Markov-Miml: A Markov chain-based multi-instance multi-label learning algorithm

Abstract

The main aim of this paper is to propose an efficient and novel Markov chain-based multi-instance multi-label (Markov-Miml) learning algorithm to evaluate the importance of a set of labels associated with objects of multiple instances. The algorithm computes ranking of labels to indicate the importance of a set of labels to an object. Our approach is to exploit the relationships between instances and labels of objects. The rank of a class label to an object depends on (i) the affinity metric between the bag of instances of this object and the bag of instances of the other objects, and (ii) the rank of a class label of similar objects. An object, which contains a bag of instances that are highly similar to bags of instances of the other objects with a high rank of a particular class label, receives a high rank of this class label. Experimental results on benchmark data have shown that the proposed algorithm is computationally efficient and effective in label ranking for MIML data. In the comparison, we find that the classification performance of the Markov-Miml algorithm is competitive with those of the three popular MIML algorithms based on boosting, support vector machine, and regularization, but the computational time required by the proposed algorithm is less than those by the other three algorithms.

Appendix

Proof of Theorem 1:

By Perron–Frobenius Theorem [22], we know that 1 is the spectral radius of $\mathbf{Q},$ that is, 1 is the maximal eigenvalue of $\mathbf{Q}$ in magnitude. The maximal eigenvalue of $(1-\alpha )\mathbf{Q}$ is equal to $1-\alpha .$ It implies that the matrix $I - (1-\alpha ) \mathbf{Q}$ is invertible. Thus, $\mathbf{P} = (I- (1-\alpha ) \mathbf{Q})^{-1} \alpha \mathbf{D}$ is well defined. It shows that there exists a unique matrix $\mathbf{P}$ satisfying $\mathbf{P} = (1-\alpha ) \mathbf{Q} \mathbf{P} + \alpha \mathbf{D}.$ By using the series of expansion of $(I- (1-\alpha ) \mathbf{Q})^{-1}$ as follows:

$$\begin{aligned} \sum _{k=0}^{\infty } (1-\alpha )^k \mathbf{Q}^k \end{aligned}$$

and the fact that the entries of $\mathbf{Q}$ are nonnegative, we have the entries of $\mathbf{P}$ must be nonnegative. We note that when $\mathbf{Q}$ is irreducible, there exists a positive integer $k^{\prime }$ such that the entries $\mathbf{Q}^k \mathbf{D}$ are even positive. Because $\sum _{k=0}^{\infty } (1-\alpha )^k \alpha = 1$ and each column of $\mathbf{Q}^k \mathbf{D}$ consists of $N$ probability distribution vectors, the results follows.

Note that $\mathbf{P}_t - \mathbf{P} = ((1-\alpha ) \mathbf{Q}) ( \mathbf{P}_{t-1} - \mathbf{P} ),$ and the spectral radius of $(1-\alpha )\mathbf{Q}$ is less than 1. The sequence converges for any initial starting vector $\mathbf{P}_0$ satisfying the required property in the theorem. $\square$