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In this work, we formulate and solve a new type of approximate nearest neighbor search (ANNS) problems called ANNS after linear transformation (ALT). In ANNS-ALT, we search for the vector (in a dataset) that, after being linearly transformed by a user-specified query matrix, is closest to a query vector. It is a very general mother problem in the sense that a wide range of baby ANNS problems that have important applications in databases and machine learning can be reduced to and solved as ANNS-ALT, or its dual that we call ANNS-ALTD. We propose a novel and computationally efficient solution, called ONe Index for All Kernels (ONIAK), to ANNS-ALT and all its baby problems when the data dimension d is not too large (say d ≤ 200). In ONIAK, a universal index is built, once and for all, for answering all future ANNS-ALT queries that can have distinct query matrices. We show by experiments that, when d is not too large, ONIAK has better query performance than linear scan on the mother problem (of ANNS-ALT), and has query performances comparable to those of the state-of-the-art solutions on the baby problems. However, the algorithmic technique behind this universal index approach suffers from a so-called dimension blowup problem that can make the indexing time prohibitively long for a large dataset. We propose a novel algorithmic technique, called fast GOE quadratic form (FGoeQF), that completely solves the (prohibitively long indexing time) fallout of the dimension blowup problem. We also propose a Johnson-Lindenstrauss transform (JLT) based ANNS-ALT (and ANNS-ALTD) solution that significantly outperforms any competitor when d is large.