  Clustering, a fundamental activity in unsupervised learning, is notoriously
difficult when the feature space is high-dimensional. Fortunately, in many
realistic scenarios, only a handful of features are relevant in distinguishing
clusters. This has motivated the development of sparse clustering techniques
that typically rely on k-means within outer algorithms of high computational
complexity. Current techniques also require careful tuning of shrinkage
parameters, further limiting their scalability. In this paper, we propose a
novel framework for sparse k-means clustering that is intuitive, simple to
implement, and competitive with state-of-the-art algorithms. We show that our
algorithm enjoys consistency and convergence guarantees. Our core method
readily generalizes to several task-specific algorithms such as clustering on
subsets of attributes and in partially observed data settings. We showcase
these contributions thoroughly via simulated experiments and real data
benchmarks, including a case study on protein expression in trisomic mice.
