297 research outputs found
Toward a unified theory of sparse dimensionality reduction in Euclidean space
Let be a sparse Johnson-Lindenstrauss
transform [KN14] with non-zeroes per column. For a subset of the unit
sphere, given, we study settings for required to
ensure i.e. so that preserves the norm of every
simultaneously and multiplicatively up to . We
introduce a new complexity parameter, which depends on the geometry of , and
show that it suffices to choose and such that this parameter is small.
Our result is a sparse analog of Gordon's theorem, which was concerned with a
dense having i.i.d. Gaussian entries. We qualitatively unify several
results related to the Johnson-Lindenstrauss lemma, subspace embeddings, and
Fourier-based restricted isometries. Our work also implies new results in using
the sparse Johnson-Lindenstrauss transform in numerical linear algebra,
classical and model-based compressed sensing, manifold learning, and
constrained least squares problems such as the Lasso
lp-Recovery of the Most Significant Subspace among Multiple Subspaces with Outliers
We assume data sampled from a mixture of d-dimensional linear subspaces with
spherically symmetric distributions within each subspace and an additional
outlier component with spherically symmetric distribution within the ambient
space (for simplicity we may assume that all distributions are uniform on their
corresponding unit spheres). We also assume mixture weights for the different
components. We say that one of the underlying subspaces of the model is most
significant if its mixture weight is higher than the sum of the mixture weights
of all other subspaces. We study the recovery of the most significant subspace
by minimizing the lp-averaged distances of data points from d-dimensional
subspaces, where p>0. Unlike other lp minimization problems, this minimization
is non-convex for all p>0 and thus requires different methods for its analysis.
We show that if 0<p<=1, then for any fraction of outliers the most significant
subspace can be recovered by lp minimization with overwhelming probability
(which depends on the generating distribution and its parameters). We show that
when adding small noise around the underlying subspaces the most significant
subspace can be nearly recovered by lp minimization for any 0<p<=1 with an
error proportional to the noise level. On the other hand, if p>1 and there is
more than one underlying subspace, then with overwhelming probability the most
significant subspace cannot be recovered or nearly recovered. This last result
does not require spherically symmetric outliers.Comment: This is a revised version of the part of 1002.1994 that deals with
single subspace recovery. V3: Improved estimates (in particular for Lemma 3.1
and for estimates relying on it), asymptotic dependence of probabilities and
constants on D and d and further clarifications; for simplicity it assumes
uniform distributions on spheres. V4: minor revision for the published
versio
Precision Tests of the Standard Model
30 páginas, 11 figuras, 11 tablas.-- Comunicación presentada al 25º Winter Meeting on Fundamental Physics celebrado del 3 al 8 de MArzo de 1997 en Formigal (España).Precision measurements of electroweak observables provide stringent tests of the Standard Model structure and an accurate determination of its parameters. An overview of the present experimental status is presented.This work has been supported in part
by CICYT (Spain) under grant No. AEN-96-1718.Peer reviewe
Low Complexity Regularization of Linear Inverse Problems
Inverse problems and regularization theory is a central theme in contemporary
signal processing, where the goal is to reconstruct an unknown signal from
partial indirect, and possibly noisy, measurements of it. A now standard method
for recovering the unknown signal is to solve a convex optimization problem
that enforces some prior knowledge about its structure. This has proved
efficient in many problems routinely encountered in imaging sciences,
statistics and machine learning. This chapter delivers a review of recent
advances in the field where the regularization prior promotes solutions
conforming to some notion of simplicity/low-complexity. These priors encompass
as popular examples sparsity and group sparsity (to capture the compressibility
of natural signals and images), total variation and analysis sparsity (to
promote piecewise regularity), and low-rank (as natural extension of sparsity
to matrix-valued data). Our aim is to provide a unified treatment of all these
regularizations under a single umbrella, namely the theory of partial
smoothness. This framework is very general and accommodates all low-complexity
regularizers just mentioned, as well as many others. Partial smoothness turns
out to be the canonical way to encode low-dimensional models that can be linear
spaces or more general smooth manifolds. This review is intended to serve as a
one stop shop toward the understanding of the theoretical properties of the
so-regularized solutions. It covers a large spectrum including: (i) recovery
guarantees and stability to noise, both in terms of -stability and
model (manifold) identification; (ii) sensitivity analysis to perturbations of
the parameters involved (in particular the observations), with applications to
unbiased risk estimation ; (iii) convergence properties of the forward-backward
proximal splitting scheme, that is particularly well suited to solve the
corresponding large-scale regularized optimization problem
Necessary and sufficient conditions of solution uniqueness in minimization
This paper shows that the solutions to various convex minimization
problems are \emph{unique} if and only if a common set of conditions are
satisfied. This result applies broadly to the basis pursuit model, basis
pursuit denoising model, Lasso model, as well as other models that
either minimize or impose the constraint , where
is a strictly convex function. For these models, this paper proves that,
given a solution and defining I=\supp(x^*) and s=\sign(x^*_I),
is the unique solution if and only if has full column rank and there
exists such that and for . This
condition is previously known to be sufficient for the basis pursuit model to
have a unique solution supported on . Indeed, it is also necessary, and
applies to a variety of other models. The paper also discusses ways to
recognize unique solutions and verify the uniqueness conditions numerically.Comment: 6 pages; revised version; submitte
Estimation in high dimensions: a geometric perspective
This tutorial provides an exposition of a flexible geometric framework for
high dimensional estimation problems with constraints. The tutorial develops
geometric intuition about high dimensional sets, justifies it with some results
of asymptotic convex geometry, and demonstrates connections between geometric
results and estimation problems. The theory is illustrated with applications to
sparse recovery, matrix completion, quantization, linear and logistic
regression and generalized linear models.Comment: 56 pages, 9 figures. Multiple minor change
Nonmonotone Barzilai-Borwein Gradient Algorithm for -Regularized Nonsmooth Minimization in Compressive Sensing
This paper is devoted to minimizing the sum of a smooth function and a
nonsmooth -regularized term. This problem as a special cases includes
the -regularized convex minimization problem in signal processing,
compressive sensing, machine learning, data mining, etc. However, the
non-differentiability of the -norm causes more challenging especially
in large problems encountered in many practical applications. This paper
proposes, analyzes, and tests a Barzilai-Borwein gradient algorithm. At each
iteration, the generated search direction enjoys descent property and can be
easily derived by minimizing a local approximal quadratic model and
simultaneously taking the favorable structure of the -norm. Moreover, a
nonmonotone line search technique is incorporated to find a suitable stepsize
along this direction. The algorithm is easily performed, where the values of
the objective function and the gradient of the smooth term are required at
per-iteration. Under some conditions, the proposed algorithm is shown to be
globally convergent. The limited experiments by using some nonconvex
unconstrained problems from CUTEr library with additive -regularization
illustrate that the proposed algorithm performs quite well. Extensive
experiments for -regularized least squares problems in compressive
sensing verify that our algorithm compares favorably with several
state-of-the-art algorithms which are specifically designed in recent years.Comment: 20 page
Efficient and feasible state tomography of quantum many-body systems
We present a novel method to perform quantum state tomography for
many-particle systems which are particularly suitable for estimating states in
lattice systems such as of ultra-cold atoms in optical lattices. We show that
the need for measuring a tomographically complete set of observables can be
overcome by letting the state evolve under some suitably chosen random circuits
followed by the measurement of a single observable. We generalize known results
about the approximation of unitary 2-designs, i.e., certain classes of random
unitary matrices, by random quantum circuits and connect our findings to the
theory of quantum compressed sensing. We show that for ultra-cold atoms in
optical lattices established techniques like optical super-lattices, laser
speckles, and time-of-flight measurements are sufficient to perform fully
certified, assumption-free tomography. Combining our approach with tensor
network methods - in particular the theory of matrix-product states - we
identify situations where the effort of reconstruction is even constant in the
number of lattice sites, allowing in principle to perform tomography on
large-scale systems readily available in present experiments.Comment: 10 pages, 3 figures, minor corrections, discussion added, emphasizing
that no single-site addressing is needed at any stage of the scheme when
implemented in optical lattice system
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