Consider the following optimization problem \begin{equation*} \min_{0\leq \sigma \leq 1} \lVert B(\sigma)\,B^{+}(\sigma)\,p -p \rVert^2 \end{equation*} where
- $\sigma\in \mathbb{R}^m$ is the decision variable, and the constraint $0\leq \sigma \leq 1$ is intended component-wise, i.e. if $\sigma\triangleq [s_1\,\,\cdots \,\,s_m]'$ (with $'$ being the transpose operator) \begin{equation*} 0\leq \sigma \leq 1 \qquad \equiv \qquad \begin{gathered} 0\leq s_1 \leq 1 \\ \vdots\\ 0\leq s_m \leq 1 \\ \end{gathered} \end{equation*}
- $B(\sigma):\mathbb{R}^m \mapsto \mathbb{R}^{m\times n}$ is a given matrix that depends non-linearly on $\sigma$, while $B^{+}(\sigma)$ is its (left) pseudo-inverse, i.e. \begin{equation*} B^{+}(\sigma)\triangleq (B'(\sigma)B(\sigma))^{-1}\,B'(\sigma) \end{equation*}
- $p\in \mathbb{R}^{m}$ is a given vector (containing the observed data)
- the norm considered is the Euclidean one
The current problem can be equivalently written as \begin{equation}\tag{1} \min_{0\leq \sigma \leq 1} \lVert B(\sigma)\,B^{+}(\sigma)\,p -p \rVert^2 = \min_{0\leq \sigma \leq 1} \lVert (B(\sigma)\,B^{+}(\sigma)-I_m)\,p \rVert^2 \end{equation} where $I_m$ is the $m\times m$ identity matrix. Now, it seems that the problem is to find the $\sigma$ that makes the product $B(\sigma)\,B^{+}(\sigma)$ as close as possible to $I_m$. Thus, I suspect that we can equivalently write the current minimization problem as \begin{equation}\tag{2} \min_{0\leq \sigma \leq 1} \lVert B'(\sigma)\,B^{+}(\sigma)-I_m\rVert_{?}^2 \end{equation} where with the symbol $\lVert \cdot \rVert_{?}^2$ I mean that probably there is a suitable (matrix) norm that makes $(2)$ and $(1)$ equivalent. If true, this would be super nice because it makes the original problem independent from the data $p$, so that it can be solved once for all depending on the definition of $B(\sigma)$.
Question
Consider the following relaxation of my initial problem \begin{equation*} \min_{0\leq \sigma \leq 1} \lVert B(\sigma)\,B^{+}(\sigma)\,p -p \rVert_a^2 \end{equation*} where the norm considered, denoted as "$a$", now is a degree of freedom. Is it possible to find a couple of norms "$a,b$" such that \begin{equation*} \min_{0\leq \sigma \leq 1} \lVert B(\sigma)\,B^{+}(\sigma)\,p -p \rVert_a^2= \min_{0\leq \sigma \leq 1} \lVert B(\sigma)\,B^{+}(\sigma)-I_m \rVert_b^2 \end{equation*} If yes, is it possible to find a norm $b$ when $a$ is the Euclidean norm?