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A Concise yet Effective model for Non-Aligned Incomplete Multi-view and Missing Multi-label Learning

Xiang Li, Songcan Chen

TL;DR

This work tackles non-aligned incomplete multi-view data with missing multi-labels by proposing NAIM$^3$L, a concise, inductive framework that uses an indicator-based regression objective and a regularizer encoding global high-rank and local low-rank label structures. A single hyper-parameter controls the regularization, and the optimization is solved via a specialized ADMM algorithm with CCCP to handle the DC form, achieving linear-time complexity in the number of samples. The authors prove a non-negativity property of the regularizer to avoid trivial solutions and provide extensive experiments on five real datasets, showing consistent superiority over state-of-the-art methods under all three challenges. The approach is also kernelizable and compatible with deep networks, suggesting strong practical impact for complex, real-world multi-view, multi-label problems.

Abstract

In reality, learning from multi-view multi-label data inevitably confronts three challenges: missing labels, incomplete views, and non-aligned views. Existing methods mainly concern the first two and commonly need multiple assumptions to attack them, making even state-of-the-arts involve at least two explicit hyper-parameters such that model selection is quite difficult. More roughly, they will fail in handling the third challenge, let alone addressing the three jointly. In this paper, we aim at meeting these under the least assumption by building a concise yet effective model with just one hyper-parameter. To ease insufficiency of available labels, we exploit not only the consensus of multiple views but also the global and local structures hidden among multiple labels. Specifically, we introduce an indicator matrix to tackle the first two challenges in a regression form while aligning the same individual labels and all labels of different views in a common label space to battle the third challenge. In aligning, we characterize the global and local structures of multiple labels to be high-rank and low-rank, respectively. Subsequently, an efficient algorithm with linear time complexity in the number of samples is established. Finally, even without view-alignment, our method substantially outperforms state-of-the-arts with view-alignment on five real datasets.

A Concise yet Effective model for Non-Aligned Incomplete Multi-view and Missing Multi-label Learning

TL;DR

This work tackles non-aligned incomplete multi-view data with missing multi-labels by proposing NAIML, a concise, inductive framework that uses an indicator-based regression objective and a regularizer encoding global high-rank and local low-rank label structures. A single hyper-parameter controls the regularization, and the optimization is solved via a specialized ADMM algorithm with CCCP to handle the DC form, achieving linear-time complexity in the number of samples. The authors prove a non-negativity property of the regularizer to avoid trivial solutions and provide extensive experiments on five real datasets, showing consistent superiority over state-of-the-art methods under all three challenges. The approach is also kernelizable and compatible with deep networks, suggesting strong practical impact for complex, real-world multi-view, multi-label problems.

Abstract

In reality, learning from multi-view multi-label data inevitably confronts three challenges: missing labels, incomplete views, and non-aligned views. Existing methods mainly concern the first two and commonly need multiple assumptions to attack them, making even state-of-the-arts involve at least two explicit hyper-parameters such that model selection is quite difficult. More roughly, they will fail in handling the third challenge, let alone addressing the three jointly. In this paper, we aim at meeting these under the least assumption by building a concise yet effective model with just one hyper-parameter. To ease insufficiency of available labels, we exploit not only the consensus of multiple views but also the global and local structures hidden among multiple labels. Specifically, we introduce an indicator matrix to tackle the first two challenges in a regression form while aligning the same individual labels and all labels of different views in a common label space to battle the third challenge. In aligning, we characterize the global and local structures of multiple labels to be high-rank and low-rank, respectively. Subsequently, an efficient algorithm with linear time complexity in the number of samples is established. Finally, even without view-alignment, our method substantially outperforms state-of-the-arts with view-alignment on five real datasets.

Paper Structure

This paper contains 31 sections, 4 theorems, 26 equations, 8 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Methods of Addressing One Challenge
  4. Methods of Addressing Two Challenges
  5. Summary
  6. The Proposed Method
  7. Problem Settings
  8. Problem Formulation
  9. Optimization
  10. ADMM Algorithm
  11. Sub-problem of $\ \mathbf{W}_{t}$
  12. Sub-problem of $\ \mathbf{Z}_{k}$
  13. Sub-problem of $\ \mathbf{\Lambda}_{k}$
  14. An Efficient Algorithm for Computing $\partial\|\bullet \|_{*}$
  15. Convergence and Complexity Analysis
  16. ...and 16 more sections

Key Result

Lemma 1

r45 Let $\mathbf{A}$ and $\mathbf{B}$ be matrices of the same row dimensions, and $[\mathbf{A}, \mathbf{B}]$ be the concatenation of $\mathbf{A}$ and $\mathbf{B}$, we have $\| [\mathbf{A}, \mathbf{B}]\left\|_{*} \leq\right\| \mathbf{A}\left\|_{*}+\right\| \mathbf{B} \|_{*}$.

Figures (8)

  • Figure 1: The global and local structures of the multiple labels. The same sample of non-aligned views is represented by the same color, and ”sky”, ”cloud” , … ,”fish” are the labels. ”1” in the label matrix means that the sample is annotated with the corresponding label whereas “-1” means not. All the label matrices are vertically concatenated. The label matrix of all samples from two views has full column rank or high rank, which refers to the global structure of multiple labels. Meanwhile, the sub-label matrix comprised of samples that share the same individual label tends to have low rank, e.g., the rank of sub-label matrix that share the same label “cloud” equals to 2, which corresponds to the local structure of multiple labels.
  • Figure 2: The high-rankness of the entire label matrix corresponding to all samples and the low-rankness of the sub-label matrices corresponding to samples that share a single label on Corel5k dataset, which has 260 labels.
  • Figure 3: The nuclear norm of the predicted entire label matrix, the mean value and the median value of the nuclear norm of the predicted sub-label matrices.
  • Figure 4: Statistical analysis of NAIM$^3$L by the Nemenyi test, the significance level is 0.05 and methods not connected are significantly different.
  • Figure 5: The performance of four metrics on the Corel5k dataset and the Pascal07 dataset under different ratios of incomplete multiple views ($\alpha$) and missing multiple labels ($\beta$). The ‘Full’ means that the multiple views and multiple labels in the training set are complete. The average value and standard deviation are shown in each sub-figures.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Lemma 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 2