We have values in the range $[a, b]$, and we apply a differential privacy mechanism to add noise to these values. After obtaining the noisy values, we employ the following truncation function: if a noisy value falls below $a$, it is truncated to $a$, and if the noisy value exceeds $b$, it is truncated to $b$. Does this truncation function ensure differential privacy, assuming that the mechanism itself provides differential privacy?
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It depends on what you mean, namely
- does it maintain the level of differential privacy one gets from adding noise, or
- does it enhance the level of differential privacy one gets.
The answer to the first question is "trivially yes". This is because differential privacy satisfies a "data-processing inequality", in the sense that if a mechanism $M(x)$ satisfies some notion of differential privacy, then $f(M(x))$ does as well, for any function $f$.
There are certain situations where one can show that natural $f$ enhance the level of differential privacy. This is generally called "privacy amplification". I don't know if clamping does, but don't see why it would. But various other types of post-processing (for example sub-sampling) do enhance the level of differential privacy one obtains.
