Incomplete multi-view learning via half-quadratic minimization

Abstract

In real applications, to deal with incomplete multi-view data, incomplete multi-view learning has experienced rapid development in recent years. Among various incomplete multi-view learning methods, a considerable number of methods were developed with the matrix factorization technique. Most of the existing matrix factorization based methods adopt the sum of squared ${\ell }_2$-norm as loss functions directly, which is known to be susceptible to value missing. To overcome this issue, we propose a new matrix factorization method, named Incomplete Multi-view Learning via Half-quadratic Minimization (IMLHM). Different from previous methods, a robust estimator based on half-quadratic minimization theory is imported to our loss function to overcome the sensitivity of ${\ell }_2$-norm to noise. The influence of bad recovered instances is decreased via the automatic weighting scheme derived from the half-quadratic minimization process, thereby improving the robustness of the proposed method. Additionally, a nuclear norm is introduced to exploit the low-rank structure of the learned representation matrix, further improving the robustness of the proposed method against to noise. An alternating iterative algorithm is developed to optimize the objective function. Comprehensive experimental results on seven data sets verify the effectiveness of the proposed method.

Introduction

In recent years, multi-view data sources are produced in various applications. For example, images can be described by Histogram of Oriented Gradient (HOG) [1], Local Binary Pattern (LBP) [2] and SIFT [3], and a film segment is represented by voice and video features. The research of multi-view learning has attracted numerous scholars’ attention [4], [5], [6], [7], [8], [9], [43]. Most traditional multi-view researches assume that the data on each view are complete. However, in real applications, the integrity of the multi-view data usually cannot be guaranteed [10], [11], [12], [44]. Practically, human error and mechanical failure occur time to time in the data collection process, which leads to incomplete multi-view data. Many traditional multi-view methods discard incomplete samples in most cases, which undoubtedly reduce the number of available samples [13], [14], [15], [42]. To make use of the information contained in the incomplete samples, a straightforward way is to fill the missing values with 0 or the mean of the survived values. Nevertheless, the performance of the filling strategy is still limited. In order to make better use of the incomplete multi-view samples, various incomplete multi-view learning methods have been proposed. They mainly fall into three categories: (1) Matrix factorization based methods with different constraints or regularization terms. This kind of methods attempt to learn a representation matrix containing as complete information as possible. Matrix factorization based methods can be easily extended to handle the case of both view missing and variable missing. (2) Kernel-based methods, which import kernel tricks into incomplete multi-view learning [16]. For instance, the within-view and between-view relationships are combined to predict the missing rows and columns of kernel matrices [17]. (3) Graph-based methods [18], [19]. When constructing graph on each view, the connecting weight between missing samples is set to 0. Then, based on the graphs of multiple views, a common representation matrix is learned. For example, Wen et al. [19] combined the self-representation model and spectral clustering to learning a common relaxed clustering indicator matrix after the graph construction. Kernel-based methods and graph-based methods achieve promising performance, but most of them are not applicable to case that there are both view and variable missing. To deal with various missing scenes in real applications, we focus on the matrix factorization based methods.

Based on the assumption that different views are generated from a common subspace, a bunch of popular incomplete multi-view learning methods are designed using the matrix decomposition technique. Concretely, based on the Nonnegative Matrix Factorization (NMF) and the ${\ell }_1$sparse regularization, Li et al. [20] proposed the Partial multi-View Clustering (PVC) method. PVC is a pioneering work to learn a full representation for incomplete two-view data. Then, Xu et al. [21] proposed the Multi-View Learning with Incomplete Views (MVL-IV) method, which factorizes the incomplete data matrix of each view as the product of a view-specific basis matrix and the common representation matrix. Almost simultaneously, Shao et al. [22] put forward the Multiple Incomplete views Clustering (MIC) method, based on NMF and the ${\ell }_{2,1}$ regularization. Concretely, MIC fills the missing views with the mean of available instances and gives smaller weights to the incomplete instances. The geometric information is not fully used in previous algorithms, Zhao et al. proposed the Incomplete Multi-modality Grouping (IMG) [23] approach to explore the compact global structure of data when learning the latent subspace. Specifically, an automatically learned graph Laplacian term is imposed on the learned latent representation to force similar samples to be close in the latent subspace. Utilizing the weighted semi-NMF, Hu et al. developed the Doubly Aligned Incomplete Multi-view Clustering (DAIMC) algorithm to align samples and the basis matrices simultaneously [24]. Albeit the above mentioned methods have greatly promoted the development of incomplete multi-view learning, the performance of incomplete multi-view learning can be further improved. It can be found that all the above mentioned methods employ squared ${\ell }_2$-norm as loss functions. As we all know that squared ${\ell }_2$-norm is not robust to noise, squared ${\ell }_2$-norm based loss functions are not ideal to deal with incomplete multi-view data, as value missing is a typical kind of noise. Therefore, to make the learned representation matrix less affected by the missing values, more robust loss functions are expected. In this paper, we propose the Incomplete Multi-view Learning via Half-quadratic Minimization (IMLHM) to solve the above issue. Specifically, along the line of using the matrix factorization technique, we attempt to learn a common representation matrix for incomplete multi-view data. Unlike the previous methods based on the sum-of-square error directly, a robust estimator based on the half-quadratic theory is imposed on the loss of each sample of each view. With the help of the robust estimator, well recovered samples are assigned with larger weights and vise versa. In this way, the robustness against missing views and missing variables is improved. Additionally, noting that incomplete multi-view learning can be seen as a special case of matrix completion, the common representation matrix is forced to be low-rank to fulfill the assumption in matrix completion [25], [26]. Finally, the objective function of IMLHM is optimized with an iterative alternating algorithm. Experimental results on several incomplete multi-view data sets verify the effectiveness of the proposed method.

The remainder of the paper is organized as follows. Section 2 introduces the problem setting and some related incomplete multi-view learning methods. In Section 3, we formulate the incomplete-view problem and introduce our proposed algorithm. The optimization strategy and the convergence behavior of the optimization method are presented in Section 4. Section 5 provides the computational complexity. We analyze the experimental results in Section 6 and conclude in Section 7.