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Retrieve default PCP parameter settings for given matrix

Source: R/get_pcp_defaults.R
get_pcp_defaults.Rd

get_pcp_defaults() calculates "default" PCP parameter settings lambda, mu (used in root_pcp()), and eta (used in rrmc()) for a given data matrix D.

The "default" values of lambda and mu offer theoretical guarantees of optimal estimation performance. Candès et al. (2011) obtained the guarantee for lambda, while Zhang et al. (2021) obtained the result for mu. It has not yet been proven whether or not eta enjoys similar properties.

In practice it is common to find different optimal parameter values after tuning these parameters in a grid search. Therefore, it is recommended to use these defaults primarily to help define a reasonable initial parameter search space to pass into grid_search_cv().

Usage

get_pcp_defaults(D)

Arguments

D

The input data matrix.

Value

A list containing:

  • lambda: The theoretically optimal lambda value used in root_pcp().

  • mu: The theoretically optimal mu value used in root_pcp().

  • eta: The default eta value used in rrmc().

The intuition behind PCP parameters

root_pcp()'s objective function is given by: $$\min_{L, S} ||L||_* + \lambda ||S||_1 + \mu ||L + S - D||_F$$

  • lambda controls the sparsity of root_pcp()'s output S matrix; larger values of lambda penalize non-zero entries in S more stringently, driving the recovery of sparser S matrices. Therefore, if you a priori expect few outlying events in your model, you might expect a grid search to recover relatively larger lambda values, and vice-versa.

  • mu adjusts root_pcp()'s sensitivity to noise; larger values of mu penalize errors between the predicted model and the observed data (i.e. noise), more severely. Environmental data subject to higher noise levels therefore require a root_pcp() model equipped with smaller mu values (since higher noise means a greater discrepancy between the observed mixture and the true underlying low-rank and sparse model). In virtually noise-free settings (e.g. simulations), larger values of mu would be appropriate.

rrmc()'s objective function is given by: $$\min_{L, S} I_{rank(L) \leq r} + \eta ||S||_0 + ||L + S - D||_F^2$$

  • eta controls the sparsity of rrmc()'s output S matrix, just as lambda does for root_pcp(). Because there are no other parameters scaling the noise term, eta can be thought of as a ratio between root_pcp()'s lambda and mu: Larger values of eta will place a greater emphasis on penalizing the non-zero entries in S over penalizing the errors between the predicted and observed data (the dense noise Z).

The calculation of the "default" PCP parameters

  • lambda is calculated as \(\lambda = 1 / \sqrt{\max(n, p)},\) where \(n\) and \(p\) are the dimensions of the input matrix \(D_{n \times p}\) Candès et al. (2011).

  • mu is calculated as \(\mu = \sqrt{\frac{\min(n, p)}{2}},\) where \(n\) and \(p\) are as above [Zhang et al. (2021)].

  • eta is simply \(\eta = \frac{\lambda}{\mu}\).

References

Candès, Emmanuel J., Xiaodong Li, Yi Ma, and John Wright. "Robust principal component analysis?." Journal of the ACM (JACM) 58, no. 3 (2011): 1-37.

Zhang, Junhui, Jingkai Yan, and John Wright. "Square root principal component pursuit: tuning-free noisy robust matrix recovery." Advances in Neural Information Processing Systems 34 (2021): 29464-29475. [available here]

See also

grid_search_cv()

Examples

# Examine the queens PM2.5 data
queens
#> # A tibble: 2,443 × 27
#>    Date            Al   NH4      As     Ba       Br     Cd      Ca      Cl
#>    <date>       <dbl> <dbl>   <dbl>  <dbl>    <dbl>  <dbl>   <dbl>   <dbl>
#>  1 2001-04-04 NA      1.62  NA      NA     NA       NA     NA      NA     
#>  2 2001-04-07  0      2.66   0       0.012  0.00488  0      0.0401  0.0079
#>  3 2001-04-13  0.0094 1.41   0.0016  0.024  0.00211  0.004  0.036   0     
#>  4 2001-04-19  0.0104 1.22   0.001   0.006  0.00422  0      0.0543  0.003 
#>  5 2001-04-25  0.0172 0.723  0.0024  0.015  0.00117  0      0.0398  0     
#>  6 2001-05-01  0.0384 3.48   0.0017  0.041  0.00873  0.001  0.136   0     
#>  7 2001-05-04  0.0964 6.22   0.0025  0.039  0.0111   0      0.137   0     
#>  8 2001-05-07  0.004  0.233  0.001   0.016  0.00263  0      0.055   0.0054
#>  9 2001-05-10  0.0547 2.04   0.001   0.055  0.00521  0      0.121   0.001 
#> 10 2001-05-13  0.0215 0.229  0       0.021  0.00122  0      0.0249  0     
#> # ℹ 2,433 more rows
#> # ℹ 18 more variables: Cr <dbl>, Cu <dbl>, EC <dbl>, Fe <dbl>, Pb <dbl>,
#> #   Mg <dbl>, Mn <dbl>, Ni <dbl>, OC <dbl>, K <dbl>, Se <dbl>, Si <dbl>,
#> #   Na <dbl>, S <dbl>, Ti <dbl>, NO3 <dbl>, V <dbl>, Zn <dbl>
# Get rid of the Date column
D <- as.matrix(queens[, 2:ncol(queens)])
# Get default PCP parameters
default_params <- get_pcp_defaults(D)
# Use default parameters to define parameter search space
scaling_factors <- sort(c(10^seq(-2, 4, 1), 2 * 10^seq(-2, 4, 1)))
etas_to_grid_search <- default_params$eta * scaling_factors
etas_to_grid_search
#>  [1] 5.611340e-05 1.122268e-04 5.611340e-04 1.122268e-03 5.611340e-03
#>  [6] 1.122268e-02 5.611340e-02 1.122268e-01 5.611340e-01 1.122268e+00
#> [11] 5.611340e+00 1.122268e+01 5.611340e+01 1.122268e+02