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Advances in Set Function Learning: A Survey of Techniques and Applications

Jiahao Xie, Guangmo Tong

TL;DR

This survey addresses learning permutation-invariant set functions, formalizing the problem with mappings $f:2^{X}\to Y$ and surveying a wide spectrum of methods from DeepSets and Set Transformer to DSPN and DSF. It categorizes approaches by architecture (CNN/RNN/FNN/attention-based) and discusses theoretical properties, including universal approximation and invariance, as well as practical aspects like scalability and datasets. The paper highlights diverse applications across point clouds, multi-label classification, molecular prediction, and recommendations, illustrating how set-based models handle unordered inputs effectively. It also identifies key challenges and future directions, emphasizing theoretical learnability, online/streaming data, hybrid architectures, and domain-specific adaptations to advance the field.

Abstract

Set function learning has emerged as a crucial area in machine learning, addressing the challenge of modeling functions that take sets as inputs. Unlike traditional machine learning that involves fixed-size input vectors where the order of features matters, set function learning demands methods that are invariant to permutations of the input set, presenting a unique and complex problem. This survey provides a comprehensive overview of the current development in set function learning, covering foundational theories, key methodologies, and diverse applications. We categorize and discuss existing approaches, focusing on deep learning approaches, such as DeepSets and Set Transformer based methods, as well as other notable alternative methods beyond deep learning, offering a complete view of current models. We also introduce various applications and relevant datasets, such as point cloud processing and multi-label classification, highlighting the significant progress achieved by set function learning methods in these domains. Finally, we conclude by summarizing the current state of set function learning approaches and identifying promising future research directions, aiming to guide and inspire further advancements in this promising field.

Advances in Set Function Learning: A Survey of Techniques and Applications

TL;DR

This survey addresses learning permutation-invariant set functions, formalizing the problem with mappings and surveying a wide spectrum of methods from DeepSets and Set Transformer to DSPN and DSF. It categorizes approaches by architecture (CNN/RNN/FNN/attention-based) and discusses theoretical properties, including universal approximation and invariance, as well as practical aspects like scalability and datasets. The paper highlights diverse applications across point clouds, multi-label classification, molecular prediction, and recommendations, illustrating how set-based models handle unordered inputs effectively. It also identifies key challenges and future directions, emphasizing theoretical learnability, online/streaming data, hybrid architectures, and domain-specific adaptations to advance the field.

Abstract

Set function learning has emerged as a crucial area in machine learning, addressing the challenge of modeling functions that take sets as inputs. Unlike traditional machine learning that involves fixed-size input vectors where the order of features matters, set function learning demands methods that are invariant to permutations of the input set, presenting a unique and complex problem. This survey provides a comprehensive overview of the current development in set function learning, covering foundational theories, key methodologies, and diverse applications. We categorize and discuss existing approaches, focusing on deep learning approaches, such as DeepSets and Set Transformer based methods, as well as other notable alternative methods beyond deep learning, offering a complete view of current models. We also introduce various applications and relevant datasets, such as point cloud processing and multi-label classification, highlighting the significant progress achieved by set function learning methods in these domains. Finally, we conclude by summarizing the current state of set function learning approaches and identifying promising future research directions, aiming to guide and inspire further advancements in this promising field.
Paper Structure (50 sections, 13 equations, 2 figures, 2 tables)

This paper contains 50 sections, 13 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Deep learning methods
  4. CNN Based Methods
  5. Extending Convolution Operation to Set
  6. Combining CNN with Symmetric Aggregation
  7. RNN Based Methods
  8. FNN Based Methods
  9. DeepSets Based Methods
  10. DeepSets
  11. Generalization of DeepSets
  12. Theoretical Analysis of DeepSets
  13. Proposing Novel Aggregating Methods
  14. Extending DeepSets to Specific Scenarios
  15. PointNet Based Methods
  16. ...and 35 more sections

Figures (2)

  • Figure 1: Visualization of examples
  • Figure 2: Structure figures. \ref{['structure figures']}(a) shows the structure of Section \ref{['section of CNN based methods']}. \ref{['structure figures']}(b) shows the structure of Section \ref{['section of DeepSets based methods']}. \ref{['structure figures']}(c) shows the structure of Section \ref{['section of PointNet based methods']}. \ref{['structure figures']}(d) shows the structure of Section \ref{['section of Set Transformer based methods']}. \ref{['structure figures']}(e) shows the structure of Section \ref{['section of Deep set prediction networks based methods']}. \ref{['structure figures']}(f) shows the structure of Section \ref{['section of Deep Submodular Function based methods']}.

Theorems & Definitions (5)

  • Definition 2.1: Set function
  • Definition 2.2: Supervised set function learning
  • Definition 2.3: Permutation-invariance
  • Definition 3.1: Permutation-equivariance
  • Definition 3.2: Submodular function