# Improving Multi-Instance Multi-Label Learning by Extreme Learning Machine
## 2 The Preliminaries
### Multi-Instance Multi-Label Learning  
In traditional supervised learning, the relationships between an object and its description and its label are always a one-to-one correspondence. That is, an object is represented by a single instance and associated with a single class label. In this sense, we refer to it as single-instance single-label learning (SISL). Formally, let \(X\) be the instance space (or say, feature space) and \(Y\) the set of class labels. The goal of SISL is to learn a function \(f_{SISL}\): \(X\)\(\rightarrow\)\(Y\) from a given dataset \(\{(x_{1},y_{1}),(x_{2},y_{2}),\ldots,(x_{m},y_{m})\}\), where \(x_{i}\)\(\in\)\(X\) is an instance and \(y_{i}\)\(\in\)\(Y\) is the label of \(x_{i}\). This formalization is prevailing and successful. However, as mentioned in Section 1, many real-world objects are complicated and ambiguous in their semantics. Representing these ambiguous objects with SISL may lose some important information and make the learning task problematic [1]. Thus, many real-world complicated objects do not fit in this framework well. In order to deal with this problem, several multi-learning frameworks have been proposed, e.g., multi-instance learning (MIL), multi-label learning (MLL) and multi-instance multi-label Learning (MIML). MIL studies the problem where a real-world object described by a number of instances is associated with a single class label. The training set for MIL is composed of many bags each containing multiple instances. In particular, a bag is labeled positively if it contains at least one positive instance and negatively otherwise. The goal is to label unseen bags correctly. Note that although the training bags are labeled, the labels of their instancesare unknown. This learning framework was formalized by [PERSON] _et al._[3] when they were investigating drug activity prediction. Formally, let \(X\) be the instance space (or say, feature space) and \(Y\) the set of class labels. The task of MIL is to learn a function \(f_{MIL}\): \(2^{X}{\rightarrow}\{-1,+1\}\) from a given dataset \(\{(X_{1},y_{1}),(X_{2},y_{2}),\ldots,(X_{m},y_{m})\}\), where \(X_{i}{\subseteq}X\) is a set of instances \(\{x_{1}^{(i)},x_{2}^{(i)},\ldots,x_{n_{i}}^{(i)}\}\), \(x_{j}^{(i)}{\in}X(j=1,2,\ldots,n_{i})\), and \(y_{i}{\in}\{-1,+1\}\) is the label of \(X_{i}\). Multi-instance learning techniques have been successfully applied to diverse applications, including image categorization [7,8], image retrieval [9,10], text categorization [11,12], web mining [13], spam detection [14], face detection [15], computer-aided medical diagnosis [16], _etc._ Differently, MLL studies the problem where a real-world object is described by one instance, but associated with a number of class labels. The goal is to learn a function \(f_{MLL}\): \(X{\rightarrow}2^{Y}\) from a given dataset \(\{(x_{1},Y_{1}),(x_{2},Y_{2}),\)\(\ldots,(x_{m},Y_{m})\}\), where \(x_{i}{\in}X\) is an instance and \(Y_{i}{\subseteq}Y\) a set of labels \(\{y_{1}^{(i)},y_{2}^{(i)},\ldots,y_{l_{i}}^{(i)}\}\), \(y_{k}^{(i)}{\in}Y(k=1,2,\ldots,l_{i})\). The existing work of MLL falls into two major categories. One attempts to divide multi-label learning to a number of two class classification problems [17,18] or to transform it into a label ranking problem [19,20]; the other tries to exploit the correlation between the labels [21,22]. MLL has been found useful in many tasks, such as text categorization [23], scene classification [24], image and video annotation [25,26], bioinformatics [27,28] and even association rule mining [29,30]. MIML is a generalization of traditional supervised learning, multi-instance learning and multi-label learning, where a real-world object may be associated with a number of instances and a number of labels simultaneously. In some cases, transforming single-instance multi-label objects to MIML objects for learning may be beneficial. Before the explanation, we first introduce how to perform such a transformation.