Estimate differentially private PC directions

Estimate principal component (PC) directions using a noisy spherical Kendall matrix, with optional ordinary PCA directions for comparison.

Display a concise overview of a dp_loading() fit, including its data dimensions, privacy parameters, and whether a non-private reference is stored.

Summarize a fitted dp_loading() object without recomputing its estimates or consuming additional privacy budget.

Usage

dp_loading(X, privacy, cpp.option = TRUE, non_private = FALSE)

# S3 method for class 'dp_loading'
print(x, ...)

# S3 method for class 'dp_loading'
summary(object, ...)

# S3 method for class 'summary.dp_loading'
print(x, digits = 3, ...)

Arguments

X: A numeric matrix or data frame with observations in rows and variables in columns. Non-numeric data-frame columns are excluded with a warning; numeric values must be finite. At least two observations and one numeric feature are required. Observations must be sufficiently separated to compute normalized pairwise differences.
privacy: A privacy_control() object allocating privacy budget to loading alone. Either specify total eps and delta or an absolute loading budget.
cpp.option: Whether to use the compiled spherical-Kendall implementation (TRUE by default); FALSE uses the R implementation.
non_private: Whether to compute and store ordinary PCA reference results (FALSE by default).
x: A summary.dp_loading object.
...: Additional arguments.
object: A dp_loading object.
digits: Number of significant digits used for printing.

Value

A dp_loading object with the following components:

directions: A p by p matrix of private PC directions.
nonprivate: A p by p matrix of ordinary PC directions, or NULL when non_private = FALSE.
metadata: Sample size, variable count and privacy parameters.

Rows represent variables, and columns represent PC1, ..., PCp. The columns are orthonormal eigenvectors, without eigenvalue or correlation scaling.

Invisibly returns x.

A summary.dp_loading object containing the stored private and nonprivate direction matrices, with sample size and privacy budget metadata. The print method displays up to six variables and four PCs; the full matrices remain stored. The non-private matrix remains NULL if it was not stored.

Invisibly returns x.

Details

The private PC directions are eigenvectors of a spherical Kendall matrix perturbed with symmetric Gaussian noise. Directions are ordered by the noisy eigenvalues; they need not be eigenvectors of the sample covariance matrix. The spherical Kendall construction uses normalized pairwise differences and is invariant to common translations.

With non_private = TRUE, ordinary covariance-based directions are stored after sign alignment with the private directions. This comparison is not a private release.

Under fixed-size replacement adjacency, the matrix has Frobenius sensitivity at most 4 / n. The diagonal Gaussian noise standard deviation is 4 * sqrt(2 * log(1.25 / delta)) / (n * eps); off-diagonal entries use this scale divided by sqrt(2). This classical calibration has a standard sufficient guarantee for 0 < eps < 1; larger eps requires an additional calibration check, which is not performed by the function.

Duplicate or numerically indistinguishable observations are rejected, but these checks do not privatize input validation. Data schema and sample size are assumed public. Data-dependent preprocessing requires separate privacy analysis and accounting.

Examples

set.seed(123)
X <- matrix(rnorm(150), nrow = 50, ncol = 3)
fit <- dp_loading(
  X, privacy = privacy_control(eps = 0.5, delta = 1e-3),
  cpp.option = FALSE
)
print(fit)
#> Differentially Private PC Directions
#> Data: 50 observations x 3 variables
#> Components: 3 PC directions
#> Total privacy budget: eps = 0.5 ; delta = 0.001 
#> Non-private reference: not stored 
summary(fit)
#> DP PC Directions Summary
#> Data: 50 observations x 3 variables
#> Components: 3 PC directions
#> 
#> Private PC directions  (first 3 variables x first 3 PCs):
#>       PC1    PC2    PC3
#> X1  0.701 -0.409 -0.584
#> X2 -0.703 -0.259 -0.663
#> X3  0.120  0.875 -0.469
#> 
#> Total privacy budget: eps = 0.5 ; delta = 0.001 
#> Non-private reference: not stored 
#> The full direction matrix is available in $directions.
fit$directions
#>           PC1        PC2        PC3
#> X1  0.7012424 -0.4092929 -0.5837280
#> X2 -0.7027776 -0.2592186 -0.6625024
#> X3  0.1198444  0.8748057 -0.4694170
fit_with_reference <- dp_loading(
  X, privacy = privacy_control(loading = c(eps = 0.5, delta = 1e-3)),
  cpp.option = FALSE, non_private = TRUE
)
fit_with_reference$nonprivate
#>           PC1          PC2       PC3
#> X1  0.1793378  0.977818990 0.1082034
#> X2 -0.5006692 -0.003963068 0.8656297
#> X3  0.8468579 -0.209414222 0.4888531
if (FALSE) { # \dontrun{
loading_plot <- plot(fit, type = c("vector", "point", "heatmap"),
                     heatmap_components = 1:3, variables = "all")
loading_plot                 # Display the saved plot
loading_plot$loading$directions # Access the stored directions
loading_plot$plot$all       # Access the combined plot
} # }