Reconstruction of depth-4 multilinear circuits with top fan-in 2

Gupta, Ankit; Kayal, Neeraj; Lokam, Satya

doi:10.1145/2213977.2214035

Cited by 15 publications

(13 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…makes sense to talk about a depth reduction for ΣΠΣΠ circuits as well.) ΣΠΣΠ(r) circuits have received significant attention for the problems of polynomial identity testing and polynomial reconstruction [KMSV10,SV11,GKL12], however prior to this work there were no nontrivial lower bounds for this class of circuits for any value of r ≥ 2.…”

Section: Our Resultsmentioning

confidence: 99%

The Limits of Depth Reduction for Arithmetic Formulas: It's All About the Top Fan-In

Kumar

Saraf²

2015

SIAM J. Comput.

View full text Add to dashboard Cite

In recent years, a very exciting and promising method for proving lower bounds for arithmetic circuits has been proposed. This method combines the method of depth reduction developed in the works of Agrawal-Vinay [AV08], Koiran [Koi12] and Tavenas [Tav13], and the use of the shifted partial derivative complexity measure developed in the works of Kayal [Kay12] and Gupta et al [GKKS13a]. These results inspired a flurry of other beautiful results and strong lower bounds for various classes of arithmetic circuits, in particular a recent work of Kayal et al [KSS13] showing superpolynomial lower bounds for regular arithmetic formulas via an improved depth reduction for these formulas. It was left as an intriguing question if these methods could prove superpolynomial lower bounds for general (homogeneous) arithmetic formulas, and if so this would indeed be a breakthrough in arithmetic circuit complexity.In this paper we study the power and limitations of depth reduction and shifted partial derivatives for arithmetic formulas. We do it via studying the class of depth 4 homogeneous arithmetic circuits. We show: (1) the first superpolynomial lower bounds for the class of homogeneous depth 4 circuits with top fan-in o(log n). The core of our result is to show improved depth reduction for these circuits. This class of circuits has received much attention for the problem of polynomial identity testing. We give the first nontrivial lower bounds for these circuits for any top fan-in ≥ 2. (2) We show that improved depth reduction is not possible when the top fan-in is Ω(log n). In particular this shows that the depth reduction procedure of Koiran and Tavenas [Koi12,Tav13] cannot be improved even for homogeneous formulas, thus strengthening the results of Fournier et al [FLMS13] who showed that depth reduction is tight for circuits, and answering some of the main open questions of [KSS13,FLMS13]. Our results in particular suggest that the method of improved depth reduction and shifted partial derivatives may not be powerful enough to prove superpolynomial lower bounds for (even homogeneous) arithmetic formulas.

show abstract

Section: Our Resultsmentioning

confidence: 99%

The Limits of Depth Reduction for Arithmetic Formulas: It's All About the Top Fan-In

Kumar

Saraf²

2015

SIAM J. Comput.

View full text Add to dashboard Cite

show abstract

“…Very recently Gupta, Kayal and Lokam [20] have used theorem 7 to solve worst-case reconstruction problem for depth-4 multilinear circuits of top fanin two. In a separate work, the authors have also found theorem 4 useful towards a certain version of the reconstruction problem for algebraic branching programs (ABPs).…”

Section: Discussionmentioning

confidence: 99%

Affine projections of polynomials

Kayal

2012

Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing

Self Cite

View full text Add to dashboard Cite

An m-variate polynomial f is said to be an affine projection of some n-variate polynomial g if there exists an n × m matrix A and an n-dimensional vector b such that f (x) = g(Ax + b). In other words, if f can be obtained by replacing each variable of g by an affine combination of the variables occurring in f , then it is said to be an affine projection of g. Given f and g can we determine whether f is an affine projection of g? Some well known problems (such as VP versus VNP and matrix multiplication for example) are instances of this problem.The intention of this paper is to understand the complexity of the corresponding computational problem: given polynomials f and g find A and b such that f = g(Ax + b), if such an (A, b) exists. We first show that this is an NP-hard problem. We then focus our attention on instances where g is a member of some fixed, well known family of polynomials so that the input consists only of the polynomial f (x) having m variables and degree d. We consider the situation where f (x) is given to us as a blackbox (i.e. for any point a ∈ F m we can query the blackbox and obtain f (a) in one step) and devise randomized algorithms with running time poly(mnd) in the following special cases:(1) when f = Permn(Ax + b) and A satisfies rank(A) = n 2 .Here Permn is the permanent polynomial.(2) when f = Detn(Ax + b) and A satisfies rank(A) = n 2 .Here Detn is the determinant polynomial.(3) when f = Pow n,d (Ax + b) and A is a random n × m matrix with d = n Ω(1) . Here Pow n,d is the power-symmetric polynomial of degree d.(4) when f = SPS n,d (Ax + b) and A is a random (nd) × m matrix with n constant. Here SPS n,d is the sum-ofproducts polynomials of degree d with n terms.

show abstract

“…. , w d−1 ), and assume 18 that w k > 1 for every k ∈ [d − 1]. Then the output of the algorithm is a full rank ABP of width w or (w d−1 , w d−2 , .…”

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

confidence: 99%

“…As the linear form pq , it must be that Z ∈ Z w k . Moreover, any Z ∈ Z w k can be used along with the relations Q k+1 = Z • Q k+1 and Q k = −Q k • Z to satisfy Equation (18) and hence also Equations (16) and (17).…”

Section: Lemma 33 (Restated) the Space Wmentioning

confidence: 99%

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

Reconstruction of depth-4 multilinear circuits with top fan-in 2

Cited by 15 publications

References 32 publications

The Limits of Depth Reduction for Arithmetic Formulas: It's All About the Top Fan-In

The Limits of Depth Reduction for Arithmetic Formulas: It's All About the Top Fan-In

Affine projections of polynomials

Reconstruction of Full Rank Algebraic Branching Programs

Contact Info

Product

Resources

About