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There isn’t any justification that would be applicable to all situations. Rather, this is a modeling choice that has to be justified in the context of your particular problem.

You can simply multiply $\Omega^{-1}$ as given by the author times $\Omega$ and see that the product is $I$. This confirms that the formula for $\Omega^{-1}$ is correct. The key here is that $(I- …

Here's a solution for the specific case where $f(x)=\| x \|_{1}$ and $g(x)=\| y-Ax \|_{2}$. The same appraoch applies to $g(x)=\| A^{T}(y-Ax) \|_{\infty}$. There are two cases. If $g(0) \leq s$, …

You need to start by by being explicit about the "equivalence" between the problems. Do you mean that "For every $s$, there is a $t$ such that if $x^{ * }$ is an optimal solution to the first problem …

The variance of the mean of $n$ random variables is $\mbox{Var}(\bar{x})=\mbox{Var}(\sum_{i=1}^{n} x_{i}/n)$ $\mbox{Var}(\bar{x})=\sum_{i=1}^{n} (1/n)^{2} \mbox{Var}(x_{i})$ $\mbox{Var}(\bar{x})=n …

A lot depends on $\hat{\eta}(\xi)$. When you convolve this with $\hat{u}$ and $l(\xi)$, you will lose the sparsity if $\hat{\eta}(\xi)$ has broad support. Just how much do you know about the spect …

In using least squares, you normally want the residuals from the various equations to be indpendent of each other. If your $q_{i}$ vectors are imprecise measurements, then this will introduce correla …

There's a very large literature on updating solutions to least squares problems as new data are added. The naive formula $\hat{\beta}=(X^{T}X)^{-1}X^{T}y$ can be problematic in practice because of nu …

Your problem might be small enough that it is within the range of polynomial optimization techniques based on SDP relaxations of sums of squares problem. This has been implemented in software package …

Now that you've provided some more information, I think I can make some useful suggestions. First, a quick review of linear transformations of multivariate normal random vectors. If $z$ is an MVN …

You haven't really told us much about the problem. Are you working in conventional single or double precision floating point arithmetic, or are you working in some obscure field? Are the element …

There's actually a much broader question that you should be asking yourself here- does it matter whether your data really is normally distributed, or will the procedures that you're going to perform o …

You're really asking the wrong question here... Let's back up a bit. You're attempting to estimate some parameters here, either by a maximum likelihood method or more likely by $\chi^2$ minimization …

"Covariance Matrix" doesn't make a lot of sense in this situation, since there's no multivariate normal distribution around the fitted parameters. You could if you wanted look at a non-ellipsoidal co …