Concentration of the Frobenius Norm of Generalized Matrix Inverses

Gribonval, Rémi

doi:10.1137/17m1145409

Cited by 4 publications

(4 citation statements)

References 27 publications

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“…This allows us to solve the class of problems in (2) that involve structured/group sparsity, namely those involving constraints on (projections onto) the mixed ∞,1 norm. The proximal mapping of the mixed 1,∞ norm is also applicable to the computation of minimax sparse pseudoinverses to underdetermined systems of linear equations [10], [11].…”

Section: Motivationmentioning

confidence: 99%

The fastest L1,oo prox in the west

Béjar

Vidal

2021

IEEE Trans. Pattern Anal. Mach. Intell.

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Proximal operators are of particular interest in optimization problems dealing with non-smooth objectives because in many practical cases they lead to optimization algorithms whose updates can be computed in closed form or very efficiently. A well-known example is the proximal operator of the vector 1 norm, which is given by the soft-thresholding operator. In this paper we study the proximal operator of the mixed 1,∞ matrix norm and show that it can be computed in closed form by applying the well-known soft-thresholding operator to each column of the matrix. However, unlike the vector 1 norm case where the threshold is constant, in the mixed 1,∞ norm case each column of the matrix might require a different threshold and all thresholds depend on the given matrix. We propose a general iterative algorithm for computing these thresholds, as well as two efficient implementations that further exploit easy to compute lower bounds for the mixed norm of the optimal solution. Experiments on large-scale synthetic and real data indicate that the proposed methods can be orders of magnitude faster than state-of-the-art methods.

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Section: Motivationmentioning

confidence: 99%

The fastest L1,oo prox in the west

Béjar

Vidal

2021

IEEE Trans. Pattern Anal. Mach. Intell.

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“…Thus, the matrix inversion problem can be expressed as finding the best approximate solution W * to minimise

C(\mathbf{W})={{\Vert}\mathbf{I}-\mathbf{W}\mathbf{X}{{\Vert}}_{F}}^{2},

where

{{\Vert}\cdot {\Vert}}_{F}

denotes the Frobenius norm. It is known that X + is the unique best approximate solution to the above equation, that is, W * = X + [34]. Obviously, with this transformation, we can relate the generalised inverse computation to FNNs.…”

Section: Methodsmentioning

confidence: 99%

“…where k⋅k F denotes the Frobenius norm. It is known that X + is the unique best approximate solution to the above equation, that is, W* = X + [34]. Obviously, with this transformation, we can relate the generalised inverse computation to FNNs.…”

Section: Constructing the Single Layer With Linear Neurons For Matrix...mentioning

confidence: 99%

An efficient second‐order neural network model for computing the Moore–Penrose inverse of matrices

2022

IET Signal Processing

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The computation of the Moore-Penrose inverse is widely encountered in science and engineering. Due to the parallel-processing nature and strong-learning ability, the neural network has become a promising approach to solving the Moore-Penrose inverse recently. However, almost all the existing neural networks for matrix inversion are based on the gradient-descent (GD) method, whose main drawbacks are slow convergence and sensitivity to learning parameters. Moreover, there is no unified neural network to compute the Moore-Penrose inverse for both the full-rank matrix and rank-deficient matrix. In this paper, an efficient second-order neural network model with the improved Newton's method is proposed to obtain the accurate Moore-Penrose inverse of an arbitrary matrix by one epoch without any learning parameter. Compared with the GD-based neural networks for Moore-Penrose inverse computation, the proposed model converges faster and has lower complexity. Furthermore, through in-depth derivation, the neural network for computing the Moore-Penrose inverse is well interpretable. Numerical studies and application to the random matrix inversion in multiple-input multiple-output detection are provided to validate the efficiency of the proposed model for solving the Moore-Penrose inverse.

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“…It is rare in practice. However, some sources, e.g., [21][22][23][24], discuss this type from a mathematical point of view, because the two types of approximation (least square and minimum-norm) are applicable here, but with constraints.…”

Section: Connection With the Electrical Calculationmentioning

confidence: 99%

Role of Voltage Control Devices in Low Voltage State Estimation Process

Táczi,

Vokony,

Barancsuk

et al. 2023

International Transactions on Electrical Energy Systems

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Megatrends, such as the proliferation of distributed generation, electrification, and the appearance of aggregator companies, put the low voltage power grids under intense pressure. Since the network infrastructure developments cannot keep up with the trends, distribution system operators turned to alternative solutions. Smart grid assets, such as on-load tap-changing distribution transformers or serial low voltage regulators, are promising solutions. However, the energy transition cannot be handled with the network expansive on the distribution level. Control centers are predicted to expand to this voltage level in the near future, and distribution system state estimation could be an enabler of all functionalities. On the low voltage level, data scarcity is a great challenge in observability; therefore, research must focus on the creation of pseudomeasurements and integration of available data sources. This paper examines the inclusion of smart assets from the conceptual point to the application on two sites, based on data from operational environments, both with a pseudomeasurement and an integrated metering point approach. The results showed that integrating smart assets could considerably mitigate voltage fluctuations, and reduce estimation errors by two magnitudes on the low voltage network.

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Concentration of the Frobenius Norm of Generalized Matrix Inverses

Cited by 4 publications

References 27 publications

The fastest L1,oo prox in the west

The fastest L1,oo prox in the west

An efficient second‐order neural network model for computing the Moore–Penrose inverse of matrices

Role of Voltage Control Devices in Low Voltage State Estimation Process

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