Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1502.07523

Cramer-Rao Bound for Sparse Signals Fitting the Low-Rank Model with Small Number of Parameters

Mahdi Shaghaghi·Sergiy A. Vorobyov
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we consider signals with a low-rank covariance matrix which reside in a low-dimensional subspace and can be written in terms of a finite (small) number of parameters. Although such signals do not necessarily have a sparse representation in a finite basis, they possess a sparse structure which makes it possible to recover the signal from compressed measurements. We study the statistical performance bound for parameter estimation in the low-rank signal model from compressed measurements. Specifically, we derive the Cramer-Rao bound (CRB) for a generic low-rank model and we show that the number of compressed samples needs to be larger than the number of sources for the existence of an unbiased estimator with finite estimation variance. We further consider the applications to direction-of-arrival (DOA) and spectral estimation which fit into the low-rank signal model. We also investigate the effect of compression on the CRB by considering numerical examples of the DOA estimation scenario, and show how the CRB increases by increasing the compression or equivalently reducing the number of compressed samples.

Explore related subjects

math.STcs.ITmath.ITstat.TH

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Mahdi Shaghaghi, Sergiy A. Vorobyov. 2015-02-26. Cramer-Rao Bound for Sparse Signals Fitting the Low-Rank Model with Small Number of Parameters. https://doi.org/10.1109/lsp.2015.2409896

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Common Drivers in Sparsely Interacting Hawkes Processes

We study a multivariate Hawkes process as a model for time-continuous relational event networks. The model does not assume the network to be known, it includes covariates, and it allows for both common drivers, parameters common to all the actors in the network, and also local parameters specific for each actor. We derive rates of convergence for all of the model parameters when both the number of actors and the time horizon tends to infinity. To prevent an exploding network, sparseness is assumed. We also discuss numerical aspects.

math.ST↗
arXiv2026

Basic Inequalities for First-Order Optimization with Applications to Statistical Risk Analysis

In this work, we introduce $\textit{basic inequalities}$ for first-order iterative optimization algorithms, forming a simple yet versatile framework which connects implicit and explicit regularization. Building on related comparison inequalities for optimization iterates that already exist in the literature, we extend and unify these arguments to produce a general framework, which can be used as a tool for statistical analysis. In more detail, let $f$ denote the objective function to be optimized. Given a first-order iterative algorithm initialized at $θ_0$, with current iterate $θ_T$, the basic inequality upper bounds $f(θ_T) - f(z)$ for any reference point $z$ in terms of the accumulated step sizes, and the distances between $θ_0$, $θ_T$, and $z$. These distances are measured in a geometry inherent to the optimization algorithm, which then translates into a notion of regularization being applied across the path of iterates. In addition to refining existing results on gradient descent, we provide new results for mirror descent and other first-order methods. We then show how to use these basic inequalities to derive elementary yet useful bounds on the prediction risk of early-stopped gradient descent and exponentiated gradient descent iterates in generalized linear models. We also supplement these findings with numerical experiments.

math.ST↗
arXiv2026

Noise-resilient penalty operators based on statistical differentiation schemes

Classical approaches to penalized smoothing often rely on basis or kernel expansions, which constrain the estimator to a fixed span and may be restrictive for discretely observed data. We instead regularize a single noisy trajectory directly on its observation grid, using difference operators that remain genuine finite-difference approximations to derivatives while being statistically normalized and mutually decorrelated under a reference noise law. We extend this white-noise construction to a parametric family of covariance-adapted reference geometries, with the parameter estimated by generalized method of moments from an independent calibration sample. A first-order plug-in expansion shows that, within this moment family, efficient calibration minimizes the leading-order discrepancy between the plug-in and oracle covariance-adapted smoothers. Numerical experiments confirm the predicted rate at which this plug-in discrepancy vanishes and compare the resulting reconstruction against conventional discrete, basis, and kernel smoothers.

math.ST↗