是最好的应用。 然而, 一些最重要的案例, 包括平均偏差的基本案例, 还没有被打开。 特别是, 它们显示 JL 变换有 $epsilon 的平均扭曲值, 嵌入 $xl- Euclideidean 空间, 其中美元= O( 1/\ eps% 2), 对于更普通的扭曲值, 美元= $qqqrum 的反差量值, 可能由美元 = 美元 美元 =xxxx 。 当我们用这些最坏的方法, 我们用这些最差的计算, 。