Complexity of Geometric programming in the Turing model and application to nonnegative tensors

We consider a geometric programming problem consisting in minimizing a function given by the supremum of finitely many log-Laplace transforms of discrete nonnegative measures on a Euclidean space. Under a coerciveness assumption, we show that a $\varepsilon$-minimizer can be computed in a time that is polynomial in the input size and in $|\log\varepsilon|$. This is obtained by establishing bit-size estimates on approximate minimizers and by applying the ellipsoid method. We also derive polynomial iteration complexity bounds for the interior point method applied to the same class of problems. We deduce that the spectral radius of a partially symmetric, weakly irreducible nonnegative tensor can be approximated within $\varepsilon$ error in poly-time. For strongly irreducible tensors, we also show that the logarithm of the positive eigenvector is poly-time computable. Our results also yield that the the maximum of a nonnegative homogeneous $d$-form in the unit ball with respect to $d$-H\"older norm can be approximated in poly-time. In particular, the spectral radius of uniform weighted hypergraphs and some known upper bounds for the clique number of uniform hypergraphs are poly-time computable.

[1]  Lek-Heng Lim,et al.  Tensors in computations , 2021, Acta Numerica.

[2]  S. Friedland,et al.  Quantum Monge-Kantorovich Problem and Transport Distance between Density Matrices. , 2021, Physical review letters.

[3]  Peter Bürgisser,et al.  Interior-point methods for unconstrained geometric programming and scaling problems , 2020, ArXiv.

[4]  S. Friedland.  Tensor optimal transport, distance between sets of measures and tensor scaling , 2020, ArXiv.

[5]  Nisheeth K. Vishnoi,et al.  On the computability of continuous maximum entropy distributions with applications , 2020, STOC.

[6]  Giuseppe Carlo Calafiore,et al.  A Universal Approximation Result for Difference of Log-Sum-Exp Neural Networks , 2019, IEEE Transactions on Neural Networks and Learning Systems.

[7]  S. Friedland,et al.  Spectral norm of a symmetric tensor and its computation , 2018, Math. Comput..

[8]  S. Gaubert,et al.  Inequalities for the spectral radii and spectral norms of nonnegative tensors , 2018, 1804.00204.

[9]  Nisheeth K. Vishnoi,et al.  Maximum Entropy Distributions: Bit Complexity and Stability , 2017, COLT.

[10]  Frank Vallentin,et al.  On the Turing Model Complexity of Interior Point Methods for Semidefinite Programming , 2015, SIAM J. Optim..

[11]  Silouanos Brazitikos.  Geometry of Isotropic Convex Bodies , 2014 .

[12]  Jiawang Nie,et al.  All Real Eigenvalues of Symmetric Tensors , 2014, SIAM J. Matrix Anal. Appl..

[13]  Mohit Singh,et al.  Entropy, optimization and counting , 2013, STOC.

[14]  J. Landsberg.  Tensors: Geometry and Applications , 2011 .

[15]  Marcello Pelillo,et al.  New Bounds on the Clique Number of Graphs Based on Spectral Hypergraph Theory , 2009, LION.

[16]  Christopher J. Hillar,et al.  Most Tensor Problems Are NP-Hard , 2009, JACM.

[17]  Michael K. Ng,et al.  Finding the Largest Eigenvalue of a Nonnegative Tensor , 2009, SIAM J. Matrix Anal. Appl..

[18]  G. Golub,et al.  A tensor higher-order singular value decomposition for integrative analysis of DNA microarray data from different studies , 2007, Proceedings of the National Academy of Sciences.

[19]  Lek-Heng Lim,et al.  Singular values and eigenvalues of tensors: a variational approach , 2005, 1st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, 2005..

[20]  Liqun Qi,et al.  Eigenvalues of a real supersymmetric tensor , 2005, J. Symb. Comput..

[21]  S. Gaubert,et al.  The Perron-Frobenius theorem for homogeneous, monotone functions , 2001, math/0105091.

[22]  J. Borwein,et al.  On the complexity of familiar functions and numbers , 1988 .

[23]  S. Gaubert,et al.  The Operator Approach to Entropy Games ∗ , 2017 .

[24]  L. Qi,et al.  The clique and coclique numbers' bounds based on the H-eigenvalues of uniform hypergraphs , 2014 .

[25]  J. Bard,et al.  Interior-Point Method , 2014 .

[26]  James Renegar,et al.  A mathematical view of interior-point methods in convex optimization , 2001, MPS-SIAM series on optimization.

[27]  Yurii Nesterov,et al.  Interior-point polynomial algorithms in convex programming , 1994, Siam studies in applied mathematics.

[28]  R. Nussbaum.  Convexity and log convexity for the spectral radius , 1986 .

[29]  Arnold Schönhage,et al.  The fundamental theorem of algebra in terms of computational complexity - preliminary report , 1982 .

[30]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

[31]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[32]  H. Wilf.  The Eigenvalues of a Graph and Its Chromatic Number , 1967 .

[33]  T. Motzkin,et al.  Maxima for Graphs and a New Proof of a Theorem of Turán , 1965, Canadian Journal of Mathematics.

[34]  J. Kingman.  A convexity property of positive matrices , 1961 .

[35]  S. Gaubert,et al.  SP ] 1 1 M ay 2 00 9 Perron-Frobenius theorem for nonnegative multilinear forms , 2022 .