Optimization Issues in Compressed Sensing

1
Optimization Issues in Compressed Sensing

• Algorithmic landscape for the most basic version
  of signal recovery still unsettled
• Variety of approaches have been proposed all
  have drawbacks. New approaches needed!
• First-order vs. second-order methods?
• Combinatorial vs. linear vs. nonlinear methods?
• Important extensions needed
• No development to date of parallel/distributed
  algorithms
• Very important for distributed sensor networks
• No development to date of algorithms that are
  stable in the presence of noise in the
  measurements

2
Optimization Issues in Compressed Sensing

• Closely Related Issues
• How to optimally design set of measurements
• Theory says choose randomly, but is the best way
  to do it?
• How many measurements are needed?
• One can get by with much fewer measurements, but
  at the expense of having to solve a tougher
  optimization problem. What is the tradeoff?
• Tangential (but still important) Issues
• Extension of Compressed-Sensing Optimization to
  Affine Rank Minimization
• Potentially very important in data mining