Generalised Mutual Information for Discriminative Clustering
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Abstract
In the last decade, recent successes in deep clustering majorly involved the mutualinformation (MI) as an unsupervised objective for training neural networks withincreasing regularisations.
While the quality of the regularisations have been largelydiscussed for improvements, little attention has been dedicated to the relevance of
MI as a clustering objective.
In this paper, we ﬁ rst highlight how the maximisationof MI does not lead to satisfying clusters.
We identi ﬁ ed the Kullback-Leiblerdivergence as the main reason of this behaviour.
Hence, we generalise the mutualinformation by changing its core distance, introducing the generalised mutualinformation (GEMINI): a set of metrics for unsupervised neural network training.
Unlike MI, some GEMINIs do not require regularisations when training.
Some ofthese metrics are geometry-aware thanks to distances or kernels in the data space.
Finally, we highlight that GEMINIs can automatically select a relevant number ofclusters, a property that has been little studied in deep clustering context where thenumber of clusters is a priori unknown.
Introduction
Clustering is a fundamental learning task which consists in separating data samples into severalcategories, each named cluster.
This task hinges on two main questions concerning the assessment ofcorrect clustering and the actual number of clusters that may be contained within the data distribution.
However, this problem is ill-posed since a cluster lacks formal de ﬁ nitions which makes it a hardproblem (Kleinberg, 2003).
Model-based algorithms make assumptions about the true distribution of the data as a result of somelatent distribution of clusters (Bouveyron et al., 2019).
These techniques are able to ﬁ nd the mostlikely cluster assignment to data points.
These models are usually generative, exhibiting an explicitassumption of the prior knowledge on the data.
Early deep models to perform clustering ﬁ rst relied on autoencoders, based on the belief that anencoding space holds satisfactory properties (Xie et al., 2016; Ghasedi Dizaji et al., 2017;
Ji et al.
However, the drawback of these architectures is that they do not guarantee that data sampleswhich should meaningfully be far apart remain so in the feature space.
Early models that droppeddecoders notably used the mutual information (MI) (Krause et al., 2010; Hu et al., 2017) as anobjective to maximise.
The MI can be written in two ways, either as measure of dependency betweentwo variables x and y , e.g. data distribution p ( x ) and cluster assignment p ( y ) :
[Equation]
or as an expected distance between implied distributions and the overall data:
[Equation]
with D KL being the Kullback-Leibler (KL) divergence.
Related works often relied on the notion of
MI as a measure of coherence between cluster assignments and data distribution (Hjelm et al., 2019).
Regularisation techniques were employed to leverage the potential of MI, mostly by specifying modelinvariances, for example with data augmentation (Ji et al., 2019).
The maximisation of MI thus gave way to contrastive learning objectives which aim at learning stablerepresentations of data through such invariance speci ﬁ cations (Chen et al., 2020; Caron et al., 2020).
The contrastive loss maximises the similarity between the features of a sample and its augmentation,while decreasing the similarity with any other sample.
Clustering methods also bene ﬁ ted from recentsuccessful deep architectures (Li et al., 2021; Tao et al., 2021; Huang et al., 2020) by encompassingregularisations in the architecture.
These methods correspond to discriminative clustering where weseek to directly infer cluster given the data distribution.
Initial methods also focused on alternateschemes, for example with curriculum learning (Chang et al., 2017) to iteratively select relevant datasamples for training, by alternating K-means cluster assignment with supervised learning using theinferred labels (Caron et al., 2018), or by proceeding to multiple distinct training steps (Van Gansbekeet al., 2020;
Dang et al., 2021; Park et al., 2021).
However, most of the methods above rarely discuss their robustness when the number of clusters toﬁ nd is different from the amount of preexisting known classes.
While previous work was essentiallymotivated by considering MI as a dependence measure, we explore in this paper the alternativede ﬁ nition of the MI as the expected distance between data distribution implied by the clusters andthe entire data.
We extend it to incorporate cluster-wise comparisons of implied distributions, andquestion the choice of the KL divergence with other possible statistical distances.
Throughout the introduction of the generalised mutual information (GEMINI), the contributions ofthis paper are:
• A demonstration of how the maxima of MI are not suf ﬁ cient criteria for clustering.
Thisextends the contribution of (Tschannen et al., 2019) to the discrete case.•
The introduction of a set of metrics called GEMINIs involving different distances betweendistributions which can incorporate prior knowledge on the geometry of the data.
Some ofthese metrics do not require regularisations.• A highlight of the implicit selection of clusters from GEMINIs which allows to select arelevant number of cluster during training.
Is mutual information a good clustering objective?
We consider in this section a dataset consisting in N unlabelled samples D
= { xxx i } Ni =1 .
We distinguish 64 two major use cases of the mutual information: one where we measure the dependence between twocontinuous variables, as is the case in representation learning, and one where the random variable isdiscrete.
In representation learning, the goal is to construct a continuous representation zzz extractedfrom the data xxx using a learnable distribution of parameters θ .
In clustering, samples xxx are assignedto the discrete variable y through another learnable distribution.
2.1 Representation learning
Representation learning consists in ﬁ nding high-level features zzz i extracted from the data
xxx i in orderto perform a downstream task , e.g. clustering or classi ﬁ cation.
MI between xxx and zzz is a commonchoice for learning features(Hjelm et al., 2019).
However, estimating correctly MI between tworandom variables in continuous domains is often intractable when p ( xxx | zzz ) or p ( zzz | xxx ) is unknown,thus lower bounds are preferred, e.g. variational estimators such as MINE (Belghazi et al., 2018), I NCE (Van den Oord et al., 2018).
Another common choice of loss function to train features arecontrastive losses such as NT-XENT (Chen et al., 2020) where the similarity between the features zzz ifrom data
xxx i is maximised with the features ˜ zzz from a data-augmented ˜
xxx i against any other features zzz j .
Recently, Do et al.
(2021) achieved excellent performances in single-stage methods by highlightingthe link between the I NCE estimator (Van den Oord et al., 2018) and contrastive learning losses.
Representation learning therefore comes at the cost of a complex lower bound estimator on MI,which often requires data augmentation.
Moreover, it was noticed that the MI is hardly predictive ofdownstream tasks (Tschannen et al., 2019) when the variable y is continuous, i.e. a high value of MI
[Figure]
x 12 N ( µ 0 , σ 2 ) + 12 N ( µ 1 , σ 2 ) .
It is clear that Figure 1a presents the best decision boundary and posterior between the two Gaussian distributions.
Yet, as p ( x | y ) converges to a dirac distribution, the MI difference converges to 0.
does not clarify whether the discovered representations are insightful with regards to the target of thedownstream task.
2.2 Discriminative clustering
[Figure]
The MI has been ﬁ rst used as an objective for learning discriminative clustering models (Bridleet al., 1992).
Associated architectures went from simple logistic regression (Krause et al., 2010)to deeper architectures (Hu et al., 2017; Ji et al., 2019).
Beyond architecture improvement, the MImaximisation was also carried with several regularisations.
These regularisations include penaltyterms such as weight decay (Krause et al., 2010) or Virtual Adversarial Training (VAT, Hu et al.
,2017; Miyato et al., 2018b).
Data augmentation was further used to provide invariances in clustering,as well as speci ﬁ c architecture designs like auxiliary clustering heads (Ji et al., 2019).
Rewriting the
MI in terms of entropies:
[Equation]
3) highlights a requirement for balanced clusters, through the cluster entropy term H ( y ) .
Indeed,a uniform distribution maximises the entropy.
This hints that an unregularised discrete mutualinformation for clustering can possibly produce uniformly distributed clusters among samples,regardless of how close they could be.
We highlight this claim in section 2.3.
As an example ofregularisation impact: maximising the MI with ℓ 2 constraint can be equivalent to a soft and regularised
K-Means in a feature space (Jabi et al., 2019).
In clustering, the number of clusters to ﬁ nd is usuallynot known in advance.
Therefore, an interesting clustering algorithm should be able to ﬁ nd a relevantnumber of clusters, i.e. perform model selection.
However, model selection for parametric deepclustering models is expensive (Ronen et al., 2022).
Cluster selection through MI maximisation hasbeen little studied in related works, since experiments usually tasked models to ﬁ nd the (supervised)classes of datasets.
Furthermore, the literature diverged towards deep learning methods focusingmainly on images, yet rarely on other type of data such as tabular data (Min et al., 2018).
2.3 Maximising the MI can lead to bad decision boundaries
Maximising the MI directly can be a poor objective: a high MI value is not necessarily predictive ofthe quality of the features regarding downstream tasks (Tschannen et al., 2019) when y is continuous.
We support a similar argument for the case where the data x is a continuous random variable and thecluster assignment y a categorical variable.
Indeed, the MI can be maximised by setting appropriatelya sharp decision boundary which partitions evenly the data.
This reasoning can be seen in theentropy-based formulation of the MI (Eq. 3): any sharp decision boundary minimises the negativeconditional entropy, while ensuring balanced clusters maximises the entropy of cluster proportions.
Consider for example Figure 1, where a mixture of Gaussian distributions with equal variances isseparated by a sharp decision boundary.
We highlight that both models will have the same mutualinformation on condition that the misplaced decision boundary of Figure 1b splits evenly the dataset(see Appendix A).
Globally, MI misses the idea in clustering that any two points close to one another may be in the samecluster according to some chosen metric.
Hence regularisations are required to ensure this constraint.
A sketch of these insights was mentionned by Bridle et al.
; Corduneanu and Jaakkola (2012).
Extending the mutual information to the generalised mutual information
Given the identi ﬁ ed limitations of MI, we now describe the discriminative clustering frameworkbased on our perception of the mutual information.
We then detail the different statistical distanceswe can use to extend MI to the generalised mutual information (GEMINI).
3.1 The discriminative clustering framework for GEMINIs
We change our view on the mutual information by seeing it as a discriminative clustering objectivethat aims at separating the data distribution given cluster assignments p ( x | y ) from the data distributionp ( x ) according to the KL divergence:
[Equation]
To highlight the discriminative clustering design, we explicitly remove our hypotheses on the datadistribution by writing p data ( xxx ) .
The only part of the model that we design is a conditional distributionp θ ( y | xxx ) that assigns a cluster y to a sample xxx using the parameters θ (Minka, 2005).
This conditionaldistribution can typically be a neural network of adequate design regarding the data, e.g. a CNN, or asimple categorical distribution.
Consequently, the cluster proportions are controlled by θ becausep θ ( y ) =
E [ p θ ( y | xxx )] and so is the conditional distribution p θ ( xxx | y ) even though intractable.
Thisquestions how Eq.
(4) can be computed.
Fortunately, well-known properties of MI can invert thedistributions on which the KL divergence is computed (Bridle et al., 1992; Krause et al., 2010) via
Bayes’ theorem:
[Equation]
which is possible to estimate. Since we highlighted earlier that the KL divergence in the MI canlead to inappropriate decision boundaries, we are interested in replacing it by other distances ordivergences.
However, changing it in Eq.
(5) would focus on the separation of cluster assignmentsfrom the cluster proportions which may be irrelevant to the data distribution.
We rather alter Eq.
(4)to clearly show that we separate data distributions given clusters from the entire data distributionbecause it allows us to take into account the data space geometry.
3.2 The GEMINI
The goal of the GEMINI is to separate data distributions according to an arbitrary distance D , i.e.changing the KL divergence for another divergence or distance in the MI.
Moreover, we questionthe evaluation of the distance between the distribution of the data given a cluster assumption p θ ( xxx | y )and the entire data distribution p data ( xxx ) .
We argue that it is intuitive in clustering to compare thedistribution of one cluster against the distribution of another cluster rather than the data distribution.
This raises the de ﬁ nition of two GEMINIs, one named one-vs-all (OvA):
[Equation]
(6) which compares the cluster distributions to the data distribution, and the one-vs-one (OvO) in whichwe independently draw cluster assignments y a and y b :
[Equation]
There exists other distances than the KL to measure how far two distributions p and q are one fromthe other.
We can make a clear distinction between two types of distances, Csiszar’s f -divergences(Csiszár, 1967) and integral probability metrics (IPM) (Sriperumbudur et al., 2009).
However,unlike f -divergences, IPM-derived distances like the Wasserstein distance or the maximum meandiscrepancy (MMD)(Gneiting and Raftery, 2007; Gretton et al., 2012) bring knowledge about thedata throughout either a distance c or a kernel κ : these distances are geometry-aware.f -divergence GEMINIs: These divergences involve a convex function f : R + → R such that 160 f (1) = 0 .
This function is applied to evaluate the ratio between two distributions p and q , as in
Eq.
[Equation]
We will focus on three f -divergences: the KL divergence, the total variation (TV) distance and thesquared Hellinger distance.
While the KL divergence is the usual divergence for the MI, the TV and
[Table]
p θ ( y b | xxx a ) p θ ( y b | xxx b ) p θ ( y b )− 2 p θ ( y a | xxx a ) p θ ( y b | xxx b ) p θ ( y a ) p θ ( y b )  1#
Wasserstein OvA
E p θ ( y ) h W c  P
Ni =1 m yi δ xxx
i ,
P
Ni =1 1 N δ xxx
i

i Wasserstein OvO
E y a ,y b ∼ p θ ( y ) h W c  P Ni =1 m y a i δ xxx
i ,
P
Ni =
[Equation]
m y b
[Equation]
xxx i  i the squared Hellinger distance present different advantages among f -divergences.
First of all, both ofthem are bounded between 0 and 1.
It is consequently easy to check when any GEMINI using thoseis maximised contrarily to the MI that is bounded by the minimum of the entropies of xxx and y (Grayand Shields, 1977).
When used as distance between data conditional distribution p θ ( xxx | y ) and datadistribution p data ( xxx ) , we can apply Bayes’ theorem in order to get an estimable equation to maximise,which only involves cluster assignment p θ ( y | xxx ) and marginals p θ ( y ) .
MMD GEMINIs: The MMD corresponds to the distance between the respective expected embedding of samples from distribution p and distribution q in a reproducible kernel hilbert space (RKHS)
H :
[Equation]
(9) where φ is the RKHS embedding.
To compute this distance we can use the kernel trick (Gretton et al.,2012) by involving the kernel function κ ( aaa,bbb ) =
⟨ φ ( aaa ) , φ ( bbb ) ⟩ .
We use Bayes’ theorem to uncovera version of the MMD that can be estimated through Monte Carlo using only the predictions p θ ( y | xxx ) .
Wasserstein GEMINIs: This distance is an optimal transport distance, de ﬁ ned as:
[Equation]
where Γ ( p, q ) is the set of all couplings between p and q and c a distance function in X .
Computingthe Wasserstein distance between two distributions p θ ( xxx | y = k 1 ) and p θ ( xxx | y
=
k 2 ) is dif ﬁ cult inour discriminative context because we only have access to a ﬁ nite set of samples N .
To achievethe Wasserstein-GEMINI, we instead use approximations of the distributions with weighted sumsof Diracs:
[Equation]
where δ xxx i is a Dirac located on sample location xxx i ∈
X .
The Wasserstein-OvA and -OvO applied to 183 Dirac sums are compatible with the emd2 function of the python optimal transport package (Flamary
[Figure]
et al., 2021) which gracefully supports automatic differentation (see Appendix D.3 for convergence).
All GEMINIs are summarised in Table 1, (see Appendix D for derivations).
Experiments
For all experiments below, we report the adjusted rand index (ARI) (Hubert and Arabie, 1985), acommon metric in clustering.
This metric is external as it requires labels for evaluation.
It ranges from0, when labels are independent from cluster assignments, to 1, when labels are equivalent to clusterassignments up to permutations.
An ARI close to 0 is equivalent to the best accuracy when votingconstantly for the majority class, e.g. 10% on a balanced 10-class dataset.
Regarding the MMD- and
Wasserstein-GEMINIs, we used by default a linear kernel and the Euclidean distance unless speci ﬁ edotherwise.
All discriminative models are trained using the Adam optimiser (Kingma and Ba, 2014).
We estimate a total of 450 hours of GPU consumption.
(See Appendix I for the details of pythonpackages for experiments and Appendix F for {+further experiments regarding model selection)4.
+}
When the MI fails because of the modelling
We ﬁ rst took the most simple discriminative clustering model, where each cluster assignmentaccording to the input datum follows a categorical distribution:
[Equation]
We generated N = 100 samples from a simple mixture of K = 3 Gaussian distributions.
Eachmodel thus only consists in NK parameters to optimise.
This is a simplistic way of describing themost ﬂ exible deep neural network.
We then maximised on the one hand the KL-OvA (MI) and onthe other hand the MMD-OvA. Both clustering results can be seen in Figure 2.
We concluded thatwithout any function, e.g. a neural network, to link the parameters of the conditional distributionwith xxx , the MI struggles to ﬁ nd the correct decision boundaries.
Indeed, the position of xxx in the 2Dspace plays no role and the decision boundary becomes only relevant with regards to cluster entropymaximisation: a uniform distribution between 3 clusters.
However, it plays a major role in the kernelof the MMD-GEMINI thus solving correctly the problem.
4.2 Resistance to outliers
To prove the strength of using neural networks for clustering trained with GEMINI, we introducedextreme samples in Gaussian mixtures by replacing a Gaussian distribution with a Student-t distribution for which the degree of freedom ρ is small.
We ﬁ xed K = 4 clusters, 3 being drawn frommultivariate Gaussian distributions and the last one from a multivariate Student-t distribution in 2dimensions for visualisation purposes with 1 degree of freedom (see AppendixE for other parametersand results).
Thus, the Student-t distribution has an unde ﬁ ned expectation and produces samples thatcan be perceived as outliers regarding a Gaussian mixture owing to its heavy tail.
We report the ARIsof multi-layered perceptron (MLP) trained 20 times with GEMINIs in Table 2.
The presence of "outliers" leads K-Means and Gaussian Mixture models to fail at grasping the 4 distributions when tring to
[Table]
[Figure]
ﬁ nd 4 clusters.
Meanwhile, GEMINIs perform better.
Note that the MMD and Wasserstein-GEMINIpresent lower standard deviation for high scores compared to f -divergence GEMINIs.
We attributethese performances to both the MLP that tries to ﬁ nd separating hyperplanes in the data space and theabsence of hypotheses regarding the data.
Moreover, as mentioned in section 2.3, the usual MI is bestmaximised when its decision boundary presents little entropy H ( p θ ( y | xxx )) .
As neural networks canbe overcon ﬁ dent (Guo et al., 2017), MI is likely to yield overcon ﬁ dent clustering by minimizing theconditional entropy.
We highlight such behaviour in Figure 3 where the Rényi entropy (Rényi, 1961)associated to each sample in the MI (Figure 3a) is much lower, if not 0, compared to MMD-OvOand Wasserstein-OvO ( ﬁ gures 3b and 3c).
We conclude that Wasserstein- and MMD-GEMINIs trainneural networks not to be overcon ﬁ dent, hence yielding more moderate distributions p θ ( y | xxx ) .
4.3 Leveraging a manifold geometry
[Figure]
We highlighted that MI can be maximised without requiring to ﬁ nd the suitable decision boundary.
Here, we show how the provided distanceto the Wasserstein-OvO GEMINI can leverageappropriate clustering when we have a good apriori on the data.
The importance of the distance c : We generated a dataset consisting of two facing moonson which we trained a MLP using either the MIor the Wasserstein-OvO GEMINI.
To construct a distance c for the Wasserstein distance, we derived adistance from the Floyd-Warshall algorithm (Warshall, 1962; Roy, 1959) on a sparse graph describingneighborhoods of samples.
This distance describes how many neighbors are in between two samples,further details are provided in appendix H. We report the different decision boundaries in Figure 4.
[Table]
We observe that the insight on the neighborhood provided by our distance c helped the MLP toconverge to the correct solution with an appropriate decision boundary unlike the MI.
Note that theusual Euclidean distance in the Wasserstein metric would have converged to a solution similar to the
MI.
Indeed for 2 clusters, the optimal transport plan has a larger value, using a distribution similarto Figure 4a, than in Figure 4b.
This toy example shows how an insightful metric provided to the
Wasserstein distance in GEMINIs can lead to correct decision boundaries while only designing adiscriminative distribution p θ ( y | xxx ) and a distance c .
[Figure]
Not using all clusters In addition, we highlight an interesting behaviour of all GEMINIs.
During optimisation, it is possible that themodel converges to using fewer clusters thanthe number to ﬁ nd.
For example in Figure 5, for5 clusters, the model can converge to 4 balancedclusters and 1 empty cluster (Figure 5b)unlike MI that produced 5 misplaced clusters(Figure 5a).
Indeed, the entropy on the clusterproportion in the MI forces to use the maximumnumber of clusters.
4.4 Fitting MNIST
We trained a neural network using either MI or GEMINIs.
Following Hu et al.
(2017), we ﬁ rst triedwith a MLP with one single hidden layer of dimension 1200.
To further illustrate the robustnessof the method and its adaptability to other architectures, we also experimented using a LeNet-5architecture (LeCun et al., 1998) since it is adequate to the MNIST dataset.
We report our results in
Table 3.
Since we are dealing with a clustering method, we may not know the number of clusters apriori in a dataset.
The only thing that can be said about MNIST is that there are at least 10 clusters,one per digit.
Indeed, writings of digits could differ leading to more clusters than the number ofclasses.
That is why we further tested the same method with 15 clusters to ﬁ nd in Table 3.
We ﬁ rstsee that the scores of the MMD and Wasserstein GEMINIs are greater than the MI, except for the
Wasserstein-OvA which systematically converged to using only 2 clusters.
We cannot explain whythis behaviour was observed in all following experiments.
Therefore we stopped experimenting withit for the remainder of this paper.
We also observe that no f -divergence-GEMINI always yield best
ARIs.
Nonetheless, we observe better performances in the case of the TV GEMINIs owing to itsbounded gradient.
This results in controlled stepsize when doing gradient descent contrarily to KLand squared Hellinger-GEMINIs.
Notice that the change of architecture from a MLP to a LeNet-
[Table]
unexpectedly halves the scores for the f -divergences.
We believe drop is the of by the
4.5 Cifar10 clustering using a SIMCLR-derived kernel
To further illustrate the bene ﬁ ts of the kernel or distance provided to GEMINIs
[Table]
we continue the sameexperiment as in section 4.4.
However
[Table]
we focus this time on the CIFAR10 dataset.
As improveddistance, we chose a linear kernel and ℓ 2 norm between features extracted from a pretrained SIMCLRmodel (Chen et al., 2020).
We provide results for two different architectures:
LeNet-5 and ResNet-18both trained from scratch on raw images, the latter being a common choice of models in deepclustering literature (Van Gansbeke et al., 2020; Tao et al., 2021).
We report the results in Table 4and provide the baseline of MI.
We also write the baselines from related works when not using dataaugmentations to make a fair comparison.
Indeed, models trained with GEMINIs do not use dataaugmentation: only the architecture and the kernel or distance function in the data space plays a role.
We observe here that the choice of kernel or distance can be critical in GEMINIs.
Indeed, while the
Euclidean norm between images does not provide insights on how images of cats and dogs are far asshown by K-Means, features derived from SIMCLR carry much more insight on the proximity ofimages.
This shows that the performances of GEMINIs depend on the quality of distance functions.
Interestingly, we observe that for the Resnet-18 using SIMCLR features to guide GEMINIs was notas successful as it has been on the LeNet-5.
We believe that the ability of this network to draw anydecision boundary makes it equivalent to a categorical distribution model as in Sec. 4.1.
Finally, to thebest of our knowledge, we are the ﬁ rst to train from scratch a standard discriminative neural networkon CIFAR raw images without using labels or direct data augmentations, while getting sensibleclustering results.
However, other recent methods achieve best scores using data augmentations whichwe do not (Park et al., 2021).
Conclusion
We highlighted that the choice of distance at the core of MI can alter the performances of deeplearning models when used as an objective for a deep discriminative clustering.
We ﬁ rst showed that
MI maximisation does not necessarily re ﬂ ect the best decision boundary in clustering.
We introducedthe GEMINI, a method which only needs the speci ﬁ cation of a neural network and a kernel ordistance in the data space.
Moreover, we showed how the notion of neighborhood built by the neuralnetwork can affect the clustering, especially for MI.
To the best of our knowledge, this is the ﬁ rstmethod that trains single-stage neural networks from scratch using neither data augmentations norregularisations, yet achieving good clustering performances.
We emphasised that GEMINIs are onlysearching for a maximum number of clusters: after convergence some may be empty.
However, wedo not have insights to explain this convergence which is part of future work.
Finally, we introducedseveral versions of GEMINIs and would encourage the MMD-OvA as a default choice, since itproves to both incorporate knowledge from the data using a kernel while remaining the less complexthan MMD-OvO and Wasserstein-OvO in time and memory.
OvO versions could be privilegedfor ﬁ ne-tuning steps.
Future works could include an optimisation of the time performances of the
Wasserstein-OvO to make it more competitive.