5 research outputs found
Truncated Sparse Approximation Property and Truncated -Norm Minimization
This paper considers approximately sparse signal and low-rank matrix's
recovery via truncated norm minimization and
from noisy measurements. We first introduce truncated
sparse approximation property, a more general robust null space property, and
establish the stable recovery of signals and matrices under the truncated
sparse approximation property. We also explore the relationship between the
restricted isometry property and truncated sparse approximation property. And
we also prove that if a measurement matrix or linear map
satisfies truncated sparse approximation property of order , then the first
inequality in restricted isometry property of order and of order can
hold for certain different constants and ,
respectively. Last, we show that if for
some , then measurement matrix and linear map
satisfy truncated sparse approximation property of order . Which should
point out is that when , our conclusion implies that sparse
approximation property of order is weaker than restricted isometry property
of order
Deterministic Analysis of Weighted BPDN With Partially Known Support Information
In this paper, with the aid of the powerful Restricted Isometry Constant
(RIC), a deterministic (or say non-stochastic) analysis, which includes a
series of sufficient conditions (related to the RIC order) and their resultant
error estimates, is established for the weighted Basis Pursuit De-Noising
(BPDN) to guarantee the robust signal recovery when Partially Known Support
Information (PKSI) of the signal is available. Specifically, the obtained
conditions extend nontrivially the ones induced recently for the traditional
constrained weighted -minimization model to those for its
unconstrained counterpart, i.e., the weighted BPDN. The obtained error
estimates are also comparable to the analogous ones induced previously for the
robust recovery of the signals with PKSI from some constrained models.
Moreover, these results to some degree may well complement the recent
investigation of the weighted BPDN which is based on the stochastic analysis
Matrix Recovery from Rank-One Projection Measurements via Nonconvex Minimization
In this paper, we consider the matrix recovery from rank-one projection
measurements proposed in [Cai and Zhang, Ann. Statist., 43(2015), 102-138], via
nonconvex minimization. We establish a sufficient identifiability condition,
which can guarantee the exact recovery of low-rank matrix via Schatten-
minimization for under affine constraint, and
stable recovery of low-rank matrix under constraint and Dantzig
selector constraint. Our condition is also sufficient to guarantee low-rank
matrix recovery via least minimization
for . And we also extend our result to Gaussian design distribution,
and show that any matrix can be stably recovered for rank-one projection from
Gaussian distributions via least minimization with high probability
Coherence-Based Performance Guarantee of Regularized -Norm Minimization and Beyond
In this paper, we consider recovering the signal
from its few noisy measurements , where
with is the measurement matrix, and
is the measurement noise/error. We first establish a
coherence-based performance guarantee for a regularized -norm
minimization model to recover such signals in the presence of the
-norm bounded noise, i.e., , and then
extend these theoretical results to guarantee the robust recovery of the
signals corrupted with the Dantzig Selector (DS) type noise, i.e.,
, and the structured block-sparse signal
recovery in the presence of the bounded noise. To the best of our knowledge, we
first extend nontrivially the sharp uniform recovery condition derived by Cai,
Wang and Xu (2010) for the constrained -norm minimization model,
which takes the form of \begin{align*} \mu<\frac{1}{2k-1}, \end{align*} where
is defined as the (mutual) coherence of , to two unconstrained
regularized -norm minimization models to guarantee the robust
recovery of any signals (not necessary to be -sparse) under the
-norm bounded noise and the DS type noise settings, respectively.
Besides, a uniform recovery condition and its two resulting error estimates are
also established for the first time to our knowledge, for the robust
block-sparse signal recovery using a regularized mixed -norm
minimization model, and these results well complement the existing theoretical
investigation on this model which focuses on the non-uniform recovery
conditions and/or the robust signal recovery in presence of the random noise.Comment: 19 page
The high-order block RIP for non-convex block-sparse compressed sensing
This paper concentrates on the recovery of block-sparse signals, which is not
only sparse but also nonzero elements are arrayed into some blocks (clusters)
rather than being arbitrary distributed all over the vector, from linear
measurements. We establish high-order sufficient conditions based on block RIP
to ensure the exact recovery of every block -sparse signal in the noiseless
case via mixed minimization method, and the stable and robust
recovery in the case that signals are not accurately block-sparse in the
presence of noise. Additionally, a lower bound on necessary number of random
Gaussian measurements is gained for the condition to be true with overwhelming
probability. Furthermore, the numerical experiments conducted demonstrate the
performance of the proposed algorithm