# Difference between revisions of "a Rank Minimization Heuristic with Application to Minimum Order System Approximation"

Line 19: | Line 19: | ||

#Showing that the new heuristic can be reduced to an SDP, and hence effictively solved. | #Showing that the new heuristic can be reduced to an SDP, and hence effictively solved. | ||

#Applying the mothod on the minimum order system approximation. | #Applying the mothod on the minimum order system approximation. | ||

+ | |||

+ | === The Heuristic === | ||

+ | This heurisitic minimizes the sum of the singular values of the matrix, i.e, the nuclear norm. | ||

+ | |||

+ | <math> | ||

+ | \begin{array}{ l l } | ||

+ | \mbox{minimize} & ||X||_* \\ | ||

+ | \mbox{subject to: } & X \in C | ||

+ | \end{array} | ||

+ | \mbox{where} | ||

+ | ||X||_*=\sum_{i=1}^{\min{m,n} \sigma_i(X) | ||

+ | </math> |

## Revision as of 21:33, 23 November 2010

Rank Minimization Problem (RMP) has application in a variety of areas such as control, system identification, statistics and signal processing. Except in some special cases RMP is known to be computationaly hard. [math] \begin{array}{ l l } \mbox{minimize} & \mbox{Rank } X \\ \mbox{subject to: } & X \in C \end{array} [/math]

If the matrix is symmetric and positive semidifinite, trace minimization is a very effective heuristic for rank minimization problem. The trace minimization results in a semidefinite problem which can be easily solved. [math] \begin{array}{ l l } \mbox{minimize} & \mbox{Tr } X \\ \mbox{subject to: } & X \in C \end{array} [/math]

This paper focuses on the following problems:

- Describing a generalization of the trace heuristic for genaral non-square matrices.
- Showing that the new heuristic can be reduced to an SDP, and hence effictively solved.
- Applying the mothod on the minimum order system approximation.

### The Heuristic

This heurisitic minimizes the sum of the singular values of the matrix, i.e, the nuclear norm.

[math] \begin{array}{ l l } \mbox{minimize} & ||X||_* \\ \mbox{subject to: } & X \in C \end{array} \mbox{where} ||X||_*=\sum_{i=1}^{\min{m,n} \sigma_i(X) [/math]