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x^=arg⁡min⁡x  ∥Ax−b∥22+λ∥Lx∥22\hat{x} = \arg\min_{x} \; \lVert Ax - b \rVert_2^2 + \lambda \lVert Lx \rVert_2^2

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Regularization is the standard mathematical answer: instead of solving the raw, ill-conditioned system, one solves a modified problem that trades a small, controlled amount of bias for a large reduction in the amplification of noise. Tikhonov regularization is the paradigmatic form, replacing the bare least-squares fit with a penalized objective x^=arg⁡min⁡x  ∥Ax−b∥22+λ∥Lx∥22\hat{x} = \arg\min_{x} \; \lVert Ax - b \rVert_2^2 + \lambda \lVert Lx \rVert_2^2. where A is the forward projection operator, b the measured data, L a penalty operator (often favoring smoothness), and λ\lambda > 0 a regularization parameter controlling the trade-off. As λ\lambda →\to 0 the solution approaches the raw, noise-amplifying least-squares fit; as λ\lambda grows the solution becomes smoother and more…

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x^=arg⁡min⁡x  ∥Ax−b∥22+λ∥Lx∥22,\hat{x} = \arg\min_{x} \; \lVert Ax - b \rVert_2^2 + \lambda \lVert Lx \rVert_2^2,

Equation 6 · Mathematics

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Regularization is the standard mathematical answer: instead of solving the raw, ill-conditioned system, one solves a modified problem that trades a small, controlled amount of bias for a large reduction in the amplification of noise. Tikhonov regularization is the paradigmatic form, replacing the bare least-squares fit with a penalized objective x^=arg⁡min⁡x  ∥Ax−b∥22+λ∥Lx∥22\hat{x} = \arg\min_{x} \; \lVert Ax - b \rVert_2^2 + \lambda \lVert Lx \rVert_2^2. where A is the forward projection operator, b the measured data, L a penalty operator (often favoring smoothness), and λ\lambda > 0 a regularization parameter controlling the trade-off. As λ\lambda →\to 0 the solution approaches the raw, noise-amplifying least-squares fit; as λ\lambda grows the solution becomes smoother and more…

Meanings in this article

  • AA: the forward projection operator.
  • bb: the measured data.
  • LL: the penalty operator (often favoring smoothness).
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