Sparse PCA

Given an integer ${\displaystyle k}$ with ${\displaystyle 1\leq k\leq p}$, the sparse PCA problem can be formulated as maximizing the variance along a direction represented by vector ${\displaystyle v\in \mathbb {R} ^{p}}$ while constraining its cardinality:

${\displaystyle {\begin{aligned}\max \quad &v^{T}\Sigma v\\{\text{subject to}}\quad &\left\Vert v\right\Vert _{2}=1\\&\left\Vert v\right\Vert _{0}\leq k.\end{aligned}}}$Eq. 1

The first constraint specifies that v is a unit vector. In the second constraint, ${\displaystyle \left\Vert v\right\Vert _{0}}$ represents the ${\displaystyle \ell _{0}}$ pseudo-norm of v, which is defined as the number of its non-zero components. So the second constraint specifies that the number of non-zero components in v is less than or equal to k, which is typically an integer that is much smaller than dimension p. The optimal value of Eq. 1 is known as the k-sparse largest eigenvalue.

If one takes k=p, the problem reduces to the ordinary PCA, and the optimal value becomes the largest eigenvalue of covariance matrix Σ.

After finding the optimal solution v, one deflates Σ to obtain a new matrix

${\displaystyle \Sigma _{1}=\Sigma -(v^{T}\Sigma v)vv^{T},}$

and iterate this process to obtain further principal components. However, unlike PCA, sparse PCA cannot guarantee that different principal components are orthogonal. In order to achieve orthogonality, additional constraints must be enforced.

The following equivalent definition is in matrix form. Let ${\displaystyle V}$ be a p×p symmetric matrix, one can rewrite the sparse PCA problem as

${\displaystyle {\begin{aligned}\max \quad &Tr(\Sigma V)\\{\text{subject to}}\quad &Tr(V)=1\\&\Vert V\Vert _{0}\leq k^{2}\\&Rank(V)=1,V\succeq 0.\end{aligned}}}$Eq. 2

Tr is the matrix trace, and ${\displaystyle \Vert V\Vert _{0}}$ represents the non-zero elements in matrix V. The last line specifies that V has matrix rank one and is positive semidefinite. The last line means that one has ${\displaystyle V=vv^{T}}$, so Eq. 2 is equivalent to Eq. 1.

Moreover, the rank constraint in this formulation is actually redundant, and therefore sparse PCA can be cast as the following mixed-integer semidefinite program[6]

${\displaystyle {\begin{aligned}\max \quad &Tr(\Sigma V)\\{\text{subject to}}\quad &Tr(V)=1\\&\vert V_{i,i}\vert \leq z_{i},\forall i\in \{1,...,p\},\vert V_{i,j}\vert \leq {\frac {1}{2}}z_{i},\forall i,j\in \{1,...,p\}:i\neq j,\\&V\succeq 0,z\in \{0,1\}^{p},\sum _{i}z_{i}\leq k\end{aligned}}}$Eq. 3

Because of the cardinality constraint, the maximization problem is hard to solve exactly, especially when dimension p is high. In fact, the sparse PCA problem in Eq. 1 is NP-hard in the strong sense.[7]

In this section the generic SPCs will be defined as ${\displaystyle t_{j}=Xv_{j},\,j=1,\ldots ,d}$, where ${\displaystyle v_{j}}$ denotes the j^th ${\displaystyle p}$-vectors of loadings.

By denoting ${\displaystyle V}$ the matrix with columns ${\displaystyle v_{j}}$ and ${\displaystyle T=XV}$, the USPCs are derived as the solutions to the minimization problem:

${\displaystyle {\begin{aligned}V={\text{arg}}\min \quad &\vert \vert X-\Pi _{T}X\vert \vert ^{2}={\text{arg}}\max \vert \vert \Pi _{T}X\vert \vert ^{2}\\{\text{subject to}}\quad &card\{v_{j}\}\leq p\\&T'T=diag{\{s_{j}\}},\end{aligned}}}$Eq. 4

where ${\displaystyle \Pi _{T}}$ is the orthogonal projector onto the column space of ${\displaystyle T}$.

Because of the orthogonality constraints, the problem in equation Eq. 4 can be solved sequentially for each ${\displaystyle v_{j},\,j=1,\ldots ,d}$, as:

${\displaystyle {\begin{aligned}v_{j}={\text{arg }}\min \quad &\vert \vert X-\Pi _{t_{j}}X\vert \vert ^{2}={\text{arg }}\max \vert \vert \Pi _{t_{j}}X\vert \vert ^{2}\\{\text{subject to}}\quad &card\{v_{j}\}\leq p\\&T_{[j-1]}'t_{j}=0,\end{aligned}}}$Eq. 5

where ${\displaystyle T_{[j-1]}}$ denotes the first ${\displaystyle j-1}$ columns of ${\displaystyle T}$. Closed form solutions to equation Eq. 5 for given ${\displaystyle p}$ are given in Merola (2015).[2]

In CSPC the loadings are obtained iteratively by minimizing the norm of the approximation error from the residuals orthogonal to the previously computed SPCs. That is, if we let ${\displaystyle X_{j-1}^{\perp }=X-\Pi _{T_{[j-1]}}X}$, the ${\displaystyle j^{\text{th}}}$ vector of loadings is obtained as the solution to:

${\displaystyle {\begin{aligned}v_{j}={\text{arg }}\min \quad &\vert \vert X_{j-1}^{\perp }-\Pi _{t_{j}}X\vert \vert ^{2}\\{\text{subject to}}\quad &card\{v_{j}\}\leq p,\end{aligned}}}$Eq. 6

Without the sparsity constraints, the problems in equations Eq. 4 - Eq. 6 are equivalent to Pearson's definition of PCA.

The PSPCA sparse loadings are obtained iteratively by projecting the first PCs of the residual matrix ${\displaystyle X_{j-1}^{\perp }}$ on subsets of the variables. that is by solving the problems

${\displaystyle {\begin{aligned}v_{j}={\text{arg }}\min \quad &\vert \vert r_{j}-\Pi _{t_{j}}r_{j}\vert \vert ^{2}\\{\text{subject to}}\quad &card\{v_{j}\}\leq p,\end{aligned}}}$Eq. 7

where ${\displaystyle r_{j}}$ is the first principal component of the residual matrix ${\displaystyle X_{j-1}^{\perp }}$

It has been proposed that sparse PCA can be approximated by semidefinite programming (SDP).[11] If one drops the rank constraint and relaxes the cardinality constraint by a 1-norm convex constraint, one gets a semidefinite programming relaxation, which can be solved efficiently in polynomial time:

${\displaystyle {\begin{aligned}\max \quad &Tr(\Sigma V)\\{\text{subject to}}\quad &Tr(V)=1\\&\mathbf {1} ^{T}|V|\mathbf {1} \leq k\\&V\succeq 0.\end{aligned}}}$Eq. 3

In the second constraint, ${\displaystyle \mathbf {1} }$ is a p×1 vector of ones, and |V| is the matrix whose elements are the absolute values of the elements of V.

The optimal solution ${\displaystyle V}$ to the relaxed problem Eq. 3 is not guaranteed to have rank one. In that case, ${\displaystyle V}$ can be truncated to retain only the dominant eigenvector.

While the semidefinite program does not scale beyond n=300 covariates, it has been shown that a second-order cone relaxation of the semidefinite relaxation is almost as tight and successfully solves problems with n=1000s of covariates [18]

LS SPCA requires that the optimal cardinality and subsets of variables forming each SPC are determined by some optimization criterion. In LS SPCA, a reasonable approach is to choose the lowest cardinality for which an SPC explains a given percentage of the cumulative variance explained by the corresponding standard PC.

Merola (2015)[19] proposed a backward elimination algorithm for variable selection in USPCA and CSPCA, which can be slow with large datasets. With PSPCA the optimal subsets can be efficiently computed by using standard algorithms for regression Feature_selection. One possible strategy for computing the LS SPCs is to use PSPCA for variable selection and then compute the SPCs with USPCA or CSPCA.

Suppose ordinary PCA is applied to a dataset where each input variable represents a different asset, it may generate principal components that are weighted combination of all the assets. In contrast, sparse PCA would produce principal components that are weighted combination of only a few input assets, so one can easily interpret its meaning. Furthermore, if one uses a trading strategy based on these principal components, fewer assets imply less transaction costs.

Consider a dataset where each input variable corresponds to a specific gene. Sparse PCA can produce a principal component that involves only a few genes, so researchers can focus on these specific genes for further analysis.

Contemporary datasets often have the number of input variables (${\displaystyle p}$) comparable with or even much larger than the number of samples (${\displaystyle n}$). It has been shown that if ${\displaystyle p/n}$ does not converge to zero, the classical PCA is not consistent. In other words, if we let ${\displaystyle k=p}$ in Eq. 1, then the optimal value does not converge to the largest eigenvalue of data population when the sample size ${\displaystyle n\rightarrow \infty }$, and the optimal solution does not converge to the direction of maximum variance. But sparse PCA can retain consistency even if ${\displaystyle p\gg n.}$[4]

The k-sparse largest eigenvalue (the optimal value of Eq. 1) can be used to discriminate an isometric model, where every direction has the same variance, from a spiked covariance model in high-dimensional setting.[20] Consider a hypothesis test where the null hypothesis specifies that data ${\displaystyle X}$ are generated from a multivariate normal distribution with mean 0 and covariance equal to an identity matrix, and the alternative hypothesis specifies that data ${\displaystyle X}$ is generated from a spiked model with signal strength ${\displaystyle \theta }$:

${\displaystyle H_{0}:X\sim N(0,I_{p}),\quad H_{1}:X\sim N(0,I_{p}+\theta vv^{T}),}$

where ${\displaystyle v\in \mathbb {R} ^{p}}$ has only k non-zero coordinates. The largest k-sparse eigenvalue can discriminate the two hypotheses if and only if ${\displaystyle \theta >\Theta ({\sqrt {k\log(p)/n}})}$.

Since computing k-sparse eigenvalue is NP-hard, one can approximate it by the optimal value of semidefinite programming relaxation (Eq. 3). If that case, we can discriminate the two hypotheses if ${\displaystyle \theta >\Theta ({\sqrt {k^{2}\log(p)/n}})}$. The additional ${\displaystyle {\sqrt {k}}}$ term cannot be improved by any other polynomial time algorithm if the planted clique conjecture holds.

• amanpg - R package for Sparse PCA using the Alternating Manifold Proximal Gradient Method [21]
• elasticnet – R package for Sparse Estimation and Sparse PCA using Elastic-Nets [22]
• nsprcomp - R package for sparse and/or non-negative PCA based on thresholded power iterations[23]
• scikit-learn – Python library for machine learning which contains Sparse PCA and other techniques in the decomposition module.[24]

An R package with routines in C++ is available on Github

With due differences, the preference of conventional SPCA for correlated variables can be observed on datasets with multiply correlated variables. As a general rule, conventional SPCA seeks subsets of highly correlated variables. This is not the case for LS SPCA, because the data reconstruction approach discourages the presence of correlated variables in the subset forming the SPCs.

As an example, Table 3[5] shows the results of USPCA, CSPCA and PSPCA applied to four row-rank deficient matrices. For each of the first three SPCs are shown the cardinality and the cumulative variance that they explain as a percentage of the cumulative variance explained by the corresponding PCs. The reduction in cardinality is striking when considering that over 96% of the variance is explained and in some cases 100% of it. The results for the three the variants of LS SPCA are quite similar.

Figure 1[5] shows the norm, relative cumulative variance explained, and correlation with the first PC for the first SPCs obtained with conventional SPCA and PSCA on four datasets. The first two datasets are full column-rank and the last two are not. More details on these datasets can be found in Merola (2019).[5] It is easy to see how the PSPCs explain more variance and reach close to one correlation with the PC with much smaller cardinality, irrespective of the much larger norm of the conventional SPCA components.

Figure 1 Comparison of the norm, relative cumulative variance explained, and correlation with the first PC for the first SPCs obtained with conventional SPCA and PSCA on four datasets.

As shown in Table 3, the datasets Khanh and Rama have much fewer rows than columns. For this reason, the PSPCs are identical to the PCs when their cardinality is equal to the rank of the matrices. Instead, the conventional SPCs are computed with a much larger cardinality than the matrices rank (hence including perfectly correlated variables) without reaching the LS SPCs performance.