In text embeddings, cosine similarity is often used to find texts similar to a search query. However, I don't want to find a text that is overall similar, but similar with regards to a specific concept (which I can also embed).
Example: Let's say you have many movie reviews that you have embedded. You choose one review and want to search for similar ones with regard to the cinematography only.
More formally stated, my problem is the following: Let $x_1, \dots, x_n \in \mathbb{R}^d$ vector embeddings of texts $t_1, \dots, t_n$. Further, let $q \in \mathbb{R}^d$ be the embedding of a search query. I want to rank the texts by similarity with the search query with regards to a specific concept/aspect with embedding $k \in \mathbb{R}^d$.
I thought about using the embedding $k$ of the concept, like "cinematography" and then projecting the text embeddings onto that direction $$S_C \propto \langle P_k q, P_k x_i\rangle$$
but that does not make sense as cosine similarity looks at the angle.
One could project just one of the text embeddings $$S_C \propto \langle P_k q, x_i\rangle = S_c(k, q) \cdot S_c(k, x_i)$$ but this amounts to multiplying the cosine similarities of the concept with both text embeddings.
I did not find any research literature on this specific problem. Is there a way to align the cosine similarity to a specific concept/context?