- 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 χ2 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 χ2
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.