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- W2050113612 abstract "Consider the matrix problem Ax = y + ε = y ̃ in the case where A is known precisely, the problem is ill conditioned, and ε is a random noise vector. Compute regularized “ridge” estimates, x ̃ λ = (A ∗ A + λI) -1 A ∗ y ̃ ,where ∗ denotes matrix transpose. Of great concern is the determination of the value of λ for which x̃ λ “best” approximates x 0 = A + y . Let Q = ‖ x ̃ λ − x 0 ‖ 2 ,and define λ 0 to be the value of λ for which Q is a minimum. We look for λ 0 among solutions of dQ / d λ = 0. Though Q is not computable (since ε is unknown), we can use this approach to study the behavior of λ 0 as a function of y and ε. Theorems involving “noise to signal ratios” determine when λ 0 exists and define the cases λ 0 > 0 and λ 0 = ∞. Estimates for λ 0 and the minimum square error Q 0 = Q (λ 0 ) are derived." @default.
- W2050113612 created "2016-06-24" @default.
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- W2050113612 date "1982-12-01" @default.
- W2050113612 modified "2023-09-28" @default.
- W2050113612 title "A theory for optimal regularization in the finite dimensional case" @default.
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- W2050113612 doi "https://doi.org/10.1016/0024-3795(82)90121-5" @default.
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