Paper: The Hidden Language of Diffusion Models (ICLR-2024)

Reviewer comment (Novelty):
lack of related previous work on image decomposition, which can be seen as a way to interpret models, such as FineGAN (Sing et al, 2019), GIRAFFE (Niemeyer etal, 2021), SlotAttention (Locatello etal, 2020), DTI Sprites (Monnier etal, 2021), GENESIS-V2 (Engelcke etal, 2021) and follow-up works. There also exists earlier/concurrent work that conducts image decomposition using text-diffusion models, so its worth discussing pros and cons but may not need comparison if works are concurrent.

Author response:
We thank the reviewer for bringing this to our attention. We have revised the paper to include the proposed works in the related works section.

To reiterate the difference between the objective of these methods and our method, we enclose a short comparison to the methods indicated above:

As a general note, all methods described below only offer decomposition for a single image and, therefore, fall short of providing concept-level explanations, which is the main objective of concept-based explainability.

**FineGAN**: disentangles background, foreground, shape, and appearance. This method is not applicable to interpret SD since (1) it’s GAN-based and therefore limited to the training domain, (2) the disentangled information provides little to no interpretable information that is not observable just by looking at the images.

**GIRAFFE**: similar to FineGAN, GIRAFFE disentangles objects, appearance, shape, and background information, with the objective of obtaining controllable generation. In addition to the points mentioned above, this method incorporates 3D scene representations, which are not applicable in our case.

 **Slot Attention**: learns object-centric representations for complex scenes, and employs them for object discovery and set prediction. The task of object discovery has some similarities to ours but is also very different, see below. Additionally, the datasets used for training (CLEVR6, Multi-dSprites, Tetrominoes) differ significantly from the distribution of images generated by SD.

**DTI Sprites**: similar to Slot Attention, DTI Sprites decomposes a scene into objects including their shape, size, and color (appearance), which allows improved controllable generation.

Inspired by the review, we conducted an additional experiment to demonstrate the difference between our objective and existing methods for image decomposition. The closest relation we observed between the proposed methods and ours is through the task of unsupervised object discovery (including shapes, colors, sizes, etc.). Importantly, our task largely differs from that of object discovery, since an interpretation should often contain elements *beyond* objects that are physically present in the image. For example, Fig. 2 shows connections such as “snake” to “gecko” + “twisted”, a “gecko” is not present in the image, and “twisted” is *not an object but an adjective*. Similarly, “cashmere” is not present in an image of a “camel”, and “winding” is not present in images of “snails”. In contrast to these semantic, profound connections, interpretation using object discovery only provides representations that are based on concrete objects and parts that are visible in the image. Therefore, these methods fall short of discovering deeper connections (e.g., reliance on exemplars, non-trivial biases, etc.).

To empirically demonstrate this point, we added exemplary results using [1], which is the state-of-the-art object discovery method based on SD, and allows to discover concepts from sets of images, as well as single images. The results of this exploration of two exemplary basic concepts, “snail”, “snake” and an additional complex concept “impression of Japanese serenity”, are presented in Appendix G of the revision. This experiment directly demonstrates our intuition above. As can be seen, the method learns to embed the concept in a single token (as expected from its task definition for images of one class) and does not employ any other learned tokens. 

[1] “Unsupervised Compositional Concepts Discovery with Text-to-Image Generative Models”, Nan Liu, Yilun Du, Shuang Li, Joshua B. Tenenbaum, Antonio Torralba, ICCV’23.

Reviewer comment (Experiments):
most of concepts shown in the paper are mainly objects, thus its ability to learn abstract concepts is not clear. For example, how does it perform on abstract concepts such as object relationships. Though, I tend to think that the model is rather limited in understanding complex concepts other than objects.

Author response:
__Re. Forming abstract connections beyond objects:__
Kindly note that our examined concepts include an extensive set of both very complex and abstract concepts (e.g., “elegance on a plate”, “happiness”, “affection”, “impression of Japanese serenity”, etc.). A full list of these concepts can be found in Appendix B (see Emotions, Actions, Complex concepts). Following the review, we have added word cloud visualizations of the learned decompositions for abstract and complex concepts (see Appendix F). The provided examples show that our method can meaningfully decompose abstract concepts and detect relations to other abstract concepts (e.g., linking “happiness” to “dream”, “emotion”, “soul”, “laughter”, “childhood”; or linking “sadness” to “grief”, “depression”, “worried”, “alone” etc.). These results indicate that Conceptor is capable of revealing connections between abstract concepts.  Examples of single-image decompositions for such abstract concepts can be found in Fig. 2 (“fear”, “impression of Japanese serenity”) and in Fig. 8 (“happiness”).

Reviewer comment (Experiments):
I'd love to see if u can optimize on electrician image and then try removing the brush concept, does it become a painter?

Author response:
__Re. The difference with word arithmetics:__
We thank the reviewer for raising this interesting point for discussion.
We wish to emphasize that our method does not perform word embedding decomposition, and in fact, significantly differs from it. In the following, we list the fundamental differences between the two:

1. **Our learned pseudo-token $w^{\ast}$ does not decompose the text embedding of the concept $w^c$, i.e.: $w^{\ast}= \sum_i w_i \alpha_i \neq w^c$:** 

      Kindly note that our method does not decompose the word embedding of the concept token(s) $w^c$. Instead, we propose to decompose the internal representation of the concept by the model. This is done by aggregating the set of features used by the model in the image generation process (see Sec. 3). The text embeddings serve merely as an intermediate language linking us to the uninterpretable inner representations of SD. To empirically prove this point, we calculated the cosine similarity between our learned pseudo-token $w^*$ and the concept token $w^c$ averaged across our entire dataset, and obtained a low similarity score of *0.61* (for reference, “cat” and “car” get a higher similarity score of 0.66). This result, combined with our Token Diversity metric (Tab. 1), our single image decomposition results (Fig. 2) and our CLIP top words ablation (Tab. 3) demonstrate that our method is able to learn connections based on visual semantics, that transcend word arithmetics, as mentioned in the paper (see Sec. 1, 6).

1. **Our single-image decomposition results do not suggest word embedding arithmetics:** 

     Note that the annotations in our single-image decomposition figures (Figs. 2,8) do not indicate word embedding arithmetics. For example: $word\textunderscore embed(painter) \neq word\textunderscore embed(electrician) + word\textunderscore embed(brush) + word\textunderscore embed(Monet)$ (the cosine similarity between “painter” and “electrician + brush + Monet” is merely 0.496). This indicates that the decomposition is not based on textual semantics but on an inner-representation decomposition.

      Additionally, note that a diffusion model generates *diverse* images for the same concept given different random seeds, and not all learned features are manifested in all of the concept images (e.g., the images of the nurses in Fig. 12 can be in black and white or in color, with scrubs or a white suit, with and without a stethoscope, etc). Our single-image decomposition scheme extracts the specific set of elements from the decomposition learned by Conceptor that are used to generate a single specific image. This is evident when different features appear for single-image decompositions of different images of the same concept, e.g., the “camel” and “snake” examples in Fig. 2 vs. Fig. 8. Thus, the arithmetic annotations in Figs. 2, 8 indicate the addition of the features of the decomposition elements to the output image.

    This ties directly to the question by the reviewer:
     > “I'd love to see if u can optimize on electrician image and then try removing the brush concept, does it become a painter?”. 

     The answer is that we are not doing word arithmetics and this manipulation is not possible, since applying Conceptor to the concept “electrician” results in a decomposition that does not include the element “brush”. This is logical: SD does not employ the feature “brush” in the generation of images of electricians, unlike the role “brush” plays for the “painter” concept.

Reviewer comment (Evaluation):
How long does this optimization process take for each image?

Author response:
__Re. Optimization length per image:__ Please note that Conceptor *does not operate on a single image but on an entire concept* (see Sec. 3). The optimization process for the entire concept takes around 6.5 minutes on a single A100 GPU with 40GB (given more memory, one could significantly increase the batch size and speed up the process).

After this optimization, manipulation of the obtained coefficients for a given image is dominated by the time it takes to generate the image (around 4 seconds, again depending on the hardware used).

We are happy to address any other questions.

Reviewer comment (Experiments):
Can the authors elaborate if the concept decomposition is an artifact of the particular CLIP text-encoder in Stable-Diffusion? Will one get similar concept-decomposition patterns if a different text-encoder is used (e.g., T5 like in DeepFloyd)? I would imagine this to be a positive answer, but might expect different patterns, so I believe it’s important to use Conceptor to understand this phenomenon.

Author response:
__Re. Analysis of other text encoders:__ Thank you for this great suggestion, which helps further substantiate Conceptor’s robustness. We note that DeepFloyd text embedding optimization suffers from significant instabilities in terms of the loss and the learning rate [see this thread from the popular `diffusers` library](https://github.com/huggingface/diffusers/issues/3356#issuecomment-1578072917). The obtained gradients for the model are very low, forcing the use of very high learning rates, and the optimization is long and expensive (around 1 hour per concept requiring 24GB, *with an unconstrained token*, see [the implementation](https://github.com/oss-roettger/T5-Textual-Inversion)). To empirically substantiate the instability claim, [here are the results we got for optimizing the concept “happiness” from Fig. 4 without constraints on the word embedding](https://i.imgur.com/iTaGPzd.jpeg). This experiment is done with the same setting as Conceptor, but using unconstrained optimization of the embedding vector, which is a much more basic task. These instabilities could stem from a number of different reasons. For example, this could stem from, [DeepFloyd predicting the variance in addition to the noise](https://github.com/huggingface/diffusers/issues/3307) which is not accounted for in the current optimization, DeepFloyd operating on the pixel space and not the latent space, or due to the fact that the model is cascaded and optimized on the lowest resolution. We believe that in general, applying text embedding optimization to DeepFloyd is an interesting and non-trivial task for future research, as the behavior of the model significantly differs from LDMs. However, this deviates from the scope of our work.

As an alternative, in order to address the reviewer’s question, we have applied Conceptor over another version of Stable Diffusion, SD 1.4, which uses a *different text encoder*. While SD 2.1 uses OpenCLIP-ViT/H, SD 1.4 uses CLIP ViT-L/14. Kindly note that these text encoders were trained on entirely different data (OpenCLIP is an open-source model trained on open-source data). The results are enclosed in Appendix L. As can be seen, Conceptor provides faithful decompositions, even for a model with a different text encoder, further substantiating our method’s robustness.

Reviewer comment (Experiments):
I am a little curious about how much the diffusion objective plays a role in concept-decomposition. For example, given the objective of reconstruction, I will expect the faithfulness metric of Conceptor to be better than other methods (e.g., PEZ). However, if you use the same idea with CLIP loss (replacing the reconstruction loss in Eq. (6) with L_clip), will you get similar decomposition? And will those decompositions transfer to diffusion models? In fact, if you use CLIP's representation for a particular token as a ground-truth with optimizing for Eq. (3), you should get a reasonable reconstruction still, which can be a cheap baseline. Did the authors run this ablation?

Author response:
__Re. Using CLIP for alternative objective functions:__ We thank the reviewer for these very interesting questions for analysis. In the following, we address each of the suggestions separately:

1. **Replacing the MSE loss with a CLIP (image) similarity loss:** The implementation of this suggestion is not trivial since a CLIP loss is not applicable to the noise vectors (unlike the MSE loss). In order to facilitate the use of CLIP loss instead of MSE loss, we employed the CLIP image-to-image cosine similarity loss on the *predicted clean image* ($pred\textunderscore x_0$) at each optimization step. The results of the decompositions obtained by this experiment (on the same concepts and random seeds as Fig. 4 of the main paper), and additional technical and mathematical details on the CLIP loss used are presented in Appendix K of the revision.

   As can be expected, the reconstructions obtained by this method fall short of the reconstructions of our full method (see Fig. 18 in the Appendix). However, importantly, the decomposition is still meaningful when replacing the MSE loss with the CLIP loss. For example, the concept “painter” still significantly relies on *the same famous artists* (e.g., “Monet”, “Picasso”), the concept “bird” still decomposes mainly into an interpolation of bird breeds, etc. We find it very reassuring that Conceptor is not sensitive to the loss function used.

2. **Using a specific word embedding as the ground truth for reconstruction:** Thank you for this great suggestion, which serves as a sanity check and helps us verify that our optimization indeed converges to a minimum over the objective function. Note that in the proposed setting, we are given a ground truth word embedding *from the vocabulary* $w\in \mathcal{V}$, and we are trying to optimize a *sparse linear combination over $\mathcal{V}$* to reconstruct $w$. The easiest solution that satisfies both perfect reconstruction and perfect sparsity is simply taking a “one-hot”-like vector of coefficients where $w$ is assigned the highest coefficient. This is exactly what happens when applying Conceptor with such a word embedding as ground truth. For example, for the concept “nurse”, the token “nurse” is assigned a coefficient of $1.3221e+01$, where the second highest coefficient is $1.1319e-03$ (virtually this is a one-hot vector of “nurse”). These results imply that when a simple optimal solution can be found, our MLP converges to that.

    Importantly, unlike word embedding decomposition (which can simply be reconstructed by the embedding itself), a diffusion model gradually adds visual features from coarse to fine. This has already been established by personalization works such as NeTI [1], which shows specifically that optimizing a reconstruction token in *different timesteps* yields *entirely different results*. Therefore, each optimization step draws a different timestep where different features are added by the model, thus Conceptor’s diversity stems from the diversity of the features added in the different denoising steps, and not from the word embeddings. This point further exemplifies the strong connection between our method and the model itself, i.e.,  *the decompositions stem from the diffusion model, rather than the text encoder*. Additionally, it provides a sanity check for our decomposition, which converges appropriately. For additional empirical results, please refer to Appendix C.1 where we show that the learned pseudo-token $w^*$ can denoise *test concept images from any timestep*, and the differences between the reconstruction loss of different timesteps.

[1] A Neural Space-Time Representation for Text-to-Image Personalization, Alaluf et al., SIGGRAPH Asia’23.

Reviewer comment (Experiments):
How can you extend your framework to more complex concept-decomposition? The current framework generates images corresponding to single concepts, but images are usually consisting of multiple concepts. In this scenario, how can one use Conceptor to understand sub-concepts? I think this is one experiment the paper is lacking.

Author response:
__Re. Complex concepts containing multiple objects:__ Please note that we incorporated a subset of very complex prompts in our tests, extracted entirely from the website [Best 30 Stable Diffusion Prompts for Great Images](https://mspoweruser.com/best-stable-diffusion-prompts/). These prompts contain integrations of multiple objects/ concepts in the same scene. For example, the prompt “impression of Japanese serenity” includes multiple concepts such as pagoda, Sakura, bonsai, etc. (see Fig. 2 for example), and the prompt “marketplace colors of Marrakesh” contains an extensive variety of elements such as “spices”, “tents”, “market”, “desert”, etc. (see Tab. 8 in Appendix K, Fig. 4 of the main paper). As can be observed in our qualitative and quantitative experiments, Conceptor faithfully captures all the concepts in the prompts, even in such complex cases.

We are happy to address any other questions.

Reviewer comment (Experiments):
How can this framework be used to mitigate some of the issues in diffusion models (e.g., bias)?

Author response:
__Re. Downstream applications (with emphasis on bias mitigation):__ First, as the reviewer kindly pointed out, Conceptor enables human-understandable interpretations for diffusion models.
Other than the useful downstream tasks (described below), we believe that interpreting diffusion models is an imperative, timely, and underexplored task on its own since these models are now being employed widely. The insights obtained by Conceptor can help both end-users and researchers to use these models responsibly while being aware of their shortcomings. Thus, we believe that interpreting generative models is, by itself, a worthy cause with much interest to our community.

With that, Conceptor also enables useful downstream tasks such as concept debiasing via semantic concept editing, that we are happy to elaborate on. As mentioned in the qualitative results section (Sec. 4.1), Conceptor allows one to control each element in the decomposition by manipulating its corresponding coefficient. The magnitude of the coefficient controls the extent to which the element is manifested in the image. Fig. 3 demonstrates such manipulations. For example, one can decouple the two meanings of a dual-meaning concept (see first two rows of Fig. 3) or remove an undesired property from the concept (e.g., “nerdy” for “professor”). 

As mentioned in Sec. 4.3, this ability is also useful for bias mitigation. Once a bias is identified by Conceptor, the user can choose to “switch off” the biased elements by simply lowering their corresponding coefficients until an equal representation is achieved. Such examples can be found in Appendix H. For better readability, we have edited the Appendix to include the biased terms of Fig. 13, see Tab. 7. As can be seen, Conceptor enables debiasing while also maintaining the other image features and the original scene, i.e., Conceptor’s decomposition can surgically remove the biased property without harming the other features.

Additionally, following the reviews, we have added an experiment showcasing another downstream task. In Appendix J, we explore Conceptor’s ability to provide insight into the representation of Out of Domain (OOD) concepts, defined by a set of image samples. As we demonstrate, our method is able to extract meaningful and profound decompositions even for completely OOD concepts (e.g., the sloth plush in Fig. 16 is linked to the concepts “plush”, “fluffy”, and “knitted”, the monster toy in Fig. 17 is linked to “woolly” due to its wool-like fur, to the character “Domo” due to the shape of its body, and to a “crab” and a “shrimp” due to its unique limbs). Kindly refer to our answer to Reviewer HDAw for more details.

Reviewer comment (Novelty):
The reviewer tested the textual similarity of several word triplets presented in Figure 2 and found that for a triplet A, B, C, the similarity of A vs. (B+C) is always higher than A vs. B or A vs. C, indicating that similar semantic correlations already exist in the CLIP text embedding space.

Author response:
We note that the inequality suggested by the reviewer, i.e., `sim(A+B, C) > max{sim(A, C), sim(B, C)}` often holds for any triplet of three words. This may be a result of increased similarity with the “blurred” word obtained by averaging two concepts. To validate this empirically, we selected 10,000 random triplets of words (A, B, C) out of the SD text encoder dictionary. In 85% of the cases, the inequality holds. This indicates that the proposed inequality does not necessarily suggest a better semantic similarity, nor that A+B is a decomposition of C.

Reviewer comment (Theory):
The reviewer notes that since CLIP text embeddings are aligned with image features, such similarities in image features will propagate into the text embedding space, making the use of T2I diffusion model–based image reconstruction unnecessary.

Author response:
We would like to point out that the CLIP text-text similarity scores (as calculated by the reviewer using CLIP ViT-B/32) appear to give abnormally high similarities for various words that are unrelated semantically. We drew 10,000 pairs of random words from SD’s vocabulary to demonstrate this point and found that the average similarity between these random words was 0.815. These high scores are somewhat surprising, especially given that a large portion of the tokens in this dictionary are not actual words but rather punctuation marks, emojis, etc. Additionally, the scores are often unintuitive. For example, the CLIP text-text similarity between "cat" and "house" is 0.79, which is higher than the CLIP text-text similarity between “cat” and “poodle” (0.74) (to reiterate the previous point, “cat” to ”house”+“poodle” is 0.81).

Reviewer comment (Novelty):
The reviewer concludes that if such triplets can be mined directly from the CLIP text embedding space, the contribution of the paper becomes quite small, leading them to suggest rejection in the original comment.

Author response:
Finally, please note that the latest version of SD (SD 2.1) employs a *different CLIP text encoder*, i.e., OpenCLIP ViT-H (see Appendix A). When examining the CLIP text-text similarities in the OpenCLIP ViT-H encoder, the phenomena described in the review do not reproduce. First, we find that the CLIP text-text similarities between the elements are *far lower* than those indicated by the reviewer (with CLIP ViT-B/32), e.g., the similarity between "reflections of earth" and "civilization" is merely 0.2817, the similarity between "snake" and "twisted" is 0.365, etc. Second, the logic indicated for CLIP ViT-B/32 is not reproduced. In 2/6 triplets mentioned by the reviewer, the sum of the two elements does not increase the similarity: 

   * For "snake": `sim("snake", "twisted") = 0.365`, `sim("snake", "gecko") = 0.639`, `sim("snake", "twisted" + "gecko") = 0.627 < 0.639`.
   * For "fear": `sim("fear", "wolf") = 0.326`, `sim("fear", "scream") = 0.638`, `sim("fear", "wolf" + "scream") = 0.60 < 0.638`.

   For this text encoder, the inequality holds for 65% of the random triplets as described in item 2.

Reviewer comment (Experiments):
The number of concepts (prototype) are limited, which can not prove whether the proposed method is effective on large-scale concepts.

Author response:
__Re. Size of test dataset:__ First, kindly note that the size of our dataset (188 concepts from diverse categories) is consistent with, and even significantly bigger than that of related works that tackle concept-based interpretability and concept analysis. For example, one of the most prominent works on concept-based explainability, ACE [1], presents a quantitative evaluation of 100 random ImageNet classes. Similarly, seminal works in concept personalization such as DreamBooth [2] performed evaluation on 30 diverse concepts, etc.
 
Importantly, note that we formed a diverse dataset including professions (to indicate biases and human-centered concepts), abstract concepts (including emotions and actions), very complex concepts (requiring hierarchical reasoning), and on top of those, we added 100 random concepts from the well-known general concept bank, ConceptNet, to ensure that our produced results are robust across a wide variety of rich and unrelated concepts. Note that ConceptNet contains a wide mixture of concepts from all categories, and is a very widely used knowledge graph. Across all these different types of concepts, Conceptor has *consistently* demonstrated an ability to produce faithful, meaningful, and robust decompositions.

[1] Towards Automatic Concept-based Explanations, Ghorbani et al. NeurIPS 2019

[2] DreamBooth: Fine Tuning Text-to-Image Diffusion Models for Subject-Driven Generation, Ruiz et al., CVPR 2023.

Reviewer comment (Novelty):
The concept decomposing can be viewed as an inner interpolation between the concepts.

Author response:
Thank you for considering our rebuttal! We share your enthusiasm for the OOD results, and deeply value the insightful suggestion that helped enhance our understanding of Conceptor.

Upon reviewing the clarification, we are still not sure we fully understand the proposed interpolation method and how it selects the condition embeddings. We derived two possible interpretations for the question which we address below, and we welcome your corrections if our understanding is not accurate.

1. Embedding interpolation as simple summation of word embeddings (i.e., summation of embeddings in the token space):

   Allow us to emphasize the following:

    * Recall that our method operates by learning a mapping function between each word embedding and a corresponding coefficient. In other words, our method can be viewed as a nonlinear mapping of the form: $w^* = \sum_i f(w_i) w_i$, where $f$ is a nonlinear function. 

       Our extensive ablations (refer to Section 4.2.1, Table 3 under "without MLP") demonstrate the critical role of this novel approach in the success of our method. Without it (i.e., learning separate coefficients) the decompositions are not faithful. This ablation highlights the non-trivial nature of assigning appropriate coefficients to word embeddings to reconstruct the learned representation, affirming that a simple summation cannot substitute for Conceptor.

   * To demonstrate the important role of our learned coefficients empirically, we enclose the results of simple word embedding summation (a+b) for examples of concepts from Fig 2. The connection “reflections of earth = sphere + civilization” is only made possible by the learned coefficients. When considering only the word embeddings, [these are the resulting images](https://i.imgur.com/6ai71Ww.jpg) (which do not resemble planet Earth whatsoever). Similarly, “snake = twisted + gecko” would yield [these images without the appropriate coefficients](https://i.imgur.com/1UYOham.jpg), for the combination “snail = winding + ladybug” [these are the resulting images](https://i.imgur.com/EB4S5ya.jpg), etc. These examples demonstrate that even given the ground truth set of decomposition concepts, a simple summation cannot match our method.

   We believe that the combination of the aforementioned points effectively demonstrates the powerful and non-trivial ability of Conceptor to establish semantic links between concepts through our novel method.

2. Encoding interpolation after applying the text encoder (i.e., summation of embeddings in the conditioning space):

   In exploring this scenario, we observed that such interpolations result in non-meaningful images (perhaps due to OOD encodings). [Here’s an example for “ladybug” + “winding”](https://i.imgur.com/QrwUgBN.jpg), and [“twisted” + “gecko”](https://i.imgur.com/0ddljfw.jpg).

We are happy to address any remaining questions, and appreciate the continued in-depth discussion.

Reviewer comment (Evaluation):
Can you provided some analysis the between the proposed method and interpolation method?

Author response:
Thank you for considering our rebuttal! We share your enthusiasm for the OOD results, and deeply value the insightful suggestion that helped enhance our understanding of Conceptor.

Upon reviewing the clarification, we are still not sure we fully understand the proposed interpolation method and how it selects the condition embeddings. We derived two possible interpretations for the question which we address below, and we welcome your corrections if our understanding is not accurate.

1. Embedding interpolation as simple summation of word embeddings (i.e., summation of embeddings in the token space):

   Allow us to emphasize the following:

    * Recall that our method operates by learning a mapping function between each word embedding and a corresponding coefficient. In other words, our method can be viewed as a nonlinear mapping of the form: $w^* = \sum_i f(w_i) w_i$, where $f$ is a nonlinear function. 

       Our extensive ablations (refer to Section 4.2.1, Table 3 under "without MLP") demonstrate the critical role of this novel approach in the success of our method. Without it (i.e., learning separate coefficients) the decompositions are not faithful. This ablation highlights the non-trivial nature of assigning appropriate coefficients to word embeddings to reconstruct the learned representation, affirming that a simple summation cannot substitute for Conceptor.

   * To demonstrate the important role of our learned coefficients empirically, we enclose the results of simple word embedding summation (a+b) for examples of concepts from Fig 2. The connection “reflections of earth = sphere + civilization” is only made possible by the learned coefficients. When considering only the word embeddings, [these are the resulting images](https://i.imgur.com/6ai71Ww.jpg) (which do not resemble planet Earth whatsoever). Similarly, “snake = twisted + gecko” would yield [these images without the appropriate coefficients](https://i.imgur.com/1UYOham.jpg), for the combination “snail = winding + ladybug” [these are the resulting images](https://i.imgur.com/EB4S5ya.jpg), etc. These examples demonstrate that even given the ground truth set of decomposition concepts, a simple summation cannot match our method.

   We believe that the combination of the aforementioned points effectively demonstrates the powerful and non-trivial ability of Conceptor to establish semantic links between concepts through our novel method.

2. Encoding interpolation after applying the text encoder (i.e., summation of embeddings in the conditioning space):

   In exploring this scenario, we observed that such interpolations result in non-meaningful images (perhaps due to OOD encodings). [Here’s an example for “ladybug” + “winding”](https://i.imgur.com/QrwUgBN.jpg), and [“twisted” + “gecko”](https://i.imgur.com/0ddljfw.jpg).

We are happy to address any remaining questions, and appreciate the continued in-depth discussion.