Deriving Private Information from Randomized Data

1
Deriving Private Information from Randomized Data

• Zhengli Huang, Wenliang Du and Biao Chen
• SIGMOD2005

2
Outline

• Motivation
• Reconstruction Algorithms
• Improve Random Perturbation
• Conclusion

3
Privacy-Preserving Data Mining
Classification Association Rules Clustering
Data Mining
Central Database
Data Collection
Data Disguising
4
Random Perturbation
Original Data X
Random Noise R
Disguised Data Y

5
Problem

• How secure is Randomization Perturbation?

6
Example

• This is the data set (x, x, x, x, x, , x)
• Random Perturbation
• (xr1, xr2,, xrm)
• rt is random number with mean 0
• So the perturbation numbers mean converges to x,
  when m becomes large.
• We know this is NOT safe.
• Observation the data set is highly correlated.

7
Summary

• We assume that the relationships among data
  attributes might be the key factor that decides
  how much privacy can be preserved.
• We introduce two methods that exploit the
  correlations among data to reconstruct the
  original data from a randomized data set.

8
Data Reconstruction (DR)
Distribution of random noise
Reconstructed Data X
Data Reconstruction
Whats their difference?
Disguised Data Y
Original Data X
9
Reconstruction Algorithms

• Principal Component Analysis (PCA)
• Bayes Estimate Method (BE)

10
PCA-Based Reconstruction
Disguised Information
Reconstructed Information
compression
Information Loss
11
PCA Introduction

• The main use of PCA reduce the dimensionality of
  a data set with interrelated variables, but still
  contain as much variance of the data set as
  possible
• PCA can transform the data set to a new data set
  with variables, which are uncorrelated
  

12
PCA Introduction

• First PC containing the greatest amount of
  variation.
• Second PC containing the next largest amount of
  variation.

13
For the Original Data

• They are correlated.
• If we remove 50 of the dimensions, the actual
  information loss might be less than 10.
• If the rest m-p non-PCs remove, it do not cause
  much information loss.

14
For the Random Noises

• They are not correlated.
• Their variance is evenly distributed to any
  direction.
• The information loss for the noise increases.

15
Summary

• If the data is highly correlated
• Then more dimensions can be reduced without
  causing too much information loss for the
  original data.

16
Bayes-Estimate-Based Reconstruction
Possible X
Possible X
Possible X
What is the Most likely X?
Random Noise
Disguised Data Y
17
The Problem Formulation

• For each possible X, there is a probability P(X
  Y).
• Find an X, s.t., P(X Y) is maximized.
• How to compute P(X Y)?

18
Computing P(X Y)?

• P(XY) P(YX) P(X) / P(Y)
• P(YX) remember Y X R
• P(Y) A constant (we dont care)
• How to get P(X)?
• This is where the correlation can be used.
• Assume Multivariate Gaussian Distribution
• The parameters are unknown.

19
Multivariate Gaussian Distribution

• A Multivariate Gaussian distribution
• Each variable is a Gaussian distribution with
  mean ?i
• Mean vector ? (?1 ,, ?m)
• Covariance matrix ?
• Both ? and ? can be estimated from Y
• So we can get P(X)

20
Improved Randomization Scheme

• From the analysis of PCA
• While most of the information of the original
  data concentrates on the PC, we discard those
  non-PCs, we can remove more noises than what we
  do to the original data.
• However, if random noises also concentrate on the
  PCs, separating original data from random noises
  becomes difficult.

21
Observation from PCA

• How to make it difficult to squeeze out noise?
• Noise now concentrates on the principal
  components.
• Make the correlation of the noise similar to the
  original data.

22
Conclusion

• Can other information affect privacy?