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Private Matrix Approximation and Geometry of Unitary Orbits

6 July 2022
Oren Mangoubi
Yikai Wu
Satyen Kale
Abhradeep Thakurta
Nisheeth K. Vishnoi
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Abstract

Consider the following optimization problem: Given n×nn \times nn×n matrices AAA and Λ\LambdaΛ, maximize ⟨A,UΛU∗⟩\langle A, U\Lambda U^*\rangle⟨A,UΛU∗⟩ where UUU varies over the unitary group U(n)\mathrm{U}(n)U(n). This problem seeks to approximate AAA by a matrix whose spectrum is the same as Λ\LambdaΛ and, by setting Λ\LambdaΛ to be appropriate diagonal matrices, one can recover matrix approximation problems such as PCA and rank-kkk approximation. We study the problem of designing differentially private algorithms for this optimization problem in settings where the matrix AAA is constructed using users' private data. We give efficient and private algorithms that come with upper and lower bounds on the approximation error. Our results unify and improve upon several prior works on private matrix approximation problems. They rely on extensions of packing/covering number bounds for Grassmannians to unitary orbits which should be of independent interest.

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