[Paper Review] A Unified Framework for Identifiability Analysis in Bilinear Inverse Problems with Applications to Subspace and Sparsity Models

Yanjun Li, Kiryung Lee+1 more|arXiv (Cornell University)|Jan 25, 2015

Image and Signal Denoising Methods23 references29 citations

TL;DR

This paper presents a unified framework for analyzing identifiability in bilinear inverse problems (BIPs), focusing on uniqueness up to transformation groups. It derives necessary and sufficient conditions for identifiability under subspace and sparsity constraints, with applications to blind gain and phase calibration (BGPC), providing the first algebraic sample complexity bounds and characterizing equivalence classes via transformation groups.

ABSTRACT

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.

Motivation & Objective

• To address the fundamental open problem of uniqueness in bilinear inverse problems (BIPs), which are typically ill-posed due to scaling and other ambiguities.
• To formalize identifiability up to a group of transformations, enabling the definition of equivalence classes for solutions.
• To derive necessary and sufficient conditions for identifiability in BIPs under subspace and sparsity constraints.
• To apply the framework to blind gain and phase calibration (BGPC), a prototypical BIP arising in inverse rendering, sensor array processing, and multichannel blind deconvolution.
• To establish tight lower bounds on sample complexity for BGPC under joint sparsity and sparsity models, demonstrating their tightness via numerical experiments.

Proposed method

• Introduce a general notion of identifiability up to a transformation group, where solutions are unique only up to equivalence under group actions.
• Define the transformation group associated with a BIP and derive necessary and sufficient conditions for identifiability in terms of the group's action on the solution space.
• Apply the framework to BGPC, where the measurement model is $ Y = \mathrm{diag}(\lambda) \Phi $, with $ \lambda $ the unknown gain and phase vector and $ \Phi $ the signal matrix.
• For joint sparsity and sparsity models, develop a procedure to compute the relevant transformation groups and equivalence classes based on the support structure of the signals.
• Derive necessary conditions in the form of tight lower bounds on the number of measurements (sample complexity), using combinatorial and algebraic arguments on index set shifts.
• Use graph-theoretic and periodicity-based analysis to characterize when identifiability fails, particularly when the joint support of signals is periodic.

Experimental results

Research questions

• RQ1 Under what conditions is a bilinear inverse problem uniquely identifiable up to a group of transformations?
• RQ2 How can the transformation group associated with a BIP be systematically computed for sparsity and subspace constraints?
• RQ3 What are the tightest possible lower bounds on sample complexity for BGPC under joint sparsity or sparsity models?
• RQ4 When does identifiability fail due to periodicity in the joint support of signals, and how can this be characterized algebraically?
• RQ5 Can the proposed framework be applied to real-world BIPs such as inverse rendering and multichannel blind deconvolution with provable uniqueness guarantees?

Key findings

• The paper establishes necessary and sufficient conditions for identifiability of BIPs up to a transformation group, providing a general theoretical foundation for analyzing uniqueness in bilinear systems.
• For BGPC with joint sparsity or sparsity constraints, the framework yields the first algebraic sample complexity bounds, which are shown to be tight through numerical validation.
• Identifiability fails when the joint support of the signal columns is periodic, and this failure occurs on a set of nonzero measure, indicating that periodicity is a significant obstruction.
• The transformation group for jointly sparse signals depends on the choice of basis, and a procedure is developed to compute this group for different bases.
• When the joint support is contiguous and non-periodic, the shifted index sets cover at least $ n-1 $ indices, ensuring identifiability under appropriate conditions.
• The framework reveals that pathological non-identifiable cases (e.g., rank-deficient or periodic supports) reside in a set of measure zero, suggesting that generic signals are identifiable under mild conditions.