vector semantics
Last edited: August 8, 2025vector semantics is a sense encoding method.
“a meaning of the word should be tied to how they are used”
we measure similarity between word vectors with cosine similarity. see also vector-space model.
motivation
idea 1
neighboring words can help infer semantic meaning of new words: “we can define a word based on its distribution in language use”
idea 2
meaning should be in a point in space, just like affective meaning (i.e. a score in each dimension).
vector space
Last edited: August 8, 2025A vector space is an object between a field and a group; it has two ops—addition and scalar multiplication. Its not quite a field and its more than a group.
constituents
- A set \(V\)
- An addition on \(V\)
- An scalar multiplication on \(V\)
such that…
requirements
- commutativity in add.: \(u+v=v+u\)
- associativity in add. and mult.: \((u+v)+w=u+(v+w)\); \((ab)v=a(bv)\): \(\forall u,v,w \in V\) and \(a,b \in \mathbb{F}\)
- distributivity: goes both ways \(a(u+v) = au+av\) AND!! \((a+b)v=av+bv\): \(\forall a,b \in \mathbb{F}\) and \(u,v \in V\)
- additive identity: \(\exists 0 \in V: v+0=v \forall v \in V\)
- additive inverse: \(\forall v \in V, \exists w \in V: v+w=0\)
- multiplicative identity: \(1v=v \forall v \in V\)
additional information
- Elements of a vector space are called vectors or points.
vector space “over” fields
Scalar multiplication is not in the set \(V\); instead, “scalars” \(\lambda\) come from this magic faraway land called \(\mathbb{F}\). The choice of \(\mathbb{F}\) for each vector space makes it different; so, when precision is needed, we can say that a vector space is “over” some \(\mathbb{F}\) which contributes its scalars.
