Finding the dictionary to get latent factors¶
Definition¶
In general, we create probabilistic assumptions on these latent variables and their connection to the inputs. Specify \(p(\mathbf{h})\) and \(p(\mathbf{x}|\mathbf{h, D})\) for some unknown \(\mathbf{D}\) with \(p(\mathbf{x}| \mathbf{D}) = \int p(\mathbf{x}|\mathbf{h, D})p(\mathbf{h})d\mathbf{h}\) Probabilistic Principal Component Analysis (PPCA)
Then use maximum likelihood estimate.
However, if we look at the simplest linear case where the goal is to find \(\mathbf{x}_{i} \approx \mathbf{h}_{i}\mathbf{D}\) for input data \(\mathbf{x}_{i}\), we use Matrix Factorization Approaches