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Feature Incremental Clustering with Generalization Bounds

Jing Zhang, Chenping Hou

Abstract

In many learning systems, such as activity recognition systems, as new data collection methods continue to emerge in various dynamic environmental applications, the attributes of instances accumulate incrementally, with data being stored in gradually expanding feature spaces. How to design theoretically guaranteed algorithms to effectively cluster this special type of data stream, commonly referred to as activity recognition, remains unexplored. Compared to traditional scenarios, we will face at least two fundamental questions in this feature incremental scenario. (i) How to design preliminary and effective algorithms to address the feature incremental clustering problem? (ii) How to analyze the generalization bounds for the proposed algorithms and under what conditions do these algorithms provide a strong generalization guarantee? To address these problems, by tailoring the most common clustering algorithm, i.e., $k$-means, as an example, we propose four types of Feature Incremental Clustering (FIC) algorithms corresponding to different situations of data access: Feature Tailoring (FT), Data Reconstruction (DR), Data Adaptation (DA), and Model Reuse (MR), abbreviated as FIC-FT, FIC-DR, FIC-DA, and FIC-MR. Subsequently, we offer a detailed analysis of the generalization error bounds for these four algorithms and highlight the critical factors influencing these bounds, such as the amounts of training data, the complexity of the hypothesis space, the quality of pre-trained models, and the discrepancy of the reconstruction feature distribution. The numerical experiments show the effectiveness of the proposed algorithms, particularly in their application to activity recognition clustering tasks.

Feature Incremental Clustering with Generalization Bounds

Abstract

In many learning systems, such as activity recognition systems, as new data collection methods continue to emerge in various dynamic environmental applications, the attributes of instances accumulate incrementally, with data being stored in gradually expanding feature spaces. How to design theoretically guaranteed algorithms to effectively cluster this special type of data stream, commonly referred to as activity recognition, remains unexplored. Compared to traditional scenarios, we will face at least two fundamental questions in this feature incremental scenario. (i) How to design preliminary and effective algorithms to address the feature incremental clustering problem? (ii) How to analyze the generalization bounds for the proposed algorithms and under what conditions do these algorithms provide a strong generalization guarantee? To address these problems, by tailoring the most common clustering algorithm, i.e., -means, as an example, we propose four types of Feature Incremental Clustering (FIC) algorithms corresponding to different situations of data access: Feature Tailoring (FT), Data Reconstruction (DR), Data Adaptation (DA), and Model Reuse (MR), abbreviated as FIC-FT, FIC-DR, FIC-DA, and FIC-MR. Subsequently, we offer a detailed analysis of the generalization error bounds for these four algorithms and highlight the critical factors influencing these bounds, such as the amounts of training data, the complexity of the hypothesis space, the quality of pre-trained models, and the discrepancy of the reconstruction feature distribution. The numerical experiments show the effectiveness of the proposed algorithms, particularly in their application to activity recognition clustering tasks.
Paper Structure (28 sections, 12 theorems, 102 equations, 5 figures, 3 tables)

This paper contains 28 sections, 12 theorems, 102 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Notations
  4. $k$-Means
  5. Feature Incremental Clustering
  6. FIC-FT Algorithm
  7. FIC-DR Algorithm
  8. FIC-DA Algorithm
  9. FIC-MR Algorithm
  10. Generalization Error bound Analysis
  11. Experiments
  12. Dataset Description and Experimental Setup
  13. Dataset Description
  14. Experimental Setup
  15. Comparison Results
  16. ...and 13 more sections

Key Result

Theorem 1

Consider training the $k$-means clustering model in (k-meansOb) using any sample set $D=\left\{\mathbf{x}_{1}, \ldots, \mathbf{x}_{n}\right\}$ of $n$ data points. Let $\mathbf{U}\in \mathcal{H}^{k}$ be all possible cluster centers. If $\lVert\mathbf{x}\rVert\leq \gamma$ for $\forall\mathbf{x}\in \ma where $\beta = \sqrt{(\mathbb{E}_{\mathbf{x}\sim \mathbb{Q}} \left[\sup _{g_{\mathbf{U}}\in\mathcal

Figures (5)

  • Figure 1: Diagram illustrating the feature incremental clustering scenario in a dynamic open environment. In the activity recognition task, let $\mathbf{X}^{(1)}_{1}$ be the old feature obtained by the previous sensors in the previous stage; In the current stage, new features $\mathbf{X}^{(2)}_{2}$ and old features $\mathbf{X}^{(1)}_{2}$ are collected by the new and the old sensors, respectively. Then, $\mathbf{X}_{2}=[\mathbf{X}^{(1)}_{2},\mathbf{X}^{(2)}_{2}]$ represents new data in the current stage.
  • Figure 2: Illustration of the four feature incremental clustering.
  • Figure 3: The influence of incremental features in the current stage with FIC-DR, FIC-DA, and FIC-MR.
  • Figure 4: Parameter sensitivity of FIC-MR on ACC results on six datasets.
  • Figure 5: The ACC, F-score, and NMI results comparison in the activity recognition clustering task.

Theorems & Definitions (29)

  • Definition 1: biau2008performance
  • Definition 2: biau2008performance
  • Definition 3: Clustering Rademacher Complexity (CRC)
  • Theorem 1
  • Remark 1
  • Theorem 2: FIC-FT
  • Remark 2
  • Definition 4: $\mathcal{Y}$-discrepancy
  • Theorem 3: FIC-DR
  • Remark 3
  • ...and 19 more