Paper: Correcting Flaws in Common Disentanglement Metrics (ICLR-2024)

Reviewer comment (Novelty):
Lack of Novelty: Despite the detailed analysis and new metrics, the work might not be considered highly novel. The concepts presented can be seen as incremental improvements over existing ideas.

Author response:
The concepts presented can be seen as incremental improvements over existing ideas. The two main idea of this paper, neuron-factor alignment and the difference between individual vs collective informativeness, are, to our knowledge, completely novel. We do not know of any existing works to explore these ideas.

Reviewer comment (Evaluation):
How do the proposed metrics compare with other existing metrics in terms of computational efficiency and scalability?

Author response:
How do the proposed metrics compare with other existing metrics in terms of computational efficiency and scalability? SNC is fast, <1 minute for the entire dataset. NK is slower, around 20minutes for the full dataset. This is still faster than DCI, which can take up to one hour.

Reviewer comment (Novelty):
A great number of related works that are important for (P1) are overlooked. If I understand correctly, the authors pointed out that existing works like MIG fail to penalize learning multiple generative factors in the same neuron, which is true. However, there are a lot of other existing metrics that are designed exactly to measure this, for example, MIG-sup [1], modularity metrics (Eqn (2) in [2]), and more that are reviewed in [3] in the category of "Modularity". The authors are encouraged to discuss and compare these metrics.

Author response:
*Discussion of MIG-sup*.
Thank for this reference, we will include it in the discussion of single neurons representing multiple factors. It is indeed a solution to this problem, but we would argue it is an imperfect solution. In the comparison between the two toy models in Section 3.1, M1 would get a higher MIG score but a lower MIG-sup, and the sum MIG+MIG-sup would be the same for both models. For our metrics, however, the sum SNC+NK would be lower, which we have argued to be more correct. There is also the problem that it is not clear how to extend MIG and MIG-sup to deal with P2, because extending MI to multiple variables becomes tricky both theoretically and computationally.

Modularity in [2] takes a similar solution to this problem, similar also to DCI, though like DCI, we do not claim that this problem occurs in [2] because it doesn't operate by explicitly aligning factors to neurons, as MIG does. Also, all three of these approaches, DCI, MIG and [2], penalize a neuron representing multiple factors by adding another metric. Our work instead employs the cleaner solution of forcing the alignment to be injective in the first place.

Reviewer comment (Evaluation):
This paper didn't clearly demonstrate how the proposed metrics solve the proposed problems. The authors are encouraged to design experiments that can clearly demonstrate that the proposed metrics can correctly evaluate the scenarios of (P1) and (P2), while the existing metrics, on the other hand, may fail and produce a "false" high disentanglement score.

Author response:
*The authors are encouraged to design experiments that can clearly demonstrate that the proposed metrics can correctly evaluate the scenarios of (P1) and (P2), while the existing metrics, on the other hand, may fail and produce a "false" high disentanglement score.*
The solution of P1 is automatic, by the definition of our metrics. For P2, it would be straightforward to show that NK will pick up on $z_{\neq i}$ representing $z_i$ by XOR or something similar, as learning a function like XOR is a simple task for an MLP. We will add an experiment of this sort.

Reviewer comment (Theory):
What perspective of disentanglement does the two proposed metrics are focused on? For example, using the terms "modularity", "compactness", and "explicitness" in [2] or universal?

Author response:
*What perspective of disentanglement does the two proposed metrics are focused on? For example, using the terms "modularity", "compactness", and "explicitness" in [2] or universal?*
They do not fit so neatly into the framework of [2]. NK corresponds to modularity, but SNC is a mixture of explicitness and compactness.

Reviewer comment (Presentation):
The example in Figure 1 right is not clear. What is the coding map for z1? I suggest the authors to show the map between factors and the code in a table, like (square,blue)→0.

Author response:
Thank you. We can reformat as you suggest.

Reviewer comment (Evaluation):
The Hungarian algorithm usually uses L2 over whichever values you use. You chose MI, which means you’re taking squared distances of bits in order to do tie-breaking or fine-grained factor-latent attribution. Is that really what you want? This is actually a strong hidden assumption in your metric. In my experience L2 is not always what you need, and situations where multiple alternatives are equally likely do not always behave well without careful consideration. See for example AlignNet [1] and especially OAT [2] for a more careful discussion of why learning the metric to align on might be necessary in more complex situations.

Author response:
*The Hungarian algorithm usually uses L2 over whichever values you use. You chose MI, which means you’re taking squared distances of bits in order to do tie-breaking or fine-grained factor-latent attribution. Is that really what you want?*  
There is no L2 in the Hungarian algorithm. It simply receives as input an $M \times N$ cost matrix $C$, $M \geq N$, and returns some permutation $s \in S_N$ that minimizes $\sum_{i=1}^N C[S_N[i],i]$ (use python-style indexing). There is no mention of L2 in the original papers by Kuhn and Munkres [1,2], nor the documentation for the scipy implementation, which is what we use. The values of our cost matrix are MI, just as we describe, with no notion of squared distances of bits.

Reviewer comment (Writing):
The presentation of the SNC metric is not as clear as it should be. What is z_i? I understand it’s the output of the Hungarian matching from G to Z, but you do not define it. The description of the binning is confusing. What happens if you had continuous factors/latents? How does this differ from computing the Discrete mutual information?

Author response:
*Presentation of metrics, what is $z_i$?*  
Thanks. We will specify that we mean the latent indices are ordered so that $z_i$ corresponds to $g_i$ for each $i$.  

*What happens if you had continuous factors/latents?*  
The latents are always continuous. If the factors were continuous, they could be binned in the same way that we bin the latents. Indeed, many of the labels in these datasets essentially are binned continuous values.  

*How does SNC differ from discrete mutual information?*  
Both accuracy and MI are roughly used for the same purpose as performance metrics, but they are computed quite differently and are different theoretical quantities. SNC is also chance adjusted.

Reviewer comment (Evaluation):
Table 1 is not discussed that precisely and Table 4 in the Appendix seems more interesting. Why did you bold all MLP values, instead of bolding across rows to indicate the best model? This was confusing / unhelpful. Why do you use an MLP despite the issues you raised? How do you counteract overfitting? The comparison of accuracies at the end of Section 5.1 just does not feel appropriate. You cannot make judgements about linearity of sum of classifier scores vs classifier score of more latents, especially given you used a nonlinear classifier. It also does not bring much.

Author response:
*Table 4 is more interesting than Table 1.*  
The only difference is that Table 4 has results for a linear classifier as well. We can add this to the main paper.  

*Bold the best model in each row*  
Thanks, we will do this.  

*The comparison of accuracies at the end of Section 5.1 just does not feel appropriate.*  
The purpose is not to imply that the collective accuracy and the sum of individual are calibrated on the same scale, but rather to show how the informativeness of each individual neuron can be low, while that of the set is high. This is also the point of the XOR example. XOR itself may not be likely in practice, but the pattern of low-individual-high-collective is.

Reviewer comment (Presentation):
Table 2 cites a CG linear which does not exist.

Author response:
*Table 2 cites a CG linear which does not exist.*  
Thanks, we will fix this.

Reviewer comment (Evaluation):
Some examples are contrived and less likely to happen in reality. For example Section 3.2 proposes a XOR latent to indicate how the metric might fail. How likely is that to happen in practice?

Author response:
*The comparison of accuracies at the end of Section 5.1 just does not feel appropriate.*  
The purpose is not to imply that the collective accuracy and the sum of individual are calibrated on the same scale, but rather to show how the informativeness of each individual neuron can be low, while that of the set is high. This is also the point of the XOR example. XOR itself may not be likely in practice, but the pattern of low-individual-high-collective is.