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Properties of the Horst Algorithm for the Multivariable Eigenvalue Problem

Properties of the Horst Algorithm for the Multivariable Eigenvalue Problem. Michael Skalak Northwestern University. Outline of Problem. Given such that and a symmetric, positive definite matrix. with. Multivariable Eigenvalue Problem

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Properties of the Horst Algorithm for the Multivariable Eigenvalue Problem

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  1. Properties of the Horst Algorithm for the Multivariable Eigenvalue Problem Michael Skalak Northwestern University

  2. Outline of Problem Given such that and a symmetric, positive definite matrix with

  3. Multivariable Eigenvalue Problem • Find real scalars and a real column vector such that where is the identity matrix of size and is partitioned into blocks with

  4. Example Given the symmetric and positive definite matrix , , the vector is a solution, as

  5. Statistical Application Find the maximum correlation coefficient of random variables, each of size Maximize subject to Hence the solution is the global maximum of for vectors in , where is a ball of radius 1 centered at the origin in dimensions.

  6. Power Method • The power method finds the eigenvector with the largest eigenvalue for the usual single-variate eigenvalue problem.

  7. Horst Algorithm Finds the which maximizes Proven to converge monotonically by Chu and Watterson [SIAM J. Sci. Comput. (14), No. 5, pp. 1089-1106]

  8. Example For that same matrix, consider the Horst algorithm with the starting point First iteration:

  9. Dependence on Initial Conditions Convergence point can depend on initial conditions: Like many other maximization algorithms, the Horst algorithm can converge to a local instead of global max.

  10. Results • For any , can have at least as many convergent points • For any m, there can be at least convergence points, and as few as one. • In at least a nontrivial special case (two convergence points, ) the portion of the region which converges to the global max can not be arbitrarily small

  11. Number of Convergence Points There exist 3 matrices, for all such that there exist convergence points. The block matrix, with meaning a matrix of size with convergence points is symmetric, positive definite, and has convergence points With a little manipulation, this proves that for any size there exist matrices with at least convergence points.

  12. Convergence to Global Max Suppose there is some transformation on the matrix that can arbitrarily move eigenvectors arbitrarily. After the transformation, the matrix is rescaled so that the largest element remains constant.

  13. Case 1 The difference of the values between the local mins and the global max is bounded . Then the derivative must increase without bound. However, since all elements of the matrix are less than a constant, this cannot happen.

  14. Case 2 The values of the local mins approach the global max as the vectors approach. Since one of the local mins is the global min, the function become closer and closer to constant, which cannot happen since the derivative is bounded below in at least one direction.

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