This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON CYBERNETICS 1 Topographic NMF for Data Representation

Abstract—Nonnegative matrix factorization (NMF) is a useful technique to explore a parts-based representation by decompos-ing the original data matrix into a few parts-based basis vectors and encodings with nonnegative constraints. It has been widely used in image processing and pattern recognition tasks due to its psychological and physiological interpretation of natural data whose representation may be parts-based in human brain. However, the nonnegative constraint for matrix factorization is generally not sufficient to produce representations that are robust to local transformations. To overcome this problem, in this paper, we proposed a topographic NMF (TNMF), which imposes a topographic constraint on the encoding factor as a reg-ularizer during matrix factorization. In essence, the topographic constraint is a two-layered network, which contains the square nonlinearity in the first layer and the square-root nonlinearity in the second layer. By pooling together the structure-correlated features belonging to the same hidden topic, the TNMF will force the encodings to be organized in a topographical map. Thus, the feature invariance can be promoted. Some experiments carried out on three standard datasets validate the effectiveness of our method in comparison to the state-of-the-art approaches. Index Terms—Data clustering, dimension reduction, feature invariance, machine learning, nonnegative matrix factorization. I