A geometric mean algorithm of symmetric positive definite matrices

• Autores: Juan Olmos, Fabio Martinez, Juan Galvis
• Localización: Revista Colombiana de Matemáticas, ISSN-e 0034-7426, Vol. 57, Nº. 2, 2023, págs. 231-255
• Idioma: varios idiomas
• Títulos paralelos:
  • Cálculo de una media geométrica en el cono de las matrices simétricas definidas positivas
• Enlaces
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• Resumen
  • español

    Este trabajo presenta un algoritmo de media geométrica para matrices positivas definidas utilizando problemas de valores propios generalizados y la factorización de Cholesky. La media geométrica de un conjunto finito de matrices positivas definidas minimiza la suma de los cuadrados de las distancias al conjunto de matrices, donde la distancia es una métrica de Riemann invariante afin en la variedad de matrices definidas positivas simétricas SN++. Para calcular aproximaciones numéricas de la media geométrica se proponen varios algoritmos. Algunos de estos algoritmos requieren el cálculo de varias diagonalizaciones en cada paso. Mostramos que al reescribir las iteraciones de estos pasos en términos de un problema de valores propios generalizados, es posible omitir algunas de las diagonalizaciones a costa de introducir problemas de valores propios que pueden resolverse utilizando factorizaciones de Cholesky. Comparamos numéricamente el rendimiento de los métodos clásicos y los algoritmos modificados que utilizan problemas de valores propios generalizados. El método resultante se aplica al análisis de vídeo utilizando la media de matrices de covarianza como un descriptor compacto para la clasificación de acciones. El descriptor medio propuesto con solo 105 valores escalares logró una precisión promedio del 75% en un conjunto de datos de video.

  • Multiple

    This work introduces a geometric mean algorithm for positive definite matrices using generalized eigenvalue problems and Cholesky factorization. The geometric mean of a finite set of positive definite matrices minimizes the sum of square distances to all the matrices where the distance is an affine-invariant Riemannian metric in the manifold of the symmetric positive definite matrices SN++. In order to compute numerical approximations of the geometric mean several algorithms have been proposed. Some of these algorithms require the computation of several diagonalizations in each iteration. We show that by rewriting the iterations in terms of generalized eigenvalue problems, it is possible to omit some of the diagonalizations at the cost of introducing much less generalized eigenvalue problems that can be solved using Cholesky factorizations. We numerically compare the performance of classical methods and the modified algorithms that use generalized eigenvalue problems. The resulting method is applied to video analysis using the mean of covariance matrices as a compact descriptor for actions classification. The proposed mean descriptor with just 105 scalar values achieved an average accuracy of 75% over a publicaction video dataset.

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