40 research outputs found
Robust sparse analysis regularization
ABSTRACT This work studies some properties of 1 -analysis regularization for the resolution of linear inverse problems. Analysis regularization minimizes the 1 norm of the correlations between the signal and the atoms in the dictionary. The corresponding variational problem includes several well-known regularizations such as the discrete total variation and the fused lasso. We give sufficient conditions such that analysis regularization is robust to noise. ANALYSIS VERSUS SYNTHESIS Regularization through variational analysis is a popular way to compute an approximation of x 0 ∈ R N from the measurements y ∈ R Q as defined by an inverse problem y = Φx 0 + w where w is some additive noise and Φ is a linear operator, for instance a super-resolution or an inpainting operator. N which is used to synthesize a signal Common examples in signal processing of dictionary include the wavelet transform or a finite-difference operator. Synthesis regularization corresponds to the following minimization problem where Ψ = ΦD, and x = Dα. Properties of synthesis prior had been studied intensively, see for instance Analysis regularization corresponds to the following minimization problem In the noiseless case, w = 0, one uses the constrained optimization which reads min x∈R N ||D * x|| 1 subject to Φx = y. This prior had been less studied than the synthesis prior, see for instance UNION OF SUBSPACES MODEL It is natural to keep track of the support of this correlation vector, as done in the following definition. A signal x such that D * x is sparse lives in a cospace G J of small dimension where G J is defined as follow. Definition 2. Given a dictionary D, and J a subset of {1 · · · P }, the cospace G J is defined as where D J is the subdictionary whose columns are indexed by J. The signal space can thus be decomposed as a union of subspaces of increasing dimensions For the 1-D total variation prior, Θ k is the set of piecewise constant signals with k − 1 steps
Sparse Support Recovery with Non-smooth Loss Functions
In this paper, we study the support recovery guarantees of underdetermined
sparse regression using the -norm as a regularizer and a non-smooth
loss function for data fidelity. More precisely, we focus in detail on the
cases of and losses, and contrast them with the usual
loss. While these losses are routinely used to account for either
sparse ( loss) or uniform ( loss) noise models, a
theoretical analysis of their performance is still lacking. In this article, we
extend the existing theory from the smooth case to these non-smooth
cases. We derive a sharp condition which ensures that the support of the vector
to recover is stable to small additive noise in the observations, as long as
the loss constraint size is tuned proportionally to the noise level. A
distinctive feature of our theory is that it also explains what happens when
the support is unstable. While the support is not stable anymore, we identify
an "extended support" and show that this extended support is stable to small
additive noise. To exemplify the usefulness of our theory, we give a detailed
numerical analysis of the support stability/instability of compressed sensing
recovery with these different losses. This highlights different parameter
regimes, ranging from total support stability to progressively increasing
support instability.Comment: in Proc. NIPS 201
Model Consistency of Partly Smooth Regularizers
This paper studies least-square regression penalized with partly smooth
convex regularizers. This class of functions is very large and versatile
allowing to promote solutions conforming to some notion of low-complexity.
Indeed, they force solutions of variational problems to belong to a
low-dimensional manifold (the so-called model) which is stable under small
perturbations of the function. This property is crucial to make the underlying
low-complexity model robust to small noise. We show that a generalized
"irrepresentable condition" implies stable model selection under small noise
perturbations in the observations and the design matrix, when the
regularization parameter is tuned proportionally to the noise level. This
condition is shown to be almost a necessary condition. We then show that this
condition implies model consistency of the regularized estimator. That is, with
a probability tending to one as the number of measurements increases, the
regularized estimator belongs to the correct low-dimensional model manifold.
This work unifies and generalizes several previous ones, where model
consistency is known to hold for sparse, group sparse, total variation and
low-rank regularizations
The degrees of freedom of the Lasso for general design matrix
In this paper, we investigate the degrees of freedom (\dof) of penalized
minimization (also known as the Lasso) for linear regression models.
We give a closed-form expression of the \dof of the Lasso response. Namely,
we show that for any given Lasso regularization parameter and any
observed data belonging to a set of full (Lebesgue) measure, the
cardinality of the support of a particular solution of the Lasso problem is an
unbiased estimator of the degrees of freedom. This is achieved without the need
of uniqueness of the Lasso solution. Thus, our result holds true for both the
underdetermined and the overdetermined case, where the latter was originally
studied in \cite{zou}. We also show, by providing a simple counterexample, that
although the \dof theorem of \cite{zou} is correct, their proof contains a
flaw since their divergence formula holds on a different set of a full measure
than the one that they claim. An effective estimator of the number of degrees
of freedom may have several applications including an objectively guided choice
of the regularization parameter in the Lasso through the \sure framework. Our
theoretical findings are illustrated through several numerical simulations.Comment: A short version appeared in SPARS'11, June 2011 Previously entitled
"The degrees of freedom of penalized l1 minimization
GSplit LBI: Taming the Procedural Bias in Neuroimaging for Disease Prediction
In voxel-based neuroimage analysis, lesion features have been the main focus
in disease prediction due to their interpretability with respect to the related
diseases. However, we observe that there exists another type of features
introduced during the preprocessing steps and we call them "\textbf{Procedural
Bias}". Besides, such bias can be leveraged to improve classification accuracy.
Nevertheless, most existing models suffer from either under-fit without
considering procedural bias or poor interpretability without differentiating
such bias from lesion ones. In this paper, a novel dual-task algorithm namely
\emph{GSplit LBI} is proposed to resolve this problem. By introducing an
augmented variable enforced to be structural sparsity with a variable splitting
term, the estimators for prediction and selecting lesion features can be
optimized separately and mutually monitored by each other following an
iterative scheme. Empirical experiments have been evaluated on the Alzheimer's
Disease Neuroimaging Initiative\thinspace(ADNI) database. The advantage of
proposed model is verified by improved stability of selected lesion features
and better classification results.Comment: Conditional Accepted by Miccai,201
Stable image reconstruction using total variation minimization
This article presents near-optimal guarantees for accurate and robust image
recovery from under-sampled noisy measurements using total variation
minimization. In particular, we show that from O(slog(N)) nonadaptive linear
measurements, an image can be reconstructed to within the best s-term
approximation of its gradient up to a logarithmic factor, and this factor can
be removed by taking slightly more measurements. Along the way, we prove a
strengthened Sobolev inequality for functions lying in the null space of
suitably incoherent matrices.Comment: 25 page