Istituto Nazionale di Alta Matematica

** R. Adler - Cosmology and (random) Topology: Is CMB Really Defined On a Sphere? **

Much of the functional data that we see seems to be defined on
relatively simple parameter spaces. For example, cosmic microwave
background data is typically seen as being defined over the
2-dimensional sphere, while galactic density data is defined over
3-dimensional space.
I will argue that, in many cases, the true dimensions of parameter
spaces are much higher (at least 5 for CMB and at least 8 for SDSS) and
the spaces themselves are structurally far more complex than one might
at first imagine, and that this calls for a topological approach to
analysing many random phenomena. I will then discuss some of the new
tools available for such an analysis. While these tools are typically
based on deep and often esoteric mathematics, understanding the
underlying ideas and learning how to apply them fortunately requires no
more than the undergraduate mathematics we have all learnt, and an open
mind.

** A. Balbi - What have we learned from the CMB? **

Over the past twenty years, the detailed study of angular
fluctuations in the intensity of the Cosmic Microwave Background has produced
an extraordinary advancement in our knowledge of the universe. I give a broad account
of the current state of affair and how it came to be.

** J. Cisewski - Mapping the Intergalactic Medium using Lyman-alpha Data and
Persistent Homology **

Light we observe from quasars has traveled through the
intergalactic medium (IGM) to reach us, and leaves an imprint of some
properties of the IGM on its spectrum. There is a particular imprint of
which cosmologists are familiar, dubbed the Lyman-alpha Forest. From this
imprint, we can infer the density of neutral hydrogen along the line of
sight from us to the quasar. The Sloan Digital Sky Survey Data Release 9
(SDSS - DR9) produced over 54,000 quasar spectra that can be used for
analysis of the Lyman alpha forest and, thus, aid cosmologists in further
understanding the IGM along with revealing or corroborating other
properties of the Universe.
With cosmological simulation output, we develop a methodology using local
polynomial smoothing to model the IGM. I will briefly describe the
modeling methodology, but focus on how to analyze the adequacy of the
modeling procedure and discuss some of the issues faced when modeling the
real data from SDSS - DR9. Finally, describing the topological features
of the IGM can aid in our understanding of the large-scale structure of
the Universe along with providing a framework for comparing cosmological
simulation output with real data beyond the standard measures. Accessing
important topological features of data can be accomplished with persistent
homology - I will introduce persistent homology, and describe an example
of how it can be used in cosmology.

** A. Dalalyan - Oracle Inequalities for Aggregation of Affine Estimators **

We consider the problem of combining a (possibly
uncountably infinite) set of affine estimators in non-parametric
regression model with heteroscedastic Gaussian noise. Focusing on the
exponentially weighted aggregate, we prove a PAC-Bayesian type
inequality that leads to sharp oracle inequalities in discrete but
also in continuous settings. The framework is general enough to cover
the combinations of various procedures such as least square
regression, kernel ridge regression, shrinking estimators and many
other estimators used in the literature on statistical inverse
problems. As a consequence, we show that the proposed aggregate
provides an adaptive estimator in the exact minimax sense without
neither discretizing the range of tuning parameters nor splitting the
set of observations. We also illustrate numerically the good
performance achieved by the exponentially weighted aggregate.
(This is a joint work with J. Salmon)

**J. Jin - Higher Criticism and Rare and Weak effects**

We are often said to be entering the era of Big Data, where massive data sets are
generated on a daily basis. A phenomenon that is frequently found in Big Data is
that the signals of interest are often rare and weak: the effects that we are interested
in are mostly very subtle and few and far between. Whether we are talking about
genome scans or tick-by-tick financial data, most of what we see is noise; the signal
is hard to find and it's easy to be fooled.
In such rare and weak effect settings, classical methods and most modern empirical
methods are simply overwhelmed. Yet, it is of great interest to develop methods that
can cope with such settings.
We introduce Higher Criticism (HC) as a method to deal with rare and weak
signals. HC is a notion that goes back to Tukey in 1976. In the past decade, HC
has evolved into an array of data-driven tools that are found to be especially useful
for analyzing rare and weak signals in \Big Data", covering an array of problems
including signal detection, classification, and spectral clustering. In this talk, we
review the development of HC (some are earlier and some are recent), and suggest a
range of problems where HC can be useful.

**S. Feeney - A case study in anomaly detection: looking for other universes in the cosmic microwave background**

With the advent of precise measurements of the cosmic microwave background (CMB),
anomaly detection has become a powerful tool in cosmology, targeting clusters of galaxies,
point-source contaminants, topological defects indicative of high-energy phase transitions, and more.
Anomaly detection encompasses many of the statistical challenges facing the field: in particular, the need to process
(in a principled and robust fashion) huge modern datasets in human timescales. It is possible to extract information
about the anomalous sources, and more interestingly the underlying physical processes governing their properties,
by viewing the population of sources as a hierarchical Bayesian model. I will demonstrate this technique by concentrating
on one particularly exciting potential anomaly: the signatures of collisions with bubble universes arising in eternal inflation.
In the picture of eternal inflation, our observable Universe resides inside a single bubble nucleated from an inflating false
vacuum. Many theories giving rise to eternal inflation predict that we have causal access to collisions with other bubble
universes, which leave characteristic localised modulations of the CMB. I will present the results from the first observational
search for the effects of bubble collisions using CMB data from the WMAP satellite.

** Chris Genovese - Hunting for Manifolds and Ridges I **

(Joint work with Marco Perone-Pacifico, Isa Verdinelli and Larry Wasserman)
We discuss the problem of finding stable, high-density regions in
point clouds. In this part of the talk we discuss the formal problem
of locating a manifold based on noisy data. We explain the
statistical model and we show how the difficulty of the problem can be
formalized using statistical minimax theory. We then find the minimax
rate under several different models for the noise. In the case of
Gaussian noise, we show that the rate is extremely slow (logarithmic).
We suggest, instead, estimating a surrogate for the manifold. This is
a set that is close to the manifold and can be estimated at a
polynomial rate.

**N. Leonenko - Monofractal and multifractal models for isotropic and spherical
random fields **

We plan to provide a brief survey of development and open problems in the
following areas:

1) Spectral theory of isotropic random fields in the 3-dimensional Euclidean
space and spherical random ?elds with Gaussian and Student distributions.

2) Rényi functions for multifractal products of isotropic random fields and
spherical random fields. Multifractal analysis based on the lognormal, loggamma
and log-negative inverted gamma scenarios. Testing for non-Gaussianity
of isotropic and spherical data based on the multifractal analysis and empirical
structure function.

3) Parametric models for statistical analysis of isotropic and spherical scalar
and vector random fields.

4) Correlation and spectral theory for tensor-valued random fields: forecasting
and interpolation.

The lecture is based on the joint papers (published, in press or in preparation)
with V.Anh, D.Denisov, D.Marinucci, A. Malyarenko, L. Sakhno and
N.-R. Shieh.

** D. Marinucci - Testing for Isotropy and Geometric Features of Needlets Excursion Sets **

In this talk, we shall be concerned with geometric functionals and excursion probabilities for some nonlinear transforms evaluated on wavelet/needlet components of spherical random fields. For such fields, we consider smoothed polynomial transforms, such as those arising from local estimates of angular power spectra and bispectra; we focus on the geometry of their excursion sets, and we study their asymptotic behaviour, in the high-frequency sense. We put particular emphasis on the analysis of Euler-Poincaré characteristics, which can be exploited to derive extremely accurate estimates for excursion probabilities. The present analysis is motivated by the statistical investigation of asymmetries and anisotropies in CMB data.

** S. Matarrese - Non-Gaussianity and Cosmology**

I'll review the role and relevance of non-Gaussian signals in
the cosmological framework, focussing on
i) their origin in the early Universe and during the recent non-linear stage
of the evolution of perturbations;
ii) their search in Cosmic Microwave Background data as well as in the
large-scale structure of the Universe;
iii) the role of rare events ("upcrossing regione" and "peaks") in the
cosmological search for primordial non-Gaussianity.

** J. McEwen - Signal processing on spherical manifolds **

Observations that live on a spherical manifold arise in many applications.
In cosmology, for example, observations of the relic radiation of the Big Bang,
the so-called cosmic microwave background (CMB), are inherently made on the
celestial sphere. To analyse such data signal processing techniques defined
on spherical manifolds are required. I will discuss recent advances in the
area of signal processing on spherical manifolds. Firstly, I will discuss
wavelet constructions on the sphere, charting the historical development from
continuous wavelet methodologies through to the scale-discretised wavelet
framework that supports the exact reconstruction of signals from a discrete
sampling of wavelet coefficients. I will then discuss the non-trivial
extension of scale-discretised wavelets to the ball, i.e. the sphere
augmented with depth. Finally, I will conclude with a cosmological
application of these techniques.

** R. Nickl - Inference for nonparametric functions on homogeneous manifolds: Lie
groups, Needlets, Rademacher processes and graph Laplacians **

Suppose one is given n observation points on a compact homogeneous
manifold, examples for which include the unit sphere but also projective
spaces and Grassmann or Stiefel manifolds. Suppose one wants to either
estimate the probability density of the observations directly, or some
underlying functional regression relationship, without any particular
parametric assumptions on the underlying function. We discuss recent
results from geometric analysis that show how one can use the Lie group
structure to define a manifold analogue of a localised wavelet (=needlet)
basis, and use it to estimate the unknown function based on the
observations. We use concentration of measure arguments for Rademacher
processes to construct non-asymptotic confidence regions for the underlying
function. For manifolds where analytical expressions of the eigenfunctions
of the Laplacian are difficult to obtain we discuss the computation of the
estimator based on graph Laplacian methods.

** G. Peccati - High-frequency asymptotics on homogeneous spaces: some explicit estimates. **

I will explore some asymptotic results concerning high-frequency central limit
theorems for random vectors, composed of harmonic coefficients associated with
isotropic random fields on a sphere. I will provide an overview of recent
findings in the area, in particular connected with entropic estimates.
The main results presented in the talk are motivated by the high-resolution
asymptotic analysis of the CMB radiation. Based on joint works with:
D. Marinucci, I. Nourdin and Y. Swan

** I. Pesenson - Shannon Sampling and Paley-Wiener Localized Frames on Riemannian Manifolds **

In the last decade, methods based on various kinds of wavelet
bases on the unit sphere S^2 and on the rotation group SO(3) have found
applications in virtually all areas where analysis of spherical data is required,
including cosmology, weather prediction, geodesy, and crystallography.

The goal of my talk is to explain constructions of Paley-Wiener localized
frames on compact and non-compact Riemannian manifolds of bounded geometry.
It is important that all our constructions produce frames which are
either Parseval or nearly Parseval.

Special consideration will be given to a particularly
important case of n-dimensional standard unit ball B^n in R^n.
Three different approaches will
be discussed. In our first approach we treat the closed B^n
as a compact submanifold of R^n with boundary. In our second approach the open ball B
is treated as a noncompact manifold which is isometric to a real hyperbolic space
(the Poincare model). In our third approach we suggest a specific identification
of B^n with a direct product of [0; 1] and unit sphere S^(n-1).

** R. Scaramella - Euclid space mission: a cosmological challenge for the next 15 years **

Euclid is the next ESA mission devoted to cosmology. It aims at covering most of the extragalactic
sky, studying both gravitational lensing and clustering over ~15000 square degrees.
The mission is expected to be launched in year 2020 and to last six years.
The sheer amount of data of different kinds, the variety of (un)known systematic effects and the
complexity of measures require efforts both in sophisticated simulations and techniques of data
analysis. We will review the mission main characteristics and mention some of the areas of interest to
this meeting. to be decided

** A. Schwartzman - Distribution of the height of local maxima of random fields **

Let f(t) be a smooth Gaussian random field over a parameter space T, where T may be a subset
of Euclidean space or, more generally, a Riemannian manifold. For any local maximum of f(t)
located at t0 in the interior of T, we provide general formulas and asymptotic approximations
for the excursion probability P{f(t0) > u | t0 is a local maximum of f(t)} and the overshoot
probability P{f(t0) > u | t0 is a local maximum of f(t) and f(t0) > v}. Assuming further that
f is isotropic, we apply the GOE techniques in random matrices to compute such conditional
probabilities explicitly when T is Euclidean or a sphere of arbitrary dimension. Such calculations
are motivated by the statistical problem of detecting peaks in the presence of smooth Gaussian noise.

**J.L. Starck - Sparsity and the Cosmic Microwave Background**

Bayesian methodology is very popular in Bayesian cosmology and is even often considered as the only way
to process properly astronomical data set.

Recent progress in harmonic analysis such as compressed sensing theory or sparsity however open us new ways to acquire/analyze data which can be hardly
understood from the Bayesian perspective.

We briefly review the concept of sparsity and its relation to Compressed Sensing, the new sampling theory, then
we show how these new concepts can help for Cosmic Microwave Background data analysis.

** J. Taylor - A significance test for adaptive linear modeling **

In this talk we consider testing the signi cance of the terms in a fitted regression, via
the lasso. We propose a novel test statistic for this problem, and show that it has a simple
asymptotic null distribution. This work builds on the least angle regression approach for
tting the lasso, and the notion of degrees of freedom for adaptive models (Efron 1986)
and for the lasso (Efron et al. 2004, Zou et al. 2007). It is also related to the distribution of the maximum of
a discrete random field. Time permitting, we will expand on this connection.
This is joint work with Richard Lockhart (Simon Fraser University) and Ryan Tibshirani
(Carnegie Mellon University).

** B. Wandelt - Cosmostatistics **

"Why cosmostatistics? Rich cosmological data sets, meaningful physical
models, and advances in computing and algorithms create the perfect
environment for principled analysis approaches. I will discuss advances
relevant to data such as the cosmic microwave background anisotropies on the
sphere, and the observed galaxy distribution in redshift space. Examples
include full physical as well as semi-blind approaches, samples from
reconstructed cosmological evolution histories, and a new general method for
fast computation of Fisher matrices for inference of covariance from data
sets on the sphere.

** L. Wasserman - Hunting for Manifolds and Ridges II **

(Joint work with Chris Genovese, Marco Perone-Pacifico and Isa Verdinelli)
In this part of the talk, we discuss a particular surrogate for a
manifold, namely, hyper-ridges in the density. These are low
dimensional sets characterized by conditions on the eigenvalues of the
Hessian. We show that the ridges can be estimated at polynomial rate
and thus serve as a good surrogate for the underlying manifold. In
fact, our methods work well even when the underlying set is not a
manifold. We then discuss the problem of ``dimensional leakage'' in
which structures can leave their imprints in several different
dimensions. Finally, we show how the bootstrap can be used to assess
the variability of the procedure.

** I. Wigman - On the geometry of random spherical harmonics **

This work is joint with D. Marinucci.

The random Gaussian spherical harmonics appear in the L2 expansion of any
Gaussian field defined on the sphere, and one is interested in their geometry
in the high degree limit. This question has numerous applications in various fields
such as spectral geometry, and Gaussianity testing for random fields originating
in Cosmology with high degree limit corresponds to high quality observations.

We address some aspects of the geometry of random Gaussian spherical harmonics,
such as the defect (or "signed measure"), and more generally, nonlinear functionals
of the spherical harminics. We were able to evaluate the asymptotic behaviour for
the defect, and prove a general Central Limit Theorem under some mild assumptions
on the functionals.

In this talk I will introduce the audience to the random spherical harmonics,
some questions regarding it and state the main results. Time permitting, I will
show some aspects of the proofs.

Last Modify - Claudio D. - 06/04/2013