# Equivariance A function $f$ is **equivariant** with respect to $T$ if $f(T(x)) = T(f(x))$ . In other words, it must *respect* a certain geometrical principle. An example of an equivariant function can be seen below. Consider the function $f$, a transformation (translation) $S$ that is our geometrical principle, and an input image of a cat, $x$ (in the upper left square below). We see that $f$ converts the cat to black and white. We see that *no matter the order* we apply $S$ and $f$, we still arrive as the same final output in the lower right hand corner. In other words, $f$ *respects* translation.  ### Equivariance vs Invariance We can summarize the distinction as follows: >In essence, **invariance** is about *ignoring changes* (the output remains constant despite changes in the input), while equivariance is **about** *mirroring changes* (the output changes in a way that reflects changes in the input) I should note that they are each applied to *functions*. --- Date: 20230204 Links to: [Invariant](Invariant.md) Tags: References: * []()