Recall that in class we talked about Gaussian mechanism which perturbs the scalar-valued query results using random noise attributed to Gaussian distribution N(0,2). Recently, Chanyaswad et al. proposed a counterpart of the Gaussian mechanism (called the matrix-valued Gaussian (MVG) mechanism) when the query from a database is matrix-valued. For example, the covariance matrix of a database is matrix-valued. Given a matrix-valued query function f(D)RMN, the MVG mechanism is defined as MVG(f(D))=f(D)+N, where N is sampled from the matrix-valued Gaussian distribution, i.e., NNM,N(0,,), and the corresponding probability density function is fX=(2)MN/2det(M/2det()N/2exp(21Tr[1XT1X]), where PDDMM and PDNN are the row-wise and column-wise full rank covariance matrix, respectively. The sufficient condition for the MVG mechanism to achieve differential privacy on matrixvalued query result is given as follows. Let (1)=[1(1),2(1),,M(1)]T and (1)=[1(1),2(1),,N(1)]T be the vectors of singular values of 1 and 1, respectively. The MVG mechanism guarantees (,) differential privacy if the following inequality holds, (1)2(1)2(+2+8)2/42. where ,,() are constants depending on the size of the query result f(D)RMN. Please draw analogy with the Gaussian mechanism and discuss how the privacy and utility of the release results, i.e., MVG (f(D)), will vary if the privacy budget decreases. Recall that in class we talked about Gaussian mechanism which perturbs the scalar-valued query results using random noise attributed to Gaussian distribution N(0,2). Recently, Chanyaswad et al. proposed a counterpart of the Gaussian mechanism (called the matrix-valued Gaussian (MVG) mechanism) when the query from a database is matrix-valued. For example, the covariance matrix of a database is matrix-valued. Given a matrix-valued query function f(D)RMN, the MVG mechanism is defined as MVG(f(D))=f(D)+N, where N is sampled from the matrix-valued Gaussian distribution, i.e., NNM,N(0,,), and the corresponding probability density function is fX=(2)MN/2det(M/2det()N/2exp(21Tr[1XT1X]), where PDDMM and PDNN are the row-wise and column-wise full rank covariance matrix, respectively. The sufficient condition for the MVG mechanism to achieve differential privacy on matrixvalued query result is given as follows. Let (1)=[1(1),2(1),,M(1)]T and (1)=[1(1),2(1),,N(1)]T be the vectors of singular values of 1 and 1, respectively. The MVG mechanism guarantees (,) differential privacy if the following inequality holds, (1)2(1)2(+2+8)2/42. where ,,() are constants depending on the size of the query result f(D)RMN. Please draw analogy with the Gaussian mechanism and discuss how the privacy and utility of the release results, i.e., MVG (f(D)), will vary if the privacy budget decreases