Assuming $\{A,Y,X\}$ are matrices, and $\{f(.),g(.),h(.)\}$ are functions with scalar outputs, and $\vec{x}_i$ is the $i$th column of $X$ and $x_{ti}$ denotes the $t$th row from the $i$th column of $X$.
I want to say that the function $f(Y)$ has its minimum if for each column $\vec{y}_i$ in $Y$ the corresponding $\vec{x}_i$ (based on the relationship $g(\vec{y}_i)\approx h(A,\vec{x}_i)$) has its non-zero entries $x_{ti}$ such that the related columns $t$ in $A$ have their non-zero entires $a_{st}$ with indices $s$ such that both of these two conditions hold:
- $\vec{h}_i=\vec{h}_s$
- $\|y_i-y_s\|_2^2$ is sufficiently small
I want to say the above in a clear, condensed mathematical way. I tried the following, but I'm not sure how good it is!
The function $f(Y)$ has its minimum if $\forall \vec{y}_i$, $g(\vec{y}_i)\approx h(A,\vec{x}_i)$ such that $\forall \vec{a}_t$, where $\vec{x}_{ti} \neq 0$ and $\forall s$, where $a_{st}\neq 0$, $\vec{h}_i=\vec{h}_s$ and $\|y_i-y_s\|_2^2\leq \epsilon$ for a sufficiently small $\epsilon >0$.