# Sparse Covariance Estimation

### Description

Provides a sparse and positive definite estimate of a covariance matrix. This function performs the majorize-minimize algorithm described in Bien & Tibshirani 2011 (see full reference below).

### Usage

1 2 3 |

### Arguments

`Sigma` |
an initial guess for Sigma (suggestions: |

`S` |
the empirical covariance matrix of the data. Must be positive definite (if it is not, add a small constant to the diagonal). |

`lambda` |
penalty parameter. Either a scalar or a matrix of the
same dimension as |

`step.size` |
the step size to use in generalized gradient descent. Affects speed of algorithm. |

`nesterov` |
indicates whether to use Nesterov's modification of
generalized gradient descent. Default: |

`n.outer.steps` |
maximum number of majorize-minimize steps to take (recall that MM is the outer loop). |

`n.inner.steps` |
maximum number of generalized gradient steps to take (recall that generalized gradient descent is the inner loop). |

`tol.outer` |
convergence threshold for outer (MM) loop. Stops
when drop in objective between steps is less than |

`thr.inner` |
convergence threshold for inner (i.e. generalized
gradient) loop. Stops when mean absolute change in |

`backtracking` |
if |

`trace` |
controls how verbose output should be. |

### Details

This is the R implementation of Algorithm 1 in Bien, J., and Tibshirani, R. (2011), "Sparse Estimation of a Covariance Matrix," Biometrika. 98(4). 807–820. The goal is to approximately minimize (over Sigma) the following non-convex optimization problem:

minimize logdet(Sigma) + trace(S Sigma^-1) + || lambda*Sigma ||_1 subject to Sigma positive definite.

Here, the L1 norm and matrix multiplication between lambda and Sigma are elementwise. The empirical covariance matrix must be positive definite for the optimization problem to have bounded objective (see Section 3.3 of paper). We suggest adding a small constant to the diagonal of S if it is not. Since the above problem is not convex, the returned matrix is not guaranteed to be a global minimum of the problem.

In Section 3.2 of the paper, we mention a
simple modification of gradient descent due to Nesterov. The argument
`nesterov`

controls whether to use this modification (we suggest
that it be used). We also strongly recommend using backtracking.
This allows the algorithm to begin by taking large steps (the initial
size is determined by the argument `step.size`

) and then to
gradually reduce the size of steps.

At the start of the algorithm, a lower bound (`delta`

in the paper) on the
eigenvalues of the solution is calculated. As shown in Equation (3) of the paper, the prox function for our generalized gradient
descent amounts to minimizing (over a matrix X) a problem of the form

minimize (1/2)|| X-A ||_F^2 + || lambda*X ||_1 subject to X >= delta I

This is implemented using an alternating direction method of multipliers approach given in Appendix 3.

### Value

`Sigma` |
the sparse covariance estimate |

`n.iter` |
a vector giving the number of generalized gradient steps taken on each step of the MM algorithm |

`obj` |
a vector giving the objective values after each step of the MM algorithm |

### Author(s)

Jacob Bien and Rob Tibshirani

### References

Bien, J., and Tibshirani, R. (2011), "Sparse Estimation of a Covariance Matrix," Biometrika. 98(4). 807–820.

### See Also

ProxADMM

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 | ```
set.seed(1)
n <- 100
p <- 20
# generate a covariance matrix:
model <- GenerateCliquesCovariance(ncliques=4, cliquesize=p / 4, 1)
# generate data matrix with x[i, ] ~ N(0, model$Sigma):
x <- matrix(rnorm(n * p), ncol=p) %*% model$A
S <- var(x)
# compute sparse, positive covariance estimator:
step.size <- 100
tol <- 1e-3
P <- matrix(1, p, p)
diag(P) <- 0
lam <- 0.06
mm <- spcov(Sigma=S, S=S, lambda=lam * P,
step.size=step.size, n.inner.steps=200,
thr.inner=0, tol.outer=tol, trace=1)
sqrt(mean((mm$Sigma - model$Sigma)^2))
sqrt(mean((S - model$Sigma)^2))
## Not run: image(mm$Sigma!=0)
``` |

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