I am working on an inverse problem of the form $Ax=b$ where $A$ and $b$ are known and I want to find $x$. I understand $L_1$ regularization and I have applied it to my work. But in addition to $L_1$ I am looking for way to restrict function which has peaks at close distance apart. Let me give an example. Consider the system where the vector I am estimating has length $N$. When we apply $L_1$ regularization consider that I get vector $x_1$ such that only two samples say sample number $n$ and sample number $m$ has high values and rest of the number of $x_1$ are negligible or small. This is exactly we expect from $L_1$ regularization. My question is how can I penalize functions where distance between $m$ and $n$ is small. Hope my question is clear.
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$\begingroup$ You might want to look at the grouped lasso statweb.stanford.edu/~tibs/ftp/sparse-grlasso.pdf. $\endgroup$– dohmatobCommented Sep 19, 2016 at 9:01
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$\begingroup$ @dohmatob Thank you for your reference. I had a look at it, but not sure it does what I want. My matrix A is fixed and is known, so I assumed the paper addresses a different problem. May I request your view? $\endgroup$– CreatorCommented Sep 20, 2016 at 20:35
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