Concept Reachability in Diffusion Models: Beyond Dataset Constraints

Thank you for taking the time to review our work. We are very happy to read your positive review! We have read through the comments and suggestions you made and addressed them below:

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• Does the underlying accuracy of the classifier not need to serve as the baseline for the reachability measure?

When evaluating on human-labelled images (5400) that were generated by prompting from two diffusion models trained on balanced data, the classifier obtains an accuracy of 96.63%, hence the results would have to consider the possible error of the classifier. We consider this accuracy to be sufficient to reflect the impact of our experiments on reachability. We will add a clarification about this to the paper.

• Paper focuses on steering, but there are other methods for reaching the concept than steering

This is fully correct: steering falls under the broader category of modifying the latent space and many variations exist on how this latent space modification is performed (which model features are being modified, how this modification is performed). Our focus here was on steering as it has achieved successful results in both diffusion models and autoregressive modelling, while still being relatively simple and efficient to implement (and hence promising also for practical use-cases).

• More complex data

This is addressed in the response to qawW (CelebA).

• Typos in Section 3.4

Thank you for pointing these out! We will correct this in the updated version of the paper.

• Does the degree to which the shapes overlap affect the reachability measure?

Originally, we encountered challenges when classifying the back shape in case these were heavily occluded. We modified the neighbourhood within which the front shape was sampled in order to ensure a reasonable portion was visible. The minimum percentage of each back shape visible in the balanced train set for the diffusion models is the following: circle 52.46%, triangle 48.97%, square 59.38% - this will be added to the Appendix in the dataset details. Overall, approximately half or more, of each shape is visible. We did not explore this further, however after this modification we didn’t notice any significant difference in reachability with the different shapes in the back.

• In Figure 4 — are the shape/color combinations at the top of figure only representative examples, or does the figure relate to these exact samples?

These are the colour-shape combinations for the accuracies displayed in the graph. For example, the first column shows a green triangle behind a red triangle, which changes no concepts with respect to the starting prompt y_s. Note that it does not refer to the exact relative position between the two shapes, only the combination (we will clarify this).

Thank you for reading through our paper! We really appreciate your positive feedback, and have addressed the questions/comments you made below:

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• How is the efficiency of steering in terms of application?

Our current work did not focus on analysing the computational efficiency of steering precisely, but overall we found that the optimisation of the steering vectors is dependent on the dataset and learning rate used for optimising, with the number of steps required to implement the optimisation varying. Analysing the efficiency of steering under different conditions in large-scale models would be an interesting extension. In Stable Diffusion, the optimisation of the steering vector required less than 20 steps.

• CLEVR and synthetic data

Although the Clevr dataset contains more complex images with greater detail and 3D structure, we believe that repeating the experiments with Clevr would yield similar results to those already observed within our synthetic framework. The underlying factor structure of both datasets is essentially the same, and as such, we anticipate that the observed outcomes would not vary significantly. In our work, we opted for a simpler dataset to ensure controllability, robustness, and realisable methods for evaluating accuracy, as well as more scalable model training at a lower computational cost. We further address experiments with more complex data in the response to Reviewer qawW (CelebA).

• Figure 7

Figure 7 shows the reachability to images containing either only circles or only blue shapes in the back, under conditions where these two concepts are tied. As the presence of images containing only blue in the back is increased (thus, reducing the bias), the reachability of images containing only blue in the back increases - this is shown in how the light-coloured X's produce a higher accuracy on the horizontal plane. However, this also leads to a general trend (with some variability) of increase in the accuracy of images containing only circles in the back - note that generally across all methods, on the vertical axis, the lightest coloured X's achieve the highest accuracy. We will clarify the description of this Figure in the paper.

Thank you for your thorough review and thoughtful comments! We address your questions below:

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• Evidence behind the claim that overly detailed prompts may not help with reachability

The starting prompt y_s used to implement the steering is as described in Section 3.5, however the accuracy displayed in Figure 6(a) for prompting is that of the prompt containing the full description of the target concepts. Hence, the graph shows that as the specification of concepts is decreased, the reachability of prompting the full description (which is overspecified with respect to the seen specification level) is severely affected due to the specification of the train set being limited. This will be clarified in the paper.

• Rationale behind the selected y_s for each experiment (e.g., how much does it deviate from y_e and why)?

In Experiment 1 we vary the deviation (number of concepts changed) between y_s and y_e to understand the behaviour between lower to higher changes in number of concepts. In the remaining sections we choose the case y_s = y_e, as it overall showed the most stable performance with respect to steering on the h-space (prompt space is mostly unaffected), and would be the natural choice when trying to reach a concept in a real data model. That is, if generating something on a model via prompting fails, in most cases one would to try to additionally steer on the prompt that describes the desired outcome.

• Quality of the classifier

This is addressed in the response to Reviewer saQX.

• Reference suggestion

Thank you for the reference! Data attribution methods are certainly related to our setup. We will discuss data attribution methods in diffusion models and the reference provided in the updated version of the paper.

• Figure 11

We will add a legend labelling the random seed to clarify the figure.

• Portion of visible back shape

This is addressed in the response to Reviewer saQX (Degree of overlap). Note that the illustrations in Figure 3 and Figure 4 are diagrams - actual samples of the train set are provided in Figure 10 in the Appendix.

• Difference between h-space and prompt space in Figure 6(b)

Figure 6(b) shows an example for one model of the behaviour of prompting and steering when trying to reach red in the back when the label c_1 is not specified in training, or reaching a square in the front when the label s_2 is not specified. We wanted to note that, particularly in the case of prompting and steering on the h-space, the generated output is close to randomly sampling the value of the unspecified label (in the case of c_1, the colour of the front shape in the target combination is green and so there are only two choices for the back colour). Steering on the prompt space, instead, produces higher accuracy on the target combination, leading to a higher value achieved on the top axis (red and square, respectively) than the one achieved by prompting, although remaining close to this value. We will update the explanation of this in the paper. The dimensionality, level of disentanglement, and the dependency of the h-space on the timestep $t$ could potentially impact the observed differences.

• CelebA

We agree that extensions to real-world data will be valuable. We are currently implementing our framework on the CelebA dataset to assess whether its structure aligns with the properties required for our analysis. As part of this, we are constructing approximately 15-20 dataset variations and training multiple models per dataset. Given the compute we have available we hope to be able to present the full results before the end of the author-reviewer discussion period and include them in the paper.

We currently have observed similar trends in a balanced dataset as in Experiment 1: reachability on the prompt space is more consistent than reachability on the h-space, which is affected by number of factors changed. Moreover, we have also observed a decrease in reachability as the level of specification of the train set is decreased. We plan to include this in the paper, as well as an analysis of scarcity and biases, with detailed examples of steered images.

We remark that while CelebA does provide attribute labels for selected factors, other latent factors such as background, lighting and pose are uncontrolled, and it is only the use of synthetic data that guarantees a fine-grained cause effect study of the impact of dataset modifications.

Thank you for reading through our work and providing your feedback! We have read through your comments and address your questions below:

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• Notation of steering vectors

We noticed that the notation $\mathbf{s}$ may also lead to confusion due to the labels $s_1$ and $s_2$ for the factors in our dataset. However, we can change the notation for the steering vectors to $\mathbf{v}$.

• Notation for $[f_{i_1}, f_{i_2}, \dotsc, f_{i_k}]_X$ and $[f_1^{(i)}, f_2^{(j)}, f_3^{(k)}]_X$

The notation $[f_1^{(i)}, f_2^{(j)}, f_3^{(k)}]_X$ refers specifically to the diagram. This diagram contains only three factors, so the notation refers to the combination of images that contain one specific value or concept for each factor in the dataset. The notation $[f_{i_1}, f_{i_2}, \dotsc, f_{i_j}]_X$ refers to the $n$-factor case, and only fixes the concept value for a subset of the factors, indexed as $i_1, i_2, \dotsc i_j$. We will modify the indexing in the diagram to avoid confusion.

• Difference between $y_e$ and $y_s$

$y_e$ describes the properties of the images we steer to (the end target). $y_s$ is the prompt from which we start, that may (if $y_s = y_e$) or may not describe the desired properties. $\mathbf{x}_0$ is the collection of images containing the target concepts that are used to steer the generation process (they may have been generated by sampling from a model using the prompt $y_e$ or obtained from a test set).

• Figure 4: why steer to such extreme combinations?

We wanted to explore the behaviour in extreme modifications under balanced conditions in order to understand the different reachability methods. Perhaps it would be expected that as the number of factors changed (with respect to $y_s$) increases, reachability via steering decreased. However, this is not the case in steering on the prompt space, which highlights structural differences in the spaces where steering is implemented. Throughout the remaining experiments presented in the main body, the steering is implemented in the case $y_s = y_e$, which we found to produce the most stable results for steering on the $h$-space. We will clarify this in the revised version of the paper.

• Examples on real datasets

This is addressed in the response to Reviewer qawW (CelebA).