The Landscape of the Spiked Tensor Model
We consider the problem of estimating a large rank‐one tensor u⊗k ∈ (ℝn)⊗k, k ≥ 3, in Gaussian noise. Earlier work characterized a critical signal‐to‐noise ratio λ Bayes = O(1) above which an ideal estimator achieves strictly positive correlation with the unknown vector of interest. Remarkably, no...
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| Vydáno v: | Communications on pure and applied mathematics Ročník 72; číslo 11; s. 2282 - 2330 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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Melbourne
John Wiley & Sons Australia, Ltd
01.11.2019
John Wiley and Sons, Limited |
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| ISSN: | 0010-3640, 1097-0312 |
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| Abstract | We consider the problem of estimating a large rank‐one tensor u⊗k ∈ (ℝn)⊗k, k ≥ 3, in Gaussian noise. Earlier work characterized a critical signal‐to‐noise ratio λ Bayes = O(1) above which an ideal estimator achieves strictly positive correlation with the unknown vector of interest. Remarkably, no polynomial‐time algorithm is known that achieved this goal unless λ ≥ Cn(k − 2)/4, and even powerful semidefinite programming relaxations appear to fail for 1 ≪ λ ≪ n(k − 2)/4.
In order to elucidate this behavior, we consider the maximum likelihood estimator, which requires maximizing a degree‐k homogeneous polynomial over the unit sphere in n dimensions. We compute the expected number of critical points and local maxima of this objective function and show that it is exponential in the dimensions n, and give exact formulas for the exponential growth rate. We show that (for λ larger than a constant) critical points are either very close to the unknown vector u or are confined in a band of width Θ(λ−1/(k − 1)) around the maximum circle that is orthogonal to u. For local maxima, this band shrinks to be of size Θ(λ−1/(k − 2)). These “uninformative” local maxima are likely to cause the failure of optimization algorithms. © 2019 Wiley Periodicals, Inc. |
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| AbstractList | We consider the problem of estimating a large rank‐one tensor u⊗k ∈ (ℝn)⊗k, k ≥ 3, in Gaussian noise. Earlier work characterized a critical signal‐to‐noise ratio λ Bayes = O(1) above which an ideal estimator achieves strictly positive correlation with the unknown vector of interest. Remarkably, no polynomial‐time algorithm is known that achieved this goal unless λ ≥ Cn(k − 2)/4, and even powerful semidefinite programming relaxations appear to fail for 1 ≪ λ ≪ n(k − 2)/4.In order to elucidate this behavior, we consider the maximum likelihood estimator, which requires maximizing a degree‐k homogeneous polynomial over the unit sphere in n dimensions. We compute the expected number of critical points and local maxima of this objective function and show that it is exponential in the dimensions n, and give exact formulas for the exponential growth rate. We show that (for λ larger than a constant) critical points are either very close to the unknown vector u or are confined in a band of width Θ(λ−1/(k − 1)) around the maximum circle that is orthogonal to u. For local maxima, this band shrinks to be of size Θ(λ−1/(k − 2)). These “uninformative” local maxima are likely to cause the failure of optimization algorithms. © 2019 Wiley Periodicals, Inc. We consider the problem of estimating a large rank‐one tensor u ⊗ k ∈ ( ℝ n ) ⊗ k , k ≥ 3 , in Gaussian noise. Earlier work characterized a critical signal‐to‐noise ratio λ Bayes = O (1) above which an ideal estimator achieves strictly positive correlation with the unknown vector of interest. Remarkably, no polynomial‐time algorithm is known that achieved this goal unless λ ≥ Cn ( k − 2)/4 , and even powerful semidefinite programming relaxations appear to fail for 1 ≪ λ ≪ n ( k − 2)/4 . In order to elucidate this behavior, we consider the maximum likelihood estimator, which requires maximizing a degree‐ k homogeneous polynomial over the unit sphere in n dimensions. We compute the expected number of critical points and local maxima of this objective function and show that it is exponential in the dimensions n , and give exact formulas for the exponential growth rate. We show that (for λ larger than a constant) critical points are either very close to the unknown vector u or are confined in a band of width Θ( λ −1/( k − 1) ) around the maximum circle that is orthogonal to u . For local maxima, this band shrinks to be of size Θ( λ −1/( k − 2) ) . These “uninformative” local maxima are likely to cause the failure of optimization algorithms. © 2019 Wiley Periodicals, Inc. We consider the problem of estimating a large rank‐one tensor u⊗k ∈ (ℝn)⊗k, k ≥ 3, in Gaussian noise. Earlier work characterized a critical signal‐to‐noise ratio λ Bayes = O(1) above which an ideal estimator achieves strictly positive correlation with the unknown vector of interest. Remarkably, no polynomial‐time algorithm is known that achieved this goal unless λ ≥ Cn(k − 2)/4, and even powerful semidefinite programming relaxations appear to fail for 1 ≪ λ ≪ n(k − 2)/4. In order to elucidate this behavior, we consider the maximum likelihood estimator, which requires maximizing a degree‐k homogeneous polynomial over the unit sphere in n dimensions. We compute the expected number of critical points and local maxima of this objective function and show that it is exponential in the dimensions n, and give exact formulas for the exponential growth rate. We show that (for λ larger than a constant) critical points are either very close to the unknown vector u or are confined in a band of width Θ(λ−1/(k − 1)) around the maximum circle that is orthogonal to u. For local maxima, this band shrinks to be of size Θ(λ−1/(k − 2)). These “uninformative” local maxima are likely to cause the failure of optimization algorithms. © 2019 Wiley Periodicals, Inc. |
| Author | Arous, Gérard Ben Mei, Song Nica, Mihai Montanari, Andrea |
| Author_xml | – sequence: 1 givenname: Gérard Ben surname: Arous fullname: Arous, Gérard Ben email: benarous@cims.nyu.edu organization: Courant Institute, 251 Mercer St – sequence: 2 givenname: Song surname: Mei fullname: Mei, Song email: songmei@stanford.edu organization: Stanford University, 475 Via Ortega – sequence: 3 givenname: Andrea surname: Montanari fullname: Montanari, Andrea email: montanari@stanford.edu organization: Stanford University, 350, Serra Mall – sequence: 4 givenname: Mihai surname: Nica fullname: Nica, Mihai email: mnica@math.utoronto.ca organization: University of Toronto, 40 St. George St. Toronto |
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| Cites_doi | 10.1103/PhysRevLett.92.240601 10.1090/surv/089 10.1214/17-AOS1637 10.1016/j.jfa.2004.09.015 10.1109/TPAMI.2012.39 10.29007/964b 10.1007/s10208-017-9365-9 10.1109/TIT.2016.2637959 10.1002/cpa.21748 10.1007/s00440-005-0433-8 10.1145/2897518.2897529 10.1051/jp1:1995164 10.1007/s00440-016-0724-2 10.1214/12-AOS1018 10.1007/BF01312184 10.1007/s10208-015-9269-5 10.1214/16-AOP1139 10.1109/TIT.2010.2046205 10.1002/cpa.21638 10.1007/PL00008774 10.1214/EJP.v12-438 10.1002/cpa.21422 10.1007/s00222-017-0726-4 10.1002/widm.1 10.1214/13-AOP862 10.1190/geo2013-0022.1 10.1088/0266-5611/27/2/025010 |
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| References | 2010; 56 2017; 63 2001; 120 2011; 1 2013; 66 2010 2013; 41 2017; 45 2009 1998 2005 2017; 210 2016; 16 2007; 12 2006; 134 1995; 5 2018; 46 2018; 18 2004; 92 2017; 70 2005; 222 2013; 35 1993; 92 2013; 78 2000; 33 2018 2017 2016 2015 2014 2018; 71 2011; 27 2017; 168 2012; 40 Bhattiprolu V. (e_1_2_1_10_1) 2016 Bhattiprolu V. (e_1_2_1_11_1) 2016 Gillin P. (e_1_2_1_19_1) 2000; 33 e_1_2_1_42_1 e_1_2_1_20_1 e_1_2_1_41_1 Hopkins S. B. (e_1_2_1_22_1) 2015 e_1_2_1_40_1 Janzamin M. (e_1_2_1_23_1) 2015 e_1_2_1_24_1 Montanari A. (e_1_2_1_34_1) 2014 e_1_2_1_21_1 e_1_2_1_44_1 e_1_2_1_43_1 Bouchaud J.‐P. (e_1_2_1_12_1) 1998 e_1_2_1_25_1 e_1_2_1_26_1 Lesieur T. (e_1_2_1_27_1) 2017 Li N. (e_1_2_1_28_1) 2010 e_1_2_1_29_1 Adler R. J. (e_1_2_1_2_1) 2009 Barak B. (e_1_2_1_7_1) 2016 Barbier J. (e_1_2_1_8_1) 2017 Zhong K. (e_1_2_1_45_1) 2017 Ge R. (e_1_2_1_18_1) 2016 e_1_2_1_31_1 e_1_2_1_30_1 e_1_2_1_5_1 e_1_2_1_6_1 e_1_2_1_35_1 e_1_2_1_13_1 e_1_2_1_33_1 e_1_2_1_32_1 Anderson G. W. (e_1_2_1_4_1) 2010 e_1_2_1_16_1 e_1_2_1_39_1 e_1_2_1_17_1 e_1_2_1_38_1 e_1_2_1_14_1 e_1_2_1_15_1 e_1_2_1_36_1 Anandkumar A. (e_1_2_1_3_1) 2016 Perry A. (e_1_2_1_37_1) 2016 e_1_2_1_9_1 |
| References_xml | – volume: 18 start-page: 1131 issue: 5 year: 2018 end-page: 1198 article-title: A geometric analysis of phase retrieval publication-title: Found. Comput. Math. – year: 2009 – year: 2016 article-title: Statistical limits of spiked tensor models publication-title: Preprint – volume: 1 start-page: 24 issue: 1 year: 2011 end-page: 40 article-title: Applications of tensor (multiway array) factorizations and decompositions in data mining publication-title: Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery – volume: 71 start-page: 2381 issue: 11 year: 2018 end-page: 2425 article-title: Spectral algorithms for tensor completion publication-title: Comm. Pure Appl. Math. – volume: 78 start-page: V273 issue: 6 year: 2013 end-page: V284 article-title: Tensor completion based on nuclear norm minimization for 5D seismic data reconstruction publication-title: Geophysics – volume: 63 start-page: 1572 issue: 3 year: 2017 end-page: 1579 article-title: On the limitation of spectral methods: from the Gaussian hidden clique problem to rank one perturbations of Gaussian tensors publication-title: IEEE Trans. Inform. Theory – year: 2005 – volume: 35 start-page: 208 issue: 1 year: 2013 end-page: 220 article-title: Tensor completion for estimating missing values in visual data publication-title: IEEE Transactions on Pattern Analysis and Machine Intelligence – volume: 222 start-page: 435 issue: 2 year: 2005 end-page: 490 article-title: A Fourier view on the R‐transform and related asymptotics of spherical integrals publication-title: J. Funct. Anal. – year: 2017 article-title: Recovery guarantees for one‐hidden‐layer neural networks publication-title: Preprint – start-page: 161 year: 1998 end-page: 223 – volume: 168 start-page: 773 issue: 3‐4 year: 2017 end-page: 820 article-title: The extremal process of critical points of the pure ‐spin spherical spin glass model publication-title: Probab. Theory Related Fields – volume: 70 start-page: 822 issue: 5 year: 2017 end-page: 883 article-title: Solving random quadratic systems of equations is nearly as easy as solving linear systems publication-title: Comm. Pure Appl. Math. – start-page: 2897 year: 2014 end-page: 2905 article-title: A statistical model for tensor PCA publication-title: Advances in Neural Information Processing Systems – volume: 16 start-page: 1031 issue: 4 year: 2016 end-page: 1068 article-title: On tensor completion via nuclear norm minimization publication-title: Found. Comput. Math. – year: 2015 article-title: Beating the perils of non‐convexity: Guaranteed training of neural networks using tensor methods publication-title: Preprint – start-page: 956 year: 2015 end-page: 1006 article-title: Tensor principal component analysis via sum‐of‐square proofs publication-title: Conference on Learning Theory – year: 2017 article-title: The layered structure of tensor estimation and its mutual information publication-title: Preprint – year: 2010 – volume: 56 start-page: 2980 issue: 6 year: 2010 end-page: 2998 article-title: Matrix completion from a few entries publication-title: IEEE Trans. Inform. Theory – volume: 210 start-page: 135 issue: 1 year: 2017 end-page: 209 article-title: The geometry of the Gibbs measure of pure spherical spin glasses publication-title: Invent. Math. – volume: 12 start-page: 1131 year: 2007 end-page: 1150 article-title: Large deviations for the largest eigenvalue of rank one deformations of Gaussian ensembles publication-title: Electron. J. Probab. – volume: 120 start-page: 1 issue: 1 year: 2001 end-page: 67 article-title: Aging of spherical spin glasses publication-title: Probab. Theory Related Fields – volume: 92 start-page: 257 issue: 2 year: 1993 end-page: 271 article-title: The spherical ‐spin interaction spin‐glass model publication-title: Zeitschrift für Physik B Condensed Matter – start-page: 2973 year: 2016 end-page: 2981 article-title: Matrix completion has no spurious local minimum publication-title: Advances in Neural Information Processing Systems – year: 2016 article-title: Multiplicative approximations for polynomial optimization over the unit sphere publication-title: Preprint – start-page: 178 year: 2016 end-page: 191 – year: 2017 article-title: Statistical and computational phase transitions in spiked tensor estimation publication-title: Preprint – year: 2010 article-title: Tensor completion for on‐board compression of hyperspectral images. 517–520 publication-title: IEEE – volume: 40 start-page: 1637 issue: 3 year: 2012 end-page: 1664 article-title: High‐dimensional regression with noisy and missing data: Provable guarantees with nonconvexity publication-title: Ann. Statist. – volume: 46 start-page: 2747 issue: 6A year: 2018 end-page: 2774 article-title: The landscape of empirical risk for non‐convex losses publication-title: Ann. Statist. – year: 2018 article-title: Complex energy landscapes in spiked‐tensor and simple glassy models: ruggedness arrangements of local minima and phase transitions publication-title: Preprint – volume: 27 start-page: 19 issue: 2 year: 2011 article-title: Tensor completion and low‐ ‐rank tensor recovery via convex optimization publication-title: Inverse Problems – volume: 41 start-page: 4214 issue: 6 year: 2013 end-page: 4247 article-title: Complexity of random smooth functions on the high‐dimensional sphere publication-title: Ann. Probab. – volume: 92 start-page: 4 issue: 24 year: 2004 article-title: Complexity of random energy landscapes, glass transition, and absolute value of the spectral determinant of random matrices publication-title: Phys. Rev. Lett – volume: 33 start-page: 3081 issue: 16 year: 2000 end-page: 3091 article-title: spin glasses with first‐order ferromagnetic transitions publication-title: J. Phys – volume: 66 start-page: 165 issue: 2 year: 2013 end-page: 201 article-title: Random matrices and complexity of spin glasses publication-title: Comm. Pure Appl. Math – volume: 134 start-page: 339 issue: 3 year: 2006 end-page: 382 article-title: Free energy of the spherical mean field model publication-title: Probab. Theory Related Fields – volume: 5 start-page: 805 issue: 7 year: 1995 end-page: 813 article-title: Thouless‐Anderson‐Palmer approach to the spherical ‐spin spin glass model publication-title: Journal de Physique I – year: 2016 article-title: Certifying random polynomials over the unit sphere via sum of squares hierarchy publication-title: Preprint – start-page: 417 year: 2016 end-page: 445 article-title: Noisy tensor completion via the sum‐of‐squares hierarchy publication-title: Conference on Learning Theory – volume: 45 start-page: 3385 issue: 5 year: 2017 end-page: 3450 article-title: The complexity of spherical ‐spin models—a second moment approach publication-title: Ann. Probab. – year: 2016 article-title: Homotopy analysis for tensor PCA publication-title: Preprint – start-page: 161 volume-title: Spin glasses and random fields year: 1998 ident: e_1_2_1_12_1 – year: 2010 ident: e_1_2_1_28_1 article-title: Tensor completion for on‐board compression of hyperspectral images. 2010 IEEE International Conference on Image Processing 517–520 publication-title: IEEE – ident: e_1_2_1_16_1 doi: 10.1103/PhysRevLett.92.240601 – ident: e_1_2_1_26_1 doi: 10.1090/surv/089 – volume-title: An introduction to random matrices year: 2010 ident: e_1_2_1_4_1 – year: 2017 ident: e_1_2_1_27_1 article-title: Statistical and computational phase transitions in spiked tensor estimation publication-title: Preprint – ident: e_1_2_1_32_1 doi: 10.1214/17-AOS1637 – ident: e_1_2_1_20_1 doi: 10.1016/j.jfa.2004.09.015 – ident: e_1_2_1_29_1 doi: 10.1109/TPAMI.2012.39 – ident: e_1_2_1_38_1 doi: 10.29007/964b – ident: e_1_2_1_42_1 doi: 10.1007/s10208-017-9365-9 – ident: e_1_2_1_33_1 doi: 10.1109/TIT.2016.2637959 – year: 2017 ident: e_1_2_1_45_1 article-title: Recovery guarantees for one‐hidden‐layer neural networks publication-title: Preprint – year: 2016 ident: e_1_2_1_37_1 article-title: Statistical limits of spiked tensor models publication-title: Preprint – ident: e_1_2_1_35_1 doi: 10.1002/cpa.21748 – volume-title: Random fields and geometry year: 2009 ident: e_1_2_1_2_1 – ident: e_1_2_1_43_1 doi: 10.1007/s00440-005-0433-8 – ident: e_1_2_1_21_1 doi: 10.1145/2897518.2897529 – start-page: 956 year: 2015 ident: e_1_2_1_22_1 article-title: Tensor principal component analysis via sum‐of‐square proofs publication-title: Conference on Learning Theory – ident: e_1_2_1_15_1 doi: 10.1051/jp1:1995164 – ident: e_1_2_1_41_1 doi: 10.1007/s00440-016-0724-2 – ident: e_1_2_1_30_1 doi: 10.1214/12-AOS1018 – ident: e_1_2_1_14_1 doi: 10.1007/BF01312184 – ident: e_1_2_1_44_1 doi: 10.1007/s10208-015-9269-5 – ident: e_1_2_1_39_1 doi: 10.1214/16-AOP1139 – ident: e_1_2_1_24_1 doi: 10.1109/TIT.2010.2046205 – year: 2016 ident: e_1_2_1_11_1 article-title: Certifying random polynomials over the unit sphere via sum of squares hierarchy publication-title: Preprint – ident: e_1_2_1_13_1 doi: 10.1002/cpa.21638 – start-page: 2897 year: 2014 ident: e_1_2_1_34_1 article-title: A statistical model for tensor PCA publication-title: Advances in Neural Information Processing Systems – ident: e_1_2_1_9_1 doi: 10.1007/PL00008774 – year: 2016 ident: e_1_2_1_10_1 article-title: Multiplicative approximations for polynomial optimization over the unit sphere publication-title: Preprint – volume: 33 start-page: 3081 issue: 16 year: 2000 ident: e_1_2_1_19_1 article-title: p > 2 spin glasses with first‐order ferromagnetic transitions publication-title: J. Phys – ident: e_1_2_1_31_1 doi: 10.1214/EJP.v12-438 – ident: e_1_2_1_6_1 doi: 10.1002/cpa.21422 – year: 2017 ident: e_1_2_1_8_1 article-title: The layered structure of tensor estimation and its mutual information publication-title: Preprint – ident: e_1_2_1_40_1 doi: 10.1007/s00222-017-0726-4 – ident: e_1_2_1_36_1 doi: 10.1002/widm.1 – ident: e_1_2_1_5_1 doi: 10.1214/13-AOP862 – start-page: 417 year: 2016 ident: e_1_2_1_7_1 article-title: Noisy tensor completion via the sum‐of‐squares hierarchy publication-title: Conference on Learning Theory – year: 2015 ident: e_1_2_1_23_1 article-title: Beating the perils of non‐convexity: Guaranteed training of neural networks using tensor methods publication-title: Preprint – ident: e_1_2_1_25_1 doi: 10.1190/geo2013-0022.1 – year: 2016 ident: e_1_2_1_3_1 article-title: Homotopy analysis for tensor PCA publication-title: Preprint – start-page: 2973 year: 2016 ident: e_1_2_1_18_1 article-title: Matrix completion has no spurious local minimum publication-title: Advances in Neural Information Processing Systems – ident: e_1_2_1_17_1 doi: 10.1088/0266-5611/27/2/025010 |
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| Snippet | We consider the problem of estimating a large rank‐one tensor u⊗k ∈ (ℝn)⊗k, k ≥ 3, in Gaussian noise. Earlier work characterized a critical signal‐to‐noise... We consider the problem of estimating a large rank‐one tensor u ⊗ k ∈ ( ℝ n ) ⊗ k , k ≥ 3 , in Gaussian noise. Earlier work characterized a critical... |
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| SubjectTerms | Algorithms Critical point Economic models Mathematical analysis Maxima Maximum likelihood estimators Optimization Polynomials Random noise Semidefinite programming Tensors |
| Title | The Landscape of the Spiked Tensor Model |
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