Geometric Means in a Novel Vector Space Structure on Symmetric Positive-Definite Matrices

By Vincent Arsigny, Pierre Fillard, Xavier Pennec, Nicholas Ayache. SIAM Journal on Matrix Analysis and Applications, 2007

967 citations recorded by the source corpus.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/bf4c1819546ec255d3997c8f5f9674282370e6eb

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In Applied Mathematics

In Astronomy and Astrophysics

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Citations in the corpus

Scene-Level Geographic Image Classification Based on a Covariance Descriptor Using Supervised Collaborative Kernel Coding (2016, background): “As is illustrated in Figure 4, covd lies in a Riemannian manifold [24], which is not a Euclidean space.”
Recent advances and trends in visual tracking: A review (2011, methodology): “Based on the Covariance Matrix descriptor and the Log-Euclidean Riemannian metric [77], Li et al. [78] presented an online subspace learning algorithm which models the appearance changes by incrementally learning an eigenspace representation for each mode of the target through adaptively updating…”
Beyond Covariance: Feature Representation with Nonlinear Kernel Matrices (2015, background): “This non-Euclidean geometric property has been effectively considered in its distance measures including the generalized-eigenvaluebased metric [22], affine-invariant Riemannian metrics [16], log-Euclidean distance [1], Stein divergence [19], etc.”
Regression on Fixed-Rank Positive Semidefinite Matrices: A Riemannian Approach (2010, methodology): “This geometry is studied in detail in the paper [Arsigny et al., 2007].”
Multiclass Brain–Computer Interface Classification by Riemannian Geometry (2012, methodology): “implementation of eigenvalue decomposition or by using other metrics approximating the Riemannian metric [23].”
A Riemannian Network for SPD Matrix Learning (2016, methodology): “In medical imaging, they are commonly used in diffusion tensor magnetic resonance imaging (Pennec, Fillard, and Ayache 2006; Arsigny et al. 2007; Jayasumana et al. 2013).”
Covariance discriminative learning: A natural and efficient approach to image set classification (2012, background): “+d Sym is the Log-Euclidean distance (LED) [2] that results in classical Euclidean computations in the domain of matrix logarithms as: 1 2 1 2 ( , ) log( ) log( ) = − C C C C where log is the ordinary matrix logarithm operator and Σ be the eigen-decomposition of SPD matrix C , its log is a…”
A dynamic 4D probabilistic atlas of the developing brain (2011, background): “The details of the definition of addition and scaling operators as well as their implementation can be found in Alexa (2002); (Aljabar et al. (2008) and Arsigny et al. (2008).”
Online action recognition using covariance of shape and motion (2014, methodology): “Besides the difference in several theoretical aspects (see [26,27] for details), the dLE and the dAI metrics differ in some practical aspects as well.”
Log-Euclidean Metric Learning on Symmetric Positive Definite Manifold with Application to Image Set Classification (2015, methodology): “Additionally, keeping the symmetry of the learned square matrix will also obtain the corresponding manifold of SPD matrices via the inverse mapping (i.e. the matrix exponential) of the matrix logarithm which is a smooth diffeomorphism (Arsigny et al., 2007).”
Riemannian Approaches in Brain-Computer Interfaces: A Review (2017, positive): “Depending on the choice of scalar product to equip the tangent spaces, one Euclidean and two different Riemannian geometries are possible and subsequently two different Riemannian distances (and one Euclidean) can be defined: • the Euclidean distance2 δe (X A, X B) = ‖X A − X B‖F = 〈X A − X B , X A ”
Matrix Backpropagation for Deep Networks with Structured Layers (2015, methodology): “It represents global high-order statistics of local descriptors inside each region by computing a covariance matrix XX then applying a tangent space mapping [2] using the matrix logarithm, which can be computed using SVD.”
Semantic Segmentation with Second-Order Pooling (2012, methodology): “Second-order feature pooling methods leveraging recent advances in computational differential geometry [8].”
A fast diffeomorphic image registration algorithm (2007, methodology): “For example, with eight time steps, the Euler integration method is equivalent to If the number of time steps is a power of two, then the solution can be determined by a scaling and squaring approach (Moler and Van Loan, 2003; Arsigny et al., 2006b,a).”
Log-Hilbert-Schmidt metric between positive definite operators on Hilbert spaces (2014, background): “From a practical viewpoint, the metric (1) tends to be computationally intensive, which is one of the main motivations for the Log-Euclidean metric of [2], in which the geodesic distance between A and B is given by dlogE(A,B) = || log(A)− log(B)||F .”
Matrix power means and the Karcher mean (2012, positive): “In fact, limt→0( A1+···+An n ) 1 t = exp( logA1+···+logAn n ) and is known as the Log-Euclidean mean [3].”