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## ‣ Sharp growth estimates for dyadic $b$-input $T(b)$ theorems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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The following deals with the $T(b)$ theorems of David, Journ\'e, and Semmes
\cite{DJS} considered in a dyadic setting. We find sharp growth estimates for a
global and a local dyadic $T(b)$ Theorem. We use multiscale analysis and Haar
wavelets in the local case.; Comment: 15 pages

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## ‣ Generalized sampling reconstruction from Fourier measurements using compactly supported shearlets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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In this paper we study the general reconstruction of a compactly supported
function from its Fourier coefficients using compactly supported shearlet
systems. We assume that only finitely many Fourier samples of the function are
accessible and based on this finite collection of measurements an approximation
is sought in a finite dimensional shearlet reconstruction space. We analyse
this sampling and reconstruction process by a recently introduced method called
generalized sampling. In particular by studying the stable sampling rate of
generalized sampling we then show stable recovery of the signal is possible
using an almost linear rate. Furthermore, we compare the result to the
previously obtained rates for wavelets.

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## ‣ Deep Convolutional Neural Networks Based on Semi-Discrete Frames

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 21/04/2015
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#Computer Science - Learning#Computer Science - Information Theory#Mathematics - Functional Analysis#Statistics - Machine Learning

Deep convolutional neural networks have led to breakthrough results in
practical feature extraction applications. The mathematical analysis of these
networks was pioneered by Mallat, 2012. Specifically, Mallat considered
so-called scattering networks based on identical semi-discrete wavelet frames
in each network layer, and proved translation-invariance as well as deformation
stability of the resulting feature extractor. The purpose of this paper is to
develop Mallat's theory further by allowing for different and, most
importantly, general semi-discrete frames (such as, e.g., Gabor frames,
wavelets, curvelets, shearlets, ridgelets) in distinct network layers. This
allows to extract wider classes of features than point singularities resolved
by the wavelet transform. Our generalized feature extractor is proven to be
translation-invariant, and we develop deformation stability results for a
larger class of deformations than those considered by Mallat. For Mallat's
wavelet-based feature extractor, we get rid of a number of technical
conditions. The mathematical engine behind our results is continuous frame
theory, which allows us to completely detach the invariance and deformation
stability proofs from the particular algebraic structure of the underlying
frames.; Comment: Proc. of IEEE International Symposium on Information Theory (ISIT)...

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## ‣ On convergence of general wavelet decompositions of nonstationary stochastic processes

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/07/2013
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The paper investigates uniform convergence of wavelet expansions of Gaussian
random processes. The convergence is obtained under simple general conditions
on processes and wavelets which can be easily verified. Applications of the
developed technique are shown for several classes of stochastic processes. In
particular, the main theorem is adjusted to the fractional Brownian motion
case. New results on the rate of convergence of the wavelet expansions in the
space $C([0,T])$ are also presented.; Comment: 24 pages, 2 figures, will appear in Electronic Journal of Probability

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## ‣ A note on grid transfer operators for multigrid methods

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/07/2008
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The Local Fourier analysis (LFA) is a classic tool to prove convergence
theorems for multigrid methods (MGMs). In particular, we are interested in
optimality that is a convergence speed independent of the size of the involved
matrices. For elliptic partial differential equations (PDEs), a well known
optimality result requires that the sum of the orders of the grid transfer
operators is not lower than the order of the PDE to solve. Analogously, when
dealing with MGMs for Toeplitz matrices in the literature an optimality
condition on the position and on the order of the zeros of the symbols of the
grid transfer operators has been found. In this work we show that in the case
of elliptic PDEs with constant coefficients, the two different approaches lead
to an equivalent condition. We argue that the analysis for Toeplitz matrices is
an algebraic generalization of the LFA, which allows to deal not only with
differential problems but also for instance with integral problems. The
equivalence of the two approaches gives the possibility of using grid transfer
operators with different orders also for MGMs for Toeplitz matrices. We give
also a class of grid transfer operators related to the B-spline's refinement
equation and we study their geometric properties. This analysis suggests
further links between wavelets and multigrid methods. A numerical
experimentation confirms the correctness of the proposed analysis.

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## ‣ Inversion of the Dual Dunkl-Sonine Transform on R Using Dunkl Wavelets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/07/2009
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We prove a Calder\'on reproducing formula for the Dunkl continuous wavelet
transform on $\mathbb{R}$. We apply this result to derive new inversion
formulas for the dual Dunkl-Sonine integral transform.

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## ‣ A dynamic hybrid model based on wavelets and fuzzy regression for time series estimation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/02/2011
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#Quantitative Finance - Statistical Finance#Mathematics - Statistics Theory#42C15, 42C40, 62A86, 62J05, 62M10, 65D15, 65K10#F.2.1#F.2.2#G.1.6#G.1.0#G.1.2#G.3#I.5.1

In the present paper, a fuzzy logic based method is combined with wavelet
decomposition to develop a step-by-step dynamic hybrid model for the estimation
of financial time series. Empirical tests on fuzzy regression, wavelet
decomposition as well as the new hybrid model are conducted on the well known
$SP500$ index financial time series. The empirical tests show an efficiency of
the hybrid model.; Comment: 15 pages, 15 figures, 2 tables

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## ‣ Asymptotic normality of wavelet estimators of the memory parameter for linear processes

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 06/05/2008
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We consider linear processes, not necessarily Gaussian, with long, short or
negative memory. The memory parameter is estimated semi-parametrically using
wavelets from a sample $X_1,...,X_n$ of the process. We treat both the
log-regression wavelet estimator and the wavelet Whittle estimator. We show
that these estimators are asymptotically normal as the sample size $n\to\infty$
and we obtain an explicit expression for the limit variance. These results are
derived from a general result on the asymptotic normality of the empirical
scalogram for linear processes, conveniently centered and normalized. The
scalogram is an array of quadratic forms of the observed sample, computed from
the wavelet coefficients of this sample. In contrast with quadratic forms
computed on the Fourier coefficients such as the periodogram, the scalogram
involves correlations which do not vanish as the sample size $n\to\infty$.

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## ‣ Central Limit Theorems for arrays of decimated linear processes

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 06/05/2008
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Linear processes are defined as a discrete-time convolution between a kernel
and an infinite sequence of i.i.d. random variables. We modify this convolution
by introducing decimation, that is, by stretching time accordingly. We then
establish central limit theorems for arrays of squares of such decimated
processes. These theorems are used to obtain the asymptotic behavior of
estimators of the spectral density at specific frequencies. Another
application, treated elsewhere, concerns the estimation of the long-memory
parameter in time-series, using wavelets.

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## ‣ Triebel-Lizorkin Spaces and Shearlets on the Cone in $\mathbb{R}^2$

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/03/2012
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The shearlets are a special case of the wavelets with composite dilation
that, among other things, have a basis-like structure and multi resolution
analysis properties. These relatively new representation systems have
encountered wide range of applications, generally surpassing the performance of
their ancestors due to their directional sensitivity. However, little is known
about their relation with spaces other than $L^2$. Here, we find a
characterization of a kind of anisotropic inhomogeneous Triebel-Lizorkin spaces
(to be defined) with the so called "shearlets on the cone" coefficients. We
first prove the boundedness of the analysis and synthesis operators with the
"traditional" shearlets coefficients. Then, with the development of the smooth
Parseval frames of shearlets of Guo and Labate we are able to prove a
reproducing identity, which was previously possible only for the $L^2$ case. We
also find some embeddings of the (classical) dyadic spaces into these highly
anisotropic spaces, and viceversa, for certain ranges of parameters. In order
to keep a concise document we develop our results in the "weightless" case
($w=1$) and give hints on how to develop the weighted case.; Comment: 30 pages

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## ‣ Bandlimited Spaces on Some 2-step Nilpotent Lie Groups With One Parseval Frame Generator

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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Let $N$ be a step two connected and simply connected non commutative
nilpotent Lie group which is square-integrable modulo the center. Let $Z$ be
the center of $N$. Assume that $N=P\rtimes M$ such that $P$, and $M$ are simply
connected, connected abelian Lie groups, $M$ acts non-trivially on $P$ by
automorphisms and $\dim P/Z=\dim M$. We study band-limited subspaces of
$L^2(N)$ which admit Parseval frames generated by discrete translates of a
single function. We also find characteristics of band-limited subspaces of
$L^2(N)$ which do not admit a single Parseval frame. We also provide some
conditions under which continuous wavelets transforms related to the left
regular representation admit discretization, by some discrete set
$\Gamma\subset N$. Finally, we show some explicit examples in the last section.

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## ‣ Locally adaptive estimation of evolutionary wavelet spectra

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 11/08/2008
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We introduce a wavelet-based model of local stationarity. This model enlarges
the class of locally stationary wavelet processes and contains processes whose
spectral density function may change very suddenly in time. A notion of
time-varying wavelet spectrum is uniquely defined as a wavelet-type transform
of the autocovariance function with respect to so-called autocorrelation
wavelets. This leads to a natural representation of the autocovariance which is
localized on scales. We propose a pointwise adaptive estimator of the
time-varying spectrum. The behavior of the estimator studied in homogeneous and
inhomogeneous regions of the wavelet spectrum.; Comment: Published in at http://dx.doi.org/10.1214/07-AOS524 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org)

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## ‣ Framings and dilations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/07/2013
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The notion of framings, recently emerging in P. G. Casazza, D. Han, and D. R.
Larson, Frames for Banach spaces, in {\em The functional and harmonic analysis
of wavelets and frames} (San Antonio, TX, 1999), {\em Contemp. Math}. {\bf 247}
(1999), 149-182 as generalization of the reconstraction formula generated by
pairs of dual frames, is in this note extended substantially. This calls on
refining the basic dilation results which still being in the flavor of {\em
th\'eor\`eme principal} of B. Sz-Nagy go much beyond it.; Comment: The final version will appear in Acta Sci. Math (Szeged)

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## ‣ An exact tree projection algorithm for wavelets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/04/2013
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We propose a dynamic programming algorithm for projection onto wavelet tree
structures. In contrast to other recently proposed algorithms which only give
approximate tree projections for a given sparsity, our algorithm is guaranteed
to calculate the projection exactly. We also prove that our algorithm has O(Nk)
complexity, where N is the signal dimension and k is the sparsity of the tree
approximation.; Comment: 4 pages, 1 figure

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## ‣ On an identity by Chaundy and Bullard. I

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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An identity by Chaundy and Bullard writes 1/(1-x)^n (n=1,2,...) as a sum of
two truncated binomial series. This identity was rediscovered many times.
Notably, a special case was rediscovered by I. Daubechies, while she was
setting up the theory of wavelets of compact support. We discuss or survey many
different proofs of the identity, and also its relationship with Gauss
hypergeometric series. We also consider the extension to complex values of the
two parameters which occur as summation bounds. The paper concludes with a
discussion of a multivariable analogue of the identity, which was first given
by Damjanovic, Klamkin and Ruehr. We give the relationship with Lauricella
hypergeometric functions and corresponding PDE's. The paper ends with a new
proof of the multivariable case by splitting up Dirichlet's multivariable beta
integral.; Comment: 20 pages; added in v3: more references to earlier occurrences of the
identity and its multivariable analogue, combinatorial proof of the identity
and extension to noninteger m,n, proof of multivariable identity by splitting
up Dirichlet's multivariable beta integral

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## ‣ Model selection for quantum homodyne tomography

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 18/12/2007
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This paper deals with a non-parametric problem coming from physics, namely
quantum tomography. That consists in determining the quantum state of a mode of
light through a homodyne measurement. We apply several model selection
procedures: penalized projection estimators, where we may use pattern functions
or wavelets, and penalized maximum likelihood estimators. In all these cases,
we get oracle inequalities. In the former we also have a polynomial rate of
convergence for the non-parametric problem. We finish the paper with
applications of similar ideas to the calibration of a photocounter, a
measurement apparatus counting the number of photons in a beam. Here the
mathematical problem reduces similarly to a non-parametric missing data
problem. We again get oracle inequalities, and better speed if the photocounter
is good.; Comment: 40 pages, 2 figures, submitted to ESAIM: Probability and Statistics

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## ‣ New Characterizations of Besov-Triebel-Lizorkin-Hausdorff Spaces Including Coorbits and Wavelets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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In this paper, the authors establish new characterizations of the recently
introduced Besov-type spaces $\dot{B}^{s,\tau}_{p,q}({\mathbb R}^n)$ and
Triebel-Lizorkin-type spaces $\dot{F}^{s,\tau}_{p,q}({\mathbb R}^n)$ with $p\in
(0,\infty]$, $s\in{\mathbb R}$, $\tau\in [0,\infty)$, and $q\in (0,\infty]$, as
well as their preduals, the Besov-Hausdorff spaces
$B\dot{H}^{s,\tau}_{p,q}(\R^n)$ and Triebel-Lizorkin-Hausdorff spaces
$F\dot{H}^{s,\tau}_{p,q}(\R^n)$, in terms of the local means, the Peetre
maximal function of local means, and the tent space (the Lusin area function)
in both discrete and continuous types. As applications, the authors then obtain
interpretations as coorbits in the sense of H. Rauhut in [Studia Math. 180
(2007), 237-253] and discretizations via the biorthogonal wavelet bases for the
full range of parameters of these function spaces. Even for some special cases
of this setting such as $\dot F^s_{\infty,q}({\mathbb R}^n)$ for $s\in{\mathbb
R}$, $q\in (0,\infty]$ (including $\mathop\mathrm{BMO} ({\mathbb R}^n)$ when
$s=0$, $q=2$), the $Q$ space $Q_\alpha ({\mathbb R}^n)$, the Hardy-Hausdorff
space $HH_{-\alpha}({\mathbb R}^n)$ for $\alpha\in (0,\min\{\frac n2,1\})$, the
Morrey space ${\mathcal M}^u_p({\mathbb R}^n)$ for $1

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## ‣ Estimation of dimension functions of band-limited wavelets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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The dimension function D_psi of a band-limited wavelet is bounded by n if the
support of its Fourier transform is contained in the interval [-{2^(n+2)/3}pi,
{2^(n+2)/3}pi]. For each positive integer n and for each epsilon > 0, we
construct a wavelet psi with support of $\hat psi$ contained in
[-{2^(n+2)/3}pi, {2^(n+2)/3}pi + epsilon] such that D_psi > n on a set of
positive measure, which proves that [-{2^(n+2)/3}pi, {2^(n+2)/3}pi] is the
largest symmetric interval for estimating the dimension function by n.; Comment: 8 pages

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## ‣ Filters and Functions in Multi-scale Constructions: Extended Abstract

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/11/2014
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We derive results about geometric means of the Fourier modulus of filters and
functions related to refinable distributions with arbitrary dilations and
translations. Then we develop multi-scale constructions for dilations by
Pisot-Vijayaraghavan numbers and translations in associated quasilattices.; Comment: 10 pages, 38 references, plenary talk at: International workshop on
Wavelets, Frames and Applications II, from December 24-30, 2014. The
conference will be held at the premises of Kirori Mal College, University of
Delhi, Delhi, India

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## ‣ Multivariate $\alpha$-molecules

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 27/04/2015
Português

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The suboptimal performance of wavelets with regard to the approximation of
multivariate data gave rise to new representation systems, specifically
designed for data with anisotropic features. Some prominent examples of these
are given by ridgelets, curvelets, and shearlets, to name a few.
The great variety of such so-called directional systems motivated the search
for a common framework, which unites many under one roof and enables a
simultaneous analysis, for example with respect to approximation properties.
Building on the concept of parabolic molecules, the recently introduced
framework of $\alpha$-molecules does in fact include the previous mentioned
systems. Until now however it is confined to the bivariate setting, whereas
nowadays one often deals with higher dimensional data. This motivates the
extension of this unifying theory to dimensions larger than 2, put forward in
this work. In particular, we generalize the central result that the
cross-Gramian of any two systems of $\alpha$-molecules will to some extent be
localized.
As an exemplary application, we investigate the sparse approximation of video
signals, which are instances of 3D data. The multivariate theory allows us to
derive almost optimal approximation rates for a large class of representation
systems.

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