a Rank Minimization Heuristic with Application to Minimum Order System Approximation: Difference between revisions
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#Applying the mothod on the minimum order system approximation. | #Applying the mothod on the minimum order system approximation. | ||
=== The Heuristic === | === A Generalization Of The Trace Heuristic === | ||
This heurisitic minimizes the sum of the singular values of the matrix, | This heurisitic minimizes the sum of the singular values of the matrix <math>X\in \real^{m\times n}</math>, which is the nuclear norm of <math>X</math> denoted by <math>|X|_*</math>. | ||
<math> | <math> | ||
\begin{array}{ l l } | \begin{array}{ l l } | ||
\mbox{minimize} & | \mbox{minimize} & |X|_* \\ | ||
\mbox{subject to: } & X \in C | \mbox{subject to: } & X \in C | ||
\end{array} | \end{array} | ||
</math> | </math> | ||
According to the definition of the nuclear norm we have <math>|X|_*=\sum_{i=1}^{\min\{m,n\} }\sigma_i(X)</math> where <math> \sigma_i(X) = \sqrt{\lambda_i (X^TX)}</math>. | |||
When the matrix variable <math>X</math> is symmetric and positive semidefinite, then its singular values are the same as its eigenvalues, and therefore the nuclear norm reduces to <math>\mbox{Tr } X</math>, and that means the heuristic reduces to the trace minimization heuristic. | |||
==== Nuclear Norm Minimization vs. Rank Minimization ==== | |||
''' Definition:''' Let <math>f:C \rightarrow\real</math> where <math>C\subseteq \real^n</math>. The convex envelope of <math>f</math> (on <math>C</math>) is defined as the largest convex function <math>g</math> such | |||
that <math>g(x)\leq f(x)</math> for all <math>x\in X</math> | |||
[[Image:Convex Envelope|thumb|100px|right|Convex Envelope of a function]] | |||
Revision as of 22:47, 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]\displaystyle{ \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]\displaystyle{ \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.
A Generalization Of The Trace Heuristic
This heurisitic minimizes the sum of the singular values of the matrix [math]\displaystyle{ X\in \real^{m\times n} }[/math], which is the nuclear norm of [math]\displaystyle{ X }[/math] denoted by [math]\displaystyle{ |X|_* }[/math].
[math]\displaystyle{ \begin{array}{ l l } \mbox{minimize} & |X|_* \\ \mbox{subject to: } & X \in C \end{array} }[/math]
According to the definition of the nuclear norm we have [math]\displaystyle{ |X|_*=\sum_{i=1}^{\min\{m,n\} }\sigma_i(X) }[/math] where [math]\displaystyle{ \sigma_i(X) = \sqrt{\lambda_i (X^TX)} }[/math].
When the matrix variable [math]\displaystyle{ X }[/math] is symmetric and positive semidefinite, then its singular values are the same as its eigenvalues, and therefore the nuclear norm reduces to [math]\displaystyle{ \mbox{Tr } X }[/math], and that means the heuristic reduces to the trace minimization heuristic.
Nuclear Norm Minimization vs. Rank Minimization
Definition: Let [math]\displaystyle{ f:C \rightarrow\real }[/math] where [math]\displaystyle{ C\subseteq \real^n }[/math]. The convex envelope of [math]\displaystyle{ f }[/math] (on [math]\displaystyle{ C }[/math]) is defined as the largest convex function [math]\displaystyle{ g }[/math] such that [math]\displaystyle{ g(x)\leq f(x) }[/math] for all [math]\displaystyle{ x\in X }[/math]