The Separating Hyperplane theorem is a fundamental result in Convex Geometry with myriad applications. Our first result, Random Separating Hyperplane Theorem (RSH), is a strengthening of this for polytopes. $\rsh$ asserts that if the distance between $a$ and a polytope $K$ with $k$ vertices and unit diameter in $\Re^d$ is at least $\delta$, where $\delta$ is a fixed constant in $(0,1)$, then a randomly chosen hyperplane separates $a$ and $K$ with probability at least $1/poly(k)$ and margin at least $\Omega \left(\delta/\sqrt{d} \right)$. An immediate consequence of our result is the first near optimal bound on the error increase in the reduction from a Separation oracle to an Optimization oracle over a polytope. RSH has algorithmic applications in learning polytopes. We consider a fundamental problem, denoted the ``Hausdorff problem'', of learning a unit diameter polytope $K$ within Hausdorff distance $\delta$, given an optimization oracle for $K$. Using RSH, we show that with polynomially many random queries to the optimization oracle, $K$ can be approximated within error $O(\delta)$. To our knowledge this is the first provable algorithm for the Hausdorff Problem. Building on this result, we show that if the vertices of $K$ are well-separated, then an optimization oracle can be used to generate a list of points, each within Hausdorff distance $O(\delta)$ of $K$, with the property that the list contains a point close to each vertex of $K$. Further, we show how to prune this list to generate a (unique) approximation to each vertex of the polytope. We prove that in many latent variable settings, e.g., topic modeling, LDA, optimization oracles do exist provided we project to a suitable SVD subspace. Thus, our work yields the first efficient algorithm for finding approximations to the vertices of the latent polytope under the well-separatedness assumption.

翻译：分离超平面定理是凸几何中一个基本结果，具有广泛的应用。本文首先提出的随机分离超平面定理（RSH）是对多面体情形下该定理的强化。RSH断言：若点$a$与顶点数为$k$、直径为单位长度、位于$\Re^d$空间中的多面体$K$之间的距离至少为$\delta$（其中$\delta$是$(0,1)$内的固定常数），则随机选择的超平面能以至少$1/poly(k)$的概率将$a$与$K$分离，且间隔至少为$\Omega(\delta/\sqrt{d})$。该结果的直接推论是：在多面体上从分离预言机到优化预言的归约过程中，误差增加首次达到近最优界。RSH在多面体学习中具有算法应用价值。我们考虑一个基本问题（称为"豪斯多夫问题"）：给定多面体$K$的优化预言机，在豪斯多夫距离$\delta$内学习单位直径多面体$K$。利用RSH，我们证明只需对优化预言机进行多项式次随机查询，即可在误差$O(\delta)$内逼近$K$。据我们所知，这是豪斯多夫问题的首个可证明算法。在此基础上，我们进一步证明：若$K$的顶点良好分离，则可利用优化预言机生成一个点列表，其中每个点与$K$的豪斯多夫距离均为$O(\delta)$，且该列表包含与$K$每个顶点接近的点。此外，我们展示了如何修剪该列表以生成多面体每个顶点的（唯一）近似。我们证明在许多潜变量设置（如主题建模、LDA）中，只要投影到合适的SVD子空间，优化预言机确实存在。因此，我们的工作在良好分离性假设下，首次给出了寻找潜多面体顶点的近似值的有效算法。