functional encryption over homomorphic encryption
The main difference between FE and HE is that:
In HE, for the decryption you need to interact with the owner of the secret-key, but in FE everybody who has access to the ciphertext and functional key can decrypt the massage.
While HE can efficiently support different kinds of computation over encrypted data, FE supports limited class of functionality (in theory it still can support general functionality).
This limited class includes: inner-product (which might be what you called weighted sum) and quadratic functionality.
Recently there has been a European Project called FENTEC, they are trying to implement the existing FE schemes.