  The notion of concept has been studied for centuries, by philosophers,
linguists, cognitive scientists, and researchers in artificial intelligence
(Margolis & Laurence, 1999). There is a large literature on formal,
mathematical models of concepts, including a whole sub-field of AI -- Formal
Concept Analysis -- devoted to this topic (Ganter & Obiedkov, 2016). Recently,
researchers in machine learning have begun to investigate how methods from
representation learning can be used to induce concepts from raw perceptual data
(Higgins, Sonnerat, et al., 2018). The goal of this report is to provide a
formal account of concepts which is compatible with this latest work in deep
learning.
  The main technical goal of this report is to show how techniques from
representation learning can be married with a lattice-theoretic formulation of
conceptual spaces. The mathematics of partial orders and lattices is a standard
tool for modelling conceptual spaces (Ch.2, Mitchell (1997), Ganter and
Obiedkov (2016)); however, there is no formal work that we are aware of which
defines a conceptual lattice on top of a representation that is induced using
unsupervised deep learning (Goodfellow et al., 2016). The advantages of
partially-ordered lattice structures are that these provide natural mechanisms
for use in concept discovery algorithms, through the meets and joins of the
lattice.
