Skip to main content

Table 1 Notations

From: Local differential privacy for unbalanced multivariate nominal attributes

A

Multiple unbalanced categorical data sets

l

Number of attributes

n

Number of participants

\(k_i\)

Number of items of the i-th attribute

d

Total number of items, \(d = \sum _ik_i\)

\({\mathbf {a}}_j\)

j-th attributes of A, the length \(|{\mathbf {a}}_j|\) of which is \(k_j\)

\({\mathbf {v}}_i\)

Private values possessed by the i-th user, the length \(|{\mathbf {v}}_i|\) of which is l

\(v_{ij}\)

j-th value of \({\mathbf {v}}_i\)

\({\mathbf {h}}_i\)

Private bit vector of i-th users, the length of which is d

H

True histogram, \({\mathbf {H}} = \sum {\{{\mathbf {h}}_1, \ldots , {\mathbf {h}}_n\}}\)

\(H'\)

Sanitized histogram of \({\mathbf {H}}\), \({\mathbf {H}}' = \sum {\{{\mathbf {h}}_1', \ldots , {\mathbf {h}}_n'\}}\)

\(H''\)

Estimated histogram of \({\mathbf {H}}'\), \({\mathbf {H}}'' = \sum {\{{\mathbf {h}}_1'', \ldots , {\mathbf {h}}_n''\}}\)

\(\epsilon _i\)

Privacy budget of the i-th attribute

CF

Cardano formula

NS

Newton-Raphson method

SE

Square error

NSE

Normalized square error, \(NSE = \frac{SE}{n}\)