- Kamalika Chaudhuri, Chuan Guo, Laurens van der Maaten, Saeed Mahloujifar,
and Mark Tygert,
"Guarantees of confidentiality via Hammersley-Chapman-Robbins bounds,"
*Transactions on Machine Learning Research*, no. 2496: 1-20, 2024: pdf.This article replaces Fisher information privacy with a nearly uniformly superior alternative, using Hammersley-Chapman-Robbins bounds rather than Cramér-Rao.

- François Charton, Kristin Lauter, Cathy Li, and Mark Tygert,
"An efficient algorithm for integer lattice reduction,"
*SIAM Journal on Matrix Analysis and Applications*, 45 (1): 353-367, 2024: pdf.This article proposes iterations to polish the results of other algorithms for lattice reduction and demonstrates the polishing on an especially efficient numerical method for Lenstra-Lenstra-Lovász (LLL) reduction in floating-point arithmetic.

- Mark Tygert,
"Calibration of P-values for calibration and for deviation of a subpopulation
from the full population,"
*Advances in Computational Mathematics*, 49 (70): 1-22, 2023: pdf.This article details calibration of attained significance levels (also known as "P-values") for formal tests of significance related to recently developed analogues of the Kolmogorov-Smirnov and Kuiper metrics. The article also details how to calculate cumulative statistics for data with scores (independent variables in a regression) that may not be unique by instead calculating the cumulative statistics for a weighted data set with scores that are unique by construction.

- Imanol Arrieta Ibarra, Paman Gujral, Jonathan Tannen, Mark Tygert,
and Cherie Xu,
"Metrics of calibration for probabilistic predictions,"
*Journal of Machine Learning Research*, 23: 1-54, 2022: pdf.This article shows the dramatic superiority of the cumulative metrics of Kuiper and of Kolmogorov and Smirnov over the canonical empirical calibration errors (also known as "estimated calibration errors").

- Aaron Defazio, Mark Tygert, Rachel Ward, and Jure Zbontar,
"Compressed sensing with a jackknife, a bootstrap, and visualization,"
*Journal of Data Science, Statistics, and Visualisation*, 2 (4): 1-29, 2022: pdf.This article simulates on a computer what could have happened with measurements that we might have taken but actually did not, given what happened with the measurements that we did in fact make.

- Mark Tygert,
"A graphical method of cumulative differences between two subpopulations,"
*Journal of Big Data*, 8 (158): 1-29, 2021: pdf.This article extends an earlier paper, "Cumulative deviation of a subpopulation from the full population," to assessing directly deviation between two subpopulations whose scores are all distinct. The previous paper is preferable when apposite.

- Mark Tygert,
"Cumulative deviation of a subpopulation from the full population,"
*Journal of Big Data*, 8 (117): 1-60, 2021: pdf.Graphs of cumulative differences and associated variants of the Kolmogorov-Smirnov and Kuiper statistics help gauge calibration or subpopulation deviation without making any particular tradeoff between resolution and statistical confidence (unlike the traditional reliability diagrams and calibration plots).

- Chuan Guo, Awni Hannun, Brian Knott, Laurens van der Maaten, Mark Tygert,
and Ruiyu Zhu,
"Secure multiparty computations in floating-point arithmetic,"
*Information and Inference: a Journal of the IMA*, iaaa038: 1-33, 2021: pdf.This article introduces secure multiparty computations for privacy-preserving machine-learning using solely standard floating-point arithmetic, with carefully controlled leakage of information less than the loss of accuracy due to roundoff, all backed by rigorous mathematical proofs of worst-case bounds on information loss and numerical stability in finite-precision arithmetic.

- Mark Tygert and Jure Zbontar,
"Simulating single-coil MRI from the responses of multiple coils,"
*Communications in Applied Mathematics and Computational Science*, 15 (2): 115-127, 2020: pdf.This article is strange, yet representative of the curious, idiosyncratic tasks that pay the bills in industry, which often require knowledgeable, special-purpose solutions.

- Cinna Wu, Mark Tygert, and Yann LeCun,
"A hierarchical loss and its problems when classifying non-hierarchically,"
*PLOS ONE*, 14 (12): 1-17, 2019: pdf.This article defines an easy-to-use metric of success or figure of merit for classification in which the classes come endowed with a hierarchical taxonomy.

- Mark Tygert,
"Regression-aware decompositions,"
*Linear Algebra and Its Applications*, 565 (6): 208-224, 2019: pdf.This article constructs matrix decompositions which leverage simultaneously a single data set's intrinsic low-rank structure as well as low-rank structure in the data's interaction with another data set.

- Huamin Li, Yuval Kluger, and Mark Tygert,
"Randomized algorithms for distributed computation of principal component
analysis and singular value decomposition,"
*Advances in Computational Mathematics*, 44 (5): 1651-1672, 2018: pdf.This article describes an implementation for Spark of principal component analysis, singular value decomposition, and low-rank approximation.

- Arthur Szlam, Andrew Tulloch, and Mark Tygert,
"Accurate low-rank approximations via a few iterations
of alternating least squares,"
*SIAM Journal on Matrix Analysis and Applications*, 38 (2): 425-433, 2017: pdf.This article points out that a few iterations of alternating least squares provably suffice to produce nearly optimal spectral- and Frobenius-norm accuracies of low-rank approximations.

- Soumith Chintala, Marc'Aurelio Ranzato, Arthur Szlam, Yuandong Tian,
Mark Tygert, and Wojciech Zaremba,
"Scale-invariant learning and convolutional networks,"
*Applied and Computational Harmonic Analysis*, 42 (1): 154-166, 2017: pdf.This article develops a classification stage specifically for use with supervised learning in scale-equivariant convolutional networks.

- Joan Bruna, Soumith Chintala, Yann LeCun, Serkan Piantino, Arthur Szlam,
and Mark Tygert,
"A mathematical motivation for complex-valued convolutional networks,"
*Neural Computation*, 28 (5): 815-825, 2016: pdf.This article embeds multiwavelet absolute values in parametric families of complex-valued convnets.

- William Perkins, Mark Tygert, and Rachel Ward,
"Some deficiencies of χ
^{2}and classical exact tests of significance,"*Applied and Computational Harmonic Analysis*, 36 (3): 361-386, 2014: pdf.This article points out that the Euclidean distance is underutilized in modern statistics.

- William Perkins, Mark Tygert, and Rachel Ward,
"Computing the confidence levels for a root-mean-square test
of goodness-of-fit,"
*Applied Mathematics and Computation*, 217 (22): 9072-9084, 2011: pdf, ps.This article works out some asymptotic distributions associated with the article, "Some deficiencies of χ

^{2}and classical exact tests of significance," available above. - Edouard Coakley, Vladimir Rokhlin, and Mark Tygert,
"A fast randomized algorithm for orthogonal projection,"
*SIAM Journal on Scientific Computing*, 33 (2): 849-868, 2011: pdf.This article can help accelerate interior-point methods for convex optimization, such as linear programming.

- Mark Tygert,
"Statistical tests for whether a given set of independent, identically
distributed draws comes from a specified probability density,"
*Proceedings of the National Academy of Sciences*, 107 (38): 16471-16476, 2010: pdf, ps.This article modifies and supplements tests of the Kolmogorov-Smirnov type (including Kuiper's).

- Mark Tygert,
"Fast algorithms for spherical harmonic expansions, III,"
*Journal of Computational Physics*, 229 (18): 6181-6192, 2010: pdf.This article simplifies the precomputations required for computing fast spherical harmonic transforms, complementing the approach taken in the article, "Fast algorithms for spherical harmonic expansions, II," available below.

- Vladimir Rokhlin, Arthur Szlam, and Mark Tygert,
"A randomized algorithm for principal component analysis,"
*SIAM Journal on Matrix Analysis and Applications*, 31 (3): 1100-1124, 2009: pdf.This article is now out-of-date; instead, please see Nathan Halko's, Per-Gunnar Martinsson's, and Joel Tropp's SIAM Review paper.

- Franco Woolfe, Edo Liberty, Vladimir Rokhlin, and Mark Tygert,
"A fast randomized algorithm for the approximation of matrices,"
*Applied and Computational Harmonic Analysis*, 25 (3): 335-366, 2008: pdf.This article provides a generally preferable alternative to the classical pivoted "

*QR*" decomposition algorithms (such as Gram-Schmidt or Householder) for the low-rank approximation of arbitrary matrices. Constructing a low-rank approximation is the core step in computing several of the greatest singular values and corresponding singular vectors of a matrix. - Vladimir Rokhlin and Mark Tygert,
"A fast randomized algorithm for overdetermined linear least-squares
regression,"
*Proceedings of the National Academy of Sciences*, 105 (36): 13212-13217, 2008: pdf.This article provides an algorithm for linear least-squares regression. When the regression is highly overdetermined, the algorithm is more efficient than the classical methods based on "

*QR*" decompositions. - Mark Tygert,
"Fast algorithms for spherical harmonic expansions, II,"
*Journal of Computational Physics*, 227 (8): 4260-4279, 2008: pdf.This article provides efficient algorithms for computing spherical harmonic transforms, largely superseding our first article on the subject, "Fast algorithms for spherical harmonic expansions."

- Edo Liberty, Franco Woolfe, Per-Gunnar Martinsson, Vladimir Rokhlin,
and Mark Tygert,
"Randomized algorithms for the low-rank approximation of matrices,"
*Proceedings of the National Academy of Sciences*, 104 (51): 20167-20172, 2007: pdf.This article surveys algorithms for the compression of matrices.

- Per-Gunnar Martinsson, Vladimir Rokhlin, and Mark Tygert,
"A randomized algorithm for the decomposition of matrices,"
*Applied and Computational Harmonic Analysis*, 30 (1): 47-68, 2011: pdf.This article provides details regarding the survey, "Randomized algorithms for the low-rank approximation of matrices," available above. "A randomized algorithm for the decomposition of matrices" is almost identical to our (better-known) technical report, "A randomized algorithm for the approximation of matrices," from 2006.

- Per-Gunnar Martinsson, Vladimir Rokhlin, and Mark Tygert,
"On interpolation and integration in finite-dimensional spaces
of bounded functions,"
*Communications in Applied Mathematics and Computational Science*, 1: 133-142, 2006: CAMCoS.This article reviews the fact that numerically stable formulae exist for interpolating any linear combination of

*n*bounded functions using the values of the linear combination at a certain collection of*n*points in the domain of the functions. The article also provides references to algorithms which determine these stable formulae at reasonably small computational expense. - Vladimir Rokhlin and Mark Tygert,
"Fast algorithms for spherical harmonic expansions,"
*SIAM Journal on Scientific Computing*, 27 (6): 1903-1928, 2006: pdf.This article is now largely (but not entirely) superseded by the paper, "Fast algorithms for spherical harmonic expansions, II," available above.