MIL-SKDE: Multiple-instance learning with supervised kernel density estimation

Abstract

Multiple-instance learning (MIL) is a variation on supervised learning. Instead of receiving a set of labeled instances, the learner receives a set of bags that are labeled. Each bag contains many instances. The aim of MIL is to classify new bags or instances. In this work, we propose a novel algorithm, MIL-SKDE (multiple-instance learning with supervised kernel density estimation), which addresses MIL problem through an extended framework of “KDE (kernel density estimation)+mean shift”. Since the KDE+mean shift framework is an unsupervised learning method, we extend KDE to its supervised version, called supervised KDE (SKDE), by considering class labels of samples. To seek the modes (local maxima) of SKDE, we also extend mean shift to a supervised version by taking into account sample labels. SKDE is an alternative of the well-known diverse density estimation (DDE) whose modes are called concepts. Comparing to DDE, SKDE is more convenient to learn multi-modal concepts and robust to labeling noise (mistakenly labeled bags). Finally, each bag is mapped into a concept space where the multi-class SVM classifiers are learned. Experimental results demonstrate that our approach outperforms the state-of-the-art MIL approaches.

Introduction

In standard supervised (or single-instance) learning, the training set is given by $\mathcal{D}\prime =\{〈{\mathbf{x}}_i,y_i〉{\}}_{i=1}^n$, where ${\mathbf{x}}_i\in {\mathbb{R}}^d$ is an instance, and for each ${\mathbf{x}}_i$, there is a known label $y_i\in \mathbb{Y}=\{0,1\}$. The task is to learn a classifier $f:{\mathbb{R}}^d\to \mathbb{Y}$. This learning usually requires much manual labor on labeling. In contrast, the multiple-instance learning (MIL) avoids labeling the individual instances and assigns only one label to a collection of instances instead. Such a collection of instances is called a bag. More formally, the training set in MIL is given by $\mathcal{D}=\{〈B_i,y_i〉{\}}_{i=1}^m$, where bag $B_i=\{{\mathbf{x}}_{i,j}{\}}_{j=1}^{\left|B_i\right|}$, ${\mathbf{x}}_{i,j}\in {\mathbb{R}}^d$ is an instance and $\left|B_i\right|$ is the number of instances in B_i. Let $y_{i,j}\in \{0,1\}$ be the latent label of instance ${\mathbf{x}}_{i,j}\in B_i$. Then, the known $y_i=\max \{y_{i,j}{\}}_{j=1}^{\left|B_i\right|}$. The aim of MIL is to learn a classifier that is able to classify new bags or new instances.

Many tasks in computer vision and machine learning can be naturally cast as an MIL problem. For instance, object categorization which involves determining whether or not an image contains a certain category of objects [1], [2], [3]. To tackle the intra-class variability, e.g., appearance diversity of vehicles according to different makes and models, MIL treats each image as a set of local regions, but only those regions that carry category-specific information are regions of interest (ROI) for purposes of classification. For example, all the wheel regions of vehicles have common circular shapes, so the wheel regions are ROI. Other regions have random features and possess no discriminative power. From this viewpoint, an image is a bag and the image regions are instances of the image categorization problem. Likewise, many other applications are also able to be treated as MIL problems, e.g., content-based image retrieval [4], [5], [6], segmentation [7], object tracking [8], human detection [9] and computer aided diagnosis [10], [11].

Loosely speaking, if an instance repeatedly occurs in different positive bags, this instance is called a concept (concepts for negative bags are not considered here because negative instances are usually randomly distributed) and the weights of concepts indicate the frequencies of which concepts occur in positive bags. Learning concepts is critical for MIL algorithms. However, MIL algorithms often confront a problem of inducing weighted concepts from a large instance space (large value of $n={\sum }_{i=1}^m\left|B_i\right|$ as each bag may contain many instances) and a large feature space (feature vectors extracted from instances are high-dimensional, e.g., 128-D SIFT features [12]). How to efficiently induce important concepts from a large instance+feature space is still challenging for MIL. Another issue is that many existing MIL algorithms follow a multiple-instance setting that a positive bag must contain at least one true positive instance, whereas a negative bag contains negative instances only. This setting is often not true in computer vision tasks because the labeling processing is quite subjective or visual features extracted from instances are distorted due to changes of scale, illumination and view angle, etc. Those bags labeled positive but containing no true positive instances and bags labeled negative but containing true positive instances are called labeling noise.

In this paper, our contributions are as follows. We propose a novel MIL algorithm, MIL-SKDE (multiple-instance learning with supervised kernel density estimation), which is able to conveniently learn concepts in large feature space and is robust to labeling noise. First, we advance a modified version of kernel density estimation (KDE) function to estimate the instance distribution. The modified KDE is named supervised kernel density estimation (SKDE) as it considers class labels of data points. SKDE is an alternative to the well-known diverse density estimation (DDE) [13] and more robust to the labeling noise. The same as DDE, the modes (local maxima) of SKDE are the concepts to be learned. As mean shift is widely used to locate modes of KDE, we extend mean shift to its supervised version (named supervised mean shift) to adapt it to SKDE. Similar to mean shift, the supervised mean shift is also steepest ascent (that only computes first order gradient) with a varying step size that is the magnitude of the gradient. Supervised mean shift is handy to seek modes in a large feature space because, from an initial point, it can quickly converge to a mode without computing Hessian matrix (second-order gradient) which is expensive in high-dimensional space.

The remainder of the paper is organized as follows. In Section 2, we review the previous work on MIL and briefly describe kernel density estimation and mean shift. In Section 3 we derive our supervised kernel density estimation and compare it with diverse density estimation. Section 4 presents the supervised mean shift for locating the modes in SKDE. Section 5 summarizes MIL-SKDE. Experiments are presented in Section 6. Finally, we conclude this paper in Section 7.