Recently, Neural Turing Machine and Memory Networks have shown that adding an external memory can greatly ameliorate a traditional recurrent neural network's tendency to forget after a long period of time. Here we present a new design of an external memory, wherein memories are stored in an Euclidean key space $\mathbb R^n$. An LSTM controller performs read and write via specialized structures called read and write heads, following the design of Neural Turing Machine. 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 is 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. As LANTM is differentiable end-to-end, training was done with RMSProp. We found the right configuration of LANTM to be capable of learning different permutation and arithmetic tasks and extrapolating to at least twice the input size, all with the number of parameters 2 orders of magnitude below that for the LSTM baseline. In particular, we trained LANTM on addition of $k$-digit numbers for $2 \le k \le 16$, but it was able to generalize almost perfectly to $17 \le k \le 32$.
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