Learning the Differential Correlation Matrix of a Smooth Function From Point Samples

Consider an open set $\mathbb{D}\subseteq\mathbb{R}^n$, equipped with a probability measure $\mu$. An important characteristic of a smooth function $f:\mathbb{D}\rightarrow\mathbb{R}$ is its $differential$ $correlation$ $matrix$ $\Sigma_{\mu}:=\int \nabla f(x) (\nabla f(x))^* \mu(dx) \in\mathbb{R}^{n\times n}$, where $\nabla f(x)\in\mathbb{R}^n$ is the gradient of $f(\cdot)$ at $x\in\mathbb{D}$. For instance, the span of the leading $r$ eigenvectors of $\Sigma_{\mu}$ forms an $active$ $subspace$ of $f(\cdot)$, thereby extending the concept of $principal$ $component$ $analysis$ to the problem of $ridge$ $approximation$. In this work, we propose a simple algorithm for estimating $\Sigma_{\mu}$ from point values of $f(\cdot)$ $without$ imposing any structural assumptions on $f(\cdot)$. Theoretical guarantees for this algorithm are provided with the aid of the same technical tools that have proved valuable in the context of covariance matrix estimation from partial measurements.