Learning sums of powers of low-degree polynomials in the non-degenerate case

by   Ankit Garg, et al.

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.



There are no comments yet.


page 1

page 2

page 3

page 4


Faster Algorithms via Waring Decompositions

We show that decompositions of certain polynomials as sums of powers of ...

Efficient reconstruction of depth three circuits with top fan-in two

We develop efficient randomized algorithms to solve the black-box recons...

On top fan-in vs formal degree for depth-3 arithmetic circuits

We show that over the field of complex numbers, every homogeneous polyno...

On the Degree of Boolean Functions as Polynomials over Z_m

Polynomial representations of Boolean functions over various rings such ...

Algorithmic Thresholds for Refuting Random Polynomial Systems

Consider a system of m polynomial equations {p_i(x) = b_i}_i ≤ m of degr...

A Quadratic Lower Bound for Algebraic Branching Programs

We show that any Algebraic Branching Program (ABP) computing the polynom...

Time-Space Tradeoffs for Learning from Small Test Spaces: Learning Low Degree Polynomial Functions

We develop an extension of recently developed methods for obtaining time...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.