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

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#Applying the mothod on the minimum order system approximation.
 
#Applying the mothod on the minimum order system approximation.
  
=== The Heuristic ===
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=== A Generalization Of The Trace Heuristic ===
This heurisitic minimizes the sum of the singular values of the matrix, i.e, the nuclear norm.
+
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} & ||X||_* \\
+
\mbox{minimize} & |X|_* \\
 
\mbox{subject to: } & X \in C
 
\mbox{subject to: } & X \in C
 
\end{array}
 
\end{array}
\mbox{where}
 
||X||_*=\sum_{i=1}^{\min{m,n} \sigma_i(X)}
 
 
</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 21:47, 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] \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:

  1. Describing a generalization of the trace heuristic for genaral non-square matrices.
  2. Showing that the new heuristic can be reduced to an SDP, and hence effictively solved.
  3. 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]X\in \real^{m\times n}[/math], which is the nuclear norm of [math]X[/math] denoted by [math]|X|_*[/math].

[math] \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]|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]

File:Convex Envelope
Convex Envelope of a function