Self-Calibration and Bilinear Inverse Problems via Linear Least Squares

Abstract

Whenever we use devices to take measurements, calibration is indispensable. While the purpose of calibration is to reduce bias and uncertainty in the measurements, it can be quite difficult, expensive, and sometimes even impossible to implement. We study a challenging problem called self-calibration, i.e., the task of designing an algorithm for devices so that the algorithm is able to perform calibration automatically. More precisely, we consider the setup y = A(d)x+ε where only partial information about the sensing matrix A(d) is known and where A(d) linearly depends on d. The goal is to estimate the calibration parameter d (resolve the uncertainty in the sensing process) and the signal/object of interests x simultaneously. For three different models of practical relevance, we show how such a bilinear inverse problem, including blind deconvolution as an important example, can be solved via a simple linear least squares approach. As a consequence, the proposed algorithms are numerically extremely efficient, thus potentially allowing for real-time deployment. We also present a variation of the least squares approach, which leads to a spectral method, where the solution to the bilinear inverse problem can be found by computing the singular vector associated with the smallest singular value of a certain matrix derived from the bilinear system. Explicit theoretical guarantees and stability theory are derived for both techniques; and the number of sampling complexity is nearly optimal (up to a poly-log factor). Applications in imaging sciences and signal processing are discussed and numerical simulations are presented to demonstrate the effectiveness and efficiency of our approach.

Blind Gain and Phase Calibration via Sparse Spectral Methods

“…In the subspace case, power iteration solves BGPC using optimal (up to log factors) numbers of sensors and snapshots. These sample complexities are comparable to the least squares method in [7]. Moreover, we show that power iteration is empirically more robust against noise than least squares.…”

“…The (k, j)-th entry in the measurement y kj has the following expression: y kj = λ k a k· x ·j + w kj . Unique Recovery [6] n > m N ≥ n−1 n−m n > 2s0 N ≥ n−1 n−2s 0 -Least Squares [7] n m N 1 -- Clearly, BGPC is a bilinear inverse problem. The solution (λ, X) suffers from scaling ambiguity, i.e., (λ/σ, σX) generates the same measurements as (λ, X), and therefore cannot be distinguished from it.…”

“…In Table I, we compare the above results with the sample complexities for unique recovery in BGPC [6], and previous guaranteed algorithms for BGPC in the subspace and sparsity case [7], [8]. In the subspace case, power iteration solves BGPC using optimal (up to log factors) numbers of sensors and snapshots.…”

“…However, such methods have not been carefully analyzed until recently. Ling and Strohmer [7] derived an error bound for the least squares solution in the subspace case of BGPC. In this paper, the power iteration method has sample complexities comparable to those of the least squares method [7], and is empirically more robust to noise than the latter.…”

“…There exists a long line of research regarding the solutions for each application. However, theoretical analysis of the problem and error bounds for its solutions have been established only recently [5], [6], [7], [8].…”