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Joint Approximate Diagonalization of Several Matrices by an Optimization Algorithm on a Matrix Manifold

Rank revealing CP decomposition is a tensor factorization which is heavily exploited in applications. In the symmetrical case, by projecting a tensor along multiple random vectors the tensor factorization reduces to the problem of joint diagonalization of several symmetric matrices. In the presence of noise it is not possible to obtain the exact joint diagonalization, and the problem is transformed to the optimization problem of finding nearly diagonal form. Existing algorithms for joint diagonalization use standard optimization algorithms and if necessary impose some additional constraint on the transformation matrix in order to secure its nonsingularity. Since the joint diagonalization algorithm represents a core of the tensor factorization, our goal was to enhance existing algorithms by combining diagonalization of the random linear combination of the matrices with the optimization algorithm (like Newton method or Conjugate Gradient method) on an appropriate matrix manifold, thus avoiding additional constraints. We are going to provide derivation and analysis of the algorithm in both the differential geometry and numerical analysis framework, and produce algorithm implementation.

optimization ; matrix manifold ; joint approximate diagonalization

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