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Kernel Representation and Similarity Measure for Incomplete Data

Yang Cao, Sikun Yang, Kai He, Wenjun Ma, Ming Liu, Yujiu Yang, Jian Weng

TL;DR

This work addresses the challenge of measuring similarity on incomplete data by introducing the proximity kernel, which encodes samples into a high-dimensional, sparse representation via data-dependent equal-frequency binning and proximity assignment. Missing values are handled without imputation in the original space through a cascading fallback, using kernel mean embeddings over progressively relaxed matches. The kernel is proven valid (PSD and symmetric) and yields linear-time complexity, enabling scalable clustering on 12 real-world datasets where it achieves superior or competitive performance versus a broad set of baselines. The approach offers interpretable, density-adaptive similarity suitable for large-scale web, recommendation, and behavior analysis tasks with incomplete data.

Abstract

Measuring similarity between incomplete data is a fundamental challenge in web mining, recommendation systems, and user behavior analysis. Traditional approaches either discard incomplete data or perform imputation as a preprocessing step, leading to information loss and biased similarity estimates. This paper presents the proximity kernel, a new similarity measure that directly computes similarity between incomplete data in kernel feature space without explicit imputation in the original space. The proposed method introduces data-dependent binning combined with proximity assignment to project data into a high-dimensional sparse representation that adapts to local density variations. For missing value handling, we propose a cascading fallback strategy to estimate missing feature distributions. We conduct clustering tasks on the proposed kernel representation across 12 real world incomplete datasets, demonstrating superior performance compared to existing methods while maintaining linear time complexity. All the code are available at https://anonymous.4open.science/r/proximity-kernel-2289.

Kernel Representation and Similarity Measure for Incomplete Data

TL;DR

This work addresses the challenge of measuring similarity on incomplete data by introducing the proximity kernel, which encodes samples into a high-dimensional, sparse representation via data-dependent equal-frequency binning and proximity assignment. Missing values are handled without imputation in the original space through a cascading fallback, using kernel mean embeddings over progressively relaxed matches. The kernel is proven valid (PSD and symmetric) and yields linear-time complexity, enabling scalable clustering on 12 real-world datasets where it achieves superior or competitive performance versus a broad set of baselines. The approach offers interpretable, density-adaptive similarity suitable for large-scale web, recommendation, and behavior analysis tasks with incomplete data.

Abstract

Measuring similarity between incomplete data is a fundamental challenge in web mining, recommendation systems, and user behavior analysis. Traditional approaches either discard incomplete data or perform imputation as a preprocessing step, leading to information loss and biased similarity estimates. This paper presents the proximity kernel, a new similarity measure that directly computes similarity between incomplete data in kernel feature space without explicit imputation in the original space. The proposed method introduces data-dependent binning combined with proximity assignment to project data into a high-dimensional sparse representation that adapts to local density variations. For missing value handling, we propose a cascading fallback strategy to estimate missing feature distributions. We conduct clustering tasks on the proposed kernel representation across 12 real world incomplete datasets, demonstrating superior performance compared to existing methods while maintaining linear time complexity. All the code are available at https://anonymous.4open.science/r/proximity-kernel-2289.
Paper Structure (28 sections, 19 equations, 6 figures, 3 tables)

This paper contains 28 sections, 19 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Traditional Imputation Methods
  4. Deep Learning and Generative Approaches
  5. Kernel-based Similarity Measure
  6. Preliminary
  7. Problem Formulation
  8. Kernel Methods
  9. Kernel Mean Embedding
  10. Methodology
  11. Data Dependent Binning and Proximity Assignment
  12. Bin Center Selection via Equal-Frequency Binning
  13. Proximity Assignment Mechanism
  14. One-Hot Encoding Construction
  15. Cascading Fallback Strategy for Missing Value
  16. ...and 13 more sections

Figures (6)

  • Figure 1: Demonstration of density-adaptive binning. Red lines denote bin centers selected through equal-frequency binning. Green and blue points represent samples in dense and sparse regions, respectively. The sparse region exhibits wider bins compared to the dense region, reflecting the natural adaptation to data distribution without explicit density estimation.
  • Figure 2: Comparison of all methods performance in terms of NMI.
  • Figure 3: Friedman-Nemenyi test at significance level 0.1. If two algorithms are connected by a CD (critical difference) line, there is no significant difference between them.
  • Figure 4: Clustering performance of top 5 methods under different missing rates on 3L and Messidor datasets.
  • Figure 5: Runtime comparison between PK, MIWAE and HI-PMK.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Definition 1: Kernel Function
  • Definition 2: Kernel Mean Embedding