  Contemporary wisdom based on empirical studies suggests that standard
recurrent neural networks (RNNs) do not perform well on tasks requiring
long-term memory. However, precise reasoning for this behavior is still
unknown. This paper provides a rigorous explanation of this property in the
special case of linear RNNs. Although this work is limited to linear RNNs, even
these systems have traditionally been difficult to analyze due to their
non-linear parameterization. Using recently-developed kernel regime analysis,
our main result shows that linear RNNs learned from random initializations are
functionally equivalent to a certain weighted 1D-convolutional network.
Importantly, the weightings in the equivalent model cause an implicit bias to
elements with smaller time lags in the convolution and hence, shorter memory.
The degree of this bias depends on the variance of the transition kernel matrix
at initialization and is related to the classic exploding and vanishing
gradients problem. The theory is validated in both synthetic and real data
experiments.
