Comparing the information content of probabilistic representation spaces
Authors: Kieran A. Murphy
Sam Dillavou
Danielle Bassett
Venue: Accepted by TMLR ()

Abstract: Probabilistic representation spaces convey information about a dataset and are shaped by factors such as the training data, network architecture, and loss function. 
Comparing the information content of such spaces is crucial for understanding the learning process, yet most existing methods assume point-based representations, neglecting the distributional nature of probabilistic spaces. 
To address this gap, we propose two information-theoretic measures to compare general probabilistic representation spaces by extending classic methods to compare the information content of hard clustering assignments.
Additionally, we introduce a lightweight method of estimation that is based on fingerprinting a representation space with a sample of the dataset, designed for scenarios where the communicated information is limited to a few bits.
We demonstrate the utility of these measures in three case studies. 
First, in the context of unsupervised disentanglement, we identify recurring information fragments within individual latent dimensions of VAE and InfoGAN ensembles. 
Second, we compare the full latent spaces of models and reveal consistent information content across datasets and methods, despite variability during training. 
Finally, we leverage the differentiability of our measures to perform model fusion, synthesizing the information content of weak learners into a single, coherent representation. 
Across these applications, the direct comparison of information content offers a natural basis for characterizing the processing of information.

Published in Transactions on Machine Learning Research (02/2025)
Comparing the information content of probabilistic representation spaces
Kieran A. Murphy I
kieronm upenn. edu
Dept. of Bioengineering, Universily of Penmsyloania
Sam Dillavou
dillavou@sas. upenn. edu
Dept. Physics & Astronomy, University of Pennsylvania
Dani S. Bassett
dsb lsbass upenn.
Depts. of Bioengineering, Electrical 8 Systems Engineering, Physica & Astronomy, Neurology, Psychiatry, University of Pennsylvonia; The Santo Fe Institute; The Neuro, Montreal Neurological Institute, McGill University
Reviewed on OpenReview: https: openreview. net/ forum? id= adahhNgUUR11
Abstract
Probabilistic representation spaces convey information about a dataset and are shaped by factors such as the training data, network architecture, and loss function. Comparing the information content of such spaces is crucial for understanding the learning process, yet most existing methods assume point- representations, neglecting the distributional nature of probabilistic spaces. To address this gap, we propose two information--hoorcti measures to compare general probabilistic representation spaces by extending classic methods to compare the information content of hard clustering assignments. Additionally, introduce lightweight method of estimation that is based fingerprinting representation space with sample of the dataset, designed for scenarios where the communicated information is limited to a few bits. We demonstrate the utility of these measures in three case studies. First, in the context of unsupervised disentanglement, we identify recurring information fragments within individual latent dimensions of VAE and InfoGAN ensembles. Second, we compare the full latent spaces of models and reveal consistent information content across datasets and methods, despite variability during training. Finally, we leverage the differentiability of our measures to perform model fusion, synthesizing the information content of weak learners into single, coherent representation ation. Across these applications, the direct comparison of information content offers natural basis for characterizing the processing of information.
1 Introduction
The comparison of representation spaces is problem that has received much attention, particularly route to deeper understanding of information processing systems (Klabunde et al., 2023; Mao et al., 2024; Huh et al., 2024; Lin & Kriegeskorte, 2024) and with applications in tasks like transfer learning, ensembling, and other forms of representational alignment (Sucholutsky et al., 2023; Muttenthaler al., 2024). Existing methods are applicable to point-based representation spaces, including centered kernel alignment (CKA) (Kornblith et al., 2019) and reprosentational similarity analysis (RSA) (Kriegeskorte et al., 2008). For representation spaces whose citizens are probability distributions, such in variational autoencoders (VAEs) in biological systems with inherent stochasticity, failure to account for the distributed nature of representations can miss important aspects of the relational structure between data points (Duong et al., 2023). Few methods account for the distributional nature of representations (Klabunde et al., 2023).
One of the few existing methods for comparing probabilistic representation spaces, stochastic shape met- rics (Duong et al., 2023), relics on geometric assumptions and fixed transformations, which can limit its flexibility. In contrast, we adopt an the information they transmit, providing method that is agnostic to properties like dimensionality
approach that evaluates representation spaces basod

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Figure 1: Similarity of representation spaces. In this work, we generalize measures compare the information content of clustering assignments apply to probabilistic representation spaces. (a) A hard clustering assignment, such as the information about the dataset (here, CIFAR-10 images). Comparing the information content of different clustering assignments enables comparative analyses between algorithms, model fusion, and benchmarking, (b) We generalize measures for comparing hard clustering assignments to be applicable to probabilistic representation spaces, by recognizing the latter as soft clustering assignments. When cast in terms of information content, there is no requirement for the dimensionality of the spaces match, and hard clusterings (e.g. labels annotations) be compared probabilistic spaces.
distinction conveyed by clustering V, communicates certain
whether the spaces are discrete or continuous. Specifically, we compare two probabilistie representation spaces using quantities derived from the mutual information between them, capturing their relational structure while ensuring broad applicability across diverse contexts.
We take as motivating example the task of unsupervised disentanglement, whose goal is to break information about dataset into useful factors of variation (Higgins et al., 2018; Locatello et al., 2019; Träuble el al., 2021; Balabin et al., 2023; Van Steenkiste et al., 2019). As an example, a representation space might be trained on images of cars so that color, orientation, and model information are separated into different latent dimensions without any supervision about such factors. When ground truth factors of variation are unavailable for evaluation, is generally the for real-world datasets, existing evaluation methods assess the degree of consensus in ensemble of trained models (Duan et al., 2020; Lin et al., 2020). However, the relatedness of representation spaces has failed to account for the probabilistic nature of the representations, reducing posterior distributions to their means and then using point-based comparisons such as correlation (Duan et al., 2020). With a direct comparison of the information content of representation spaces, we stand to improve the characterization consensus and of unsupervised disentanglement more generally.
In this work, we generalize measures of similarity of the information content of hard clustering assignments (Fig. la) to be applicable to probabilistic representation spaces (Fig. 1b). To motivate the generalization, treat probabilistic representation spaces soft clustering assigunents of data, whereby partial distin- guishability between data points is expressed by the overlap between posterior distributions, We then assess the information content of learned representation spaces produced by models an ensemble random initializations, and study the effects method, training progress, and datasct on information processing.
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Related work
The proposed method builds upon classic means of comparing the information content of different cluster assignments (clusterings) of a dataset and generalizes them to compare probabilistic representation spaces. We then use the method to empirically study generative models designed to fragment information about a dataset a learned latent space, i.e, for unsupervised disentanglement.
Similarity of clusterings and of representation spaces. The capacity compare transformations of data produced by different machine learning models enables ensemble learning, deeper understanding of
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methodology, and benchmarking (Punera & Ghosh, 2007). Strehl & Ghosh (2002) used outputs of clustering algorithms to perform ensemble learning, based on measure of similarity between clustering assignments that we will extend in this work: the normalized mutual information (NMI). Referred to as consensus clustering or ensemble clustering, efforts to combine multiple clustcring outputs can leverage any of a varicty of similarity measures (Wagner& & Wagner, 2007; Vinh et al., 2009; 2010; Vega-Pons & Ruiz-Shulcloper, 2011; Huang et al., 2017). Another measure backed by information theory the variation of information (VT), coined by Meilä (2003) for clustering but recognized as a metric distance between information sources least twice before (Shannon, 1953; Crutchfield, 1990).
Commonly referred to simply as 'clustering'. hard clusterings assign every input datum to one and only one output cluster and have been generalized to multiple forms of soft clustering (Campagner et al., 2023). We focus specifically on fuzzy clustering, where membership is assigned to multiple clusters and must sum to one for each datum (Zadch, 1965; Dunn, 1973; Ruspini et al., 2019). We observe that a probabilistic representation space communicates for each datum a soft assignment over latent vectors, with the degree of membership expressed by the probability density of posterior distributions. Comparisons of hard clustering have been extended to fuzzy clustering for measures that indirectly assess information content, such the Rand index (Punera & Ghosh, 2007; Hullermeier et al., 2011; D' D'Ambrosio et al., 2021; Andrews et al., 2022; Wang et al., 2022). Information- based measures have been extended for specific types of soft clusters (Campagner & Ciucci, 2019; Campagner et al., 2023) but none, to our awareness, for comparing the information content. of fuzzy clusterings over a continuous representation space.
A rich area of research compares point- based representation spaces via the pairwise geometric similarity of a common set of data points in the space (Kornblith et al., 2019; Hermann & Lampinen, 2020; Klabunde et al., 2023), building upon representational similarity analysis from neuroscience (Kriegeskorte et al., 2008). In place of geometric similarity, topological similarity more closely probes the information available for downstream processing by employing specific similarity functions with tunable parameters (Lin & Kriegeskorte, 2024; Williams, 2024). For comparing probabilistic representation spaces, stochastic shape metrics (Duong et al., 2023) exlend a distance metric between point-based representation spaces based on aligning one with another through prescribed transformations (e.g., rotations) (Williams et al., 2021). By contrast, the information- theoretic lens we adopt requires no enumcration of transformations nor tuning of parameters, and directly assesses the information available for processing by downstream neural networks.
Unsupervised disentanglement. Disentanglement is the problem of splitting information into useful pieces, possibly for interpretability, compositionality, or stronger representations for downstream tasks. Shown to be impossible in the fully unsupervised case (Locatello et al., 2019; Khemakhem et al., 2020), research has moved to investigate how to utilize weak supervision (Khemakhem et al., 2020; Sanchez et al., 2020; Vowels et al., 2020; Murphy et al., 2022) and incorporate inductive biases (Balabin et al., 2023; Chen et al., 2018; Rolinek et al., 2019; Zietlow et al., 2021; Hsu et al., 2024).
A significant challenge for disentanglement is evaluation when no ground truth factorization is available. While one proposed route to evaluation relics on charactcrization of the posterior distributions in model (PIPE, Estermann & Wattenhofer (2023)), a more common approach assesses consensus among trained models in an ensemble of randomly initialized repeats. The motivation, first proposed in Duan et al. (2020), is that discntangled models are more similar to each other than are entanglod ones because there are intuitively more ways to entangle information than to disentangle it. There, the similarity between two models was evaluated with an ad hoc function of dimension-wise similarity, which was computed as the rank correlation or the weights of a linear model between embeddings in different one-dimensional spaces. ModelCentrality (Lin et al., 2020) quantified similarity between models with the FactorVAE score (Kim & Mnih, 2018) where one model's embeddings serve as the labels for another model. However, none of these met hods fully account for the distributional nature of representations, instead relying on point estimates such as posterior means or single embeddings. In contrast, our approach focuses on capturing the probabilistic structure of representation spaces, and additionally shifts the emphasis from model similarity to channel (or latent subspace) similarity under the premise that the fragmentation of information, central to disentanglement, is more naturally studied via the information fragments themselves.
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3 Method
Our goal is to compare the information transmitted about a dataset by different representation spaces, and we will use the comparison of hard clustering assigmments (e.g., the output of k-means) as a point of reference. Analogously to hard clustering (Fig, la), probabilistic representation spaces communicate a soft assignment over embeddings that is expressed by a probability distribution the space for each data point (Fig. 1b).
While the proposed method can be applied to any representation space with probabilistic embeddings, we will focus primarily variational autoencoders (VAEs) (Kingma & Welling, 2014). Let the random variable X represent a sample I p(x) from the dataset under study. X is transformed by stochastic encoder parameterized by neural network to a variable U = f(X,e), where e is source of stochasticity. We will make use of two fundamental quantities in information theory (Cover & Thomas, 1999), the entropy of a random variable ((6) = [==-(=)[--oo p(>] and the mutual information between two random variables I(Y; = H(Y + (2) Z). The encoder maps each data point æ to a posterior distribution in the latent space, p(u[z), after which a point p(u(z)) sampled for downstream processing. Commonly, the posterior distributions are parameterized normal distributions with diagonal covariance matricos, which will facilitate many of the involved measurements but is not required for the method.
3.1 Comparing representation spaces as soft clusterings
Consider hard clustering of data as communicating certain information about the data (Fig. la). By observing the cluster assignment U instead of sample X from the dataset, information I(X;U) will have been conveyed. For hard clustering, every data point is assigned unambiguously to cluster -ie., (U 0- which makes the communicated information equal to the entropy of the clustering, I(X; = HU). We note that maximizing communicated information, such as by assigning each data point to its own cluster, does not yield a useful representation. The value of a representation lies in the balance between the information preserved and the irrelevant variation discarded, highlighting the importance of assessing its specific information content.
Given two hard clustering assignments U and V for the same data X, the mutual information I(U;V) measures the amount of shared information content they express about the data. Previous works have found it useful to relate the mutual information to functions of the entropies HU)) and HVV.. Strehl & Ghosh (2002) proposed the normalized mutual information (NMI) as the ratio of the mutual information and the geometric mean of the entropies,
NMI(U,V) I(U.V)
(1)
VH(U)H(V)
Meilá (2003) proposed the variation of information (VI), a metric distance between clusterings that is proportional to the difference of the mutual information and the arithmetic mean of the entropies,
V)(U, - (I(::)) (UU)
H(U) + HH())
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Soft clustering extends hard clustering to allow each datum to have partial membership in multiple clusters, allowing partial distinguishability between data points to be communicated (Zadeh, 1965; Ruspini et al., 2019) (Fig. 1b). While soft clustering is predominantly performed over a discrete set of clusters, here we view each point u in a continuous latent space as a cluster, with the posterior distribution p(u[ær assigning membership over the continuum.
The utility of NMI or VI over raw mutual information largely resides in the standardization of values. For two identical hard clustering assignments, U and U', NMI(U, = 1 and VI(U,U") = O. In contrast, soft assignments include additional entropy stemming from uncertainty that is unrelated to the information with H(UIX) > 0.
communicated about the dataset, as is evident from the relation HU)) (IX:U))HHUU) As a result, the entropy terms in Eqns. 1 and 2 that ground the mutual information [U; have components that disrupt the standardization of values. To address this issuc, we propose replacing the entropy of a clustering assignment with the mutual information between two copies of that assignment: specifically, substituting HU) with (U;U"), and similarly for V. For hard clustering, H(U) T(X:U) I(U:U")
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making these quantities interchangeable. Only the third quantity, I(U maintains the standardization of NMT = 1 and VI = 0 for identical soft clustering assignments. The generalized forms then become
I(U.V)
NMI(U.V) VIUUUUNI(VVVV) and
(3)
VI(U,V) - - (1(:V)
- 10U00) - +
(4)
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The generalization leaves NMI and VI unchanged for hard clustering, where I(U HU),, and enables the measures to be applicd to both soft and hard assignments. We note that the generalized VI is no longer proper metric, as the triangle inequality is not guaranteed (Appx. D).
The conditional independence of clustering assignments given the data i.e., I(U; VIX) = 0 allows each mutual information term in Eqns. 3 and to be rewritten with regard to information communicated about the data. Namely, T(U: V) = T(; U) + T(X;V) T(X; U, V), and T(U = 2T(X;U) - (X);UU"), Because we have access the posterior distributions, is easier to estimate the mutual information between the data X and a representation space (or a combination thereof) than it is to estimate the information between two representation spaces (Poole et al., 2019).
The extended NMI and VI benefit from the generality of information theory: the information content of two probabilistic representation spaces can be compared regardless of dimensionality or parameterization of posteriors, and a soft clustering can be compared to a hard clustering, whether from a quantized latent space or discrete labelling process (Fig. 1b).
3.2 Routes to estimation
We propose two routes to estimating NMI and VI that offer a tradeoff between precision and speed; both leverage the known posterior distributions to calculate the information transmitted about the dataset by combinations of representation spaces, T(X;-). The first route to compute r(X:-) :) with a straightforward Monte Carlo estimate using the aggregated posterior over the entire dataset of size L,
p(u[x)
I(X:U)- = Benp(a)
Eurp(ale) (arnpt))eurpuupee
log
(5)
Σ p ((/a))))
For the second route, we can use a measure of statistical similarity between posteriors p(u(m)) and p(u(22) in given representation space- space-the Bhattacharyya coefficient (Kailath, 1967). BC(p, q) = - V p(())(()dd= to quickly "fingerprint" the information content of spaces with the pairwisc distinguishability of a sample of data points (Murphy & Bassett, 2023). The BC between two multivariate normal distributions can be efficiently computed in bulk via array operations. Once a matrix BC,y == BC(p(u p()l((()))))) of the pairwise values for a random sample of data points is obtained, a lower bound for the information transmitted by the channel can be estimated with I(X; U) 2 N E. log 1 ΣS BC,, (Kolchinsky & Traccy, 2017). For fast estimation of N the information content of a representation space with respect to a ground truth generative factor, we can treat the labels as an effective hard clustering according to that factor, where BC values are either Finally, the matrix BC,j for the combination of spaces U, and U2 is their elementwise product (Appx. C). In other words, receiving both of the messages from U1 and U2 leads to distinguishability between data points that is simply the product of the distinguishabilltiss under U, and U2 separately. Together, the properties allow us to fingerprint each representation space through Bhattacharyya matrices, and then perform all subsequent analysis with only the matrices ie., without having to load the models into memory again.
one.
3.3 Discovering consistently learned information fragments via OPTICS clustering of latent dimensions
In the context of disentanglement, we are interested in the similarity of information contained in individual dimensions across an ensemble of models. We compute the pairwise similarities between all dimensions of all modcls-for example, 10 dimensions each from 50 models an ensemble for a 500x500 matrix of similarity values in Sec. 4.2-and then use density- based clustering to identify information content that is found consistently. OPTICS (Ankerst et al., 1999) is an algorithm that orders elements in a set according to
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similarity and produces "reachability" profile where natural groupings of elements are indicated through valleys. An OPTICS profile is analogous to dendrogram produced by hierarchical clustering, but can be more comprehensible for large sets of points (Sander et al., 2003). The consistency of information fragments in the latent dimensions of an ensemble of representation spaces can then be visualized with the OPTICS reachability profile, the reordered pairwise similarity matrix which will have block diagonal form the fragments are consistent-and the information contained about ground truth generative factors, if available.
3.4 Model fusion
Consider a set of representation spaces found by an ensemble of weak learners. In the spirit of "knowledge reuse" (Strehl & Ghosh, 2002) originally applied to ensembles of hard clusters, we might obtain a superior representation space from the synthesis of the set. In contrast to other assessments of representation space similarity (Duan et al., 2020; Kim & Mnih, 2018; Duong et al., 2023), the proposed measures of similarity are composed of mutual information terms, which can be optimized with differentiable operations by variety of means (Poole et al., 2019). We optimize synthesis space by maximizing its average similarity with a set of reference spaces, performing gradient descent directly on the encoding of the data points used for the Bhattacharyya matrices. Instead of requiring many models to remain in memory during training of new encoder, the Bhattacharyya matricos are computed for the ensemble once and then used for comparison during training, and training is fast because of the involved array operations.
4 Experiments
4.1
Comparison of related methods on synthetic spaces
We first evaluated the similarity of synthetic representation spaces using ours and related methods (Fig. 2). A dataset of 64 points was embedded to one- and two-dimensional representation spaces, each communicating different information about the dataset. To assess the information available for downstream processing, we trained a classification head for each of the nine latent spaces to predict the input x from a sample of the corresponding posterior, u ~ p(u]zr) As shown in Fig. 2b, the classification head's average predictions, Eu--ptlleslD((tylll))) [p(}ylu)], reflected the overlap of posterior distributions in latent space by assigning non-zero probabilities to multiple outputs. The structure of these output probabilities closely matched the similarity patterns in the distinguishability matrices of panel c, supporting our use of the latter as fingerprints of the information in the latent space. Unlike the classification head's outputs, the distinguishability matrices were dircetly accessible from the latent space without additional network training.
p=;)) = Eu-p(ula.)
We evaluated the pairwise similarity between the nine latent spaces of panel a using several methods (Fig. 2d). To assess the information content available for downstream processing, we compared the average predictions of the trained classification heads for cach input 2; through the Jensen-Shannon divergence, (JSD (I) If classfficatoo heads for latent spaces a and ,3 produce similar output distributions, the latent spaces likely share similar information content. Next, CKA (Kornblith et al., 2019) can use different measures of representation similarity as its basis of comparison; we used the inner product between mcans of the posterior distributions for the representational similarity in linear CKA, an exponential function of the Fuclidean distance for nonlinear CKA, and the Bhattacharyya coefficient between posteriors as statistical basis of representational similarity. Nonlinear CKA requires selecting a distance parameter for each space: we used the arithmetic mean of the standard deviation of every representation. Next, we computed similarity according to stochastic shape metrics (Duong et al., 2023). Finally, we computed VT and NMT, both with a Monte Carlo approach and the significantly faster Bhattacharyya fingerprint- approach.
Spearman's 's rank correlation of the similarity values between all pairs of synthetic spaces reveals multiple relationships between our proposed measures and the baselines. First, for this small dataset, the Bhattacharyya estimates of NMI and VI are highly consistent with the Monte Carlo alternatives while offering a speed up of 100x. Second, VI captures similarity much the same as when comparing the outputs of downstream classification heads (JSD) even though the former can be evaluated directly from the latent space. Third, NMI relates the latent spaces much the same way as the Bhattacharyya variant of CKA. While this ad hoc variant of CKA lacks the information-thowe theorrtic underpinnings of NMI, it offers an easy-to-use alternative by simply replacing a point-based distance with statistical distance between representations.
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Figure 2: Comparing similarity measures for synthetic embedding spaces. (a) A dataset of 64 points, x = =, 64, is transformed into nine representation spaces marked i-ix. Fach posterior distribution p(u|z)) a Gaussian with diagonal covariance matrix and standard deviations indicated by the colored ellipses. (b) We trained a classification head on top of the latent spaces a to predict the input x, given a sample from the posterior distribution p(u|z). The predicted probability distributions (x;) are displayed as a matrix, with corresponding to the input I;- (c) The pairwise distinguishability of data points i; and zy, computed by the Bhattacharyya coefficient, serves fingerprint of the information content of the latent space. (d) The pairwise similarity of the representation spaces in panel a, found by a variely of methods. Runtimes to calculate the full matrix are shown above each method, except for Jensen (JSD) because required training an additional classification network on top of cach latent spacc. The stochastic shape metric requires the dimensionality of the compared spaces match; undefined ent entries are grayed out. The Spearman rank correlation between similarity measures is shown in the bottom right.
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We additionally observe differences between the methods in relative similarities between the latent spaces. All methods except linear CKA and stochastic shape metrics detect. the similarity of the quasi-one-dimeenionaal encodings (representation spaces i, iv, and vi). These encodings are not related by simple transformations, though a downstream neural network can extract similar information from cach (sce JSD). Representation spaces vii and ix are similar by information ion content, as variations of splitting the data inlo two groups, but. they are deemed different by all but NMI, VI, and the Bhattacharyya- basod CKA. While stochastic shape metrics satisfy the desirable properties of a metric, the generalized NMI/VI offer a measure of similarity more relevant for downstream processing owing to their utilization of mutual information.
4.2 Unsupervised detection of structure: channel similarity
We next analyzed the consistency of information fragmentation in ensembles of generative models trained image datasets. Using fifty models with ten latent dimensions (channels) each, for variety of datasets, methods, and hyperparameters (some released with Locatello et al. (2019)*), we assessed structure in an ensemble's channels. Every latent dimension of every model in an ensemble was compared pairwise to every other; we used the Bhattacharyya fingerprint approach with a sample of 1000 images randomly selected from the training set. Before using OPTICS to group latent dimensions by similarity (described in Soc. 3.3), we removed dimensions transmitting less than 0.01 bits of information. We found NMI to more successfully detect channels with similar information content, and use it in this section (comparison with VI in Appx. E).
In Fig. 3, the matrices display the pairwise NMI between all informative dimensions in the ensemble, and have been reorganized by OPTICS such that highly similar latent dimensions are close logether and appear as blocks along the diagonal. On the left of each similarity matrix is the NMT with the ground truth generative factors (or other label information), and above is the OPTICS reachability profile where identified groupings of consistently learned channels are indicated with shading and Roman numerals.
In Fig. 3a, the regularization of a 3VV is increased for the dsprites dataset (Higgins et al., 2017). Although there are more channels that convey information for lower regularization (3=4), there is little discernible structure in the population of channels aside from a group of channels that communicate roughly the same information about scale. With B = 16, block diagonal structure is found, and channels with the same xpos, ypos, and scale information are found repeatedly across runs in the ensemble. We applied the same analysis to an ensemble of InfoGAN-CR models (Lin et al., 2020), whose models additionally encoded shape information consistently. We approximated the latent distribution for an image by first encoding it, then re-generating 256 new images with the predicted latent dimensions fixed and the remaining (unconstrained) dimensions randomly resampled, and finally using the moments of the newly predicted latent representations as parameters for a Gaussian distribution. We note that in any scenario where natural probability distribution exists per datun in the representation space, the method can be used.
Fig. 3b shows remarkable consistency of information fragmentation by an ensemble of 3-VAEs trained on the cars3d dataset (Reed et al., 2015), though not with regards to the provided generative factors. The object factor is broken repeatedly into the same set of information fragments, which is sensible given that it is a fine-grained factor of variation that compriscs many lower-entropy factors such as color and height. More surprising is that three of the fragments contain mix of camera pose information and object information that is consistently learned across the ensemble. The majority of existing disentanglement metries rely on ground truth generative factors including the FactorVAE score (Kim & Mnih, 2018), mutual information gap (Chen et al., 2018), DCI (Eastwood & Williams, 2018), and InfoMEC (Hsu et al., 2023)-aarn and would miss the consistency found when comparing the information content of individual latent dimensions.
Given a group of latent dimensions identified by OPTICS, we take as a representative the dimension that maximizes the average similarity to others in the group. Latent traversals visualize the fragmented information: groups and iii communicate different partial information about both pose and color, while group vi conveys information about the color of the car with no pose information.
Is this particular way of fragmenting information about the cars3d dataset reproduced across different. methods? In Fig. 3c we compared the information content of the seven representative latent dimensions from the 8- (8= 16) to the ooorCCS-rorganizee latent dimensions for different methods (3-TCVAE (Chen
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Figure 3: Assessing the consistency of channel information in ensembles of models. We used NMI as similarity measure for OPTICS to detect fragments of information that are consistently stored in individual channels in an ensemble of trained models. (a) The channel consistency of models trained on dsprites, for VAVA and InfoGAN-CR. The information with respect to generative factors shown on the left of each similarity matrix. The 3 = 3-VAE fragmented information inconsistently compared to the other ensembles. (b) For 3-VAE ensemble trained on cars3d, the information content of channols was highly consistent, with seven distinct combinations of the three generative factors. Latent traversals for representative channel from each grouping visualize the information content. (c) We compare the information content of the representatives from panel b to that of channels in 3-TCVAE and FactorVAB ensembles. (d,e) Channel similarity and latent traversals for 3-VAB ensembles trained on fashi on ceshionn-nistt and celebA. Additional channel similarity analyses and latent traversals can be found in Appx. A.

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(NM)) T (U:X) (H(XX) e
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Figure 4: Comparing full latent spaces. (a) We compare trained models across several methods and six hyperparameters each, all from Locatello et al. (2019). (b) We compare 3-VAE models over the course of training. / X))H(X) is the fraction of total information about the dataset contained in the latent space; a value of one means all data points are well-separated in the latent space. (NMI) and (VT) denote the average pairwise NMT and VT values over five models in an ensemble. All mutual information terms were estimated via Monte Carlo, and the displayed error bars are the standard error after accounting for the uncertainty the constituent mutual information terms.
et al., 2018), 8 10 and FactorVAE (Kim & Mnih, 2018), 7 30). The consistent fragments for the other methods are recognizable from those of the 3-VAE ensemble, though interestingly the PactorVAB conveyed comparatively less azimuth information in several of the fragments. Channels of group vi for the 8 3-VAE jointly encoded information about the tint of the windows and the color of the car, and this information was encoded separately by the 3-TCVAE and FactorVAE (traversals in Appx. A).
Finally, we studied the manner of information fragmentation on datasets which are not simply an exhaustive set of combinations of generative factors (Fig. 3d.e). For 3-VAE ensembles trained on fashion-mnist (Xiao et al., 2017) and celebA (Liu al., 2015), some information fragments are more consistently learned than others. A particular piece of information that distinguishes between shoes and clothing (group iif) repeatedly learned for fashion iatiion-mnistt for celebA, remarkably consistent fragment of information conveyed background color (group v). The remaining information was less consistently fragmented across the ensemble.
4.3 Assessing the content of the full latent space
If an ensemble of models fragments information into channels inconsistently (e.g., Fig. 3a), is it because the information content of the full latent space varies across runs? We compared the information contained in the full 10-dimensional latent spaces of five models from each VAE ensemble (Fig. 4). The amount of information was generally too large for estimation via Bhattacharyya matrices, which saturates at the logarithm of the fingerprint size, we used Monte Carlo estimates of the mutual information. Error bars in Fig. 4 are the propagated uncertainty from the mutual information measurements, with additional details in Appx. C. Whereas NMI proved more useful for the channel similarity analysis, VI tended to be more revealing about the heterogeneity of full latent spaces. Due to the form of its normalization, NMI becomes unreliable ie., is plagued by large uncertainty- when the information contained in the latent space T(U; X) is small.
when
Across mothods and hyperparameters for cach method, the information contained in the full latent space was largely consistent over an ensemble (Fig. 4a; models from Locatello et al. (2019)). For the cars3d dataset, most latent spaces conveyed the full entropy of the dataset, meaning all representations were well-separated
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for the dataset of almost 18,000 images, making the information content trivially similar. By contrast, none of the models trained on smallnorb contained more than around 80% of the information about the dataset, and the similarity of latent spaces across members of an ensemble was more dependent on method and hyperparamcter. Ensembles of 3-VAE and 3-TCVAE were fairly consistent even though the transmitted information dropped considerably with increasing 8; DIP-I (Kumar et al., 2018) showed the opposite behavior, where consisteney varied strongly with Aod but transmitted information did not.
How does the consistency of latent space information evolve over the course of training? For 3-VAE ensembles, again compared the full latent spaces for five models with random initializations (Fig. 4b). For all datasets considered (cars3d, smallnorb, celebA), consistency across the models dropped at around the same point in training that total information increased dramatically; after this point, all models encapsulated nearly the same information in their latent spaces up to convergence.
4.4 Model fusion in a toy example
Finally, we demonstrate the capacity for model fusion with the proposed measures of representation space similarity. Consider a dataset with a single generative factor with S0(2) symmetry, such as an object's 's huc. Difficulties can arisc when the global structure of a degree of freedom is incompatible with the latent space (Falorsi et al., 2018; Zhou et al., 2019; Esmaeili et al., 2024). In this example, a one one-dimensional latent space cannot represent the global circular topology of SO(2), leading to discontimuities in the representation.
To clearly demonstrate the synthesis of information from multiple representation spaces, we trained an ensemble of B-VAEs, each with a one-dimensional latent space that was insufficient to represent the global structure of the generative factor. Fig. 5a shows an example latent space and its associated distinguishability matrix, showing pairwise Bhattacharyya coefficients between posterior distributions. The matrix reveals flaws in the latent space where similar values of the generative factor have dissimilar representations.
Assuming the flaws randomly distributed from one training run to the fusion of multiple such latent spaces might yield an improved representation of the generative factor. We performed gradient descent directly on posterior distributions in a two-dimensional latent space, with the objective to maximize average similarity with the ensemble. Namely, we computed either NMI or the cXI(-VI) using the Bhattacharyya matrix corresponding to the optimizable every training step and those of the ensemble. As shown in Fig. 5b,c, the synthesized latent space more closely captured the global structure of the generative factor as the ensemble size grew. To quantify the performance, we employed the continuity metric used in Esmaeili et al. (2024) and adapted from Falorsi et al. (2018), substituting the Bhattacharyya distance between posteriors for the Euclidean distance typically used for boset-based representations (Appx. F).
need not be truc- the
Both NMI and VI proved effective in boosting an ensemble of weak representation spaces to represent the generative factor, with fidelity (measured by continuity) improving for larger ensembles. By comparison, directly maximizing the mutual information between the synthesis space and the ensemble, (I(U; Vi)), resulted in scattered representations with no overlap (Fig. 5c, right). The normalization terms of NMI and VI were essential for preserving the relational structure between data points. Finally, it is worth noting that after training the ensemble of weak models, neither the original data nor the models were needed to train the synthesis space: the latent representations were optimized directly from the Bhattacharyya matrices.
5 Discussion
The processing of information can be directly assessed when representation spaces are probabilistic because information theoretic quantities are well-defined. In this work, generalized two classic measures for comparing the information content of clustering assignments to make them applicable to probabilistic spaces. By focusing on the information content of a representation space, we can assess what information is available for downstream processing, while remaining agnostic to aspects of the spaces such as their dimensionality, their discrete or continuous nature, and even whether a space serves as auxiliary information like labels or annotations (Newman & Clauset, 2016; Savić, 2018; Bazinet et al., 2023). While the examples in this work focused on relatively low-dimensional latent spaces that are common in practice, scaling to higher-dimenssonal
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101 b
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arg muKK (NMI(U, Vi))
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Figure 5: Fusing weak representation spaces. (a) Example of one-dimensional latent space of 3-VAE trained on dataset generated from single periodic factor (color hue), which has SO(2) symmetry. The latent space exhibits flaws where similar values of the generative factor are mapped to dissimilar representations, as seen in the posterior distributions (left) and the distinguishaaiility matrix of Bhattacharyya coefficients between posteriors, BC (right). (b) We optimized a synthosis representation space to maximize similarity with an ensemble of such one-dimensional latent spaces. The continuity statistical distances between neighboring points, an assessment of the fidelity of the global structure of the generative factor, improved as the ensemble size grew. Error bars show the standard deviation over five experiments, and values offsct horizontally for visibility. (c) Synthesized -wo-dimensional representation spaces (posterior means shown points; covariances as shaded ellipses) and their corresponding distinguishabiiity matrices. Panels compare results when maximizing average NMI (left, middle) and mutual information (right).
representation spaces may face challenges related to the reliable estimation of mutual information (MeAllester & Stratos, 2020).
With differentiable formulations for NMI and VI, model fusion and the more general problem of reprosentational alignment (Sucholuisky el al., 2023; Mullenthaler et al., 2024) can be effectively approached from an information theoretic perspective. Potential applications include aligning representation spaces across models trained on different subsets of data, improving ensemble methods, and evaluating consistency of reprosontations in multitask learning or domain adaptation, where reconciling heterogeneous latent spaces is often crucial.
A more subtle contribution of this work is to shift the current focus of unsupervised disentanglement evaluation. As is clear from the cars3d information fragmentation (Fig. 3b), cxisting metrics that compare latent dimensions to ground truth generative factors can completely miss consistent information fragmentation. Unsupervised methods of evaluation (Duan et al., 2020; Lin et al., 2020) assess consistency of fragmentation at the scale of models, which can obscure much about the manner of fragmentation. Consider the fragmentation of information about the celebA dataset in Fig. 3e: an assessment of the similarity of models would find middling values across the ensemble and miss that two fragments of information are remarkably consistent. We argue that more fine-grained inspection of information fragmentation essential for a deeper understanding of disentanglement in practice.
The current work largely focused on comparing repeat training runs in an ensemble, making computational costs consideration. We found 50 models per ensemble to be sufficient to assess channel similarity, and 5 models per ensemble for the full latent space. We also found that there is much to learn from ensembles of relatively simple models. On our machine, training a single model from a recently proposed method (QLAE, Hsu et al. (2023)) took more than five hours, and in the same amount of time we could train ten V-VAEs with a simpler architecture. Model fusion with weak models that are inexpensive to train, as in Sec. 4.4, might offer a promising alternative to representation learning with more computationally expensive models.
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8-VAE, 1616 16
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Figure 6: Latent traversals for cars3d groups from Fig. 3b,c. We traverse the representative channels for the groups found by OPTICS, and reorder them to align with the ordering for 3-VAE (left). Note that group vi splits in Iwo groups for both the 3-TCVAE and the FaclorVAE. The tint of the windows and the color of the were encoded jointly for the B- B-VAE, and then separately for the other two methods. Traversals are over the range [-2,2].
A Appendix: Extended channel similarity results
In Fig. 6 we present additional latent traversals for the cars3d channel groupings presented in Fig, 3c. With the channels most centrally located in each group (as before), also display latent traversals for 3-TCVAE (8=10) and FactorVAB (7- = 30).
Fig. 7 visualizes the channel similarity structure on mnist (LeCun et al., 2010) and fashion-mnist, with latent (raversals for four channels per grouping to show the consistency of the encoded information.
In Figs. 8, 9, and 10, we repeat the structural analysis of Sec. 4.2 with all B-VAE hyperparameters explored by the authors of Locatello al. (2019). The effect of increasing 3 is clearly observed by the emergence of block diagonal channel similarity matrices, though with fewer informative channels for increased B.
Consider the cars3d ensembles in Fig. 8. With 1,2, the information fragmentation is not consistent across repeat runs, even though the amount of information shared with the three generative factors is fairly consistent. This highlights the value of directly comparing the information content of channels with NMT or V1 instead of comparing indirectly via information content about known generative factors, with the added benefit of being fully unsupervised. For 8 the learned fragments of information start to coalesce, with information regularization breaking the degeneracy that plagues unsupervised disentanglement (Locatello et al., 2019). For 3 = 8, 16, the regularization is strong enough to form consistent fragments of information across random initializations.
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Figure 8: Channel similarity structure for cars3d, VAE,,, assessed with NMI and VI. (a-f) Ensembles with values of 8 from 1 to 16, with channels reordered using OPTICS and either NMI (left) or V1 (right) for the method of comparison. Regardless of the measure used for comparison, the NMI with the generative factors shown on the left of cach pairwise similarity matrix. All matrices are sized 500 x 500, with uninformative channels (IUU; X) < 0.01 bits) displayed as white.
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Published in Transactions on Machine Learning Research (02/2025)
OF110 mahape 2CC.0 Xppos 7pos
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d
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NMKUU C,)
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NMIOU CJ)
NMI(U.C))
Figure 9: Channel similarity structure for dsprites, 8- V-VAE, assessed with NMI and VI. Everything
in this figure mirrors Fig. 8.
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Published in Transactions on Machine Learning Research (02/2025)
N
instance cat molev m1igat
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500 /50 channels ->
00
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NMIUU C2)
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expl. V10L,
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NMIIU C2)
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Figure 10: Channel similarity structure for smallnorb, 3- VAE,, assessed with NMI and VI.
Everything in this figure mirrors Fig.
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Published in Transactions on Machine Learning Research (02/202))
Latent space 1) x(e)=w6))
Bhsttacharyva Clpssifeaation fnnerprin
head predictions
Latent space
Bisttacbaryya Ingerprinl
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Bbattacharyya Ennerrriint
head predictions
Classification
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Figure 11: Full comparison of Bhattacharyya fingerprints and classification outputs from latent spaces of Fig. 2.
B Appendix: Extended results synthesized latent space comparison (Sec. 4.1)
We include in Fig. 11 the full set of comparisons between the Bhattacharyya fingerprint and the average predictions of trained classification head for the nine synthetic latent spaces of Sec. 4.1. Fig. 12 shows the pairwisc scatter plots for the methods of comparing the latent spaces.
C Appendix: Information estimation using Bhattacharyya distinguishability matrices and Monte Carlo estimator
To estimate the mutual information XX) from the Bhattacharyya distinguishability matrices, we have employed the lower bound derived Kolchinsky & Tracey (2017) for the information communicated through channel about mixture distribution (following the most updated version on arXiv?). The bound simplifies greatly when the empirical distribution is assumed to be reasonable approximation for the data distribution, and then we further assume the sample of data used for the fingerprint allows for an adequate approximation of the marginal distribution in latent space. First we reproduce the bound from Sec. V of Kolchinsky & Traccy (2017) using the notation of this work, and then we describe our assumptions to apply the bound as an estimate the information contained in a probabilistic representation space.
Let X be the input channel, following a mixture distribution with N components, p(x) = EN (pp())) and U the output of the same channel, ~ p(u)= = Em (fx p( (()))))(())dz)) Then we have
I(x:)) - an E BC, + H(UC)
(6)
where C is random variable representing the component identity.
In this work, we assume that the data distribution p(x) = ΣN 5(x- 2,N/N, simplifying Eqn. 6 so that e; 1/N and H(UCC) HU)) because the identity of the component is equivalent to the identity of the datum. Finally, we assume that the set of posterior distributions for a representative sample of size M of the dataset, taken for the fingerprint, adcquately approximates the empirical distribution. Larger samples may be necessary in different scenarios when the
be approximated by the empirical distribution,
Phttps:/ httppp/://axtivoog//aa/////////
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JSD
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CKA, noulin.
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Figure 12: Pairwise scatter plot comparison of similarity measures from Fig. 2.
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Published in Transactions on Machine Learning Research (02/2025)
Fingerprint size: 10
Fingerprint size: 30 Time: msee
Fingerpriut size: 100 Time: msen.
Fingerprint size: 300
Fingerprint size: 1000 Timo: msec.
Time: meec
Time: msec
[(U:)) X)) Bhat. (lait)
T(U: X), MC (bits)
T(U: X), MC (bits) Monte Carlo samples: 2 x 10*
1(7; x), MC (bits)
12
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1(U: XV, Mc (bits)
Time: 1300 mase
Figure 13: Information estimates with Bhattacharyya fingerprints and Monte Carlo. For 500 channels randomly sampled from across all models released by Locatello et al. (2019) for the cars3d dataset (including all methods and all hyperparametens) we estimated the amount of information transmitted by each channel (U;;)) using the Bhattacharyya matrix fingerprints and Monte Carlo (MC) sampling. Error bars are displayed for the MC estimates, though they are generally smaller than the markers. The dashed black line represents equality between the two estimates, and the solid blue line is the logarithm of the fingerprint size, which is the saturation point for the Bhattacharyya estimate. Listed run times for single channel, excluding the time to load models but including inference and the calculation necessary for I(U;))
amount of information transmitted by channels is larger than handful of bits, but M 1000 appeared sufficient for the analyses of this work.
The generalizations of NMI and VI required the information conveyed by two measurements from different channels, U(X;U, V) as well as from the same channel, [(X;U,U), U'). The matrix of Bhattacharyya coefficients given measurements and V is simply the elementwise product of the coefficients given and the coefficients given V. The posterior in the joint space of U and V factorizable given x Le., p(u, v/z) = p(u ((v())) because the stochasticity in each channel is independent. The same is true for the joint space of and U' two draws from the same channel. The Bhattacharyya coefficient of the joint variable simplifies,
BC = -. L Velu, vlx,)((, v ddude
Volulai)p(ulrs)ddu I, Volola.ippllos.ddd
(7)
= BCy x BC
For the Monte Carlo estimator, we sampled random data points from the dataset p(æ) and then random embedding vector from each posterior distribution,
p(ulz). Then computed the log ratio of the likelihood under the corresponding posterior p(u(x) and the aggregated posterior p(u), which was gotten by iterating over the entire dataset. We repeated N times to estimate the following expectation:
I(X:U) (e log p(ulx)
(8)
p(u)
For the analysis of Fig. 4, N x x 10 for all datascts except dsprites and celebA, where N = 10 the standard error of the estimate was on the order of 0.01 bits or less.
Computing the fully aggregated posterior is time consuming; could subsample of the dataset be used instead? In Fig. 14, we varied the fraction of the datasct used to compute the (partially) aggregated posterior for full 10-dimensional latent space for three models that Transmit an intermediale amount of information about the dataset on which they were trained. Each point represents the mutual information estimate following Eqn. 8 for a random subset the datasct, and the extent of its error bar is given by the standard error: the standard deviation of the set of values divided by the square root of the size of the set. We find that the transmitted information is relatively robust. to using half of the dataset when computing the aggregated posterior, but
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In Fig. 4, the NMI and VI values are averaged over all pairwise comparisons for the five representation spaces, with weights given by the inverse squared uncertainty on each pairwise value. The uncertainty was calculatod by propagating the uncertainty of Monte Carlo estimates of (X):U) according the expression for VI I(X;U,U) - - (X;V,VV) the propagated uncertainty is Published in Transactions on Machine Learning Research (02/2025)
celebA, B-VAE B 8, Step /97/300,000
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Figure 14: Partially aggregated posterior for the Monte Carlo estimator. For three models selected from those of Fig. 4, we seleced random subsets of the full dataset to use when computing the aggregated posterior for estimating the transmitted information X) of the full 10-dimensional latent space. The points are offset horizontally by random amounts for visibility, and the error bars show the standard error on the mean of the Monte Carlo samples.
deleterious effects grow when reducing the dataset by a factor of 10 or more. Interestingly, the dsprites estimate was largely insensitive to using 3% of the dataset, perhaps due to its highly structured nature.
NMT. To be specific, VI(U,V) UV,; given by
AVI(U, VIAT(X;U.V) -
with AT(X;U,V) the standard error from the Monte Carlo estimate. The expression for the uncertainty the NMI estimate contains many more terms, and we will not reproduce them here. Finally, the error bars in Fig. 4 are the standard error of the weighted mean,
AQ = Σ (A
(10)
for quantity Q.
D Appendix: Broken triangle inequality for the generalized VI
Here provide an example of three clusterings that breaks the triangle inequality for the generalized VI. Thus, while VI for hard clusters satisfies the properties of metric (Crutchfield, 1990), the generalization to soft clusters does not.
In Fig. 15, there are three clusterings (U, V, and W) of four equally likely data points (x E {a, b, d)). U, V, and W each communicate one bit information about X, but U and are hard clusterings while V soft clustering. We have I(U; V) = I(V: 0.5 bits, [UU; W) bits, I(V:V") 0.5 bits, and I(U; U") I(W; w') H(U) bit. The generalized VI from U and then from to W less than from U to W directly.
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W
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Figure 15: Example that breaks the triangle inequality for generalized VI. Four datapoints a, b, d are clustered according to scheme U, V, or W. The total VI from U to V and then V to W is less than the VI from U to W.
VI(U,W) I(U:U)) I(W:W') W)
= 2 bits
VI(UV) VIV,,) = -
=1 bit.
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8-VAE, dsprites: 3
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Figure 16: Direct comparison of NMI and VI. We convert both measures to distance measure (black) and to a similarity mcasure (blue gray) and compare them for the pairwise channel comparisons from the ensembles of Fig. 3.
E
Appendix: Are NMI and VI interchangeable?
NMI and VI, aside from the inversion required convert from similarity to distance, can both be seen as a normalized mutual information. Are they interchangeable, do they assess structure differently?
In Fig. 16, compare NMI and VI (estimated via Bhattacharyya matrices) similarity or distance measures for the pairwise comparisons between channels used in Fig. 3. Specifically, as measures of similarity, we plot. NMI against exp(-VT), and for distance we plot - IoM(NMT) against VI. We find that NMI and VI are tri-trivially related, shown clearly by the horizontal and vertical swaths of points where one of the two measures roughly constant while the other varies considerably. Interestingly, the corresponding NMI and VI comparisons Fig. 16 show multiple distinct ares of channel similarity, as well clear vertical bands where NMI has discerning power and VT does not.
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F
Appendix: Implementation specifics
Code to reproduce the experiments of Sec. 4 can be found at the following repository: https:/ http://gttthh..o/// www//oooooyaaaaeeeeeoooooo http://uuhyyyk/rppreetaaaiiiiiie wwtprreenaaaoooppee--ooorooopppo utils.py. py, containing the Bhattacharyya and Monte Carlo calculations of I (U; and the NMI/VI calcula- tions.
The heart of the codebase is in
All experiments were implemented in TensorFlow and run on a single computer with a GB GeForce RTX 3060 GPU.
Models: For the daprites, smallnorb, and cars3d datasets, used the trained models that were publicly released by the authors of Locatello et al. (2019). Thus, all of the model and channel numbers recorded above the latent traversals in Fig. 3b correspond to models that can be downloaded from that paper's github page?.
Simply add the model offset corresponding to the = 16 3-VAE for cars3d, 9250 (e.g., for the traversal labeled with model 31 ch 3, download model 9281 and traverse latent dimension 3, indexed).
For the InfoGAN-CR models on dsprites, used the trained models that were uploaded with Lin et al. (
For results on the training progression of smallnorb, celebA, and cars3d, we used the same architecture and training details from Locatello et al. (2019).
For the MNIST and Fashion-MNIST ensembles, we trained 50 3-VAEs with a 10-dimensional latent space. The encoder had the following architecture:
Conv2D: 32 4x4 ReLU kernels, stride 2, padding 'same'
Conv2D: 64 4x4 ReLU kernels, stride 2, padding 'same'
Reshape(I-1))
Dense: 256 ReLU
Dense: 20.
The decoder had the following architecture:
Dense: 7 x 7 x 32 ReLU
Reshape([7, 7, 32))
Cons2DTranspose: 64 4x4 ReLU kernels, stride 2, padding 'same"
Conv2DTranspose: 32 4x4 ReLU kernels, stride 2, padding 'same"
Cony2DTranspose: 1 4x4 ReLU kernels, stride padding 'same.
The models were trained for 2 x 10s steps, with a Bernoulli loss on the pixels, the Adam optimizer with a learning rate of 10-1, and a batch size of 64.
Clustering analysis: We used the OPTICS implementation from sklearn with 'precomputed' distance metric and min_ sin_samples- 20 (and all other parameters their default values). For distance matrices we converted NMT to distance with log mNx(NMI, 10-1)
Ensemble learning: For the ensemble learning toy problem (Sec. 4.4), we trained 250 simple 3 V-VAEs (8=0.03) whose encoder and decoder were each fully connected networks with two layers of 256 tanh activation. The input was two-dimensional, the latent space was one-dimensional, and the output was two-ddmennsonall.
Shttpss/ggiaubb.co//oogggg woggeeeeeoogggooorrerrrrc.c. http://rcceerrrr/ssssnaaannnnooo disentanggemment /bsss/aalllaaeetttt//////////sse lib/trce/master hhttps///gthub..mmffmmb/ http:////////o///anaaeaaa 5 http:///wtt..oooooooomm///////ee
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The loss was MSE, the optimizer was Adam with learning rate 10-3, and the batch size was 2048, trained for 3000 steps. Data was sampled anew each batch, uniformly at random from the unit circle.
To perform ensemble learning, we evaluated the Bhattacharyya matrices for 200 evenly spaced points around the unit circle for each model the ensemble. Then we directly optimized the parameters for 200 posterior distributions (Gaussians with diagonal covariance matrices) a. two-dimensional latent. space, so as maximize the average similarity (NMI, exponentiated negative VI, or mutual information) between the Bhattacharyya matrix for the trainable embeddings and those of the ensemble. We used SGD with learning rate of 3 for 20,000 iterations, and repeated for 5 trials for each ensemble size.
Stochastic shape metrics: We used publicly released code on github", using the CaussianStochhhscchtttiicic with = and the parallelized pairwise distances for the timing cal- culation.
CKA: We replaced the dot-product similarity matrices K and in the Hilbert-Schmidt Independence Criterion (HSIC) with the Bhatlacharyya matrices,
HSIC(BC(), (B(2)) = (n Tr(BC() H. BBC H),
(11)
and then followed the prescribed normalization Kornblith et al. (2019).
Continuity metric: Falorsi et al. (2018) used the ralio of neighbor distances in representation space to the corresponding distances data space, with neighbors taken along continuous paths in data space, as the basis for discrete continuity metric. It indicated whether any ratios were above some multiplicative factor of some percentile value in the distribution, and thus depended on two parameter choices. Esmaeili et al. (2024) removed one of the parameters -thee multiplicative factor-yiee victoryyeeldig. a. continuous continuity metrie that reports the maximum ratio value over the 90th percentile value.
The central premise this work is to respect the nature of representations as probability distributions, so we used the Bhattacharyya distance (i.e., Dú - log BC,j)) between posteriors instead of Fuclidean distances between postcrior mcans representation space. Other than this modification, we left the continuity metric in Esmaeili et al. (2024): the maximum ratio value over the 90th percentile value.
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Reviews:
[{"review_id": "Xia1Zkcq9V", "reviewer": "Reviewer_1uJU", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths\n\nThe paper is clearly written and generally easy to follow.  The intuition behind the proposed approach is well-articulated, and the authors provide supporting evidence for most of their claims. The paper explores a diverse set of datasets, including a balanced combination of real-world and synthetic data, to evaluate the properties of the proposed method. Results on unsupervised detection of structure and model fusion look very promising and interesting.\n\n### Weaknesses  \nWhile I remain positive about the paper, one critical weakness, in my view, is the limited amount of quantitative evidence and comparisons to existing methods — particularly regarding the claims that the proposed approach is *fast* and *lightweight*.\n\n- For instance, while the paper claims that the proposed measures are *lightweight* (p.2) and *fast* (p.5), the evidence for these assertions is primarily qualitative. Experiment 4.1 estimates runtimes for individual methods, but the supporting data lacks quantitative rigor.  \n- Additionally, on page 8, the authors state that \"NMI and VI are nearly the same as the Monte Carlo alternatives\". However, this claim is assessed only through visual comparison of matrices in Figure 2.  \n\nWhile I find the proposed method promising, I recommend a more rigorous and quantitative demonstration of the following:  \n1. **Runtime Comparison** – Vary key parameters and quantitatively show that the proposed method consistently outperforms alternatives in terms of speed.  \n2. **Result Similarity** – Compare the proposed method with other approaches, such as CKA, and provide quantitative evaluations (e.g., norm differences between similarity matrices?). \n\n**Question to the Authors:**  \nWhile it is understandable that evaluating methods in unsupervised settings can be challenging, a potential solution could be to demonstrate the stated claims in a supervised setting. By using a scenario where some ground truth is available, the performance of various methods could be quantitatively assessed on this (potentially toy) example —(for example, [1]), or in the setting similar to the Experiment 4.1. Would the authors consider this as a viable path to strengthen the empirical validation of the proposed method?\n\n[1] Bykov, Kirill, et al. \"DORA: Exploring Outlier Representations in Deep Neural Networks.\" Transactions on Machine Learning Research.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper investigates the problem of comparing the information content of probabilistic representation spaces and introduces a generalization of two information-theoretic methods for such comparisons. The authors propose a novel, lightweight approach based on fingerprinting the representation space using the Bhattacharyya coefficient. The paper further demonstrates the applicability of this method through a series of experiments, highlighting its potential for model fusion and unsupervised detection of model structures.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "**Critical Requested Changes:**  \n- Address the issues raised in the \"Weaknesses\" section.\n\n**Would Strengthen the Work:**  \n1. **Figure 3** – Additional description and details would improve clarity regarding the OPTICS reachability profile and the clustering process. Expanding on how the clustering was performed would enhance the reader’s understanding.  \n2. **Experiment 4.1 and Figure Placement** – Experiment 4.1 (page 8) references Figure 2 (page 6) for the first time, which disrupts the reading flow. Consider adjusting the figure placement to improve readability.  \n3. **Applications to Representational Alignment** – Discussing potential applications of the proposed method for representational alignment could add further value to the paper [2].  \n\n[2] Muttenthaler, Lukas, et al. \"Aligning Machine and Human Visual Representations across Abstraction Levels.\" arXiv preprint arXiv:2409.06509 (2024).", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "UeDWQkId11", "reviewer": "Reviewer_ouLB", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n* Few papers deal with comparing probabilistic or stochastic representations. One of the only other examples is the recent work of [Duong et al.](https://arxiv.org/abs/2211.11665). Thus, the paper is addressing a key gap in the literature.\n\n* The paper leverages information theoretic concepts, which is also novel in this area as far as I am aware.\n\nWeaknesses:\n\n* While I like creative ideas in this paper, the exposition is confusing and I am ultimately unsure of the motivation behind the method. I'll expand on this in the points below.\n\n* One point of confusion is the notion that probabilistic representations can be thought of as \"soft clusterings\" of datapoints. What the authors mean by this isn't really spelled out in the paper, and to me this doesn't comport with the usual meaning of \"soft clusters.\" My understanding of k-means clustering (see e.g. [Zhu et al.](https://arxiv.org/abs/2403.15700)) typically involves assigning each data point, $\\mathbf{x}_i$ for $i \\in \\{1, \\dots, n\\}$, to a probability vector $\\mathbf{p}_i$ that takes values on a the $k$-simplex (for k clusters). The natural probabilistic representation that corresponds to this would be a random variable following a Dirichlet distribution, but typically the latent space of a VAE is defined by an isotropic Gaussian prior. Interestingly, some papers have been published on Dirichlet VAEs (such as this paper by [Joo et al.](https://arxiv.org/abs/1901.02739)). Here, I can see how the latent space would correspond to a soft clustering, and it could be a very useful way for the authors to apply their method!\n\n* A similar point of confusion for me is why clustering is a good framework to ground comparisons of neural representations. The authors say at the beginning of section 3 that \"Our goal is to compare the information transmitted about a dataset by different representation space\" but then later say that the information transmitted for hard clustering is simply the entropy of the cluster labels. Thus, the optimal representation for $n$ datapoints is just a representation that puts every datapoint into a singleton cluster, which doesn't seem like a particularly useful representation. I suspect that I'm missing the point, but I find it hard to understand the rest of the paper unless the authors flesh out their reasoning here.\n\n* The Monte Carlo estimator of mutual information seems critical, but is only described briefly in appendix E. I feel pretty strongly that it should be put into the main text / methods section.\n\n* I expect that the Monte Carlo estimator will perform poorly for high dimensional latent spaces due to standard curse of dimensionality arguments. I gather that this is why the authors look at relatively low-dimensional latent spaces in their experiments (up to ten dimensions are considered for Fig 4). The authors should be more up front about these limitations and discuss the statistical estimation properties (e.g. how are the error bars in Fig 4 computed?).\n\n* Moreover, the Monte Carlo estimator only gives an estimate for the mutual information between the empirical distribution over $\\mathbf{x}$ and the latent representation. I would be interested in a plot that shows whether this estimator is converging to something reasonable as a function of the dataset size (i.e. number of images). In particular if you take random 50-50 splits of the data and estimate the mutual information on each subset, how much disagreement is there?\n\n* In figure 2, I am not sure whether the proposed method is directly comparable to the baselines the authors compare to: CKA and stochastic shape distances. Neither of these baselines adopt the clustering perspective (either hard or soft clustering), so I don't know how I'm supposed to compare with NMI? Perhaps the authors could re-word this section more neutrally. For example, the shape metric and (to a lesser extent) CKA results in Fig 2 show that these methods do a good job of separating out representations into three clusters based on their geometry---i.e. they are behaving as they were intended to behave. The authors seem to prefer their approach, which is a \"more flexible sense of [representational] similarity\". By this, I guess that they want a method that is more sensitive to the topology (rather than geometry) of the probabilistic point cloud? If that is the case, I think it could help to state that and explain why certain circumstances might call for a geometric vs topological measure. Furthermore, the authors should discuss nonlinear CKA and topological RSA, which may be more fair baselines to compare against (see the papers by [Lin & Kriegeskorte](https://doi.org/10.1073/pnas.2317881121) and [Williams, 2024](https://www.biorxiv.org/content/10.1101/2024.10.23.619871v1.abstract))", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "Quantifying geometric similarity in neural representations is an active topic of research in machine learning (see e.g. [Klabunde et al.](https://arxiv.org/abs/2305.06329) for a recent review). This paper proposes a novel approach to quantify similarity in the information content of probabilistic representation spaces, which occur (for example) in the bottleneck layer of variational autoencoder. The authors claim the method will be helpful in evaluating the level of disentanglement in learned representations, which is also an important and challenging topic (see e.g. [Locatello et al.](https://proceedings.mlr.press/v97/locatello19a.html)).", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I covered major points under weaknesses above. In general, I don't have any strong feelings against this paper. I would just like to see an honest effort made at clarifying the points listed above.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "ZiSCgtB05Y", "reviewer": "Reviewer_gsLP", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\n* (S1) There is only one other metric applicable to stochastic representation spaces and the proposed metrics seem to be more flexible in their application.\n* (S2) The metrics themselves are well presented and easy to understand.\n* (S3) Understanding of neural network representations is an important problem.\n\n## Weaknesses\n* (W1) The biggest problem of the paper lies with its presentation. In general, I found significant parts of the manuscript hard to follow. I think the reason is that the authors assume the reader is familiar with disentangled representation learning, but the paper proposes generally usable similarity metrics and thus should be understandable to a general machine learning audience. A few examples of parts that can be improved:\n  * Introduction paragraph 3 and 4: unsupervised evaluation of representation disentanglement is introduced as a motivation but remains vague. Giving an example of what disentangled representations are could help.\n  * Contribution 2: “fingerprinting each representation space with the distinguishability of a sample from the dataset”. What does distinguishability mean in this context?\n  * Figures are sometimes not self-contained and could benefit from extended captions. For example, for Fig 2, what is the interpretation of the heatmaps? This particularly important as the figures and text, in which they are referenced, are not close to each other. Similarly, for Fig3, there is a ton of information, but the reader is left guessing what it could mean. \t\n  * Section 3.3 has no results?\n  * Section 4.4: Please explain the problem better.\n* (W2) The metrics are “applicable to general probabilistic representation spaces”. Can you give some general guidelines how to use them? It appears the applications in the paper are limited to low dimensional representations.\n* (W3) Comparison with existing metrics. Fig2 shows comparisons to prior metrics CKA and Stochastic Shape Metric. In particular, CKA seems find similarity just as well as NMI, given pairwise distances via Bhattacharyya fingerprint. Does it make sense to use NMI over CKA? What are the pros/cons? Similar for Stochastic Shape Metrics, the similarity estimates seem reasonable. And can the dimensionality issue not be resolved by adding some zero dimensions?\n\nMinor:\n* The color scheme in Fig2 goes counter to how similarity is presented popular works such as [1,2].\n\n[1] Kornblith et al., 2019. Similarity of Neural Network Representations Revisited  \n[2] Raghu et al., 2017. SVCCA: Singular Vector Canonical Correlation Analysis for Deep Learning Dynamics and Interpretability", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors propose a two new similarity metrics between representation spaces. In contrast to typical metrics, the proposed metrics are applicable to compare non-deterministic systems, with representation distributions per input, instead of one deterministic representation per input. The metric is based on estimating mutual information and thus generally applicable regardless of representation space characteristics. They apply the metrics to study representations of VAEs and GANs in the context of disentangled representation learning. The metric is differentiable and can thus be used to directly influence similarity.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "The following points are critical for acceptance:\n* Improved presentation of the paper, e.g., by improving figure captions and taking a bit more space to explain the experiments. \n* Add discussion on pros/cons of the new metrics over existing ones. \n* Add discussion on the limitations of the proposed metrics and how someone would use them for a new problem.", "broader_impact_concerns": "I do not have any concerns with respect to ethical implications of this work.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: After the authors' revisions, the paper has improved substantially. All three reviewers think that their main concerns have been addressed and that the paper meets the acceptance criteria.
Decision: Accept (poster)