算子融合

Symmetry in the parameter space of deep neural networks (DNNs) has proven beneficial for various deep learning applications. A well-known example is the permutation symmetry in Multi-Layer Perceptrons (MLPs), where permuting the rows of weight matrices in one layer and applying the inverse permutation to adjacent layers yields a functionally equivalent model. While permutation symmetry fully characterizes the equivalence set for MLPs, its discrete nature limits its utility for transformers. In this paper, we introduce rotation symmetry, a novel form of parameter space symmetry for transformers that generalizes permutation symmetry by rotating parameter matrices in self-attention layers. Unlike permutation symmetry, rotation symmetry operates in a continuous domain, thereby significantly expanding the equivalence set for transformers. Based on this property, we propose a theoretically optimal parameter matching algorithm as a plug-and-play module to enhance model fusion. We evaluate our approach using pre-trained transformers across diverse natural language and vision tasks. Experimental results demonstrate that our rotation symmetry-based matching algorithm substantially improves model fusion, highlighting the potential of parameter space symmetry to facilitate model fusion. Our code is available on https://github.com/zhengzaiyi/RotationSymmetry

Little is known about the training dynamics of equivariant neural networks, in particular how it compares to data augmented training of their non-equivariant counterparts. Recently, neural tangent kernels (NTKs) have emerged as a powerful tool to analytically study the training dynamics of wide neural networks. In this work, we take an important step towards a theoretical understanding of training dynamics of equivariant models by deriving neural tangent kernels for a broad class of equivariant architectures based on group convolutions. As a demonstration of the capabilities of our framework, we show an interesting relationship between data augmentation and group convolutional networks. Specifically, we prove that they share the same expected prediction over initializations at all training times and even off the data manifold. In this sense, they have the same training dynamics. We demonstrate in numerical experiments that this still holds approximately for finite-width ensembles. By implementing equivariant NTKs for roto-translations in the plane ($G=C_{n}\ltimes\mathbb{R}^{2}$) and 3d rotations ($G=\mathrm{SO}(3)$), we show that equivariant NTKs outperform their non-equivariant counterparts as kernel predictors for histological image classification and quantum mechanical property prediction.

Decentralized federated learning (DFL) is a collaborative machine learning framework for training a model across participants without a central server or raw data exchange. DFL faces challenges due to statistical heterogeneity, as participants often possess data of different distributions reflecting local environments and user behaviors. Recent work has shown that the neural tangent kernel (NTK) approach, when applied to federated learning in a centralized framework, can lead to improved performance. We propose an approach leveraging the NTK to train client models in the decentralized setting, while introducing a synergy between NTK-based evolution and model averaging. This synergy exploits inter-client model deviation and improves both accuracy and convergence in heterogeneous settings. Empirical results demonstrate that our approach consistently achieves higher accuracy than baselines in highly heterogeneous settings, where other approaches often underperform. Additionally, it reaches target performance in 4.6 times fewer communication rounds. We validate our approach across multiple datasets, network topologies, and heterogeneity settings to ensure robustness and generalization. Source code for NTK-DFL is available at https://github.com/Gabe-Thomp/ntk-dfl}{https://github.com/Gabe-Thomp/ntk-dfl

Kernel-based hypothesis tests offer a flexible, non-parametric tool to detect high-order interactions in multivariate data, beyond pairwise relationships. Yet the scalability of such tests is limited by the computationally demanding permutation schemes used to generate null approximations. Here we introduce a family of permutation-free high-order tests for joint independence and partial factorisations of $d$ variables. Our tests eliminate the need for permutation-based approximations by leveraging V-statistics and a novel cross-centring technique to yield test statistics with a standard normal limiting distribution under the null. We present implementations of the tests and showcase their efficacy and scalability through synthetic datasets. We also show applications inspired by causal discovery and feature selection, which highlight both the importance of high-order interactions in data and the need for efficient computational methods.

Structured Kernel Interpolation (SKI) scales Gaussian Processes (GPs) by approximating the kernel matrix via inducing point interpolation, achieving linear computational complexity. However, it lacks rigorous theoretical error analysis. This paper bridges this gap by proving error bounds for the SKI Gram matrix and examining their effect on hyperparameter estimation and posterior inference. We further provide a practical guide to selecting the number of inducing points under convolutional cubic interpolation: they should grow as \(n^{d/3}\) for error control. Crucially, we identify two dimensionality regimes for the SKI Gram matrix spectral norm error vs. complexity trade-off. For \(d<3\), \textit{any} error tolerance can achieve linear time for sufficiently large sample size. For \(d\geq 3\), the error must \textit{increase} with sample size for our guarantees to hold. Our analysis provides key insights into SKI's scalability-accuracy trade-offs, establishing precise conditions for achieving linear-time GP inference with controlled error.

Kernel method-based intensity estimators, formulated within reproducing kernel Hilbert spaces (RKHSs), and classical kernel intensity estimators (KIEs) have been among the most easy-to-implement and feasible methods for estimating the intensity functions of inhomogeneous Poisson processes. While both approaches share the term "kernel", they are founded on distinct theoretical principles, each with its own strengths and limitations. In this paper, we propose a novel regularized kernel method for Poisson processes based on the least squares loss and show that the resulting intensity estimator involves a specialized variant of the representer theorem: it has the dual coefficient of unity and coincides with classical KIEs. This result provides new theoretical insights into the connection between classical KIEs and kernel method-based intensity estimators, while enabling us to develop an efficient KIE by leveraging advanced techniques from RKHS theory. We refer to the proposed model as the *kernel method-based kernel intensity estimator* (K$^2$IE). Through experiments on synthetic datasets, we show that K$^2$IE achieves comparable predictive performance while significantly surpassing the state-of-the-art kernel method-based estimator in computational efficiency.

This paper considers a canonical problem in kernel regression: how good are the model performances when it is trained by the popular online first-order algorithms, compared to the offline ones, such as ridge and ridgeless regression? In this paper, we analyze the foundational single-pass Stochastic Gradient Descent (SGD) in kernel regression under source condition where the optimal predictor can even not belong to the RKHS, i.e. the model is misspecified. Specifically, we focus on the inner product kernel over the sphere and characterize the exact orders of the excess risk curves under different scales of sample sizes $n$ concerning the input dimension $d$. Surprisingly, we show that SGD achieves min-max optimal rates up to constants among all the scales, $without$ suffering the saturation, a prevalent phenomenon observed in (ridge) regression, except when the model is highly misspecified and the learning is in a final stage where $n\gg d^\gamma$ with any constant $\gamma >0$. The main reason for SGD to overcome the curse of saturation is the exponentially decaying step size schedule, a common practice in deep neural network training. As a byproduct, we provide the $first$ provable advantage of the scheme over the iterative averaging method in the common setting.

Multimodal learning faces challenges in effectively fusing information from diverse modalities, especially when modality quality varies across samples. Dynamic fusion strategies, such as attention mechanism in Transformers, aim to address such challenge by adaptively emphasizing modalities based on the characteristics of input data. However, through amounts of carefully designed experiments, we surprisingly observed that the dynamic adaptability of widely-used self-attention models diminishes. Model tends to prefer one modality regardless of data characteristics. This bias triggers a self-reinforcing cycle that progressively overemphasizes the favored modality, widening the distribution gap in attention keys across modalities and deactivating attention mechanism's dynamic properties. To revive adaptability, we propose a simple yet effective method Rolling Query (RollingQ), which balances attention allocation by rotating the query to break the self-reinforcing cycle and mitigate the key distribution gap. Extensive experiments on various multimodal scenarios validate the effectiveness of RollingQ and the restoration of cooperation dynamics is pivotal for enhancing the broader capabilities of widely deployed multimodal Transformers. The source code is available at https://github.com/GeWu-Lab/RollingQ_ICML2025.

This paper considers the challenging computational task of estimating nested expectations. Existing algorithms, such as nested Monte Carlo or multilevel Monte Carlo, are known to be consistent but require a large number of samples at both inner and outer levels to converge. Instead, we propose a novel estimator consisting of nested kernel quadrature estimators and we prove that it has a faster convergence rate than all baseline methods when the integrands have sufficient smoothness. We then demonstrate empirically that our proposed method does indeed require the fewest number of samples to estimate nested expectations over a range of real-world application areas from Bayesian optimisation to option pricing and health economics.

Applying Gaussian processes (GPs) to very large datasets remains a challenge due to limited computational scalability. Matrix structures, such as the Kronecker product, can accelerate operations significantly, but their application commonly entails approximations or unrealistic assumptions. In particular, the most common path to creating a Kronecker-structured kernel matrix is by evaluating a product kernel on gridded inputs that can be expressed as a Cartesian product. However, this structure is lost if any observation is missing, breaking the Cartesian product structure, which frequently occurs in real-world data such as time series. To address this limitation, we propose leveraging latent Kronecker structure, by expressing the kernel matrix of observed values as the projection of a latent Kronecker product. In combination with iterative linear system solvers and pathwise conditioning, our method facilitates inference of exact GPs while requiring substantially fewer computational resources than standard iterative methods. We demonstrate that our method outperforms state-of-the-art sparse and variational GPs on real-world datasets with up to five million examples, including robotics, automated machine learning, and climate applications.

Previous influential work showed that infinite width limits of neural networks in the lazy training regime are described by kernel machines. Here, we show that neural networks trained in the rich infinite-width regime in two different settings are also described by kernel machines, but with data-dependent kernels. For both cases, we provide explicit expressions for the kernel predictors and prescriptions to numerically calculate them. To derive the first predictor, we study the large-width limit of feature-learning Bayesian networks, showing how feature learning leads to task-relevant adaptation of layer kernels and preactivation densities. The saddle point equations governing this limit result in a min-max optimization problem that defines the kernel predictor. To derive the second predictor, we study gradient flow training of randomly initialized networks trained with weight decay in the infinite-width limit using dynamical mean field theory (DMFT). The fixed point equations of the arising DMFT defines the task-adapted internal representations and the kernel predictor. We compare our kernel predictors to kernels derived from lazy regime and demonstrate that our adaptive kernels achieve lower test loss on benchmark datasets.

As deep learning continues to be driven by ever-larger datasets, understanding which examples are most important for generalization has become a critical question. While progress in data selection continues, emerging applications require studying this problem in dynamic contexts. To bridge this gap, we pose the Incremental Data Selection (IDS) problem, where examples arrive as a continuous stream, and need to be selected without access to the full data source. In this setting, the learner must incrementally build a training dataset of predefined size while simultaneously learning the underlying task. We find that in IDS, the impact of a new sample on the model state depends fundamentally on both its geometric relationship in the feature space and its prediction error. Leveraging this insight, we propose PEAKS (Prediction Error Anchored by Kernel Similarity), an efficient data selection method tailored for IDS. Our comprehensive evaluations demonstrate that PEAKS consistently outperforms existing selection strategies. Furthermore, PEAKS yields increasingly better performance returns than random selection as training data size grows on real-world datasets. The code is available at https://github.com/BurakGurbuz97/PEAKS.

Implicit Neural Representations (INRs) rely heavily on architectural choices for good generalization. Developing theoretically grounded approaches for architecture design remains an active area of research. Via theoretical analysis of the infinite-width limit, we establish a methodology that characterizes INR's generalization by means of kernel alignment. We first formulate the optimal kernel that minimizes pointwise expected squared error, then demonstrate that the Neural Tangent Kernel of the composed function (INR with input encoding) can approximate any positive semidefinite dot-product kernels through input feature mapping adjustments. Building upon these insights, we propose a Kernel Alignment Regularizer (KAR) that naturally integrates with existing INR systems to enhance kernel alignment. We further develop Plug-in Encoding for Aligned Kernels (PEAK) to refine INR models with KAR using learnable input encoding. This work contributes to the ongoing research efforts in bridging theory and practice for principled INR architecture design. Code is available at https://github.com/lizhemin15/KAR.

Inspired by the well-known coreset in clustering algorithms, we introduce the definition of the core kernel for multiple kernel clustering (MKC) algorithms. The core kernel refers to running MKC algorithms on smaller-scale base kernel matrices to obtain kernel weights similar to those obtained from the original full-scale kernel matrices. Specifically, the core kernel refers to a set of kernel matrices of size $\widetilde{\mathcal{O}}(1/\varepsilon^2)$ that perform MKC algorithms on them can achieve a $(1+\varepsilon)$-approximation for the kernel weights. Subsequently, we can leverage approximated kernel weights to obtain a theoretically guaranteed large-scale extension of MKC algorithms. In this paper, we propose a core kernel construction method based on singular value decomposition and prove that it satisfies the definition of the core kernel for three mainstream MKC algorithms. Finally, we conduct experiments on several benchmark datasets to verify the correctness of theoretical results and the efficiency of the proposed method.