  This paper considers the optimization problem in the form of $\min_{X \in
\mathcal{F}_v} f(x) + \lambda \|X\|_1,$ where $f$ is smooth, $\mathcal{F}_v =
\{X \in \mathbb{R}^{n \times q} : X^T X = I_q, v \in \mathrm{span}(X)\}$, and
$v$ is a given positive vector. The clustering models including but not limited
to the models used by $k$-means, community detection, and normalized cut can be
reformulated as such optimization problems. It is proven that the domain
$\mathcal{F}_v$ forms a compact embedded submanifold of $\mathbb{R}^{n \times
q}$ and optimization-related tools including a family of computationally
efficient retractions and an orthonormal basis of any normal space of
$\mathcal{F}_v$ are derived. An inexact accelerated Riemannian proximal
gradient method that allows adaptive step size is proposed and its global
convergence is established. Numerical experiments on community detection in
networks and normalized cut for image segmentation are used to demonstrate the
performance of the proposed method.
