Learning the Coefficients: A Presentable Version of Border Complexity and Applications to Circuit Factoring
Pages 130 - 140
Abstract
The border, or the approximative, model of algebraic computation (VP) is quite popular due to the Geometric Complexity Theory (GCT) approach to P≠NP conjecture, and its complex analytic origins. On the flip side, the definition of the border is inherently existential in the field constants that the model employs. In particular, a poly-size border circuit C(ε, x) cannot be compactly presented in reality, as the limit parameter ε may require exponential precision. In this work we resolve this issue by giving a constructive, or a presentable, version of border circuits and state its applications. We make border presentable by restricting the circuit C to use only those constants, in the function field Fq(ε), that it can generate by the ring operations on {ε}∪Fq, and their division, within poly-size circuit. This model is more expressive than VP as it affords exponential-degree in ε; and analogous to the usual border, we define new border classes called VPε and VNPε. We prove that both these (now called presentable border) classes lie in VNP. Such a ’debordering’ result is not known for the classical border classes VP and respectively for VNP. We pose VPε=VP as a new conjecture to study the border. The heart of our technique is a newly formulated exponential interpolation over a finite field, to bound the Boolean complexity of the coefficients before deducing the algebraic complexity. It attacks two factorization problems which were open before. We make progress on (Conj.8.3 in Bürgisser 2000, FOCS 2001) and solve (Conj.2.1 in Bürgisser 2000; Chou,Kumar,Solomon CCC 2018) over all finite fields: 1. Each poly-degree irreducible factor, with multiplicity coprime to field characteristic, of a poly-size circuit (of possibly exponential-degree), is in VNP. 2. For all finite fields, and all factors, VNP is closed under factoring. Consequently, factors of VP are always in VNP. The prime characteristic cases were open before due to the inseparability obstruction (i.e. when the multiplicity is not coprime to q).
References
[1]
Manindra Agrawal, Sumanta Ghosh, and Nitin Saxena. 2019. Bootstrapping variables in algebraic circuits. Proc. Natl. Acad. Sci. USA, 116, 17 (2019), 8107–8118. issn:0027-8424 https://doi.org/10.1073/pnas.1901272116
[2]
Eric Allender and Fengming Wang. 2016. On the power of algebraic branching programs of width two. Comput. Complexity, 25, 1 (2016), 217–253. issn:1016-3328 https://doi.org/10.1007/s00037-015-0114-7
[3]
Michael Ben-Or and Richard Cleve. 1992. Computing algebraic formulas using a constant number of registers. SIAM J. Comput., 21, 1 (1992), 54–58. issn:0097-5397 https://doi.org/10.1137/0221006
[4]
C.S. Bhargav, Prateek Dwivedi, and Nitin Saxena. 2024. Learning the Coefficients: A Presentable Version of Border Complexity and Applications to Circuit Factoring. https://cse.iitk.ac.in/users/nitin/papers/PresentableVNP.pdf
[5]
Vishwas Bhargava, Shubhangi Saraf, and Ilya Volkovich. 2020. Deterministic factorization of sparse polynomials with bounded individual degree. J. ACM, 67, 2 (2020), Art. 8, 28. issn:0004-5411 https://doi.org/10.1145/3365667
[6]
D. Bini. 1980. Relations between exact and approximate bilinear algorithms. Applications. Calcolo. A Quarterly on Numerical Analysis and Theory of Computation, 17, 1 (1980), 87–97. issn:0008-0624 https://doi.org/10.1007/BF02575865
[7]
Dario Bini, Milvio Capovani, Francesco Romani, and Grazia Lotti. 1979. O(n^ 2.7799) complexity for n × n approximate matrix multiplication. Inform. Process. Lett., 8, 5 (1979), 234–235. issn:0020-0190,1872-6119 https://doi.org/10.1016/0020-0190(79)90113-3
[8]
Pranav Bisht and Nitin Saxena. 2021. Blackbox identity testing for sum of special ROABPs and its border class. Comput. Complexity, 30, 1 (2021), Paper No. 8, 48. issn:1016-3328 https://doi.org/10.1007/s00037-021-00209-y
[9]
Markus Bläser, Christian Ikenmeyer, Meena Mahajan, Anurag Pandey, and Nitin Saurabh. 2020. Algebraic Branching Programs, Border Complexity, and Tangent Spaces. In 2020. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 21:1–21:24. https://doi.org/10.4230/LIPICS.CCC.2020.21
[10]
Markus Bläser and Christian Ikenmeyer. 2018. Introduction to geometric complexity theory. https://www.dcs.warwick.ac.uk/~u2270030/teaching_sb/summer17/introtogct/gct.pdf Lecture Notes
[11]
Karl Bringmann, Christian Ikenmeyer, and Jeroen Zuiddam. 2018. On algebraic branching programs of small width. J. ACM, 65, 5 (2018), Art. 32, 29. issn:0004-5411 https://doi.org/10.1145/3209663
[12]
Peter Bürgisser, Michael Clausen, and Mohammad Amin Shokrollahi. 1997. Algebraic complexity theory (Grundlehren der mathematischen Wissenschaften, Vol. 315). Springer. isbn:3-540-60582-7 https://doi.org/10.1007/978-3-662-03338-8
[13]
Peter Bürgisser. 2000. Completeness and reduction in algebraic complexity theory (Algorithms and computation in mathematics, Vol. 7). Springer-Verlag. isbn:3-540-66752-0 https://doi.org/10.1007/978-3-662-04179-6
[14]
Peter Bürgisser. 2004. The complexity of factors of multivariate polynomials. Foundations of Computational Mathematics, 4, 4 (2004), 369–396. issn:1615-3375,1615-3383 https://doi.org/10.1007/s10208-002-0059-5 2001
[15]
Peter Bürgisser. 2020. Correction to: The complexity of factors of multivariate polynomials. Foundations of Computational Mathematics, 20, 6 (2020), 1667–1668. issn:1615-3375,1615-3383 https://doi.org/10.1007/s10208-020-09477-6
[16]
Prasad Chaugule and Nutan Limaye. 2022. On the closures of monotone algebraic classes and variants of the determinant. In LATIN 2022: theoretical informatics (Lecture Notes in Comput. Sci., Vol. 13568). Springer, 610–625. https://doi.org/10.1007/978-3-031-20624-5_37
[17]
Xi Chen, Neeraj Kayal, and Avi Wigderson. 2010. Partial derivatives in arithmetic complexity and beyond. Found. Trends Theor. Comput. Sci., 6, 1-2 (2010), issn:1551-305X,1551-3068 https://doi.org/10.1561/0400000043
[18]
Mahdi Cheraghchi, Elena Grigorescu, Brendan Juba, Karl Wimmer, and Ning Xie. 2018. AC^0∘ MOD_2 lower bounds for the Boolean inner product. J. Comput. System Sci., 97 (2018), 45–59. issn:0022-0000 https://doi.org/10.1016/j.jcss.2018.04.006
[19]
Benny Chor and Ronald L. Rivest. 1988. A knapsack-type public key cryptosystem based on arithmetic in finite fields. IEEE Trans. Inform. Theory, 34, 5, part 1 (1988), 901–909. issn:0018-9448 https://doi.org/10.1109/18.21214
[20]
Chi-Ning Chou, Mrinal Kumar, and Noam Solomon. 2019. Closure of VP under taking factors: a short and simple proof. arxiv:1903.02366. arxiv:1903.02366
[21]
Chi-Ning Chou, Mrinal Kumar, and Noam Solomon. 2019. Closure results for polynomial factorization. Theory of Computing. An Open Access Journal, 15 (2019), Paper No. 13, 34. issn:1557-2862 https://doi.org/10.4086/toc.2019.v015a013 2018
[22]
Don Coppersmith and Shmuel Winograd. 1990. Matrix multiplication via arithmetic progressions. Journal of Symbolic Computation, 9, 3 (1990), 251–280. issn:0747-7171,1095-855X https://doi.org/10.1016/S0747-7171(08)80013-2
[23]
Pranjal Dutta, Prateek Dwivedi, and Nitin Saxena. 2021. Demystifying the border of depth-3 algebraic circuits. In 2021. IEEE, 92–103. https://doi.org/10.1109/FOCS52979.2021.00018
[24]
Pranjal Dutta and Nitin Saxena. 2022. Separated borders: Exponential-gap fanin-hierarchy theorem for approximative depth-3 circuits. In 2022. IEEE, 200–211. https://doi.org/10.1109/FOCS54457.2022.00026
[25]
Pranjal Dutta, Nitin Saxena, and Amit Sinhababu. 2022. Discovering the Roots: Uniform Closure Results for Algebraic Classes Under Factoring. J. ACM, 69, 3 (2022), 18:1–18:39. https://doi.org/10.1145/3510359
[26]
Zeev Dvir, Amir Shpilka, and Amir Yehudayoff. 2009/10. Hardness-randomness tradeoffs for bounded depth arithmetic circuits. SIAM J. Comput., 39, 4 (2009/10), 1279–1293. issn:0097-5397,1095-7111 https://doi.org/10.1137/080735850
[27]
Michael Forbes. 2016. Some Concrete Questions on the Border Complexity of Polynomials. https://www.youtube.com/watch?v=1HMogQIHT6Q Talk at the Workshop on Algebraic Complexity Theory (WACT) 2016 in Tel Aviv.
[28]
Michael Forbes and Amir Shpilka. 2015. Complexity Theory Column 88: Challenges in Polynomial Factorization1. SIGACT News, 46, 4 (2015), Dec, 32–49. issn:0163-5700 https://doi.org/10.1145/2852040.2852051
[29]
Michael A. Forbes. 2014. Polynomial identity testing of read-once oblivious algebraic branching programs. Ph. D. Dissertation. Massachusetts Institute of Technology, Cambridge, MA, USA. https://hdl.handle.net/1721.1/89843
[30]
Michael A. Forbes and Amir Shpilka. 2018. A PSPACE construction of a hitting set for the closure of small algebraic circuits. In 2018. ACM. https://doi.org/10.1145/3188745.3188792
[31]
Michael A. Forbes, Amir Shpilka, Iddo Tzameret, and Avi Wigderson. 2021. Proof complexity lower bounds from algebraic circuit complexity. Theory Comput., 17 (2021), Paper No. 10, 88. https://doi.org/10.4086/toc.2021.v017a010
[32]
Bruno Grenet. 2016. Bounded-degree factors of lacunary multivariate polynomials. J. Symbolic Comput., 75 (2016), 171–192. issn:0747-7171 https://doi.org/10.1016/j.jsc.2015.11.013
[33]
Joshua A. Grochow, Ketan D. Mulmuley, and Youming Qiao. 2016. Boundaries of VP and VNP. In 2016 (LIPIcs, Vol. 55). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 34:1–34:14. https://doi.org/10.4230/LIPICS.ICALP.2016.34
[34]
Zeyu Guo, Mrinal Kumar, Ramprasad Saptharishi, and Noam Solomon. 2022. Derandomization from algebraic hardness. SIAM J. Comput., 51, 2 (2022), 315–335. issn:0097-5397 https://doi.org/10.1137/20M1347395
[35]
Venkatesan Guruswami and Madhu Sudan. 1999. Improved decoding of Reed-Solomon and algebraic-geometry codes. IEEE Trans. Inform. Theory, 45, 6 (1999), 1757–1767. issn:0018-9448 https://doi.org/10.1109/18.782097
[36]
Maurice J. Jansen. 2011. Extracting Roots of Arithmetic Circuits by Adapting Numerical Methods. In Innovations in Computer Science - ICS. Tsinghua University Press, 87–100. http://conference.iiis.tsinghua.edu.cn/ICS2011/content/papers/4.html
[37]
Valentine Kabanets and Russell Impagliazzo. 2004. Derandomizing polynomial identity tests means proving circuit lower bounds. Comput. Complexity, 13, 1-2 (2004), 1–46. issn:1016-3328 https://doi.org/10.1007/s00037-004-0182-6
[38]
Erich Kaltofen. 1985. Polynomial-time reductions from multivariate to bi- and univariate integral polynomial factorization. SIAM J. Comput., 14, 2 (1985), 469–489. issn:0097-5397 https://doi.org/10.1137/0214035
[39]
Erich Kaltofen. 1987. Single-Factor Hensel Lifting and Its Application to the Straight-Line Complexity of Certain Polynomials. In 1987. Association for Computing Machinery, 443–452. isbn:0897912217 https://doi.org/10.1145/28395.28443
[40]
Erich Kaltofen. 1989. Factorization of Polynomials Given by Straight-Line Programs. Adv. Comput. Res., 5 (1989), 375–412. https://users.cs.duke.edu/~elk27/bibliography/89/Ka89_slpfac.pdf
[41]
Erich Kaltofen and Pascal Koiran. 2008. Expressing a fraction of two determinants as a determinant. In 2008. ACM, 141–146. https://doi.org/10.1145/1390768.1390790
[42]
Erich Kaltofen and Barry M. Trager. 1990. Computing with polynomials given by black boxes for their evaluations: greatest common divisors, factorization, separation of numerators and denominators. J. Symbolic Comput., 9, 3 (1990), 301–320. issn:0747-7171 https://doi.org/10.1016/S0747-7171(08)80015-6
[43]
Erich L. Kaltofen. 1986. Uniform Closure Properties of P-Computable Functions. In 1986. ACM, 330–337. https://doi.org/10.1145/12130.12163
[44]
Pascal Koiran and Sylvain Perifel. 2011. Interpolation in Valiant’s theory. Comput. Complexity, 20, 1 (2011), 1–20. issn:1016-3328,1420-8954 https://doi.org/10.1007/s00037-011-0002-8
[45]
Swastik Kopparty, Shubhangi Saraf, and Amir Shpilka. 2015. Equivalence of polynomial identity testing and polynomial factorization. Computational Complexity, 24, 2 (2015), 295–331. issn:1016-3328,1420-8954 https://doi.org/10.1007/s00037-015-0102-y
[46]
Mrinal Kumar. 2020. On the Power of Border of Depth-3 Arithmetic Circuits. ACM Trans. Comput. Theory, 12, 1 (2020), 5:1–5:8. https://doi.org/10.1145/3371506
[47]
Mrinal Kumar and Ramprasad Saptharishi. 2019. Hardness-Randomness Tradeoffs for Algebraic Computation. Bull. EATCS, 129 (2019), http://bulletin.eatcs.org/index.php/beatcs/article/view/591/599
[48]
Mrinal Kumar, Ramprasad Saptharishi, and Anamay Tengse. 2019. Near-optimal bootstrapping of hitting sets for algebraic circuits. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM, 639–646. https://doi.org/10.1137/1.9781611975482.40
[49]
J. M. Landsberg. 2017. Geometry and Complexity Theory. Cambridge University Press. https://doi.org/10.1017/9781108183192
[50]
Joseph M. Landsberg and Giorgio Ottaviani. 2015. New lower bounds for the border rank of matrix multiplication. Theory of Computing. An Open Access Journal, 11 (2015), 285–298. issn:1557-2862 https://doi.org/10.4086/toc.2015.v011a011
[51]
H. W. Lenstra, Jr. 1999. Finding small degree factors of lacunary polynomials. In Number theory in progress, Vol. 1 (Zakopane-Kościelisko, 1997). de Gruyter, 267–276. https://doi.org/10.1515/9783110285581.267
[52]
Meena Mahajan. 2014. Algebraic Complexity Classes. Springer International Publishing, 51–75. isbn:978-3-319-05446-9 https://doi.org/10.1007/978-3-319-05446-9_4
[53]
Guillaume Malod. 2003. Polynômes et coefficients. (Polynomials and coefficients). Ph. D. Dissertation. Claude Bernard University Lyon, France. https://tel.archives-ouvertes.fr/tel-00087399
[54]
Guillaume Malod and Natacha Portier. 2008. Characterizing Valiant’s algebraic complexity classes. J. Complex., 24, 1 (2008), 16–38. https://doi.org/10.1016/J.JCO.2006.09.006
[55]
Gary L. Mullen and Daniel Panario. 2013. Introduction to finite fields : Basic properties of finite fields. In Handbook of Finite Fields. CRC Press, 13–31. https://doi.org/10.1201/b15006
[56]
Ketan D. Mulmuley. 2011. On P vs. NP and geometric complexity theory. J. ACM, 58, 2 (2011), Art. 5, 26. issn:0004-5411,1557-735X https://doi.org/10.1145/1944345.1944346
[57]
Ketan D. Mulmuley. 2012. The GCT Program toward the P vs. NP Problem. Commun. ACM, 55, 6 (2012), 98–107. issn:0001-0782 https://doi.org/10.1145/2184319.2184341
[58]
Ketan D. Mulmuley and Milind Sohoni. 2001. Geometric complexity theory. I. An approach to the P vs. NP and related problems. Siam Journal On Computing, 31, 2 (2001), 496–526. issn:0097-5397,1095-7111 https://doi.org/10.1137/S009753970038715X
[59]
Ketan D. Mulmuley and Milind Sohoni. 2008. Geometric complexity theory. II. Towards explicit obstructions for embeddings among class varieties. SIAM J. Comput., 38, 3 (2008), 1175–1206. issn:0097-5397,1095-7111 https://doi.org/10.1137/080718115
[60]
David Mumford. 1976. Algebraic geometry. I. Springer-Verlag. https://link.springer.com/book/9783540586579 Complex projective varieties
[61]
Rafael Oliveira. 2016. Factors of low individual degree polynomials. Comput. Complexity, 25, 2 (2016), 507–561. issn:1016-3328 https://doi.org/10.1007/s00037-016-0130-2
[62]
Rafael Oliveira. 2020. Conditional lower bounds on the spectrahedral representation of explicit hyperbolicity cones. In 2020. ACM, 396–401. https://doi.org/10.1145/3373207.3404010
[63]
Christos H. Papadimitriou. 1994. Computational complexity. Addison-Wesley Publishing Company, Reading, MA. isbn:0-201-53082-1 https://dl.acm.org/doi/abs/10.5555/1074100.1074233
[64]
Sylvain Périfel. 2004. Polynômes donnés par des circuits algébriques et généralisation du modèle de Valiant. École Normal Supérieure de Lyon, France. Master’s Thesis
[65]
Kenneth W. Regan. 2002. Understanding the Mulmuley-Sohoni approach to P vs. NP. Bull. Eur. Assoc. Theor. Comput. Sci. EATCS, 78 (2002), 86–99. issn:0252-9742 https://cse.buffalo.edu/faculty/regan/papers/pdf/Reg02MSFD.pdf
[66]
Ramprasad Saptharishi. 2015. A survey of lower bounds in arithmetic circuit complexity. https://github.com/dasarpmar/lowerbounds-survey Github Survey
[67]
Nitin Saxena. 2009. Progress on polynomial identity testing. Bull. Eur. Assoc. Theor. Comput. Sci. EATCS, 99 (2009), 49–79. issn:0252-9742 https://cse.iitk.ac.in/users/nitin/papers/pit-survey09.pdf
[68]
Nitin Saxena. 2014. Progress on polynomial identity testing-II. In Perspectives in computational complexity (Progr. Comput. Sci. Appl. Logic, Vol. 26). Birkhäuser/Springer, 131–146. isbn:978-3-319-05445-2; 978-3-319-05446-9 https://cse.iitk.ac.in/users/nitin/papers/pit-survey13.pdf
[69]
Nitin Saxena. 2023. Closure of algebraic classes under factoring. https://www.cse.iitk.ac.in/users/nitin/talks/Sep2023-paris.pdf Talk at Recent Trends in Computer Algebra (2023) in Institut Henri Poincaré, Paris.
[70]
Jayalal Sharma and Dinesh K. 2012. Advanced Complexity Theory. https://www.cse.iitm.ac.in/~jayalal/teaching/CS6840/2012/lecture04.pdf Lecture Notes
[71]
Victor Shoup. 2009. A computational introduction to number theory and algebra (second ed.). Cambridge University Press. isbn:978-0-521-51644-0 https://shoup.net/ntb/
[72]
Amir Shpilka and Ilya Volkovich. 2010. On the relation between polynomial identity testing and finding variable disjoint factors. In Automata, languages and programming. Part I (Lecture Notes in Comput. Sci., Vol. 6198). Springer, 408–419. https://doi.org/10.1007/978-3-642-14165-2_35
[73]
Amir Shpilka and Amir Yehudayoff. 2010. Arithmetic Circuits: A Survey of Recent Results and Open Questions. Found. Trends Theor. Comput. Sci., 5, 3–4 (2010), 207–388. issn:1551-305X https://doi.org/10.1561/0400000039
[74]
Amit Sinhababu and Thomas Thierauf. 2021. Factorization of Polynomials Given by Arithmetic Branching Programs. Comput. Complex., 30, 2 (2021), 15. https://doi.org/10.1007/S00037-021-00215-0
[75]
Volker Strassen. 1974. Polynomials with rational coefficients which are hard to compute. Siam Journal On Computing, 3 (1974), 128–149. issn:0097-5397 https://doi.org/10.1137/0203010
[76]
Madhu Sudan. 1997. Decoding of Reed Solomon codes beyond the error-correction bound. J. Complexity, 13, 1 (1997), 180–193. issn:0885-064X https://doi.org/10.1006/jcom.1997.0439
[77]
Madhu Sudan. 1998. Algebra and Computation. https://people.csail.mit.edu/madhu/FT98/ Lecture Notes
[78]
Leslie G. Valiant. 1979. Completeness Classes in Algebra. In 1979. ACM, 249–261. https://doi.org/10.1145/800135.804419
[79]
L. G. Valiant. 1982. Reducibility by algebraic projections. In Logic and algorithmic (Monogr. Enseign. Math., Vol. 30). Univ. Genève, Geneva, 365–380.
[80]
Joachim von zur Gathen. 1984. Hensel and Newton methods in valuation rings. Math. Comp., 42, 166 (1984), 637–661. issn:0025-5718 https://doi.org/10.2307/2007608
[81]
Joachim von zur Gathen and Jürgen Gerhard. 2013. Modern computer algebra (third ed.). Cambridge University Press, Cambridge. isbn:978-1-107-03903-2 https://doi.org/10.1017/CBO9781139856065
[82]
Joachim von zur Gathen and Erich L. Kaltofen. 1985. Factoring Sparse Multivariate Polynomials. J. Comput. Syst. Sci., 31, 2 (1985), 265–287. https://doi.org/10.1016/0022-0000(85)90044-3
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- Learning the Coefficients: A Presentable Version of Border Complexity and Applications to Circuit Factoring
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