Memory-efficient scaling on clustering problems without sacrificing statistical accuracy is of central interest for large-scale data analysis and machine learning problems. Nonnegative low-rank (NLR) matrix factorization for $K$-means is a scalable clustering method, which connects to semidefinite relaxations with optimal average-case exact recovery guarantees. However, a direct GPU implementation of NLR requires multiple large factor-sized buffers and substantial data movements that are essentially memory-bound. In this paper, we introduce GEM-KMeans, a spectrally normalized yet mathematically equivalent NLR formulation that fuses the gradient update, nonnegative projection, and sufficient statistics for normalization and iterate movement into a matrix-multiplication epilogue. Instead of retaining three massive factor-sized arrays, our IO-aware GPU implementation materializes only one single factor with small tile-reduction arrays as additional storage in the High Bandwidth Memory (HBM). We derive explicit memory costs and spectrally normalized smoothness bounds for optimizing the clustering objective function. Accurate clustering is demonstrated at massive scales on synthetic and real datasets, where performance gains of GEM-KMeans over existing GPU-accelerated Lloyd's algorithms involve data-dependent runtime tradeoffs.

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