536 research outputs found
Asymptotic Analysis of Inpainting via Universal Shearlet Systems
Recently introduced inpainting algorithms using a combination of applied
harmonic analysis and compressed sensing have turned out to be very successful.
One key ingredient is a carefully chosen representation system which provides
(optimally) sparse approximations of the original image. Due to the common
assumption that images are typically governed by anisotropic features,
directional representation systems have often been utilized. One prominent
example of this class are shearlets, which have the additional benefitallowing
faithful implementations. Numerical results show that shearlets significantly
outperform wavelets in inpainting tasks. One of those software packages,
www.shearlab.org, even offers the flexibility of usingdifferent parameter for
each scale, which is not yet covered by shearlet theory.
In this paper, we first introduce universal shearlet systems which are
associated with an arbitrary scaling sequence, thereby modeling the previously
mentioned flexibility. In addition, this novel construction allows for a smooth
transition between wavelets and shearlets and therefore enables us to analyze
them in a uniform fashion. For a large class of such scaling sequences, we
first prove that the associated universal shearlet systems form band-limited
Parseval frames for consisting of Schwartz functions.
Secondly, we analyze the performance for inpainting of this class of universal
shearlet systems within a distributional model situation using an
-analysis minimization algorithm for reconstruction. Our main result in
this part states that, provided the scaling sequence is comparable to the size
of the (scale-dependent) gap, nearly-perfect inpainting is achieved at
sufficiently fine scales
Robust 1-Bit Compressed Sensing via Hinge Loss Minimization
This work theoretically studies the problem of estimating a structured
high-dimensional signal from noisy -bit Gaussian
measurements. Our recovery approach is based on a simple convex program which
uses the hinge loss function as data fidelity term. While such a risk
minimization strategy is very natural to learn binary output models, such as in
classification, its capacity to estimate a specific signal vector is largely
unexplored. A major difficulty is that the hinge loss is just piecewise linear,
so that its "curvature energy" is concentrated in a single point. This is
substantially different from other popular loss functions considered in signal
estimation, e.g., the square or logistic loss, which are at least locally
strongly convex. It is therefore somewhat unexpected that we can still prove
very similar types of recovery guarantees for the hinge loss estimator, even in
the presence of strong noise. More specifically, our non-asymptotic error
bounds show that stable and robust reconstruction of can be achieved with
the optimal oversampling rate in terms of the number of
measurements . Moreover, we permit a wide class of structural assumptions on
the ground truth signal, in the sense that can belong to an arbitrary
bounded convex set . The proofs of our main results
rely on some recent advances in statistical learning theory due to Mendelson.
In particular, we invoke an adapted version of Mendelson's small ball method
that allows us to establish a quadratic lower bound on the error of the first
order Taylor approximation of the empirical hinge loss function
-Analysis Minimization and Generalized (Co-)Sparsity: When Does Recovery Succeed?
This paper investigates the problem of signal estimation from undersampled
noisy sub-Gaussian measurements under the assumption of a cosparse model. Based
on generalized notions of sparsity, we derive novel recovery guarantees for the
-analysis basis pursuit, enabling highly accurate predictions of its
sample complexity. The corresponding bounds on the number of required
measurements do explicitly depend on the Gram matrix of the analysis operator
and therefore particularly account for its mutual coherence structure. Our
findings defy conventional wisdom which promotes the sparsity of analysis
coefficients as the crucial quantity to study. In fact, this common paradigm
breaks down completely in many situations of practical interest, for instance,
when applying a redundant (multilevel) frame as analysis prior. By extensive
numerical experiments, we demonstrate that, in contrast, our theoretical
sampling-rate bounds reliably capture the recovery capability of various
examples, such as redundant Haar wavelets systems, total variation, or random
frames. The proofs of our main results build upon recent achievements in the
convex geometry of data mining problems. More precisely, we establish a
sophisticated upper bound on the conic Gaussian mean width that is associated
with the underlying -analysis polytope. Due to a novel localization
argument, it turns out that the presented framework naturally extends to stable
recovery, allowing us to incorporate compressible coefficient sequences as
well
Generic Error Bounds for the Generalized Lasso with Sub-Exponential Data
This work performs a non-asymptotic analysis of the generalized Lasso under
the assumption of sub-exponential data. Our main results continue recent
research on the benchmark case of (sub-)Gaussian sample distributions and
thereby explore what conclusions are still valid when going beyond. While many
statistical features of the generalized Lasso remain unaffected (e.g.,
consistency), the key difference becomes manifested in the way how the
complexity of the hypothesis set is measured. It turns out that the estimation
error can be controlled by means of two complexity parameters that arise
naturally from a generic-chaining-based proof strategy. The output model can be
non-realizable, while the only requirement for the input vector is a generic
concentration inequality of Bernstein-type, which can be implemented for a
variety of sub-exponential distributions. This abstract approach allows us to
reproduce, unify, and extend previously known guarantees for the generalized
Lasso. In particular, we present applications to semi-parametric output models
and phase retrieval via the lifted Lasso. Moreover, our findings are discussed
in the context of sparse recovery and high-dimensional estimation problems
Compressed Sensing with 1D Total Variation: Breaking Sample Complexity Barriers via Non-Uniform Recovery (iTWIST'20)
This paper investigates total variation minimization in one spatial dimension
for the recovery of gradient-sparse signals from undersampled Gaussian
measurements. Recently established bounds for the required sampling rate state
that uniform recovery of all -gradient-sparse signals in is
only possible with measurements.
Such a condition is especially prohibitive for high-dimensional problems, where
is much smaller than . However, previous empirical findings seem to
indicate that the latter sampling rate does not reflect the typical behavior of
total variation minimization. Indeed, this work provides a rigorous analysis
that breaks the -bottleneck for a large class of natural signals.
The main result shows that non-uniform recovery succeeds with high probability
for measurements if the jump
discontinuities of the signal vector are sufficiently well separated. In
particular, this guarantee allows for signals arising from a discretization of
piecewise constant functions defined on an interval. The present paper serves
as a short summary of the main results in our recent work [arxiv:2001.09952].Comment: in Proceedings of iTWIST'20, Paper-ID: 32, Nantes, France, December,
2-4, 2020. arXiv admin note: substantial text overlap with arXiv:2001.0995
Sparse Proteomics Analysis - A compressed sensing-based approach for feature selection and classification of high-dimensional proteomics mass spectrometry data
Background: High-throughput proteomics techniques, such as mass spectrometry
(MS)-based approaches, produce very high-dimensional data-sets. In a clinical
setting one is often interested in how mass spectra differ between patients of
different classes, for example spectra from healthy patients vs. spectra from
patients having a particular disease. Machine learning algorithms are needed to
(a) identify these discriminating features and (b) classify unknown spectra
based on this feature set. Since the acquired data is usually noisy, the
algorithms should be robust against noise and outliers, while the identified
feature set should be as small as possible.
Results: We present a new algorithm, Sparse Proteomics Analysis (SPA), based
on the theory of compressed sensing that allows us to identify a minimal
discriminating set of features from mass spectrometry data-sets. We show (1)
how our method performs on artificial and real-world data-sets, (2) that its
performance is competitive with standard (and widely used) algorithms for
analyzing proteomics data, and (3) that it is robust against random and
systematic noise. We further demonstrate the applicability of our algorithm to
two previously published clinical data-sets
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