Learning sums of powers of low-degree polynomials in the non-degenerate case
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We develop algorithms for writing a polynomial as sums of powers of low degree polynomials. Consider an n-variate degree-d polynomial f which can be written as f = c_1Q_1^m + … + c_s Q_s^m, where each c_i∈𝔽^×, Q_i is a homogeneous polynomial of degree t, and t m = d. In this paper, we give a poly((ns)^t)-time learning algorithm for finding the Q_i's given (black-box access to) f, if the Q_i's satisfy certain non-degeneracy conditions and n is larger than d^2. The set of degenerate Q_i's (i.e., inputs for which the algorithm does not work) form a non-trivial variety and hence if the Q_i's are chosen according to any reasonable (full-dimensional) distribution, then they are non-degenerate with high probability (if s is not too large). Our algorithm is based on a scheme for obtaining a learning algorithm for an arithmetic circuit model from a lower bound for the same model, provided certain non-degeneracy conditions hold. The scheme reduces the learning problem to the problem of decomposing two vector spaces under the action of a set of linear operators, where the spaces and the operators are derived from the input circuit and the complexity measure used in a typical lower bound proof. The non-degeneracy conditions are certain restrictions on how the spaces decompose.
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