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Consider we have matrix $S$, all we want is:

select some columns of $S$ that have maximum reconstruction of all columns of $S$ and the number of selected columns is minimal (or semi minimal).

the mean of reconstruction is linear combination.

For example: If the $S$ has 10 columns, the goal is select 3 or 4 columns that can reconstruct all 10 columns of $S$.

Convention: the goal matrix called $S^*$

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The compressive sensing theory basically says, a signal can be (even fully perhaps) recovered even when it is extremely under-sampled.

Normally, it is an inverse problem, meaning that if you have a signal $x$, then what you actually collect is some sorta forms of transformed $x$, say $y$. The transform can be represented as

$y=Fx$

where $F$ here is a transformation. For example, Fourier transform.

Say, $x$ is an image, and if you apply 2D Fourier transform to this image, you get its 2D spectrum $y$. Then you undersample $y$ with some mask function to get $y_u$ (this is what you mentioned to pick only some columns of $y$ ). Obviously, now if you apply direct inverse Fourier transform on $y_u$, you are not going to get a perfect $x$, because information has lost.

Now if you do compressive sensing, you can pretty much recover $x$ almost as good as original. The basic idea is to construct a cost function, and iteratively solve the problem. There are many ways to do so, but something like following:

$Minimize (WT (x))$

$S.t. \qquad y=Fx$

where $WT$ means wavelet transform.

There are many literature on this topic. Google is will give a lot of results. Hope this helps.

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    But this only works for certain special sampling patterns, such as random sampling patterns. That's one of the key points of compressed sensing. – 2017-07-11
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    @littleO You are definitely right. This is called incoherent sampling. In many cases, you can control how you sample, but all you want is to sample faster to shorten the lengthy sampling process. – 2017-07-11