  We provide descriptions of the derived categories of degree $d$ hypersurface
fibrations which generalize a result of Kuznetsov for quadric fibrations and
give a relative version of a well-known theorem of Orlov. Using a local
generator and Morita theory, we re-interpret the resulting matrix factorization
category as a derived-equivalent sheaf of dg-algebras on the base. Then,
applying homological perturbation methods, we obtain a sheaf of
$A_\infty$-algebras which gives a new description of homological projective
duals for (relative) $d$-Veronese embeddings, recovering the sheaf of Clifford
algebras obtained by Kuznetsov in the case when $d=2$.
