Built on

P. Bickel and E. Levina, Regularized estimation of large covariance matrices , Ann. Statist. 36
Original
1909
Earlier work this paper cites.
A. Walther, The question of phase retrieval in optics , J. Mod. Opt. 10
1963
Earlier work this paper cites.
A. Jamiolkowski, Linear transformations which preserve trace and positive semidefiniteness of operators , Rep. Math. Phys. 3
1972
Earlier work this paper cites.
M.-D. Choi, Completely positive linear maps on complex matrices , Lin. Alg. App. 10
1975
Earlier work this paper cites.
P. Delsarte, J. Goethals, and J. Seidel, Spherical codes and designs , Geom. Dedicata 6
1977
Earlier work this paper cites.
B. K. Natarajan, Sparse approximate solutions to linear systems , SIAM J. Comp. 24
1995
Earlier work this paper cites.
M. Fazel, H. Hindi, and S. Boyd, A rank minimization heuristic with application to minimum order system approximation , in Proceedings American Control Conference , Vol. 6 (2001) pp. 4734–4739
2001
Earlier work this paper cites.
A. Y. Kitaev, A. Shen, and M. N. Vyalyi, Classical and quantum computation , Vol. 47 (American Mathematical Society, 2002)
2002
Earlier work this paper cites.
V. Paulsen, Completely bounded maps and operator algebras , Vol. 78 (Cambridge University Press, 2002)
2002
Earlier work this paper cites.
A. Barvinok, A course in convexity , Graduate Studies in Mathematics, Vol. 54 (American Mathematical Society, 2002)
2002
Earlier work this paper cites.
R. Basri and D. W. Jacobs, Lambertian reflectance and linear sub-spaces , IEEE Trans. Pattern Anal. Mach. Intell. 25
2003
Earlier work this paper cites.
S. Boyd and L. Vandenberghe, Convex optimization (Cambridge University Press, New York, 2004)
2004
Earlier work this paper cites.
J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, Symmetric informationally complete quantum measurements , J. Math. Phys. 45
2004
Earlier work this paper cites.
J. Emerson, R. Alicki, and K. Życzkowski, Scalable noise estimation with random unitary operators , J. Opt. B 7
2005
Earlier work this paper cites.
A. Ambainis and J. Emerson, Quantum t-designs: t-wise Independence in the Quantum World , in Computational Complexity, 2007. CCC ’07. Twenty-Second Annual IEEE Conference on (2007) pp. 129–140, quant-ph/0701126
2007
Earlier work this paper cites.
D. Gross, K. M. R. Audenaert, and J. Eisert, Evenly distributed unitaries: on the structure of unitary designs , J. Math. Phys. 48
2007
Earlier work this paper cites.
M. Grant and S. Boyd, in Recent advances in learning and control , Lecture Notes in Control and Information Sciences, edited by V. Blondel, S. Boyd, and H. Kimura (Springer-Verlag Limited, 2008) pp. 95–110, http://stanford.edu/˜boyd/graph_dcp.html
2008
Earlier work this paper cites.
M. Ziman, Process positive-operator-valued measure: A mathematical framework for the description of process tomography experiments , Phys. Rev. A 77
Original
2008
Earlier work this paper cites.
Y. Koren, R. Bell, and C. Volinsky, Matrix Factorization Techniques for Recommender Systems , Computer 42
2009
Earlier work this paper cites.
E. Candès and B. Recht, Exact Matrix Completion via Convex Optimization , Found. Comput. Math. 9
Original
2009
Earlier work this paper cites.
J. Watrous, Semidefinite Programs for Completely Bounded Norms , Theory of Computing 5
Original
2009
Earlier work this paper cites.
C. Dankert, R. Cleve, J. Emerson, and E. Livine, Exact and approximate unitary 2-designs and their application to fidelity estimation , Phys. Rev. A 80
2009
Earlier work this paper cites.

Similar

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, Quantum State Tomography via Compressed Sensing , Phys. Rev. Lett. 105
Original
2010
Cited alongside, same era.
E. J. Candès and T. Tao, The power of convex relaxation: near-optimal matrix completion , IEEE Trans. Inf. Th. 56
Original
2010
Cited alongside, same era.
B. Recht, M. Fazel, and P. A. Parrilo, Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization , SIAM Rev. 52
Original
2010
Cited alongside, same era.
A. Ben-Aroya and A. Ta-Shma, On the Complexity of Approximating the Diamond Norm , Quantum Info. Comput. 10
Original
2010
Cited alongside, same era.
M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge university press, 2010)

Then

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2015

Improving compressed sensing with the diamond norm

Kliesch, Martin, Kueng, Richard, Eisert, Jens et al.

Understand

In low-rank matrix recovery, one aims to reconstruct a low-rank matrix from a minimal number of linear measurements.

Within the paradigm of compressed sensing, this is made computationally efficient by minimizing the nuclear norm as a convex surrogate for rank.
In this work, we identify an improved regularizer based on the so-called diamond norm, a concept imported from quantum information theory.
We show that -for a class of matrices saturating a certain norm inequality- the descent cone of the diamond norm is contained in that of the nuclear norm.

Built on

P. Bickel and E. Levina, Regularized estimation of large covariance matrices , Ann. Statist. 36
Original
1909
Earlier work this paper cites.
A. Walther, The question of phase retrieval in optics , J. Mod. Opt. 10
1963
Earlier work this paper cites.
A. Jamiolkowski, Linear transformations which preserve trace and positive semidefiniteness of operators , Rep. Math. Phys. 3
1972
Earlier work this paper cites.
M.-D. Choi, Completely positive linear maps on complex matrices , Lin. Alg. App. 10
1975
Earlier work this paper cites.
P. Delsarte, J. Goethals, and J. Seidel, Spherical codes and designs , Geom. Dedicata 6
1977
Earlier work this paper cites.
B. K. Natarajan, Sparse approximate solutions to linear systems , SIAM J. Comp. 24
1995
Earlier work this paper cites.
M. Fazel, H. Hindi, and S. Boyd, A rank minimization heuristic with application to minimum order system approximation , in Proceedings American Control Conference , Vol. 6 (2001) pp. 4734–4739
2001
Earlier work this paper cites.
A. Y. Kitaev, A. Shen, and M. N. Vyalyi, Classical and quantum computation , Vol. 47 (American Mathematical Society, 2002)
2002
Earlier work this paper cites.
V. Paulsen, Completely bounded maps and operator algebras , Vol. 78 (Cambridge University Press, 2002)
2002
Earlier work this paper cites.
A. Barvinok, A course in convexity , Graduate Studies in Mathematics, Vol. 54 (American Mathematical Society, 2002)
2002
Earlier work this paper cites.
R. Basri and D. W. Jacobs, Lambertian reflectance and linear sub-spaces , IEEE Trans. Pattern Anal. Mach. Intell. 25
2003
Earlier work this paper cites.
S. Boyd and L. Vandenberghe, Convex optimization (Cambridge University Press, New York, 2004)
2004
Earlier work this paper cites.
J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, Symmetric informationally complete quantum measurements , J. Math. Phys. 45
2004
Earlier work this paper cites.
J. Emerson, R. Alicki, and K. Życzkowski, Scalable noise estimation with random unitary operators , J. Opt. B 7
2005
Earlier work this paper cites.
A. Ambainis and J. Emerson, Quantum t-designs: t-wise Independence in the Quantum World , in Computational Complexity, 2007. CCC ’07. Twenty-Second Annual IEEE Conference on (2007) pp. 129–140, quant-ph/0701126
2007
Earlier work this paper cites.
D. Gross, K. M. R. Audenaert, and J. Eisert, Evenly distributed unitaries: on the structure of unitary designs , J. Math. Phys. 48
2007
Earlier work this paper cites.
M. Grant and S. Boyd, in Recent advances in learning and control , Lecture Notes in Control and Information Sciences, edited by V. Blondel, S. Boyd, and H. Kimura (Springer-Verlag Limited, 2008) pp. 95–110, http://stanford.edu/˜boyd/graph_dcp.html
2008
Earlier work this paper cites.
M. Ziman, Process positive-operator-valued measure: A mathematical framework for the description of process tomography experiments , Phys. Rev. A 77
Original
2008
Earlier work this paper cites.
Y. Koren, R. Bell, and C. Volinsky, Matrix Factorization Techniques for Recommender Systems , Computer 42
2009
Earlier work this paper cites.
E. Candès and B. Recht, Exact Matrix Completion via Convex Optimization , Found. Comput. Math. 9
Original
2009
Earlier work this paper cites.
J. Watrous, Semidefinite Programs for Completely Bounded Norms , Theory of Computing 5
Original
2009
Earlier work this paper cites.
C. Dankert, R. Cleve, J. Emerson, and E. Livine, Exact and approximate unitary 2-designs and their application to fidelity estimation , Phys. Rev. A 80
2009
Earlier work this paper cites.

Similar

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, Quantum State Tomography via Compressed Sensing , Phys. Rev. Lett. 105
Original
2010
Cited alongside, same era.
E. J. Candès and T. Tao, The power of convex relaxation: near-optimal matrix completion , IEEE Trans. Inf. Th. 56
Original
2010
Cited alongside, same era.
B. Recht, M. Fazel, and P. A. Parrilo, Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization , SIAM Rev. 52
Original
2010
Cited alongside, same era.
A. Ben-Aroya and A. Ta-Shma, On the Complexity of Approximating the Diamond Norm , Quantum Info. Comput. 10
Original
2010
Cited alongside, same era.
M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge university press, 2010)

Then

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…

2010

Cited alongside, same era.

D. Gross, Recovering Low-Rank Matrices From Few Coefficients in Any Basis , IEEE Trans. Inf. Th. 57

Original

2011

Cited alongside, same era.

E. J. Candès and Y. Plan, Tight oracle bounds for low-rank matrix recovery from a minimal number of random measurements , IEEE Trans. Inform. Theory 57

Original

2011

Cited alongside, same era.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, Efficient Measurement of Quantum Dynamics via Compressive Sensing , Phys. Rev. Lett. 106

Original

2011

Cited alongside, same era.

J. Watrous, CS 766 Theory of Quantum Information , Available online at https://cs.uwaterloo.ca/˜watrous/LectureNotes.html (2011)

2011

Cited alongside, same era.

S. T. Flammia, D. Gross, Y.-K. Liu, and J. Eisert, Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators , New J. Phys. 14

Original

2012

Cited alongside, same era.

J. Watrous, Simpler semidefinite programs for completely bounded norms , arXiv:1207.5726 [quant-ph]

Original

2012

Cited alongside, same era.

V. Chandrasekaran, B. Recht, P. Parrilo, and A. Willsky, The Convex Geometry of Linear Inverse Problems , Found. Comput. Math. 12

Original

2012

Cited alongside, same era.

Original

2015

Closest in time.

V. Koltchinskii and S. Mendelson, Bounding the smallest singular value of a random matrix without concentration , International Mathematics Research Notices , rnv096 (2015) , arXiv:1312.3580 [math.PR]

Original

2015

Closest in time.

R. Kueng, Low rank matrix recovery from few orthonormal basis measurements , in Sampling Theory and Applications (SampTA), 2015 International Conference on (2015) pp. 402–406

2015

Closest in time.

E. J. Candès, X. Li, and M. Soltanolkotabi, Phase retrieval from coded diffraction patterns , Appl. Comput. Harmon. Anal. 39

Original

2015

Closest in time.

D. Gross, F. Krahmer, and R. Kueng, A Partial Derandomization of PhaseLift Using Spherical Designs , J. Fourier Anal. Appl. 21

Original

2015

Closest in time.

H. Rauhut and Ž. Stojanac, Tensor theta norms and low rank recovery , arXiv:1505.05175 [cs.IT]

Original

2015

Closest in time.

M. Kabanava, H. Rauhut, and U. Terstiege, Analysis of low rank matrix recovery via Mendelson’s small ball method , in Sampling Theory and Applications (SampTA), 2015 International Conference on (2015) pp. 387–391

2015

Closest in time.

M. Kabanava, H. Rauhut, and U. Terstiege, On the minimal number of measurements in low-rank matrix recovery , in Sampling Theory and Applications (SampTA), 2015 International Conference on (2015) pp. 382–386

2015

Closest in time.

S. Ling and T. Strohmer, Self-Calibration and Biconvex Compressive Sensing , arXiv:1501.06864 [cs.IT]

Original

2015

Closest in time.

S. Kimmel and Y.-K. Liu, Quantum Compressed Sensing Using 2-Designs , arXiv:1510.08887 [quant-ph]

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2015

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C. Granade, C. Ferrie, and D. G. Cory, Accelerated randomized benchmarking , New J. Phys. 17

Original