Generalized Inverse Matrices

By H. Yanai, K. Takeuchi, Yoshio Takane. 2011

1,885 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/c39cf7057efe6d47fd2aea2dd3c9d1772b807841

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In Computational Theory and Mathematics

In Statistical and Nonlinear Physics

Citations in the corpus

Vital nodes identification in complex networks (2016, methodology): “To calculate I i st, we firstly construct the Laplacian matrix of the network as L = D − A, (27) and then compute the Moore-Penrose pseudoinverse [85, 86] of the L, say T = L.”
Generalized Gaussian quadrature rules on arbitrary polygons (2010, background): “Once we have an appropriate initial guess Xk, (4) is used to get a better solution Xk+1 until convergence is attained within the acceptable error range: Xk+1 = Xk − J(Xk)F(Xk), (4) where J(Xk) is the pseudo-inverse [21] of the Jacobian J : R n′ → R defined by (5) for m basis functions and n degrees ”
CatMAP: A Software Package for Descriptor-Based Microkinetic Mapping of Catalytic Trends (2015, neutral): “C 1⁄4 RDþ ð9Þ where D is the Moore–Penrose pseudo-inverse of D [47, 48].”
Characterizations and Representations of Core and Dual Core Inverses (2016, background): “An element a ∈ R is calledMoore–Penrose invertible [7] if there exists x ∈ R satisfying the following equations (i) axa = a, (ii) xax = x, (iii) (ax)∗ = ax, (iv) (xa)∗ = xa.”
Ten Pairs to Tag – Multilingual POS Tagging via Coarse Mapping between Embeddings (2016, methodology): “In other words, we find P that minimizes ||ΣtP − Σs||(2) via the Moore–Penrose pseudoinverse (Moore, 1920; Penrose, 1955).”
DINGO: Distributed Newton-Type Method for Gradient-Norm Optimization (2019, methodology): “The Moore–Penrose inverse [12] of A is denoted by A†.”
ppcor: An R Package for a Fast Calculation to Semi-partial Correlation Coefficients (2015, methodology): “Note that when the determinant of variance-covariance matrix is numerically zero, the R package ppcor computes its pseudo-inverse using the Moore-Penrose generalized matrix inverse (Penrose, 1995).”
Analyzing Transformers in Embedding Space (2022, background): “…the embedding space: given two vectors x, y ∈ R d , their inner product x T y can be considered in the embedding space by multiplying with E and then by one of its right inverses (e.g., its pseudo-inverse E + [Moore, 1920; Bjerhammar, 1951; Penrose, 1955]): x T y = x T EE + y = ( x T E )( E + y ) .”
Morphing shell structures: A generalised modelling approach (2015, methodology): “the Moore-Penrose pseudo-inverse [34, 35] of the extended matrix, which is rectangular.”
Online and adaptive pseudoinverse solutions for ELM weights (2014, background): “The first incremental solutions to the pseudoinverse of a matrix started appearing not long after Penrose's rediscovery of the pseudoinverse concept [3], and were probably a response to the paucity of computational power available at that time.”
On the constraints violation in forward dynamics of multibody systems (2017, methodology): “However, the concept of the Moore-Penrose inverse matrix, D, can be employed as [133-135]”
Real-Time Profiling of Fine-Grained Air Quality Index Distribution Using UAV Sensing (2017, background): “Note that J is both row-column full rank matrix, which has a corresponding generalized inverse matrix [37].”
Lump solutions to nonlinear partial differential equations via Hirota bilinear forms (2016, methodology): “2) which uniquely defines A+ for any given matrix A [24].”
Causality-driven slow-down and speed-up of diffusion in non-Markovian temporal networks (2013, background): “Some of these works show that compared to systems where, similar to static networks, most or all links are available concurrently, dynamical processes like epidemic spreading or diffusion are slowed down by the continuously switching topologies of temporal networks [1, 4, 5].”
Open-Ended Learning Leads to Generally Capable Agents (2021, background): “For a given w we define ρ(w) as a distribution of resistance distances (Klein and Randić, 1993) between every pair of vertices in Gw, where a resistance distance between vi and v j is given by Γii + Γj j − Γi j − Γji for Γ = (L + 1 w·h1w·h×w·h) †, L being the Laplacian matrix of Gw and † being the M”
Randomized Iterative Methods for Linear Systems: Momentum, Inexactness and Gossip (2019, background): “Using the properties of Moore-Penrose pseudoinverse, this implies that C†i C †.”
Comparing Statistical Methods for Constructing Large Scale Gene Networks (2012, methodology): “GeneNet uses Moore-Penrose pseudoinverse [33] and bootstrap methods to obtain a shrunk estimate of the concentration matrix.”
New characterizations for core inverses in rings with involution (2015, background): “The problem of the Moore-Penrose inverse of the sum of two MoorePenrose invertible elements in complex matrix ring was first considered by Penrose in [18].”