LAPLEX: The FFT of Learnable Laplace Kernels

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.

The quadratic complexity of softmax attention presents a major obstacle for scaling Transformers to high-resolution vision tasks. Existing linear attention variants often replace the softmax with Gaussian kernels to reduce complexity, but such approximations lack theoretical grounding and tend to oversuppress mid-range token interactions. We propose LaplacianFormer, a Transformer variant that employs a Laplacian kernel as a principled alternative to softmax, motivated by empirical observations and theoretical analysis. To address expressiveness degradation under low-rank approximations, we introduce a provably injective feature map that retains fine-grained token information. For efficient computation, we adopt a Nyström approximation of the kernel matrix and solve the resulting system using Newton--Schulz iteration, avoiding costly matrix inversion and SVD. We further develop custom CUDA implementations for both the kernel and solver, enabling high-throughput forward and backward passes suitable for edge deployment. Experiments on ImageNet show that LaplacianFormer achieves strong performance-efficiency trade-offs while improving attention expressiveness.

Recent GPU generations deliver significantly higher FLOPs using lower-precision arithmetic, such as FP8. While successfully applied to large language models (LLMs), its adoption in large recommendation models (LRMs) has been limited. This is because LRMs are numerically sensitive, dominated by small matrix multiplications (GEMMs) followed by normalization, and trained in communication-intensive environments. Applying FP8 directly to LRMs often degrades model quality and prolongs training time. These challenges are inherent to LRM workloads and cannot be resolved merely by introducing better FP8 kernels. Instead, a system-model co-design approach is needed to successfully integrate FP8. We present LoKA (Low-precision Kernel Applications), a framework that makes FP8 practical for LRMs through three principles: profile under realistic distributions to know where low precision is safe, co-design model components with hardware to expand where it is safe, and orchestrate across kernel libraries to maximize the gains. Concretely, LoKA Probe is a statistically grounded, online benchmarking method that learns activation and weight statistics, and quantifies per-layer errors. This process pinpoints safe and unsafe, fast and slow sites for FP8 adoption. LoKA Mods is a set of reusable model adaptations that improve both numerical stability and execution efficiency with FP8. LoKA Dispatch is a runtime that leverages the statistical insights from LoKA Probe to select the fastest FP8 kernel that satisfies the accuracy requirements.