Kernel Fusion

In this paper, we study the angle testing problem in high-dimensional Euclidean spaces and propose two projection-based probabilistic kernel functions, one designed for angle comparison and the other for angle thresholding. Unlike existing approaches that rely on random projection vectors drawn from Gaussian distributions, our approach leverages reference angles and employs a deterministic structure for the projection vectors. Notably, our kernel functions do not require asymptotic assumptions, such as the number of projection vectors tending to infinity, and can be both theoretically and experimentally shown to outperform Gaussian-distribution-based kernel functions. We further apply the proposed kernel function to Approximate Nearest Neighbor Search (ANNS) and demonstrate that our approach achieves a 2.5X-3X higher query-per-second (QPS) throughput compared to the state-of-the-art graph-based search algorithm HNSW.

The attention mechanism in Transformers is an important primitive for accurate and scalable sequence modeling. Its quadratic-compute and linear-memory complexity however remain significant bottlenecks. Linear attention and state-space models enable linear-time, constant-memory sequence modeling and can moreover be trained efficiently through matmul-rich parallelization across sequence length. However, at their core these models are still RNNs, and thus their use of a fixed-size hidden state to model the context is a fundamental limitation. This paper develops log-linear attention, an attention mechanism that balances linear attention's efficiency and the expressiveness of softmax attention. Log-linear attention replaces the fixed-size hidden state with a logarithmically growing set of hidden states. We show that with a particular growth function, log-linear attention admits a similarly matmul-rich parallel form whose compute cost is log-linear in sequence length. Log-linear attention is a general framework and can be applied on top of existing linear attention variants. As case studies, we instantiate log-linear variants of two recent architectures---Mamba-2 and Gated DeltaNet---and find they perform well compared to their linear-time variants.

Learning from structured multi-way data, represented as higher-order tensors, requires capturing complex interactions across tensor modes while remaining computationally efficient. We introduce Uncertainty-driven Kernel Tensor Learning (UKTL), a novel kernel framework for $M$-mode tensors that compares mode-wise subspaces derived from tensor unfoldings, enabling expressive and robust similarity measures. To handle large-scale tensor data, we propose a scalable Nystr\"{o}m kernel linearization with dynamically learned pivot tensors obtained via soft $k$-means clustering. A key innovation of UKTL is its uncertainty-aware subspace weighting, which adaptively down-weights unreliable mode components based on estimated confidence, improving robustness and interpretability in comparisons between input and pivot tensors. Our framework is fully end-to-end trainable and naturally incorporates both multi-way and multi-mode interactions through structured kernel compositions. Extensive evaluations on action recognition benchmarks (NTU-60, NTU-120, Kinetics-Skeleton) show that UKTL achieves state-of-the-art performance, superior generalization, and meaningful mode-wise insights. This work establishes a principled, scalable, and interpretable kernel learning paradigm for structured multi-way and multi-modal tensor sequences.

Modeling non-stationary processes, where statistical properties vary across the input domain, is a critical challenge in machine learning; yet most scalable methods rely on a simplifying assumption of stationarity. This forces a difficult trade-off: use expressive but computationally demanding models like Deep Gaussian Processes, or scalable but limited methods like Random Fourier Features (RFF). We close this gap by introducing Random Wavelet Features (RWF), a framework that constructs scalable, non-stationary kernel approximations by sampling from wavelet families. By harnessing the inherent localization and multi-resolution structure of wavelets, RWF generates an explicit feature map that captures complex, input-dependent patterns. Our framework provides a principled way to generalize RFF to the non-stationary setting and comes with a comprehensive theoretical analysis, including positive definiteness, unbiasedness, and uniform convergence guarantees. We demonstrate empirically on a range of challenging synthetic and real-world datasets that RWF outperforms stationary random features and offers a compelling accuracy-efficiency trade-off against more complex models, unlocking scalable and expressive kernel methods for a broad class of real-world non-stationary problems.

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. Our anonymous repository is at \href{https://anonymous.4open.science/r/sdfasfsdgsfgdrf}{\textcolor{black}{https://anonymous.4open.science/r/sdfasfsdgsfgdrf}}.

Large language models (LLMs) do not preserve privacy at inference-time. The LLM's outputs can inadvertently reveal information about the model's context, which presents a privacy challenge when the LLM is augmented via tools or databases containing sensitive information. Existing privacy-preserving methods at inference-time have significant limitations since they (i) lack provable guarantees or (ii) have a poor utility/privacy trade-off. We propose DP-Fusion, a Differentially Private Inference (DPI) mechanism for LLMs that provably bounds the influence a set of tokens in the context can have on the LLM's output. DP-Fusion works as follows: (1) label a subset of sensitive tokens, (2) infer the LLM without any sensitive tokens to obtain a baseline, (3) infer the LLM with the sensitive tokens, and (4) blend distributions so that the final output remains within a bounded distance of the baseline distribution. While this per-token influence bound also mitigates jailbreak-style prompt injection, we focus on document privatization, where the goal is to paraphrase a document containing sensitive tokens, e.g., personally identifiable information, so that no attacker can reliably infer them from the paraphrased document while preserving high text quality. The privacy/utility trade-off is controlled by $\epsilon$, where $\epsilon=0$ hides sensitive tokens entirely, while higher values trade off privacy for improved text quality. We show that our method creates token-level provably privatized documents with substantially improved theoretical and empirical privacy, achieving $6\times$ lower perplexity than related DPI methods.

Integrating multimodal screening data is challenging because biological signals only partially overlap and cell-level pairing is frequently unavailable. Existing approaches either require pairing or fail to capture both shared and modality-specific information in an end-to-end manner. We present PETRI, an early-fusion transformer that learns a unified cell embedding from unpaired cellular images and gene expression profiles. PETRI groups cells by shared experimental context into multimodal “documents” and performs masked joint reconstruction with cross-modal attention, permitting information sharing while preserving modality-specific capacity. The resulting latent space supports construction of perturbation-level profiles by simple averaging across modalities. Applying sparse autoencoders to the embeddings reveals learned concepts that are biologically meaningful, multimodal, and retain perturbation-specific effects. To support further machine learning research, we release a blinded, matched optical pooled screen (OPS) and Perturb-seq dataset in HepG2 cells.

Mixture of Experts (MoE) models have emerged as the de-facto architecture for scaling up language models without significantly increasing the computational cost. Existing MoE methods optimize system efficiency or model architecture independently. We show that as MoE models get more granular and sparser, they become more memory-bound, and jointly optimizing the algorithms and the kernel design leads to a major improvement in MoE training throughput. We first propose a memory-efficient algorithm to compute the forward and backward of MoE with minimal activation saved. We then design GPU kernels that overlap memory IO latency with compute, benefiting all MoE architectures. Finally, we propose a novel "token rounding" method that minimizes the wasted compute brought by tile quantization. As a result, our method SNaX reduces activation memory by 45% and has 1.80x throughput improvement on NVidia H100 GPUs compared to ScatterMoE for a fine-grained 7B MoE. Moreover, SNaX on 64 H100s achieves a training throughput of 213 billion tokens a day comparable to ScatterMoE's 225 billion tokens a day on 96 H100s for a 7B MoE model training with token-choice routing while training with FSDP-2. Under high MoE sparsity settings, our tile-aware token rounding algorithm yields an additional 1.18x speedup on kernel execution time compared to vanilla top-$K$ routing while maintaining similar downstream performance.

Determining an appropriate size for an artificial neural network under computational constraints is a fundamental challenge. This paper introduces a practical metric, derived from Neural Tangent Kernel (NTK), for estimating the minimum necessary network width with respect to test loss -- prior to training. We provide both theoretical and empirical evidence that the smallest eigenvalue of the NTK strongly influences test loss in wide but finite-width neural networks. Based on this observation, we define an NTK-based metric computed at initialization to identify what we call cardinal width, i.e., the width of a network at which generalization performance saturates. Our experiments across multiple datasets and architectures demonstrate the effectiveness of this metric in estimating the cardinal width.