# Construction of Differentially Private Empirical Distributions from a low-order Marginals Set through Solving Linear Equations with l2 Regularization

## TL;DR

This paper presents CIPHER, a simple and efficient algorithm for constructing differentially private empirical distributions from low-order marginals by solving regularized linear equations, outperforming existing methods in utility and computational efficiency.

## Contribution

CIPHER introduces a novel linear equation approach with l2 regularization for differentially private distribution construction, reducing computational requirements compared to full-dimensional histograms.

## Key findings

- CIPHER outperforms the exponential mechanism in information preservation.
- CIPHER achieves similar or better utility than FDH sanitization at the same privacy level.
- CIPHER requires significantly less computational storage and memory.

## Abstract

We introduce a new algorithm, Construction of dIfferentially Private Empirical Distributions from a low-order marginals set tHrough solving linear Equations with l2 Regularization (CIPHER), that produces differentially private empirical joint distributions from a set of low-order marginals. CIPHER is conceptually simple and requires no more than decomposing joint probabilities via basic probability rules to construct a linear equation set and subsequently solving the equations. Compared to the full-dimensional histogram (FDH) sanitization, CIPHER has drastic\-ally lower requirements on computational storage and memory, which is practically attractive especially considering that the high-order signals preserved by the FDH sanitization are likely just sample randomness and rarely of interest. Our experiments demonstrate that CIPHER outperforms the multiplicative weighting exponential mechanism in preserving original information and has similar or superior cost-normalized utility to FDH sanitization at the same privacy budget.