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Unsupervised Deep Distance Metric Learning (UDML) aims to learn sample similarities in the embedding space from an unlabeled dataset. Traditional UDML methods usually use the triplet loss or pairwise loss which requires the mining of positive and negative samples w.r.t. anchor data points. This is, however, challenging in an unsupervised setting as the label information is not available. In this paper, we propose a new UDML method that overcomes that challenge. In particular, we propose to use a deep clustering loss to learn centroids, i.e., pseudo labels, that represent semantic classes. During learning, these centroids are also used to reconstruct the input samples. It hence ensures the representativeness of centroids - each centroid represents visually similar samples. Therefore, the centroids give information about positive (visually similar) and negative (visually dissimilar) samples. Based on pseudo labels, we propose a novel unsupervised metric loss which enforces the positive concentration and negative separation of samples in the embedding space. Experimental results on benchmarking datasets show that the proposed approach outperforms other UDML methods.