GEM-KMeans: Memory-Efficient and Accurate Clustering on Massive Scale with GPU Optimization

We present Flash-GMM, a fused Triton kernel for efficient computation of Gaussian Mixture Models (GMMs) over large-scale data in a single GPU pass. By eliminating the need to materialize the full responsibility matrix in GPU memory, Flash-GMM achieves a \textbf{20$\times$} speedup over existing implementations and enables training on datasets more than \textbf{100$\times$} larger than previously feasible on one device. To demonstrate its impact, we integrate Flash-GMM into the IVF coarse quantizer for approximate nearest-neighbor (ANN) search. We show that soft GMM clustering is now a viable drop-in replacement for $k$-means, and that GMM responsibilities can be leveraged to assign border vectors to multiple clusters. Our approach reaches fixed recall targets with up to $1.7\times$ fewer distance computations, or equivalently, yields $+2$--$12$ recall@10 at matched computational cost. We release the kernel as an open-source project.

Symmetric non-negative matrix factorization (SymNMF) recovers latent group structure from a dependence matrix, but its dense, quadratic-memory objective has confined prior work to moderate sizes. We present a large-scale GPU study of seven algorithm families (over 30 configurations) on absolute Pearson correlation and tail pairwise dependence matrices from Extreme Value Theory, two proxies for empirical risk-factor estimation on large portfolios. A trace-identity reformulation eliminates all $n \times n$ intermediates, so a single GPU reaches $n \approx 10^5$ and multi-node distribution scales to $n = 10^6$ and beyond. Under a two-phase protocol, eleven methods converge at moderate scale; six remain efficient enough at $n = 10^5$ (five AdaGrad-family plus ADMM), and five AdaGrad-family methods still converge at $n = 10^6$: AdaGrad, RMSprop, and three we introduce (Piecewise AdaGrad, Row-Stochastic SVRG, Block-SVRG AdaptGrow). At $n = 10^6$ the fastest solver tracks the matrix spectrum: Block-SVRG AdaptGrow wins on the flat, ill-conditioned tail-dependence spectrum, where its lower per-iteration cost decides a long factorization, and full-batch AdaGrad wins on the dominant-low-rank correlation spectrum, where the run is short. We also benchmark spherical K-means as a hard-label baseline: cheaper when angular cluster structure is present, yet provably degenerate once the matrix collapses toward a single common factor, where the soft factorization remains necessary.

Symmetric nonnegative matrix factorization (Symmetric NMF) approximates a matrix as $WW^T$ with nonnegative rectangular factor $W$. It has broad applications in graph clustering and machine learning. In contrast to the NMF, projected gradient methods for the symmetric problem had been associated with slow convergence. To address this, we introduce SNMPBB, the first adaptation of nonmonotone projected Barzilai-Borwein methods to Symmetric NMF, demonstrating that gradient algorithms are significantly more effective than previously understood. We further extend SNMPBB to graph clustering using the graph Laplacian regularization (Graph-SNMPBB) and to large problems with low-rank approximations (LAI-SNMPBB). For all variants we prove global convergence to first-order stationary points and also that Barzilai-Borwein curvature information is preserved with randomized approximations. On synthetic data, SNMPBB achieves 6 times speedup over the alternative SymANLS for similar residuals, with advantages growing at higher ranks. Across six real-world clustering benchmarks, Graph-SNMPBB matches or exceeds SymANLS accuracy. Lastly, LAI-SNMPBB outperforms state-of-the-art LAI-SymPGNCG on 34 SuiteSparse matrices in both runtime and residual quality.