Operator scaling via geodesically convex optimization, invariant theory and polynomial identity testing
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Avi Wigderson | Yuanzhi Li | Ankit Garg | Zeyuan Allen-Zhu | Rafael Mendes de Oliveira | A. Wigderson | Yuanzhi Li | Zeyuan Allen-Zhu | A. Garg | R. Oliveira
[1] F. Kirwan. Cohomology of Quotients in Symplectic and Algebraic Geometry. (MN-31), Volume 31 , 1984 .
[2] Herbert Busemann,et al. The geometry of geodesics , 1955 .
[3] Roger A. Horn,et al. Specht's criterion for systems of linear mappings , 2017, 1701.08826.
[4] Levent Tunçel,et al. Optimization algorithms on matrix manifolds , 2009, Math. Comput..
[5] Peter Bürgisser,et al. Alternating minimization, scaling algorithms, and the null-cone problem from invariant theory , 2017, ITCS.
[6] Zeev Dvir,et al. Locally Decodable Codes with Two Queries and Polynomial Identity Testing for Depth 3 Circuits , 2007, SIAM J. Comput..
[7] Yair Carmon,et al. Accelerated Methods for Non-Convex Optimization , 2016, SIAM J. Optim..
[8] Eugene M. Luks,et al. Testing isomorphism of modules , 2008 .
[9] Avi Wigderson,et al. Much Faster Algorithms for Matrix Scaling , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).
[10] Aleksander Madry,et al. Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).
[11] Alex Samorodnitsky,et al. A Deterministic Strongly Polynomial Algorithm for Matrix Scaling and Approximate Permanents , 1998, STOC '98.
[12] Tengyu Ma,et al. Finding Approximate Local Minima for Nonconvex Optimization in Linear Time , 2016, ArXiv.
[13] R. Bishop,et al. Manifolds of negative curvature , 1969 .
[14] Youming Qiao,et al. Algorithms based on *-algebras, and their applications to isomorphism of polynomials with one secret, group isomorphism, and polynomial identity testing , 2018, SODA.
[15] Ilya Volkovich,et al. Black-Box Identity Testing of Depth-4 Multilinear Circuits , 2011, Combinatorica.
[16] Marek Karpinski,et al. Deterministic Polynomial Time Algorithms for Matrix Completion Problems , 2010, SIAM J. Comput..
[17] Harm Derksen,et al. Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients , 2000 .
[18] Ran Raz,et al. Deterministic polynomial identity testing in non commutative models , 2004 .
[19] Tosio Kato. A Short Introduction to Perturbation Theory for Linear Operators , 1982 .
[20] Amir Shpilka,et al. Explicit Noether Normalization for Simultaneous Conjugation via Polynomial Identity Testing , 2013, APPROX-RANDOM.
[21] Nicholas I. M. Gould,et al. Trust Region Methods , 2000, MOS-SIAM Series on Optimization.
[22] Marek Karpinski,et al. Polynomial time algorithms for modules over finite dimensional algebras , 1997, ISSAC.
[23] Harm Derksen,et al. Polynomial bounds for rings of invariants , 2000 .
[24] C. Woodward,et al. Moment maps and geometric invariant theory , 2009, 0912.1132.
[25] Alexander J. Smola,et al. Stochastic Variance Reduction for Nonconvex Optimization , 2016, ICML.
[26] László Lovász,et al. On determinants, matchings, and random algorithms , 1979, International Symposium on Fundamentals of Computation Theory.
[27] Zeyuan Allen Zhu,et al. Variance Reduction for Faster Non-Convex Optimization , 2016, ICML.
[29] Joel W. Robbin,et al. The Moment-Weight Inequality and the Hilbert–Mumford Criterion , 2013, Lecture Notes in Mathematics.
[30] Zeev Dvir,et al. Locally decodable codes with 2 queries and polynomial identity testing for depth 3 circuits , 2005, STOC '05.
[31] Youming Qiao,et al. Non-commutative Edmonds’ problem and matrix semi-invariants , 2015, computational complexity.
[32] Leslie G. Valiant,et al. The Complexity of Computing the Permanent , 1979, Theor. Comput. Sci..
[33] Yair Carmon,et al. Accelerated Methods for NonConvex Optimization , 2018, SIAM J. Optim..
[34] Amir Shpilka,et al. Quasipolynomial-Time Identity Testing of Non-commutative and Read-Once Oblivious Algebraic Branching Programs , 2013, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.
[35] Avi Wigderson,et al. Operator Scaling: Theory and Applications , 2015, Found. Comput. Math..
[36] Youming Qiao,et al. Constructive noncommutative rank computation in deterministic polynomial time over fields of arbitrary characteristics , 2015, ArXiv.
[37] T. Andô,et al. Means of positive linear operators , 1980 .
[38] D. M.,et al. SPACES OF MATRICES WITH SEVERAL ZERO EIGENVALUES , 2006 .
[39] L. Gurvits,et al. The Deeation-innation Method for Certain Semideenite Programming and Maximum Determinant Completion Problems , 1998 .
[40] Marek Karpinski,et al. Generalized Wong sequences and their applications to Edmonds' problems , 2013, J. Comput. Syst. Sci..
[41] Alexander J. Smola,et al. Stochastic Frank-Wolfe methods for nonconvex optimization , 2016, 2016 54th Annual Allerton Conference on Communication, Control, and Computing (Allerton).
[42] Suvrit Sra,et al. Fast stochastic optimization on Riemannian manifolds , 2016, ArXiv.
[43] Helene Shapiro,et al. A survey of canonical forms and invariants for unitary similarity , 1991 .
[44] Naihuan Jing. Unitary and orthogonal equivalence of sets of matrices , 2015 .
[45] Ami Wiesel,et al. Geodesic Convexity and Covariance Estimation , 2012, IEEE Transactions on Signal Processing.
[46] Neeraj Kayal,et al. Polynomial Identity Testing for Depth 3 Circuits , 2006, 21st Annual IEEE Conference on Computational Complexity (CCC'06).
[47] Suvrit Sra,et al. Conic Geometric Optimization on the Manifold of Positive Definite Matrices , 2013, SIAM J. Optim..
[48] Yin Tat Lee,et al. The Paulsen problem, continuous operator scaling, and smoothed analysis , 2017, STOC.
[49] D. Hilbert,et al. Ueber die vollen Invariantensysteme , 1893 .
[50] Marek Karpinski,et al. Generalized Wong sequences and their applications to Edmonds' problems , 2015, J. Comput. Syst. Sci..
[51] Harm Derksen,et al. On non-commutative rank and tensor rank , 2016, 1606.06701.
[52] Tamás Rapcsák,et al. Smooth nonlinear optimization in Rn.. (Nonconvex optimization and its applications, 19.) , 1997 .
[53] C. Udriste,et al. Convex Functions and Optimization Methods on Riemannian Manifolds , 1994 .
[54] Michel Van den Bergh,et al. Semi-invariants of quivers for arbitrary dimension vectors , 1999 .
[55] Russell Impagliazzo,et al. Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds , 2003, STOC '03.
[56] Avi Wigderson,et al. Algorithmic and optimization aspects of Brascamp-Lieb inequalities, via Operator Scaling , 2016, Geometric and Functional Analysis.
[57] Yurii Nesterov,et al. Cubic regularization of Newton method and its global performance , 2006, Math. Program..
[58] Mike D. Atkinson,et al. LARGE SPACES OF MATRICES OF BOUNDED RANK , 1980 .
[59] Suvrit Sra,et al. First-order Methods for Geodesically Convex Optimization , 2016, COLT.
[60] Richard Sinkhorn. A Relationship Between Arbitrary Positive Matrices and Doubly Stochastic Matrices , 1964 .
[61] Maher Moakher,et al. A Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices , 2005, SIAM J. Matrix Anal. Appl..
[62] Tengyu Ma,et al. Finding approximate local minima faster than gradient descent , 2016, STOC.
[63] J. Edmonds. Systems of distinct representatives and linear algebra , 1967 .
[64] Tamás Rapcsák,et al. Smooth Nonlinear Optimization in Rn , 1997 .
[65] Zeyuan Allen-Zhu,et al. Natasha 2: Faster Non-Convex Optimization Than SGD , 2017, NeurIPS.
[66] Daniel A. Spielman,et al. Randomness efficient identity testing of multivariate polynomials , 2001, STOC '01.
[67] Ketan Mulmuley,et al. Geometric Complexity Theory V: Efficient algorithms for Noether Normalization , 2012 .
[68] Roy Meshulam,et al. Spaces of Singular Matrices and Matroid Parity , 2002, Eur. J. Comb..
[69] Mátyás Domokos,et al. Semi-invariants of quivers as determinants , 2001 .
[70] P. Wedin. Perturbation bounds in connection with singular value decomposition , 1972 .
[71] Michael Walter,et al. Multipartite Quantum States and their Marginals , 2014, 1410.6820.
[72] G. Kempf,et al. The length of vectors in representation spaces , 1979 .
[73] Harm Derksen,et al. Algorithms for orbit closure separation for invariants and semi-invariants of matrices , 2018, ArXiv.
[74] N. Wiegmann,et al. Necessary and sufficient conditions for unitary similarity , 1961 .
[75] Joe W. Harris,et al. Vector spaces of matrices of low rank , 1988 .
[76] Bernd Sturmfels,et al. Algorithms in Invariant Theory (Texts and Monographs in Symbolic Computation) , 2008 .
[77] George Szeto,et al. A Generalization of the Artin-Procesi Theorem , 1977 .
[78] D. Hilbert. Ueber die Theorie der algebraischen Formen , 1890 .
[79] Alex Samorodnitsky,et al. A Deterministic Algorithm for Approximating the Mixed Discriminant and Mixed Volume, and a Combinatorial Corollary , 2002, Discret. Comput. Geom..
[80] Zeyuan Allen Zhu,et al. Natasha: Faster Non-Convex Stochastic Optimization Via Strongly Non-Convex Parameter , 2017, ArXiv.
[81] Mike D. Atkinson,et al. SPACES OF MATRICES OF BOUNDED RANK , 1978 .
[82] R. Phillips,et al. Linear Transformations , 1940, Essential Mathematics for Engineers and Scientists.
[83] Carlo Tomasi,et al. ON LINEAR TRANSFORMATIONS , 2010 .
[84] L. Khachiyan,et al. ON THE COMPLEXITY OF NONNEGATIVE-MATRIX SCALING , 1996 .
[85] P. Absil,et al. Erratum to: ``Global rates of convergence for nonconvex optimization on manifolds'' , 2016, IMA Journal of Numerical Analysis.
[86] Harm Derksen,et al. Polynomial degree bounds for matrix semi-invariants , 2015, ArXiv.
[87] Ketan Mulmuley,et al. Geometric Complexity Theory I: An Approach to the P vs. NP and Related Problems , 2002, SIAM J. Comput..
[88] Leonid Gurvits,et al. Classical complexity and quantum entanglement , 2004, J. Comput. Syst. Sci..
[89] P. Newstead. Moduli Spaces and Vector Bundles: Geometric Invariant Theory , 2009 .
[90] Bernd Sturmfels,et al. Algorithms in invariant theory , 1993, Texts and monographs in symbolic computation.
[91] F. Hiai,et al. Riemannian metrics on positive definite matrices related to means , 2008, 0809.4974.