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			@article{boumal2018deterministic,
			title={Deterministic guarantees for Burer-Monteiro factorizations of smooth semidefinite programs},
			author={Boumal, Nicolas and Voroninski, Vladislav and Bandeira, Afonso S},
			journal={arXiv preprint arXiv:1804.02008},
			year={2018}
			}

			@inproceedings{Samangouei2018,
			title={Defense-{GAN}: Protecting Classifiers Against Adversarial Attacks Using Generative Models},
			author={Pouya Samangouei and Maya Kabkab and Rama Chellappa},
			booktitle={International Conference on Learning Representations},
			year={2018},
			url={https://openreview.net/forum?id=BkJ3ibb0-},
			}


			@ARTICLE{Kingma2014,
			author = {{Kingma}, Diederik P. and {Ba}, Jimmy},
			title = "{Adam: A Method for Stochastic Optimization}",
			journal = {arXiv e-prints},
			keywords = {Computer Science - Machine Learning},
			year = 2014,
			month = Dec,
			eid = {arXiv:1412.6980},
			pages = {arXiv:1412.6980},
			archivePrefix = {arXiv},
			eprint = {1412.6980},
			primaryClass = {cs.LG},
			adsurl = {https://ui.adsabs.harvard.edu/\#abs/2014arXiv1412.6980K},
			adsnote = {Provided by the SAO/NASA Astrophysics Data System}
			}


			@article{Radford2015,
			author = {{Radford}, A. and {Metz}, L. and {Chintala}, S.},
			title = "{Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks}",
			journal = {ArXiv e-prints},
			archivePrefix = "arXiv",
			eprint = {1511.06434},
			keywords = {Computer Science - Machine Learning, Computer Science - Computer Vision and Pattern Recognition},
			year = 2015,
			month = nov,
			adsurl = {http://adsabs.harvard.edu/abs/2015arXiv151106434R},
			adsnote = {Provided by the SAO/NASA Astrophysics Data System}
			}

			@article{ferreira2018semidefinite,
			title={Semidefinite programming approach for the quadratic assignment problem with a sparse graph},
			author={Ferreira, Jos{\'e} FS Bravo and Khoo, Yuehaw and Singer, Amit},
			journal={Computational Optimization and Applications},
			volume={69},
			number={3},
			pages={677--712},
			year={2018},
			publisher={Springer}
			}


			@article{rockafellar1993lagrange,
			title={Lagrange multipliers and optimality},
			author={Rockafellar, R Tyrrell},
			journal={SIAM review},
			volume={35},
			number={2},
			pages={183--238},
			year={1993},
			publisher={SIAM}
			}

			@book{birgin2014practical,
			title={Practical augmented Lagrangian methods for constrained optimization},
			author={Birgin, Ernesto G and Mart\_nez, Jos{\^a} Mario},
			volume={10},
			year={2014},
			publisher={SIAM}
			}

			@article{bertsekas1982constrained,
			title={Constrained optimization and Lagrange Multiplier methods},
			author={Bertsekas, Dimitri P},
			journal={Computer Science and Applied Mathematics, Boston: Academic Press, 1982},
			year={1982}
			}

			@inproceedings{karimi2016linear,
			title={Linear convergence of gradient and proximal-gradient methods under the polyak-{\l}ojasiewicz condition},
			author={Karimi, Hamed and Nutini, Julie and Schmidt, Mark},
			booktitle={Joint European Conference on Machine Learning and Knowledge Discovery in Databases},
			pages={795--811},
			year={2016},
			organization={Springer}
			}

			@article{zhao1998semidefinite,
			title={Semidefinite programming relaxations for the quadratic assignment problem},
			author={Zhao, Qing and Karisch, Stefan E and Rendl, Franz and Wolkowicz, Henry},
			journal={Journal of Combinatorial Optimization},
			volume={2},
			number={1},
			pages={71--109},
			year={1998},
			publisher={Springer}
			}
			@article{Goodfellow2014,
			author = {{Goodfellow}, I.~J. and {Pouget-Abadie}, J. and {Mirza}, M. and
			{Xu}, B. and {Warde-Farley}, D. and {Ozair}, S. and {Courville}, A. and
			{Bengio}, Y.},
			url = {https://arxiv.org/abs/1406.2661},
			title = "{Generative Adversarial Networks}",
			journal = {ArXiv e-prints},
			archivePrefix = "arXiv",
			eprint = {1406.2661},
			primaryClass = "stat.ML",
			keywords = {Statistics - Machine Learning, Computer Science - Machine Learning},
			year = 2014,
			month = jun,
			adsurl = {http://adsabs.harvard.edu/abs/2014arXiv1406.2661G},
			adsnote = {Provided by the SAO/NASA Astrophysics Data System}
			}

			@ARTICLE{Ilyas2017,
			author = {{Ilyas}, Andrew and {Jalal}, Ajil and {Asteri}, Eirini and {Daskalakis},
			Constantinos and {Dimakis}, Alexandros G.},
			title = "{The Robust Manifold Defense: Adversarial Training using Generative Models}",
			journal = {arXiv e-prints},
			keywords = {Computer Science - Computer Vision and Pattern Recognition, Computer Science - Cryptography and Security, Computer Science - Machine Learning, Statistics - Machine Learning},
			year = 2017,
			month = Dec,
			eid = {arXiv:1712.09196},
			pages = {arXiv:1712.09196},
			archivePrefix = {arXiv},
			eprint = {1712.09196},
			primaryClass = {cs.CV},
			adsurl = {https://ui.adsabs.harvard.edu/\#abs/2017arXiv171209196I},
			adsnote = {Provided by the SAO/NASA Astrophysics Data System}
			}

			@ARTICLE{Szegedy2013,
			author = {{Szegedy}, Christian and {Zaremba}, Wojciech and {Sutskever}, Ilya and
			{Bruna}, Joan and {Erhan}, Dumitru and {Goodfellow}, Ian and
			{Fergus}, Rob},
			title = "{Intriguing properties of neural networks}",
			journal = {arXiv e-prints},
			keywords = {Computer Science - Computer Vision and Pattern Recognition, Computer Science - Machine Learning, Computer Science - Neural and Evolutionary Computing},
			year = 2013,
			month = Dec,
			eid = {arXiv:1312.6199},
			pages = {arXiv:1312.6199},
			archivePrefix = {arXiv},
			eprint = {1312.6199},
			primaryClass = {cs.CV},
			adsurl = {https://ui.adsabs.harvard.edu/\#abs/2013arXiv1312.6199S},
			adsnote = {Provided by the SAO/NASA Astrophysics Data System}
			}



			@article{tepper2018clustering,
			title={Clustering is semidefinitely not that hard: Nonnegative SDP for manifold disentangling},
			author={Tepper, Mariano and Sengupta, Anirvan M and Chklovskii, Dmitri},
			journal={Journal of Machine Learning Research},
			volume={19},
			number={82},
			year={2018}
			}

			@article{beck2009fast,
			title={A fast iterative shrinkage-thresholding algorithm for linear inverse problems},
			author={Beck, Amir and Teboulle, Marc},
			journal={SIAM journal on imaging sciences},
			volume={2},
			number={1},
			pages={183--202},
			year={2009},
			publisher={SIAM}
			}

			@article{tibshirani1996regression,
			title={Regression shrinkage and selection via the lasso},
			author={Tibshirani, Robert},
			journal={Journal of the Royal Statistical Society. Series B (Methodological)},
			pages={267--288},
			year={1996},
			publisher={JSTOR}
			}


			@book{bauschke2011convex,
			title={Convex analysis and monotone operator theory in Hilbert spaces},
			author={Bauschke, Heinz H and Combettes, Patrick L and others},
			volume={408},
			year={2011},
			publisher={Springer}
			}


			@article{waldspurger2018rank,
			title={Rank optimality for the Burer-Monteiro factorization},
			author={Waldspurger, Ir{\`e}ne and Waters, Alden},
			journal={arXiv preprint arXiv:1812.03046},
			year={2018}
			}


			@inproceedings{jaggi2013revisiting,
			title={Revisiting Frank-Wolfe: Projection-Free Sparse Convex Optimization.},
			author={Jaggi, Martin},
			booktitle={ICML (1)},
			pages={427--435},
			year={2013}
			}



			@inproceedings{nesterov1983method,
			title={A method for solving the convex programming problem with convergence rate O (1/k\^{} 2)},
			author={Nesterov, Yurii E},
			booktitle={Dokl. Akad. Nauk SSSR},
			volume={269},
			pages={543--547},
			year={1983}
			}


			@inproceedings{raghavendra2008optimal,
			title={Optimal algorithms and inapproximability results for every CSP?},
			author={Raghavendra, Prasad},
			booktitle={Proceedings of the fortieth annual ACM symposium on Theory of computing},
			pages={245--254},
			year={2008},
			organization={ACM}
			}



			@online{xiao2017/online,
			author = {Han Xiao and Kashif Rasul and Roland Vollgraf},
			title = {Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms},
			date = {2017-08-28},
			year = {2017},
			eprintclass = {cs.LG},
			eprinttype = {arXiv},
			eprint = {cs.LG/1708.07747},
			}

			@article{cartis2011evaluation,
			title={On the evaluation complexity of composite function minimization with applications to nonconvex nonlinear programming},
			author={Cartis, Coralia and Gould, Nicholas IM and Toint, Philippe L},
			journal={SIAM Journal on Optimization},
			volume={21},
			number={4},
			pages={1721--1739},
			year={2011},
			publisher={SIAM}
			}

			@article{bolte2017error,
			title={From error bounds to the complexity of first-order descent methods for convex functions},
			author={Bolte, J{\'e}r{\^o}me and Nguyen, Trong Phong and Peypouquet, Juan and Suter, Bruce W},
			journal={Mathematical Programming},
			volume={165},
			number={2},
			pages={471--507},
			year={2017},
			publisher={Springer}
			}



			@article{obozinski2011group,
			title={Group lasso with overlaps: the latent group lasso approach},
			author={Obozinski, Guillaume and Jacob, Laurent and Vert, Jean-Philippe},
			journal={arXiv preprint arXiv:1110.0413},
			year={2011}
			}

			@article{tran2018smooth,
			title={A smooth primal-dual optimization framework for nonsmooth composite convex minimization},
			author={Tran-Dinh, Quoc and Fercoq, Olivier and Cevher, Volkan},
			journal={SIAM Journal on Optimization},
			volume={28},
			number={1},
			pages={96--134},
			year={2018},
			publisher={SIAM}
			}

			@article{tran2018adaptive,
			title={An Adaptive Primal-Dual Framework for Nonsmooth Convex Minimization},
			author={Tran-Dinh, Quoc and Alacaoglu, Ahmet and Fercoq, Olivier and Cevher, Volkan},
			journal={arXiv preprint arXiv:1808.04648},
			year={2018}
			}

			@article{arora2018compressed,
			title={A compressed sensing view of unsupervised text embeddings, bag-of-n-grams, and LSTMs},
			author={Arora, Sanjeev and Khodak, Mikhail and Saunshi, Nikunj and Vodrahalli, Kiran},
			year={2018}
			}

			@article{candes2008introduction,
			title={An introduction to compressive sampling},
			author={Cand{\`e}s, Emmanuel J and Wakin, Michael B},
			journal={IEEE signal processing magazine},
			volume={25},
			number={2},
			pages={21--30},
			year={2008},
			publisher={IEEE}
			}

			@article{chen2001atomic,
			title={Atomic decomposition by basis pursuit},
			author={Chen, Scott Shaobing and Donoho, David L and Saunders, Michael A},
			journal={SIAM review},
			volume={43},
			number={1},
			pages={129--159},
			year={2001},
			publisher={SIAM}
			}


			@article{flores2012complete,
			title={A complete characterization of strong duality in nonconvex optimization with a single constraint},
			author={Flores-Baz{\'a}n, Fabi{\'a}n and Flores-Baz{\'a}n, Fernando and Vera, Cristi{\'a}n},
			journal={Journal of Global Optimization},
			volume={53},
			number={2},
			pages={185--201},
			year={2012},
			publisher={Springer}
			}

			@article{dai2002convergence,
			title={Convergence properties of the BFGS algoritm},
			author={Dai, Yu-Hong},
			journal={SIAM Journal on Optimization},
			volume={13},
			number={3},
			pages={693--701},
			year={2002},
			publisher={SIAM}
			}

			@article{yang2015sdpnal,
			title={SDPNAL+: a majorized semismooth Newton-CG augmented Lagrangian method for semidefinite programming with nonnegative constraints},
			author={Yang, Liuqin and Sun, Defeng and Toh, Kim-Chuan},
			journal={Mathematical Programming Computation},
			volume={7},
			number={3},
			pages={331--366},
			year={2015},
			publisher={Springer}
			}


			@article{mascarenhas2004bfgs,
			title={The BFGS method with exact line searches fails for non-convex objective functions},
			author={Mascarenhas, Walter F},
			journal={Mathematical Programming},
			volume={99},
			number={1},
			pages={49--61},
			year={2004},
			publisher={Springer}
			}

			@inproceedings{boumal2016non,
			title={The non-convex Burer-Monteiro approach works on smooth semidefinite programs},
			author={Boumal, Nicolas and Voroninski, Vlad and Bandeira, Afonso},
			booktitle={Advances in Neural Information Processing Systems},
			pages={2757--2765},
			year={2016}
			}




			@inproceedings{bhojanapalli2016dropping,
			title={Dropping convexity for faster semi-definite optimization},
			author={Bhojanapalli, Srinadh and Kyrillidis, Anastasios and Sanghavi, Sujay},
			booktitle={Conference on Learning Theory},
			pages={530--582},
			year={2016}
			}


			@inproceedings{nesterov1983method,
			title={A method for solving the convex programming problem with convergence rate O (1/k\^{} 2)},
			author={Nesterov, Yurii E},
			booktitle={Dokl. Akad. Nauk SSSR},
			volume={269},
			pages={543--547},
			year={1983}
			}



			@article{shi2017penalty,
			title={Penalty dual decomposition method for nonsmooth nonconvex optimization},
			author={Shi, Qingjiang and Hong, Mingyi and Fu, Xiao and Chang, Tsung-Hui},
			journal={arXiv preprint arXiv:1712.04767},
			year={2017}
			}



			@article{fernandez2012local,
			title={Local convergence of exact and inexact augmented Lagrangian methods under the second-order sufficient optimality condition},
			author={Fernandez, Damian and Solodov, Mikhail V},
			journal={SIAM Journal on Optimization},
			volume={22},
			number={2},
			pages={384--407},
			year={2012},
			publisher={SIAM}
			}

			@article{bolte2018nonconvex,
			title={Nonconvex Lagrangian-based optimization: monitoring schemes and global convergence},
			author={Bolte, Jerome and Sabach, Shoham and Teboulle, Marc},
			journal={Mathematics of Operations Research},
			year={2018},
			publisher={INFORMS}
			}


			@article{nouiehed2018convergence,
			title={Convergence to Second-Order Stationarity for Constrained Non-Convex Optimization},
			author={Nouiehed, Maher and Lee, Jason D and Razaviyayn, Meisam},
			journal={arXiv preprint arXiv:1810.02024},
			year={2018}
			}



			@article{cartis2018optimality,
			title={Optimality of orders one to three and beyond: characterization and evaluation complexity in constrained nonconvex optimization},
			author={Cartis, Coralia and Gould, Nicholas IM and Toint, Ph L},
			journal={Journal of Complexity},
			year={2018},
			publisher={Elsevier}
			}



			@article{nesterov2009primal,
			title={Primal-dual subgradient methods for convex problems},
			author={Nesterov, Yurii},
			journal={Mathematical programming},
			volume={120},
			number={1},
			pages={221--259},
			year={2009},
			publisher={Springer}
			}

			@inproceedings{yurtsever2015universal,
			title={A universal primal-dual convex optimization framework},
			author={Yurtsever, Alp and Dinh, Quoc Tran and Cevher, Volkan},
			booktitle={Advances in Neural Information Processing Systems},
			pages={3150--3158},
			year={2015}
			}

			@article{yurtsever2018conditional,
			title={A Conditional Gradient Framework for Composite Convex Minimization with Applications to Semidefinite Programming},
			author={Yurtsever, Alp and Fercoq, Olivier and Locatello, Francesco and Cevher, Volkan},
			journal={arXiv preprint arXiv:1804.08544},
			year={2018}
			}


			@article{bertsekas1976penalty,
			title={On penalty and multiplier methods for constrained minimization},
			author={Bertsekas, Dimitri P},
			journal={SIAM Journal on Control and Optimization},
			volume={14},
			number={2},
			pages={216--235},
			year={1976},
			publisher={SIAM}
			}

			@article{tran2018adaptive,
			title={An Adaptive Primal-Dual Framework for Nonsmooth Convex Minimization},
			author={Tran-Dinh, Quoc and Alacaoglu, Ahmet and Fercoq, Olivier and Cevher, Volkan},
			journal={arXiv preprint arXiv:1808.04648},
			year={2018}
			}


			@article{hestenes1969multiplier,
			title={Multiplier and gradient methods},
			author={Hestenes, Magnus R},
			journal={Journal of optimization theory and applications},
			volume={4},
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			pages={303--320},
			year={1969},
			publisher={Springer}
			}


			@article{powell1969method,
			title={A method for nonlinear constraints in minimization problems},
			author={Powell, Michael JD},
			journal={Optimization},
			pages={283--298},
			year={1969},
			publisher={Academic Press}
			}



			@inproceedings{kulis2007fast,
			title={Fast low-rank semidefinite programming for embedding and clustering},
			author={Kulis, Brian and Surendran, Arun C and Platt, John C},
			booktitle={Artificial Intelligence and Statistics},
			pages={235--242},
			year={2007}
			}


			@article{xu2017inexact,
			title={Iteration complexity of inexact augmented Lagrangian methods for constrained convex programming},
			author={Xu, Yangyang},
			journal={arXiv preprint arXiv:1711.05812v2},
			year={2017}
			}



			@article{nedelcu2014computational,
			title={Computational complexity of inexact gradient augmented Lagrangian methods: application to constrained MPC},
			author={Nedelcu, Valentin and Necoara, Ion and Tran-Dinh, Quoc},
			journal={SIAM Journal on Control and Optimization},
			volume={52},
			number={5},
			pages={3109--3134},
			year={2014},
			publisher={SIAM}
			}


			@article{pataki1998rank,
			title={On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues},
			author={Pataki, G{\'a}bor},
			journal={Mathematics of operations research},
			volume={23},
			number={2},
			pages={339--358},
			year={1998},
			publisher={INFORMS}
			}


			@article{Barvinok1995problems,
			title={Problems of distance geometry and convex properties of quadratic maps},
			author={Barvinok, Alexander I.},
			journal={Discrete \& Computational Geometry},
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			number={2},
			pages={189--202},
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			publisher={Springer}
			}


			@article{cartis2012complexity,
			title={Complexity bounds for second-order optimality in unconstrained optimization},
			author={Cartis, Coralia and Gould, Nicholas IM and Toint, Ph L},
			journal={Journal of Complexity},
			volume={28},
			number={1},
			pages={93--108},
			year={2012},
			publisher={Elsevier}
			}



			@article{ghadimi2016accelerated,
			title={Accelerated gradient methods for nonconvex nonlinear and stochastic programming},
			author={Ghadimi, Saeed and Lan, Guanghui},
			journal={Mathematical Programming},
			volume={156},
			number={1-2},
			pages={59--99},
			year={2016},
			publisher={Springer}
			}

			@article{boumal2014manopt,
			Author = {Boumal, Nicolas and Mishra, Bamdev and Absil, P-A and Sepulchre, Rodolphe},
			Journal = {The Journal of Machine Learning Research},
			Number = {1},
			Pages = {1455--1459},
			Publisher = {JMLR. org},
			Title = {Manopt, a Matlab toolbox for optimization on manifolds},
			Volume = {15},
			Year = {2014}}

			@article{boumal2016global,
			Author = {Boumal, Nicolas and Absil, P-A and Cartis, Coralia},
			Journal = {arXiv preprint arXiv:1605.08101},
			Title = {Global rates of convergence for nonconvex optimization on manifolds},
			Year = {2016}}

			@inproceedings{ge2016efficient,
			Author = {Ge, Rong and Jin, Chi and Netrapalli, Praneeth and Sidford, Aaron and others},
			Booktitle = {International Conference on Machine Learning},
			Pages = {2741--2750},
			Title = {Efficient algorithms for large-scale generalized eigenvector computation and canonical correlation analysis},
			Year = {2016}}

			@inproceedings{boumal2016non,
			Author = {Boumal, Nicolas and Voroninski, Vlad and Bandeira, Afonso},
			Booktitle = {Advances in Neural Information Processing Systems},
			Pages = {2757--2765},
			Title = {The non-convex Burer-Monteiro approach works on smooth semidefinite programs},
			Year = {2016}}

			@article{davis2011university,
			Author = {Davis, Timothy A and Hu, Yifan},
			Journal = {ACM Transactions on Mathematical Software (TOMS)},
			Number = {1},
			Pages = {1},
			Publisher = {ACM},
			Title = {The University of Florida sparse matrix collection},
			Volume = {38},
			Year = {2011}}

			@article{clason2018acceleration,
			Author = {Clason, Christian and Mazurenko, Stanislav and Valkonen, Tuomo},
			Journal = {arXiv preprint arXiv:1802.03347},
			Title = {Acceleration and global convergence of a first-order primal--dual method for nonconvex problems},
			Year = {2018}}

			@article{chambolle2011first,
			Author = {Chambolle, Antonin and Pock, Thomas},
			Journal = {Journal of mathematical imaging and vision},
			Number = {1},
			Pages = {120--145},
			Publisher = {Springer},
			Title = {A first-order primal-dual algorithm for convex problems with applications to imaging},
			Volume = {40},
			Year = {2011}}

			@article{tran2018smooth,
			Author = {Tran-Dinh, Quoc and Fercoq, Olivier and Cevher, Volkan},
			Journal = {SIAM Journal on Optimization},
			Number = {1},
			Pages = {96--134},
			Publisher = {SIAM},
			Title = {A smooth primal-dual optimization framework for nonsmooth composite convex minimization},
			Volume = {28},
			Year = {2018}}

			@article{bhojanapalli2018smoothed,
			Author = {Bhojanapalli, Srinadh and Boumal, Nicolas and Jain, Prateek and Netrapalli, Praneeth},
			Journal = {arXiv preprint arXiv:1803.00186},
			Title = {Smoothed analysis for low-rank solutions to semidefinite programs in quadratic penalty form},
			Year = {2018}}

			@article{park2016provable,
			Author = {Park, Dohyung and Kyrillidis, Anastasios and Bhojanapalli, Srinadh and Caramanis, Constantine and Sanghavi, Sujay},
			Journal = {arXiv preprint arXiv:1606.01316},
			Title = {Provable Burer-Monteiro factorization for a class of norm-constrained matrix problems},
			Year = {2016}}

			@article{mixon2016clustering,
			Author = {Mixon, Dustin G and Villar, Soledad and Ward, Rachel},
			Journal = {arXiv preprint arXiv:1602.06612},
			Title = {Clustering subgaussian mixtures by semidefinite programming},
			Year = {2016}}

			@article{yurtsever2018conditional,
			Author = {Yurtsever, Alp and Fercoq, Olivier and Locatello, Francesco and Cevher, Volkan},
			Journal = {arXiv preprint arXiv:1804.08544},
			Title = {A Conditional Gradient Framework for Composite Convex Minimization with Applications to Semidefinite Programming},
			Year = {2018}}

			@inproceedings{jin2017escape,
			Author = {Jin, Chi and Ge, Rong and Netrapalli, Praneeth and Kakade, Sham M and Jordan, Michael I},
			Booktitle = {International Conference on Machine Learning},
			Pages = {1724--1732},
			Title = {How to Escape Saddle Points Efficiently},
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