---
title: "Differentially Private PC Directions"
description: >
  Private principal component directions based on the spherical Kendall matrix,
  Gaussian perturbation, and loading plot visualization in dppca.
output:
  rmarkdown::html_vignette:
    toc: true
    toc_depth: 2
vignette: >
  %\VignetteIndexEntry{Differentially Private PC Directions}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r loading-theory-setup, include=FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>", fig.align = "center")
```

In the `dppca` package, a first step in DP-PCA is to estimate the PC directions privately. For this purpose, [Kim and Jung (2025)](#ref-Kim2025) proposed **g-DPPCA**, which applies a matrix Gaussian mechanism to the generalized multivariate Kendall's tau matrix based on the generalized spatial-sign transformations introduced by [Raymaekers and Rousseeuw (2019)](#ref-Raymaekers2019). The `dppca` package implements g-DPPCA using the spherical transformation. The resulting DP PC directions are used in subsequent PCA visualizations and can be directly visualized through loading plots.

For the definitions of ordinary PCA, see [PCA Background](method-pca-background.html). For R examples, arguments, and plot customization, see the [Tutorial of DP Loading Plot](tutorials-loading-plots.html). 

## 1. Spherical Kendall matrix and PC directions

Let $X\in\mathbb R^{n\times p}$ contain observations
$x_1,\ldots,x_n\in\mathbb R^p$ in its rows. As described in the [PCA Background](method-pca-background.html), ordinary sample PC directions are the eigenvectors of the sample
covariance matrix
\[
\widehat\Sigma
=\frac{1}{n-1}\sum_{i=1}^n
(x_i-\bar x)(x_i-\bar x)^\top,
\qquad \bar x=\frac1n\sum_{i=1}^n x_i.
\]

Equivalently, the sample covariance can be expressed using pairwise differences:

\[
\widehat\Sigma
=\frac{1}{n(n-1)}\sum_{i<j}
(x_i-x_j)(x_i-x_j)^\top.
\]

The **sample spherical Kendall matrix** similarly uses pairwise differences, but normalizes each difference to unit length:
\[
\displaystyle
\widehat K_{\mathrm{sph}}
=\frac{2}{n(n-1)}\sum_{i<j}
\frac{(x_i-x_j)(x_i-x_j)^\top}
{\|x_i-x_j\|_2^2}.
\]

This expression is defined when the pairwise differences are nonzero. For numerical stability, the implementation requires the stronger condition $\|x_i-x_j\|_2>\sqrt{\texttt{.Machine\$double.eps}}$ for every pair, and rejects nonfinite norms. Thus, distinct observations can still be rejected when their separation is too small in the chosen measurement units.

The matrices $\widehat K_{\mathrm{sph}}$ and $\widehat\Sigma$ generally have different sample eigenvectors. However, under suitable elliptical distributional assumptions, their population counterparts share principal eigendirections. Spherical Kendall PCA therefore provides an alternative
way to estimate PC directions, rather than an exact reconstruction of the sample covariance eigenvectors.

More generally, a generalized spatial sign can be written as

\[
g_h(z)=h(\|z\|_2)\frac{z}{\|z\|_2},
\qquad z\ne0,
\]

where \(h\) is a radial weight function. For a bounded transformation \(g_h\), the corresponding **generalized multivariate Kendall's tau matrix** is estimated by

\[
\widehat K_{g_h}
=\frac{2}{n(n-1)}\sum_{i<j}
g_h\!\left(\frac{x_i-x_j}{\sqrt2}\right)
g_h\!\left(\frac{x_i-x_j}{\sqrt2}\right)^\top.
\]

The spherical case corresponds to \(h(r)=1\), giving \(g_h(z)=z/\|z\|_2\) for \(z\ne0\). The `dp_loading()` function implements this spherical case.

### Translation invariance

The spherical Kendall matrix is invariant to a common translation of the observations. For any $c\in\mathbb R^p$,
\[
(x_i-c)-(x_j-c)=x_i-x_j,
\qquad
\widehat K_{\mathrm{sph}}(X-\mathbf1_n c^\top)
=\widehat K_{\mathrm{sph}}(X).
\]

Consequently, estimating spherical Kendall directions does not require computing or releasing the sample mean. 


## 2. DP PC directions {#dp-private-directions}

We consider fixed-size replacement adjacency, in which two neighboring datasets have the same number of observations and differ in one row. 
Since changing one observation affects at most $n-1$ terms in the pairwise sum, the Frobenius sensitivity of the spherical Kendall matrix satisfies
\[
\Delta_F
=\sup_{X\sim X'}
\bigl\|\widehat K_{\mathrm{sph}}(X)
-\widehat K_{\mathrm{sph}}(X')\bigr\|_F
\le\frac4n.
\]

The mechanism adds symmetric Gaussian noise to the spherical Kendall matrix. 
For privacy parameters $\varepsilon_L>0$ and $0<\delta_L<1$, define
\[
\sigma_K
=\frac{4\sqrt{2\log(1.25/\delta_L)}}{n\varepsilon_L}.
\]

This classical Gaussian calibration has a standard sufficient privacy guarantee for $0<\varepsilon_L<1$. The implementation also accepts larger positive values, but a separate calibration check is required to establish their privacy guarantee; the function does not perform that check.

A symmetric noise matrix $E\in\mathbb R^{p\times p}$ is generated with independent entries on and below the diagonal, such that

\[
E_{jj}\sim N(0,\sigma_K^2),
\qquad
E_{j\ell}\sim N(0,\sigma_K^2/2)\quad(j<\ell),
\qquad E_{\ell j}=E_{j\ell}.
\]

The perturbed matrix is $\widetilde{K}=\widehat{K}_{\mathrm{sph}}+E$.
The **private PC directions** are obtained from the eigen-decomposition

\[
\widetilde{K}
=\widetilde{V} \widetilde{\Theta}\widetilde{V}^\top,
\qquad
\widetilde{V}=[\widetilde{v}_1,\ldots,\widetilde{v}_p],
\]

where the columns of $\widetilde{V}$ are orthonormal eigenvectors ordered by decreasing eigenvalues of $\widetilde{K}$.


Importantly, the eigenvalues of $\widehat{K}$ are not same as the covariance eigenvalues $\widehat\lambda_\ell$ defined in the non-private ordinary PCA.
The [DP Scree](method-dp-scree.html) procedure estimates variance along private PC directions separately.


## 3. DP loading plots in `dppca` {#dp-loading-plots}

A **DP loading plot** displays the coefficients of the private PC direction
vectors. Let $\widetilde{v}_{j\ell}$ denote the coefficient of variable $j$
in private PC direction $\ell$. For two selected components $a$ and $b$,
with distinct $a,b\in\{1,\ldots,p\}$, variable $j$ is represented by

\[
L_j^{\mathrm{DP}}
=(\widetilde{v}_{ja},\widetilde{v}_{jb}).
\]

Large absolute coefficients indicate that the corresponding variables have larger weights in a selected PC direction. 
The signs distinguish opposing contributions along an axis, but eigenvector signs are arbitrary:
replacing $\widetilde{v}_\ell$ with $-\widetilde{v}_\ell$ does not change the underlying direction. 
For comparison, the non-private ordinary PCA loading coordinates are $L_j^{\mathrm{NP}}=(\widehat v_{ja},\widehat v_{jb})$ where $\widehat V$ consists of eigenvectors of $\widehat\Sigma$.


The package supports **vector**, **point**, and **heatmap** displays of loading coefficients. For a selected pair of PCs, vector and point plots encode the same coordinates $L_j^{\mathrm{DP}}$. A **vector plot** draws an arrow from $(0,0)$ to $L_j^{\mathrm{DP}}$; a **point plot** places a point at that coordinate. A **heatmap** displays the signed entries $\widetilde{v}_{j\ell}$ for selected variables and PCs.

For examples and plotting options, see the [DP Loading Plot Tutorial](tutorials-loading-plots.html).


## References

- <span id="ref-Kim2025"></span>Kim, M. and Jung, S. (2025). *Robust and differentially private principal component analysis*. Statistical Analysis and Data Mining, 18(6), e70053. https://doi.org/10.1002/sam.70053

- <span id="ref-Raymaekers2019"></span>Raymaekers, J. and Rousseeuw, P. J. (2019). *A generalized spatial sign covariance matrix*. Journal of Multivariate Analysis, 171, 94--111. https://doi.org/10.1016/j.jmva.2018.11.010
- Dwork, C. and Roth, A. (2014). *The Algorithmic Foundations of Differential Privacy*. Foundations and Trends in Theoretical Computer Science, 9(3--4), 211--407. https://doi.org/10.1561/0400000042