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?
