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A common problem in many types of databases is retrieving the most similar matches to a query object. Finding those matches in a large database can be too slow to be practical, especially in domains where objects are compared using computationally expensive similarity (or distance) measures. This paper proposes a novel method for approximate nearest neighbor retrieval in such spaces. Our method is embedding-based, meaning that it constructs a function that maps objects into a real vector space. The mapping preserves a large amount of the proximity structure of the original space, and it can be used to rapidly obtain a short list of likely matches to the query. The main novelty of our method is that it constructs, together with the embedding, a query-sensitive distance measure that should be used when measuring distances in the vector space. The term "query-sensitive" means that the distance measure changes depending on the current query object. We report experiments with an image database of handwritten digits, and a time-series database. In both cases, the proposed method outperforms existing state-of-the-art embedding methods, meaning that it provides significantly better trade-offs between efficiency and retrieval accuracy.