  Self-attention (SA) based models have recently achieved significant
performance improvements in hybrid and end-to-end automatic speech recognition
(ASR) systems owing to their flexible context modeling capability. However, it
is also known that the accuracy degrades when applying SA to long sequence
data. This is mainly due to the length mismatch between the inference and
training data because the training data are usually divided into short segments
for efficient training. To mitigate this mismatch, we propose a new
architecture, which is a variant of the Gaussian kernel, which itself is a
shift-invariant kernel. First, we mathematically demonstrate that
self-attention with shared weight parameters for queries and keys is equivalent
to a normalized kernel function. By replacing this kernel function with the
proposed Gaussian kernel, the architecture becomes completely shift-invariant
with the relative position information embedded using a frame indexing
technique. The proposed Gaussian kernelized SA was applied to connectionist
temporal classification (CTC) based ASR. An experimental evaluation with the
Corpus of Spontaneous Japanese (CSJ) and TEDLIUM 3 benchmarks shows that the
proposed SA achieves a significant improvement in accuracy (e.g., from 24.0%
WER to 6.0% in CSJ) in long sequence data without any windowing techniques.
