scholarly journals Riemannian Metric and Geometric Mean for Positive Semidefinite Matrices of Fixed Rank

2010 ◽  
Vol 31 (3) ◽  
pp. 1055-1070 ◽  
Author(s):  
Silvère Bonnabel ◽  
Rodolphe Sepulchre
Keyword(s):  
Geometric Mean ◽  
Positive Semidefinite ◽  
Riemannian Metric ◽  
Positive Semidefinite Matrices ◽  
Fixed Rank

scholarly journals Fall Detection of Elderly People Using the Manifold of Positive Semidefinite Matrices

2021 ◽  
Vol 7 (7) ◽  
pp. 109
Author(s):  
Abdessamad Youssfi Alaoui ◽  
Youness Tabii ◽  
Rachid Oulad Haj Thami ◽  
Mohamed Daoudi ◽  
Stefano Berretti ◽  
...  
Keyword(s):  
Computer Vision ◽  
Elderly People ◽  
Similarity Measure ◽  
Fall Detection ◽  
Positive Semidefinite ◽  
The Body ◽  
Positive Semidefinite Matrices ◽  
Build Time ◽  
Fixed Rank ◽  
Context Models

Falls are one of the most critical health care risks for elderly people, being, in some adverse circumstances, an indirect cause of death. Furthermore, demographic forecasts for the future show a growing elderly population worldwide. In this context, models for automatic fall detection and prediction are of paramount relevance, especially AI applications that use ambient, sensors or computer vision. In this paper, we present an approach for fall detection using computer vision techniques. Video sequences of a person in a closed environment are used as inputs to our algorithm. In our approach, we first apply the V2V-PoseNet model to detect 2D body skeleton in every frame. Specifically, our approach involves four steps: (1) the body skeleton is detected by V2V-PoseNet in each frame; (2) joints of skeleton are first mapped into the Riemannian manifold of positive semidefinite matrices of fixed-rank 2 to build time-parameterized trajectories; (3) a temporal warping is performed on the trajectories, providing a (dis-)similarity measure between them; (4) finally, a pairwise proximity function SVM is used to classify them into fall or non-fall, incorporating the (dis-)similarity measure into the kernel function. We evaluated our approach on two publicly available datasets URFD and Charfi. The results of the proposed approach are competitive with respect to state-of-the-art methods, while only involving 2D body skeletons.

scholarly journals A Geometric Mean of Parameterized Arithmetic and Harmonic Means of Convex Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Sangho Kum ◽  
Yongdo Lim
Keyword(s):  
Convex Functions ◽  
Convex Analysis ◽  
Geometric Mean ◽  
Positive Semidefinite ◽  
Dual Operator ◽  
Positive Semidefinite Matrices ◽  
Essential Properties ◽  
Harmonic Means ◽  
Further Development ◽  
Interesting Generalization

The notion of the geometric mean of two positive reals is extended by Ando (1978) to the case of positive semidefinite matricesAandB. Moreover, an interesting generalization of the geometric meanA # BofAandBto convex functions was introduced by Atteia and Raïssouli (2001) with a different viewpoint of convex analysis. The present work aims at providing a further development of the geometric mean of convex functions due to Atteia and Raïssouli (2001). A new algorithmic self-dual operator for convex functions named “the geometric mean of parameterized arithmetic and harmonic means of convex functions” is proposed, and its essential properties are investigated.

scholarly journals A trace bound for integer-diagonal positive semidefinite matrices

2020 ◽  
Vol 8 (1) ◽  
pp. 14-16
Author(s):  
Lon Mitchell
Keyword(s):  
Positive Semidefinite ◽  
Positive Semidefinite Matrix ◽  
Positive Semidefinite Matrices ◽  
Semidefinite Matrix

AbstractWe prove that an n-by-n complex positive semidefinite matrix of rank r whose graph is connected, whose diagonal entries are integers, and whose non-zero off-diagonal entries have modulus at least one, has trace at least n + r − 1.

scholarly journals Fischer Type Log-Majorization of Singular Values on Partitioned Positive Semidefinite Matrices

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Benju Wang ◽  
Yun Zhang
Keyword(s):  
Singular Values ◽  
Positive Semidefinite ◽  
Positive Semidefinite Matrices ◽  
Fischer’S Inequality ◽  
New Inequalities

In this paper, we establish a Fischer type log-majorization of singular values on partitioned positive semidefinite matrices, which generalizes the classical Fischer's inequality. Meanwhile, some related and new inequalities are also obtained.

scholarly journals A Characterization of Polynomial Density on Curves via Matrix Algebra

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1231
Author(s):  
Carmen Escribano ◽  
Raquel Gonzalo ◽  
Emilio Torrano
Keyword(s):  
Matrix Algebra ◽  
Optimization Problems ◽  
Positive Semidefinite ◽  
Point Of View ◽  
Alternative Proof ◽  
General Context ◽  
Positive Semidefinite Matrices ◽  
Jordan Curves ◽  
Point Evaluations

In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces L 2 ( μ ) , with μ a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix with the measure μ . To do it, in the more general context of Hermitian positive semidefinite matrices, we introduce two indexes, γ ( M ) and λ ( M ) , associated with different optimization problems concerning theses matrices. Our main result is a characterization of density of polynomials in the case of measures supported on Jordan curves with non-empty interior using the index γ and other specific index related to it. Moreover, we provide a new point of view of bounded point evaluations associated with a measure in terms of the index γ that will allow us to give an alternative proof of Thomson’s theorem, by using these matrix indexes. We point out that our techniques are based in matrix algebra tools in the framework of Hermitian positive definite matrices and in the computation of certain indexes related to some optimization problems for infinite matrices.

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