Affine Superposition

linear-logic

Definition

Affine Superposition

Affine superposition is forming an affine combination of elements (states/inputs) in a space with addition and scalar multiplication:

$$ax+by\text{for scalars }a,b\text{ with }a+b=1$$

(More generally: ${\sum }_{i=1}^n{\alpha }_i{x}_i$ with ${\sum }_{i=1}^n{\alpha }_i=1$.)

A function/system $F$ respects affine superposition (is affine linear) iff, for all $(x,y)$ and scalars $(a,b)$ with $a+b=1$,

$$F(ax+by)=aF(x)+bF(y).$$

Equivalently: There exists a linear map $L$ and a fixed vector $c$ such that

$$\forall x:F(x)=L(x)+c$$

Principle of Affine Superposition

If a problem/operator $F$ is affine linear, then affine combinations of solutions give affine combinations of outputs:

$$F(x_1)=y_1,F(x_2)=y_2,a+b=1\Rightarrow F(a{x}_1+b{x}_2)=a{y}_1+b{y}_2.$$

In particular, if $F(x_1)=F(x_2)=y$ and $a+b=1$, then any affine combination

$$x=a{x}_1+b{x}_2$$

is again a solution of $F(x)=y$.