a Direct Formulation For Sparse PCA Using Semidefinite Programming

Still under construction

Introduction

Principle Component Analysis is a popular technique which finds a transformation from correlated variables to uncorrelated ones which correspond to the direction of maximal variance in the data. The Principal component can be representative of the whole data with minimum information loss. In sparse PCA, the embedded variables are linear combination of the input variables subject to the constraint that the number of non-zero elements in this combination is limited. Sparse PCA decomposition interpretation is facilitated since the number of non-zero elements are not all variables if the coordinate axes have a physical meaning. It is also helpful in expressing a space of a set of low-dimensional vectors with minimum loss of information.

Semidefinite programs are convex optimization. It optimize a linear function by constraining it to an affine combination of symmetric matrices is positive semidefinite. Semidefinite programs can be solved in polynomial time by interior-point solvers like SEDUMI or SDPT3. Unfortunately it is not viable and practical for high dimensional data sets.

This paper suggests a direct approach for formulation of sparse PCA via semidefinite programming which is convex. Since interior-point solvers cannot handle large data sets, Nesterov’s smoothing technique is used efficiently to help solving large dimensional problems. First order methods require less memory which is an important issue in interior-point solvers. On the other hand their convergence is slower but it is not a major concern in the certain case.So The optimal first-order minimization algorithm is going to be applied for solving the optimization problem.

The content of the paper could be summarized as below:

First they tried to maximize the variance projection by limiting the number of non-zero elements via semidefinite programming. then they show the robustness of their method and how this method could be used for decomposing a matrix in to limited number of variables. As their problem size is large and can not be solved by common techniques which are used for solving the convex optimization problems; they show Nesterov's smoothing method is helping to achieve the solution.

Semidefinite relaxation

Sparse variance maximization

A is assumed to be a symmetric matrix and WLOG, A is a covariance matrix and we are going to maximize the variance of vector [math]\displaystyle{ x \in \textbf{R}^n }[/math] while it is sparse.

maximize [math]\displaystyle{ x^{T}{A}x }[/math]

subject to [math]\displaystyle{ \|x\|_2=1 }[/math]

[math]\displaystyle{ \textbf{Card}(x)\leq k }[/math]

Semidefinite relaxation

Rewriting the above formulas, we will have:

maximize [math]\displaystyle{ \textbf{Tr}({A}{X}) }[/math]

subject to [math]\displaystyle{ \textbf{Tr}({X})=1 }[/math],

[math]\displaystyle{ \textbf{Card}({X})\leq k^2 }[/math],

[math]\displaystyle{ {X}\geq 0 }[/math]

[math]\displaystyle{ \textbf{Rank}({X})=1 }[/math]

where [math]\displaystyle{ X=xx^T }[/math] and it is the solution to the above problem. Finally by replacing the only non-convex constraint, [math]\displaystyle{ \textbf{Card}({X})\leq k^2 }[/math], to a weaker but convex one, we come to a semidefinite relaxation of the sparse PCA for square matrices.

maximize [math]\displaystyle{ \textbf{Tr}({A}{X}) }[/math]

subject to [math]\displaystyle{ \textbf{Tr}({X})=1 }[/math],

[math]\displaystyle{ \textbf{1}^{T}|{X}|\textbf{1}\leq k }[/math],

[math]\displaystyle{ {X}\geq 0 }[/math]

The above formula can be extended to non-square cases. the sparse variance maximization problem can be written as followed:

maximize [math]\displaystyle{ u^{T}{A}v }[/math]

subject to [math]\displaystyle{ \|u\|_{2}=\|v\|_{2}=1 }[/math],

[math]\displaystyle{ \textbf{Card}(u)\leq k_{1},\textbf{Card}(u)\leq k_{1} }[/math],

Changing the constraint to the convex one teh same as what we did for square cases, the following formulas will be achieved:

maximize [math]\displaystyle{ \textbf{Tr}({A}^{T}{X}^{12}) }[/math]

subject to [math]\displaystyle{ {X}\leq0, \textbf{Tr}({X}^{ii})=1 }[/math],

[math]\displaystyle{ \textbf{1}^{T}|{X}^{ii}|\textbf{1}\leq k_{i}, i=1,2 }[/math],

[math]\displaystyle{ \textbf{1}^{T}|{X}^{12}|\textbf{1}\leq \sqrt{k_1k_2} }[/math]

Sparse decomposition

Here, we are going to apply our formula for obtaining component of sparse PCA. Given matrix [math]\displaystyle{ A_1 \in \textbf{S}^n }[/math], our goal is to decompose it in components which are sparse.

maximize [math]\displaystyle{ \textbf{Tr}({A}_1{X}) }[/math]

subject to [math]\displaystyle{ \textbf{Tr}({X})=1 }[/math],

[math]\displaystyle{ \textbf{1}^{T}|{X}|\textbf{1}\leq k }[/math],

[math]\displaystyle{ {X}\geq 0 }[/math]

[math]\displaystyle{ X_1 }[/math] will e truncated as below and the only first eigenvector [math]\displaystyle{ x_1 }[/math] which is corespond to the largest eigenvalue will be kept.

[math]\displaystyle{ {A}_2={A}_1-(x_{1}^{T}{A}_1x_1)x_1x_1^T }[/math]

And repeat the algorithm to obtain more components. But the issue is when should we stop?

When the values in [math]\displaystyle{ A_i }[/math] are all less than a constant [math]\displaystyle{ \rho }[/math], we must stop. [math]\displaystyle{ \rho }[/math] acn be interpreted as noise level and when the matrix can not be distinguished from noise, we will terminate the iterations.

Algorithm

SEDUMI and SDPT3 are used for small size problems of semidefinite programming. But for larger problems those methods are not practical because it is impossible to store any second order information of size [math]\displaystyle{ n }[/math], which is necessary in interior-point SDP solvers. SO, the first order methods would be popular. These methods need a smaller size memory cost per iteration but their convergence time is [math]\displaystyle{ O(1/\epsilon) }[/math] while interior-point method will converge in [math]\displaystyle{ O(log(1/\epsilon)) }[/math]. Fortunately convergence time is not important in our application.

Smoothing technique

There are two important issue for large scale semidefinite programming. One is the memory difficulty and for solving this problem we 'll go toward first -order techniques. The other one is smoothness. As [math]\displaystyle{ X\geq 0 }[/math] is not smooth, as a result the number of iteration using first-order methodsto an accuracy of [math]\displaystyle{ \epsilon }[/math] is given by [math]\displaystyle{ O(1/\epsilon ^2) }[/math]. this bound is tight without any further information about the structure. since we have available information this complexity will be brought to [math]\displaystyle{ O(1/\epsilon) }[/math].

[math]\displaystyle{ f(x)=\hat{f}(x)+max_{u}\{\lt \textbf{T}x,u\gt -\hat{\phi}(u) : u \in \textbf{Q}_2\} }[/math]

[math]\displaystyle{ min _{x\in \textbf{Q}_1}f(x) }[/math]

Application o sparse PCA

maximize [math]\displaystyle{ \textbf{Tr}({A}{X})-\textbf{1}^{T}|{X}|\textbf{1} }[/math]

subject to [math]\displaystyle{ \textbf{Tr}({X})=1 }[/math],

[math]\displaystyle{ {X}\geq 0 }[/math]

[math]\displaystyle{ min_{{U}\in{Q}_1}f({U}) }[/math]

[math]\displaystyle{ {Q}_1=\{{U}\in \textbf{S}^{n}:|{U}_{ij}|\leq 1,i,j=1,...,n\},{Q}_2=\{{X}\in\textbf{S}^n:\textbf{Tr} X=1,X\geq0\} }[/math]

[math]\displaystyle{ f(U):=max_{X \in Q_2}\lt TU,X\gt -\hat{\phi}(X) }[/math] with [math]\displaystyle{ T=I_{n^2}, \hat{\phi}(X)=-\textbf{Tr}(AX) }[/math]

[math]\displaystyle{ d_1(U)=\frac{1}{2}U^T U }[/math]

[math]\displaystyle{ U_0:=arg min_{U\in Q_{1}}d_1(U) }[/math]

[math]\displaystyle{ D_1:=max_{U \in Q_1}d_1(U)=n^2/2 }[/math]

[math]\displaystyle{ d_2(X)=\textbf{Tr}(XlogX)+log(n) }[/math]

[math]\displaystyle{ max_{X\in Q_2}d_2(X)\leq log n:=D_2 }[/math]

[math]\displaystyle{ \| T\| _{1,2}:= max_{X,U}\lt TX,U\gt :\| X\| _{F}=1,\|U\|^*_2=1 }[/math]

[math]\displaystyle{ =max_X\|X\|_2:\|X\|_F\leq 1 }[/math]

[math]\displaystyle{ =1 }[/math]

[math]\displaystyle{ D_1=n^2/2,\sigma _1=1,D_2=log(n),\sigma_2=1/2,\|T\|_{1,2}=1 }[/math]

Regularization

[math]\displaystyle{ \mu:=\frac{\epsilon}{2D_2} }[/math]

[math]\displaystyle{ N=\frac{4\|T\|_{1,2}}{\epsilon}\sqrt{\frac{D_1D_2}{\sigma_1\sigma_2}} }[/math]

[math]\displaystyle{ min_{U \in Q_1}f_{\mu}(U) }[/math]

[math]\displaystyle{ f_{\mu}(U):=max_{X \in Q_2}\lt TU,X\gt -\hat{\phi}(X)-\mu d_2(X) }[/math]

[math]\displaystyle{ L:=\frac{D_2\|T\|_{1,2}^2}{\epsilon 2\sigma_2} }[/math]

[math]\displaystyle{ f_\mu(U)=\mu log(\textbf{Tr}exp((A+U)/\mu))-\mu log n }[/math]

References

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