Maximum Likelihood Covariance Estimation with a Condition Number Constraint

Won, Joong Ho; Kim, Seung-Jean

doi:10.1109/acssc.2006.354997

Cited by 28 publications

(31 citation statements)

References 5 publications

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“…4 reveals that FNE and SNE are basically equivalent and outperform in terms of average SINR all the counterparts, included those specific for compound Gaussian disturbance 10 . Precisely, in the matched condition, as already observed in Fig.…”

Section: A Spatial Processingmentioning

confidence: 99%

“…Without surprise, LRE-6 outperforms LRE-7 reflecting the presence of a mismatch loss. Finally, in the mismatched scenario, FNE and SNE also grant better performance than FML and CML with 10 As to the LRE, the clutter covariance matrix rank is evaluated as the number of the eigenvalues greater than tr (M s)/10 5 ≥ 10 −4 . Hence, the estimators exploiting the true rank, i.e., 6 (LRE-6) and rank 7 (LRE-7) are displayed.…”

Section: A Spatial Processingmentioning

confidence: 99%

“…Whereas, the latter context mainly considers clutter power variations within the sample support. Thus, assuming homogeneity, in [9] the Maximum Likelihood (ML) covariance matrix estimator is derived modeling the disturbance as the sum of a coloured interference plus white disturbance; in [10], the ML estimation of an unstructured covariance matrix with a condition number upper bound requirement is considered; in [11], using the same covariance structure as in [9], the ML estimator is derived when a constraint on the condition number is imposed too; in [12], a rank-constrained ML estimator is developed; furthermore, relying on a Mean Square Error (MSE) design criterion, in [13] and [14], some shrinkage estimators are proposed. With reference to heterogeneous scenarios, compound Gaussian statistical models (such as Kdistributed or Gamma amplitudes) are usually exploited to account for clutter returns spikiness.…”

Section: Introductionmentioning

confidence: 99%

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A Geometric Approach to Covariance Matrix Estimation and its Applications to Radar Problems

Aubry

Maio

Pallotta

2018

IEEE Trans. Signal Process.

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Abstract-A new class of disturbance covariance matrix estimators for radar signal processing applications is introduced following a geometric paradigm. Each estimator is associated with a given unitary invariant norm and performs the sample covariance matrix projection into a specific set of structured covariance matrices. Regardless of the considered norm, an efficient solution technique to handle the resulting constrained optimization problem is developed. Specifically, it is shown that the new family of distribution-free estimators shares a shrinkagetype form; besides, the eigenvalues estimate just requires the solution of a one-dimensional convex problem whose objective function depends on the considered unitary norm. For the two most common norm instances, i.e., Frobenius and spectral, very efficient algorithms are developed to solve the aforementioned one-dimensional optimization leading to almost closed form covariance estimates. At the analysis stage, the performance of the new estimators is assessed in terms of achievable Signal to Interference plus Noise Ratio (SINR) both for a spatial and a Doppler processing assuming different data statistical characterizations. The results show that interesting SINR improvements with respect to some counterparts available in the open literature can be achieved especially in training starved regimes.

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Section: A Spatial Processingmentioning

confidence: 99%

Section: A Spatial Processingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Geometric Approach to Covariance Matrix Estimation and its Applications to Radar Problems

Aubry

Maio

Pallotta

2018

IEEE Trans. Signal Process.

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“…Methods which are more convenient for these applications are presented in Ledoit and Wolf (2004a,b); Huang et al (2006); Warton (2008); Won et al (2009) and Deng and Tsui (2013) along with our approach presented in this article. Huang et al (2006) were among the first to parallel the penalized log-likelihood estimation with l 2 (and l 1 ) regularization with the common ridge (Lasso) regression.…”

Section: Introductionmentioning

confidence: 99%

“…The penalized log-likelihood functions and other constrained optimization techniques are used to gain better estimates for the matrices. Examples of these include the graphical Lasso algorithm and its extensions (Friedman et al (2008); Fan et al (2009);Witten et al (2011); Bien and Tibshirani (2011)) and other regularization driven approaches for the log-likelihood function (see e.g., Won et al (2009) ;Yuan and Wang (2013) ;Deng and Tsui (2013)). Examples of optimization-based approaches which do not deal with the likelihood-based inference are presented in Ledoit and Wolf (2004a,b) and Cai et al (2011).…”

Section: Introductionmentioning

confidence: 99%

Precision Matrix Estimation With ROPE

Kuismin

Kemppainen

Sillanpää

2017

Journal of Computational and Graphical Statistics

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It is known that the accuracy of the maximum likelihood based covariance and precision matrix estimates can be improved by penalized log-likelihood estimation. In this article we propose a Ridge-type Operator for the Precision matrix Estimation, ROPE for short, to maximize a penalized likelihood function where the Frobenius norm is used as the penalty function. We show that there is an explicit closed form representation of a shrinkage estimator for the precision matrix when using a penalized log-likelihood, which is analogous to ridge regression in a regression context.

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A solution to the ill‐conditioning of gradient‐enhanced covariance matrices for Gaussian processes

Marchildon,

Zingg

2024

Numerical Meth Engineering

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Gaussian processes provide probabilistic surrogates for various applications including classification, uncertainty quantification, and optimization. Using a gradient‐enhanced covariance matrix can be beneficial since it provides a more accurate surrogate relative to its gradient‐free counterpart. An acute problem for Gaussian processes, particularly those that use gradients, is the ill‐conditioning of their covariance matrices. Several methods have been developed to address this problem for gradient‐enhanced Gaussian processes but they have various drawbacks such as limiting the data that can be used, imposing a minimum distance between evaluation points in the parameter space, or constraining the hyperparameters. In this paper a diagonal preconditioner is applied to the covariance matrix along with a modest nugget to ensure that the condition number of the covariance matrix is bounded, while avoiding the drawbacks listed above. The method can be applied with any twice‐differentiable kernel and when there are noisy function and gradient evaluations. Optimization results for a gradient‐enhanced Bayesian optimizer with the Gaussian kernel are compared with the use of the preconditioning method, a baseline method that constrains the hyperparameters, and a rescaling method that increases the distance between evaluation points. The Bayesian optimizer with the preconditioning method converges the optimality, that is, the norm of the gradient, an additional 5 to 9 orders of magnitude relative to when the baseline method is used and it does so in fewer iterations than with the rescaling method. The preconditioning method is available in the open source Python library GpGradPy, which can be found at https://github.com/marchildon/gpgradpy/tree/paper_precon.

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Maximum Likelihood Covariance Estimation with a Condition Number Constraint

Cited by 28 publications

References 5 publications

A Geometric Approach to Covariance Matrix Estimation and its Applications to Radar Problems

A Geometric Approach to Covariance Matrix Estimation and its Applications to Radar Problems

Precision Matrix Estimation With ROPE

A solution to the ill‐conditioning of gradient‐enhanced covariance matrices for Gaussian processes

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