  With the resurgence of interest in neural networks, representation learning
has re-emerged as a central focus in artificial intelligence. Representation
learning refers to the discovery of useful encodings of data that make
domain-relevant information explicit. Factorial representations identify
underlying independent causal factors of variation in data. A factorial
representation is compact and faithful, makes the causal factors explicit, and
facilitates human interpretation of data. Factorial representations support a
variety of applications, including the generation of novel examples, indexing
and search, novelty detection, and transfer learning.
  This article surveys various constraints that encourage a learning algorithm
to discover factorial representations. I dichotomize the constraints in terms
of unsupervised and supervised inductive bias. Unsupervised inductive biases
exploit assumptions about the environment, such as the statistical distribution
of factor coefficients, assumptions about the perturbations a factor should be
invariant to (e.g. a representation of an object can be invariant to rotation,
translation or scaling), and assumptions about how factors are combined to
synthesize an observation. Supervised inductive biases are constraints on the
representations based on additional information connected to observations.
Supervisory labels come in variety of types, which vary in how strongly they
constrain the representation, how many factors are labeled, how many
observations are labeled, and whether or not we know the associations between
the constraints and the factors they are related to.
  This survey brings together a wide variety of models that all touch on the
problem of learning factorial representations and lays out a framework for
comparing these models based on the strengths of the underlying supervised and
unsupervised inductive biases.
