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SUMMARY:Tensor decomposition and structured matrices
DTSTART;VALUE=DATE-TIME:20180514T130000Z
DTEND;VALUE=DATE-TIME:20180514T140000Z
DTSTAMP;VALUE=DATE-TIME:20200706T102324Z
UID:indico-contribution-2828-1603@indico.math.cnrs.fr
DESCRIPTION:Speakers: Bernard Mourrain (Inria Sophia-Antipolis)\nTensor de
composition problems appear in many areas such as Signal Processing\, Quan
tum Information Theory\, Algebraic Statistics\, Biology\, Complexity Analy
sis\, … as a way to recover hidden structures from data. The decompositi
on is a representation of the tensor as a weighted sum of a minimal number
of terms\, which are tensor products of vectors. We present an algebraic
approach to address this problem\, which involves duality\, Gorenstein Art
inian algebras and Hankel operators. We show the connection with low rank
decomposition of Hankel matrices\, discuss algebraic and optimization tech
niques to solve it and illustrate the methods on some examples.\n\nhttps:/
/indico.math.cnrs.fr/event/2828/contributions/1603/
LOCATION:ENS de Lyon Amphi A
URL:https://indico.math.cnrs.fr/event/2828/contributions/1603/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Structured matrix polynomials and their sign characteristics: clas
sical results and recent developments
DTSTART;VALUE=DATE-TIME:20180515T073000Z
DTEND;VALUE=DATE-TIME:20180515T083000Z
DTSTAMP;VALUE=DATE-TIME:20200706T102324Z
UID:indico-contribution-2828-1602@indico.math.cnrs.fr
DESCRIPTION:Speakers: Françoise Tisseur (University of Manchester)\nThe s
ign characteristic is an invariant associated with particular eigenvalues
of structured matrices\, matrix pencils\, or matrix polynomials. The conce
pt of sign characteristic arises in different forms in many scientific fie
lds\, and is essential for the stability analysis in Hamiltonian systems o
r the perturbation behaviour of eigenvalues under structured perturbations
. We will start by discussing the sign characteristics of Hermitian matrix
polynomials\, and show how to extend its definition to eigenvalues at inf
inity. We will discuss applications of the sign characteristic in particul
ar in control systems\, in the solution of structured inverse polynomial e
igenvalue problems and in the characterization of special structured matri
x polynomials such as overdamped quadratics\, hyperbolic and quasidefinite
matrix polynomials.\n\nhttps://indico.math.cnrs.fr/event/2828/contributio
ns/1602/
LOCATION:ENS de Lyon Amphi A
URL:https://indico.math.cnrs.fr/event/2828/contributions/1602/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Exploiting off-diagonal rank structures in the solution of linear
matrix equations
DTSTART;VALUE=DATE-TIME:20180515T113000Z
DTEND;VALUE=DATE-TIME:20180515T123000Z
DTSTAMP;VALUE=DATE-TIME:20200706T102324Z
UID:indico-contribution-2828-1601@indico.math.cnrs.fr
DESCRIPTION:Speakers: Stefano Massei (École Polytechnique Fédérale Laus
anne)\nLinear matrix equations\, namely Sylvester and Lyapunov equations\,
play an important role in several applications arising in control theory
and PDEs. In the large scale scenario\, it is crucial to exploit the struc
ture in the data in order to carry on the computations and store the final
solution. We focus on the case in which the coefficients have off-diagona
l blocks with low-rank and we study when this property is numerically pres
erved in the solution. Then\, we propose a divide and conquer scheme able
to exploit the structure\, reaching a linear-polylogarithmic complexity in
both time and memory consumption.\n\nhttps://indico.math.cnrs.fr/event/28
28/contributions/1601/
LOCATION:ENS de Lyon Amphi A
URL:https://indico.math.cnrs.fr/event/2828/contributions/1601/
END:VEVENT
BEGIN:VEVENT
SUMMARY:On minimal ranks and the approximate block-term tensor decompositi
on
DTSTART;VALUE=DATE-TIME:20180514T143000Z
DTEND;VALUE=DATE-TIME:20180514T153000Z
DTSTAMP;VALUE=DATE-TIME:20200706T102324Z
UID:indico-contribution-2828-1600@indico.math.cnrs.fr
DESCRIPTION:Speakers: José Henrique DE MORAIS GOULART (GIPSA-lab\, CNRS\,
Grenoble)\nThe block-term tensor decomposition (BTD) is a generalization
of the tensor rank (or canonical polyadic) decomposition which is well sui
ted for certain source separation problems. In this talk\, I will discuss
the existence of a best approximate block-term tensor decomposition (BTD)
consisting of a sum of low-rank matrix-vector tensor products. This invest
igation is motivated by the fact that a tensor might not admit an exact BT
D with a given structure (number of blocks and their ranks).\nAfter a brie
f introduction\, we will proceed by exploring the isomorphism between thir
d-order tensors and matrix polynomials. To every matrix polynomial one can
associate a sequence of minimal ranks\, which is unique up to permutation
and invariant under the action of the general linear (or tensor equivalen
ce) group. This notion is a key ingredient of our problem\, since it induc
es a natural hierarchy of sets of third-order tensors corresponding to dif
ferent choices of ranks for the blocks of the BTD.\nIn the particular case
of matrix pencils\, I will explain how the minimal ranks of a pencil can
be directly determined from its Kronecker canonical form. By relying on th
is result\, one can show that no real pencil of dimensions 2k x 2k having
full minimal ranks admits a best approximation on the set of real pencils
whose minimal ranks are bounded by 2k-1. From the tensor viewpoint\, this
means that there exist third-order tensors which do not have a best approx
imation on a certain set of two-block BTDs. These tensors form a non-empty
open subset of the space of 2k x 2k x 2 tensors\, which is therefore of p
ositive volume. I shall sketch the proof of this result and then discuss s
ome possible extensions of this work and open problems.\n\nhttps://indico.
math.cnrs.fr/event/2828/contributions/1600/
LOCATION:ENS de Lyon Amphi A
URL:https://indico.math.cnrs.fr/event/2828/contributions/1600/
END:VEVENT
BEGIN:VEVENT
SUMMARY:About Diagonalisation of Para-Hermitian Matrix
DTSTART;VALUE=DATE-TIME:20180515T090000Z
DTEND;VALUE=DATE-TIME:20180515T093000Z
DTSTAMP;VALUE=DATE-TIME:20200706T102324Z
UID:indico-contribution-2828-1599@indico.math.cnrs.fr
DESCRIPTION:Speakers: Sylvie Icart (I3S Université Côte d’Azur)\nIt is
well-known that a Hermitian matrix can be diagonalized by means of a\nuni
tary matrix. The aim of this talk is to present the extension of this resu
lt\nto polynomial matrices\, known as PEVD (Polynomial Eigen Value Decompo
sition)\n[1]\, occurring e.g. in blind equalization.\n\nIn this context\,
the eigenvalues are polynomials instead of scalars. Moreover\,\npolynomial
s are Laurent polynomials\, this means with positive and negative\nexponen
ts. We will show that in this framework\, one can still define unimodular\
nmatrices\, Smith form and invariant polynomials. We will present the diff
erence\nbetween the order (or length) and the degree of a polynomial matri
x.\n\nExtending polynomial paraconjugation to polynomial matrix\, one defi
nes\npara-hermitianity. We give some properties of these matrices. Then\,
we will\nshow that diagonalization of para-hermitian matrices is not alway
s possible.\nFurthermore we will present some results found in the literat
ure on how to\napproximate a PEVD.\n\n[1] J. G. McWhirter\, P. D. Baxter\,
T. Cooper\, S. Redif\, and J. Foster\, An\nEVD algorithm for Para-Hermiti
an polynomial matrices\, IEEE Transactions\non Signal Processing\, vol. 55
\, no. 6\, June 2007\, pp. 2158-2169.\n\nhttps://indico.math.cnrs.fr/event
/2828/contributions/1599/
LOCATION:ENS de Lyon Amphi A
URL:https://indico.math.cnrs.fr/event/2828/contributions/1599/
END:VEVENT
BEGIN:VEVENT
SUMMARY:A Newton-like Validation Method for Chebyshev Approximate Solution
s of Linear Ordinary Differential Equations
DTSTART;VALUE=DATE-TIME:20180514T123000Z
DTEND;VALUE=DATE-TIME:20180514T130000Z
DTSTAMP;VALUE=DATE-TIME:20200706T102324Z
UID:indico-contribution-2828-1598@indico.math.cnrs.fr
DESCRIPTION:Speakers: Florent Bréhard (LAAS-CNRS Toulouse and CNRS\, LIP
- INRIA AriC / Plume - ENS de Lyon)\nA wide range of efficient numerical r
outines exist for solving function space\nproblems (ODEs\, PDEs\, optimiza
tion\, etc.) when no closed form is known for the\nsolution. While most ap
plications prioritize efficiency\, some safety-critical\ntasks\, as well a
s computer assisted mathematics\, need rigorous guarantees on\nthe compute
d result. For that\, rigorous numerics aims at providing numerical\napprox
imations together with rigorous mathematical statements about them\,\nwith
out sacrificing (too much) efficiency and automation.\n\nIn the spirit of
Newton-like validation methods (see for example [2])\, we\npropose a fully
automated algorithm which computes both a numerical approximate\nsolution
in Chebyshev basis and a rigorous uniform error bound for a restricted\nc
lass of differential equations\, namely Linear ODEs (LODEs). Functions are
\nrigorously represented using Chebyshev models [1]\, which are a generali
zation\nof Taylor models [3] with better convergence properties. Broadly s
peaking\, the\nalgorithm works in two steps: (i) After applying an integra
l transform on the\nLODE\, an infinite-dimensional linear almost-banded sy
stem is obtained. Its\ntruncation at a given order N is solved with the fa
st algorithm of [4].\n(ii) This solution is validated using a specific New
ton-like fixed-point\noperator. This is obtained by approximating the inte
gral operator with a\nfinite-dimensional truncation\, whose inverse Jacobi
an is in turn approximated\nby an almost-banded matrix\, obtained with a m
odified version of the algorithm\nof [4].\n\nA C library implementing this
validation method is freely available online at\nhttps://gforge.inria.fr/
projects/tchebyapprox/\n\n[1] N. Brisebarre and M. Joldeş. Chebyshev inte
rpolation polynomial-based tools\nfor rigorous computing. In Proceedings o
f the 2010 International Symposium on\nSymbolic and Algebraic Computation\
, pages 147-154. ACM\, 2010.\n[2] J.-P. Lessard and C. Reinhardt. Rigorous
numerics for nonlinear\ndifferential equations using Chebyshev series. SI
AM J. Numer. Anal.\,\n52(1):1-22\, 2014.\n[3] K. Makino and M. Berz. Taylo
r models and other validated functional\ninclusion methods. International
Journal of Pure and Applied Mathematics\,\n4(4):379-456\, 2003.\n[4] S. Ol
ver and A. Townsend. A fast and well-conditioned spectral method. SIAM\nRe
view\, 55(3):462-489\, 2013.\n\nhttps://indico.math.cnrs.fr/event/2828/con
tributions/1598/
LOCATION:ENS de Lyon Amphi A
URL:https://indico.math.cnrs.fr/event/2828/contributions/1598/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Algorithms for Structured Linear Systems Solving and their Impleme
ntation
DTSTART;VALUE=DATE-TIME:20180515T130000Z
DTEND;VALUE=DATE-TIME:20180515T134500Z
DTSTAMP;VALUE=DATE-TIME:20200706T102324Z
UID:indico-contribution-2828-1597@indico.math.cnrs.fr
DESCRIPTION:Speakers: Romain Lebreton (LIRMM - Université de Montpellier)
\nThere exists a vast literature dedicated to algorithms for structured ma
trices\,\nbut relatively few descriptions of actual implementations and th
eir practical\nperformance in symbolic computation. In this talk\, we cons
ider the problem of\nsolving Cauchy-like systems\, and its application to
mosaic Toeplitz systems\, in\ntwo contexts: first over finite fields where
basic operations have unit cost\,\nthen over Q. We introduce new variants
of previous algorithms and describe an\nimplementation of these technique
s and its practical behavior. We pay a special\nattention to particular ca
ses such as the computation of algebraic\napproximants.\n\nhttps://indico.
math.cnrs.fr/event/2828/contributions/1597/
LOCATION:ENS de Lyon Amphi A
URL:https://indico.math.cnrs.fr/event/2828/contributions/1597/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Certified and Fast computation of SVD
DTSTART;VALUE=DATE-TIME:20180514T120000Z
DTEND;VALUE=DATE-TIME:20180514T123000Z
DTSTAMP;VALUE=DATE-TIME:20200706T102324Z
UID:indico-contribution-2828-1596@indico.math.cnrs.fr
DESCRIPTION:Speakers: Jean-Claude Yakoubsohn (Institut de Mathématiques d
e Toulouse)\nThis work is in progress with the collaboration Joris Van Der
Hoeven. We\ndefine and study a new method to approximate locally the SVD
of a general\nmatrix: this method has a local quadratic convergence. We
also give result of\ncomplexity of this approximation using a such type ho
motopy method.\n\nhttps://indico.math.cnrs.fr/event/2828/contributions/159
6/
LOCATION:ENS de Lyon Amphi A
URL:https://indico.math.cnrs.fr/event/2828/contributions/1596/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Connecting geodesics for the Stiefel manifold
DTSTART;VALUE=DATE-TIME:20180514T093000Z
DTEND;VALUE=DATE-TIME:20180514T100000Z
DTSTAMP;VALUE=DATE-TIME:20200706T102324Z
UID:indico-contribution-2828-1595@indico.math.cnrs.fr
DESCRIPTION:Speakers: Marco Sutti (University of Geneva)\nSeveral applicat
ions in optimization\, image and signal processing deal with\ndata that be
long to the Stiefel manifold St(n\, p)\, that is\, the set of n × p\nmatr
ices with orthonormal columns. Some applications\, for example\, the\ncomp
utation of the Karcher mean\, require evaluating the geodesic distance\nbe
tween two arbitrary points on St(n\, p). This can be done by explicitly\n
constructing the geodesic connecting these two points. An existing method
for\nfinding geodesics is the leapfrog algorithm introduced by J. L. Noa
kes\, which\nenjoys global convergence properties. We reinterpret this alg
orithm as a\nnonlinear block Gauss-Seidel process and propose a new conver
gence proof based\non this framework for the case of St(n\, p).\n\nhttps:/
/indico.math.cnrs.fr/event/2828/contributions/1595/
LOCATION:ENS de Lyon Amphi A
URL:https://indico.math.cnrs.fr/event/2828/contributions/1595/
END:VEVENT
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SUMMARY:Bridging the gap between flat and hierarchical low-rank matrix for
mats
DTSTART;VALUE=DATE-TIME:20180515T123000Z
DTEND;VALUE=DATE-TIME:20180515T130000Z
DTSTAMP;VALUE=DATE-TIME:20200706T102324Z
UID:indico-contribution-2828-1594@indico.math.cnrs.fr
DESCRIPTION:Speakers: Theo Mary (University of Manchester)\nMatrices posse
ssing a low-rank property arise in numerous scientific\napplications. This
property can be exploited to provide a substantial reduction\nof the comp
lexity of their LU or LDLT factorization. Among the possible\nlow-rank mat
rix formats\, the flat Block Low-Rank (BLR) format is easy to use\nbut ach
ieves superlinear complexity. Alternatively\, the hierarchical formats\na
chieve linear complexity at the price of a much more complex hierarchical\
nmatrix representation. In this talk\, we propose a new format based on\nm
ultilevel BLR approximations. Contrarily to hierarchical matrices\, the nu
mber\nof levels in the block hierarchy is fixed to a given constant\; we p
rove that\nthis provides a simple way to finely control the desired comple
xity of the\ndense multilevel BLR factorization. By striking a balance bet
ween the\nsimplicity of the BLR format and the low complexity of the hiera
rchical ones\,\nthe multilevel BLR format bridges the gap between flat and
hierarchical\nlow-rank formats. The multilevel BLR format is of particula
r relevance in the\ncontext of sparse (e.g. multifrontal) solvers\, for wh
ich it is able to trade\noff the optimal dense complexity of the hierarchi
cal formats to benefit from\nthe simplicity of the BLR format while still
achieving O(n) sparse complexity.\n\nhttps://indico.math.cnrs.fr/event/282
8/contributions/1594/
LOCATION:ENS de Lyon Amphi A
URL:https://indico.math.cnrs.fr/event/2828/contributions/1594/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Axis-alignment in low rank and other structures
DTSTART;VALUE=DATE-TIME:20180514T083000Z
DTEND;VALUE=DATE-TIME:20180514T093000Z
DTSTAMP;VALUE=DATE-TIME:20200706T102324Z
UID:indico-contribution-2828-1592@indico.math.cnrs.fr
DESCRIPTION:Speakers: Lloyd N. Trefethen (University of Oxford and LIP - E
NS de Lyon)\nIn two or more dimensions\, various methods have been develop
ed to represent matrices or functions more compactly. The efficacy of such
methods often depends on alignment of the data with the axes. We shall di
scuss four cases: low-rank approximations\, sparse grids\, quasi-Monte Car
lo\, and multivariate polynomials.\n\nhttps://indico.math.cnrs.fr/event/28
28/contributions/1592/
LOCATION:ENS de Lyon Amphi A
URL:https://indico.math.cnrs.fr/event/2828/contributions/1592/
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