---
title: "任意维度学习：通过采样实现不同大小输入的统一泛化理论"
source: "arXiv AI/CL/LG（论文）"
source_tier: "t2"
url: "https://arxiv.org/abs/2607.07680v1"
permalink: "https://feed.astralor.com/items/itm_0d94e5621a5b09c9a9af8372"
published: "2026-07-08T17:38:29.000Z"
score: 74
channel: "tech"
tags: ["Transformer","任意维度学习","图神经网络","泛化理论","草图化","随机采样"]
---

# 任意维度学习：通过采样实现不同大小输入的统一泛化理论

## AI 点评

> 这篇论文给研究不同大小输入泛化的同行提供了一个很强的理论工具箱，尤其是那些做图神经网络、点云和序列建模的朋友。它把之前零散的技巧（比如随机下采样、分箱）统一到数学框架下，并给出了误差界，这对于理解模型行为、设计更高效的推理方案很有价值。

## 速览

这篇论文提出了一种基于随机采样映射的统一框架，用于比较不同大小的输入（如点云、序列、图），并回答两个核心问题：小尺寸输入的模型如何泛化到大尺寸输入？如何对大输入进行草图化以获得近似结果？作者系统分析了三种采样方式（放回采样、随机分箱、物种采样），并刻画了每种方式适用的函数类，基于对称性和问题实例间关系。框架显式给出了函数类的泛化率和草图化率，涵盖序列、图和张量上的大量函数，包括矩多项式、同态密度、置换不变Transformer和图神经网络。

## 关键点

- 提出使用随机采样映射比较不同大小输入的统一方法
- 分类三种采样方式并给出适用条件
- 导出泛化率和草图化率的显式边界
- 理论覆盖序列、图和张量上的多种机器学习模型

## 全文

Many machine learning models are defined for inputs of different sizes, such as point clouds containing different numbers of points, sequences of tokens of different lengths, and graphs on different numbers of nodes. Such models are trained on finitely-many examples of necessarily limited sizes. How well do these models generalize from inputs of small size to larger inputs of size not seen during training? Furthermore, evaluating such models on large inputs is often expensive. How can we sketch large inputs to obtain smaller ones on which the model takes similar values? At the heart of both questions is the need to compare inputs of different sizes and to approximate large inputs by small ones. We present a unified approach to address these questions by using random sampling maps to compare inputs of different sizes. The sampling maps we consider are generalizations of sampling with replacement, random binning, and species sampling. We characterize the application domains in which each type of sampling is appropriate in terms of the symmetries and relations between problem instances of different sizes in the domain. Our framework yields explicit generalization and sketching rates for function classes continuous with respect to a chosen notion of sampling, encompassing large families of functions defined on sequences, graphs, and tensors of different sizes. Specific examples include moment polynomials on measures, homomorphism densities and numbers of graphs, permutation-invariant transformers, and graph neural networks.

---

引用前请核对原文;本文由星际流动整理:https://feed.astralor.com/items/itm_0d94e5621a5b09c9a9af8372