655 research outputs found
A Unified Framework for Identifiability Analysis in Bilinear Inverse Problems with Applications to Subspace and Sparsity Models
Bilinear inverse problems (BIPs), the resolution of two vectors given their
image under a bilinear mapping, arise in many applications. Without further
constraints, BIPs are usually ill-posed. In practice, properties of natural
signals are exploited to solve BIPs. For example, subspace constraints or
sparsity constraints are imposed to reduce the search space. These approaches
have shown some success in practice. However, there are few results on
uniqueness in BIPs. For most BIPs, the fundamental question of under what
condition the problem admits a unique solution, is yet to be answered. For
example, blind gain and phase calibration (BGPC) is a structured bilinear
inverse problem, which arises in many applications, including inverse rendering
in computational relighting (albedo estimation with unknown lighting), blind
phase and gain calibration in sensor array processing, and multichannel blind
deconvolution (MBD). It is interesting to study the uniqueness of such
problems.
In this paper, we define identifiability of a BIP up to a group of
transformations. We derive necessary and sufficient conditions for such
identifiability, i.e., the conditions under which the solutions can be uniquely
determined up to the transformation group. Applying these results to BGPC, we
derive sufficient conditions for unique recovery under several scenarios,
including subspace, joint sparsity, and sparsity models. For BGPC with joint
sparsity or sparsity constraints, we develop a procedure to compute the
relevant transformation groups. We also give necessary conditions in the form
of tight lower bounds on sample complexities, and demonstrate the tightness of
these bounds by numerical experiments. The results for BGPC not only
demonstrate the application of the proposed general framework for
identifiability analysis, but are also of interest in their own right.Comment: 40 pages, 3 figure
Optimal Injectivity Conditions for Bilinear Inverse Problems with Applications to Identifiability of Deconvolution Problems
We study identifiability for bilinear inverse problems under sparsity and
subspace constraints. We show that, up to a global scaling ambiguity, almost
all such maps are injective on the set of pairs of sparse vectors if the number
of measurements exceeds , where and denote the
sparsity of the two input vectors, and injective on the set of pairs of vectors
lying in known subspaces of dimensions and if .
We also prove that both these bounds are tight in the sense that one cannot
have injectivity for a smaller number of measurements. Our proof technique
draws from algebraic geometry. As an application we derive optimal
identifiability conditions for the deconvolution problem, thus improving on
recent work of Li et al. [1]
Identifiability Scaling Laws in Bilinear Inverse Problems
A number of ill-posed inverse problems in signal processing, like blind
deconvolution, matrix factorization, dictionary learning and blind source
separation share the common characteristic of being bilinear inverse problems
(BIPs), i.e. the observation model is a function of two variables and
conditioned on one variable being known, the observation is a linear function
of the other variable. A key issue that arises for such inverse problems is
that of identifiability, i.e. whether the observation is sufficient to
unambiguously determine the pair of inputs that generated the observation.
Identifiability is a key concern for applications like blind equalization in
wireless communications and data mining in machine learning. Herein, a unifying
and flexible approach to identifiability analysis for general conic prior
constrained BIPs is presented, exploiting a connection to low-rank matrix
recovery via lifting. We develop deterministic identifiability conditions on
the input signals and examine their satisfiability in practice for three
classes of signal distributions, viz. dependent but uncorrelated, independent
Gaussian, and independent Bernoulli. In each case, scaling laws are developed
that trade-off probability of robust identifiability with the complexity of the
rank two null space. An added appeal of our approach is that the rank two null
space can be partly or fully characterized for many bilinear problems of
interest (e.g. blind deconvolution). We present numerical experiments involving
variations on the blind deconvolution problem that exploit a characterization
of the rank two null space and demonstrate that the scaling laws offer good
estimates of identifiability.Comment: 25 pages, 5 figure
Identifiability and Stability in Blind Deconvolution under Minimal Assumptions
Blind deconvolution (BD) arises in many applications. Without assumptions on
the signal and the filter, BD does not admit a unique solution. In practice,
subspace or sparsity assumptions have shown the ability to reduce the search
space and yield the unique solution. However, existing theoretical analysis on
uniqueness in BD is rather limited. In an earlier paper, we provided the first
algebraic sample complexities for BD that hold for almost all bases or frames.
We showed that for BD of a pair of vectors in , with subspace
constraints of dimensions and , respectively, a sample complexity of
is sufficient. This result is suboptimal, since the number of
degrees of freedom is merely . We provided analogus results, with
similar suboptimality, for BD with sparsity or mixed subspace and sparsity
constraints. In this paper, taking advantage of the recent progress on the
information-theoretic limits of unique low-rank matrix recovery, we finally
bridge this gap, and derive an optimal sample complexity result for BD with
generic bases or frames. We show that for BD of an arbitrary pair (resp. all
pairs) of vectors in , with sparsity constraints of sparsity
levels and , a sample complexity of (resp. ) is sufficient. We also present analogous results for BD with
subspace constraints or mixed constraints, with the subspace dimension
replacing the sparsity level. Last but not least, in all the above scenarios,
if the bases or frames follow a probabilistic distribution specified in the
paper, the recovery is not only unique, but also stable against small
perturbations in the measurements, under the same sample complexities.Comment: 32 page
Optimal Sample Complexity for Blind Gain and Phase Calibration
Blind gain and phase calibration (BGPC) is a structured bilinear inverse
problem, which arises in many applications, including inverse rendering in
computational relighting (albedo estimation with unknown lighting), blind phase
and gain calibration in sensor array processing, and multichannel blind
deconvolution. The fundamental question of the uniqueness of the solutions to
such problems has been addressed only recently. In a previous paper, we
proposed studying the identifiability in bilinear inverse problems up to
transformation groups. In particular, we studied several special cases of blind
gain and phase calibration, including the cases of subspace and joint sparsity
models on the signals, and gave sufficient and necessary conditions for
identifiability up to certain transformation groups. However, there were gaps
between the sample complexities in the sufficient conditions and the necessary
conditions. In this paper, under a mild assumption that the signals and models
are generic, we bridge the gaps by deriving tight sufficient conditions with
optimal sample complexities.Comment: 17 pages, 1 figure. arXiv admin note: text overlap with
arXiv:1501.0612
Identifiability in Blind Deconvolution with Subspace or Sparsity Constraints
Blind deconvolution (BD), the resolution of a signal and a filter given their
convolution, arises in many applications. Without further constraints, BD is
ill-posed. In practice, subspace or sparsity constraints have been imposed to
reduce the search space, and have shown some empirical success. However,
existing theoretical analysis on uniqueness in BD is rather limited. As an
effort to address the still mysterious question, we derive sufficient
conditions under which two vectors can be uniquely identified from their
circular convolution, subject to subspace or sparsity constraints. These
sufficient conditions provide the first algebraic sample complexities for BD.
We first derive a sufficient condition that applies to almost all bases or
frames. For blind deconvolution of vectors in , with two subspace
constraints of dimensions and , the required sample complexity is
. Then we impose a sub-band structure on one basis, and derive a
sufficient condition that involves a relaxed sample complexity , which we show to be optimal. We present the extensions of these
results to BD with sparsity constraints or mixed constraints, with the sparsity
level replacing the subspace dimension. The cost for the unknown support in
this case is an extra factor of 2 in the sample complexity.Comment: 17 pages, 3 figures. Some of these results will be presented at SPARS
201
Blind Recovery of Sparse Signals from Subsampled Convolution
Subsampled blind deconvolution is the recovery of two unknown signals from
samples of their convolution. To overcome the ill-posedness of this problem,
solutions based on priors tailored to specific application have been developed
in practical applications. In particular, sparsity models have provided
promising priors. However, in spite of empirical success of these methods in
many applications, existing analyses are rather limited in two main ways: by
disparity between the theoretical assumptions on the signal and/or measurement
model versus practical setups; or by failure to provide a performance guarantee
for parameter values within the optimal regime defined by the information
theoretic limits. In particular, it has been shown that a naive sparsity model
is not a strong enough prior for identifiability in the blind deconvolution
problem. Instead, in addition to sparsity, we adopt a conic constraint, which
enforces spectral flatness of the signals. Under this prior, we provide an
iterative algorithm that achieves guaranteed performance in blind deconvolution
at near optimal sample complexity. Numerical results show the empirical
performance of the iterative algorithm agrees with the performance guarantee
Blind Identification of Graph Filters
Network processes are often represented as signals defined on the vertices of
a graph. To untangle the latent structure of such signals, one can view them as
outputs of linear graph filters modeling underlying network dynamics. This
paper deals with the problem of joint identification of a graph filter and its
input signal, thus broadening the scope of classical blind deconvolution of
temporal and spatial signals to the less-structured graph domain. Given a graph
signal modeled as the output of a graph filter, the goal is to
recover the vector of filter coefficients , and the input signal
which is assumed to be sparse. While is a bilinear
function of and , the filtered graph signal is also a
linear combination of the entries of the lifted rank-one, row-sparse matrix
. The blind graph-filter identification problem can
thus be tackled via rank and sparsity minimization subject to linear
constraints, an inverse problem amenable to convex relaxations offering
provable recovery guarantees under simplifying assumptions. Numerical tests
using both synthetic and real-world networks illustrate the merits of the
proposed algorithms, as well as the benefits of leveraging multiple signals to
aid the blind identification task
Blind Gain and Phase Calibration via Sparse Spectral Methods
Blind gain and phase calibration (BGPC) is a bilinear inverse problem
involving the determination of unknown gains and phases of the sensing system,
and the unknown signal, jointly. BGPC arises in numerous applications, e.g.,
blind albedo estimation in inverse rendering, synthetic aperture radar
autofocus, and sensor array auto-calibration. In some cases, sparse structure
in the unknown signal alleviates the ill-posedness of BGPC. Recently there has
been renewed interest in solutions to BGPC with careful analysis of error
bounds. In this paper, we formulate BGPC as an eigenvalue/eigenvector problem,
and propose to solve it via power iteration, or in the sparsity or joint
sparsity case, via truncated power iteration. Under certain assumptions, the
unknown gains, phases, and the unknown signal can be recovered simultaneously.
Numerical experiments show that power iteration algorithms work not only in the
regime predicted by our main results, but also in regimes where theoretical
analysis is limited. We also show that our power iteration algorithms for BGPC
compare favorably with competing algorithms in adversarial conditions, e.g.,
with noisy measurement or with a bad initial estimate.Comment: 28 pages, 11 figure
A Bayesian framework for molecular strain identification from mixed diagnostic samples
We provide a mathematical formulation and develop a computational framework
for identifying multiple strains of microorganisms from mixed samples of DNA.
Our method is applicable in public health domains where efficient
identification of pathogens is paramount, e.g., for the monitoring of disease
outbreaks. We formulate strain identification as an inverse problem that aims
at simultaneously estimating a binary matrix (encoding presence or absence of
mutations in each strain) and a real-valued vector (representing the mixture of
strains) such that their product is approximately equal to the measured data
vector. The problem at hand has a similar structure to blind deconvolution,
except for the presence of binary constraints, which we enforce in our
approach. Following a Bayesian approach, we derive a posterior density. We
present two computational methods for solving the non-convex maximum a
posteriori estimation problem. The first one is a local optimization method
that is made efficient and scalable by decoupling the problem into smaller
independent subproblems, whereas the second one yields a global minimizer by
converting the problem into a convex mixed-integer quadratic programming
problem. The decoupling approach also provides an efficient way to integrate
over the posterior. This provides useful information about the ambiguity of the
underdetermined problem and, thus, the uncertainty associated with numerical
solutions. We evaluate the potential and limitations of our framework in silico
using synthetic and experimental data with available ground truths.Comment: 25 pages, 4 figure
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