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May
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awarded  Commentator
May
4
comment Decomposing a discrete signal into a sum of rectangle functions
Thanks for the comment. To my understanding the problem can be formulated as a linear programming, but it cannot be solved as linear programming since it is too big to be tractable. In my case the number of features/bases would be $\sim 10^6$. Please see "option 2" in the question.
May
1
comment Decomposing a discrete signal into a sum of rectangle functions
brute_force behaves differently than explore_a because it solves $\left(w - w'\right)^2$ instead of $\left|w - w'\right|$
May
1
comment Decomposing a discrete signal into a sum of rectangle functions
I agree that the test signal is not ideal. However I wanted to evaluate the methods with a signal that is "not obviously easy to approximate by rectangles" to have kinda of a "worst case" evaluation. Now that I see that the problem seems tractable, I am working on creating the real signal I want to approximate (which will take about a week of work).
May
1
comment Decomposing a discrete signal into a sum of rectangle functions
wow! I am impressed! You can send me the new version at profiles.google.com/rodrigo.benenson or simply put the file online using letscrate.com and add a link on a comment.
May
1
accepted Decomposing a discrete signal into a sum of rectangle functions
Apr
30
awarded  Teacher
Apr
30
revised Decomposing a discrete signal into a sum of rectangle functions
minor edits
Apr
30
awarded  Scholar
Apr
30
answered Decomposing a discrete signal into a sum of rectangle functions
Apr
29
awarded  Supporter
Apr
29
comment Decomposing a discrete signal into a sum of rectangle functions
Thanks for this proposal. I will look into it in detail and post here my findings.
Apr
28
comment Decomposing a discrete signal into a sum of rectangle functions
Yes $a_i$ can be negative since $w$ can be negative. I am curious about your idea...
Apr
28
comment Decomposing a discrete signal into a sum of rectangle functions
mmm.... simulated annealing is indeed a possibility. But is not that a fancy name for "let us search randomly for a solution" ? I would expect that more "targeted" optimization techniques exist (at least some kind of greedy approach).
Apr
27
revised Decomposing a discrete signal into a sum of rectangle functions
added size of $w$
Apr
27
awarded  Editor
Apr
27
revised Decomposing a discrete signal into a sum of rectangle functions
clarifying the question, adding numbering to bullets
Apr
27
comment Decomposing a discrete signal into a sum of rectangle functions
mmm... yes you are right. Thanks for pointing out that the Haar-wavelet will provide a solution. However, to my understanding, for a given $N$ this wavelet will be do a poor job at minimizing $\left\Vert w-w'\right\Vert _{1}$ since it is ignoring the degrees of freedom provided by $b_i,\ c_i$. In that sense it seems not to be a proper solution to my problem. Any idea of how to side-step this issue ?
Apr
27
comment Decomposing a discrete signal into a sum of rectangle functions
Haar wavelet does not solves the problem as formulated, since it has fixed scaling and shift, which in my problem are free parameters.
Apr
27
comment Decomposing a discrete signal into a sum of rectangle functions
Yes the decomposition is not unique. To start with I would like a method to find a solution. After that we can look into finding a "good" solution, given for instance a $l_1$ regularization term.