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# Symmetry-Based Representations for Artificial and Biological General
Intelligence

Irina Higgins  DeepMind, London  Sébastien Racanière  DeepMind, London  Danilo
Rezende  DeepMind, London

###### Abstract

Biological intelligence is remarkable in its ability to produce complex
behaviour in many diverse situations through data efficient, generalisable and
transferable skill acquisition. It is believed that learning “good” sensory
representations is important for enabling this, however there is little
agreement as to what a good representation should look like. In this review
article we are going to argue that symmetry transformations are a fundamental
principle that can guide our search for what makes a good representation. The
idea that there exist transformations (symmetries) that affect some aspects of
the system but not others, and their relationship to conserved quantities has
become central in modern physics, resulting in a more unified theoretical
framework and even ability to predict the existence of new particles.
Recently, symmetries have started to gain prominence in machine learning too,
resulting in more data efficient and generalisable algorithms that can mimic
some of the complex behaviours produced by biological intelligence. Finally,
first demonstrations of the importance of symmetry transformations for
representation learning in the brain are starting to arise in neuroscience.
Taken together, the overwhelming positive effect that symmetries bring to
these disciplines suggest that they may be an important general framework that
determines the structure of the universe, constrains the nature of natural
tasks and consequently shapes both biological and artificial intelligence.

###### keywords:

machine learning, representation learning, symmetries, physics, neuroscience,
vision

##  1 Introduction

Neuroscience and machine learning (ML) have a long history of mutually
beneficial interactions (Hassabis et al., 2017), with neuroscience inspiring
algorithmic and architectural improvements in ML (Rosenblatt, 1958; LeCun et
al., 1989), and new ML approaches serving as computational models of the brain
(Yamins et al., 2014; Yamins and DiCarlo, 2016; Wang et al., 2018; Dabney et
al., 2020). The two disciplines are also interested in answering the same
fundamental question: what makes a “good” representation of the often high-
dimensional, non-linear and multiplexed sensory signals to support general
intelligence (Niv, 2019; Bengio et al., 2013). In the same way as the adoption
of the decimal system for representing numbers has produced an explosion in
the quantity of numerical tasks that humans could solve efficiently (note that
the information content remained unaffected by this change in the
representational form), finding a “good” representation of the sensory inputs
is likely to be a fundamental computational step for enabling data efficient,
generalisable and transferrable skill acquisition. While neuroscientists go
about trying to answer this question by studying the only working
instantiation of general intelligence - the brain, ML scientists approach the
same problem from the engineering perspective, by testing different
representational forms in the context of task learning through supervised or
reinforcement learning (RL), which allows faster iteration. In this review we
will discuss how bringing the idea of symmetry transformations from physics
into neural architecture design has enabled more data efficient and
generalisable task learning, and how this may be of value to neuroscience.

The reason why it makes sense to turn to physics when it comes to
understanding the goal of perception in artificial or biological intelligence,
is because intelligence evolved within the constraints of our physical world,
and likewise, the tasks that we find interesting or useful to solve are
similarly constrained by physics. For example, it is useful to know how to
manipulate physical objects, like rocks, water or electricity, but it is less
useful to know how to manipulate arbitrary regions of space (which also do not
have a word to describe them, further highlighting their lack of relevance).
Hence, a representation that reflects the fundamental physical properties of
the world is likely to be useful for solving natural tasks expressed in terms
of the same physical objects and properties. _Symmetry transformations_ are a
simple but fundamental idea that allows physicists to discover and categorise
physical objects - the “stubborn cores that remain unaltered even under
transformations that could change them” (Livio, 2012), and hence symmetries
are a good candidate for being the target of representation learning.

The study of symmetries in physics (that is, the transformations that leave
the physical “action” invariant) in its modern form originates with Noether’s
Theorem (Noether, 1915), which proved that every conservation law is grounded
in a corresponding continuous symmetry transformation. For example, the
conservation of energy arises from the time translation symmetry, the
conservation of momentum arises from the space translation symmetry, and the
conservation of angular momentum arises due to the rotational symmetry. This
insight, that transformations (the joints of the world) and conserved
properties (the invariant cores of the world that words often tend to refer to
(Tegmark, 2008)) are tightly related, has led to a paradigm shift in the
field, as the emphasis in theoretical physics changed from studying objects
directly to studying _transformations_ in order to discover and understand
objects. Since the introduction of Noether’s theorem, symmetry transformations
have permeated the field at every level of abstraction, from microscopic
quantum models to macroscopic astrophysics models.

In this paper we are going to argue that, similarly to physics, a change in
emphasis in neuroscience from studying representations in terms of static
objects to studying representations in terms of what natural symmetry
transformations they reflect can be impactful, and we will use the recent
advances in ML brought about by the introduction of symmetries to neural
networks to support our argument. By introducing the mathematical language of
group theory used to describe symmetries, we hope to provide the tools to the
neuroscience community to help in the search for symmetries in the brain.
While ML research has demonstrated the importance of symmetries in the context
of different data domains, here we will mainly concentrate on vision, since it
is one of the most prominent and most studied sensory systems in both ML and
neuroscience. For this reason, topics like the importance of symmetries in RL
will be largely left out (although see Anand et al. (2016); Madan et al.
(2018); Agostini and Celaya (2009); van der Pol et al. (2020); Kirsch et al.
(2021)). We will finish the review by describing some of the existing evidence
from the neuroscience community that hints at symmetry-based representations
in the ventral visual stream.

##  2 What are symmetries?

###  2.1 Invariant and equivariant representations

Given a task, there often exist transformations of the inputs that should not
affect it. For example, if one wants to count the number of objects on a
table, the outcome should not depend on the colours of those objects, their
location or the illumination of the scene. In that case, we say the output
produced by an intelligent system when solving the task is invariant with
respect to those transformations. Since the sensory input changes with
transformations, while the output is invariant, we need to decide what should
happen to the intermediate representations. Should they be invariant like the
output or should they somehow transform similarly to the input?

Much of the research on perception and representation learning, both in ML and
neuroscience, has focused on object recognition. In ML, this line of research
has historically emphasised the importance of learning representations that
are invariant to transformations like pose or illumination (Lowe, 1999; Dalal
and Triggs, 2005; Sundaramoorthi et al., 2009; Krizhevsky et al., 2012;
Soatto, 2010). In this framework, transformations are considered nuisance
variables to be thrown away (Fig. 1a and Fig. 2b). Some of the most successful
deep learning methods (Krizhevsky et al., 2012; Hu et al., 2018; Dai et al.,
2021; Mnih et al., 2015; Silver et al., 2016; Espeholt et al., 2018) end up
learning such invariant representations (see Tishby et al. (1999); Tishby and
Zaslavsky (2015) for a potential explanation of why this happens in the
context of supervised learning). This is not a problem for narrow
intelligence, which only needs to be good at solving the few tasks it is
explicitly trained for, however, discarding “nuisance” information can be
problematic for general intelligence which needs to reuse its representations
to solve many different tasks, and it is not known ahead of time which
transformations may be safe to discard. It is not surprising then that despite
the enormous success of the recent deep learning methods trained on single
tasks, they still struggle with data efficiency, transfer and generalisation
when exposed to new learning problems (Lake et al., 2016; Higgins et al.,
2017b; Garnelo et al., 2016; Marcus, 2018; Kansky et al., 2017; Cobbe et al.,
2019).

(a) Inputs transform with symmetries, but hidden features and outputs are
invariant.

(b) Inputs and hidden features transform with symmetries, only outputs are
invariant.

Figure 1: Different approaches to dealing with symmetries in neural networks.
Both figures represent a neural network transforming an image, and a rotated
image. The grey $3\times 3$ squares are activations of the neural networks.

Similarly to ML, in neuroscience ventral visual stream is traditionally seen
to be progressively discarding information about the identity preserving
transformations of objects (Tanaka, 1996; Poggio and Bizzi, 2004; Fukushima,
1980; Yamins et al., 2014). While neurons in the early processing stages, like
V1, are meant to represent all information about the input stimuli and their
transformations in high-dimensional “entangled” manifolds, where the
identities of the different objects are hard to separate (Fig. 2a), later in
the hierarchy such manifolds are meant to collapse into easily separable
points corresponding to individual recognisable objects, where all the
information about the identity preserving transformations is lost, resulting
in the so called “exemplar” neurons111These are also referred to as
“grandmother cells” in the literature. following the naming convention of
Chang and Tsao (2017). In this view, every neuron has a preferred stimulus
identity in response to which the neuron fires maximally, while its response
to other stimuli decreases proportionally to their distance from the preferred
stimulus (Fig. 2b).

Figure 2: Different hypothesised coding properties of neurons at the start (A)
and end (B-C) of visual processing in neural networks and the brain. Top row,
schematic representation of manifolds representing two classes of robots: blue
manifold contains robots that vary in the shape of their head, arms, legs and
body; red manifold contains robots that have no body and vary in the shape of
their head, arms and legs only. Bottom row, schematic representation of the
activations of a single idealised real or artificial neuron in response to
variations in the visual stimulus. A: Top: early processing stages have
entangled high-dimensional manifolds. All information about the two robot
categories, and their identity preserving transformations is present, but is
hard to read out from the neural activations. Arrows represent the high-
dimensional space spanned by all V1 neurons. Bottom: line plot shows the
activation of a single idealised neuron in response to different robot
variations. Neuron responds to robots from both classes. B: “Exemplar” or
invariant representation at the end of visual processing. Top: Single neurons
have maximal firing for the prototype example of their preferred robot class
(blue - with, red - without body). All information about the identity
preserving transformations has been collapsed (illustrated by red or blue
points), which makes object classification easy, but any other task (like
clustering robots based on their arm variation) impossible. Arrows represent
the high-dimensional space spanned by the higher-order neurons, arrow colour
represents preferred stimulus of the neuron. Bottom left: activation of a
single idealised neuron to all robot variations from both classes. Lighter,
higher activation. Big blue circle indicates preferred stimulus for neuron,
resulting in highest activation, smaller blue circles indicate other robots
resulting in lower to no activation. Blue arrow, cross section of robot
variations shown in the line plot. Bottom right: line plot shows activation of
the same idealised neuron as on the left but in the cross section of robot
variations spanned by the blue arrow. Response declines proportionally to the
distance from the preferred stimulus (big blue circle). C: “Axis” or
equivariant representation at the end of visual processing. Top: two robot
classes have been separated into different representational manifolds, which
are also aligned in terms of the shared transformations (e.g. both robot
classes have similar identity preserving transformations in head shape,
spanned by green axis; and arm shape, spanned by purple axis). This makes it
easy to classify the robots, and solve other tasks, like clustering robots
based on their arm variations. Bottom left: activation of a single idealised
neuron to robot variations along the head shape change axis (green) and arm
shape change axis (purple). Ligher, higher activation. Neuron has a ramped
response proportional to changes in its preferred transformation (changes in
head shape, green), and no change in firing to other transformations (e.g.
changes in arm shape, blue). Bottom right: as in B, but the cross section is
spanned by the purple axis. Green dot indicates higher neural activation in
response to a change in the robot head shape.

An alternative point of view in both disciplines has advocated that instead of
discarding information about the identity preserving transformations,
information about these factors should be preserved but reformatted in such a
way that aligns transformations within the representations with the
transformations observed in the physical world (Fig. 1b and Fig. 2c),
resulting in the so called equivariant representations (Hinton et al., 2012;
Bengio et al., 2013; DiCarlo and Cox, 2007). In the equivariant approach to
perception, certain subsets of features may be invariant to specific
transformations, but the overall representation is still likely to preserve
more overall information than an invariant representation, making them more
conducive of diverse task learning (Fig. 1b). For example, some hidden units
may be invariant to changes in the object colour, but will preserve
information about object position, while other hidden units may have an
opposite pattern of responses, which means that information about both
transformations will be preserved across the whole hidden layer, while each
individual subspace in the hidden representation will be invariant to all but
one transformation. Researchers in both neuroscience and ML communities have
independently hypothesised that equivariant representations are likely to be
important to support general intelligence, using the terms “untangling”
(DiCarlo and Cox, 2007; DiCarlo et al., 2012) and “disentangling” (Bengio et
al., 2009; Bengio, 2011; Bengio et al., 2013) respectively. We are next going
to introduce the mathematical language for describing symmetry transformations
and use it to discuss how adding neural network modules that are equivariant
to such symmetry transformations can improve data efficiency, generalisation
and transfer performance in ML models.

###  2.2 Defining symmetries and actions

Symmetries are sets of _transformations_ of objects, and the same abstract set
of symmetries can transform different objects. For example, consider the set
of rotations by multiple of $90^{\circ}$ and reflections along both horizontal
and vertical axis, known as the dihedral group $D_{4}$ (Dummit and Foote,
1991). By rotating images, symmetries from $D_{4}$ can be applied to images of
cats or tea pots, either $32\times 32$ or $1024\times 1024$, colour or black
and white. In mathematics, the concept of symmetries, that is transformations
that are invertible and can be composed, is abstracted into the concept of
groups. For example, $D_{4}$ is a group with $8$ elements (Fig. 3).

More formally, a group $G$ is defined as a set with a binary operation (also
called composition or multiplication)

$\displaystyle\begin{array}[]{ccc}G\times G&\rightarrow&G\\\
(g_{1},g_{2})&\mapsto&g_{1}\cdot g_{2},\end{array}$ (3)

such that

  1. 1.

the operation is associative: $(g_{1}\cdot g_{2})\cdot
g_{3}=g_{1}\cdot(g_{2}\cdot g_{3})$;

  2. 2.

there exists an identity element $e\in G$ such that $e\cdot g=g\cdot
e=g,\forall g\in G$;

  3. 3.

all elements are invertible: for any $g\in G$, there exists $g^{-1}\in G$ such
that $g\cdot g^{-1}=g^{-1}\cdot g=e$.

Note how we defined a group as a set of symmetries, without explicitly saying
what these are symmetries of. That’s because the concept of group in
mathematics seeks to study properties of symmetries that are independent of
the objects being transformed. In practice though, we will of course want to
apply symmetries to objects. This is formally defined as an action222This kind
of action is distinct from the action in physics; here, it just refers to the
action of an operator.. For example, the group $D_{4}$ can act on both
$32\times 32$ grey-scale images, that is $\mathbb{R}^{32\times 32}$, and on
$1024\times 1024$ colour images, that is $\mathbb{R}^{1024\times 1024\times
3}$.

(a) While the image is transformed, some properties, such as the teapot
identity or colour, are invariant with respect to the applied transformations.

(b) Symmetric images are left invariant by some elements of $D_{4}$, and
modified by others.

Figure 3: All $8$ transformations of an image under the dihedral group
$D_{4}$. Rotations by $90^{\circ}$ are applied along the inner and outer
circles. Reflections are applied along straight lines.

More formally, given a group $G$ and a set $X$, an action333To be precise, we
are defining here a left action. of $G$ on $X$ is a map

$\displaystyle\begin{array}[]{ccc}G\times X&\rightarrow&X\\\
(g,x)&\mapsto&g\cdot x,\end{array}$ (6)

such that

  1. 1.

the multiplication of the group and the action are compatible:
$g_{1}\cdot(g_{2}\cdot x)=(g_{1}\cdot g_{2})\cdot x$;

  2. 2.

the identity of the group leaves elements of $X$ invariant: $e\cdot x=x$.

Note how we overloaded the symbol $\cdot$ to define both a multiplication in
the group, and an action on a set. This makes sense because multiplication of
the group defines an action of that group on itself. The identity $e$ leaves
all elements in $X$ invariant $e\cdot x=x$, but for a given $x$, there can
exist $g\neq e$ such that $g\cdot x=x$, for example in Fig. 3b.

Two elements of a group are said to commute if the order in which we multiply
them does not matter. Formally, we say that $g_{1},g_{2}\in G$ commute if
$g_{1}\cdot g_{2}=g_{2}\cdot g_{1}$. If all the elements in the groups commute
with each other, the group itself is called commutative444The term Abelian is
also used in the literature.. Even if a group is not commutative, it might
still be a product of two subgroups that commute with each other. For example,
assume you have three cubes of different sizes and colours, and three pyramids
of different sizes and colours. If these objects are put on three different
tables, each with a cube and a pyramid, we can move the cubes around while
leaving the pyramids where they are, or we can move the pyramids and leave the
cubes untouched. The action of re-ordering the cubes is an action of the group
of permutations over three elements $\mathcal{S}_{3}$. Here we are making that
group act on our arrangement of cubes and pyramids, by leaving the pyramids
invariant. The action of re-ordering the pyramids is also an action of
$\mathcal{S}_{3}$. So, overall, we have an action of
$\mathcal{S}_{3}\times\mathcal{S}_{3}$. The group as a whole is not
commutative, since each of the $\mathcal{S}_{3}$ is not, but it does not
matter if we reorder the pyramids first, or the cubes first. Formally, this
means that as a set $G=G_{1}\times G_{2}$, where $G_{1}$ and $G_{2}$ are
themselves groups, and all elements of $G_{1}$ commute with all elements of
$G_{2}$. This last commutation requirement is important. Indeed, consider once
again the case of $D_{4}$. Let $F$ be the subgroup made of the identity and
the reflection along the vertical axis. And let $R$ be the group made of
rotations by $0^{\circ},90^{\circ},180^{\circ}$ and $270^{\circ}$. Any element
of $D_{4}$ can be written in a unique way as $f\cdot r$ for $(f,r)\in F\times
R$, but since $f\cdot r\neq r\cdot f$, it is not true that $D_{4}$ is equal to
$F\times R$ as a group.

We just mentioned the idea that some properties are preserved by symmetries.
Indeed, while a group action defines how elements of a set are transformed, it
is often useful to also consider what is being preserved under the action. For
example, consider a Rubik’s cube. Algorithms on how to solve a Rubik’s cube
use steps described by simple transformations such as "rotate left face
clockwise" or "rotate front face anti-clockwise". The set of all
transformations built by such simple rotations of faces forms a group, and
that group acts on the Rubik’s cube by modifying the colours on faces. But
what is being preserved here? The answer is the structure of the cube. Indeed,
after any of these transformations, we still have a cube with faces, each made
of $9$ squares arranged in a regular $3\times 3$ grid. In the case of our
dihedral group $D_{4}$ in Fig 3, colours but also relative distances are being
preserved: two pixels in the original image will move to a new location in a
rotated image, but their distance from each other is unchanged, thus
preserving the object identity.

We are now ready to define the concepts of invariant and equivariant maps -
the building blocks for obtaining the invariant and equivariant
representations we introduced earlier. Lets start with invariance. Formally,
if a group $G$ acts on a space $X$, and if $F:X\rightarrow Y$ is a map between
sets $X$ and $Y$, then $F$ is invariant if $F(g\cdot x)=F(x),\forall(g,x)\in
G\times X$. In words, this means that applying $F$ to a point or to a
transformed point will give the same result. For example, in Fig. 3, the map
that recognises a tea pot in the input picture should not depend on the
orientation of the picture. Invariant maps delete information since knowing
$y=F(x)$ does not allow to distinguish between $x$ and $g\cdot x$. If the
invariant features required to solve a task are highly non-linear with respect
to the inputs, then we might want to first transform the inputs before
extracting any invariant information. And here we need to be careful, because
if $H$ is any map while $F$ is invariant, it will not be true in general that
$F(H(x))$ is invariant. On the other hand, we will see that if $H$ is
equivariant, then $F(H(x))$ will indeed be invariant. Let us now define
equivariance: if $G$ is a group acting on both spaces $X$ and $Y$, and
$H:X\rightarrow Y$ is a map between these spaces, then $H$ is said to be
equivariant if for any $g\in G$ and any $x\in X$, we have $H(g\cdot x)=g\cdot
H(x)$. In words, it does not matter in which order we apply the group
transformation and the map $H$. We can now verify our earlier claim: if $H$ is
equivariant and $F$ is invariant, then $F(H(g\cdot x))=F(g\cdot
H(x))=F(H(x))$, and $F\circ H$ is indeed invariant. As we will see later, this
recipe of stacking equivariant maps followed by an invariant map, as shown in
Fig.1b, is a commonly used recipe in ML (Bronstein et al., 2021).

So far we have considered discrete symmetries. However, many of the symmetries
encountered in the real world are continuous. A group of symmetries is said to
be continuous if there exist continuous paths between symmetries. For example,
in the group of $2D$ rotations, we can create paths by smoothly varying the
angle of the rotations. On the other hand, if we only allow rotations by
multiple of $90^{\circ}$, then it is not possible to move smoothly from a
rotation by $180^{\circ}$ to a rotation by $270^{\circ}$. In that case, the
group is said to be discrete555Some groups will have both continuous and
discrete aspects. For example, the group of all invertible matrices of a given
size has a clear continuous aspect, but it also has a discrete aspect as we
cannot move continuously from a matrix with positive determinant to a matrix
with negative determinant without hitting a matrix with determinant $0$.. A
simple approach to handle continuous symmetries used in practice in ML is to
fall back to the discrete case by approximating the full group of continuous
symmetries by a subgroup of discrete ones. For example, the group of rotations
of the 2D plane can be approximated by only considering rotations by
$\frac{360}{N}^{\circ}$, although this can become computationally expensive
for very large groups (Finzi et al., 2020). While other approaches that truly
handle a full group of continuous symmetries do exist (Katsman et al., 2021;
Papamakarios et al., 2019; Köhler et al., 2020; Rezende et al., 2020; Huang et
al., 2020; Rezende et al., 2019; Pfau et al., 2020a; Cohen et al., 2021;
Rezende and Racanière, 2021), we will concentrate on discrete symmetries in
this paper for simplicity.

##  3 Implementation and utility of symmetries in ML

Although not always explicitly acknowledged, symmetries have been at the core
of some of the most successful deep neural network architectures. For example,
convolutional layers (CNNs) (LeCun et al., 1995) responsible for the success
of the deep classifiers that are able to outperform humans in their ability to
categorise objects in images (Dai et al., 2021; Hu et al., 2018) are
equivariant to translation symmetries characteristic of image classification
tasks, while graph neural networks (GNNs) (Battaglia et al., 2018) and
attention blocks commonly used in transformer architectures (Vaswani et al.,
2017) are equivariant to the full group of permutations. While there are
several reasons, including optimisation considerations, why these
architectural choices have been so successful compared to MLPs (Rosenblatt,
1958) \- the original neural networks, one of the reasons is that these
architectures reflect the prevalent symmetry groups of their respective data
domains, while the linear layers used in MLPs are not compatible with any
particular symmetry (Haykin, 1994), despite being theoretically proven
universal function approximators (Hornik et al., 1989; Cybenko, 1989).
Architectures like CNNs and GNNs reflect single type of symmetries
(translations and permutations respectively), but active research is also
looking into building techniques to incorporate larger groups of symmetries
into neural networks (Cohen and Welling, 2016; Gens and Domingos, 2014;
Anselmi et al., 2013; Cohen et al., 2018).

(a) The original problem domain.

(b) With one symmetry, a reflection along a plane, we can half the domain on
which we need to learn.

(c) Further symmetries keep on reducing the volume of problem domain.

Figure 4: Symmetries let us reduce the volume of the domain on which our
models need to learn.

One of the main reasons why incorporating symmetries into neural networks
helps is due to improvements in data efficiency. Indeed, incorporating
symmetries can reduce the volume of the problem space, as illustrated in Fig.
4. If we assume that the data processed by our model are points in a $3D$ cube
(Fig. 4a), when symmetries can be exploited, the models only need to work with
a subset of the cube (Fig. 4b and Fig. 4c), which reduces the volume of the
input space. Provided the model respects symmetries by construction, learning
on this reduced space is enough to learn on the entire cube. This naturally
also leads to improvements in generalisation and transfer, since new points
outside of the training data distribution that can be obtained by applying the
known symmetries to the observed data will be automatically recognisable. This
principle has been exploited in scientific applications of ML, such as free
energy estimation (Wirnsberger et al., 2020), protein folding (Baek et al.,
2021; Fuchs et al., 2020), or quantum chemistry (Pfau et al., 2020b; Batzner
et al., 2021).

An alternative to building symmetries into the model, is to use data-
augmentation and let the model learn the symmetries. This is achieved by
augmenting the training dataset (for example images) with the relevant
transformations of this data (for example, all rotations and reflections of
these images). This principle has been used as a source of augmentations for
self-supervised contrastive learning approaches (Chen et al., 2020; Grill et
al., 2020). While these approaches have been shown to be very effective in
improving data efficiency on image classification tasks, other research has
shown that learning symmetries from data augmentations is usually less
effective than building them into the model architecture (Veeling et al.,
2018; Rezende et al., 2019; Cohen and Welling, 2016; Satorras et al., 2021; Qi
et al., 2017; Köhler et al., 2020).

An alternative to hard wiring inductive biases into the network architecture
is to instead adjust the model’s learning objective to make sure that its
representations are equivariant to certain symmetries. This can be done
implicitly by adding (unsupervised) regularisers to the main learning
objective (Jaderberg et al., 2017; Bellemare et al., 2017), or explicitly by
deciding on what a “good” representation should look like and directly
optimising for those properties. One example of the latter line of research is
the work on disentangled 666Although the term “disentanglement” and its
opposite “entanglement” are also used in quantum mechanics (QM), and indeed
the term “entanglement” refers to a mixing of factors in both ML (through any
diffeomorphism) and QM (through a linear combinations), there is no deeper
connection between the two. representation learning (Bengio et al., 2009,
2013) (also see related ideas in Schmidhuber (1992); Hyvärinen (1999)). While
originally proposed as an intuitive framework that suggested that the world
can be described using a small number of independent generative factors, and
the role of representation learning is to discover what these are and
represent each generative factor in a separate representational dimension
(Bengio et al., 2013), disentangling has recently been re-defined through a
formal connection to symmetries (Higgins et al., 2019). In this view, a vector
representation is seen as disentangled with respect to a particular
decomposition of a symmetry group into a product of subgroups, if it can be
decomposed into independent subspaces where each subspace is affected by the
action of a single subgroup, and the actions of all the other subgroups leave
the subspace unaffected.

Figure 5: A: Simplified schematic showing discrete approximation of continuous
translation and scale symmetries of 3D objects. Translations are applied along
the inner and outer circles. Scale transformations are applied along straight
lines. B: Translation and scale transformation commute with each other. They
can be applied in permuted order without affecting the final state. C:
Disentangling neural networks learn to infer a representation of an image that
is a concatenation of independent subspaces, each one being (approximately)
equivariant to a single symmetry transformation. The model uses inference to
obtain a low-dimensional representation of an image, and generation to
reconstruct the original image from the representation. Two example latent
traversals demonstrate the effect on the image reconstruction of smoothly
varying the value of the position and size subspaces.

To understand this definition better, let’s consider a concrete example of an
object classification task (Fig. 5a). Transformations like changes in the
position or size of an object are symmetry transformations that keep the
object identity invariant. These transformations also commute with each other,
since they can be applied in random order without affecting the final state of
the world (Fig. 5b). This implies that the symmetry group used to describe the
natural transformations in this world can be decomposed into a product of
separate subgroups, including one subgroup that affects the position of an
object, and another one affecting its size.

Assuming that the symmetry transformations act on a set of hypothetical ground
truth abstract states of our world, and the disentangling model observes high-
dimensional image renderings of such states, in which all the information
about object identity, size and position among other factors is entangled, the
goal of disentangled representation learning is to infer a representation
which is decomposed into independent subspaces, where each subspace is
affected only by a single subgroup of our original group of symmetry
transformations. In other words, the vector space of such a representation
would be a concatenation of independent subspaces, such that, for example, a
change in size only affects the “size subspace”, but not the “position
subspace” or any other subspace (Fig. 5c). This definition of disentangled
representations is very general – it does not assume any particular
dimensionality or basis for each subspace. The changes along each of the
subspaces in the representation may also be implemented by an arbitrary,
potentially non-linear mapping, although if this mapping is linear, it can
provide additional nice properties to the representation (Higgins et al.
(2018) call such a representation a _linear disentangled representations_),
since it means that the task relevant information (e.g. the “stable cores” of
colour or position attributes of the object) can be read out using linear
decoders, and “nuisance” information can be easily ignored using a linear
projection.

While the early approaches to disentangled representation learning (including
related ideas from nonlinear dimensionality reduction literature, e.g. Belkin
and Niyogi (2002); Coifman and Lafon (2006); Hyvärinen and Pajunen (1999);
Hyvärinen (1999); Tenenbaum et al. (2000)) either struggled to scale
(Tenenbaum et al., 2000; Desjardins et al., 2012; Tang et al., 2013; Cohen and
Welling, 2014, 2015) or relied on a form of supervision (Hinton et al., 2011;
Reed et al., 2014; Zhu et al., 2014; Yang et al., 2015; Goroshin et al., 2015;
Kulkarni et al., 2015; Cheung et al., 2015; Whitney et al., 2016; Karaletsos
et al., 2016), most of the modern methods for successful unsupervised
disentangling (Higgins et al., 2017a; Ridgeway and Mozer, 2018; Chen et al.,
2018; Dupont, 2018; Kim and Mnih, 2018; Esmaeili et al., 2019; Kumar et al.,
2018; Achille et al., 2018; Ansari and Soh, 2019; Mathieu et al., 2019;
Detlefsen and Hauberg, 2019; Dezfouli et al., 2019; Lorenz et al., 2019; Lee
et al., 2020; Caselles-Dupré et al., 2019; Quessard et al., 2020; Ramesh et
al., 2019) are based on the Variational AutoEncoder (VAE) architecture (Kingma
and Welling, 2014; Rezende et al., 2014) \- a generative network that learns
by predicting its own inputs. The base VAE framework learns a compressed
representation that maximises the marginal likelihood of the data and are
related to the idea of “mean field approximation” from physics. In this
framework no explicit desiderata are made about the representational form – as
long as the distribution of the learnt data representation is close to the
chosen prior (which often consists of independent unit Gaussians), it is
considered to be acceptable. Disentangling VAEs, on the other hand, aim to
learn a representation of a very particular form – it has to decompose into
independent subspaces, each one reflecting the action of a single symmetry
transformation. Disentangling VAEs typically work by adjusting the VAE
learning objective to restrict the capacity of the representational
bottleneck. This is usually done by encouraging the representation to be as
close to the isotropic unit Gaussian distribution as possible, hence also
encouraging factorisation. Although it has been proven that unsupervised
disentangled representation learning in this setting should be theoretically
impossible (Locatello et al., 2018), these approaches work in practice by
exploiting the interactions of the implicit biases in the data and the
learning dynamics (Burgess et al., 2018; Rolinek et al., 2019; Locatello et
al., 2019; Mathieu et al., 2019). Since these approaches are not optimising
for symmetry-based disentanglement directly, they are not principled and
struggle to scale. However, they have been shown to learn an approximate
symmetry-based disentangled representation (for example they often lose the
cyclical aspect of the underlying symmetry) that still preserves much of the
group structure (e.g. the commutativity of the symmetries) and hence serves as
a useful tool for both understanding the benefits of symmetry-based
representations in ML models, and as a computational model for studying
representations in the brain (Higgins et al., 2021a; Soulos and Isik, 2020).
In the meantime, new promising approaches to more scalable and/or principled
disentanglement are starting to appear in the ML literature (Pfau et al.,
2020a; Wang et al., 2021; Higgins et al., 2021b; Besserve et al., 2018).

In order to generalise learnt skills to new situations, it is helpful to base
learning only on the smallest relevant subset of sensory variables, while
ignoring everything else (Niv, 2019; Bengio et al., 2013; Niv et al., 2015;
Leong et al., 2017; Canas and Jones, 2010; Jones and Canas, 2010). Symmetry-
based representations make such attentional attenuation very easy, since
meaningful sensory variables get separated into independent representational
subspaces, as was demonstrated in a number of ML papers (Higgins et al.,
2017b; Locatello et al., 2020). Following the reasoning described earlier,
disentangled representations have also been shown to help with data efficiency
when learning new tasks (Wulfmeier et al., 2021; Locatello et al., 2020).
Finally, disentangled representations have also been shown to be a useful
source of intrinsically motivated transferable skill learning. By learning how
to control their own disentangled subspaces (e.g. how to control the position
of an object), it has been shown that RL agents with disentangled
representations could discover generally useful skills that could be readily
re-used for solving new tasks (e.g. how to stack objects) in a more data
efficient manner (Wulfmeier et al., 2021; Grimm et al., 2019; Laversanne-Finot
et al., 2018; Achille et al., 2018).

##  4 Symmetries in Neuroscience

Although psychology and cognitive science picked up the mathematical framework
of group theory to describe invariances and symmetry in vision a long time ago
(Dodwell, 1983), this framework was not broadly adopted and progress in this
direction quickly stalled (although see Leibo et al. (2017); Liao et al.
(2013)). However, circumstantial evidence from work investigating the geometry
of neural representations suggests the possibility that the brain may be
learning symmetry-based representations. For example, factorised
representations of independent attributes, such as orientation and spatial
frequency (Mazer et al., 2002; Gáspár et al., 2019; Hubel and Wiesel, 1959) or
motion and direction tuning (Grunewald and Skoumbourdis, 2004) have long been
known to exist at the start of the ventral visual stream in V1. Going further
along the visual hierarchy, Kayaert et al. (2005) demonstrated that many of
the primate IT neurons had monotonic tuning to the generative dimensions of
toy visual stimuli, such as curvature, tapering or aspect ratio, known to be
discriminated independently from each other by humans in psychophysical
studies (Arguin and Saumier, 2000; de Beeck et al., 2003; Stankiewicz, 2002).
In particular, they found that the firing of each neuron was modulated
strongly by its preferred generative attribute but significantly less so by
the other generative attributes (Fig. 6a).

Figure 6: Examples of axis-based coding at the end of the ventral visual
stream. A: Single IT cell shows preference to a single transformation (change
in positive curvature) regardless of the geometric shape (triangle or
rectangle). Single IT cell responds to changes in curvature of the triangle
while being invariant to changes in length, changes in tapering of the
rectangle while being invariant to changes in curvature, and changes in
negative curvature of the rectangle while being invariant to changes in
tapering. Bars are standard errors in response to multiple stimulus
presentations (14 on average). Adapted form Kayaert et al. (2005). B: Single
IT cells have ramped responses proportional to changes along their preferred
axis of variation in the generative face space, and no changes in their
responses to orthogonal directions in the face space. Adapted from (Chang and
Tsao, 2017). C: Single cells in the IT have strong one-to-one alignment to
single subspaces discovered through disentangled representation learning.
Adapted from Higgins et al. (2021a). D: Different representation geometries
have different trade-offs in terms of how much they support generalisation as
measured by the abstraction scores (green), and how expressive they are as
measured by the shattering dimensionality score (orange). Representations in
the prefrontal cortex (PFC) and hippocampus (HPS) of primates, as well as in
the final layer of a neural network trained to solve multiple tasks in the
reinforcement learning framework were found to exhibit disentangling-like
geometry highlighted in blue that scores well on to both metrics. Adapted from
(Bernardi et al., 2020).

More recently, Chang and Tsao (2017) investigated the coding properties of
single IT neurons in the primate face patches. By parametrising the space of
faces using a low-dimensional code, they were able to show that each neuron
was sensitive to a specific axis in the space of faces spanned by as few as 6
generative dimensions on average, with different cells preferring different
axes. Moreover, the recorded IT cells were found to be insensitive to changes
in directions orthogonal to their preferred axis, suggesting a low-dimensional
factorised representation reminiscent of disentangled representations from ML
(Fig. 6b). To directly test whether the two representations resembled each
other, Higgins et al. (2021a) compared the responses of single cells in the IT
face patches to disentangled latent units discovered by a model exposed to the
same faces as the primates (Fig. 6c). By measuring the alignment between the
two manifolds, the authors were able to compare the two representational forms
in a way that was sensitive to linear transformations (unlike the traditional
measures of similarity used in the neuroscience literature, like explained
variance (Khaligh-Razavi and Kriegeskorte, 2014; Yamins and DiCarlo, 2016;
Cadieu et al., 2007; Güçlü and van Gerven, 2015; Cadena et al., 2019) or
Representational Similarity Analysis (Kriegeskorte et al., 2008; Khaligh-
Razavi and Kriegeskorte, 2014), which are invariant to linear transformations)
- any rotation or shear of one manifold with respect to the other would result
in reduced scores. The authors found that there was a strong one-to-one
alignment between IT neurons and disentangled units to the point where the
small number of disentangled dimensions discovered by the model were
statistically equivalent to a similarly sized subset of real neurons, and the
alignment was significantly stronger than that with supervised classifiers
(which learn an invariant representation) or the generative model used in
Chang and Tsao (2017). Furthermore, it was possible to visualise novel faces
viewed by the primates from the decoded activity of just twelve neurons
through their best matched disentangled units. This result established the
first direct link between coding in single IT neurons and disentangled
representations, suggesting that the brain may be learning representations
that reflect the symmetries of the world. Other recent work showed that
disentangled representations can also predict fMRI activation in the ventral
visual stream (Soulos and Isik, 2020).

While many of the existing approaches to disentangled representation learning
are generative models, thus fitting well within the predictive coding and free
energy principle (Clark, 2013; Friston, 2010; Rao and Ballard, 1999; Elias,
1955; Srinivasan et al., 1982) hypotheses of brain function, an alternative
biologically plausible way to learn disentangled representations was recently
proposed by Johnston and Fusi (2021). The authors showed that disentangled
representations can arise from learning to solve numerous diverse tasks in a
supervised manner, which would be required to produce the complex behaviors
that biological intelligence exhibits in the natural world. A similar result
was also demonstrated by Bernardi et al. (2020), who looked into the geometry
of neural representations for solving tasks in the RL framework in both
primates and neural networks. They found that the final layer of an MLP
trained through RL supervision to solve a number of tasks, as well as the
dorsolateral prefrontal cortex, the anterior cingulate cortex and the
hippocampus of primates exhibited disentangled-like qualities. Although the
representations of the underlying task variables were rotated in the space of
neural activation (unlike the axis aligned codes described in Higgins et al.
(2021a)), the underlying geometry was in line with what would be expected from
disentangled representations. The authors found that the degree to which such
geometry was present correlated with the primates success on the tasks (no
such correlation existed for the more traditional decoding methods that do not
take the geometry of the representation into account), and that such
representations supported both strong generalisation (as measured by the
abstraction scores) and high representational capacity (as measured by the
shattering dimensionality scores) (Fig. 6d).

Figure 7: Similarities between various aspects of disentangled representation
learning in ML (right column) and visual representation learning in the brain
(left column). A: The properties of the visual data obtained through a head
camera from toddlers (Slone et al., 2019; Smith et al., 2018) is similar to
the properties of the visual data that allows ML approaches to discover
disentangled representations. The scenes are uncluttered, and contain many
continuous transformations of few objects at a time. B: Perceptual
straightening of natural image trajectories observed in human vision (Hénaff
et al., 2019) is similar to the “Euclidenisation” of the latent space learnt
by disentangled ML models. C: Factorised representations that align with
semantically meaningful attributes hypothesised to be important for further
processing in the hippocampus (Bellmund et al., 2018; Behrens et al., 2018)
resembles the factorised representations learnt by disentangled ML models.

Further validation of the biological plausibility of disentangled
representation learning comes from comparing the data distribution that many
modern ML approaches require for optimal disentangling to the early visual
experiences of infants (Wood and Wood, 2018; Slone et al., 2019; Smith et al.,
2018). It appears that the two are similar, with smooth transformations of
single objects dominating both (Fig. 7a). Disentangled representation also
have properties that are believed to be true of the visual brain, such as
“Euclideanisation" or straightening of complex non-linear trajectories in the
representation space compared to the input observation space (Hénaff et al.,
2019) (Fig. 7b), and factorisation into semantically interpretable axes, such
as colour or shape of objects (Fig. 7c), which are hypothesised to be
important for more data efficient and generalisable learning (Behrens et al.,
2018), and for supporting abstract reasoning (Bellmund et al., 2018). It is
hypothesised that the same principles that allow biological intelligence to
navigate the physical space using the place and grid cells may also support
navigation in cognitive spaces of concepts, where concepts are seen as convex
regions in a geometric space spanned by meaningful axes like engine power and
car weight (Gardenfors, 2014; Gärdenfors, 2004; Balkenius and Gärdenfors,
2016). Learning disentangled representations that reflect the symmetry
structure of the world could be a plausible mechanism for discovering such
axes. Evidence from the ML literature has already demonstrated the utility of
disentangled representations for basic visual concept learning, imagination
and abstract reasoning (Higgins et al., 2018; Locatello et al., 2020;
Steenbrugge et al., 2018; van Steenkiste et al., 2019).

##  5 Discussion

The question of what makes a good representation has been historically central
to both ML and neuroscience, and both disciplines have faced the same debate:
whether the best representation to support intelligent behaviour should be
low-dimensional and interpretable or high-dimensional and multiplexed. While
the former dominated both early neuroscience (Barlow, 1972; Hubel and Wiesel,
1959) and ML (early success of feature engineering), recent development of
high-throughput recording methods in neuroscience (Saxena and Cunningham,
2019; Eichenbaum, 2018; Yuste, 2015) and the success of large black-box deep
learning models in ML (Hu et al., 2018; Vaswani et al., 2017) have shifted the
preference in both fields towards the latter. As a consequence, this led to
deep classifiers emerging as the main computational models for the ventral
visual stream (Yamins et al., 2014; Yamins and DiCarlo, 2016), and a belief
that higher-level sensory representations that can support diverse tasks are
too complex to interpret at a single neuron level. This pessimism was
compounded by the fact that designing stimuli for discovering interpretable
tuning in single cells at the end of the sensory processing pathways is hard.
While it is easy to systematically vary stimulus identity, it is hard to know
what the other generative attributes of complex natural stimuli may be, and
hence to create stimuli that systematically vary along those dimensions.
Furthermore, new representation comparison techniques between computational
models and the brain became progressively population-based and insensitive to
linear transformations (Yamins and DiCarlo, 2016; Kriegeskorte et al., 2008;
Khaligh-Razavi and Kriegeskorte, 2014), thus further stalling progress towards
gaining a more fine-grained understanding of the representational form
utilised by the brain (Thompson et al., 2016; Higgins et al., 2021a). At the
same time, it is becoming increasingly unlikely that high-dimensional,
multiplexed, uninterpretable population-based representations like those
learnt by deep classifiers are the answer to what makes a “good”
representation to support general intelligence, since ML research has shown
that models with such representations suffer from problems in terms of data
efficiency, generalisation, transfer, and robustness - all the properties that
are characteristic of biological general intelligence. In this paper we have
argued that representations which reflect the natural symmetry transformations
of the world may be a plausible alternative. This is because both the nature
of the tasks, and the evolutionary development of biological intelligence are
constrained by physics, and physicists have been using symmetry
transformations to discover and study the “joints” and the “stable cores” of
the world for the last century. By studying symmetry transformations,
physicists have been able to reconcile explanatory frameworks, systematically
describe physical objects and even discover new ones. Representations that are
equivariant to symmetry transformations are therefore likely to expose the
relevant invariants of our world that are useful for solving natural tasks.
From the information theory perspective, such representations can be viewed as
the simplest (in the context of Solomonoff induction (Solomonoff, 1964)) and
the most informative representations of the input to support the most likely
future tasks (Schmidhuber, 2010; MacKay, 1995; MacKay et al., 2003; Hutter,
2004; Wallace and Dowe, 1999).

We have introduced the basic mathematical language for describing symmetries,
and discussed evidence from ML literature that demonstrates the power of
symmetry-based representations in bringing better data efficiency,
generalisation and transfer when included into ML systems. Furthermore,
emerging evidence from the neuroscience community suggests that sensory
representations in the brain may also be symmetry-based. We hope that our
review will give the neuroscience community the necessary motivation and tools
to look further into how symmetries can explain representation learning in the
brain, and to consider them as an important general framework that determines
the structure of the universe, constrains the nature of natural tasks and
consequently shapes both biological and artificial intelligence.

## Conflict of Interest Statement

Authors are employed by DeepMind.

## Author Contributions

I.H and S.R. contributed to writing the review, D.R. contributed comments,
discussions and pointers that shaped the paper.

## Funding

Not applicable.

## Acknowledgments

We would like to thank Fabian Fuchs, David Pfau and Christopher Summerfield
for interesting discussions, useful comments and providing references.

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