  Deep learning has been widely used for supervised learning and
classification/regression problems. Recently, a novel area of research has
applied this paradigm to unsupervised tasks; indeed, a gradient-based approach
extracts, efficiently and autonomously, the relevant features for handling
input data. However, state-of-the-art techniques focus mostly on algorithmic
efficiency and accuracy rather than mimic the input manifold. On the contrary,
competitive learning is a powerful tool for replicating the input distribution
topology. This paper introduces a novel perspective in this area by combining
these two techniques: unsupervised gradient-based and competitive learning. The
theory is based on the intuition that neural networks are able to learn
topological structures by working directly on the transpose of the input
matrix. At this purpose, the vanilla competitive layer and its dual are
presented. The former is just an adaptation of a standard competitive layer for
deep clustering, while the latter is trained on the transposed matrix. Their
equivalence is extensively proven both theoretically and experimentally.
However, the dual layer is better suited for handling very high-dimensional
datasets. The proposed approach has a great potential as it can be generalized
to a vast selection of topological learning tasks, such as non-stationary and
hierarchical clustering; furthermore, it can also be integrated within more
complex architectures such as autoencoders and generative adversarial networks.
