A unified framework for identifiability analysis in bilinear inverse problems

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. As an effort to address the question, we propose a unified framework for identifiability analysis in BIPs. We define identifiability of a BIP up to a group of transformations. Then 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. 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). Applying our unified framework 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 transformation groups corresponding to inherent ambiguities. 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. Blind deconvolution (BD), the resolution of a signal and a filter given their convolution, is another bilinear inverse problem routinely encountered in signal processing and communications. Existing theoretical analysis on uniqueness in BD is rather limited. 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. Then we impose a sub-band structure on one basis, and derive a less demanding sufficient condition, which is essentially optimal, using our unified framework. 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

Bilinear inverse problems with sparsity: Optimal identifiability conditions and efficient recovery

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, parsimonious structures of natural signals (e.g., subspace or sparsity) are exploited. However, there are few theoretical justifications for using such structures for BIPs. We consider two types of BIPs, blind deconvolution (BD) and blind gain and phase calibration (BGPC), with subspace or sparsity structures. Our contributions are twofold: we derive optimal identifiability conditions, and propose efficient algorithms that solve these problems. In previous work, we provided the first algebraic sample complexities for BD that hold for Lebesgue almost all bases or frames. We showed that for BD of a pair of vectors in \bbC^n, with subspace constraints of dimensions $m_1$ and $m_2$, respectively, a sample complexity of $n\geq m_1m_2$ is sufficient. This result is suboptimal, since the number of degrees of freedom is merely $m_1+m_2-1$. We provided analogous results, with similar suboptimality, for BD with sparsity or mixed subspace and sparsity constraints. In Chapter 2, 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 \bbC^n, with sparsity constraints of sparsity levels $s_1$ and $s_2$, a sample complexity of $n > s_1+s_2$ (resp. $n > 2(s_1+s_2)$) 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 Chapter 2, the recovery is not only unique, but also stable against small perturbations in the measurements, under the same sample complexities. In previous work, 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 Chapter 3, under a mild assumption that the signals and models are generic, we bridge the gaps by deriving tight sufficient conditions with optimal or near optimal sample complexities. Recently there has been renewed interest in solutions to BGPC with careful analysis of error bounds. In Chapter 4, 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 (which we show is equivalent to a sparsity-projected gradient descent). 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. A problem related to BGPC is multichannel blind deconvolution (MBD) with a circular convolution model, i.e., the recovery of an unknown signal $f$ and multiple unknown filters $x_i$ from circular convolutions $y_i=x_i \circledast f$ ($i=1,2,\dots,N$). In Chapter 5, we consider the case where the $x_i$'s are sparse, and convolution with $f$ is invertible. Our nonconvex optimization formulation solves for a filter $h$ on the unit sphere that produces sparse outputs $y_i\circledast h$. Under some technical assumptions, we show that all local minima of the objective function correspond to the inverse filter of $f$ up to an inherent sign and shift ambiguity, and all saddle points have strictly negative curvatures. This geometric structure allows successful recovery of $f$ and $x_i$ using a simple manifold gradient descent algorithm with random initialization. Our theoretical findings are complemented by numerical experiments, which demonstrate superior performance of the proposed approach over the previous methods

Identifiability Conditions for Compressive Multichannel Blind Deconvolution

In applications such as multi-receiver radars and ultrasound array systems, the observed signals can often be modeled as a linear convolution of an unknown signal which represents the transmit pulse and sparse filters which describe the sparse target scenario. The problem of identifying the unknown signal and the sparse filters is a sparse multichannel blind deconvolution (MBD) problem and is in general ill-posed. In this paper, we consider the identifiability problem of sparse-MBD and show that, similar to compressive sensing, it is possible to identify the sparse filters from compressive measurements of the output sequences. Specifically, we consider compressible measurements in the Fourier domain and derive identifiability conditions in a deterministic setup. Our main results demonstrate that $L$-sparse filters can be identified from $2L^2$ Fourier measurements from only two coprime channels. We also show that $2L$ measurements per channel are necessary. The sufficient condition sharpens as the number of channels increases asymptotically in the number of channels, it suffices to acquire on the order of $L$ Fourier samples per channel. We also propose a kernel-based sampling scheme that acquires Fourier measurements from a commensurate number of time samples. We discuss the gap between the sufficient and necessary conditions through numerical experiments including comparing practical reconstruction algorithms. The proposed compressive MBD results require fewer measurements and fewer channels for identifiability compared to previous results, which aids in building cost-effective receivers.Comment: 13 pages, 5 figure