This is a local page for the invited Minisymposium on "Linear Algebra and
Positivity with Applications to Data Science", at the 2017 meeting of the
International Linear Algebra Society
(ILAS). The Minisymposium takes place on
July 24, 25, and 27.
The organizers are: Dominique Guillot, Apoorva Khare, and Bala
Rajaratnam.
For information on hotels, and registration,
click on the ILAS 2017 logo at the top of this page.
Making sense of vast amounts of data has become one of the great
challenges of the 21st century. This minisymposium will highlight how
recent advances in analysis (in particular positivity and related
topics), linear algebra, algebraic geometry, and statistics provide new
tools to analyze data in areas such as covariance estimation and the
theory of graphical models.
Click on the name of a speaker to go to their abstract below.
Monday, July
24 
Room 202, Carver
Hall 
 
 
14:45–15:10 
Mahya Ghandehari 
Geometric graphs and uniform
embeddings 
 
 
15:15–15:40 
Alfred Hero 
Continuum limits for shortest
paths 
 
 
15:45–16:10 
Peter Diao 
DistributionFree Consistency of Graph
Clustering 
 
 
 
 
 
 
 
 
Tuesday, July
25 
Room 305, Carver
Hall 
 
 
10:10–10:35 
Nikolas Stott 
Minimal upper bounds in the Loewner order:
characterizations and parametrization 
 
 
10:40–11:05 
Tanvi Jain 
Hadamard powers of two classes of positive
matrices 
 
 
11:10–11:35 
Shaun Fallat 
Hadamard Powers, Critical Exponents, and Total
Positivity 
 
 
11:40–12:05 
Alexander Belton 
A quantitative form of Schoenberg's theorem in
fixed dimension 
 
 
 
 
 
 
 
 
Thursday, July
27 
Room 232, Carver
Hall 
 
 
10:10–10:35 
Ilse Ipsen 
Randomized matrixfree trace and
logdeterminant estimators 
 
 
10:40–11:05 
Helene Massam 
Precision matrix estimation and sampling in
coloured graphical Gaussian models 
 
 
11:10–11:35 
Caroline Uhler 
Your dreams may come true with
MTP2 
 
 
11:40–12:05 
Cynthia Vinzant 
Hyperbolicity and reciprocal linear
spaces 
 
 
 
 
 
 
 
 


Abstract.
The Hadamard product of two matrices is formed by multiplying
corresponding entries, and the Schur product theorem states that this
operation preserves positive semidefiniteness.
It follows immediately that every analytic function with nonnegative
Maclaurin coefficients, when applied entrywise, preserves positive
semidefiniteness for matrices of any order. The converse is due to
Schoenberg: a function which preserves positive semidefiniteness for
matrices of arbitrary order is necessarily analytic and has nonnegative
Maclaurin coefficients.
For matrices of fixed order, the situation is more interesting. This talk
will present recent work which shows the existence of polynomials with
negative leading term which preserve positive semidefiniteness, and
characterises precisely how large this term may be.
This is joint work with D. Guillot (Delaware), A. Khare (Stanford) and M.
Putinar (UC Santa Barbara and Newcastle). This work was supported by the
American Institute of Mathematics and the International Centre for
Mathematical Sciences, Edinburgh.


Abstract.
The theory of dense graph limits shows how to embed arbitrary sized
matrices of the form $M \in [0,1]_{m \times m}$ in a common graphon space
consisting of symmetric measurable functions of the form $W : [0,1]^2 \to
[0,1]$. The space of such functions is equipped with a norm called the
cutnorm, which is canonical for dense matrices. In our paper, we prove
the continuity of top eigenvectors of the Laplacian associated to such
matrices with respect to the cut norm. As a consequence, we derive
distributionfree consistency results for spectral clustering. In this
talk, we will discuss the cutnorm, the novel framework we have developed
for the analysis of graph clustering, and the technical results required
to derive our results.
This is joint work with Apoorva Khare, Dominique Guillot, and Bala
Rajaratnam.


Abstract.
An $m \times n$ matrix $A$ is called totally nonnegative, TN (resp.
totally positive, TP) of every minor of $A$ is nonnegative (resp.
positive). For an entrywise nonnegative matrix $B=[b_{ij}]$ and $t \gt
0$, $B^{\circ t}$ is defined to be the matrix with entries $b_{ij}^t$
($t$th Hadamard power of $B$). It is known that $A^{\circ t}$ need not be
TN nor TP whenever $A$ is TN or TP. However, if $A$ is TP, then $A^{\circ
t}$ is eventually TP.
On the other hand, if $B$ is a $n \times n$ positive semidefinite and
entrywise nonnegative matrix, then $B^{\circ t}$ is positive
semidefinite for all $t \geq n2$. The number $n2$ is referred to as a
Hadamard critical exponent. In this talk we will show that for $n \geq
5$, there is no Hadamard critical exponent in the TP setting. We will
also explore similarities between the positive semidefinite case and the
class of TP Hankel matrices.
This work is joint with Profs. A. Sokal and C.R. Johnson.


Abstract.
Many reallife networks can be modelled by stochastic processes with a
spatial embedding. The spatial reality can be used to represent
attributes of the vertices which are inaccessible or unknown, but which
are assumed to inform link formation. For example, in a social network,
vertices may be considered as members of a social space, where the
coordinates represent the interests and background of the users. The
graph formation is modelled as a stochastic process, where the
probability of a link occurring between two vertices decreases as their
metric distance increases. A fundamental question is to determine whether
a given network is compatible with a spatial model. That is, given a
graph how can we judge whether the graph is likely generated by a spatial
model, and if so whether the model is uniform in nature?
Using the theory of graph limits, we show how to recognize graph
sequences produced by random graph processes with a linear embedding (a
natural embedding into real line). We then discuss whether a linear
embedding is uniform in nature, that is whether it is possible to
"transform" the linear embedding into one in which the probability of a
link between two vertices depends only on the distance between them. We
give necessary and sufficient conditions for the existence of a uniform
linear embedding for random graphs with finite number of probability
values. Our findings show that for a general linear embedding the answer
is negative in most cases.
This talk is based on joint articles with H. Chuangpishit, M. Hurshman,
J. Janssen, and N. Kalyaniwalia.


Abstract.
Many applications involve computing shortest paths over the nodes of a
graph relative to a measure of pairwise node dissimilarity. When the node
attributes are real valued random vectors and the dissimilarity is an
increasing function of Euclidean distance these shortest paths can have
continuum limits as the number of nodes approaches infinity. Such
continuum limits can lead to low complexity continuous diffusion
approximations to the combinatorial shortest path problem. This work is
joint with Sung Jin Hwang and Steven Damelin and was supported in part by
NSF Grant CCF1217880 and ARO grant W911NF1510479.


Abstract.
We present randomized algorithms for estimating the trace and determinant
of Hermitian positive semidefinite matrices. The algorithms are based on
subspace iteration, and access the matrix only through matrix vector
products. We analyse the error due to randomization, for starting guesses
whose elements are Gaussian or Rademacher random variables. The analysis
is cleanly separated into a structural (deterministic) part followed by a
probabilistic part. Our absolute bounds for the expectation and
concentration of the estimators are nonasymptotic and informative even
for matrices of low dimension. For the trace estimators, we also present
asymptotic bounds on the number of samples (columns of the starting
guess) required to achieve a userspecified relative error.
This is joint work with Alen Alexanderian and Arvind Saibaba. The work is
supported in part by the XDATA Program of the Defense Advanced Research
Projects Agency.


Abstract.
It is known that $n2$ is the least number for which the $r$th Hadamard
power of an $n\times n$ doubly nonnegative matrix is doubly nonnegative
for all $r\ge n2.$ An analogous result holds for positive semidefinite
matrices with real (not necessarily nonnegative) entries. We shall
discuss two interesting classes of positive semidefinite matrices whose
$r$th Hadamard powers need not be positive semidefinite for $r \lt n2$.
In particular we shall focus on the $n\times n$ matrices,
$\left[(1+x_ix_j)^r\right]$ and $\left[\cos((ij)\pi/n)^r\right]$ where
$r$ is a positive number and $x_1,\ldots,x_n$ are distinct positive real
numbers.


Abstract.
We consider graphical Gaussian models with edge and vertex symmetries on
the precision matrix, as introduced by Hojsgaard and Lauritzen
(2008).
First, we identify the DiaconisYlvisaker conjugate prior for these
models and develop a scheme to sample from the prior and posterior
distributions. We thus obtain estimates of the posterior mean of the
precision and covariance matrices. We verify the accuracy of our
estimates on simulated data for graphs with up to thirty vertices and
various symmetries.
Second, for larger graph, we develop a distributed estimation method for
the precision matrix and give the asymptotic distribution of the estimate
thus obtained.
This is joint work with Qiong Li and Xin Gao, York University. This work
was supported by NSERC grant 14094.


Abstract.
We study the set of minimal upper bounds of finitely many symmetric
matrices, in the Loewner order. In the case of two matrices, we provide a
parametrization of this set in terms of the quotient of the indefinite
orthogonal group $O(p,q)$ by the maximal compact subgroup $O(p) \times
O(q)$. Moreover, we show that a matrix is a minimal upper bound of $A_1,
\dots A_p$ if and only if it is positively exposed, meaning that it is
the unique upper bound that minimizes a function of the form $X \mapsto
\text{trace}(CX)$ where $C$ is a positive definite matrix. Finally, when
only two matrices are involved, we show that most usual selections of
minimal upper bounds can be computed analytically.


Abstract.
We study maximum likelihood estimation for exponential families that are
multivariate totally positive of order two (MTP2). Such distributions
appear in the context of ferromagnetism in the Ising model and various
latent models, as for example Brownian motion tree models used in
phylogenetics. We show that maximum likelihood estimation for MTP2
exponential families is a convex optimization problem. For quadratic
exponential families such as Ising models and Gaussian graphical models,
we show that MTP2 implies sparsity of the underlying graph without the
need of a tuning parameter. Moreover, we show that the MLE always exists
even in the highdimensional setting. These properties make MTP2
constraints an intriguing alternative to methods for learning sparse
graphical models such as the graphical lasso.


Abstract.
A reciprocal linear space is the image of a linear space under
coordinatewise inversion. This nice algebraic variety appears in many
contexts and its structure is governed by the combinatorics of an
underlying hyperplane arrangement. A reciprocal linear space is also an
example of a hyperbolic variety, meaning that there is a family of linear
spaces all of whose intersections with it are real. This special real
structure is witnessed by a determinantal representation of its Chow form
in the Grassmannian. For generic linear spaces, these determinantal
formulas are closely related to the Laplacian of the complete graph and
generalizations to simplicial matroids. In this talk, I will introduce
reciprocal linear spaces and discuss the relation of their algebraic
properties to their combinatorial and real structure.

