admm.spca {ADMM} | R Documentation |

## Sparse PCA

### Description

Sparse Principal Component Analysis aims at finding a sparse vector by solving

`\textrm{max}_x~x^T\Sigma x \quad \textrm{s.t.} \quad \|x\|_2\le 1,~\|x\|_0\le K`

where `\|x\|_0`

is the number of non-zero elements in a vector `x`

. A convex relaxation
of this problem was proposed to solve the following problem,

`\textrm{max}_X~<\Sigma,X> ~\textrm{s.t.} \quad Tr(X)=1,~\|X\|_0 \le K^2, ~X\ge 0,~\textrm{rank}(X)=1`

where `X=xx^T`

is a `(p\times p)`

matrix that is outer product of a vector `x`

by itself,
and `X\ge 0`

means the matrix `X`

is positive semidefinite.
With the rank condition dropped, it can be restated as

`\textrm{max}_X~ <\Sigma,X>-\rho\|X\|_1 \quad \textrm{s.t.}\quad Tr(X)=1,X\ge 0.`

After acquiring each principal component vector, an iterative step based on Schur complement deflation method
is applied to regress out the impact of previously-computed projection vectors. It should be noted that
those sparse basis may *not be orthonormal*.

### Usage

```
admm.spca(
Sigma,
numpc,
mu = 1,
rho = 1,
abstol = 1e-04,
reltol = 0.01,
maxiter = 1000
)
```

### Arguments

`Sigma` |
a |

`numpc` |
number of principal components to be extracted. |

`mu` |
an augmented Lagrangian parameter. |

`rho` |
a regularization parameter for sparsity. |

`abstol` |
absolute tolerance stopping criterion. |

`reltol` |
relative tolerance stopping criterion. |

`maxiter` |
maximum number of iterations. |

### Value

a named list containing

- basis
a

`(p\times numpc)`

matrix whose columns are sparse principal components.- history
a length-

`numpc`

list of dataframes recording iteration numerics. See the section for more details.

### Iteration History

For SPCA implementation, main computation is sequentially performed for each projection vector. The `history`

field is a list of length `numpc`

, where each element is a data frame containing iteration history recording
following fields over iterates,

- r_norm
norm of primal residual

- s_norm
norm of dual residual

- eps_pri
feasibility tolerance for primal feasibility condition

- eps_dual
feasibility tolerance for dual feasibility condition

In accordance with the paper, iteration stops when both `r_norm`

and `s_norm`

values
become smaller than `eps_pri`

and `eps_dual`

, respectively.

### References

Ma S (2013).
“Alternating Direction Method of Multipliers for Sparse Principal Component Analysis.”
*Journal of the Operations Research Society of China*, **1**(2), 253–274.
ISSN 2194-668X, 2194-6698, doi: 10.1007/s40305-013-0016-9.

### Examples

```
## generate a random matrix and compute its sample covariance
X = matrix(rnorm(1000*5),nrow=1000)
covX = stats::cov(X)
## compute 3 sparse basis
output = admm.spca(covX, 3)
```

*ADMM*version 0.3.3 Index]