328 research outputs found
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
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 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
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
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
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
Fundamental Limits of Blind Deconvolution Part I: Ambiguity Kernel
Blind deconvolution is an ubiquitous non-linear inverse problem in
applications like wireless communications and image processing. This problem is
generally ill-posed, and there have been efforts to use sparse models for
regularizing blind deconvolution to promote signal identifiability. Part I of
this two-part paper characterizes the ambiguity space of blind deconvolution
and shows unidentifiability of this inverse problem for almost every pair of
unconstrained input signals. The approach involves lifting the deconvolution
problem to a rank one matrix recovery problem and analyzing the rank two null
space of the resultant linear operator. A measure theoretically tight
(parametric and recursive) representation of the key rank two null space is
stated and proved. This representation is a novel foundational result for
signal and code design strategies promoting identifiability under convolutive
observation models. Part II of this paper analyzes the identifiability of
sparsity constrained blind deconvolution and establishes surprisingly strong
negative results on scaling laws for the sparsity-ambiguity trade-off.Comment: 20 pages, 4 figure
Sparse Model Uncertainties in Compressed Sensing with Application to Convolutions and Sporadic Communication
The success of the compressed sensing paradigm has shown that a substantial
reduction in sampling and storage complexity can be achieved in certain linear
and non-adaptive estimation problems. It is therefore an advisable strategy for
noncoherent information retrieval in, for example, sporadic blind and
semi-blind communication and sampling problems. But, the conventional model is
not practical here since the compressible signals have to be estimated from
samples taken solely on the output of an un-calibrated system which is unknown
during measurement but often compressible. Conventionally, one has either to
operate at suboptimal sampling rates or the recovery performance substantially
suffers from the dominance of model mismatch. In this work we discuss such type
of estimation problems and we focus on bilinear inverse problems. We link this
problem to the recovery of low-rank and sparse matrices and establish stable
low-dimensional embeddings of the uncalibrated receive signals whereby
addressing also efficient communication-oriented methods like universal random
demodulation. Exemplary, we investigate in more detail sparse convolutions
serving as a basic communication channel model. In using some recent results
from additive combinatorics we show that such type of signals can be
efficiently low-rate sampled by semi-blind methods. Finally, we present a
further application of these results in the field of phase retrieval from
intensity Fourier measurements.Comment: Book chapter, submitted to "Compressed Sensing and its Applications",
31 pages, revised versio
Parametric Bilinear Generalized Approximate Message Passing
We propose a scheme to estimate the parameters and of the
bilinear form from noisy measurements
, where and are related through an arbitrary
likelihood function and are known. Our scheme is based on
generalized approximate message passing (G-AMP): it treats and as
random variables and as an i.i.d.\ Gaussian 3-way tensor in order
to derive a tractable simplification of the sum-product algorithm in the
large-system limit. It generalizes previous instances of bilinear G-AMP, such
as those that estimate matrices and from a
noisy measurement of , allowing the application
of AMP methods to problems such as self-calibration, blind deconvolution, and
matrix compressive sensing. Numerical experiments confirm the accuracy and
computational efficiency of the proposed approach
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
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