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CREATED:20210602T082154Z
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SUMMARY:Ομιλία κ. Antonios Varvitsiotis " No
n-commutative Extension of Lee-Seung
's Algorithm for Positive Semidefini
te Factorizations"
LOCATION:Η ομιλία θα γίνει με τηλεδιάσκεψη
DESCRIPTION:https://www.tuc.gr/index.php?id=9160
&L=928%27%27A%3D0&tx_tucevents2_tuce
ventsdisplay%5Bevent%5D=2439&tx_tuce
vents2_tuceventsdisplay%5Baction%5D=
show&tx_tucevents2_tuceventsdisplay%
5Bcontroller%5D=Event\nAntonios Varv
itsiotis, Assistant Professor at the
Singapore University of Technology
and Design\n\nTitle\n A Non-commutat
ive Extension of Lee-Seung's Algorit
hm for Positive Semidefinite Factori
zations\n Abstract\n Given a data ma
trix $X\in \mathbb{R}_+^{m\times n}$
with non-negative entries, a Positi
ve Semidefinite (PSD) factorization
of $X$ is a collection of $r \times
r$-dimensional PSD matrices $\{A_i\}
$ and $\{B_j\}$ satisfying the condi
tion $X_{ij}= \mathrm{tr}(A_i B_j)$
for all $\ i\in [m],\ j\in [n]$. PS
D factorizations are fundamentally l
inked to understanding the expressiv
eness of semidefinite programs as we
ll as the power and limitations of q
uantum resources in information theo
ry. The PSD factorization task gene
ralizes the Non-negative Matrix Fact
orization (NMF) problem in which we
seek a collection of $r$-dimensional
non-negative vectors $\{a_i\}$ and
$\{b_j\}$ satisfying $X_{ij}= a_i^T
b_j$, for all $i\in [m],\ j\in [n]$
-- one can recover the latter probl
em by choosing matrices in the PSD f
actorization to be diagonal. The mo
st widely used algorithm for computi
ng NMFs of a matrix is the Multiplic
ative Update algorithm developed by
Lee and Seung, in which non-negativi
ty of the updates is preserved by sc
aling with positive diagonal matrice
s. In this paper, we describe a non
-commutative extension of Lee-Seung'
s algorithm, which we call the Matri
x Multiplicative Update (MMU) algori
thm, for computing PSD factorization
s. The MMU algorithm ensures that u
pdates remain PSD by congruence scal
ing with the matrix geometric mean o
f appropriate PSD matrices, and it r
etains the simplicity of implementat
ion that the multiplicative update a
lgorithm for NMF enjoys. Building o
n the Majorization-Minimization fram
ework, we show that under our update
scheme the squared loss objective i
s non-increasing and fixed points co
rrespond to critical points. The an
alysis relies on Lieb's Concavity T
heorem. Beyond PSD factorizations,
we show that the MMU algorithm can b
e also used as a primitive to calcul
ate block-diagonal PSD factorization
s and tensor PSD factorizations. We
demonstrate the utility of our meth
od with experiments on real and synt
hetic data. \n Joint work with Yong
Sheng Soh, National University of S
ingapore. \n About the speaker\nhttp
s://sites.google.com/site/antoniosva
rvitsiotis/ \n Assistant Professor V
arvitsiotis received a PhD in Mathem
atical Optimization from the Dutch N
ational Research Institute for Mathe
matics and Computer Science (CWI). P
rior to joining the Singapore Univer
sity of Technology and Design he hel
d Research Fellow positions at the C
entre for Quantum Technologies (Comp
uter Science group) and the National
University of Singapore (Department
of Electrical and Computer Engineer
ing and Department of Industrial and
Systems Engineering). He also holds
a MSc degree in Theoretical Compute
r Science and a BSc in Applied and T
heoretical Mathematics, both from th
e National University of Athens in G
reece. Dr. Varvitsiotis’ research is
focused on fundamental aspects of c
ontinuous and discrete optimisation,
motivated by real-life applications
in data science, engineering, and q
uantum information.\n \n Meeting ID
: 923 5451 8920\n Password: 953377\n
STATUS:CONFIRMED
ORGANIZER;RSVP=FALSE;CN=TUC;CUTYPE=TUC:mailto:webmaster@tuc.gr
DTSTART:20210602T130000
DTEND:20210602T140000
TRANSP:OPAQUE
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