In batch Kernel Density Estimation (KDE) for a kernel function $f$, we are given as input $2n$ points $x^{(1)}, \cdots, x^{(n)}, y^{(1)}, \cdots, y^{(n)}$ in dimension $m$, as well as a vector $v \in \mathbb{R}^n$. These inputs implicitly define the $n \times n$ kernel matrix $K$ given by $K[i,j] = f(x^{(i)}, y^{(j)})$. The goal is to compute a vector $v$ which approximates $K w$ with $|| Kw - v||_\infty < \varepsilon ||w||_1$. A recent line of work has proved fine-grained lower bounds conditioned on SETH. Backurs et al. first showed the hardness of KDE for Gaussian-like kernels with high dimension $m = \Omega(\log n)$ and large scale $B = \Omega(\log n)$. Alman et al. later developed new reductions in roughly this same parameter regime, leading to lower bounds for more general kernels, but only for very small error $\varepsilon < 2^{- \log^{\Omega(1)} (n)}$. In this paper, we refine the approach of Alman et al. to show new lower bounds in all parameter regimes, closing gaps between the known algorithms and lower bounds. In the setting where $m = C\log n$ and $B = o(\log n)$, we prove Gaussian KDE requires $n^{2-o(1)}$ time to achieve additive error $\varepsilon < \Omega(m/B)^{-m}$, matching the performance of the polynomial method up to low-order terms. In the low dimensional setting $m = o(\log n)$, we show that Gaussian KDE requires $n^{2-o(1)}$ time to achieve $\varepsilon$ such that $\log \log (\varepsilon^{-1}) > \tilde \Omega ((\log n)/m)$, matching the error bound achievable by FMM up to low-order terms. To our knowledge, no nontrivial lower bound was previously known in this regime. Our new lower bounds make use of an intricate analysis of a special case of the kernel matrix -- the `counting matrix'. As a key technical lemma, we give a novel approach to bounding the entries of its inverse by using Schur polynomials from algebraic combinatorics.

翻译：在针对核函数$f$的批量核密度估计中，我们输入维度$m$下的$2n$个点$x^{(1)}, \cdots, x^{(n)}, y^{(1)}, \cdots, y^{(n)}$以及向量$v \in \mathbb{R}^n$。这些输入隐式定义了$n \times n$核矩阵$K$，其中$K[i,j] = f(x^{(i)}, y^{(j)})$。目标是计算向量$v$以近似$K w$，满足$|| Kw - v||_\infty < \varepsilon ||w||_1$。近期一系列研究基于SETH假设证明了细粒度的计算下界。Backurs等人首次证明了高维度$m = \Omega(\log n)$与大尺度$B = \Omega(\log n)$条件下类高斯核KDE的困难性。Alman等人随后在近似参数范围内提出了新的归约方法，得到了更一般核函数的下界，但仅适用于极小的误差$\varepsilon < 2^{- \log^{\Omega(1)} (n)}$。本文通过改进Alman等人的方法，在所有参数范围内建立了新的下界，从而填补了现有算法与下界之间的空白。在$m = C\log n$且$B = o(\log n)$的设置中，我们证明高斯核KDE需要$n^{2-o(1)}$时间才能达到加性误差$\varepsilon < \Omega(m/B)^{-m}$，该结果与多项式方法的性能仅相差低阶项。在低维设置$m = o(\log n)$中，我们证明高斯核KDE需要$n^{2-o(1)}$时间才能达到满足$\log \log (\varepsilon^{-1}) > \tilde \Omega ((\log n)/m)$的$\varepsilon$值，该误差界与快速多极方法所能达到的界限仅相差低阶项。据我们所知，此前在该参数范围内尚未发现非平凡的下界。我们的新下界通过对核矩阵的特殊情况——"计数矩阵"进行精细分析而实现。作为关键的技术引理，我们提出了一种利用代数组合学中的Schur多项式来界定其逆矩阵元素的新方法。