13,827 research outputs found
On the solution uniqueness characterization in the L1 norm and polyhedral gauge recovery
Paper to appear.International audienceThis paper first proposes another proof of the necessary and sufficient conditions of solution uniqueness in 1-norm minimization given recently by H. Zhang, W. Yin, and L. Cheng. The analysis avoids the need of the surjectivity assumption made by these authors and should be mainly appealing by its short length (it can therefore be proposed to students exercising in convex optimization). In the second part of the paper, the previous existence and uniqueness characterization is extended to the recovery problem where the β 1 norm is substituted by a polyhedral gauge. In addition to present interest for a number of practical problems, this extension clarifies the geometrical aspect of the previous uniqueness characterization. Numerical techniques are proposed to compute a solution to the polyhedral gauge recovery problem in polynomial time and to check its possible uniqueness by a simple linear algebra test
RSP-Based Analysis for Sparsest and Least -Norm Solutions to Underdetermined Linear Systems
Recently, the worse-case analysis, probabilistic analysis and empirical
justification have been employed to address the fundamental question: When does
-minimization find the sparsest solution to an underdetermined linear
system? In this paper, a deterministic analysis, rooted in the classic linear
programming theory, is carried out to further address this question. We first
identify a necessary and sufficient condition for the uniqueness of least
-norm solutions to linear systems. From this condition, we deduce that
a sparsest solution coincides with the unique least -norm solution to a
linear system if and only if the so-called \emph{range space property} (RSP)
holds at this solution. This yields a broad understanding of the relationship
between - and -minimization problems. Our analysis indicates
that the RSP truly lies at the heart of the relationship between these two
problems. Through RSP-based analysis, several important questions in this field
can be largely addressed. For instance, how to efficiently interpret the gap
between the current theory and the actual numerical performance of
-minimization by a deterministic analysis, and if a linear system has
multiple sparsest solutions, when does -minimization guarantee to find
one of them? Moreover, new matrix properties (such as the \emph{RSP of order
} and the \emph{Weak-RSP of order }) are introduced in this paper, and a
new theory for sparse signal recovery based on the RSP of order is
established
The Lasso Problem and Uniqueness
The lasso is a popular tool for sparse linear regression, especially for
problems in which the number of variables p exceeds the number of observations
n. But when p>n, the lasso criterion is not strictly convex, and hence it may
not have a unique minimum. An important question is: when is the lasso solution
well-defined (unique)? We review results from the literature, which show that
if the predictor variables are drawn from a continuous probability
distribution, then there is a unique lasso solution with probability one,
regardless of the sizes of n and p. We also show that this result extends
easily to penalized minimization problems over a wide range of loss
functions.
A second important question is: how can we deal with the case of
non-uniqueness in lasso solutions? In light of the aforementioned result, this
case really only arises when some of the predictor variables are discrete, or
when some post-processing has been performed on continuous predictor
measurements. Though we certainly cannot claim to provide a complete answer to
such a broad question, we do present progress towards understanding some
aspects of non-uniqueness. First, we extend the LARS algorithm for computing
the lasso solution path to cover the non-unique case, so that this path
algorithm works for any predictor matrix. Next, we derive a simple method for
computing the component-wise uncertainty in lasso solutions of any given
problem instance, based on linear programming. Finally, we review results from
the literature on some of the unifying properties of lasso solutions, and also
point out particular forms of solutions that have distinctive properties.Comment: 25 pages, 0 figure
A Unique "Nonnegative" Solution to an Underdetermined System: from Vectors to Matrices
This paper investigates the uniqueness of a nonnegative vector solution and
the uniqueness of a positive semidefinite matrix solution to underdetermined
linear systems. A vector solution is the unique solution to an underdetermined
linear system only if the measurement matrix has a row-span intersecting the
positive orthant. Focusing on two types of binary measurement matrices,
Bernoulli 0-1 matrices and adjacency matrices of general expander graphs, we
show that, in both cases, the support size of a unique nonnegative solution can
grow linearly, namely O(n), with the problem dimension n. We also provide
closed-form characterizations of the ratio of this support size to the signal
dimension. For the matrix case, we show that under a necessary and sufficient
condition for the linear compressed observations operator, there will be a
unique positive semidefinite matrix solution to the compressed linear
observations. We further show that a randomly generated Gaussian linear
compressed observations operator will satisfy this condition with
overwhelmingly high probability
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