We propose new differential privacy solutions for when external invariants and integer constraints are simultaneously enforced on the data product. These requirements arise in real world applications of private curation, including public release 2020 U.S. Decennial Census. They pose a great challenge to production provably products with adequate statistical usability. subspace rigorously articu...

Given the matrices A and E in Cn×n, we consider, for the family A(t) = A+ tE, t ∈ C, questions such as i) existence and analyticity of t 7→ R(t, z) = (A(t) − zI)) , and ii) limit as |t| → ∞ of σ(A(t)), the spectrum of A(t). The answer depends on the Jordan structure of 0 ∈ σ(E), more precisely on the existence of trivial Jordan blocks (of size 1). The results of the theory of Homotopic Deviatio...

A concept of virtual subspace is introduced for realizing a robust face recognition independent of the lighting conditions. The virtual subspace is a paradoxical concept because it can be constructed even if only one image is taken. Furthermore, the virtual subspace is gradually converged to the real subspace when face images are subsequently taken. The virtual subspace is defined as an eigensp...