One-class classification (OCC) needs samples from only a single class to train the classifier. Recently, an auto-associative kernel extreme learning machine was developed for the OCC task. This paper introduces a novel extension of this classifier by embedding minimum variance information within its architecture and is referred to as VAAKELM. The minimum variance embedding forces the network output weights to focus in regions of low variance and reduces the intra-class variance. This leads to a better separation of target samples and outliers, resulting in an improvement in the generalization performance of the classifier. The proposed classifier follows a reconstruction-based approach to OCC and minimizes the reconstruction error by using the kernel extreme learning machine as the base classifier. It uses the deviation in reconstruction error to identify the outliers. We perform experiments on 15 small-size and 10 medium-size one-class benchmark datasets to demonstrate the efficiency of the proposed classifier. We compare the results with 13 existing one-class classifiers by considering the mean F1 score as the comparison metric. The experimental results show that VAAKELM consistently performs better than the existing classifiers, making it a viable alternative for the OCC task.