Let $A$ and $B$ be $m \times n$ matrices with low-rank structures: $$ A = U_{A}\Sigma_{A}V_{A}^{T},\quad B= U_{B}\Sigma_{B}V_{B}^{T}, $$
Prove that Hadamard product $A\circ B$ admits the following representation $$ A\circ B = (U_{A}^T\odot U_{B}^T)^T (\Sigma_{A}\otimes\Sigma_{B})(V_{A}^{T}\odot V_{B}^{T}), $$ where $\odot$ represents the Khatri-Rao product, and $\otimes$ the Kronecker product.
We will use the following properties and definitions: \begin{align} (A\odot^T B)(C\odot D) = (AC)\circ(BD),\label{eq:khr-had}\\ (A\otimes B)(C\odot D) = (AC)\odot(BD),\label{eq:khr-kro} \end{align} It is easy to prove that Hadamard product of A and B admits the following representation: \begin{align} A \circ B &= (U_{A}\Sigma_{A}V_{A}^{T})\circ(U_{B}\Sigma_{B}V_{B}^{T})\nonumber\\ &= (U_{A}^T\odot U_{B}^T)^T (\Sigma_{A}V_{A}^{T}\odot \Sigma_{B}V_{B}^{T})\nonumber\\ &=(U_{A}^T\odot U_{B}^T)^T (\Sigma_{A}\otimes\Sigma_{B})(V_{A}^{T}\odot V_{B}^{T})\label{eq:repres} \end{align} where $\odot$ represents the Khatri-Rao product, and $\otimes$ the Kronecker product.