- NEW: Textbook in High Dimensional Probability, ongoing project. Feedback welcome!
- R. Vershynin,
Estimation in high dimensions: a geometric perspective.
*Sampling Theory, a Renaissance*, 3--66, Birkhauser Basel, 2015. [arXiv:1405.5103, May 2014] - R. Vershynin,
Introduction to the non-asymptotic analysis of random matrices.
*Compressed sensing*, 210--268, Cambridge Univ. Press, Cambridge, 2012. [arXiv:1011.3027, Aug 2010]This tutorial partially supersedes my old lecture notes, except for some geometric topics which are going to be included in the

*Lecture notes in geometric functional analysis*(below). - M. Rudelson, R. Vershynin,
Non-asymptotic theory of random matrices: extreme singular values,
*Proceedings of the International Congress of Mathematicians*. Volume III, 1576--1602, Hindustan Book Agency, New Delhi, 2010. - Lectures in functional analysis (2010).
- Lectures in geometric functional analysis (2009).
- Lecture notes in probability theory (2007-2008).
- R. Vershynin,
On the role of sparsity in Compressed Sensing and Random Matrix Theory,
*CAMSAP'09 (3rd International Workshop on Computational Advances in Multi-Sensor Adaptive Processing)*, 2009, 189--192. Commentary. - D. Needell, J. A. Tropp, and R. Vershynin,
Greedy signal recovery review,
*42nd Asilomar Conference on Signals, Systems and Computers*, 2008, 1048--1050. - R. Vershynin,
Some problems in asymptotic convex geometry
and random matrices motivated by numerical algorithms.
*Banach spaces and their applications in analysis*, 209--218, Walter de Gruyter, Berlin, 2007. - M. Rudelson, R. Vershynin, Random processes via the combinatorial dimension: introductory notes. Unpublished, 2004.

- E. Rebrova, R. Vershynin, Norms of random matrices: local and global problems, submitted. [arXiv:1608.06953].
- J. Banks, C. Moore, R. Vershynin, J. Xu, Information-theoretic bounds and phase transitions in clustering, sparse PCA, and submatrix localization, submitted. [arXiv:1607.05222].
- C. Liaw, A. Mehrabian, Y. Plan, R. Vershynin,
A simple tool for bounding the deviation of random matrices on geometric sets,
*Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics*, Springer, Berlin, to appear. [arXiv:1603.00897]. - M. Rudelson, R. Vershynin,
No-gaps delocalization for general random matrices,
*Geometric and Functional Analysis*, to appear. [arXiv:1506.04012]. - C. Le, E. Levina, R. Vershynin,
Concentration and regularization of random graphs,
*Random Structures and Algorithms*, to appear. [arXiv:1506.00669]. - Y. Plan, R. Vershynin,
The generalized Lasso with non-linear observations,
*IEEE Transactions on Information Theory*, to appear. [arXiv:1502.04071]. - C. Le, E. Levina, R. Vershynin, Sparse random graphs: regularization and concentration of the Laplacian, submitted. [arXiv:1502.03049]. This manuscript was not published since it was superseded by our newer paper.
- O. Guedon, R. Vershynin,
Community detection in sparse networks via Grothendieck's inequality,
*Probability Theory and Related Fields*165 (2016), 1025--1049. [arXiv:1411.4686]. - C. Le, E. Levina, R. Vershynin,
Optimization via low-rank approximation, with applications to
community detection in networks,
*Annals of Statistics*44 (2016), 373--400. [arXiv:1406.0067]. Supplement. - B. Farrell, R. Vershynin,
Smoothed analysis of symmetric random matrices with continuous distributions,
*Proceedings of the AMS*144 (2016), 2257--2261. [arXiv:1212.3531]. Hagelstein's weak triangle inequality for weak L1 norm used in this paper is discussed here: - R. Vershynin, Weak triangle inequalities for weak L1 norm, not intended for publication.
- R. Giryes, Y. Plan, R. Vershynin,
On the effective measure of dimension in analysis cosparse model,
*IEEE Transactions on Information Theory*61 (2015), 5745--5753. [arXiv:1410.0989].Conference version: On the effective measure of dimension in total variation minimization, in:

*Sampling Theory and Applications (SampTA)*, 2015, IEEE, 593--597. - Y. Plan, R. Vershynin, E. Yudovina,
High-dimensional estimation with geometric constraints,
*Information and Inference*0 (2016), 1--40. [arXiv:1404.3749]. - M. Rudelson, R. Vershynin,
Small ball probabilities for linear images of high dimensional distributions,
*International Mathematics Research Notices*2015, no. 19, 9594--9617. [arXiv:1402.4492]. - M. Rudelson, R. Vershynin,
Delocalization of eigenvectors of random matrices with independent entries,
*Duke Mathematical Journal*164 (2015), 2507--2538. [arXiv:1306.2887]. - A. Ai, A. Lapanowski, Y. Plan and R. Vershynin,
One-bit compressed sensing with non-Gaussian measurements,
*Linear Algebra and Applications*441 (2014), 222--239. [arXiv:1208.6279]. - M. Rudelson, R. Vershynin, Invertibility of random matrices: unitary and orthogonal perturbations,
*Journal of the AMS*27 (2014), 293--338. [arXiv:1202.1212]. - Y. Plan, R. Vershynin, Dimension reduction by random hyperplane tessellations,
*Discrete and Computational Geometry*51 (2014), 438–-461. [arXiv:1111.4452]. - R. Vershynin,
Invertibility of symmetric random matrices,
*Random Structures and Algorithms*44 (2014), 135--182. [arXiv:1102.0300]. - M. Rudelson, R. Vershynin,
Hanson-Wright inequality and sub-gaussian concentration,
*Electronic Communications in Probability*18 (2013), 1-9. [arXiv:1306.2872]. - Y. Plan, R. Vershynin, Robust 1-bit compressed sensing and sparse logistic regression:
a convex programming approach,
*IEEE Transactions on Information Theory*59 (2013), 482--494. [arXiv:1202.1212]. - Y. Plan, R. Vershynin, One-bit compressed sensing by linear programming,
*Communications on Pure and Applied Mathematics*66 (2013), 1275--1297. [arXiv:1109.4299]. See also 1BitCompressiveSensing page at Rice. - N. Srivastava, R. Vershynin,
Covariance estimation for distributions with 2+epsilon moments,
*Annals of Probability*41 (2013), 3081--3111. - R. Vershynin, A tribute to Nigel J. Kalton (1946-2010),
Peter Casazza (coordinating editor), Alexander Pelczynski, Joe Diestel, Roman Vershynin, Gilles Godefroy,
*Notices of the AMS*59 (7), August 2012, 942--951. - E. Levina, R. Vershynin,
Partial estimation of covariance matrices,
*Probability Theory and Related Fields*153 (2012), 405--419. [arXiv:1008.1716]. - R. Vershynin,
How close is the sample covariance matrix
to the actual covariance matrix?
*Journal of Theoretical Probability*25 (2012), 655--686. [arXiv:1004.3484]. - R. Vershynin,
A simple decoupling inequality in probability theory,
*not intended for publication*(2011). - R. Vershynin,
Approximating the moments of marginals of high dimensional distributions,
*Annals of Probability*39 (2011), 1591--1606. - R. Vershynin,
Spectral norm of products of random and deterministic matrices,
*Probability Theory and Related Fields*150 (2011), 471--509. DOI: 10.1007/s00440-010-0281-z. Commentary. -
Yu. Lyubarskii, R. Vershynin,
Uncertainty principles and vector quantization,
*IEEE Transactions on Information Theory*56 (2010), 3491--3501. Commentary. Slides. - T. Strohmer, R. Vershynin,
Comments on the randomized Kaczmarz method,
*Journal of Fourier Analysis and Applications*15 (2009), 437--440. Commentary. - M. Rudelson, R. Vershynin,
Smallest singular value of a random rectangular matrix,
*Communications on Pure and Applied Mathematics*62 (2009), 1707--1739. Commentary. Slides. -
D. Needell, R. Vershynin,
Signal recovery from incomplete and inaccurate measurements
via Regularized Orthogonal Matching Pursuit,
*IEEE Journal of Selected Topics in Signal Processing*4 (2010), 310--316. Commentary. A conference version: D. Needell, R. Vershynin, Signal recovery from incomplete and inaccurate measurements via ROMP,*SAMPTA'09 (8th international conference on Sampling Theory and Applications)*, 2009.One more conference version: D. Needell, R. Vershynin, Greedy signal recovery and uncertainty principles,

*Computational Imaging VI, IS&T/SPIE 19th Annual Symposium*, 2008. Proc. SPIE Vol. 6814, 68140J. -
D. Needell, R. Vershynin,
Uniform Uncertainty Principle and signal recovery
via Regularized Orthogonal Matching Pursuit,
*Foundations of Computational Mathematics*9 (2009), 317--334. Commentary. - M. Rudelson, R. Vershynin,
The least singular value of a random square matrix is O(n^{-1/2}),
*Comptes rendus de l'Académie des sciences - Mathématique*346 (2008), 893--896. Commentary. - M. Rudelson, R. Vershynin,
The Littlewood-Offord Problem and invertibility of random matrices,
*Advances in Mathematics*218 (2008), 600--633. Commentary. - M. Rudelson, R. Vershynin,
Sampling from large matrices: an approach through geometric functional analysis,
*Journal of the ACM*(2007), Art. 21, 19 pp. Commentary. - A. Gilbert, M. Strauss, J. Tropp, R. Vershynin,
One sketch for all: fast algorithms for compressed sensing,
*STOC'07--Proceedings of the 39th Annual ACM Symposium on Theory of Computing*, 237--246, ACM, New York, 2007. Complete version of the paper. - B. Klartag, R. Vershynin,
Small ball probability and Dvoretzky theorem,
*Israel Journal of Mathematics*157 (2007), 193--207. Commentary. - T. Strohmer, R. Vershynin,
A randomized solver for linear systems with exponential convergence,
*Approximation, randomization and combinatorial optimization*, 499--507, Lecture Notes in Comput. Sci., 4110, Springer, Berlin, 2006. CommentaryJournal version: A randomized Kaczmarz algorithm with exponential convergence,

*Journal of Fourier Analysis and Applications*15 (2009), 262--278. - R. Vershynin,
Beyond Hirsch Conjecture: walks on random polytopes
and smoothed complexity of the simplex method,
*FOCS 2006 (47th Annual Symposium on Foundations of Computer Science)*, 133--142. CommentaryJournal version: Beyond Hirsch Conjecture: walks on random polytopes and smoothed complexity of the simplex method,

*SIAM Journal on Computing*39 (2009), 646--678 (Invited paper). -
A. Gilbert, M. Strauss, J. Tropp, R. Vershynin,
Algorithmic linear dimension reduction in the ell-1 norm for sparse vectors,
*unpublished*. Commentary.Conference version: Algorithmic Linear Dimension Reduction in the L1 Norm for Sparse Vectors,

*Allerton 2006 (44th Annual Allerton Conference on Communication, Control, and Computing)*, invited paper. - M. Rudelson, R. Vershynin,
Sparse reconstruction by convex relaxation: Fourier and Gaussian measurements,
*CISS'06 (40th Annual Conference on Information Sciences and Systems)*, 2006, 207--212 Commentary.Journal version: On sparse reconstruction from Fourier and Gaussian measurements,

*Communications on Pure and Applied Mathematics*61 (2008), 1025-1045. Commentary. - R. Vershynin,
Random sets of isomorphism of linear operators on Hilbert space,
*High dimensional probability*, 148--154, IMS Lecture Notes Monogr. Ser., 51, Inst. Math. Statist., Beachwood, OH, 2006. Commentary. - R. Vershynin,
Isoperimetry of waists and local versus global asymptotic convex geometries
(with an appendix by M. Rudelson and R. Vershynin),
*Duke Mathematical Journal*131 (2006), 1--16. Commentary. - M. Rudelson, R. Vershynin,
Combinatorics of random processes and sections of convex bodies,
*Annals of Mathematics*164 (2006), 603--648. Commentary. Introductory Notes. - M. Rudelson, R. Vershynin,
Geometric approach to error correcting codes and reconstruction of signals,
*International Mathematical Research Notices*64 (2005), 4019--4041. Commentary.Conference version: E. Candes, M. Rudelson, T. Tao, R. Vershynin, Error correction via Linear Programming,

*FOCS 2005 (46th Annual Symposium on Foundations of Computer Science)*, 668--681. - R. Vershynin,
Frame expansions with erasures: an approach through the non-commutative
operator theory,
*Applied and Computational Harmonic Analysis*18 (2005), 167--176 Commentary. - R. Vershynin,
Integer cells in convex sets,
*Advances in Mathematics*197 (2005), 248--273. - P. G. Casazza, O. Christensen, A. M. Lindner, R. Vershynin,
Frames and the Feichtinger conjecture,
*Proceedings of the AMS*133 (2005), 1025--1033. - A. Litvak, A. Pajor, M. Rudelson, N. Tomczak-Jaegermann, R. Vershynin,
Euclidean embeddings in spaces of finite volume ratio via random matrices,
*Journal für die Reine und Angewandte Mathematik*589 (2005), 1--19. - A. Litvak, A. Pajor, M. Rudelson, N. Tomczak-Jaegermann, R. Vershynin,
Random Euclidean embeddings in spaces of bounded volume ratio,
*Comptes Rendus de l'Académie des sciences - Mathématique*339 (2004), 33--38. - S. Mendelson, R. Vershynin,
Remarks on the geometry of coordinate projections in R^n,
*Israel Journal of Mathematics*140 (2004), 203--220. - S. Mendelson, R. Vershynin,
Entropy and the combinatorial dimension,
*Inventiones Mathematicae*152 (2003), 37--55. - G. Schechtman, N. Tomczak-Jaegermann, R. Vershynin,
Maximal l_p^n-structures in spaces with extremal parameters,
Geometric aspects of functional analysis, 223--240,
*Lecture Notes in Mathematics*, 1807, Springer, Berlin, 2003. - M. Rudelson, R. Vershynin,
Embedding Levy families into Banach spaces,
*Geometric and Functional Analysis*12 (2002), 183--198. - R. Vershynin,
John's decompositions: selecting a large part,
*Israel Journal of Mathematics*122 (2001), 253--277. - S. Mendelson, R. Vershynin,
Entropy, combinatorial dimensions and random averages.
In:
*COLT 2002 (15th Annual Conference on Computational Learning Theory)*, edited by J. Kivinen and R. Sloan, Springer 2002, p. 14--28. - N. Tomczak-Jaegermann, R. Vershynin, Sizes of projections of symmetric convex bodies, unpublished (2001).
- R. Vershynin,
Subsequences of frames,
*Studia Mathematica*145 (2001), 185--197. - R. Vershynin,
On large random almost Euclidean bases,
*Acta Mathematica Universitatis Comenianae*69 (2000), no. 2, 137--144. - R. Vershynin,
Absolutely representing systems, uniform smoothness and type,
*Questionae Mathematica*23 (2000), 87--98. - R. Vershynin,
On constructions of strong and uniformly minimal M-bases in Banach spaces,
*Archiv der Mathematik (Basel)*74 (2000), 50--60. - R. Vershynin,
Imbedding of the images of operators and reflexivity of Banach spaces (Russian),
*Ukrainian Mathematical Journal*51 (1999), 293--296. - R. Vershynin,
On 1+epsilon_n-bounded M-bases (Russian),
*Russian Mathematics (Izvestiya VUZ. Matematika)*43 (1999), 22--25. - R. Vershynin,
On representing and absolutely representing systems in Banach spaces (Russian),
*Matematicheskaya Fizika, Analiz, Geometriya*5 (1998), 3--14. - V. Kadets, R. Vershynin,
The selection problem for bases with brackets and for strong M-bases,
*Rendiconti Istituto Lombardo Sez. A*131 (1997), 169--177.