Topic models have been extensively used to organize and interpret the contents of large, unstructured corpora of text documents. Although topic models often perform well on traditional training vs. test set evaluations, it is often the case that the results of a topic model do not align with human interpretation. This interpretability fallacy is largely due to the unsupervised nature of topic models, which prohibits any user guidance on the results of a model. In this paper, we introduce a semi-supervised method called topic supervised non-negative matrix factorization (TS-NMF) that enables the user to provide labeled example documents to promote the discovery of more meaningful semantic structure of a corpus. In this way, the results of TS-NMF better match the intuition and desired labeling of the user. The core of TS-NMF relies on solving a non-convex optimization problem for which we derive an iterative algorithm that is shown to be monotonic and convergent to a local optimum. We demonstrate the practical utility of TS-NMF on the Reuters and PubMed corpora, and find that TS-NMF is especially useful for conceptual or broad topics, where topic key terms are not well understood. Although identifying an optimal latent structure for the data is not a primary objective of the proposed approach, we find that TS-NMF achieves higher weighted Jaccard similarity scores than the contemporary methods, (unsupervised) NMF and latent Dirichlet allocation, at supervision rates as low as 10 to 20 .
Other related extensions of NMF include constrained NMF @cite_11 , and semi-supervised NMF @cite_19 . Constrained NMF assumes that some subset of columns of @math have class labels that dictate their location in latent space. In this formulation, one constrains the matrix @math so as to enforce the relation that if the @math th and @math th columns of @math have the same class labels, then the @math th and @math th rows of @math are equal. Semi-supervised NMF was developed with the objective of identifying the clusters of each column of @math . With this method, the user can provide pairwise constraints on the columns of @math , specifying whether they must or cannot be clustered together. The minimization problem in ) is reformulated as a non-negative tri-factorization of the similarity matrix that provides the clustering information. As we will see in the next section, each of these methods permits much stronger supervision than TS-NMF in that our proposed method constrains only which of a subset of topics are not allowed to be in some subset of documents.
@cite_19 @cite_11
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We present a methodology for analyzing polyphonic musical passages comprised of notes that exhibit a harmonically fixed spectral profile (such as piano notes). Taking advantage of this unique note structure, we can model the audio content of the musical passage by a linear basis transform and use non-negative matrix decomposition methods to estimate the spectral profile and the temporal information of every note. This approach results in a very simple and compact system that is not knowledge-based, but rather learns notes by observation. Non-negative matrix factorization (NMF), as a useful decomposition method for multivariate data, has been widely used in pattern recognition, information retrieval and computer vision. NMF is an effective algorithm to find the latent structure of the data and leads to a parts-based representation. However, NMF is essentially an unsupervised method and can not make use of label information. In this paper, we propose a novel semi-supervised matrix decomposition method, called Constrained Non-negative Matrix Factorization, which takes the label information as additional constraints. Specifically, we require that the data points sharing the same label have the same coordinate in the new representation space. This way, the learned representations can have more discriminating power. We demonstrate the effectiveness of this novel algorithm through a set of evaluations on real world applications.