The paper Greedy and Gumble attack defines the input space for the text clasifier in the following form, "We assume a model in the form of a conditional distribution, $p_m = (Y|X), X= x \in W^d$ where $W := ${$w_0,w_1,...w_m$} is a discrete space such as a dictionary of words or space of characters." This is the common way in which input to word based as well as character based CNNs are defined Now, in case of word based CNNs , I can think of the input to be having $d$ words each along the row, with each word ($w_i$) having an embedding of length $m$. Similarly , in case of character based CNNs, all I can think of is $d$ characters, each having embedding length $m$ (26?) .
I am just wondering if my assumptions are correct and I would also want to clarify on the meaning of the symbol $:=$ while defining $W$. Does it simply mean that $W$ is a $m$ length vector?