Supposing $ {A, Y, X } $ are matrices, and $ {f (.), g (.), h (.) } $ are functions with scalar outputs, and $ vec {x} _i $ is the $ i $e column of $ X $ and $ x_ {ti} $ denotes the $ t $row of $ i $e column of $ X $.

I mean that the function $ f (Y) $ at its minimum if for each column $ vec {y} _i $ in $ Y $ the correspondent $ vec {x} _i $ (based on the relationship $ g ( vec {y} _i) about h (A, vec {x} _i) $) has non-zero entries $ x_ {ti} $ such as linked columns $ t $ in $ A $ have their non-zero entries $ a_ {st} $

with clues $ s $ so that these two conditions are met:

- $ vec {h} _i = vec {h} _s $
- $ | y_i-y_s | _2 ^ 2 is small enough

I mean the above in a clear and condensed mathematical way. I've tried the following, but I'm not sure of its quality!

Function $ f (Y) $ at its minimum if

$ forall vec {y} _i $, $ g ( vec {y} _i) about h (A, vec {x} _i) $

such as

$ forall vec {a} _t $, or $ vec {x} _ {ti} neq $ 0 and $ forall s $, or $ 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.