Random arithmetic formulas can be reconstructed efficiently

Gupta, Ankit; Kayal, Neeraj; Qiao, Youming

doi:10.1007/s00037-014-0085-0

Cited by 10 publications

(31 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We obtain hitting sets for both classes and an interpolating set for the second. We also observe that the reconstruction algorithm of [GKQ14] works for the polynomials in the orbit of ROANFs. Although the results that we obtain for the subclass defined by the continuant polynomial are stronger, we think that every such dense subclass can shed more light on VP e and may eventually be used in order to obtain new lower bounds.…”

Section: Introductionmentioning

confidence: 69%

Hitting Sets and Reconstruction for Dense Orbits in $\text{VP}_e$ and $ΣΠΣ$ Circuits

Medini,

Shpilka

2021

Preprint

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 69%

Hitting Sets and Reconstruction for Dense Orbits in $\text{VP}_e$ and $ΣΠΣ$ Circuits

Medini,

Shpilka

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…It has found applications to the elimination of redundant variables [36], the computation of the Lie algebra of a polynomial [37], the reconstruction of random arithmetic formulas [25], full rank algebraic programs [38] and nondegenerate depth 3 circuits [39].…”

Section: From Black Box Pit To Linear Dependenciesmentioning

confidence: 99%

Derandomization and absolute reconstruction for sums of powers of linear forms

Koiran¹,

Skomra²

2019

Preprint

View full text Add to dashboard Cite

We study the decomposition of multivariate polynomials as sums of powers of linear forms. As one of our main results, we give an algorithm for the following problem: given a homogeneous polynomial f (x 1 , . . . , x n ) of degree 3, decide whether it can be written as a sum of cubes of linearly independent linear forms with complex coefficients. Compared to previous algorithms for the same problem, the two main novel features of this algorithm are:(i) It is an algebraic algorithm, i.e., it performs only arithmetic operations and equality tests on the coefficients of the input polynomial f . In particular, it does not make any appeal to polynomial factorization.

show abstract

“…Another way to moderate the reconstruction setup is given by average-case reconstruction. Here the input polynomial f is picked according to some 'natural' distribution on circuits from a class C. This relaxation led to the development of randomized polynomial time reconstruction algorithm for some powerful circuit classes [17,19] (albeit on average), including arithmetic formulas for which we do not know of any super-polynomial lower bound. The notion of average-case reconstruction is related to pseudo-random polynomial Average-case ABP reconstruction: In order to study average-case complexity of the reconstruction problem for ABPs, we need to define a distribution on polynomials computed by ABPs.…”

Section: Circuit Reconstructionmentioning

confidence: 99%

“…, w 1 ), with probability at least 1 − 1 poly(w,d) , where w = max k∈ [d−1] {w k }. In fact, any full rank ABP computing f is 'unique' up to the symmetries 19 of iterated matrix multiplication which we study in Section 6.…”

Section: Definition 4 (Full Rank Algebraic Branching Program)mentioning

confidence: 99%

See 1 more Smart Citation

Reconstruction of Full Rank Algebraic Branching Programs

Kayal

Nair

Saha

et al. 2018

ACM Trans. Comput. Theory

Self Cite

View full text Add to dashboard Cite

An algebraic branching program (ABP) A can be modelled as a product expression X 1 • X 2. .. X d , where X 1 and X d are 1×w and w ×1 matrices respectively, and every other X k is a w ×w matrix; the entries of these matrices are linear forms in m variables over a field F (which we assume to be either Q or a field of characteristic poly(m)). The polynomial computed by A is the entry of the 1 × 1 matrix obtained from the product d k=1 X k. We say A is a full rank ABP if the w 2 (d − 2) + 2w linear forms occurring in the matrices X 1 , X 2 ,. .. , X d are F-linearly independent. Our main result is a randomized reconstruction algorithm for full rank ABPs: Given blackbox access to an m-variate polynomial f of degree at most m, the algorithm outputs a full rank ABP computing f if such an ABP exists, or outputs 'no full rank ABP exists' (with high probability). The running time of the algorithm is polynomial in m and β, where β is the bit length of the coefficients of f. The algorithm works even if X k is a w k−1 × w k matrix (with w 0 = w d = 1), and w = (w 1 ,. .. , w d−1) is unknown.

show abstract

Random arithmetic formulas can be reconstructed efficiently

Cited by 10 publications

References 35 publications

Hitting Sets and Reconstruction for Dense Orbits in $\text{VP}_e$ and $ΣΠΣ$ Circuits

Hitting Sets and Reconstruction for Dense Orbits in $\text{VP}_e$ and $ΣΠΣ$ Circuits

Derandomization and absolute reconstruction for sums of powers of linear forms

Reconstruction of Full Rank Algebraic Branching Programs

Contact Info

Product

Resources

About