We establish new exponential in dimension lower bounds for the Maximum Halfspace Discrepancy problem, which models linear classification. Both are fundamental problems in computational geometry and machine learning in their exact and approximate forms. However, only $O(n^d)$ and respectively $\tilde O(1/\varepsilon^d)$ upper bounds are known and complemented by polynomial lower bounds that do not support the exponential in dimension dependence. We close this gap up to polylogarithmic terms by reduction from widely-believed hardness conjectures for Affine Degeneracy testing and $k$-Sum problems. Our reductions yield matching lower bounds of $\tildeΩ(n^d)$ and respectively $\tildeΩ(1/\varepsilon^d)$ based on Affine Degeneracy testing, and $\tildeΩ(n^{d/2})$ and respectively $\tildeΩ(1/\varepsilon^{d/2})$ conditioned on $k$-Sum. The first bound also holds unconditionally if the computational model is restricted to make sidedness queries, which corresponds to a widely spread setting implemented and optimized in many contemporary algorithms and computing paradigms.

翻译：我们针对最大半空间偏差问题（该问题刻画了线性分类）建立了新的指数级（以维度为底）下界。这两个问题在精确与近似形式下均是计算几何与机器学习中的基础问题。然而，目前仅已知$O(n^d)$和$\tilde O(1/\varepsilon^d)$的上界，并辅以多项式下界（该下界不支持指数级维度依赖）。通过从广泛接受的仿射退化检测问题和$k$-Sum难题假设进行归约，我们在多对数因子范围内填补了这一间隙。基于仿射退化检测假设，我们的归约给出了$\tildeΩ(n^d)$和$\tildeΩ(1/\varepsilon^d)$的匹配下界；而基于$k$-Sum假设，则得到$\tildeΩ(n^{d/2})$和$\tildeΩ(1/\varepsilon^{d/2})$的下界。此外，若计算模型限制为仅能进行有向性查询（该设置对应于众多当代算法与计算范式中实现与优化的广泛场景），第一个下界亦无条件成立。