  Determining whether a subspace spanned by certain quantum states is entangled
and its entanglement dimensionality remains a fundamental challenge in quantum
information science. This paper introduces a geometric measure of $r$-bounded
rank, $E_r(\mathcal{S})$, for a given subspace $\mathcal{S}$. Derived from the
established geometric measure of entanglement, this measure is specifically
designed to assess the entanglement within $\mathcal{S}$. It not only serves as
a tool for determining the entanglement dimensionality but also illuminates the
subspace's capacity to preserve such entanglement. By employing developed
non-convex optimization techniques utilized in machine learning area, we can
accurately calculate $E_r(\mathcal{S})$ within the manifold optimization
framework. Our approach demonstrates notable advantages over existing
hierarchical methods, PPT relaxation techniques, and the seesaw strategy,
particularly by combining computational efficiency with broad applicability.
More importantly, it paves the way for high-dimensional entanglement
certification, which is crucial for numerous quantum information tasks. We
showcase its effectiveness in validating high-dimensional entangled subspaces
in bipartite systems, determining the border rank of multipartite pure states,
and identifying genuinely or completely entangled subspaces.
