Gaussian random projections of convex cones: approximate kinematic formulae and applications

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 $GK\equiv \{G\mu: \mu \in K\}$ behaves like a randomly rotated cone in $\mathbb{R}^m$ with statistical dimension $\min\{\delta(K),m\}$, in the following kinematic sense: for any fixed closed convex cone $L$ in $\mathbb{R}^m$, \begin{align*} &\delta(L)+\delta(K)\ll m\, \Rightarrow\, L\cap GK = \{0\} \hbox{ with high probability},\\ &\delta(L)+\delta(K)\gg m\, \Rightarrow\, L\cap GK \neq \{0\} \hbox{ with high probability}. \end{align*} A similar kinematic description is obtained for $G^{-1}L\equiv \{\mu \in \mathbb{R}^n: G\mu \in L\}$. 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.