DCI-ES: An Extended Disentanglement Framework with Connections to Identifiability
Authors: Cian Eastwood
Andrei Liviu Nicolicioiu
Julius Von Kügelgen
Armin Kekić
Frederik Träuble
Andrea Dittadi
Bernhard Schölkopf
Venue: ICLR 2023 poster (2023)
Keywords: disentanglement
identifiability
representation learning

Abstract: In representation learning, a common approach is to seek representations which disentangle the underlying factors of variation. Eastwood & Williams (2018) proposed three metrics for quantifying the quality of such disentangled representations: disentanglement (D), completeness (C) and informativeness (I). In this work, we first connect this DCI framework to two common notions of linear and nonlinear identifiability, thereby establishing a formal link between disentanglement and the closely-related field of independent component analysis. We then propose an extended DCI-ES framework with two new measures of representation quality—explicitness (E) and size (S)—and point out how D and C can be computed for black-box predictors. Our main idea is that the functional capacity required to use a representation is an important but thus-far neglected aspect of representation quality, which we quantify using explicitness or ease-of-use (E). We illustrate the relevance of our extensions on the MPI3D and Cars3D datasets.

Published as a conference paper at ICLR 2023
DCI-ES: AN EXTENDED DISENTANGLEMENT FRAME- WORK WITH CONNECTIONS TO IDENTIFIABILITY
Cian Eastwood Andrei Liviu Nicolicioiu"', Julius von Kügelgenn*:; 1133 Armin Kekić', Frederik Träuble', Andrea Dittadi 14 and Bernhard Schölkopf'
Julius
Max Planck Institute for Intelligent Systems, Tübingen, Germany School of Informatics, University of Edinburgh
DDepartment of Engineering. University of Cambridge *Technical University of Denmark
ABSTRACT
In representation learning, common approach is to seek representations which disentangle the underlying factors of variation. Eastwood & Williams (2018) proposed three metrics for quantifying the quality of such disentangled representations: disentanglement (D), completeness (C) and informativeness (I). In this work, we first connect this DCI framework to two common notions of linear and nonlinear identifiability, thereby establishing a formal link between disentanglement and the closely-related field of independent component analysis. We then propose extended DCI-ES framework with sentation expily-meels (E) and size (S)-and and point out how D and C can be computed for black-box predictors. Our main idea is that the functional capacity required to use a representation is an important but thus-far neglected aspect representation quality, which quantify using explicitness or ease-of-use (E). We illustrate the relevance of our extensions on the MP I3D and Ca datasets.
measures of repre-
INTRODUCTION
A primary goal of representation learning is to learn representations r(x) of complex data that "make it easier extract useful information when building classifiers or other predictors" (Bengio et al., 2013). Disentangled representations, which aim to recover and separate (or, more formally, identify) the underlying factors of variation z that generate the data as x = g(z), are promising step in this direction. In particular, it has been argued that such representations are not only inter- pretable (Kulkarni al., 2015; Chen et al., 2016) but also make easier extract useful information for downstream tasks by recombining previously-lears factors in novel ways (Lake et al., 2017).
While there is no single, widely definition, many evaluation protocols have been proposed to capture different notions of disentanglement based on the relationship between the learnt representa- tion code r(x) and the ground-truth fata-generative factors (Higgins et al., 2017; Eastwood & Williams, 2018; Ridgeway & Mozer, 2018; Kim & Mnih, 2018; Chen et al., 2018; Suter et al., 2019; Shu al., 2020). In particular, the metrics of Eastwood & Williams (2018 dise-dddsnnattnlemmmen (D), completeness (C) and informativeness (I) estimatee this relationship by learning a probe ff to predict from c and can be used to relate many other notions of disentanglement (see Locatello et al. 2020, & 6).
In this work, we extend this DCI framework in several ways. Our main idea that the functional capacity required to recover z from c is an important but thus-far neglected aspect of representation quality. For example, consider the case recovering from: (i) a noisy version thereof; (ii) raw, high- dimensional data (c.g. images); and (iii) a linearly-mixed version thereof, with each C; containing the same amount of information about each 2, (precise definition in 6.1.. The noisy version (i) will do quite well with just linear capacity, but is fundamentally limited by the noise corruption; the raw data (ii) will likely do quite poorly with linear capacity, but eventually outperform (i) given sufficient capac- ity: and the linearly-mixed version (iii) will perfectly recover with just linear capacity, yet achieve the porst-possible disentanglement score D = O. Motivated by this observation, we introduce a measure of explicitness or ease-of-use based a representation's 's loss-capacity curve (see Fig. 1).
* Equul contribution.

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MLP Probes
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Figure 1: Loss-capacity curves. Empirical loss-capacity curves (see $ 4.1) for various representations (see logend), datascts (top: MP 33D-Rell bottom; 3D), and probe types (left: multi-layer pereeptrons / MLPs, middle: Random Fourier Features / RFFs, right: Random Forests RFs). The loss was first averaged over factors 2j. and then means and 95% confidence intervals were computed over 3 random seeds. Details in $ 6.
Structure and contributions. First, we connect the DCI metrics to two common notions of linear and nonlinear identifiability (5 3). Next, we propose an extended DCI-ES framework ($ 4) in which we: (i) introduce two new complementary measures of representation
(E), derived from a representation's 's loss-capacity curve, and size (S); and then (ii) elucidate a means to compute the D and C scores for arbitrary black-box probes (e.g., MLPs). Finally, in our experiments ($ 6), we use our extended framework to compare different representations on the MP 3DD-eeal (Gondal et al., 2019) and Cars3D (Reed et al., 2015) datasets, illustrating the practical usefulness of our E score through its strong correlation with downstream performance.
aeality--xxpiccin
BACKGROUND
Given a synthetic dataset of observations x x= along with the corresponding K-dimensional date-gener factors E RK, the DCI framework quantitatively evaluates an L-dimensional data representation code r=)(x) E R- using two steps: (i) train a probe f to predict from c, i.e. 2 f(c = fr((x) = f(r(g(z))) and then (ii) quantify f's prediction error and its deviation from the ideal one-to-one mapping, namely a permutation matrix (with extra "dead" units in c whenever L > K)." For step (i), Eastwood & Williams (2018) use Lasso (Tibshirani, 1996) or Random Forests (RFs, Breiman 2001) as linear nonlinear predictors, respectively, for which it is straightforward to read-off suitable "relative feature importances".
Definition 2.1. R RL*K is matrix of relative importances for predicting from via f(c) if Rij captures some notion of the contribution of c; to predicting 2j s.t. Vi, Rij- Z 0 and _ Rij
For step (ii), Eastwood & Williams use R and the prediction error to define and quantify three desider- ata of disentangled representations: disentanglement (D), completeness (C), and informativeness (I).
Disentanglement. Disentanglement (D) measures the average number of data cata-generating factors zj that are captured by any single code G, The score D; is given by D; = 1 (P,),,) where (P.).) ΣK Pix logx Pix denotes the entropy of the distribution P. over row i of R, with P3 Py=Bi / ΣK RR- If c; is only important for predicting a single 2,, we get perfect score of D; = If ci is equally important for predicting all zj (for == K). we get the worst score of D; = 0. The overall score D is then given by the weighted average D Σ P;D;, with p; ΣK R
Completeness. Completeness (C) measures the average number of code variables C; required to capture any single zj, has also been called compactesss (Ridgeway & Mozer, 2018). The score Cj in capturing 2, is given by C, = (1 - (P,)) where Hr, = - E Poş log, Pej denotes the
W.L.o.g.. it can assumed that 2; and cj are normalised to have mean zero and variance one for all i, for otherwise such normalisation can be "absorbed" into g(.) and r(-).
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entropy of the distribution P. over column j of R, with Ptj = Rij. If single contributes 2j pre- diction, we get perfect score of C;=1. 1. If all c equally contribute to zj's prediction (for i i=1,..,, we get the worst score of 0 The overall completeness score given by C k C
Remark 2.2. Together, and C quantify the degree of "mixing" between C and z, i.., the deviation from one-to-one mapping. They are reported separately they capture distinct criteria.
Informativeness. The informativeness (I) of representation about data- uuta-generative factor 2j is quantified by the prediction error, i... 1j = E (f where & is an appropriate loss function.2 2 capture the informativeness of about 2j. fj must have sufficient capacity to extract all of the information in с about 2j. This capacityinfoommativenss dependency motivates separate measure of representation explictness in $ 4.1. The overall informativeness score is given by K E
Note that 1 depends on the capacity of fj. as depicted in Fig. 1. Thus, for I; to accurately
CONNECTION TO IDENTIFIABILITY
The goal of learning data representation which recovers the underlying data-generating factors is closely related blind source separation and independent component analysis (ICA, Comon 1994; Hyvärinen & Pajunen 1999; Hyvarinen et al. 2019). Whether a given learning algorithm provably achieves this goal up to acceptable ambiguities, subject certain assumptions on the data-generating process, is typically formalised using the notion of identifiability. Two common types of identifiability for linear and nonlinear settings, respectively, are the following.
Definition 3.1. We say that r(x)= r(((z)) identifies z up to sign and permutation if Pz for some signed permutation matrix (i.e., [P| is a permutation).
Definition 3.2. We say e identifies z up to permutation and element-wise reparametrisation if there exists a permutation π of {1,..,, K} and invertible scalar falatr-fnstiion {hx) k=1 s.t. Vj: =)((-(())
We now establish theoretical connections between the DCI framework and these identifiability types. Proposition 3.3. D = C C and K = L (i.e, dim(c) = (z)) then R is a permutation matrix.
All proofs are provided in Appendix A. Using Prop. 3.3, we can establish links identifiability, provided the inferred representation e perfectly predicts the true data-generating factors z, i.e., I = 1.
Corollary 3.4. Under the same conditions Prop. 3.3, if WTe (so that I for some W
Iwyl then identifies z up to permutation and sign (Defn. 3.1).
with Rij- E lagl"
For nonlinear f, we give a more general statement for suitably-chosen feature-importance matrices R.
Corollary 3.5. Under the same conditions as Prop. 3.3, let z = f(c) (so that 1 = 1) with f an invertible and differentiable nonlinear function, and let R be matrix of relative feature importances for f (Defn. 2.1) with the property that R;j
and only if fj does not depend on Cj i.e., ||affil| = 0. Then c identifies z up to permutation and element-wise reparametrisation (Defn. 3.2).
Remark 3.6. While the if part of Corollary 3.5 holds for most feature importance measures, the only if part, in general, does not: not using feature Cj is typically sufficient condition for R,j but it need not be necessary condition (as required for Corollary 3.5). E.g., measures based on average performance may not satisfy this since a feature may not contribute on average, but still be used sometimes helping and sometimes hurting performance (see 7 for further discussion). In contrast, Gini importances, as used in random forests, do satisfy the necessary condition. While the non-invertibility of random forests prevents an explicit link to identifiability (typically studied for con- tinuous features), they can still be a principled choice in practice (where features are often categorical).
Summary. We have established that the learnt representation c identifies the ground-truie tttnd-htuthh- z up to:
sign and permutation if D = C= and f is linear;
permutation and element-wise reparametrisation if D D-CC -- 1-1 and Rij = 0 + ||ah = 0.
HHee deviate from Fastwood & Williams (who had Ij (F(((z, such that the hest score.
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4 EXTENDED DCI-ES FRAMEWORK
Motivated by our theoretical insights from $ considerrin different probe function classes provides links to different types of identifiability and the empirically-observe performance differences between representations trained with different-capacity probes shown in Fig. 1, we now propose several extensions of the DCI framework.
4.1 EXPLICITNESS (E)
We first introduce a new complementary notion of disentanglement based on the functional capacity required to recover predict from c. The key idea is measure the explicimess ease-of-use (E) of a representation using its loss-capacity curve.
Notation. Let F be probe function class (e.g., MLPs or RFs), let ff E arg fer B(((z,,
be a minimum-loss probe for factor 2j held-out data split,, and let Cap(-) be suitable capacity measure F F-c.g.. for RFs, Cap(f) could correspond the maximum tree-depth of f-
Loss-capacity curves. A loss-capacity curve for representation c, factor 2j. and probe class F displays test-set loss against probe capacity for plot such curve, we must train T predictors with capacities K1. Ki,...,KK predict 2j. with
probes f F (see Fig. 1). To
fj E arg migmiee E [E f(c))]
s.t. Cap(f) = K₁.
(4.1)
Here Ki,...,KT is suitable choices for K1 and Xr depending on both F and the dataset. For example, we may choose KT to be large enough for all representations to achieve their lowest loss and, for random forest fs, we may choose an initial tree depth of K1 1 and then
list of T increasing probe capacities, ideally* shared by all representations, with
tree depths between 1 and Ky.
AULCC. We next define the Area Under the Loss- Capacity Curve (AULCC) for representation c, factor 2j. and probe class F as the (approximate) area between the corresponding loss-capacity curve and the loss-line of our best predictor f EE E[{(z), fj (c))]- To compute this area, depicted in Fig. we use the trapezoidal rule
AULCC Normalizer
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Loss
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1
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gr.
8"
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where fer denotes the index of c's lowest-loss capac- ity Ka,ci e.ce ff(()) the test-set loss with predictor f/. see Eq. (4.1); and Ax; k the size of the capacity interval step If the lowest loss is achieved at the lowest capacity, i.c. f* = 1, we set AULCC=0, O.
Capacity
Figure 2: Explicitness via the area under the loss-capacity curve (AULCC), Here, Kj,-., (x-axis) are sequence of increas- ing function-capacities and fle d.s,...,TT. (y-axis) are the losses achieved by corresponding optimal predictors for c. The lowest loss fo is achieved at capacity K,,e- while e and are suitable baseline and best-possible losses for the probe class.
Explicitness. We define the explicitness (E) of represen- tation c for predicting factor 2j with predictor class F as
(U,CC(Z),;FF))
=1 1( xxx K1) e7)
where eg is a suitable baseline loss (c.g., that of E[2z)) and f a suitable lowest loss (c.g., 0) for F. Here, the denominator represents the area of the light-blue triangle in Fig. 2, normalizing the AULCC such that E, E [-1,1] so long as Ej < <Eg. The best score E, 1 means that the best loss was achieved with the lowest-capacity probe f/. i.c., E7f ==f=0h =eee and K.,c = K1, and thus our representation c was explicit or easy-to-use for predicting 2j with f E F since there was no surplus
In practice, all expectations are taken w.r.t. the corresponding empirical (train/validatiinnttss distributions. 4Tree for RFs but input-size dependent ML.Ps (see 6).
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capacity required (beyond K1) to achieve lowest loss. In contrast, Ej means that the loss reduced linearly from E to éj with increased probe capacity, i.e., AULCC Normalizer in Fig. 2. More generally, if g& = f* i.c. the lowest loss for F can be reached with representation then Ej < 0 implics that the loss decreased sub-linearly with increased capacity while Ej > 0 implies it decreased super-linearly. The overall explicitness score is given by E x E.
vs. I. While the informativeness score Ij captures the (total) amount of information in about 2,, the explicitness score E; captures the ease-of-use of this information. In particular, while I is quantified by the lowest prediction error with any capacity (* corresponding single point c's loss-capacity curve, Ej is quantified by the area under this curve.
the commonly-un used mean correlation coefficient (MCC) or Amari distance (Amari et al., 1996; Yang & Amari. 1997). the D, P,C, scores represent empirical measures which: (i) casily extend to mismatches in dimensionalitiis. i.c., L K; and (ii) provide a more fine-grained picture of identifiability (violations), for if the initial probe capacity K1 is linear and R satisfies Corollary 3.5, we have that:
fine-grained picture of identifiability. Compared
D= --C--I--a- = identified up to sign and permutation (Defn. 3.1);
D 1---==11 = identified up permutation and element wise reparametrisation (Defn. 3.2);
1-E-1 = identified up to invertible linear transformation (ef. Khemakhem et al., 2020).
Thus, if D= 1 does not hold exactly, which score deviates the most from 1 may provide valuable insight into the type of identifiability violation.
Probe classes. As emphasized above, whether not representation is explicit or easy-to-use for predicting factor 2j depends on the class of probe F used, e.g., MLPs or RFs. More generally, the explicitness of representation depends with different downstream uses or probe classes resulting in different definitions of explicit or easy-to-use information. We thus conduct experiments with different probe classes in $ 6.
the way in which it is used in downstream applications,
4.2 SIZE (S)
We next introduce measure of representation size (S), motivated by the observation that larger representations tend to be both more informative and more explicit (see Tab. I, more details below). Reporting S thus allows size-informativeness and size-explicitness trade-offs to be analysed.
measure of size. We measure representation size (S) relative the ground-truth as:
K dim(z) S = L
dim(c)
When L> K, as often the case, we have Se (0, 1] with the perfect score being $ However, if we also consider the I. < K case, which would likely sacrifice some informativeness, have SE (1, K].
Larger representations are often more informative. When L. < K, it is intuitive that larger representations are more informative-thh they simply preserve more information about When L> K, however, is also common for larger representations to be more informative, perhaps due to an easier optimization landscape (Frankle & Carbin, 2019; Golubeva et al., 2021). Tab. 1 illustrates this point, where AE-5 denotes an autoencoder with =5 5. Note that = for MP I3D-Real Real (see $ 6).
Larger representations are often more explicit. The explicitness of representation also depends on its size: larger representations tend be more explicit, as apparent from the second column of Tab. 1. To explain this, we plot the corresponding loss-capacity curves in Fig, 3. Here we see that the increased explicitness (i.c., smaller AULLC) larger representations stems from a substantially lower initial loss when using a linear- capacity MLP probe. The fact that larger representations perform better with linear-capacity MLPs is unsurprising since they have more parameters.
4.3 PROBE-AGNOSTIC FEATURE IMPORTANCES
Finally, to meaningfully discuss more flexible probe-function choices within the DCI-ES frame- work, point that the D and C scores can be computed for arbitrary black-box probes by using probe-agnostic feature-importance measures. In particular, in our experiments ($ 6), we use SAGE (Covert et al., 2020) which summarises each feature's importance based its contribution
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0.7
Representation
0.6
AE_10
AL.
AE-5
0.75
0.74
1.4
0.5
AE 100
AE-7
0.92
0.71
1.0
0.4
AE 500
AE-10
0.99
0.72
0.7
AE-100
1.0
0.90
0.07
AE-500
1.0
0.93
0.01
Table 1: E and S scores for auto-encoders of vari- sizes MP ID--Real with MLP probes.
Figure 3:
Loss-capacity curves for auto-encoders of various sizes MP I 3D-Real with MLP probes.
predictive performance, making use of Shapley values (Shapley, 1953) to account for complex feature interactions. Such probe-agnostic measures allow the D and C scores to be computed for probes with no inherent or built-in notion of feature importance (e.g., MLPs), thereby generalising the Lasso and RF examples of Eastwood & Williams (2018, $ 4.3). While SAGE has several practical advantages over other probe-agnostic methods (see, e.g., Covert et al., 2020, Table 1). it may not satisfy the conditions required to link the D and C scores to different identifiability equivalence classes (see Remark 3.6). Future work may explore alternative methods which do, e.g., by looking at feature's mean absolute attribution value (Lundberg & Lee, 2017) since, intuitively, absolute contributions do not allow for cancellation of positive and negative attribution average (cf. Remark 3.6).
5 RELATED WORK
Eastwood Williams (2018, $ 2) noted that the informativeness score with linear probe quantifies the amount of information in about that is "explicitly represented", while Ridgeway & Mozer (2018, $ 3) proposed measure of "explicitness" which simply reports the infor- mativeness score with linear probe. In contrast, our DCI-ES framework differentiates between the amount of information in about z (informativeness) and the case-of-use of this information (explicit- ness). This allows a more fine- fine-grained analysis of the relationship between c and z, both theoretically (distinguishing between more identifiability equivalence classes; $ 3) and empirically ($ 6).
Explicit representations.
Loss-capacity curves. Plotting loss against model complexity or capacity has long been used in statistical learning theory, e.g., for studying the mlas-varianee trade-off (Hastie al., 2009, Fig. 7.1). More recently, such loss-capacity curves have been used to study the double-descent phenomenon of neural networks (Belkin et al., 2019; Nakkiran al., 2021) as well the scaling laws of large language models (Kaplan et al., 2020). However, they have yet to used for assessing the quality or explicitness of representations.
Loss-data curves. Whitney et (2020) use loss-data curves, which plot loss against dataset size, to assess representations. They measure the quality of representation by the sample complexity of learning probes that achieve low loss on task of interest. Loss-data curves are also studied under the term learning curves in standard/purelly supervised-leaniie for a recent review). contrast, we focus on functional complexity and the task of predicting the data-generative factors z, and then discuss the functional complexity for other tasks in 7.
settings (see, e,g., Viering & Loog, 2021,
EXPERIMENTS
6.1
SETUP
Data. We perform our analysis of loss-capacity curves on the MPI3D-Real (Gondal et al., 2019) and Cars3D (Reed al., 2015) datasets. MP Realleeea contains 1M real-world images of robotic arm holding different objects with seven annotated ground-truth factors: object colour (6), object shape (6), object size (2), camera height (3), background colour (3) and two degrees rotations of the arm (40 x 40); numbers in brackets indicate the number of possible values for each factor. Cars3D contains 17.5k rendered images of cars with three annotated ground factors: camera elevation (4), azimuth (24) and car type (183).
Representations. We the following synthetic baselines and standard models as representations:

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Noisy labels: c= = + e, with (0,00.011 Ix).
Linedd labels: = Wz, with W,j IR + Eij and (0, 0.001) to achieve "uniform mixing" (each 2; evenly-distributed across the c;s) while also ensuring the invertibility of W a.s.
Raw data (pixels): c X = 8(z).
Others: We also use VAEs (Kingma & Welling, 2014) with 10 latents (L=10), (L=10),--VAEs B-VAEs (Higgins et al. 2017, L=10); and an ImageNet-pretrainel ResNet18 (He et al. 2016, L-512).
Probes. We use MLPs, RFs and Random Fourier Features (RFFs, Rahimi & Recht 2007) to predict z from with RFFs having linear classifier top. For MLPs, we start with linear probes (no hidden layers) then increase capacity by adding two hidden layers and varying their widths from 2 x K to 512 x K. We then measure capacity based on the number of "extra" parameters beyond that of the linear probe, and compute feature importances using SAGE with permutation-sampligg estimators and marginal sampling of masked values (see http://gghuuu.co/aooooeeeee sembles of 100 trees, control capacity by varying the maximum depth between and 32, and compute feature importances using Gini importance. For RFFs, we control capacity by exponentially increasing the number of random features from 2 to 2 and compute feature importances using SAGE.
For RFs, we use en-
Implementation details. We split the data into training, validation and test sets of size 295k, 16k, and 726k respectively for MP 1:33-Rell and 12.6k, 1.4k, 3.4k for Cars3d. We use the validation split for hyperparameter selection and report results on the test split. We train MLP probes using the Adam (Kingma & Ba, 2015) optimizer for 100 epochs. We use mean-square error and cross-entropy losses for continuous and discrete factors 2j. respectively. To compute Ej, we use the baseline losses of IE [[zz]] and a random classifier for continuous and discrete 2,, respectively. Further details can be found in our open-source code: http:///tthb..ommmaateenicccccco
6.2 EVALUATION RESULTS: CURVES AND SCORES
Loss-capacity curves. Fig. 1 depicts loss-capacity curves for the three probes and two datasets, averaged over factors zj. In all six plots, the noisy-labels baseline performs well with low-capacity and then surpassed by other representations given sufficient capacity, as expected. Note that the linearly- mixed-labels baseline immediately achieves = 0 loss with MLP probes but not with RFF or RF probes, supporting the idea that the explicitness or ease-of-use of representation depends on the way in which it is used. Also note that, with MLP probes and log (exeess #params) as the capacity measure, larger input representations are afforded more parameters with linear probe and thus are more expressive. This further explains why larger representations are often more explicit, and highlights the difficulty of measuring the capacity of MLPs-an active area of research in its own right, which we discuss in $ 7. Finally, in Appendix B.2, we investigate the effect of dataset size by plotting loss-capacity curves for different dataset sizes, observing that larger datasets have smaller performance gaps between: (i) synthetic and learned representations; and (ii) small and large representations (see Fig. 10).
DCI-ES scores. Tab. reports the corresponding DCI-ES scores, along with some oracle scores for MLPs. Note that: (i) the GT labels z get perfect scores of 1 for all metrics; (ii) by attaining very low D and C scores but near-perfect E scores, the linearly-mixe labels expose the key difference between mixing-based (D,C) and aunctionl--capacity-ooe
(E) measures of the simplicity of the e-z relationship; (iii) larger representations (ImgNet-pretr, raw data) tend to be more explicit than smaller ones (VAE, f- VAE), with S and E together capturing this exzleexplisssness trade-off; and (iv) 6- achieves better mixing- based scores (D,C) but similar E scores compared to the VAE, illustrating that these two "disentanglement" notions are indeed orthogonal and complementary.
6.3 DOWNSTREAM RESULTS: SCORE CORRELATIONS
Setup. To illustrate the practical usefulness of our explictness score, we calculate its correlation with downstream performance when using low-capacity probes. Using MP I-D,, we create 14 synthetic downstream tasks: 7 regression tasks with y' M'z and M U(0, 1), and 7 classification tasks
with y' 11 [=:-m:]] and m. the median value of factor Z;. For representations, we use AEs, VAEs and B-VAEs, 2 latent dimensionalities (i.e. (c) of 10 and 50, and 5 random seeds-resulting in total of 30 different representations c. To compute the correlations, we first compute the DCIE scores as before, training MLP and RF probes f to predict from c, i.e. 2, = ff(c), and then compute the down-
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Table 2: DCI-ES scores for different probes, datasets representations. Empirical scores using MLP. RFF RF probes trained on the MP 30-Real and Cars3D datasets, as well as theoretical/oracle scores for some simple representations with MLPs (MLP*). We show averages over 3 random seeds; standard deviations were all 0.05. Note that which representation is deemed "best" depends on the application of interest- some are more disentangled, some more informative, some more explicit, etc.
MPI3D
CARS3D
Representation
Probe
D
C
S
E
GT Labels
MLP
1
1
MLP
1
0.9
1.0
0.9
1
1.0
MLP
0.97
0.97
0.89
0.99
1.0
0.99
0.99
0.92
0.99
1.0
Noisy labels
RFF
0.97
0.97
0.88
0.99
1.0
1.0
1.0
0.91
1.0
RF
0.93
0.93
0.89
0.98
1.0
0.95
0.95
0.92
0.99
1.0
MLP
0
1.0
Linearly- mixed labels
MLP
0.13
0.22
1.0
1.0
1.0
0.21
0.22
1.0
1.0
1.0
RFF
0.11
0.21
1.0
0.94
1.0
0.19
0.19
1.0
1.0
1.0
RF
0.17
0.21
1.0
0.72
1.0
0.08
0.12
0.99
0.78
1.0
ML.P
0.15
0.14
0.99
0.71
0.7
0.18
0.11
0.95
0.80
0.3
VAE
RFF
0.13
0.14
0.97
0.69
0.7
0.16
0.11
0.91
0.87
0.3
RF
0.10
0.10
0.93
0.65
0.7
0.14
0.09
0.80
0.81
0.3
MLP
0.46
0.41
0.96
0.74
0.7
0.27
0.23
0.94
0.78
0.3
V-E
RFF
0.41
0.38
0.71
0.7
0.31
0.23
0.86
0.86
RF
0.39
0.35
0.94
0.76
0.7
0.20
0.17
0.84
0.83
0.3
MLP
0.16
0.10
0.99
0.82
0.01
0.22
0.07
0.88
0.86
0.006
ImgNet-preet
RFF
0.15
0.13
0.96
0.58
0.01
0.24
0.10
0.83
0.65
0.006
RF
0.35
0.20
0.89
0.78
0.01
0.20
0.09
0.62
0.83
0.006
MLP
0.22
0.16
0.99
0.82
0.001
0.39
0.27
0.95
0.84
0.0002
Raw data
RFF
0.37
0.14
0.97
0.44
0.001
0.32
0.24
0.87
0.64
0.0002
RF
0.84
0.41
0.96
0.80
0.001
0.53
0.31
0.86
0.82
0.0002
MLP (p=0.96)
MLP
1.0
Downstream avg. pert
Z
Pearson corr. Coetr.
0.7
h
0.8
0.6
0.6
x
y
0.0
0.75
0.80
0.85
Explicitness
(a)
(b)
(c)
Figure 4: (a) Correlation coefficients between DCIE scores and downstream performance with low-capacity probes. (b) downstream performance with linear MLPs. (c) DCIE scores computed predicting from
e with probes f. then downstream tasks y = 11(z) are solved by predicting y from e with low-capacity probes f.
stream performance by training new low-capacity MLP and RF probes f to predict y from i... g fi(c) (see Fig. 4c). For MLP probes, low capacity means linear. For RF probes, low capacity means the maximum tree depth is 10. Next, we average the downstream performances across all 14 tasks before computing the correlation coefficient between this average and each of the D, C, I, and E scores.
Analysis. Figs. 4a and 4b show that E is strongly correlated with downstream performance when and RF probes (p- 0p=00 0.88, p p= 2=-10... In contrast, mixing-based disentanglement scores (D. C) exhibit much weaker correlations with MLP probes, corroborating the results of Träuble et al. (2022, Fig. 8) who also found a weak correlation between D and downstream performance reinforcement learning tasks with MLPs. See App. B.1 for further details and results.
using both MLP (p=00 1pp=0.96 0.96, p= == 8e-18)))))))
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MLP probes
RFF probes
RF probes
1.00
11.00
1.00
-1.00
scores log
0.90
0.90
0.80
0.98
0.90
0.80
0.60
Scores linear scores line
0.80
0.80
0.99
0.96
0.40
0.70
0.98
0.20
1.00
1.00
1.00
-1.00
1.00
scores log
0.90
0.90
0.98
0.80
-0.9
0.90
0.00
0.00
0.97
0.70
0.40
linear
linear
log
scaling
scaling
scaling
ImageNet
Linearl-mixeed labels
Noisy Labels
Raw Data
WAE
3-VAE
Figure 5: Explicitness scores on MP I 3D-Real (top) and 3D (hottom) for different representations (sec legend). Each pair of points represents the score with logarithmic (left) and linear (right) capacity-scaling
7 DISCUSSION
Why connect disentanglement and identifiability? Connecting rrdictioon-bsse evaluation in the disentanglement literature to the more theoretical notion of identifiability has several benefits. Firstly, it provides concrete link between two often-separate communities. Secondly, it endows the often empirically-driven or practice-focused disentanglement metrics with a solid and well-studied theoretical foundation. Thirdly, compared to the commonly-used MCC or Amari distance, it provides the ICA or identifiability community with more fine-grained empirical measures, as discussed in $ 4.1.
Measuring probe capacity. Our measure of explicitness E depends strongly on the choice of capacity measure for a probe function class. For some probes like RFs or RFFs, there exist natural measures of capacity. However, for other probes like MLPs, coming up with a good capacity measure is itself an important and active area of research (Jiang et al., 2020; Dziugaite et al., 2020). Another difficulty arises from choosing a capacity scale, with different scales (c.g., log, linear, etc.) leading to loss-capacity curves with different shapes, areas and thus explicitness scores. To investigate the extent of this issue, i.e., the sensitivity of our explicitness measure to the choice of capacity scale, Fig. 5 compares the explicitness scores when using logarithmic and linear scaling. Here we that the ranking essentially remains the same except for the raw-data representation with MLP probes.
Mcasuring feature importance. Similarly, the choice of feature-importance measure has strong influence on the D and scores, with some probes having natural or in-built measures (e.g.. random forests) and others not (e.g.. MLPs). For the latter, we proposed the use of probe-agnostic feature- importance measures like SAGE, and specified the conditions (Corollary 3.5) that importance measures must satisfy if the resulting D and C scores are to be connected to identifiability. As with probe capacity, coming up with good measures of feature importance is its own orthogonal field of study (e.g., model explainability), with future ad advances likely to improve the DCI-ES framework.
What about explicitness for other tasks y? While we focused on the explicitness case-of-use of representation for predicting the data-generative factors z, one may also be interested in its ease-of-use for other tasks/labels y. While is often implicitly assumed that the ease-of-use for predicting z corre- lates with the ease-of-use for common tasks of interest (e.g., object classification, segmentation, etc.), future work could directly evaluate the explicitness of representation for particular tasks y. For ex- ample, one could consider the entire loss-capacity capacity curve when benchmarking self superviedd represen- tations on ImageNet, rather than just linear- linear-probe performance (a single slice). Future work could also explore the trade-off between explicit but mask-specific and implicit but task-agnostic representations.
CONCLUSION
We have presented DCI-ES an extended disentanglement framework with two new complementary measures of representation quality- and proven connections identifiability. In particular, we have advocated for additionally measuring the explicitness (E) of a representation by the functional capacity required to it, and proposed quantify this explicitness using representation' 's loss- capacity curve. Together with the size (S) of a representation, we believe that our extended DCI-ES framework allows for more fine-grained and nuanced benchmarking of representation quality.

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ACKNOWLEDGMENTS
The authors would like to thank Chris Williams, Francesco Locatello, Nasim Rahaman, Sidak Singh and Yash Sharma for helpful discussions and comments. This work was supported by the German Federal Ministry of Education and Research (BMBF): Tübingen Al Center, FKZ: 011S18039A, OUIS I8039B; and by the Machine Learning Cluster of Excellence, EXC number 2064/1 Project number 390727645.
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A PROOFS
A. PROOF OF PROPOSITION 3.3
Proposition 3.3. D C= and K = L (i.e., dim(c) dim(2)) 2)), then R is a permutation matrix.
Proof. First, by Defin 2.1, we have 0 < Rij and Σ Rúj = 1 Vi,j, so S Rúj S 1. It follows that vi.j: Pi., P.j.P. E Áx-1 where Ax-1 denotes the K-dim. probability simplex, i.e., P. and P.; are valid probability vectors. Hence, the Shannon entropies Hk HK(P,,), Hx Hx(,)) are well-defined Vi,j. and, due to using logx in the definition of Hx (see 2), are bounded in [0, 1]. follows that vi, S D; S and 0 S Cj S 1. Since D and C are convex combinations of the D, and Cj. we have
D=1 vi: Dj =1 + vi: (g(),,) =0,
C= Vj: 1- = -
Vj: Hx 0.
Now for any = (p
E A--1, we have that
Hk(p) = - - Pk: logx Pk - + vk: Pk logx Pk = 0 + vk: Pk E {0,1}
where Pt log Pk := O for pk = O, consistent with lim, *0+ x xlog 0. Together with the simplex constraint, this implies that p must be standard basis vector p e; for some 1, i.e. pt 1 and
Px for k # Hence, P.,, P., must be standard basis vectors for all i, and so each row and column of R contains exactly one non-zero element. Since columns of R sum one, these non-zero elements must all be one.
A.2 PROOF OF COROLLARY 3.4
Corollary 3.4. Under the same conditions Prop. 3.3, if WTc (so that I for some W [umgl then identifies up to permutation and sign (Defin. 3.1).
with Rij E. lagll
Jaryl
is well-defined feature importance matrix. Suppose for Proof. First, we show that Rij- - mm:
contradiction, that Σ |wal for some 1. Since |zyl| 2 0, this implies wy for all i. Consider = E Wn Taking the covariance, we obtain Var[z]] Σز wa Cov(g, cy) = O, which is a contradiction since 2 has positive (unit) variance by the normalisation assumption (see footnote 1). Hence, Σ |wyl > 0 for all I. Thus R is well-defined, with its elements being non-negative and a valid feature importance matrix.
its columns summing to one by construction,
Next, note that we can write R
W where D is the invertible diagonal matrix with positive
diagonal entries Djj
0.
By Prop. 3.3, multiplication by D-1 yields PD-1 WW] that WW| has exactly one non-zero positive element each row and each column (and zeros elsewhere). Thus W and therefore also W are generalised permutation matrices. Hence (WT) exists and is also generalised permutation matrix.
permutation matrix, R P W for some permutation matrix P. Right
Finally, consider e (WT) 1z. Since all but ony element in each row of (WT) are zero, we have for any i: c; = Wyjz, for some j. where æ denotes the (i, element of (W (WT)-1 By considering the variances of both sides and recalling that all c;'s and z 's are normalised unit variance, it follows that 1 = Var(c) = 2 Var(z;) = : Hence, w = 1 and so [WT)-1 is, in fact, a signed permutation matrix, which concludes the proof.
A.3 PROOF OF COROLLARY 3.5
Corollary 3.5. Under the same conditions as Prop. 3.3, let z f(c) (so that 1 = 1) with f an invertible and differentiable nonlinear function, and let R be a matrix of relative feature importances
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for (Defn. 2.1) with the property that Rij if and only if f/ does not depend C₁ i.e., |@āff 0. Then c identifies z up to permutation and element-wise reparametrisation (Defn. 3.2).
Proof. For any consider 2, = fí(c). By Prop. 3.3, R is a permutation matrix, column j of R contains exactly one non-zero entry in row π(j) for some permutation of (1, (1,-.. K). Hence, by the assumed property of R. fj(e) does not depend on e; for all * π(j). and thus 2j = fj((x()) Vj. By and j' π (f).
invertibility of f, we obtain Cj = hz(zy) with hj
B ADDITIONAL EXPERIMENTAL RESULTS
B.1 DOWNSTREAM CORRELATIONS
Here we present the full results of the correlations between the DCIE scores and downstream performance, the latter with low-capacity probes (as discussed in 6.3).
In Tab. and Tab. 4 we show the values of the Pearson and Spearman correlations alongside the corresponding p-values Note that some of the assumptions behind these p-values, e.g. that the DCIE scores and downstream performances are normally distributed, likely do not hold. Thus, these p-values should not be interpreted precise probabilities but rather rough indications of statistical significance. Tab. 5 we show the correlations for regression and classification tasks separately, with both task types exhibiting similar correlations. We note that E has the strongest correlation with the downstream performance (when using low-capacity probes for the downstream task).
Table 3:
Pearson correlation coefficient, between the D, C, I, and E scores and downstream performance, along with the corresponding p-values (in parentheses). See 6.3 for experimental details.
Probe f
D
MLP
0.15 (4 10-1) 0.28 (9 x 10 0.47 (9 x 10-3) 0.96 (8 10 (0-18)
RF
0.8 (1 x 10 0.70 (2 x 10 5) 0.4 (3 x 10 0.88 (2 x 10 10-10)
Table 4: Spcarman rank correlation between the D, C, I, and scores and downstrcam performance, along with the corresponding p-values (in parentheses).
Probe f
E
MLP
0.12 (5 10-1) -0.07 10-1)
0.55 (2 10 10-3) 0.94 (1 x 10 10-14)
RF
0.81 (6 x 10 8) 0.75 (2 x 10 6) 0.28 (1 x 10 10-1) 0.78 (3 x 10-7)
Table 5: Pearson correlation coefficient, p between the D,C.T, E scores and downstream performance for each task type (regression and classification). Correlations similar across both task types.
Probe
Task
D
I
E
Regression
0.16
0.04
0.46
0.96
MLP
Classification
0.14
0.00
0.48
0.96
Regression
0.76
0.66
0.42
0.84
RF
Classification
0.81
0.72
0.35
0.89
Score-by-score analysis. To get a deeper insight into the correlations reported in Tabs. 3 and 4, we plot each of the D, C, I and scores against downstream performance for each of the 30 models considered in $ 6.3. As shown in Figs. 6 to only E correlates strongly with downstream performance
*Computed using hitps:/ / /ows www.osssssp....ooooooooo.oow.p.e scipy. http://......../////////////eeee http://eeeeeeeeeeeeee/////////// stats. pearsonr. htm1
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for both probe types, again highlighting: (i) the value that E adds to the existing DCI framework: and (ii) the practical usefulness of reporting E when comparing/evaluating learned representations.
MLP (p=0.96)
RF (p=0.88)
0.75
Downstream avg. perf
Downstream avg. perf
0.74
0.70
0.72
0.65
0.70
0.60
0.68
ae dim_mimm _0
0.55
ae dim 50
0.66
beta vae dim, 10 beta va_ dim_ 50 vae dim
0.50
0.64
vae, dim 50
0.45
0.62
0.75
0.80 Explicitness
0.85
0.60
0.70 Explicitness
0.75
0.80
Figure 6: Explicitness (E) vs. downstream performance. Scatter plots show 30 data points: models (AEs, VAEs, B- VA)s)) x 2 latent dimensionalities (L = 10 and L = 50) x random seeds.
MLP (p=0.15)
RF (p=0.80)
0.75
Downstream avg. perf
Downstream avg. perf
0.74
0.70
0.72
0.65
0.70
0.60
0.68
0.55
0.66
0.64
0.45
0.62
0.1
0.2
0.3
0.4
0.5
0.1
0.2
0.3
0.4
0.5
Disentanglement
Disentanglement
Figure 7: Disentanglement (D) vs. downstream performance. Scatter plots show 30 data points: models (AEs, VAEs, B-VAEs) 2 latent dimensionalities (L 10 and 50) random seeds.
B.2 DIFFERENT AMOUNTS OF DATA
In Fig. 10 present loss-capacity curves obtained when using different amounts of data to train the MLP probes. As shown, larger datasets have smaller performance gaps between (i) synthetic and leamned representations; and (ii) small and large representations.
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MLP (p=0.03)
RF (p=0.70)
0.75
Downstream avg. perf
Downstream avg. perf
0.74
0.70
0.72
0.65
0.70
0.60
0.68
0.55
0.66
0.50
0.64
0.45
0.62
0.1
0.2
0.3
0.4
0.1
0.3 Completeness
0.2
Completeness
Figure 8: Completeness (C) vs. downstream performance. Scatter plots show 30 data points: models (AEs, VAEs, BA))s)) x 2 latent dimensionalities (L = 10 and L = 50) x random seeds.
MLP (p=0.47)
RF (p=0.40)
0.75
Downstream avg. pert
Downstream avg. perf
0.74
0.70
0.72
0.65
0.70
0.60
0.68
0.55
0.66
0.50
0.64
0.45
0.62
0.96
0.97 0.98
0.99
1.00
0.93
0.94
0.95 0.96
Informativeness
Informativeness
Figure 9: Informativeness (I) vs. downstream performance. Scatter plots show 30 data points: models (AEs, VAEs, B- VAEs) latent dimensionalities (L 10 and 50) random seeds.
Hul Probcs
NL2 P5245
0.3
3.3
03
12
63
65
Figure 10: Loss-capacity curves for -I3--Real subsets size 100, 1000, 5000 and 10000 respectively.
16


Reviews:
[{"review_id": "xj0RnYvk2Sj", "reviewer": "Reviewer_wQrr", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "main_review": "", "summary_of_the_paper": "This work draws connections between the DCI metrics, commonly used in the deep learning literature on disentanglement, and notions of representation identifiability, commonly used in the literature on independent component analysis (ICA). More precisely, Corollary 3.5 provides conditions under which D=C=1 and K=L implies equivalence between c and z, up to permutation and element-wise reparameterization. In addition, the DCI metrics are extended by adding \"explicitness\" (E) and \"size\" (S). Basically, the former measures how a given probe class can predict the latent variables with various capacity levels and the latter is simply the ratio of the dimensionality of the ground-truth latent vector over the dimensionality of the learned representation. These novel metrics are then evaluated on various learned representation and dataset.", "summary_of_the_review": "I believe this paper makes valuable contributions, but (i) Corollary 3.4 contains a mistake (although I am optimistic that the issue can be resolved) and (ii) I do not see the added value in using E instead of simply reporting the loss-capacity curves more transparently. Mainly for these reasons, I recommend weak rejection for now, but I am very open to increasing it to a weak accept once the problem in Corollary 3.4 is fixed.\n\n--- Post-rebuttal update ---\nI have updated my score to 6, see my comment below.", "strength_and_weaknesses": "Strengths:\n- I believe both the deep learning community and the (more theoretically oriented) nonlinear ICA community will benefit from the connections made in this work.\n- I very much agree with the authors that one should take into account the capacity of a probe when evaluating a representation and that exploring different classes of probes with different capacities gives a more complete picture of the learned representation.\n- The \"size\" metric, although very simple and obvious, is important if one wants to make fair comparisons.\n- The paper is clearly written and easy to follow.\n- The experiments are clearly presented and sufficient for the argument.\n\nWeaknesses:\n- Mathematical precision is sometimes lacking. See my comment below, especially about Corollary 3.4.\n- As I said, I agree with the author that one should take into account the capacity of a probe when evaluating a representation and that exploring different classes of probes with different capacities gives a more complete picture of the learned representation. However, I am a bit skeptical that the proposed explicitness score (E) brings more value than just transparently reporting the Informativeness score (I) for a couple of different probes with different capacities (i.e. the loss-capacity curves). E is basically a less transparent way of communicating this information. Moreover, Figure 4 shows that the ranking can change when simple rescaling are applied to the capacity measures. I find this worrying.\n- I do not understand why the DCI metrics are restricted to R matrices with columns that sum to one (Definition 2.1). This is counterintuitive to me, since one could have a learned representation c that is completely useless to predict one of the ground-truth latent variable z_j, but since the columns of R must sum to one, it is impossible for the corresponding column (R_{.,j} to be filled with zeros (as it should, to represent the fact that the representation is useless to predict that ground-truth factor). ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "qOnE-hJmohg", "reviewer": "Reviewer_WGrh", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper proposes an extended Disentanglement Framework to evaluate the disentanglement in representation learning. Specifically, the paper first links the disentanglement to independent component analysis, then propose two new measures of representation quality: explicitness (E) and size (S). The paper uses MPI3D and Cars3D datasets to evaluate the method.\n", "summary_of_the_review": "The reviewer thinks the disentanglement evaluation should consider relatively orthogonal and important evaluation metrics. Motivation and necessity are the primary concerns. \n\n\n\n--------------------------------------------------------\nThanks for the author's feedback! After reading the feedback, some concerns are addressed, e.g., the motivation for measuring explicitness(E) and size(S). While the reviewer's second question was not well addressed, \"If the capacity of probing needs to be evaluated, then the training time, cost, and learning rate may also be considered because they may influence the final value of DCI.\". After reading other reviewers' comments, the reviewer has similar concerns about the proposed method's novelty (technical contribution) compared to DCI and practical usefulness. Therefore, the reviewer will keep the original score and vote for borderline.", "strength_and_weaknesses": "**Strength**\n\n1 The new measurement of explicitness (E) (easy-to-use) and size (E) is interesting for disentanglement evaluation.\n\n2 The paper writing is clear and easy to follow.\n\n**Weaknesses**\n\n1 The traditional DCI framework may already be considered explicitness(E) and size(S). For instance, to evaluate the disentanglement (D) of different representation methods, you may need to use a fixed capacity of probing (f), and the latent size should also be fixed. DCI and ES may be entangled with each other. For instance, if you change the capacity of probing or the latent size, then the DCI evaluation also changes correspondingly. The reviewer still needs clarification on the motivation for considering explicitness(E) and size(S) as extra evaluation.\n\n2 Intuitively, explicitness(E) and size(S) may be highly related to the given dataset. The different capacity requirements in the 3rd paragraph may be due to the input modality difference. Given a fixed dataset, the evaluation of disentanglement should provide enough capacity and training time which is powerful enough to achieve the DCI evaluation. If the capacity of probing needs to be evaluated, then the training time, cost, and learning rate may also be considered because they may influence the final value of DCI.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "jWgCBKHfQDd", "reviewer": "Reviewer_R81A", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper introduces two quantifiers of disentanglement (measures of how good a latent representation is, measurable when we have the ground truth latents), complementary to Eastwood & Williams' DCI, Disentanglement, Completeness, and Informativeness. The first is _explicitness_, which is related to a (normalized) area under the curve of informativeness as a function of capacity, and illustrates how accessible the true latents are from a code. The second is _size_, which simply is the ratio of the dimensionality of the code to the dimensionality of the true latents.", "summary_of_the_review": "I'm leaning reject but still quite on the fence.\n\nThis paper provides a nice intuition and provides a decent amount of justification for it, but where the work is lacking is in showing where it leads. The authors propose a quantity, but it's not too clear how useful or predictive it is, nor if it has some empirical regularities which make it desirable in practice (if I wrote a paper on a new kind of VAE tomorrow, would I want to use this? I'm not convinced).\n\nUpdate: The authors have addressed most of my concerns, and going through the updated version of the paper, I think it is now a much better work.", "strength_and_weaknesses": "### Strengths\n- The paper is certainly well written\n- The work touches on an important open problem, how do we evaluate disentanglement?\n- The proposed ideas are intuitively appealing\n\n### Weaknesses\n- The main proposed quantity, explicitness, relies heavily on other quantities for which it is still an open problem how to measure\n- The empirical validation is a bit lacking, what are these measures predictive of? What are they a model of?\n\n\n### Comments\n\n> Note that $I_j$ depends on the capacity of $f_j$, as depicted in Fig. 1.\n\nShouldn't D and C also depend on the capacity of $f$ through $R$?\n\n> Corollary 3.5, [..] with $f$ an invertible nonlinear function\n\nThis seems to be a really strong requirement, which greatly reduces the class of functions that the results of section 3 can give us intuition, or \"theoretical insights\", about. I wonder if I am missing something deeper here?\n\n\n> Larger representations are often more informative. When dim$(c)<$dim$(z)$.\n\nI know that this is just meant to be an intuitive passage, but I'm not sure this is the right threshold for dim$(c)$. Perhaps appealing to compression-based generalization theory here might make more sense, if the number of bits one can encode in $c$ is smaller than the number of bits of information in $z$, then there is necessarily degradation/lossy compression. Some phase transition occurs at that threshold, although as the authors already argue, regardless of the threshold more capacity is generally accepted to be more informative.\n\n\n> larger representations (ImgNet-pretr, raw data) tend to be more explicit than smaller ones (VAE, $\\beta$-VAE)\n\nThere's an experiment that's missing here, that might make this claim much easier to validate. What happens to a [$\\beta$-]VAE as $L$ increases, i.e. S decreases?\n\nThe downside of asking this question is that, as the authors point out, changing $L$ changes the architecture of the model, which inherently changes the explicitness. Are two models with slightly different architectures comparable? Would it even make sense to plot E as a function of $L$? \n\nAs Figure 4 shows, even the choice of scaling on the AULCC affects the ranking of methods for which we have a poor measure of capacity, MLPs. This work leans heavily on a choice of quantification of capacity, which is still an open problem for DNNs. \n\nIt's obvious from recent research on generalization that we cannot hope to get interesting quantities that speak to generalization and disentanglement without taking into account (effective) capacity and data. This is why what is proposed is quite appealing intuitively. But I wonder what this quantity currently predicts, i.e. what it is a model of, what other quantities it correlates with and/or predicts.\n\nIn the face of this brittleness, one way to reassure ourselves is to gather more data. Perhaps such a plot as I'm proposing above would reveal some interesting (if noisy) patterns.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "06ZjgXVE4Ph", "reviewer": "Reviewer_xBj7", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper connects the DCI framework [1] for disentanglement to identifiability and proposes an extension of the framework that incorporates two new measures of representation quality which are explicitness (E) and size (S).", "summary_of_the_review": "This paper is well written. However, its novelty and significance are limited. It seems to complicate things by introducing two minor, ad-hoc metrics besides DCI instead of designing better, more elegant metrics. Thus, I think the paper is below the acceptance bar of ICLR.\n\n### Post rebuttal\n******************************************\nI would like to thank the authors for your comments. I think most of my concerns are well addressed by the authors. Thus, I raise my score to 6 though I still worry about the soundness and the correctness of the metric for practical use.  ", "strength_and_weaknesses": "### Strengths\n1) Overall, the writing of the paper is good \n\n### Weaknesses\n1) The novelty and significance of the work are low as it is mainly developed on an existing disentanglement metric which is DCI. \n2) The DCI metric was proposed long ago and has several drawbacks such as i) only supporting continuous factors, ii) using a not-well-normalized importance matrix R, …  as discussed in a more recent work by Do & Tran ICLR-2020 [1] (Appendix A.8). The current work inherits all these drawbacks from DCI. It even complicates things by introducing 2 new minor metrics, while in my opinion, it should address the limitations of DCI and propose a simpler yet better metric for disentanglement.\n3) Lack of discussion of several related works about metrics for disentanglement [1, 2].\n4) Linking DCI to identifiability is straightforward and can be proven quite easily. Thus, Proposition 3.3 in the paper does not sound interesting to me. Besides, the DCI only achieves identifiability in the extreme case when D=C=1. But in practice, we rarely achieve such perfect values of D and C. Normally, D, and C will be between 0 and 1. How can the authors quantify the “amount” of identifiability in this case? I think answering this question is more worthwhile than introducing 2 complementary minor metrics.\n5) The explicitness metric seems complex and ad-hoc to me. Since the capacity Cap(.) is defined differently for different kinds of models (e.g., Cap(.) is the maximum tree depth for a random forest while it is the number of neurons in neural networks), explicitness will be different if different models are used, which is not desirable. There is no consistent strategy to choose the “probe capacities” $\\kappa\\_1$, …, $\\kappa\\_T$. In addition, we need to train a new model for each “probe capacity” which can be very costly if the model size or the number of probe capacities is large.  The suitable base loss $\\ell^b_j$ and the suitable lowest loss $\\ell^{*}_j$ are not well defined and not consistent between models. I don’t understand what $\\mathbb{E}[z_j]$ really means for $\\ell^b_j$. Moreover, the loss $\\ell^{t, c}$ for each probe capacity $\\kappa_t$ can be noisy and incorrect, which greatly affects the accuracy of the metric. The explicitness E=1 means that AULCC=0, which is almost unlikely to happen.\n6) I don’t see any point in using Size (S) as a metric.\n7) The results in Table 2 are very hard to interpret as I don’t understand which representations are the best. Besides, there is no clear evaluation target for the representations. Looking at Table 2, how can we know the two proposed metrics E and S are correct and reasonable?\n\n[1] Theory and Evaluation Metrics for Learning Disentangled Representations, Kien Do and Truyen Tran, ICLR-2020.\n\n[2] Weakly Supervised Disentanglement with Guarantees, Shu et al., ICLR-2020.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Accept: poster