In my defense, I will talk about three relative topics relative to sparse recovery and representation of signals. I will start with the topic of recovering the low-rank matrix from incomplete measurements with prior information. Signal recovery assumes that we know the sensing matrix i.e. the linear transformation. But sometimes, we want the sparse representation of signals without knowing the transformation between the signal and its representation. Thus in the second topic, I'll talk about a novel algorithm that allows us to learn the linear transformation as well as the sparse representation and admits the transformation in complexity $O(n\log n)$ for a $n$ dimensional input signal. In the third topic, I'll introduce the usage of representation learning assuming that the transformation is a more complex function. I'll propose a deep neural network structure that can be used in image generation and introduce a specific application about it raised in computer game industry.