  It is well known that recurrent neural networks (RNNs) faced limitations in
learning long-term dependencies that have been addressed by memory structures
in long short-term memory (LSTM) networks. Matrix neural networks feature
matrix representation which inherently preserves the spatial structure of data
and has the potential to provide better memory structures when compared to
canonical neural networks that use vector representation. Neural Turing
machines (NTMs) are novel RNNs that implement notion of programmable computers
with neural network controllers to feature algorithms that have copying,
sorting, and associative recall tasks. In this paper, we study the augmentation
of memory capacity with a matrix representation of RNNs and NTMs (MatNTMs). We
investigate if matrix representation has a better memory capacity than the
vector representations in conventional neural networks. We use a probabilistic
model of the memory capacity using Fisher information and investigate how the
memory capacity for matrix representation networks are limited under various
constraints, and in general, without any constraints. In the case of memory
capacity without any constraints, we found that the upper bound on memory
capacity to be $N^2$ for an $N\times N$ state matrix. The results from our
experiments using synthetic algorithmic tasks show that MatNTMs have a better
learning capacity when compared to its counterparts.
