The sparse Johnson-Lindenstrauss transform is one of the central techniques in dimensionality reduction. It supports embedding a set of $n$ points in $\mathbb{R}^d$ into $m=O(\varepsilon^{-2} \lg n)$ dimensions while preserving all pairwise distances to within $1 \pm \varepsilon$. Each input point $x$ is embedded to $Ax$, where $A$ is an $m \times d$ matrix having $s$ non-zeros per column, allowing for an embedding time of $O(s \|x\|_0)$. Since the sparsity of $A$ governs the embedding time, much work has gone into improving the sparsity $s$. The current state-of-the-art by Kane and Nelson (JACM'14) shows that $s = O(\varepsilon ^{-1} \lg n)$ suffices. This is almost matched by a lower bound of $s = \Omega(\varepsilon ^{-1} \lg n/\lg(1/\varepsilon))$ by Nelson and Nguyen (STOC'13). Previous work thus suggests that we have near-optimal embeddings. In this work, we revisit sparse embeddings and identify a loophole in the lower bound. Concretely, it requires $d \geq n$, which in many applications is unrealistic. We exploit this loophole to give a sparser embedding when $d = o(n)$, achieving $s = O(\varepsilon^{-1}(\lg n/\lg(1/\varepsilon)+\lg^{2/3}n \lg^{1/3} d))$. We also complement our analysis by strengthening the lower bound of Nelson and Nguyen to hold also when $d \ll n$, thereby matching the first term in our new sparsity upper bound. Finally, we also improve the sparsity of the best oblivious subspace embeddings for optimal embedding dimensionality.

翻译：稀疏Johnson-Lindenstrauss变换是降维领域的核心技术之一。该变换支持将$\mathbb{R}^d$中的$n$个点嵌入到$m=O(\varepsilon^{-2} \lg n)$维空间中，同时保持所有成对距离在$1 \pm \varepsilon$范围内。每个输入点$x$被嵌入为$Ax$，其中$A$是一个$m \times d$矩阵，每列有$s$个非零元素，嵌入时间为$O(s \|x\|_0)$。由于$A$的稀疏性决定嵌入时间，大量研究致力于提升稀疏度$s$。目前Kane和Nelson（JACM'14）提出的最优结果表明$s = O(\varepsilon ^{-1} \lg n)$即可满足要求。这一结果几乎与Nelson和Nguyen（STOC'13）的下界$s = \Omega(\varepsilon ^{-1} \lg n/\lg(1/\varepsilon))$相匹配。现有工作因此暗示我们已接近最优嵌入方案。本文重新审视稀疏嵌入问题，并发现该下界存在一个漏洞：它要求$d \geq n$，这在许多实际应用中并不成立。我们利用这一漏洞，在$d = o(n)$条件下实现了更稀疏的嵌入，达到$s = O(\varepsilon^{-1}(\lg n/\lg(1/\varepsilon)+\lg^{2/3}n \lg^{1/3} d))$。同时，我们通过加强Nelson-Nguyen下界使其在$d \ll n$时仍然成立，从而与新型稀疏上界中的第一项相匹配。最后，我们还改进了最优嵌入维度下最佳无偏子空间嵌入的稀疏性。