## Machine learning for circular sequences

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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?

1I 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 – Cameron Chandler – 2020-10-17T09:06:56.607

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Have a look into https://arxiv.org/pdf/1602.07576.pdf

– Graph4Me Consultant – 2020-10-18T15:05:17.147