In the machine learning literature, I often see it said that something is "embedded" in some space. For instance, that something is "embedded" in feature space, or that our data are "embedded" in dot product space, etc. However, I've never actually seen an explanation of what this is supposed to mean. So what does it actually mean to say that something is "embedded" in some space?
Embedding is the process of representing data (from a source domain) in a new (or target) domain. Usually, the source domain is discrete, and the target domain is continuous. For example, embedding words into the continuous vector space can be done by the word2vec method.
The main reason behind using the embedding is doing meaningful mathematical computations in the target domain, which is not possible or straightforward in the source domain. For example, summing two words
"brother" - "man" + "woman" not meaningful in the word and character levels. However, when using word2vec,
embedding("brother") - embedding("man") + embedding("woman") can be meaningful and comparable with other embedded vectors; It should be near the embedded vector of "sister".