Extracting Certainty from Uncertainty: Transductive Pairwise Classification from Pairwise Similarities
In this work, we study the problem of transductive pairwise classification from pairwise similarities 1 . The goal of transductive pairwise classification from pairwise similarities is to infer the pairwise class relationships, to which we refer as pairwise labels, between all examples given a subset of class relationships for a small set of examples, to which we refer as labeled examples. We propose a very simple yet effective algorithm that consists of two simple steps: the first step is to complete the sub-matrix corresponding to the labeled examples and the second step is to reconstruct the label matrix from the completed sub-matrix and the provided similarity matrix. Our analysis exhibits that under several mild preconditions we can recover the label matrix with a small error, if the top eigen-space that corresponds to the largest eigenvalues of the similarity matrix covers well the column space of label matrix and is subject to a low coherence, and the number of observed pairwise labels is sufficiently enough. We demonstrate the effectiveness of the proposed algorithm by several experiments.
The authors study problem of transductive pairwise classification and propose a simple algorithm to address the task by using two step approach: 1) sub-matrix completion corresponding to the labeled examples 2) label matrix reconstruction from the provided similarity matrix and the completed sub-matrix. Furthermore, authors give theoretical result on reconstruction error and empirically evaluate performance of the proposed algorithm. On one hand, it is certainly useful and interesting to consider transductive pairwise classification as a matrix completion problem. On the other hand, a natural setting for such task could be a semi-supervised or transductive preference learning. A large body of research exists on (ss) preference learning and ranking (e.g. work of Nir Ailon, Chu Wei, Johannes Fürnkranz, Eyke Hüllermeier, and others). These algorithms are frequently applied for link prediction, protein sequence ranking, etc, - basically all of the examples authors use to motivate their method. Why is this "setting" missing from the manuscript? Short description/reference to the relevant pairwise classification/preference learning algorithms as well as empirical comparison (e.g. to semi-supervised gaussian processes for preference learning) would further strengthen the paper.The authors suggest notable differences of their approach (in particular transductive  setting) compared to the algorithms presented in [25] and [23], however, a clear motivation for the proposed algorithm is lacking. While theoretical contribution of the paper is impressive, a related analysis has been described in [25] for semi-supervised setting and the strategy for perfect matrix recovery is quite similar to the one presented by the authors.   The experimental results section could be improved. For example, a comparison of several baseline methods on a synthetic data would be better than reporting the TPCMC result only. Also, it is not clear to me why "comparable" methods that can use side information (as in [25]) are not included in the evaluation. In fact, comparison is done only with two clustering techniques. Finally, performance of the methods such as SC or SKL greatly depends on a number of parameters, choice of kernel, they way Laplacian matrix is construed, etc. How were these parameters estimated? Were the experiments performed multiple times and the average results reported as in [25]?  Unless some of these questions are addressed it is difficult to agree with the last statement in the manuscript, that is "empirical evaluations have verified the effectiveness of the proposed algorithm".