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A Dynamic Low-Rank Fast Gaussian Transform

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 5.6Download Report PDF
Fast Gaussian TransformTheorykernal-density estimation
Abstract:

The Fast Gaussian Transform (FGT) enables subquadratic-time multiplication of an n×nn\times n Gaussian kernel matrix Ki,j=exp⁡(−∥xi−xj∥22)\mathsf{K}_{i,j}= \exp ( - \lVert x_i - x_j \rVert_2^2 ) with an arbitrary vector h∈Rnh \in \mathbb{R}^n, where x1,…,xn∈Rdx_1,\dots, x_n \in \mathbb{R}^d are a set of fixed source points. This kernel plays a central role in machine learning and random feature maps. Nevertheless, in most modern data analysis applications, datasets are dynamically changing (yet often have low rank), and recomputing the FGT from scratch in (kernel-based) algorithms incurs a major computational overhead (≳n\gtrsim n time for a single source update ∈Rd\in \mathbb{R}^d). These applications motivate a dynamic FGT algorithm, which maintains a dynamic set of sources under kernel-density estimation (KDE) queries in sublinear time while retaining Mat-Vec multiplication accuracy and speed.

Assuming the dynamic data-points xix_i lie in a (possibly changing) kk-dimensional subspace (k≤dk\leq d), our main result is an efficient dynamic FGT algorithm, supporting the following operations in log⁡O(k)(n/ε)\log^{O(k)}(n/\varepsilon) time: (1) Adding or deleting a source point, and (2) Estimating the "kernel-density" of a query point with respect to sources with ε\varepsilon additive accuracy. The core of the algorithm is a dynamic data structure for maintaining the projected "interaction rank" between source and target boxes, decoupled into finite truncation of Taylor and Hermite expansions.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
NOTE that some papers exist in multiple, slightly different versions (e.g., with different titles or URLs). The system may retrieve several versions of the same underlying work. The current automated pipeline does not reliably align or distinguish these cases, so human reviewers will need to disambiguate them manually.
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Overview

Overall Novelty Assessment

The paper proposes a dynamic Fast Gaussian Transform algorithm that maintains kernel density estimates under point insertions and deletions in logarithmic time, assuming data lie in a low-dimensional subspace. It resides in the 'Dynamic Data Structures for Kernel Density Estimation' leaf, which contains only one sibling paper among the ten total papers in the taxonomy. This sparse leaf suggests the specific combination of dynamic updates and low-rank structure for kernel density estimation is relatively underexplored compared to broader algorithmic frameworks or domain applications.

The taxonomy reveals neighboring leaves focused on tensor-based kernel regression and reduced-order modeling for dynamical systems, both exploiting low-rank structure but not emphasizing dynamic point-set updates. The broader 'Algorithmic Frameworks' branch includes methods for static kernel approximation and tensor decompositions, while the sibling 'Domain-Specific Applications' branch addresses signal processing and geometric data denoising. The original paper's focus on sublinear-time updates for changing point sets distinguishes it from these related directions, which typically assume batch or static settings.

Among fifteen candidates examined across three contributions, none were flagged as clearly refuting the proposed work. The 'Dynamic Low-Rank Fast Gaussian Transform Algorithm' examined seven candidates with zero refutable matches, while the 'Dynamic Data Structure for Projected Interaction Rank' examined eight candidates, also with zero refutations. The 'Low-Dimensional Subspace Extension' contribution had no candidates examined. This limited search scope suggests that within the top-fifteen semantic matches, no prior work directly overlaps with the combination of dynamic updates and low-rank Gaussian kernel computations.

Based on the sparse taxonomy leaf and absence of refuting candidates among fifteen examined papers, the work appears to occupy a relatively novel niche. However, the small search scope and single sibling paper limit confidence in assessing broader field coverage. The analysis captures top semantic matches but does not guarantee exhaustive review of all kernel density or dynamic data structure literature.

Taxonomy

Core-task Taxonomy Papers
10
3
Claimed Contributions
15
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: dynamic kernel density estimation with low-rank data. The field divides into two main branches: algorithmic frameworks that develop efficient computational methods for kernel-based and low-rank techniques, and domain-specific applications that deploy these methods in areas such as point cloud processing, wireless channel estimation, and large-scale machine learning. The algorithmic branch encompasses dynamic data structures for kernel density estimation, low-rank tensor decompositions (e.g., Bayesian Tucker Decomposition[4], Tensor Train Regression[6]), and hybrid approaches that exploit both kernel approximations and low-rank structure (e.g., Kernel Low-Rank Modeling[5]). Meanwhile, the applications branch demonstrates how these techniques address real-world challenges, from Point Cloud Denoising[2] to Millimeter Wave Estimation[7], often leveraging geometric or probabilistic insights to handle streaming or high-dimensional data. Within the algorithmic frameworks, a particularly active line of work focuses on dynamic data structures that maintain kernel density estimates as data evolves over time, balancing computational speed with approximation accuracy. Dynamic Fast Gaussian Transform[0] sits squarely in this cluster, emphasizing efficient updates for kernel sums under changing point sets. Nearby methods such as Fast Metric Learning[10] and Kernel Clustering Approximation[8] share the goal of accelerating kernel computations, yet they differ in whether they prioritize metric adaptation or clustering-based approximations. In contrast, works like Adaptive Density Ratio[9] explore probabilistic density estimation with adaptive bandwidth selection, highlighting a trade-off between structural assumptions (low-rank factorizations versus kernel bandwidth tuning) and the ability to track non-stationary distributions. The original paper's focus on dynamic updates and low-rank representations places it at the intersection of these themes, offering a computational strategy that complements both classical kernel methods and modern tensor-based approaches.

Claimed Contributions

Dynamic Low-Rank Fast Gaussian Transform Algorithm

The authors introduce a fully-dynamic data structure for the Fast Gaussian Transform that supports sublinear-time insertion and deletion of source points, as well as kernel-density estimation queries, while maintaining accuracy and speed for matrix-vector multiplication operations.

7 retrieved papers
Dynamic Data Structure for Projected Interaction Rank

The authors develop a novel dynamic data structure that maintains the interaction rank between source and target boxes by decoupling it into truncated Taylor and Hermite expansions, enabling efficient updates and queries in the dynamic setting.

8 retrieved papers
Low-Dimensional Subspace Extension

The authors extend their dynamic FGT algorithm to handle data points lying in low-dimensional subspaces, achieving runtime that depends on the intrinsic dimension k rather than the ambient dimension d, thereby mitigating the curse of dimensionality.

0 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Dynamic Low-Rank Fast Gaussian Transform Algorithm

The authors introduce a fully-dynamic data structure for the Fast Gaussian Transform that supports sublinear-time insertion and deletion of source points, as well as kernel-density estimation queries, while maintaining accuracy and speed for matrix-vector multiplication operations.

Contribution

Dynamic Data Structure for Projected Interaction Rank

The authors develop a novel dynamic data structure that maintains the interaction rank between source and target boxes by decoupling it into truncated Taylor and Hermite expansions, enabling efficient updates and queries in the dynamic setting.

Contribution

Low-Dimensional Subspace Extension

The authors extend their dynamic FGT algorithm to handle data points lying in low-dimensional subspaces, achieving runtime that depends on the intrinsic dimension k rather than the ambient dimension d, thereby mitigating the curse of dimensionality.

Table of Contents

A Dynamic Low-Rank Fast Gaussian Transform | Novelty Validation