# Machine learning for circular sequences

My data are sequences of real numbers $$a_0,a_1,...,a_{n-1}$$. The length of a sequence is fixed and equals $$n$$. Each sequence is mapped to a real number $$y$$ and I want to predict $$y$$ given the sequence.

The arrangement of the elements within a sequence is important. However, the sequences are circular, meaning that $$a_0$$ is not the first element, and $$a_{n-1}$$ is not the last one. The sequence $$a_0,a_1,...,a_{n-1}$$ is indistinguishable from the sequence $$a_k, a_{k+1}, ..., a_{n-1}, a_0, ..., a_{k-1}$$: they are mapped to the same $$y$$. Moreover, one can circle them in the opposite direction, thus $$a_{n-1}, a_{n-2},..., a_0$$ maps to the same answer.

I know that recurrent neural networks (RNN) are used for sequences where it is important that the inputs are fed into the network in a specific order. How to make such a network invariant to circular transformations and the change of direction described above?

I don't insist on RNN. Are there any supervised learning algorithms that work with such sequences?

• I don't know of any algorithms that specifically cater to such data, however, you could always feed in the data to a RNN with each batch starting at a random point Oct 17, 2020 at 9:06
• Have a look into arxiv.org/pdf/1602.07576.pdf Oct 18, 2020 at 15:05