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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

We show that for every homogeneous polynomial of degree d, if it has determinantal complexity at most s, then it can be computed by a homogeneous algebraic branching program (ABP) of size at most O(d⁵s). Moreover, we show that for most homogeneous polynomials, the width of the resulting homogeneous ABP is just s-1 and the size is at most O(ds).
Thus, for constant degree homogeneous polynomials, their determinantal complexity and ABP complexity are within a constant factor of each other and hence, a super-linear lower bound for ABPs for any constant degree polynomial implies a super-linear lower bound on determinantal complexity; this relates two open problems of great interest in algebraic complexity. As of now, super-linear lower bounds for ABPs are known only for polynomials of growing degree [Mrinal Kumar, 2019; Prerona Chatterjee et al., 2022], and for determinantal complexity the best lower bounds are larger than the number of variables only by a constant factor [Mrinal Kumar and Ben Lee Volk, 2022].
While determinantal complexity and ABP complexity are classically known to be polynomially equivalent [Meena Mahajan and V. Vinay, 1997], the standard transformation from the former to the latter incurs a polynomial blow up in size in the process, and thus, it was unclear if a super-linear lower bound for ABPs implies a super-linear lower bound on determinantal complexity. In particular, a size preserving transformation from determinantal complexity to ABPs does not appear to have been known prior to this work, even for constant degree polynomials.

Abhranil Chatterjee, Mrinal Kumar, and Ben Lee Volk. Determinants vs. Algebraic Branching Programs. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 27:1-27:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chatterjee_et_al:LIPIcs.ITCS.2024.27, author = {Chatterjee, Abhranil and Kumar, Mrinal and Volk, Ben Lee}, title = {{Determinants vs. Algebraic Branching Programs}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {27:1--27:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.27}, URN = {urn:nbn:de:0030-drops-195550}, doi = {10.4230/LIPIcs.ITCS.2024.27}, annote = {Keywords: Determinant, Algebraic Branching Program, Lower Bounds, Singular Variety} }

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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

We give reconstruction algorithms for subclasses of depth-3 arithmetic circuits. In particular, we obtain the first efficient algorithm for finding tensor rank, and an optimal tensor decomposition as a sum of rank-one tensors, when given black-box access to a tensor of super-constant rank. Specifically, we obtain the following results:
1) A deterministic algorithm that reconstructs polynomials computed by Σ^{[k]}⋀^{[d]}Σ circuits in time poly(n,d,c) ⋅ poly(k)^{k^{k^{10}}},
2) A randomized algorithm that reconstructs polynomials computed by multilinear Σ^{[k]}∏^{[d]}Σ circuits in time poly(n,d,c) ⋅ k^{k^{k^{k^{O(k)}}}},
3) A randomized algorithm that reconstructs polynomials computed by set-multilinear Σ^{[k]}∏^{[d]}Σ circuits in time poly(n,d,c) ⋅ k^{k^{k^{k^{O(k)}}}},
where c = log q if 𝔽 = 𝔽_q is a finite field, and c equals the maximum bit complexity of any coefficient of f if 𝔽 is infinite.
Prior to our work, polynomial time algorithms for the case when the rank, k, is constant, were given by Bhargava, Saraf and Volkovich [Vishwas Bhargava et al., 2021].
Another contribution of this work is correcting an error from a paper of Karnin and Shpilka [Zohar Shay Karnin and Amir Shpilka, 2009] (with some loss in parameters) that also affected Theorem 1.6 of [Vishwas Bhargava et al., 2021]. Consequently, the results of [Zohar Shay Karnin and Amir Shpilka, 2009; Vishwas Bhargava et al., 2021] continue to hold, with a slightly worse setting of parameters. For fixing the error we systematically study the relation between syntactic and semantic notions of rank of Σ Π Σ circuits, and the corresponding partitions of such circuits.
We obtain our improved running time by introducing a technique for learning rank preserving coordinate-subspaces. Both [Zohar Shay Karnin and Amir Shpilka, 2009] and [Vishwas Bhargava et al., 2021] tried all choices of finding the "correct" coordinates, which, due to the size of the set, led to having a fast growing function of k at the exponent of n. We manage to find these spaces in time that is still growing fast with k, yet it is only a fixed polynomial in n.

Shir Peleg, Amir Shpilka, and Ben Lee Volk. Tensor Reconstruction Beyond Constant Rank. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 87:1-87:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{peleg_et_al:LIPIcs.ITCS.2024.87, author = {Peleg, Shir and Shpilka, Amir and Volk, Ben Lee}, title = {{Tensor Reconstruction Beyond Constant Rank}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {87:1--87:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.87}, URN = {urn:nbn:de:0030-drops-196157}, doi = {10.4230/LIPIcs.ITCS.2024.87}, annote = {Keywords: Algebraic circuits, reconstruction, tensor decomposition, tensor rank} }

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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

The stabilizer rank of a quantum state ψ is the minimal r such that |ψ⟩ = ∑_{j = 1}^r c_j |φ_j⟩ for c_j ∈ ℂ and stabilizer states φ_j. The running time of several classical simulation methods for quantum circuits is determined by the stabilizer rank of the n-th tensor power of single-qubit magic states.
We prove a lower bound of Ω(n) on the stabilizer rank of such states, improving a previous lower bound of Ω(√n) of Bravyi, Smith and Smolin [Bravyi et al., 2016]. Further, we prove that for a sufficiently small constant δ, the stabilizer rank of any state which is δ-close to those states is Ω(√n/log n). This is the first non-trivial lower bound for approximate stabilizer rank.
Our techniques rely on the representation of stabilizer states as quadratic functions over affine subspaces of 𝔽₂ⁿ, and we use tools from analysis of boolean functions and complexity theory. The proof of the first result involves a careful analysis of directional derivatives of quadratic polynomials, whereas the proof of the second result uses Razborov-Smolensky low degree polynomial approximations and correlation bounds against the majority function.

Shir Peleg, Ben Lee Volk, and Amir Shpilka. Lower Bounds on Stabilizer Rank. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 110:1-110:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{peleg_et_al:LIPIcs.ITCS.2022.110, author = {Peleg, Shir and Volk, Ben Lee and Shpilka, Amir}, title = {{Lower Bounds on Stabilizer Rank}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {110:1--110:4}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.110}, URN = {urn:nbn:de:0030-drops-157063}, doi = {10.4230/LIPIcs.ITCS.2022.110}, annote = {Keywords: Quantum Computation, Lower Bounds, Stabilizer rank, Simulation of Quantum computers} }

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**Published in:** LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

The determinantal complexity of a polynomial P ∈ 𝔽[x₁, …, x_n] over a field 𝔽 is the dimension of the smallest matrix M whose entries are affine functions in 𝔽[x₁, …, x_n] such that P = Det(M). We prove that the determinantal complexity of the polynomial ∑_{i = 1}^n x_i^n is at least 1.5n - 3.
For every n-variate polynomial of degree d, the determinantal complexity is trivially at least d, and it is a long standing open problem to prove a lower bound which is super linear in max{n,d}. Our result is the first lower bound for any explicit polynomial which is bigger by a constant factor than max{n,d}, and improves upon the prior best bound of n + 1, proved by Alper, Bogart and Velasco [Jarod Alper et al., 2017] for the same polynomial.

Mrinal Kumar and Ben Lee Volk. A Lower Bound on Determinantal Complexity. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 4:1-4:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kumar_et_al:LIPIcs.CCC.2021.4, author = {Kumar, Mrinal and Volk, Ben Lee}, title = {{A Lower Bound on Determinantal Complexity}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {4:1--4:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-193-1}, ISSN = {1868-8969}, year = {2021}, volume = {200}, editor = {Kabanets, Valentine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.4}, URN = {urn:nbn:de:0030-drops-142781}, doi = {10.4230/LIPIcs.CCC.2021.4}, annote = {Keywords: Determinantal Complexity, Algebraic Circuits, Lower Bounds, Singular Variety} }

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**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

We show that there is an equation of degree at most poly(n) for the (Zariski closure of the) set of the non-rigid matrices: that is, we show that for every large enough field 𝔽, there is a non-zero n²-variate polynomial P ∈ 𝔽[x_{1, 1}, …, x_{n, n}] of degree at most poly(n) such that every matrix M which can be written as a sum of a matrix of rank at most n/100 and a matrix of sparsity at most n²/100 satisfies P(M) = 0. This confirms a conjecture of Gesmundo, Hauenstein, Ikenmeyer and Landsberg [Fulvio Gesmundo et al., 2016] and improves the best upper bound known for this problem down from exp(n²) [Abhinav Kumar et al., 2014; Fulvio Gesmundo et al., 2016] to poly(n).
We also show a similar polynomial degree bound for the (Zariski closure of the) set of all matrices M such that the linear transformation represented by M can be computed by an algebraic circuit with at most n²/200 edges (without any restriction on the depth). As far as we are aware, no such bound was known prior to this work when the depth of the circuits is unbounded.
Our methods are elementary and short and rely on a polynomial map of Shpilka and Volkovich [Amir Shpilka and Ilya Volkovich, 2015] to construct low degree "universal" maps for non-rigid matrices and small linear circuits. Combining this construction with a simple dimension counting argument to show that any such polynomial map has a low degree annihilating polynomial completes the proof.
As a corollary, we show that any derandomization of the polynomial identity testing problem will imply new circuit lower bounds. A similar (but incomparable) theorem was proved by Kabanets and Impagliazzo [Valentine Kabanets and Russell Impagliazzo, 2004].

Mrinal Kumar and Ben Lee Volk. A Polynomial Degree Bound on Equations for Non-Rigid Matrices and Small Linear Circuits. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 9:1-9:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kumar_et_al:LIPIcs.ITCS.2021.9, author = {Kumar, Mrinal and Volk, Ben Lee}, title = {{A Polynomial Degree Bound on Equations for Non-Rigid Matrices and Small Linear Circuits}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {9:1--9:9}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.9}, URN = {urn:nbn:de:0030-drops-135486}, doi = {10.4230/LIPIcs.ITCS.2021.9}, annote = {Keywords: Rigid Matrices, Linear Circuits, Degree Bounds, Circuit Lower Bounds} }

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**Published in:** LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

We show that any Algebraic Branching Program (ABP) computing the polynomial ∑_{i=1}^n xⁿ_i has at least Ω(n²) vertices. This improves upon the lower bound of Ω(nlog n), which follows from the classical result of Baur and Strassen [Volker Strassen, 1973; Walter Baur and Volker Strassen, 1983], and extends the results of Kumar [Mrinal Kumar, 2019], which showed a quadratic lower bound for homogeneous ABPs computing the same polynomial.
Our proof relies on a notion of depth reduction which is reminiscent of similar statements in the context of matrix rigidity, and shows that any small enough ABP computing the polynomial ∑_{i=1}^n xⁿ_i can be depth reduced to essentially a homogeneous ABP of the same size which computes the polynomial ∑_{i=1}^n xⁿ_i + ε(𝐱), for a structured "error polynomial" ε(𝐱). To complete the proof, we then observe that the lower bound in [Mrinal Kumar, 2019] is robust enough and continues to hold for all polynomials ∑_{i=1}^n xⁿ_i + ε(𝐱), where ε(𝐱) has the appropriate structure.

Prerona Chatterjee, Mrinal Kumar, Adrian She, and Ben Lee Volk. A Quadratic Lower Bound for Algebraic Branching Programs. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 2:1-2:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chatterjee_et_al:LIPIcs.CCC.2020.2, author = {Chatterjee, Prerona and Kumar, Mrinal and She, Adrian and Volk, Ben Lee}, title = {{A Quadratic Lower Bound for Algebraic Branching Programs}}, booktitle = {35th Computational Complexity Conference (CCC 2020)}, pages = {2:1--2:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-156-6}, ISSN = {1868-8969}, year = {2020}, volume = {169}, editor = {Saraf, Shubhangi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.2}, URN = {urn:nbn:de:0030-drops-125546}, doi = {10.4230/LIPIcs.CCC.2020.2}, annote = {Keywords: Algebraic Branching Programs, Lower Bound} }

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**Published in:** LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

We study the problem of constructing explicit families of matrices which cannot be expressed as a product of a few sparse matrices. In addition to being a natural mathematical question on its own, this problem appears in various incarnations in computer science; the most significant being in the context of lower bounds for algebraic circuits which compute linear transformations, matrix rigidity and data structure lower bounds.
We first show, for every constant d, a deterministic construction in time exp(n^(1-Ω(1/d))) of a family {M_n} of n × n matrices which cannot be expressed as a product M_n = A_1 ⋯ A_d where the total sparsity of A_1,…,A_d is less than n^(1+1/(2d)). In other words, any depth-d linear circuit computing the linear transformation M_n⋅ 𝐱 has size at least n^(1+Ω(1/d)). This improves upon the prior best lower bounds for this problem, which are barely super-linear, and were obtained by a long line of research based on the study of super-concentrators (albeit at the cost of a blow up in the time required to construct these matrices).
We then outline an approach for proving improved lower bounds through a certain derandomization problem, and use this approach to prove asymptotically optimal quadratic lower bounds for natural special cases, which generalize many of the common matrix decompositions.

Mrinal Kumar and Ben Lee Volk. Lower Bounds for Matrix Factorization. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kumar_et_al:LIPIcs.CCC.2020.5, author = {Kumar, Mrinal and Volk, Ben Lee}, title = {{Lower Bounds for Matrix Factorization}}, booktitle = {35th Computational Complexity Conference (CCC 2020)}, pages = {5:1--5:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-156-6}, ISSN = {1868-8969}, year = {2020}, volume = {169}, editor = {Saraf, Shubhangi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.5}, URN = {urn:nbn:de:0030-drops-125578}, doi = {10.4230/LIPIcs.CCC.2020.5}, annote = {Keywords: Algebraic Complexity, Linear Circuits, Matrix Factorization, Lower Bounds} }

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**Published in:** LIPIcs, Volume 102, 33rd Computational Complexity Conference (CCC 2018)

We prove a lower bound of Omega(n^2/log^2 n) on the size of any syntactically multilinear arithmetic circuit computing some explicit multilinear polynomial f(x_1, ..., x_n). Our approach expands and improves upon a result of Raz, Shpilka and Yehudayoff ([Ran Raz et al., 2008]), who proved a lower bound of Omega(n^{4/3}/log^2 n) for the same polynomial. Our improvement follows from an asymptotically optimal lower bound for a generalized version of Galvin's problem in extremal set theory.

Noga Alon, Mrinal Kumar, and Ben Lee Volk. Unbalancing Sets and an Almost Quadratic Lower Bound for Syntactically Multilinear Arithmetic Circuits. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 11:1-11:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{alon_et_al:LIPIcs.CCC.2018.11, author = {Alon, Noga and Kumar, Mrinal and Volk, Ben Lee}, title = {{Unbalancing Sets and an Almost Quadratic Lower Bound for Syntactically Multilinear Arithmetic Circuits}}, booktitle = {33rd Computational Complexity Conference (CCC 2018)}, pages = {11:1--11:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-069-9}, ISSN = {1868-8969}, year = {2018}, volume = {102}, editor = {Servedio, Rocco A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.11}, URN = {urn:nbn:de:0030-drops-88799}, doi = {10.4230/LIPIcs.CCC.2018.11}, annote = {Keywords: Algebraic Complexity, Multilinear Circuits, Circuit Lower Bounds} }

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**Published in:** LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016)

Read-k oblivious algebraic branching programs are a natural generalization of the well-studied model of read-once oblivious algebraic branching program (ROABPs). In this work, we give an exponential lower bound of exp(n/k^{O(k)}) on the width of any read-k oblivious ABP computing some explicit multilinear polynomial f that is computed by a polynomial size depth-3 circuit. We also study the polynomial identity testing (PIT) problem for this model and obtain a white-box subexponential-time PIT algorithm. The algorithm runs in time 2^{~O(n^{1-1/2^{k-1}})} and needs white box access only to know the order in which the variables appear in the ABP.

Matthew Anderson, Michael A. Forbes, Ramprasad Saptharishi, Amir Shpilka, and Ben Lee Volk. Identity Testing and Lower Bounds for Read-k Oblivious Algebraic Branching Programs. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 30:1-30:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{anderson_et_al:LIPIcs.CCC.2016.30, author = {Anderson, Matthew and Forbes, Michael A. and Saptharishi, Ramprasad and Shpilka, Amir and Volk, Ben Lee}, title = {{Identity Testing and Lower Bounds for Read-k Oblivious Algebraic Branching Programs}}, booktitle = {31st Conference on Computational Complexity (CCC 2016)}, pages = {30:1--30:25}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-008-8}, ISSN = {1868-8969}, year = {2016}, volume = {50}, editor = {Raz, Ran}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.30}, URN = {urn:nbn:de:0030-drops-58255}, doi = {10.4230/LIPIcs.CCC.2016.30}, annote = {Keywords: Algebraic Complexity, Lower Bounds, Derandomization, Polynomial Identity Testing} }

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**Published in:** LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)

In this paper we give subexponential size hitting sets for bounded depth multilinear arithmetic formulas. Using the known relation
between black-box PIT and lower bounds we obtain lower bounds for these models.
For depth-3 multilinear formulas, of size exp(n^delta), we give a hitting set of size exp(~O(n^(2/3 + 2*delta/3))). This implies a lower bound of exp(~Omega(n^(1/2))) for depth-3 multilinear formulas, for some explicit polynomial.
For depth-4 multilinear formulas, of size exp(n^delta), we give a hitting set of size exp(~O(n^(2/3 + 4*delta/3)). This implies a lower bound of exp(~Omega(n^(1/4))) for depth-4 multilinear formulas, for some explicit polynomial.
A regular formula consists of alternating layers of +,* gates, where all gates at layer i have the same fan-in. We give a
hitting set of size (roughly) exp(n^(1-delta)), for regular depth-d multilinear formulas of size exp(n^delta), where delta = O(1/sqrt(5)^d)). This result implies a lower bound of roughly exp(~Omega(n^(1/sqrt(5)^d))) for such formulas.
We note that better lower bounds are known for these models, but also that none of these bounds was achieved via construction of
a hitting set. Moreover, no lower bound that implies such PIT results, even in the white-box model, is currently known.
Our results are combinatorial in nature and rely on reducing the underlying formula, first to a depth-4 formula, and then to a
read-once algebraic branching program (from depth-3 formulas we go straight to read-once algebraic branching programs).

Rafael Oliveira, Amir Shpilka, and Ben Lee Volk. Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 304-322, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{oliveira_et_al:LIPIcs.CCC.2015.304, author = {Oliveira, Rafael and Shpilka, Amir and Volk, Ben Lee}, title = {{Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas}}, booktitle = {30th Conference on Computational Complexity (CCC 2015)}, pages = {304--322}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-81-1}, ISSN = {1868-8969}, year = {2015}, volume = {33}, editor = {Zuckerman, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.304}, URN = {urn:nbn:de:0030-drops-50548}, doi = {10.4230/LIPIcs.CCC.2015.304}, annote = {Keywords: Arithmetic Circuits, Derandomization, Polynomial Identity Testing} }

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