Analyzing Facial Features through Non-Negative Matrix Factorization from Ground Up in R Programming Language
In the realm of computer vision, Non-negative Matrix Factorization (NMF) has emerged as a popular method for learning the features of objects, particularly in the case of facial images. This technique was first introduced in 1994 by Paatero and Tapper and gained further prominence in 1999 by Daniel D. Lee and H. Sebastian Seung.
NMF operates by factorizing a data matrix V into two components: a basis matrix W and an encoding matrix H. The product of W and H provides a close approximation of the original matrix V. This process is crucial in understanding the underlying structure of the data.
However, NMF is not without its limitations when applied to learning parts from object databases, particularly facial images. One of the main constraints is the additivity and non-negativity requirements. These limitations prevent NMF from effectively modelling complex interactions that are subtractive or involve negative correlations, potentially leading to an incomplete representation of the structure of objects if the underlying data has complex, non-additive relationships.
Another challenge faced by NMF is its sensitivity to initialization and local minima. The optimization process, typically done via iterative updates, can converge to local minima, affecting the interpretability and stability of the learned parts, especially in large or noisy databases. This can result in inconsistent decompositions across runs.
Moreover, standard NMF does not incorporate external or contextual knowledge, relying solely on the data matrix without leveraging side information such as metadata, labels, or known interactions. This limits the richness and accuracy of the learned parts from complex object datasets unless extended with graph regularization or other constraints.
The dimensionality and computational complexity of NMF also pose challenges. While NMF reduces dimensionality, its computational complexity can increase significantly with data size or feature dimension, especially without sparsity or regularization. This can limit scalability to very large object databases.
Despite these limitations, NMF is favoured for its interpretability compared to other factorization methods. However, the parts learned may not always correspond to meaningful or easily human-interpretable components, depending on the dataset and factorization rank chosen.
In the context of facial recognition, it is crucial to acknowledge the ethical ramifications of such technology. A system that can remotely identify or classify people without their knowledge is fundamentally dangerous.
The LFW (Labeled Faces in the Wild) dataset is commonly used for learning facial features in NMF. The dataset contains more than 13,000 facial images collected from the web. Each image is scaled to 150 x 150 pixels and each pixel's value is standardized so that the mean and standard deviation become equal to 0.25.
Each column of the encoding matrix H indicates which feature is present in which image with what intensity. The non-negativity constraints in NMF are compatible with the intuitive notion of combining parts to form a whole. Each image pixel is flattened to become a column vector, and combining all the flattened images as columns results in the data matrix V.
NMF performs both learning and inference simultaneously, learning a set of basis images and inferring values for the hidden variables from the visible variables. The minimization problem in NMF is an NP-Hard problem, meaning it is difficult to find the global minimum. The iteration number is set to 500 by default, but 50 iterations are enough to reach an appreciable convergence.
The gradient descent method is used to slowly approach the minimum of the objective function in NMF. The underlying NMF objective is to minimize the distance between V and WH with respect to W and H while preserving the non-negativity of W and H. The W and H matrix in NMF is initialized by values from a uniform distribution.
In summary, while NMF is effective for unsupervised parts-based learning due to the non-negativity constraint, its limitations include sensitivity to non-additive patterns, local minima, lack of external knowledge incorporation, and potentially high computational cost, all of which affect its performance and interpretability for object databases. Extensions like graph-regularized NMF seek to alleviate some of these issues by integrating additional structure or prior knowledge.
- Data-and-cloud-computing can play a significant role in improving the performance and scalability of Non-negative Matrix Factorization (NMF) algorithms, particularly in reducing the computational complexity associated with large object databases.
- The education-and-self-development sector can benefit from technology-driven learning platforms that incorporate the principles of NMF, allowing learners to discover hidden structures and relationships in vast amounts of data, fostering a deeper understanding of various concepts.
