2,758 research outputs found
Phase Retrieval of Quaternion Signal via Wirtinger Flow
The main aim of this paper is to study quaternion phase retrieval (QPR),
i.e., the recovery of quaternion signal from the magnitude of quaternion linear
measurements. We show that all -dimensional quaternion signals can be
reconstructed up to a global right quaternion phase factor from
phaseless measurements. We also develop the scalable algorithm quaternion
Wirtinger flow (QWF) for solving QPR, and establish its linear convergence
guarantee. Compared with the analysis of complex Wirtinger flow, a series of
different treatments are employed to overcome the difficulties of the
non-commutativity of quaternion multiplication. Moreover, we develop a variant
of QWF that can effectively utilize a pure quaternion priori (e.g., for color
images) by incorporating a quaternion phase factor estimate into QWF
iterations. The estimate can be computed efficiently as it amounts to finding a
singular vector of a real matrix. Motivated by the variants of
Wirtinger flow in prior work, we further propose quaternion truncated Wirtinger
flow (QTWF), quaternion truncated amplitude flow (QTAF) and their pure
quaternion versions. Experimental results on synthetic data and color images
are presented to validate our theoretical results. In particular, for pure
quaternion signal recovery, our quaternion method often succeeds with
measurements notably fewer than real methods based on monochromatic model or
concatenation model.Comment: 21 pages (paper+supplemental), 6 figure
Uniform Exact Reconstruction of Sparse Signals and Low-rank Matrices from Phase-Only Measurements
In phase-only compressive sensing (PO-CS), our goal is to recover
low-complexity signals (e.g., sparse signals, low-rank matrices) from the phase
of complex linear measurements. While perfect recovery of signal direction in
PO-CS was observed quite early, the exact reconstruction guarantee for a fixed,
real signal was recently done by Jacques and Feuillen [IEEE Trans. Inf. Theory,
67 (2021), pp. 4150-4161]. However, two questions remain open: the uniform
recovery guarantee and exact recovery of complex signal. In this paper, we
almost completely address these two open questions. We prove that, all complex
sparse signals or low-rank matrices can be uniformly, exactly recovered from a
near optimal number of complex Gaussian measurement phases. By recasting PO-CS
as a linear compressive sensing problem, the exact recovery follows from
restricted isometry property (RIP). Our approach to uniform recovery guarantee
is based on covering arguments that involve a delicate control of the (original
linear) measurements with overly small magnitude. To work with complex signal,
a different sign-product embedding property and a careful rescaling of the
sensing matrix are employed. In addition, we show an extension that the uniform
recovery is stable under moderate bounded noise. We also propose to add
Gaussian dither before capturing the phases to achieve full reconstruction with
norm information. Experimental results are reported to corroborate and
demonstrate our theoretical results.Comment: 39 pages, 1 figur
Quantized Low-Rank Multivariate Regression with Random Dithering
Low-rank multivariate regression (LRMR) is an important statistical learning
model that combines highly correlated tasks as a multiresponse regression
problem with low-rank priori on the coefficient matrix. In this paper, we study
quantized LRMR, a practical setting where the responses and/or the covariates
are discretized to finite precision. We focus on the estimation of the
underlying coefficient matrix. To make consistent estimator that could achieve
arbitrarily small error possible, we employ uniform quantization with random
dithering, i.e., we add appropriate random noise to the data before
quantization. Specifically, uniform dither and triangular dither are used for
responses and covariates, respectively. Based on the quantized data, we propose
the constrained Lasso and regularized Lasso estimators, and derive the
non-asymptotic error bounds. With the aid of dithering, the estimators achieve
minimax optimal rate, while quantization only slightly worsens the
multiplicative factor in the error rate. Moreover, we extend our results to a
low-rank regression model with matrix responses. We corroborate and demonstrate
our theoretical results via simulations on synthetic data or image restoration.Comment: 16 pages (Submitted
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