78 research outputs found
Guarantees of Total Variation Minimization for Signal Recovery
In this paper, we consider using total variation minimization to recover
signals whose gradients have a sparse support, from a small number of
measurements. We establish the proof for the performance guarantee of total
variation (TV) minimization in recovering \emph{one-dimensional} signal with
sparse gradient support. This partially answers the open problem of proving the
fidelity of total variation minimization in such a setting \cite{TVMulti}. In
particular, we have shown that the recoverable gradient sparsity can grow
linearly with the signal dimension when TV minimization is used. Recoverable
sparsity thresholds of TV minimization are explicitly computed for
1-dimensional signal by using the Grassmann angle framework. We also extend our
results to TV minimization for multidimensional signals. Stability of
recovering signal itself using 1-D TV minimization has also been established
through a property called "almost Euclidean property for 1-dimensional TV
norm". We further give a lower bound on the number of random Gaussian
measurements for recovering 1-dimensional signal vectors with elements and
-sparse gradients. Interestingly, the number of needed measurements is lower
bounded by , rather than the bound
frequently appearing in recovering -sparse signal vectors.Comment: lower bounds added; version with Gaussian width, improved bounds;
stability results adde
Precise Phase Transition of Total Variation Minimization
Characterizing the phase transitions of convex optimizations in recovering
structured signals or data is of central importance in compressed sensing,
machine learning and statistics. The phase transitions of many convex
optimization signal recovery methods such as minimization and nuclear
norm minimization are well understood through recent years' research. However,
rigorously characterizing the phase transition of total variation (TV)
minimization in recovering sparse-gradient signal is still open. In this paper,
we fully characterize the phase transition curve of the TV minimization. Our
proof builds on Donoho, Johnstone and Montanari's conjectured phase transition
curve for the TV approximate message passing algorithm (AMP), together with the
linkage between the minmax Mean Square Error of a denoising problem and the
high-dimensional convex geometry for TV minimization.Comment: 6 page
Precise Semidefinite Programming Formulation of Atomic Norm Minimization for Recovering d-Dimensional () Off-the-Grid Frequencies
Recent research in off-the-grid compressed sensing (CS) has demonstrated
that, under certain conditions, one can successfully recover a spectrally
sparse signal from a few time-domain samples even though the dictionary is
continuous. In particular, atomic norm minimization was proposed in
\cite{tang2012csotg} to recover -dimensional spectrally sparse signal.
However, in spite of existing research efforts \cite{chi2013compressive}, it
was still an open problem how to formulate an equivalent positive semidefinite
program for atomic norm minimization in recovering signals with -dimensional
() off-the-grid frequencies. In this paper, we settle this problem by
proposing equivalent semidefinite programming formulations of atomic norm
minimization to recover signals with -dimensional () off-the-grid
frequencies.Comment: 4 pages, double-column,1 Figur
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