Solving classic unsupervised learning problems with deep neural networks

Unsupervised learning methods summarize data or transform it such that some desirable properties are enforced. These properties are often easily achieved analytically but are harder to enforce when working in a stochastic optimization (e.g. Before a model is created or a method is defined, some groundwork needs to be laid. What assumptions do we make about the data or the model? How do we know that the model we end up with is good and what do we exactly mean by good?