NMF

Non-negative matrix factorization (NMF), also non-negative matrix approximation[1][2] is a group of algorithms in multivariate analysis and linear algebra where a matrix V is factorized into two matrices W and H, with the property that all three matrices have no negative elements. This non-negativity makes the resulting matrices easier to inspect. The following figure[3] shows the main idea of the NMF:

NMF generates two factors (W, H) so that each image X(:, j) is approximated using a linear combination of the columns of W. Since W is nonnegative, the columns of W can be interpreted as images (that is, vectors of pixel intensities) which we refer to as the basis images. As the weights in the linear combinations are nonnegative (H ≥ 0), these basis images can only be summed up to reconstruct each original image. Moreover, the large number of images in the data set must be reconstructed approximately with only a few basis images, hence the latter should be localized features (hence sparse) found simultaneously in several images. Besides, NMF is more robust to occlusion than PCA (which generates dense factors)[4].

Reference:

1. Inderjit S. Dhillon; Suvrit Sra (2005). Generalized Nonnegative Matrix Approximations with Bregman Divergences. NIPS.
2. Tandon, Rashish; Suvrit Sra (2010). Sparse nonnegative matrix approximation: new formulations and algorithms. TR.
3. https://en.wikipedia.org/wiki/Non-negative_matrix_factorization
4. Gillis, N. (2014). The why and how of nonnegative matrix factorization. Regularization, Optimization, Kernels, and Support Vector Machines, 12(257).