Hitting sets and reconstruction for dense orbits in VPe and ΣΠΣ circuits

Abstract

In this paper we study polynomials in VPe (polynomial-sized formulas) and in ΣΠΣ (polynomial-size depth-3 circuits) whose orbits, under the action of the affine group GLaffn (F) (the action of (A, b) ∈ GLaffn (F) on a polynomial f ∈ F[x] is defined as (A, b) ο f = f (AT x + b)), are dense in their ambient class. We construct hitting sets and interpolating sets for these orbits as well as give reconstruction algorithms. Specifically, we obtain the following results: 1. For [EQUATION], where the ℓis are linearly independent linear functions, we construct a polynomial-sized interpolating set, and give a polynomial-time reconstruction algorithm. By a result of Bringmann, Ikenmeyer and Zuiddam, the set of all such polynomials is dense in VPe [14], thus our construction gives the first polynomial-size interpolating set for a dense subclass of VPe. 2. For polynomials of the form ANFΔ (ℓ1(x),..., ℓ4Δ(x)) where ANFΔ(x) is the canonical read-once formula in alternating normal form, of depth 2Δ, and the ℓis are linearly independent linear functions, we provide a quasipolynomial-size interpolating set. We also observe that the reconstruction algorithm of [35] works for all polynomials in this class. This class is also dense in VPe. 3. Similarly, we give a quasipolynomial-sized hitting set for read-once formulas (not necessarily in alternating normal form) composed with a set of linearly independent linear functions. This gives another dense class in VPe. 4. We give a quasipolynomial-sized hitting set for polynomials of the form f(ℓ1(x),..., ℓm(x)), where f is an m-variate s-sparse polynomial. and the ℓis are linearly independent linear functions in n ≥ m variables. This class is dense in ΣΠΣ. 5. For polynomials of the form Σsi=1 Πdj=1 ℓi,j (x), where the ℓi,js are linearly independent linear functions, we construct a polynomial-sized interpolating set. We also observe that the reconstruction algorithm of [45] works for every polynomial in the class. This class is dense in ΣΠΣ. As VP = VNC2, our results for VPe translate immediately to VP with a quasipolynomial blow up in parameters. If any of our hitting or interpolating sets could be made robust then this would immediately yield a hitting set for the superclass in which the relevant class is dense, and as a consequence also a lower bound for the superclass. Unfortunately, we also prove that the kind of constructions that we have found (which are defined in terms of k-independent polynomial maps) do not necessarily yield robust hitting sets.