Understanding the stochastic behavior of random projections of geometric sets constitutes a fundamental problem in high dimension probability that finds wide applications in diverse fields. This paper provides a kinematic description for the behavior of Gaussian random projections of closed convex cones, in analogy to that of randomly rotated cones studied in [ALMT14]. Formally, let $K$ be a closed convex cone in $\mathbb{R}^n$, and $G\in \mathbb{R}^{m\times n}$ be a Gaussian matrix with i.i.d. $\mathcal{N}(0,1)$ entries. We show that $GK\equiv \{Gμ: μ\in K\}$ behaves like a randomly rotated cone in $\mathbb{R}^m$ with statistical dimension $\min\{δ(K),m\}$, in the following kinematic sense: for any fixed closed convex cone $L$ in $\mathbb{R}^m$, \begin{align*} &δ(L)+δ(K)\ll m\, \Rightarrow\, L\cap GK = \{0\} \hbox{ with high probability},\\ &δ(L)+δ(K)\gg m\, \Rightarrow\, L\cap GK \neq \{0\} \hbox{ with high probability}. \end{align*} A similar kinematic description is obtained for $G^{-1}L\equiv \{μ\in \mathbb{R}^n: Gμ\in L\}$. The practical utility and broad applicability of the prescribed approximate kinematic formulae are demonstrated in a number of distinct problems arising from statistical learning, mathematical programming and asymptotic geometric analysis. In particular, we prove (i) new phase transitions of the existence of cone constrained maximum likelihood estimators in logistic regression, (ii) new phase transitions of the cost optimum of deterministic conic programs with random constraints, and (iii) a local version of the Gaussian Dvoretzky-Milman theorem that describes almost deterministic, low-dimensional behaviors of subspace sections of randomly projected convex sets.

翻译：理解几何集合随机投影的随机行为是高维概率论中的一个基本问题，在众多领域具有广泛应用。本文对闭凸锥的高斯随机投影行为给出了运动学描述，类比于[ALMT14]中研究的随机旋转锥。形式化地，设$K$为$\mathbb{R}^n$中的闭凸锥，$G\in \mathbb{R}^{m\times n}$为具有独立同分布$\mathcal{N}(0,1)$条目的高斯矩阵。我们证明$GK\equiv \{Gμ: μ\in K\}$在以下运动学意义上类似于$\mathbb{R}^m$中统计维度为$\min\{δ(K),m\}$的随机旋转锥：对于$\mathbb{R}^m$中任意固定闭凸锥$L$，\begin{align*} &δ(L)+δ(K)\ll m\, \Rightarrow\, L\cap GK = \{0\} \hbox{ 以高概率成立},\\ &δ(L)+δ(K)\gg m\, \Rightarrow\, L\cap GK \neq \{0\} \hbox{ 以高概率成立}. \end{align*} 对于$G^{-1}L\equiv \{μ\in \mathbb{R}^n: Gμ\in L\}$也可获得类似的运动学描述。所建立的近似运动公式的实用性与广泛适用性通过统计学习、数学规划及渐近几何分析中产生的多个不同问题得到验证。特别地，我们证明了：（i）逻辑回归中锥约束极大似然估计存在性的新相变现象；（ii）具有随机约束的确定性锥规划成本最优值的新相变现象；（iii）高斯Dvoretzky-Milman定理的局部形式，该定理描述了随机投影凸集子空间截面的近乎确定性低维行为。