{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,5]],"date-time":"2025-10-05T04:19:52Z","timestamp":1759637992057,"version":"3.41.0"},"reference-count":42,"publisher":"Association for Computing Machinery (ACM)","issue":"1","license":[{"start":{"date-parts":[[2018,11,21]],"date-time":"2018-11-21T00:00:00Z","timestamp":1542758400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Comput. Theory"],"published-print":{"date-parts":[[2019,3,31]]},"abstract":"\n An algebraic branching program (ABP) A can be modelled as a product expression\n X<\/jats:italic>\n 1<\/jats:sub>\n amp;middot;\n X<\/jats:italic>\n 2<\/jats:sub>\n \u2026\n \n X\n d<\/jats:sub>\n <\/jats:italic>\n , where\n X<\/jats:italic>\n 1<\/jats:sub>\n and\n \n X\n d<\/jats:sub>\n <\/jats:italic>\n are 1 \u00d7\n w<\/jats:italic>\n and\n w<\/jats:italic>\n \u00d7 1 matrices, respectively, and every other\n \n X\n k<\/jats:sub>\n <\/jats:italic>\n is a\n w<\/jats:italic>\n \u00d7\n w<\/jats:italic>\n matrix; the entries of these matrices are linear forms in\n m<\/jats:italic>\n variables over a field F (which we assume to be either Q or a field of characteristic poly(\n m<\/jats:italic>\n )). The polynomial computed by A is the entry of the 1 \u00d7 1 matrix obtained from the product \u220f\n \n k=1<\/jats:sub>\n <\/jats:italic>\n \n d<\/jats:sup>\n <\/jats:italic>\n \n X\n k<\/jats:sub>\n <\/jats:italic>\n . We say A is a\n full rank<\/jats:italic>\n ABP if the\n w<\/jats:italic>\n 2<\/jats:sup>\n (\n d<\/jats:italic>\n \u2212 2) + 2\n w<\/jats:italic>\n linear forms occurring in the matrices\n X<\/jats:italic>\n 1<\/jats:sub>\n ,\n X<\/jats:italic>\n 2<\/jats:sub>\n , \u2026,\n \n X\n d<\/jats:sub>\n <\/jats:italic>\n are F-linearly independent. Our main result is a randomized reconstruction algorithm for full rank ABPs: Given blackbox access to an\n m<\/jats:italic>\n -variate polynomial\n f<\/jats:italic>\n of degree at most\n m<\/jats:italic>\n , the algorithm outputs a full rank ABP computing\n f<\/jats:italic>\n if such an ABP exists, or outputs \u201cno full rank ABP exists\u201d (with high probability). The running time of the algorithm is polynomial in\n m<\/jats:italic>\n and \u03b2, where \u03b2 is the bit length of the coefficients of\n f<\/jats:italic>\n . The algorithm works even if\n \n X\n k<\/jats:sub>\n <\/jats:italic>\n is a\n \n w\n k\u22121<\/jats:sub>\n <\/jats:italic>\n \u00d7\n \n w\n k<\/jats:sub>\n <\/jats:italic>\n matrix (with\n \n w\n 0<\/jats:sub>\n <\/jats:italic>\n =\n \n w\n d<\/jats:sub>\n <\/jats:italic>\n = 1), and w = (\n w<\/jats:italic>\n 1<\/jats:sub>\n , \u2026,\n \n w\n d \u2212 1<\/jats:sub>\n <\/jats:italic>\n ) is\n unknown<\/jats:italic>\n . The result is obtained by designing a randomized polynomial time equivalence test for the family of iterated matrix multiplication polynomial IMM\n \n w,\n d<\/jats:italic>\n <\/jats:sub>\n , the (1, 1)-th entry of a product of\n d<\/jats:italic>\n rectangular symbolic matrices whose dimensions are according to w \u2208 N\n \n d<\/jats:italic>\n \u22121\n <\/jats:sup>\n . At its core, the algorithm exploits a connection between the\n irreducible invariant subspaces<\/jats:italic>\n of the Lie algebra of the group of symmetries of a polynomial\n f<\/jats:italic>\n that is equivalent to IMM\n \n w,\n d<\/jats:italic>\n <\/jats:sub>\n and the \u201clayer spaces\u201d of a full rank ABP computing\n f<\/jats:italic>\n . This connection also helps determine the group of symmetries of IMM\n \n w,\n d<\/jats:italic>\n <\/jats:sub>\n and show that IMM\n \n w,\n d<\/jats:italic>\n <\/jats:sub>\n is characterized by its group of symmetries.\n <\/jats:p>","DOI":"10.1145\/3282427","type":"journal-article","created":{"date-parts":[[2018,11,26]],"date-time":"2018-11-26T13:07:25Z","timestamp":1543237645000},"page":"1-56","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":["Reconstruction of Full Rank Algebraic Branching Programs"],"prefix":"10.1145","volume":"11","author":[{"given":"Neeraj","family":"Kayal","sequence":"first","affiliation":[{"name":"Microsoft Research, Vigyan, Lavelle Road, Shanthala Nagar, Ashok Nagar, Bengaluru, Karnataka, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vineet","family":"Nair","sequence":"additional","affiliation":[{"name":"Indian Institute of Science, Bengaluru, Karnataka, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chandan","family":"Saha","sequence":"additional","affiliation":[{"name":"Indian Institute of Science, Bengaluru, Karnataka, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"S\u00e9bastien","family":"Tavenas","sequence":"additional","affiliation":[{"name":"Microsoft Research, India and Univ. Savoie Mont Blanc, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2018,11,21]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1007\/11590156_6"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1137\/140975103"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1007\/11672142_8"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1023\/A:1022821128753"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1109\/CCC.2008.22"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1145\/337244.337257"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1002\/j.1538-7305.1967.tb03174.x"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1109\/SFCS.1988.21955"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1981-0606517-5"},{"volume-title":"Algebraic Geometry and Geometric Modelling, Mathematics and Visualization","author":"Carlini Enrico","key":"e_1_2_1_10_1","unstructured":"Enrico Carlini . 2006. Reducing the number of variables of a polynomial . In Algebraic Geometry and Geometric Modelling, Mathematics and Visualization . Springer , 237--247. Enrico Carlini. 2006. Reducing the number of variables of a polynomial. In Algebraic Geometry and Geometric Modelling, Mathematics and Visualization. Springer, 237--247."},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.5555\/2982445.2982455"},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-43948-7_35"},{"key":"e_1_2_1_13_1","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2013.34"},{"key":"e_1_2_1_14_1","doi-asserted-by":"publisher","DOI":"10.1145\/3055399.3055496"},{"key":"e_1_2_1_15_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2016.04.028"},{"key":"e_1_2_1_17_1","volume-title":"Towards an algebraic natural proofs barrier via polynomial identity testing. CoRR abs\/1701.01717","author":"Grochow Joshua A.","year":"2017","unstructured":"Joshua A. 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An exponential lower bound for the sum of powers of bounded degree polynomials . Proceedings of the Electronic Colloquium on Computational Complexity (ECCC\u201912) , vol. 19 , 81. Neeraj Kayal. 2012. An exponential lower bound for the sum of powers of bounded degree polynomials. Proceedings of the Electronic Colloquium on Computational Complexity (ECCC\u201912), vol. 19, 81."},{"key":"e_1_2_1_28_1","volume-title":"Proceedings of the Electronic Colloquium on Computational Complexity (ECCC\u201918)","volume":"25","author":"Kayal Neeraj","year":"2018","unstructured":"Neeraj Kayal , Vineet Nair , and Chandan Saha . 2018 . Average-case linear matrix factorization and reconstruction of low width algebraic branching programs . Proceedings of the Electronic Colloquium on Computational Complexity (ECCC\u201918) , vol. 25 , 29. Retrieved from https:\/\/eccc.weizmann.ac.il\/report\/ 2018\/029. Neeraj Kayal, Vineet Nair, and Chandan Saha. 2018. Average-case linear matrix factorization and reconstruction of low width algebraic branching programs. Proceedings of the Electronic Colloquium on Computational Complexity (ECCC\u201918), vol. 25, 29. Retrieved from https:\/\/eccc.weizmann.ac.il\/report\/2018\/029."},{"key":"e_1_2_1_29_1","doi-asserted-by":"publisher","DOI":"10.1109\/CCC.2013.18"},{"key":"e_1_2_1_30_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-45167-9_34"},{"key":"e_1_2_1_31_1","doi-asserted-by":"publisher","DOI":"10.1145\/380752.380801"},{"key":"e_1_2_1_32_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01457454"},{"key":"e_1_2_1_33_1","article-title":"Determinant: Combinatorics, algorithms, and complexity","volume":"1997","author":"Mahajan Meena","year":"1997","unstructured":"Meena Mahajan and V. Vinay . 1997 . Determinant: Combinatorics, algorithms, and complexity . Chicago J. Theor. Comput. Sci. 1997 , 5 (1997). Meena Mahajan and V. Vinay. 1997. 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