A set of high dimensional points $X=\{x_1, x_2,\ldots, x_n\} \subset R^d$ in isotropic position is said to be $\delta$-anti concentrated if for every direction $v$, the fraction of points in $X$ satisfying $|\langle x_i,v \rangle |\leq \delta$ is at most $O(\delta)$. Motivated by applications to list-decodable learning and clustering, recent works have considered the problem of constructing efficient certificates of anti-concentration in the average case, when the set of points $X$ corresponds to samples from a Gaussian distribution. Their certificates played a crucial role in several subsequent works in algorithmic robust statistics on list-decodable learning and settling the robust learnability of arbitrary Gaussian mixtures, yet remain limited to rotationally invariant distributions. This work presents a new (and arguably the most natural) formulation for anti-concentration. Using this formulation, we give quasi-polynomial time verifiable sum-of-squares certificates of anti-concentration that hold for a wide class of non-Gaussian distributions including anti-concentrated bounded product distributions and uniform distributions over $L_p$ balls (and their affine transformations). Consequently, our method upgrades and extends results in algorithmic robust statistics e.g., list-decodable learning and clustering, to such distributions. Our approach constructs a canonical integer program for anti-concentration and analysis a sum-of-squares relaxation of it, independent of the intended application. We rely on duality and analyze a pseudo-expectation on large subsets of the input points that take a small value in some direction. Our analysis uses the method of polynomial reweightings to reduce the problem to analyzing only analytically dense or sparse directions.

翻译：若一组处于各向同性位置的高维点集 $X=\{x_1, x_2,\ldots, x_n\} \subset R^d$ 满足：对于任意方向 $v$，满足 $|\langle x_i,v \rangle |\leq \delta$ 的点在 $X$ 中所占比例至多为 $O(\delta)$，则称该点集是 $\delta$-反集中的。受列表可解码学习与聚类应用的推动，近期研究关注在平均情况下（即点集 $X$ 对应于高斯分布样本时）构建高效的反集中性证明。这些证明在后续算法鲁棒统计的多个工作中发挥了关键作用，特别是在列表可解码学习领域以及确定任意高斯混合模型的鲁棒可学习性方面，但其适用范围仍局限于旋转不变分布。本研究提出了一种新的（且可论证为最自然的）反集中性表述形式。基于此表述，我们给出了准多项式时间可验证的和平方（sum-of-squares）反集中性证明，该证明适用于包括反集中有界乘积分布、$L_p$ 球面均匀分布（及其仿射变换）在内的广泛非高斯分布。因此，我们的方法将算法鲁棒统计中的相关结果（如列表可解码学习与聚类）升级并扩展至此类分布。我们的方法构建了一个规范的反集中性整数规划，并分析了其和平方松弛形式，该过程独立于具体应用场景。我们依托对偶理论，分析了输入点子集中在某个方向上取值较小的伪期望。通过采用多项式重加权方法，我们的分析将问题简化为仅需处理解析稠密或稀疏方向的情形。