  Session based model is widely used in recommend system. It use the user click
sequence as input of a Recurrent Neural Network (RNN), and get the output of
the RNN network as the vector embedding of the session, and use the inner
product of the vector embedding of session and the vector embedding of the next
item as the score that is the metric of the interest to the next item. This
method can be used for the "match" stage for the recommendation system whose
item number is very big by using some index method like KD-Tree or Ball-Tree
and etc.. But this method repudiate the variousness of the interest of user in
a session. We generated the model to modify the vector embedding of session to
a symmetric matrix embedding, that is equivalent to a quadratic form on the
vector space of items. The score is builded as the value of the vector
embedding of next item under the quadratic form. The eigenvectors of the
symmetric matrix embedding corresponding to the positive eigenvalues are
conjectured to represent the interests of user in the session. This method can
be used for the "match" stage also. The experiments show that this method is
better than the method of vector embedding.
