We present an I O-efficient algorithm for computing similarity joins based on locality-sensitive hashing (LSH). In contrast to the filtering methods commonly suggested our method has provable sub-quadratic dependency on the data size. Further, in contrast to straightforward implementations of known LSH-based algorithms on external memory, our approach is able to take significant advantage of the available internal memory: Whereas the time complexity of classical algorithms includes a factor of @math , where @math is a parameter of the LSH used, the I O complexity of our algorithm merely includes a factor @math , where @math is the data size and @math is the size of internal memory. Our algorithm is randomized and outputs the correct result with high probability. It is a simple, recursive, cache-oblivious procedure, and we believe that it will be useful also in other computational settings such as parallel computation.
We study algorithms for similarity join in the , which has been widely adopted in the literature (see, e.g., the survey by Vitter @cite_22 ). The external memory model consists of an internal memory of @math words and an external memory of unbounded size. The processor can only access data stored in the internal memory and move data between the two memories in blocks of size @math . For simplicity we will here measure block and internal memory size in units of points from @math , such that they can contain @math points and @math points, respectively.
@cite_22
2290588870
Following the recent trend in explicit neural memory structures, we present a new design of an external memory, wherein memories are stored in an Euclidean key space @math . An LSTM controller performs read and write via specialized read and write heads. It can move a head by either providing a new address in the key space (aka random access) or moving from its previous position via a Lie group action (aka Lie access). In this way, the "L" and "R" instructions of a traditional Turing Machine are generalized to arbitrary elements of a fixed Lie group action. For this reason, we name this new model the Lie Access Neural Turing Machine, or LANTM. We tested two different configurations of LANTM against an LSTM baseline in several basic experiments. We found the right configuration of LANTM to outperform the baseline in all of our experiments. In particular, we trained LANTM on addition of @math -digit numbers for @math , but it was able to generalize almost perfectly to @math , all with the number of parameters 2 orders of magnitude below the LSTM baseline.