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Reconstruction of Depth 3 Arithmetic Circuits with Top Fan-In 3

Authors Shubhangi Saraf , Devansh Shringi

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ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Circuit complexity
Keywords
  • arithmetic circuits
  • learning
  • reconstruction

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Abstract

In this paper, we give the first subexponential (and in fact quasi-polynomial time) reconstruction algorithm for depth 3 circuits of top fan-in 3 (ΣΠΣ(3) circuits) over the fields ℝ and C. Concretely, we show that given blackbox access to an n-variate polynomial f computed by a ΣΠΣ(3) circuit of size s, there is a randomized algorithm that runs in time quasi-poly(n,s) and outputs a generalized ΣΠΣ(3) circuit computing f. The size s includes the bit complexity of coefficients appearing in the circuit.
Depth 3 circuits of constant fan-in (ΣΠΣ(k) circuits) and closely related models have been extensively studied in the context of polynomial identity testing (PIT). The study of PIT for these models led to an understanding of the structure of identically zero ΣΠΣ(3) circuits and ΣΠΣ(k) circuits using some very elegant connections to discrete geometry, specifically the Sylvester-Gallai Theorem, and colorful and high dimensional variants of them. Despite a lot of progress on PIT for ΣΠΣ(k) circuits and more recently on PIT for depth 4 circuits of bounded top and bottom fan-in, reconstruction algorithms for ΣΠΣ(k) circuits has proven to be extremely challenging. 
In this paper, we build upon the structural results for identically zero ΣΠΣ(3) circuits that bound their rank, and prove stronger structural properties of ΣΠΣ(3) circuits (again using connections to discrete geometry). One such result is a bound on the number of codimension 3 subspaces on which a polynomial computed by an ΣΠΣ(3) circuit can vanish on. Armed with the new structural results, we provide the first reconstruction algorithms for ΣΠΣ(3) circuits over ℝ and C.
Our work extends the work of [Sinha, CCC 2016] who provided a reconstruction algorithm for ΣΠΣ(2) circuits over ℝ and C as well as the works of [Shpilka, STOC 2007] who provided a reconstruction algorithms for ΣΠΣ(2) circuits in the setting of small finite fields, and [Karnin-Shpilka, CCC 2009] who provided reconstruction algorithms for ΣΠΣ(k) circuits in the setting of small finite fields.

Cite As Get BibTex

Shubhangi Saraf and Devansh Shringi. Reconstruction of Depth 3 Arithmetic Circuits with Top Fan-In 3. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 21:1-21:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CCC.2025.21

Author Details

Shubhangi Saraf
  • Department of Mathematics & Department of Computer Science, University of Toronto, Canada
Devansh Shringi
  • Department of Computer Science, University of Toronto, Canada

Funding

  • Saraf, Shubhangi: Research partially supported by an NSERC Discovery Grant and a McLean Award.

Acknowledgements

The authors want to thank Nitin Saxena for helpful discussions.

References

  1. M. Agrawal and V. Vinay. Arithmetic circuits: A chasm at depth four. In Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 67-75, 2008. Google Scholar
  2. D. Angluin. Queries and concept learning. Machine Learning, 2:319-342, 1988. Google Scholar
  3. Boaz Barak, Zeev Dvir, Avi Wigderson, and Amir Yehudayoff. Fractional sylvester-gallai theorems. Proceedings of the National Academy of Sciences, 110(48):19213-19219, 2013. Google Scholar
  4. A. Beimel, F. Bergadano, N. H. Bshouty, E. Kushilevitz, and S. Varricchio. Learning functions represented as multiplicity automata. J. ACM, 47(3):506-530, 2000. URL: https://doi.org/10.1145/337244.337257.
  5. M. Ben-Or and P. Tiwari. A deterministic algorithm for sparse multivariate polynominal interpolation. In Proceedings of the 20th Annual ACM Symposium on Theory of Computing (STOC), pages 301-309, 1988. Google Scholar
  6. Vishwas Bhargava, Ankit Garg, Neeraj Kayal, and Chandan Saha. Learning generalized depth three arithmetic circuits in the non-degenerate case. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022), volume 245 of LIPIcs, pages 21:1-21:22. Schloss-Dagstuhl-Leibniz Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.APPROX/RANDOM.2022.21.
  7. Vishwas Bhargava, Shubhangi Saraf, and Ilya Volkovich. Reconstruction of depth-4 multilinear circuits. SODA 2020, 2020. Google Scholar
  8. Vishwas Bhargava, Shubhangi Saraf, and Ilya Volkovich. Reconstruction algorithms for low-rank tensors and depth-3 multilinear circuits. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 809-822, 2021. URL: https://doi.org/10.1145/3406325.3451096.
  9. Vishwas Bhargava and Devansh Shringi. Faster & deterministic FPT algorithm for worst-case tensor decomposition. Electron. Colloquium Comput. Complex., TR24-123, 2024. URL: https://eccc.weizmann.ac.il/report/2024/123.
  10. William E. Bonnice and Michael Edelstein. Flats associated with finite sets in P^d. Niew. Arch. Wisk, 15:11-14, 1967. Google Scholar
  11. Enrico Carlini. Reducing the number of variables of a polynomial. In Algebraic geometry and geometric modeling, pages 237-247. Springer, 2006. URL: https://doi.org/10.1007/978-3-540-33275-6_15.
  12. Pranjal Dutta, Prateek Dwivedi, and Nitin Saxena. Deterministic Identity Testing Paradigms for Bounded Top-Fanin Depth-4 Circuits. In Valentine Kabanets, editor, 36th Computational Complexity Conference (CCC 2021), volume 200 of Leibniz International Proceedings in Informatics (LIPIcs), pages 11:1-11:27, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2021.11.
  13. Zeev Dvir, Shubhangi Saraf, and Avi Wigderson. Improved rank bounds for design matrices and a new proof of kelly’s theorem. In Forum of Mathematics, Sigma, volume 2, page e4. Cambridge University Press, 2014. Google Scholar
  14. Zeev Dvir and Amir Shpilka. Locally decodable codes with 2 queries and polynomial identity testing for depth 3 circuits. In Proceedings of the Thirty-Seventh Annual ACM Symposium on Theory of Computing, STOC '05, pages 592-601, New York, NY, USA, 2005. Association for Computing Machinery. URL: https://doi.org/10.1145/1060590.1060678.
  15. M. A. Forbes and A. Shpilka. Quasipolynomial-time identity testing of non-commutative and read-once oblivious algebraic branching programs. Electronic Colloquium on Computational Complexity (ECCC), 19:115, 2012. Google Scholar
  16. 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), pages 20:1-20:30. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.CCC.2023.20.
  17. Abhibhav Garg, Rafael Oliveira, and Akash Kumar Sengupta. Robust Radical Sylvester-Gallai Theorem for Quadratics. In Xavier Goaoc and Michael Kerber, editors, 38th International Symposium on Computational Geometry (SoCG 2022), volume 224 of Leibniz International Proceedings in Informatics (LIPIcs), pages 42:1-42:13, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.SoCG.2022.42.
  18. Ankit Garg, Neeraj Kayal, and Chandan Saha. Learning sums of powers of low-degree polynomials in the non-degenerate case. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 889-899. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00087.
  19. A. Gupta, P. Kamath, N. Kayal, and R. Saptharishi. Arithmetic circuits: A chasm at depth three. In Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 578-587, 2013. URL: https://doi.org/10.1109/FOCS.2013.68.
  20. A. Gupta, N. Kayal, and S. V. Lokam. Efficient reconstruction of random multilinear formulas. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS, pages 778-787, 2011. URL: https://doi.org/10.1109/FOCS.2011.70.
  21. A. Gupta, N. Kayal, and S. V. Lokam. Reconstruction of depth-4 multilinear circuits with top fanin 2. In Proceedings of the 44th Annual ACM Symposium on Theory of Computing (STOC), pages 625-642, 2012. Full version at https://eccc.weizmann.ac.il/report/2011/153. Google Scholar
  22. A. Gupta, N. Kayal, and Y. Qiao. Random arithmetic formulas can be reconstructed efficiently. Computational Complexity, 23(2):207-303, 2014. URL: https://doi.org/10.1007/s00037-014-0085-0.
  23. Sten Hansen. A generalization of a theorem of sylvester on the lines determined by a finite point set. Mathematica Scandinavica, 16(2):175-180, 1965. Google Scholar
  24. Zohar S Karnin and Amir Shpilka. Black box polynomial identity testing of generalized depth-3 arithmetic circuits with bounded top fan-in. In 2008 23rd Annual IEEE Conference on Computational Complexity, pages 280-291. IEEE, 2008. Google Scholar
  25. Zohar S Karnin and Amir Shpilka. Reconstruction of generalized depth-3 arithmetic circuits with bounded top fan-in. In 2009 24th Annual IEEE Conference on Computational Complexity, pages 274-285. IEEE, 2009. Google Scholar
  26. N. Kayal, V. Nair, C. Saha, and S. Tavenas. Reconstruction of full rank algebraic branching programs. In 32nd Computational Complexity Conference, CCC 2017., pages 21:1-21:61, 2017. URL: https://doi.org/10.4230/LIPIcs.CCC.2017.21.
  27. N. Kayal and S. Saraf. Blackbox polynomial identity testing for depth 3 circuits. In Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 198-207, 2009. Full version at https://eccc.weizmann.ac.il/report/2009/032. Google Scholar
  28. Neeraj Kayal. Efficient algorithms for some special cases of the polynomial equivalence problem. In Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete algorithms, pages 1409-1421. SIAM, 2011. URL: https://doi.org/10.1137/1.9781611973082.108.
  29. Neeraj Kayal, Vineet Nair, and Chandan Saha. Average-case linear matrix factorization and reconstruction of low width algebraic branching programs. computational complexity, 28:749-828, 2019. URL: https://doi.org/10.1007/S00037-019-00189-0.
  30. Neeraj Kayal and Chandan Saha. Reconstruction of non-degenerate homogeneous depth three circuits. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 413-424, 2019. URL: https://doi.org/10.1145/3313276.3316360.
  31. A. Klivans and A. Shpilka. Learning restricted models of arithmetic circuits. Theory of computing, 2(10):185-206, 2006. URL: https://doi.org/10.4086/TOC.2006.V002A010.
  32. A. Klivans and D. Spielman. Randomness efficient identity testing of multivariate polynomials. In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing (STOC), pages 216-223, 2001. Google Scholar
  33. P. Koiran. Arithmetic circuits: the chasm at depth four gets wider. CoRR, abs/1006.4700, 2010. URL: https://arxiv.org/abs/1006.4700.
  34. Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas. Superpolynomial lower bounds against low-depth algebraic circuits. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 804-814, 2022. URL: https://doi.org/10.1109/FOCS52979.2021.00083.
  35. Rafael Oliveira and Akash Kumar Sengupta. Radical sylvester-gallai theorem for cubics. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 212-220. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00027.
  36. Rafael Oliveira and Akash Kumar Sengupta. Strong algebras and radical sylvester-gallai configurations. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, pages 95-105, 2024. URL: https://doi.org/10.1145/3618260.3649617.
  37. Shir Peleg and Amir Shpilka. Polynomial time deterministic identity testing algorithm for Σ^[3]ΠΣΠ^[2] circuits via Edelstein-Kelly type theorem for quadratic polynomials. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 259-271, 2021. URL: https://doi.org/10.1145/3406325.3451013.
  38. Shir Peleg and Amir Shpilka. A generalized sylvester-gallai-type theorem for quadratic polynomials. In Forum of Mathematics, Sigma, volume 10, page e112. Cambridge University Press, 2022. Google Scholar
  39. Shir Peleg and Amir Shpilka. Robust Sylvester-Gallai Type Theorem for Quadratic Polynomials. In Xavier Goaoc and Michael Kerber, editors, 38th International Symposium on Computational Geometry (SoCG 2022), volume 224 of Leibniz International Proceedings in Informatics (LIPIcs), pages 43:1-43:15, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.SoCG.2022.43.
  40. Shir Peleg, Amir Shpilka, and Ben Lee Volk. Tensor Reconstruction Beyond Constant Rank. In Venkatesan Guruswami, editor, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024), volume 287 of Leibniz International Proceedings in Informatics (LIPIcs), pages 87:1-87:20, Dagstuhl, Germany, 2024. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2024.87.
  41. Shubhangi Saraf and Devansh Shringi. Reconstruction of depth 3 arithmetic circuits with top fan-in 3. Electron. Colloquium Comput. Complex., TR25-008, 2025. URL: https://eccc.weizmann.ac.il/report/2025/008.
  42. Nitin Saxena and Comandur Seshadhri. Blackbox identity testing for bounded top fanin depth-3 circuits: the field doesn't matter. In Proceedings of the forty-third annual ACM symposium on Theory of computing, pages 431-440, 2011. Google Scholar
  43. Nitin Saxena and Comandur Seshadhri. From sylvester-gallai configurations to rank bounds: Improved blackbox identity test for depth-3 circuits. Journal of the ACM (JACM), 60(5):1-33, 2013. URL: https://doi.org/10.1145/2528403.
  44. Jacob T Schwartz. Fast probabilistic algorithms for verification of polynomial identities. Journal of the ACM (JACM), 27(4):701-717, 1980. URL: https://doi.org/10.1145/322217.322225.
  45. Amir Shpilka. Interpolation of depth-3 arithmetic circuits with two multiplication gates. In Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pages 284-293, 2007. URL: https://doi.org/10.1145/1250790.1250833.
  46. Amir Shpilka. Sylvester-gallai type theorems for quadratic polynomials. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 1203-1214, 2019. URL: https://doi.org/10.1145/3313276.3316341.
  47. Gaurav Sinha. Reconstruction of Real Depth-3 Circuits with Top Fan-In 2. In Ran Raz, editor, 31st Conference on Computational Complexity (CCC 2016), volume 50 of Leibniz International Proceedings in Informatics (LIPIcs), pages 31:1-31:53, Dagstuhl, Germany, 2016. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2016.31.
  48. Gaurav Sinha. Efficient reconstruction of depth three arithmetic circuits with top fan-in two. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022), pages 118:1-118:33. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ITCS.2022.118.
  49. S. Tavenas. Improved bounds for reduction to depth 4 and depth 3. In MFCS, pages 813-824, 2013. URL: https://doi.org/10.1007/978-3-642-40313-2_71.
  50. Richard Zippel. Probabilistic algorithms for sparse polynomials. In International symposium on symbolic and algebraic manipulation, pages 216-226. Springer, 1979. URL: https://doi.org/10.1007/3-540-09519-5_73.
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