Sparse optimization with applications Sparse optimization sv gles

Sparse optimization with applications

Sparse optimization (sv. gles optimering) is a tool for solving problems of the type Find the smallest number of. . . ’ ’Find the simplest possible. . . ’ • Sparse optimization problems are in general hard (combinatorial). • There exists efficient methods for solving sparse optimization problems approximately.

Image reconstruction Compressive sensing

Model reduction Trend filtering

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Application: Sparse DFT A. k. a. breaking the Nyquist-Shannon sampling theorem

Nyquist-Shannon sampling theorem With uniform sampling, to be able to reconstruct a signal, the sampling frequency should be atleast twice as high as the highest frequency component in the signal.

Compressive sensing With random sampling, we can (with high probability) do better!

Sparse optimization using CVX in MATLAB cvx_begin variable x(100, 1) complex minimize ( norm(x, 1) ) subject to y(ind)==A*x; cvx_end • Here ind is a vector of indices, indicating the time stamps of our samples.

Application: Trend filtering

Trend filtering (somethimes called smoothing) is the problem of separating trends from noise in data • Sometimes trends are the interesting parts of a signal – Stock market analysis weather forecasting, tracking of energy demand… • Sometimes trends masks the interesting parts of data – Slowly drifting process noise – External unknown disturbances on an experiment

Assignment •

4. Get an academic license 1. Download CVX 2. Install CVX by running ’cvx_setup. m’ 3. Run command cvx_version in MATLAB 5. Fill in the form to get a license. You’ll need the username and Host ID from Step 3. 6. Follow instructions in the e-mail.