FloatSOM: GPU-Accelerated, Distributed, Topology-Flexible Self-Organizing Maps

We present \textbf{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.

The top-$k$ operation is a fundamental building block of modern sparse computation, enabling token routing, expert activation, memory selection, and attention pruning. Yet standard hard top-$k$ blocks gradients, while existing continuous (soft) relaxations remain too costly for large-scale models. We introduce Fast LapSum, an exact-budget soft top-$k$ primitive whose GPU solver runs in linear time after sorting. Unlike prior linear-time methods such as DFTopK, which relax the normalization constraint, Fast LapSum is, to our knowledge, the first method to preserve an exact selection mass of $k$ while remaining fully differentiable end-to-end. Our solver combines a linear-time threshold computation with an analytical vector--Jacobian product, and for extreme scales employs probabilistic bracketing to sort only the uncertain middle band of kernel-noised scores. The resulting overhead is almost negligible: the solver processes $10^6$, $10^7$, and $10^8$ scores in $0.41$, $1.15$, and $5.23$,ms, respectively. This makes exact soft top-$k$ practical for sparse routing, retrieval, and large-scale optimization. We demonstrate Fast LapSum on two demanding applications operating over millions of coordinates inside the training loop: generating megapixel sparse adversarial examples with an exact soft budget of ${\sim}0.02\%$ of an image's pixels, achieving an order-of-magnitude speedup over state-of-the-art methods, and training a fully differentiable sparse image coder from scratch.

Large language model (LLM) training today runs on clusters spanning thousands of GPUs. While this scale enables rapid model advances, developing, debugging, and performance-tuning the training framework inevitably becomes complex and costly. This is because engineers often need to reproduce production behaviors to diagnose failures or evaluate optimizations, thereby demanding frequent and even exclusive access to production-scale clusters -- which becomes increasingly hard given that the majority of GPUs are already committed to production workloads. Simulation relies on complex performance models that are difficult to maintain, and downscaled experiments often fail to capture scale-dependent behaviors. We present PrismLLM to decouple large-scale execution from the need to access large clusters, enabling engineers to run and observe ranks of interest under faithful large-scale behavior using only a few GPUs. PrismLLM constructs a high-fidelity execution graph via a slicing-based approach that captures computation, communication, and dependencies of the target scale. Then, PrismLLM performs hybrid emulation where selected ranks execute the original program while the remaining ranks are replayed as virtual participants. Experiments on large-scale LLM training workloads show that PrismLLM accurately reproduces performance and memory behavior, achieving only 0.58% average error in iteration time and less than 0.01% error in peak GPU memory usage. PrismLLM can emulate clusters of up to 8192 GPUs using fewer than 1% of the physical GPUs required by the original deployment.