2006.14304.pdf

Unsupervised deep learning identifies

semantic disentanglement

in single inferotemporal neurons

Irina Higgins

†∗

, Le Chang

†

, Victoria Langston

, Demis Hassabis

Christopher Summerfield

‡

, Doris Tsao

‡

, Matthew Botvinick

‡

DeepMind, London, UK,

Caltech, Pasadena, USA

Chinese Academy of Sciences, Shanghai, China

University College London, London, UK,

University of Oxford, Oxford, UK

Howard Hughes Medical Institute, Pasadena, USA

∗

To whom correspondence should be addressed; E-mail: irinah@google.com

†

Equal contribution,

‡

Equal contribution

Deep supervised neural networks trained to classify objects have emerged as popular models

of computation in the primate ventral stream. These models represent information with a

high-dimensional distributed population code, implying that inferotemporal (IT) responses

are also too complex to interpret at the single-neuron level. We challenge this view by mod-

elling neural responses to faces in the macaque IT with a deep unsupervised generative model,

-VAE. Unlike deep classifiers,

-VAE “disentangles” sensory data into interpretable latent

factors, such as gender or hair length. We found a remarkable correspondence between the

generative factors discovered by the model and those coded by single IT neurons. Moreover,

we were able to reconstruct face images using the signals from just a handful of cells. This sug-

gests that the ventral visual stream may be optimising the disentangling objective, producing

a neural code that is low-dimensional and semantically interpretable at the single-unit level.

Introduction

In search of a basic unit of representation in the neocortex.

What is the basic unit of repre-

sentation in the neocortex, and what computational objective gives rise to it? The foundational

“neuron doctrine” argues that single cells are the key building blocks of brain function

, and decades

of extracellular single neuron recordings have defined canonical coding principles, such as the

sensitivity of early visual neurons to oriented contours and more anterior ventral stream neurons to

complex objects and faces

2, 3

. More recently however, advances in recording methods have permitted

the simultaneous recording of large populations of neurons

. Hand-in-hand with this innovation

has come the idea that meaningful variables (e.g. the gender of face) are encoded not in single

neurons but in neural populations

4–6

, and that previously reported one-to-one mappings between the

declarative aspects of the external world and single neurons may be spurious or misleading

In parallel, visual neuroscience moved beyond handcrafted computational models towards

arXiv:2006.14304v1 [q-bio.NC] 25 Jun 2020

theories that emphasise representation learning through end-to-end optimization

7, 8

. When trained

with high-density teaching signals, contemporary deep networks can outperform humans on mul-

tiway object recognition tasks

, and in doing so form high-dimensional representations that are

multiplexed over many simulated neurons. Examined at the population level, these tuning distribu-

tions closely resemble those in biological systems

, especially in higher-performing networks

allowing deep learning networks to make accurate predictions about neural responses to synthesised

images

. A natural synergy has thus arisen between new tools for multivariate population encoding

and new computational theories that assume that the tuning properties of a single unit are all but

uninterpretable

5–7

Disentangled representation learning through self-supervision.

An important challenge for

theories that rely on deep supervised networks, however, is that external teaching signals are scarce

in the natural world, and visual development relies heavily on untutored statistical learning

13–15

Building on this intuition, one longstanding hypothesis

16, 17

is that the visual system uses self-

supervision to recover the semantically interpretable latent structure of sensory signals, such as the

shape or size of an object, or the gender or age of a face image. While appearing deceptively simple

and intuitive to humans, such interpretable structure has proven hard to recover in practice, since it

forms a highly complex non-linear transformation of pixel-level inputs. Recent advances in machine

learning, however, have offered an implementational blueprint for this theory with the advent of

deep self-supervised generative models that learn to “disentangle” high-dimensional sensory signals

into meaningful factors of variation. One such model, known as the beta-variational autoencoder (

VAE), learns to faithfully reconstruct sensory data from a low-dimensional embedding whilst being

additionally regularised in a way that encourages individual network units to code for semantically

meaningful variables, such as the colour of an object, the gender of a face, or the arrangement of

a scene (Fig. 1a-c)

18–20

. These deep generative models thus continue the longstanding tradition

from the neuroscience community of building self-supervised models of vision

21, 22

, while moving

in a new direction that allows strong generalisation, imagination, abstract reasoning, compositional

inference and other hallmarks of biological visual cognition

19, 23–25

Results

How well do single disentangled latent units explain the responses of single neurons?

If the

computations employed in biological sensory systems resemble those employed by this class of deep

generative model to disentangle the visual world, then contrary to the “population doctrine”

4–6

, the

tuning properties of single neurons should map readily onto the meaningful latent units discovered

by the

-VAE. Here, we tested this hypothesis, drawing on a previously published dataset

neural recordings from 159 neurons in macaque face area AM, made whilst the animals viewed

2,100 natural face images (Fig. 2a, see Online Methods). Using face perception as the test

domain for understanding whether IT neurons may be employing similar disentangling learning

mechanisms to the deep generative models has unique advantages. Specifically, both neural

Preprint. Under review.

responses and image statistics in this domain have been particularly well studied compared to other

visual stimulus classes. This allows for comparisons with strong hand-engineered baselines

using

relatively densely sampled neural data

. Furthermore, although faces make up a small subset

of all possible visual objects, and neurons that preferentially respond to faces tend to cluster in

particular patches of the inferotemporal (IT) cortex

, the computational mechanisms and basic

units of representation employed for face processing may in fact generalise more broadly within the

ventral visual stream

27, 28

We first investigated whether the variation in average spike rates of any of the individual

recorded neurons was strongly explained by the activity in single units of a trained

-VAE that

learnt to “disentangle” the same face dataset that was presented to the primates. For illustration, in

Fig. 1c we show faces that were generated (or “imagined”) by such a

-VAE. Each row of faces is

produced by gradually varying the output of a single network unit (we call these “latent units”), and

it can be seen that they learnt to encode interpretable variables – e.g. hairstyle, age, face shape or

emotional variables such as the presence of a smile. Individual disentangled units discovered by the

-VAE were also able to explain the response variance in single recorded neurons, as shown in Fig.

2b. For example, neuron 95 is shown to be sensitive to the thickness of the hair, and neuron 136 is

shown to respond differentially to the presence of a smile.

To quantify this effect, we used a metric recently proposed in the machine learning literature,

referred to as neural “alignment”

in this work, which measures the extent to which variance

in each neuron’s firing rate can be explained by a single latent unit

, but is insensitive to the

converse, i.e. whether a single unit predicts the response of many biological neurons (Fig. 3a,

see Online Methods). High alignment scores thus indicate that a neural population is intrinsically

low-dimensional, with the factors of variation mapping onto the variables discovered by the latent

units of the neural network. We first compared alignment scores between the

-VAE and the

monkey data to a theoretical ceiling which was obtained by subsampling the neural data to match the

intrinsic dimensionality of the

-VAE latent representation (see Online Methods) and computing its

alignment with itself (Fig. 3b). Remarkably, alignment scores in the

-VAE met this ceiling, with no

reliable difference between the two estimates obtained when the analysis was repeated on multiple

subsamples and with multiple network instances (

= 0

, Welch’s t-test). Furthermore, when

we repeated this analysis while computing alignment against fictitious neural responses obtained

by linearly recombining the original neural data, we found a significant drop in scores for both

the

-VAE and neural subsets (Fig. 3c,

p <

, Welch’s t-test), indicating that the individual

disentangled units discovered by the

-VAE map significantly better onto the responses of single

neurons recorded from macaque IT, rather than onto their linear combinations.

The extent to which the

-VAE is effective in disentangling a dataset into its latent factors

can vary substantially with the way it is regularised, as well as with randomness in its initialisation

and training conditions

. The parameter after which the network class is named determines the

weight of a regularisation term that aims to keep the latent factors independent. Networks with

Two versions of the same measure were simultaneously and independently proposed in the machine learning

literature, referred to as “completeness”

or “compactness”

. We choose to refer to the same measure as “alignment”

for more intuitive exposition.

Preprint. Under review.

higher values of

thus typically give rise to more disentangled representations, as measured by a

metric known as the unsupervised disentanglement ranking (UDR, see Online Methods)

, a finding

we replicate here. However, we also found that networks with higher UDR scores additionally had

higher alignment scores with the neural data (Fig. 3d), and that this relationship held for networks

with the same and different values of

(Fig. 3e). In other words, the better the network was able to

disentangle the latent factors in the face dataset, the more those factors were expressed in single

neurons recorded from macaque IT.

No single aspect of the disentanglement objective is sufficient to achieve high alignment with

neural responses

Next, we compared the

-VAE alignment scores with a number of rival models.

These baseline models were carefully chosen to disambiguate the role played by the different

aspects of the

-VAE design and training in explaining the coding of neurally aligned variables in

its single latent units (see Online Methods). We included a state-of-the-art deep supervised network

(VGG,

) that has previously been proposed as a good model for comparison against neural data

in face recognition tasks

34, 35

, other generative models, such as a basic autoencoder (AE)

and a

variational autoencoder (VAE)

, as well as baselines provided by ICA, PCA and a classifier which

used only the encoder from the

-VAE. We defined “latent units” as those emerging in the deepest

layers of these networks and, where appropriate, used PCA or feature subsampling (e.g. for VGG

raw) to equate the dimensionality of the latent units (to

≤

) to provide a fair comparison with

the

-VAE. We also compared

-VAE to the “gold standard” provided by the previously published

active appearance model (AAM)

, which produced a low-dimensional code that explained the

responses of single neurons to face images well

3, 26

. Unlike the

-VAE, which relied on a general

learning mechanism to discover its latent units, AAM relied on a manual process idiosyncratic to

the face domain. Hence,

-VAE provides a

learning

-based counterpart to the handcrafted AAM

units that could generalise beyond the domain of faces. Although the baselines considered varied in

their average alignment scores (Fig. 4a), none approached those of the

-VAE, for which alignment

was statistically higher than every other model (all p-values

, Welch’s t-test). The alignment

scores broken down by individual neurons are plotted in Fig. 4b for the

-VAE and its baselines.

We validated the findings above using a more direct metric for the coding of latent factors in

single neurons, which compared the ratio between the maximum correlation between spike rates

and activations in each latent unit, and the sum of such correlations over the model units (average

correlation ratio in Fig. 5a, see Online Methods). This ratio was higher for the

-VAE than for other

models, confirming the results with alignment scores (Fig. 5b). Interestingly, different neurons

did not tend to covary with the same

-VAE latent unit. In fact, there was more heterogeneity

among

-VAE units that achieved maximum correlation with the neural responses than among the

equivalent units for other models (Fig. 5c). Rich heterogeneity in response properties of single

neurons (or latent units) is exactly what would be desired to enable a population of computational

units to encode the rich variation in the image dataset.

Taken together these results suggest that no one feature of the

-VAE – its architecture

(baselined by AE, VAE and classifier), training data distribution (baselined by VGG) or isolated

aspects of its learning objective (baselined by PCA and ICA) – was sufficient to explain the coding

Preprint. Under review.

of neurally aligned latent variables in single units. Rather, it was all of these design choices together

that allowed the

-VAE to learn a set of disentangled latent units that explained the responses to

single neurons so well.

Disentangled units carry sufficient information to decode previously unseen faces from as few

as twelve neurons

Finally, we conducted an analysis that sought to link the virtues of the

-VAE

as a tool in machine learning – its capacity to make strong inferences about held out data – with its

qualities emphasised here as a theory of visual cognition – strong one-to-one alignment between

individual neural and individual disentangled latent units. During training we omitted 62 faces that

had been viewed by the monkeys from the training set of the

-VAE, allowing us to verify that

these were reconstructed more faithfully by the

-VAE than by other networks. Critically, in order

to reconstruct these faces, we applied the decoder of the

-VAE not to its latent units as inferred

by its encoder, but rather to the latent unit responses predicted from the activity of a small subset

of single neurons (as few as twelve) that best aligned with each model unit on a different subset

of data (Fig. 6a, see Online Methods). We found that such one-to-one decoding of latent units

from the corresponding single neurons was significantly more accurate for the disentangled latent

units learnt by the

-VAE compared to the latent units learnt by other baseline models (all p-values

, Welch’s t-test) (Fig. 6b). Furthermore, we visualised the

-VAE reconstructions decoded

from just twelve matching neurons (Fig. 6c). Qualitatively, these appeared both more identifiable

and of higher image quality than those produced by the latent units decoded from the nearest rival

models, the AE and the basic VAE, which required twice as many neurons for decoding (Fig. 6c). It

should be noted that the AE was explicitly optimised for reconstruction quality, while the

-VAE

was optimised for disentangling. These results suggest that both the small subset of just twelve

neurons and the corresponding twelve disentangled units carried sufficient information to decode

previously unseen faces, the capacity that is required for effective vision in an unpredictable and

ever-changing natural world.

Discussion

Disentangled representation learning as a predictive computational model of vision.

The

results we have presented here provide evidence that the code for facial identity in the primate IT

may in fact be low-dimensional and interpretable at a single neuron level. In particular, we showed

that the axes of variation represented by single IT neurons align with single semantically meaningful

“disentangled” latent units discovered by the

-VAE, a recent class of self-supervised deep neural

networks proposed in the machine learning community.

Our work extends recent studies of the coding properties of single neurons in the primate

face patch area, reporting finding one-to-one correspondences between model units and neurons, as

opposed to few-to-one as previously reported

. Moreover, we show that disentangling may occur at

the end of the ventral visual stream (IT), extending results recently reported for V1

. Past studies

have proposed that the ventral visual cortex may disentangle

17, 38

and represent visual information

Preprint. Under review.

with a low-dimensional code

3, 39, 40

. However, this work did not ask how these representations

emerge via learning. Here, we propose a theoretically grounded

computational model (the

-VAE)

for how disentangled, low-dimensional codes may be learnt from the statistics of visual inputs

An important aspect of our proposed learning mechanism is that it generalises beyond the

domain of faces

18–20

. We believe that the difficulty in identifying interpretable codes in the IT

encountered in the past may have been due to the fact that semantically meaningful axes of variation

of complex visual objects are more challenging for humans to define (and hence use as visual

probes) compared to simple features, such as visual edges

. A computational model like the

-VAE, on the other hand, is able to automatically discover disentangled latent units that align with

such axes, as was demonstrated for the domain of faces in this work. Hence, assuming that the

computational mechanisms underlying face perception in the brain generalise to the broader set

visual domains

27, 28

-VAE may serve as a promising tool to understand IT codes at a single neuron

level even for rich and complex visual stimuli in the future.

One contribution of this paper is the introduction of novel measures for comparing neural

and model representations. Unlike other often used representation comparison methods which are

insensitive to invertible linear transformations

11, 42

, our methods measure the alignment between

individual neurons and model units. Hence, they do not abstract away the representational form and

preserve the ability to discriminate between alternative computational models that may otherwise

score similarly

While the development of

-VAE for learning disentangled representations was originally

guided by high level neuroscience principles

43–45

, subsequent work in demonstrating the utility

of such representations for intelligent behaviour was primarily done in the machine learning

community

23–25

. In line with the rich history of mutually beneficial interactions between neuro-

science and machine learning

, we hope that the latest insights from machine learning may now

feed back to the neuroscience community to investigate the merit of disentangled representations for

supporting intelligence in the biological systems, in particular as the basis for abstract reasoning

or generalisable and efficient task learning

Acknowledgments

We would like to thank Raia Hadsell, Zeb Kurth-Nelson and Koray Kavukcouglu for comments on

the manuscript.

Preprint. Under review.

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Online methods

Dataset

We used a dataset of 2,162 natural grayscaled, centered and cropped images of frontal views of faces

with neutral facial expressions pasted on a gray 200x200 pixel background as described in

. The

face images were collated from multiple publicly available datasets

49–55

. 62 held out face images

were randomly chosen. These faces were among the 2,100 faces presented to the primates, but not

among the 2,100 faces used to train the models. All models (apart from VGG) were trained on the

same set of faces, which were mirror flipped with respect to the images presented to the primates.

This ensured that the train and test data distributions were similar, but not identical. To train the

Classifier baseline, we augmented the data with 5x5 pixel translations of each face to ensure that

multiple instances were present for each unique face identity. The data was split into 80%/10%/10%

train/validation/test sets.

Neurophysiological data

All neurophysiological data was re-used from

. The data was collected from two male rhesus

macaques (Macaca mulatta) of 7-10 years old. Face patches were determined by identifying regions

responding significantly more to faces than to non-face stimuli while passively viewing images on a

screen in a 3T TIM (Siemens, Munich, Germany) magnet. Tungsten electrodes (18–20 Mohm at

1 kHz, FHC) were used for single-unit recording. Spikes were sampled at 40 kHz. All spike data

were re-sorted with offline spike sorting clustering algorithms (Plexon). Only well-isolated units

were considered for further analysis. Monkeys were head fixed and passively viewed the screen in

a dark room. Eye position was monitored using an infrared eye tracking system (ISCAN). Juice

reward was delivered every 2–4 s if fixation was properly maintained. Images were presented in

random order. All images were presented for 150 ms interleaved by 180 ms of a gray screen. Each

image was presented 3–5 times. The number of spikes in a time window of 50-350 ms after stimulus

onset was counted for each stimulus. See

for further details.

Artificial neurophysiological data

In order to investigate whether the responses of

-VAE units

encoded linear combinations of neural responses, we created artificial neural data by linearly

recombining the responses of the real neurons. We first standardised the responses of the 159

recorded neurons across the 2,100 face images. We then multiplied the original matrix of neural

responses with a random projection matrix

. Each value

of the projection matrix was sampled

from the unit Gaussian distribution. The absolute value of the matrix was then taken, and each

column was normalised to sum to 1.

Neuron subsets

For fairer comparison with the models, which learnt latent representations of size

∈

[10

50]

as will be described below, we sampled neural subsets with fifty or fewer neurons. To

do this, we first uniformly sampled five values from

∈

[10

50]

without replacement to indicate

Preprint. Under review.

the size of the subsets. Then, for each size value we sampled ten random neuron subsets without

replacement, resulting in 50 neuron subsets in total.

Model details

-VAE model

We used the standard architecture and optimisation parameters introduced in

for training the

-VAE (Fig. 7a). The encoder consisted of four convolutional layers (32x4x4 stride 2, 32x4x4

stride 2, 64x4x4 stride 2, and 64x4x4 stride 2), followed by a 256-d fully connected layer and

a 50-d latent representation. The decoder architecture was the reverse of the encoder. We used

ReLU activations throughout. The decoder parametrised a Bernoulli distribution. We used Adam

optimiser with

−

learning rate and trained the models for 1 mln iterations using batch size of

16, which was enough to achieve convergence. The models were trained to optimise the following

disentangling objective:

−

V AE

E

(

)

[

E

φ

(

)

[log

(

)]

−

βKL

(

)

(

)) ]

(1)

where

(

)

is the probability of the image data,

(

)

is the learnt posterior over the latent

units given the data, and

(

)

is the unit Gaussian prior with a diagonal covariance matrix.

Baseline models

We compared

-VAE to a number of baselines to test whether any individual aspects of

-VAE

training could account for the quality of its learnt latent units. To disambiguate the role of the

learning objective, we compared

-VAE to a traditional

autoencoder (AE)

and a basic

variational

autoencoder (VAE)

37, 56

. These models had the same architecture, training data, and optimisation

parameters as the

-VAE (Fig. 7a), but different learning objectives. The AE optimsed the following

objective that tried to optimise the quality of its reconstructions:

E

(

)

(

;

θ,φ

)

−

(2)

where

(

;

θ,φ

)

is the image reconstruction produced by putting the original image through

the encoder and decoder networks parametrised by

and

respectively. The VAE optimised the

variational lower bound on the data distrbution

(

)

V AE

E

(

)

[

E

φ

(

)

[log

(

)]

−

(

)

(

)) ]

(3)

where

(

)

is the learnt posterior over the latent units given the data, and

(

)

is the isotropic

unit Gaussian prior.

To test whether the supervised classification objective could be a good alternative to the

self-supervised disentangling objective, we compared

-VAE to two classifier neural network

baselines. One of these baselines, referred to as the

Classifier

in all figures and text, shared the

encoder architecture, the data distribution and the optimisation parameters with the

-VAE (Fig.

Preprint. Under review.

5b), but instead of disentangling, it was trained to differentiate between the 2,100 faces using a

supervised objective. In particular, the four convolutional layers and the fully connected layer of the

encoder fed into an N-dimensional representation, which was followed by 2,100 logits that were

trained to recognise the unique 2,100 face identities. In order to avoid overfitting, we used early

stopping. The final models trained for between 300 k and 1 mln training steps.

The other classifier baseline was the

VGG-Face

model

(referred to as the VGG in all

figures and text), a more powerful deep network developed for state-of-the-art face recognition

performance and previously chosen as an appropriate computational model for comparison against

neural data in face recognition tasks

34, 35, 57

(Fig. 5c). Similarly to other works

26, 34, 35, 57

, we used a

standard pre-trained VGG network, trained to differentiate between 2,622 unique individuals using

a dataset of 982,803 images

. Note that the data used for VGG training was unrelated to the 2,100

face images presented to the primates. The VGG therefore had a different architecture, training

data distribution and optimisation parameters compared to the

-VAE. The model consisted of 16

convolutional layers, followed by 3 fully connected layers (see

for more details). The last hidden

layer before the classification logits contained 4,096 units. Following the precedent set by

and

we used PCA to reduce the dimensionality of the VGG representation by projecting the activations

in its last hidden layer in response to the 2,100 test faces to the top N principal components (PCs)

(Fig. 5c, referred to as VGG (PCA) in figures). Alternatively, we also randomly subsampled the

units in the last hidden layer of VGG (without replacement) to control for any potential linear

mixing of their responses which PCA could plausibly introduce (Fig. 7c, referred to as VGG (raw)

in figures).

To rule out that the responses of single neurons could be modelled by simply explaining the

variance in the data we compared

-VAE to N PCs produced by applying

principal component

analysis (PCA)

to the 2,100 faces. To rule out the role of simply finding the independent compo-

nents of the data during

-VAE training, we compared

-VAE to the N independent components

discovered by

independent component analysis (ICA)

applied to the 2,100 face images.

Finally, we also compared

-VAE to the

active appearance model (AAM)

. Linear combi-

nations of small numbers of its latent units (six on average) was previously reported to explain

the responses of single neurons in the primate AM area well

3, 26

. We re-used the AAM latent

units from

. These were obtained by setting 80 landmarks on each of the 2,100 facial images

presented to the primates. The positions of landmarks were normalised to calculate the average

shape template. Each face was warped to the average shape using spline interpolation. The warped

image was normalised and reshaped to a 1-d vector. PCA was carried out on landmark positions

and shape-free intensity independently. The first N/2 shape PCs and the first N/2 appearance PCs

were concatenated to produce the N-dimensional AAM representations (Fig. 7d).

Training procedure and model selection

To ensure that all models had a fair chance of learning a useful representation, we trained multiple

instances of each model class using different hyperparameter settings. The choice of hyperparam-

eters and their values were dependent on the model class. However, all models went through the

Preprint. Under review.

same model selection pipeline: 1)

model instances with different hyperparameter settings were

obtained as appropriate; 2)

⊆

models with the best performance on the training objective were

selected; 3) models that did not discover any latent units that shared information with the neural

responses were excluded, resulting in

⊆

models retained for the final analyses. These steps

are expanded below for each model class.

Hyperparameter sweep

For the

-VAE model the main hyperparameter of interest that affects

the quality of the learnt latent units is the value of

. The

hyperparameter controls the degree of

disentangling achieved during training, as well as the intrinsic dimensionality of the learnt latent

representation

. Typically a

β >

is necessary to achieve good disentangling, however the exact

value differs for different datasets. Hence, we trained 400 models with different values of

uniformly sampling 40 values of

in the

20]

range. Another factor that affects the quality

of disentangled representation is the random initialisation seed for training the models

. Hence,

for each

value, we trained 10 models from different random initialisation seeds, resulting in

the total pool of 400 trained

-VAE model instances to choose from. All

-VAE models were

initialised to have

= 50

latent units, however due to the variability in the values of

, the intrinsic

dimensionality of the trained models varied between ten and fifty.

In order to isolate the role of disentangling within the

-VAE optimisation objective from

the self-supervision aspect of training, we kept as many choices as possible unchanged between

the

-VAE and the AE/VAE baselines: the model architecture, optimiser, learning rate, batch size

and number of training steps. The remaining free hyperparameters that could affect the quality

of the AE/VAE learnt latent units were the random initialisation seeds, and the number of latent

units

. The latter was necessary to sweep over explicitly, since AE and VAE models do not

have an equivalent to the

hyperparameter that affects the intrinsic dimensionality of the learnt

representation. Hence, we trained 100 model instances for each of the AE and VAE model classes,

with five values of

sampled uniformly without replacement from

∈

[10

50]

, each trained from

twenty random initialisation seed values.

For the Classifier baseline we used the following hyperparameters for the initial selection:

five values of

∈

[10

50]

sampled uniformly without replacement, as well as a number of learning

rate values

{

−

}

and batch sizes

{

128

256

}

, resulting in

100 model instances. We trained the models with early stopping to avoid overfitting, and used the

classification performance on the validation set to choose the settings for the learning rate and batch

size. We found that the values used for training

-VAE, AE and VAE (learning rate

−

, batch

size 16) were also reasonable for training the Classifier, achieving

95%

classification accuracy.

Hence, we trained the final set of 450 Classifier model instances with fixed learning rate and batch

size, five values of

∈

[10

50]

sampled uniformly without replacement and fifty random seeds.

We used FastICA algorithm

to extract ICA units, which is dependent on the random

initialisation seed. Hence, we extracted

∈

[10

50]

independent components with ten random

initialisation seeds each, resulting in 41 ICA model instances.

The remaining baseline models relied on using a single canonical model instance (VGG and

AAM) and/or on a deterministic dimensionality reduction process (PCA, AAM, VGG). Hence, the

Preprint. Under review.

random seed hyperparameter did not apply to them. In order to make a fairer comparison with the

other baselines, we therefore created different model instances by extracting different numbers of

representation dimensions with

∈

[10

50]

, resulting in 41 PCA and VGG (PCA) model instances,

and 21 AAM instances (since

needs to split evenly into shape and appearance related units).

For the VGG (raw) variant, we first uniformly sampled five values from

∈

[10

50]

without

replacement to indicate the size of the hidden unit subsets. Then, for each size value we sampled

ten random hidden unit subsets without replacement, resulting in 50 VGG (raw) model instances in

total.

Model selection based on training performance

For each model class, apart from the deter-

ministic baselines (PCA, AAM and VGG), we selected a subset of model instances based on

their training performance. For the

-VAEs, we used the recently proposed Unsupervised Dis-

entanglement Ranking (UDR) score

to select 51 model instances with the most disentangled

representations (within the top 15% of UDR scores) for further analysis. For AE baseline, we

selected 50 model instances with the lowest reconstruction error per chosen value of

. For the

VAE baseline we selected 50 model instances with the highest lower bound on the training data

distribution per chosen value of

. Finally, for the Classifier baseline, we selected 81 models which

achieved

95%

classification accuracy on the test set.

Filtering out uninformative models

To ensure that all models used in the final analyses shared

at least some information with the recorded neural population, we performed the following filtering

procedure. First, we trained Lasso regressors as per Variance Explained section below, to predict the

responses of each neuron across the 2,100 faces from the population of latent units extracted from

each trained model. We then calculated the mean amount of Variance Explained (VE) averaged

across all neurons for each of the models. We filtered out all models where

V E <

V E

−

(

V E

)

with

V E

and

(

V E

)

represent the mean and standard deviation of VE scores across all models

respectively.

The full model selection pipeline resulted in 51

-VAE model instances, 50 AE, VAE and

ICA model instances, 41 PCA and VGG (PCA) model instances, 22 VGG (raw) model instances,

21 AAM model instances and 64 Classifier model instances that were used for further analyses.

Analysis methods

Variance explained

We used Lasso regression to predict the response of each neuron

from

model units. We used 10-fold cross validation using standardised units and neural responses to find

the sparsest weight matrix that produced mean squared error (MSE) results between the predicted

neural responses

ˆn

and the real neural responses

no more than one standard error away from the

smallest MSE obtained using 100 lambda values. The learnt weight vectors were used to predict the

neural responses from model units on the test set of images. Variance explained (VE) was calculated

on the test set according to the following:

Preprint. Under review.

= 1

−

∑

(

ˆn

−

)

∑

(

−

)

where

is the neuron index,

is the test image index, and

is the mean response magnitude

for neuron

across all test images. In order to speed up the Lasso regression calculations, we

manually zeroed out the responses of those model units that did not carry much information about

the face images. We defined units as “uninformative” if their standardised responses had low

variance

across the dataset of 2,100 faces. We verified that this did not affect the sparsity

of the resulting Lasso regression weights.

Alignment score

We used the completeness score from

, referred to as alignment score in text

for more intuitive exposition. First we obtained the matrix

necessary for calculating the score

by training Lasso regressors to predict the responses of each neuron from the population of model

latent units. When calculating completeness against the original neural responses, we followed

the same procedure as per the variance explained calculations. When calculating completeness

against the artificial (linearly recombined) neural responses, we did not zero out the responses of

the “uninformative” units, since in this case this procedure affected the sparsity of the resulting

Lasso regression weights. Instead, in order to speed up calculations, we reduced the number of

cross validation splits from ten to three. The completeness score

for neuron

was calculated

according to the following:

−

(

))

(4)

(

) =

−

∑

log

(5)

∑

(6)

∑

(7)

where

indexes over neurons,

indexes over model units, and

is the total number of model

units. The overall completeness score per model is equal to the sum of all per-neuron completeness

scores

∑

. See

for more details.

Unsupervised Disentanglement Ranking (UDR) score

The UDR score

measures the quality

of disentanglement achieved by trained

-VAE models by performing pairwise comparisons between

the representations learnt by models trained using the same hyperparameter setting but with different

seeds. This approach requires no access to labels or neural data. We used the Spearman version of

the UDR score described in

. For each trained

-VAE model we performed 9 pairwise comparisons

with all other models trained with the same

value and calculated the corresponding UDR

score,

Preprint. Under review.

where

and

index the two

-VAE models. Each UDR

score is calculated by computing the

similarity matrix

, where each entry is the Spearman correlation between the responses of

individual latent units of the two models. The absolute value of the similarity matrix is then taken

and the final score for each pair of models is calculated according to:

[

∑

∗

(

)

∑

(

a,b

)

∑

∗

(

)

∑

(

a,b

)

]

(8)

where

and

index into the latent units of models

and

respectively,

= max

(

a,b

)

and

= max

(

a,b

)

indicate the “informative” latent units within each model, and

is the

number of such latent units. The final score for model

is calculated by taking the median of UDR

across all

Average correlation ratio and average unit proportion

For each neuron we calculated the

absolute magnitude of Pearson correlation with each of the “informative” model units. We then

calculated the ratio between the highest correlation and the sum of all correlations per neuron. The

ratio scores were then averaged (mean) across the set of unique model units with the highest ratios,

and this formed the

average correlation ratio

score per model. The number of unique model units

with the highest ratios divided by the total number of informative model units formed the

average

unit proportion

score.

Decoding novel faces from single neurons

We first found the best one-to-one match between

single model units and corresponding single neurons. To do this, we calculated a correlation matrix

Corr

(

)

between the responses of each model unit

and the responses of each neuron

over the subset of 2,038 face images that were seen by both the models and the primates, where

Corr

stands for Pearson correlation. We then used Hopcroft-Karp

algorithm to find the best

one-to-one assignment between each model unit and a unique neuron based on the lowest overall

−

)

score across all matchings. We used the resulting one-to-one assignments to regress the

responses of single latent units from the responses of their corresponding single neurons to the

heldout 62 faces, using the same subset of 2,038 face images that were seen by both the models

and the primates for estimating the regression parameters. We standardised both model units and

neural responses for the regression. The resulting predicted latent unit responses were fed into

the pre-trained model decoder to obtain reconstructions of the novel faces. We calculated the

cosine distance between the standardised predicted and real latent unit responses for each face (after

filtering out the “uninformative” units), and presented the mean scores across the 62 held out faces

for each model.

Statistical tests

We used a two-tailed Welsch’s t-test for all pairwise model comparisons.

Preprint. Under review.

CNN

reconstructed

image

input

image

latent units

decoder

encoder

-3

age

ethnicity

skin

shade

fringe

hair

length

hair

thickness

-3

horizontal

position

vertical

position

size

unit 1

unit 2

unit 3

face

shape

smile

-3

age

hair

colour

gender

unit 1

unit 2

unit 3

unit 1

-3

rotation

chair

type

leg

type

unit 2

unit 3

-3

nose

prominence

eye

distance

eyebrow

thickness

unit 1

unit 2

unit 3

Figure 1:

Disentangled representation learning. a.

Latent traversals used to visualise the semantic

meaning encoded by single disentangled latent units of a trained model. In each row the value of a

single latent unit is varied between -3 and 3, while the other units are fixed. The resulting effect on

the reconstruction is visualised. Each column represents a different model trained to disentangle a

different dataset. Aspects of some sub-figures are reproduced with the permission of Burgess et

and Lee et al

Schematic representation of a self-supervised deep neural network. The

encoder maps the input image into a low dimensional latent representation, which is used by the

decoder to reconstruct the original image. Blue indicates trainable neural network units that are

free to represent anything. Pink indicates latent representation units that are compared to neurons.

CNN, convolutional neural network. FC, fully connected neural network.

Latent traversals of

eight units of a

-VAE model trained to disentangle 2,100 natural face images. The initial values of

all latent units were obtained by encoding the same input image.

Preprint. Under review.

neuron 95

neuron 117

neuron 140

unit 2 - hair thickness

unit 3 - hair length

unit 4 - face shape

neuron 151

unit 5 - ethnicity

neuron 136

unit 6 - smile

neuron 68

unit 10 - age

variance explained

Figure 2:

Responses of single neurons are well explained by single disentangled latent units.

Coronal section showing the location of fMRI-identified face patches in two primates, with patch

AM circled in red. Dark black lines, electrodes.

Explained variance of single neuron responses to

2,100 faces. Response variance in single neurons is explained primarily by single disentangled units

encoding different semantically meaningful information (insets, latent traversals as in Fig. 1a,c).

Preprint. Under review.