Suppose $X_1, \dots, X_n \sim^{\text{iid}} \mathcal{N}(\theta, \sigma^2 I_p)$ and $\sigma^2$ is known. Define the risk as follows $$ \mathcal{R}(\theta_1, \theta_2) = \mathbb{E} \|\theta_1 - \theta_2\|_2^2. $$ Then, it can be shown that $$ \sup_{\overline{\theta} \in \Theta} \mathcal{R}(\overline...

Suppose $$ A = \left( \begin{array}{cc} 1 & 4 \\ 5 & 6 \end{array}\right) $$ How do I calculate $\|A\|_{\text{OP}}$? I know the definition of operator norm, but I am clueless on how to calculate it for real example like this. Can somebody please give me a step-by-step instruction on how to d...

I was wondering if the spectral norm is a Lipschitz function with respect to the spectral norm. How can we prove whether it is or not? In other words, is $$\big| \|X\| - \|Y\| \big| \le L \|X-Y\|$$ for some $L$?

Given symmetric matrices $A_0, A_1, \dots, A_n \in \mathbb R^{m \times m}$, let $A(x) := A_0 + x_1 A_1 +\cdots + x_n A_n$. How to formulate the following unconstrained spectral minimization problem as a semidefinite program? $$\min_{x \in \mathbb R^n} \|A(x)\|_2$$ Can anyone please help on this...

I wonder if the $2$-norm or spectral norm is also submultiplicative for non-square matrices, i.e., $$\| A B \|_2 \leq \| A \|_2 \cdot \| B \|_2$$ if the number of columns of $A$ coincides with the number of rows of $B$. In the literature I can only find a statement about square matrices. Thank...

I was reading the use of semidefinite programs to formulate the matrix norm minimization but am having trouble trying to understand it. I'd also like to understand it at a more intuitive level. [Boyd and Vandenberghe: Convex optimization $\S$ 4.6.3] Matrix norm minimization Let $A(x...

This may be a trivial question yet I was unable to find an answer: $$\left \| A \right \| _2=\sqrt{\lambda_{\text{max}}(A^{^*}A)}=\sigma_{\text{max}}(A)$$ where the spectral norm $\left \| A \right \| _2$ of a complex matrix $A$ is defined as $$\text{max} \left\{ \|Ax\|_2 : \|x\| = 1 \right\}$$...

Is there an intuitive meaning for the spectral norm of a matrix? Why would an algorithm calculate the relative recovery in spectral norm between two images (i.e. one before the algorithm and the other after)? Thanks