  Many data applications have certain invariant constraints due to practical
needs. Data curators who employ differential privacy need to respect such
constraints on the sanitized data product as a primary utility requirement.
Invariants challenge the formulation, implementation, and interpretation of
privacy guarantees.
  We propose subspace differential privacy, to honestly characterize the
dependence of the sanitized output on confidential aspects of the data. We
discuss two design frameworks that convert well-known differentially private
mechanisms, such as the Gaussian and the Laplace mechanisms, to subspace
differentially private ones that respect the invariants specified by the
curator. For linear queries, we discuss the design of near-optimal mechanisms
that minimize the mean squared error. Subspace differentially private
mechanisms rid the need for post-processing due to invariants, preserve
transparency and statistical intelligibility of the output, and can be suitable
for distributed implementation. We showcase the proposed mechanisms on the 2020
Census Disclosure Avoidance demonstration data, and a spatio-temporal dataset
of mobile access point connections on a large university campus.
