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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.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 264, 38th Computational Complexity Conference (CCC 2023)

We prove a higher codimensional radical Sylvester-Gallai type theorem for quadratic polynomials, simultaneously generalizing [Hansen, 1965; Shpilka, 2020]. Hansen’s theorem is a high-dimensional version of the classical Sylvester-Gallai theorem in which the incidence condition is given by high-dimensional flats instead of lines. We generalize Hansen’s theorem to the setting of quadratic forms in a polynomial ring, where the incidence condition is given by radical membership in a high-codimensional ideal. Our main theorem is also a generalization of the quadratic Sylvester-Gallai Theorem of [Shpilka, 2020].
Our work is the first to prove a radical Sylvester-Gallai type theorem for arbitrary codimension k ≥ 2, whereas previous works [Shpilka, 2020; Shir Peleg and Amir Shpilka, 2020; Shir Peleg and Amir Shpilka, 2021; Garg et al., 2022] considered the case of codimension 2 ideals. Our techniques combine algebraic geometric and combinatorial arguments. A key ingredient is a structural result for ideals generated by a constant number of quadratics, showing that such ideals must be radical whenever the quadratic forms are far apart. Using the wide algebras defined in [Garg et al., 2022], combined with results about integral ring extensions and dimension theory, we develop new techniques for studying such ideals generated by quadratic forms. One advantage of our approach is that it does not need the finer classification theorems for codimension 2 complete intersection of quadratics proved in [Shpilka, 2020; Garg et al., 2022].

Abhibhav Garg, Rafael Oliveira, Shir Peleg, and Akash Kumar Sengupta. Radical Sylvester-Gallai Theorem for Tuples of Quadratics. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 20:1-20:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{garg_et_al:LIPIcs.CCC.2023.20, author = {Garg, Abhibhav and Oliveira, Rafael and Peleg, Shir and Sengupta, Akash Kumar}, title = {{Radical Sylvester-Gallai Theorem for Tuples of Quadratics}}, booktitle = {38th Computational Complexity Conference (CCC 2023)}, pages = {20:1--20:30}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-282-2}, ISSN = {1868-8969}, year = {2023}, volume = {264}, editor = {Ta-Shma, Amnon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.20}, URN = {urn:nbn:de:0030-drops-182903}, doi = {10.4230/LIPIcs.CCC.2023.20}, annote = {Keywords: Sylvester-Gallai theorem, arrangements of hypersurfaces, algebraic complexity, polynomial identity testing, algebraic geometry, commutative algebra} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Cohen, Peri and Ta-Shma [Gil Cohen et al., 2021] considered the following question: Assume the vertices of an expander graph are labelled by ± 1. What "test" functions f : {±1}^t → {±1} can or cannot distinguish t independent samples from those obtained by a random walk? [Gil Cohen et al., 2021] considered only balanced labellings, and proved that for all symmetric functions the distinguishability goes down to zero with the spectral gap λ of the expander G. In addition, [Gil Cohen et al., 2021] show that functions computable by AC⁰ circuits are fooled by expanders with vanishing spectral expansion.
We continue the study of this question. We generalize the result to all labelling, not merely balanced ones. We also improve the upper bound on the error of symmetric functions. More importantly, we give a matching lower bound and show a symmetric function with distinguishability going down to zero with λ but not with t. Moreover, we prove a lower bound on the error of functions in AC⁰ in particular, we prove that a random walk on expanders with constant spectral gap does not fool AC⁰.

Gil Cohen, Dor Minzer, Shir Peleg, Aaron Potechin, and Amnon Ta-Shma. Expander Random Walks: The General Case and Limitations. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 43:1-43:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{cohen_et_al:LIPIcs.ICALP.2022.43, author = {Cohen, Gil and Minzer, Dor and Peleg, Shir and Potechin, Aaron and Ta-Shma, Amnon}, title = {{Expander Random Walks: The General Case and Limitations}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {43:1--43:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.43}, URN = {urn:nbn:de:0030-drops-163849}, doi = {10.4230/LIPIcs.ICALP.2022.43}, annote = {Keywords: Expander Graphs, Random Walks, Lower Bounds} }

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**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

In this work we extend the robust version of the Sylvester-Gallai theorem, obtained by Barak, Dvir, Wigderson and Yehudayoff, and by Dvir, Saraf and Wigderson, to the case of quadratic polynomials. Specifically, we prove that if {𝒬} ⊂ ℂ[x₁.…,x_n] is a finite set, |{𝒬}| = m, of irreducible quadratic polynomials that satisfy the following condition
There is δ > 0 such that for every Q ∈ {𝒬} there are at least δ m polynomials P ∈ {𝒬} such that whenever Q and P vanish then so does a third polynomial in {𝒬}⧵{Q,P}.
then dim(span) = Poly(1/δ).
The work of Barak et al. and Dvir et al. studied the case of linear polynomials and proved an upper bound of O(1/δ) on the dimension (in the first work an upper bound of O(1/δ²) was given, which was improved to O(1/δ) in the second work).

Shir Peleg and Amir Shpilka. Robust Sylvester-Gallai Type Theorem for Quadratic Polynomials. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 43:1-43:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{peleg_et_al:LIPIcs.SoCG.2022.43, author = {Peleg, Shir and Shpilka, Amir}, title = {{Robust Sylvester-Gallai Type Theorem for Quadratic Polynomials}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {43:1--43:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.43}, URN = {urn:nbn:de:0030-drops-160515}, doi = {10.4230/LIPIcs.SoCG.2022.43}, annote = {Keywords: Sylvester-Gallai theorem, quadratic polynomials, Algebraic computation} }

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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.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 169, 35th Computational Complexity Conference (CCC 2020)

In this work we prove a version of the Sylvester-Gallai theorem for quadratic polynomials that takes us one step closer to obtaining a deterministic polynomial time algorithm for testing zeroness of Σ^{[3]}ΠΣΠ^{[2]} circuits. Specifically, we prove that if a finite set of irreducible quadratic polynomials 𝒬 satisfy that for every two polynomials Q₁,Q₂ ∈ 𝒬 there is a subset 𝒦 ⊂ 𝒬, such that Q₁,Q₂ ∉ 𝒦 and whenever Q₁ and Q₂ vanish then ∏_{Q_i∈𝒦} Q_i vanishes, then the linear span of the polynomials in 𝒬 has dimension O(1). This extends the earlier result [Amir Shpilka, 2019] that showed a similar conclusion when |𝒦| = 1.
An important technical step in our proof is a theorem classifying all the possible cases in which a product of quadratic polynomials can vanish when two other quadratic polynomials vanish. I.e., when the product is in the radical of the ideal generated by the two quadratics. This step extends a result from [Amir Shpilka, 2019] that studied the case when one quadratic polynomial is in the radical of two other quadratics.

Shir Peleg and Amir Shpilka. A Generalized Sylvester-Gallai Type Theorem for Quadratic Polynomials. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 8:1-8:33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{peleg_et_al:LIPIcs.CCC.2020.8, author = {Peleg, Shir and Shpilka, Amir}, title = {{A Generalized Sylvester-Gallai Type Theorem for Quadratic Polynomials}}, booktitle = {35th Computational Complexity Conference (CCC 2020)}, pages = {8:1--8:33}, 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.2020.8}, URN = {urn:nbn:de:0030-drops-125606}, doi = {10.4230/LIPIcs.CCC.2020.8}, annote = {Keywords: Algebraic computation, Computational complexity, Computational geometry} }

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