A Category-theoretical Meta-analysis
of Deﬁnitions of Disentanglement
Yivan Zhang1, 21

Masashi Sugiyama2, 1

The University of Tokyo
2

RIKEN AIP

July 2023

International Conference on Machine Learning 2023

What is disentanglement?
Disentanglement : the process of identifying and separating the underlying
factors of variation in data.
sour?

(a) Composition

taste?
sweet(b) Supervision(c) Generalization

Factors: colors, shapes, and tastes
Task: taste prediction based on colors and shapes
We can predict unseen candies without observing all combinations.
Can a neural network do this?
1/8Existing deﬁnitions
Algebraic approach: group theory, representation theory
Group actions capture the symmetries of an object [Cohen and Welling, 2014,2015]. A disentangled encoder should be equivariant to group actions of adirect product of groups [Higgins et al., 2018].
Statistical approach: probability, statistics, information theory
Probabilistic models capture the relationships and uncertainty of variables.
A disentangled encoder should satisfy certain statistical independence
conditions [Higgins et al., 2017, Chen et al., 2018, Suter et al., 2019].
What do direct product of groups and independent random variables
have in common?
2/8

A uniﬁed perspective?

Questions
What are the deﬁning properties of disentanglement?
Can we deﬁne disentanglement using only sets and functions?
Are the existing algebraic and statistical approaches compatible?
Category theory : cartesian/monoidal product underlies many existingdeﬁnitions of disentanglement.

3/8

Product: core of disentanglement
composite function
identity function
setfunctionset(red, triangle)(blue, triangle)(green, circle)(red, circle)..
.

function

setsoursweet

Set: category of sets and functionsA function C Ñ A ˆ B
to a Cartesian product of sets
is just two component functions
C Ñ A and C Ñ B.
A function A ˆ B Ñ Cfrom a Cartesian product of sets
can depend on both components.

triangle
circleWhen is A ˆ B Ñ C just A Ñ C?

4/8

Deﬁning properties of disentangled representations
Modularity : factor Y Ñ code Z is a product morphism.
red

p1(red, circle)

p2

circleY1

p1

Y1 ˆ Y2

p2

Y2

gg
m1

m2
fcolor„ ȷ
p1

m1

fshapef1

f„ ȷ „ ȷl(, △ )ll△lm2Xff2

„ ȷp2

Z1

p1

Z1 ˆ Z2

p2

Z2

Informativeness : factor Y Ñ code Z is a split monomorphism.
5/8

Equivariant maps

˚a˚ˆˇ
ˇ

aX
paˆbqX
bX

aˆbb

Z1aZ

Z

paˆbqZ“aZ ˆbZ

Z2bZ

ϕ˚X

f :“ϕ˚ ˆ ˇϕˇ

algebra, action, and equivariance ⇝ functors and natural transformations

6/8

Stochastic maps
Z1

Z2“

mY1

Z1

Y2

Z2
mm

Y1

Y2

“

Z1Z2

m1

m2

Y1

Y2

measure, joint, and independence ⇝ Markov category of stochastic maps

7/8

Conclusion

Modularity, direct product, independence ⇝ product in a category
Formulation of disentanglement in more complex problems
More structures and operations beyond product!

8/8

A Category-theoretical Meta-analysis
of Deﬁnitions of Disentanglement
Yivan Zhang1, 21

Masashi Sugiyama2, 1

The University of Tokyo
2

RIKEN AIP

July 2023

International Conference on Machine Learning 2023

References
Ricky TQ Chen, Xuechen Li, Roger B Grosse, and David K Duvenaud. Isolating sources of disentanglement invariational autoencoders. In Neural Information Processing Systems, 2018. URL https://proceedings.ne
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https://arxiv.org/abs/1812.02230. 2
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