What should we optimize?

Least-squares problems often present themselves as fitting goals:

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To balance our possibly contradictory goals we need weighting functions. The quadratic form that we should minimize is

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where is the inverse multivariate spectrum of the noise (data-space residuals) and is the inverse multivariate spectrum of the model. In other words, is a leveler on the data fitting error and is a leveler on the model. There is a curious unresolved issue: What is the most suitable constant scaling ratio of to ?