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+# CYCLIP: Cyclic Contrastive Language-Image Pretraining
+
+Shashank Goel∗ UCLA shashankgoel@ucla.edu
+
+Hritik Bansal∗ UCLA hbansal@ucla.edu
+
+Sumit Bhatia MDSR Lab, Adobe Systems sumit.bhatia@adobe.com
+
+Ryan A. Rossi Adobe Research ryrossi@adobe.com
+
+Vishwa Vinay Adobe Research vinay@adobe.com
+
+Aditya Grover UCLA adityag@cs.ucla.edu
+
+# Abstract
+
+Recent advances in contrastive representation learning over paired image-text data have led to models such as CLIP [44] that achieve state-of-the-art performance for zero-shot classification and distributional robustness. Such models typically require joint reasoning in the image and text representation spaces for downstream inference tasks. Contrary to prior beliefs, we demonstrate that the image and text representations learned via a standard contrastive objective are not interchangeable and can lead to inconsistent downstream predictions. To mitigate this issue, we formalize consistency and propose CYCLIP, a framework for contrastive representation learning that explicitly optimizes for the learned representations to be geometrically consistent in the image and text space. In particular, we show that consistent representations can be learned by explicitly symmetrizing (a) the similarity between the two mismatched image-text pairs (cross-modal consistency); and (b) the similarity between the image-image pair and the text-text pair (in-modal consistency). Empirically, we show that the improved consistency in CYCLIP translates to significant gains over CLIP, with gains ranging from $1 \dot { 0 } \% - 2 4 \%$ for zero-shot classification accuracy on standard benchmarks (CIFAR-10, CIFAR-100, ImageNet1K) and $1 0 \% - 2 7 \%$ for robustness to various natural distribution shifts. The code is available at https://github.com/goel-shashank/CyCLIP.
+
+# 1 Introduction
+
+The ability to learn general-purpose representations from diverse data modalities is a long-standing goal of artificial intelligence (AI) [4, 32]. In this regard, recent instantiations such as CLIP [44], ALIGN [29], and BASIC [41] have scaled up vision-language contrastive pretraining to jointly learn image and text embeddings, by exploiting an enormous amount of paired image-text data on the web. Post pretraining, these embeddings exhibit impressive zero-shot classification performance [13] and robustness to natural distribution shifts [48, 57, 24, 26]. Recently, these embeddings have been extended to text-guided generation of natural images [47, 12, 38, 46] and transferred to modalities such as 3-D shapes [50] by emphasizing the interchangeability of the image and text embeddings.
+
+In the context of vision-language pretraining, the standard contrastive learning objective aims to maximize the similarity between matched image-text pairs (“positives") against all the mismatched image-text pairs (“negatives") [45, 7, 40, 22]. While such an objective aligns the true image-text pairs, it poses no constraints on the overall geometry of all data pairs, including the mismatched pairs and pairs within the same modality. In Figure 1 (a), we illustrate this effect where matched image-text pairs, $( I _ { \mathrm { d o g } } , T _ { \mathrm { d o g } } )$ and $( I _ { \mathrm { c a t } } , T _ { \mathrm { c a t } } )$ , get close to each other but the overall geometry of pairwise distances can be highly irregular (see e.g., $( I _ { \mathrm { d o g } } , T _ { \mathrm { c a t } } )$ and $( I _ { \mathrm { c a t } } , T _ { \mathrm { d o g } } ) )$ . If we use such representations for downstream inference, such irregularities can translate into inconsistent reasoning in the image and text spaces. For example, CLIP designs proxy captions for class labels and uses the most similar class caption to perform zero-shot classification for images; using the default captions in Figure 1 (a), this would imply that a test image $I _ { \mathrm { t e s t } }$ gets classified as a dog in the image space even when a simple nearest neighbor classifier in the text space would correctly infer the label to be a cat.
+
+![](images/113dedcfa7f5e22fc188dc3390117ab472deeaa1e0b6883316da098994bce5ea.jpg)  
+Figure 1: An illustration of the planar geometry of the learned representations of image-text pairs by (a) CLIP and (b) CYCLIP. The edges indicate the distance between the representations i.e., $\dot { d ( e _ { 1 } , e _ { 2 } ) } = 1 - \langle e _ { 1 } , e _ { 2 } \rangle$ , where $\langle \cdot , \cdot \rangle$ is the inner product. CYCLIP is cyclic consistent between image-text pairs as the in-modal distances, $d ( T _ { \mathrm { c a t } } , T _ { \mathrm { d o g } } ) \sim d ( I _ { \mathrm { c a t } } , I _ { \mathrm { d o g } } )$ , and the cross-modal distances, $d ( \bar { T _ { \mathrm { c a t } } } , \bar { I _ { \mathrm { d o g } } } ) ~ \sim ~ d ( \bar { I _ { \mathrm { c a t } } } , \bar { T _ { \mathrm { d o g } } } )$ , are similar to each other unlike CLIP. Due to explicit consistency constraints, the test image of a cat is classified as a cat in the image as well as the text space.
+
+To mitigate these challenges, we propose Cyclic Contrastive Language-Image Pretraining (CYCLIP), a framework that imposes additional geometric structure on the learned representations. Specifically, given two image-text pairs, we augment the contrastive learning objective with two symmetrization terms. The first term provides for in-modal consistency by encouraging the distance between the two image embeddings to be close to the distance between the corresponding text embeddings. The second term for the cross-modal consistency that encourages the distance between the image and text embedding from the first and second pairs respectively to be close to the distance between the text and image embeddings from the first and second pairs respectively. As shown in Figure 1 (b), if representations of any two image-text pairs, $( I _ { \mathrm { d o g } } , T _ { \mathrm { d o g } } )$ and $( \dot { I } _ { \mathrm { c a t } } , T _ { \mathrm { c a t } } )$ exactly satisfy both forms of cyclic consistency, then we can guarantee that any test image $I _ { \mathrm { t e s t } }$ respects the ordering of distances in both image and text spaces (i.e., if $d ( I _ { \mathrm { t e s t } } , I _ { \mathrm { d o g } } ) > d ( I _ { \mathrm { t e s t } } , I _ { \mathrm { c a t } } )$ , then $d ( I _ { \mathrm { t e s t } } , T _ { \mathrm { d o g } } ) > d ( I _ { t e s t } , T _ { \mathrm { c a t } } ) )$ .
+
+Empirically, we demonstrate that the improved consistency in CYCLIP translates to improvements over CLIP. In all cases, we pre-train our models on the Conceptual Captions 3M dataset[52]. On zero-shot classification, we observe that CYCLIP improves over CLIP by $1 0 . 2 \%$ on ImageNet1K, $1 0 . 6 \%$ on CIFAR-10 and $2 3 . 9 \%$ on CIFAR-100 respectively. Further, CYCLIP outperforms CLIP with an average relative gain of $+ 1 7 \%$ on ImageNet natural distribution shift benchmarks. We further analyze the improved performance of CYCLIP and find that the additional geometric structure in the representation space better captures the coarse and fine-grained concept hierarchies of datasets.
+
+Our contributions are as follows:
+
+1. We analyze contrastive learning for representation learning jointly over image and text modalities. We identify a critical shortcoming in the geometry of the learned representation space that can lead to inconsistent predictions in image and text domains.
+
+2. We propose CYCLIP, a simple and effective framework for contrastive representation learning with two additional cycle consistency constraints for mitigating the above issue. 3. We demonstrate that CYCLIP achieves significant empirical improvements over CLIP on zero-shot classification and robustness benchmarks. We further explain these improvements by analyzing the impact of consistency on the hierarchical structure of datasets.
+
+# 2 Cycle Consistent Representation Learning
+
+# 2.1 Preliminaries
+
+We are interested in using text supervision to learn general-purpose visual representations that can be generalized to downstream predictive tasks. To this end, there have been several recent advances in language-image pretraining concerning model architectures, training objectives, and sources of supervision. Our work is most closely related to Contrastive Language-Image Pretraining (CLIP) [44] which combines many such advances in a highly scalable and generalizable learning framework.
+
+CLIP is trained on millions of images with their captions scraped from the web. Formally, we consider a dataset $S \subset \mathcal { T } \times \mathcal { T }$ consisting of pairs $( I _ { j } , T _ { j } )$ where $I _ { j }$ is a raw image and $T _ { j }$ is a text caption. We use $\mathcal { T }$ and $\tau$ to denote the domain of images and text, respectively. The CLIP architecture consists of 3 components: (i) an image encoder network, $f _ { I } : \mathcal { T } \mapsto \mathbb { R } ^ { d }$ , to encode the raw image into an embedding vector of dimension $d$ , (ii) a text encoder network, $f _ { T } : T \mapsto \mathbb { R } ^ { d }$ , to encode the raw text into an embedding vector of dimension $d$ , (iii) a contrastive objective that pulls the embeddings of paired image-caption pairs together while pushing apart embeddings of unmatched pairs.
+
+Formally, during training, consider a batch of $N$ image-captions pairs, $\{ I _ { j } , T _ { j } \} _ { j = 1 } ^ { N }$ , where $I _ { j }$ and $T _ { j }$ represent the raw image and text pair, respectively. The image embedding $\bar { I } _ { j } ^ { e } \in \mathbb { R } ^ { d }$ and text embedding $T _ { j } ^ { e } \in \mathbb { R } ^ { d }$ are obtained by passing $I _ { j }$ and $T _ { j }$ through the image encoder $f _ { I }$ and text encoder $f _ { T }$ , respectively; i.e. $I _ { j } ^ { e } = f _ { I } ( I _ { j } )$ and $T _ { j } ^ { e } = f _ { T } ( T _ { j } )$ . Further, we assume they are normalized to have unit $\ell _ { 2 }$ -norm. The contrastive objective in CLIP aims to align the image and text representations by minimizing the loss function ${ \mathcal { L } } _ { \mathrm { C L I P } }$ shown below:
+
+$$
+\mathcal { L } _ { \mathrm { C L I P } } = - \frac { 1 } { 2 N } \sum _ { j = 1 } ^ { N } \log \left[ \frac { \exp \left( \langle I _ { j } ^ { e } , T _ { j } ^ { e } \rangle / \tau \right) } { \displaystyle \sum _ { k = 1 } ^ { N } \exp \left( \langle I _ { j } ^ { e } , T _ { k } ^ { e } \rangle / \tau \right) } \right] - \frac { 1 } { 2 N } \sum _ { k = 1 } ^ { N } \log \left[ \frac { \exp \left( \langle I _ { k } ^ { e } , T _ { k } ^ { e } \rangle / \tau \right) } { \displaystyle \sum _ { j = 1 } ^ { N } \exp \left( \langle I _ { j } ^ { e } , T _ { k } ^ { e } \rangle / \tau \right) } \right]
+$$
+
+where $\langle \cdot , \cdot \rangle$ represents the inner product, and $\tau$ is a trainable temperature parameter. CLIP and its variants can be used to perform zero-shot image classification, i.e., classifying test images into categories not seen at training time. We first transform each category into a suitable caption (e.g., the airplane category in CIFAR-10 can be expressed as ‘a photo of an airplane’). Then, the similarity of the test image to each caption is computed (e.g., cosine distance), and the model predicts the category for which the image-caption similarity is the highest.
+
+# 2.2 Inconsistent Representation Learning in CLIP
+
+As illustrated in Figure 1 (a), the standard contrastive objective in CLIP can learn image-text representations such that the predicted labels for the test image are different in the image and text spaces. Here, we reason about such inconsistencies more formally in the context of downstream classification. As discussed above, we can predict a label in the text embedding space (zero-shot setting) by selecting the label that is closest to the test image $( P _ { T } )$ . Additionally, for classification in the image embedding space, if we had access to a labeled training set, then one natural way to infer the predicted label $( { \dot { P } } _ { I } ^ { k } )$ of a test image $I _ { t e s t }$ is by taking a majority vote from the true labels associated with the $\mathbf { k }$ -nearest training images. Formally, we define a consistency score that measures the synchrony between the predicted labels in the image and text spaces as:
+
+![](images/a1fca37a434834539ae005b54f2227219c3c73d2e5bd566f8fe8abde7bd51ec3.jpg)  
+Figure 2: Illustrative overview for CYCLIP $N = 2 \AA$ ). It consists of 3 major components: (a) cross-modal contrastive alignment, (b) cross-modal consistency, and (c) in-modal consistency. Only (a) is present in CLIP, whereas our proposed regularizers in (b) and (c) mitigate inconsistency.
+
+$$
+{ \mathrm { C o n s i s t e n c y ~ S c o r e } } _ { k } = { \frac { 1 } { N } } \sum _ { j = 1 } ^ { N } \mathbb { 1 } \left[ P _ { I } ^ { k } ( I _ { j } ) = P _ { T } ( I _ { j } ) \right]
+$$
+
+where $_ \mathrm { N }$ is the number of test images. In our experiments (discussed in detail in $\ S 3$ ), we found the CLIP’s consistency score $k = 1$ ) to be $44 \%$ , $16 \%$ , and $16 \%$ on the standard benchmarks CIFAR-10, CIFAR-100, and ImageNet1K, respectively, showing a very high degree of disagreement in the image and text spaces. In the following section, we describe our approach to alleviate the inconsistent inference problem and quantitatively show that our solution improves the consistency score in $\ S 4 . 1$ .
+
+# 2.3 Cycle Consistent Representation Learning via CYCLIP
+
+We showed that the visual representations learned by CLIP could be inconsistent when used for inference in the image and text spaces. To mitigate this problem, we propose CYCLIP, a learning framework that builds upon CLIP by augmenting the contrastive loss in Eq. 1 with additional geometric consistency regularizers. The intuition follows directly from Figure 1 (b), where we showed that inconsistency in the image and text spaces could be eliminated if we symmetrize the similarity between the two mismatched image-text pairs and the similarity between the image-image pair and the text-text pair. We formalize this intuition with two consistency regularizers.
+
+(1) The cross-modal consistency regularizer reduces the gap in the similarity scores between the embeddings of all the mismatched image-text pairs in a batch, two at a time:
+
+$$
+\mathcal { L } _ { \mathrm { C - C y c l i c } } = \frac { 1 } { N } \sum _ { j = 1 } ^ { N } \sum _ { k = 1 } ^ { N } \left( \langle I _ { j } ^ { e } , T _ { k } ^ { e } \rangle - \langle I _ { k } ^ { e } , T _ { j } ^ { e } \rangle \right) ^ { 2 } .
+$$
+
+(2) The in-modal consistency regularizer reduces the gap in the similarity scores between the embeddings of all combinations of image pairs and their corresponding text pairs in a batch:
+
+$$
+\mathcal { L } _ { \mathrm { I - C y c l i c } } = \frac { 1 } { N } \mathrm {sum _ { \substack { j = 1 } } ^ { N } } \mathrm { \sum _ { \substack { k = 1 } } ^ { N } } \left( \langle I _ { j } ^ { e } , I _ { k } ^ { e } \rangle - \langle T _ { k } ^ { e } , T _ { j } ^ { e } \rangle \right) ^ { 2 } .
+$$
+
+Hence, our overall loss for CYCLIP is given as:
+
+$$
+{ \mathcal { L } } _ { \mathrm { C Y C L I P } } = { \mathcal { L } } _ { \mathrm { C L I P } } + \lambda _ { 1 } { \mathcal { L } } _ { \mathrm { I - C y c l i c } } + \lambda _ { 2 } { \mathcal { L } } _ { \mathrm { C - C y c l i c } }
+$$
+
+where $\lambda _ { 1 } > 0$ and $\lambda _ { 2 } > 0$ are hyperparameters controlling the importance of the in-modal and cross-modal cyclic consistency regularizers relative to the contrastive loss in CLIP. We can also characterize the effect of the regularizers in terms of symmetrizing the in-modal and cross-modal similarity matrices, as illustrated in Figure 2. Note that the optimal solution to the contrastive loss formulation would push the similarity between the normalized embeddings of the matched pairs towards 1 while forcing all other pairs of similarities to 0, thereby also symmetrizing the cross-modal similarity matrix and minimizing the cross-modal consistency loss. However, this idealized scenario does not occur in practice, and we find that explicit regularization via cycle-consistency in CYCLIP facilitates improved learning, as we show in our experiments.
+
+# 3 Experiments
+
+Setup: We use Conceptual Captions 3M [52] (CC3M) image-caption pairs as the source of multimodal pretraining data for all our models. Note while this dataset is smaller than the custom dataset (400 million pairs) used in the original work on CLIP [44], it is suitable for our available data and compute and has been used for benchmark evaluations in many subsequent works on language-image pretraining [5, 33, 37, 56]. Following prior work [44], our CLIP models use ResNet-50 as the image encoder and a transformer architecture as the text encoder. Further, we train our models from scratch for 64 epochs on 4 V100 GPUs with a batch size of 128 and an initial learning rate of 0.0005 with cosine scheduling and 10000 warmup steps. The dimension of the image and text embeddings is 1024. For CYCLIP, we use $\lambda _ { 1 } = 0 . 2 5$ and $\lambda _ { 2 } = 0 . 2 5$ across all our experiments.
+
+# 3.1 Zero-Shot Transfer
+
+We compare the zero-shot performance of CLIP and CYCLIP on standard image classification datasets: CIFAR-10, CIFAR-100 [31], and ImageNet1K [49]. We follow the evaluation strategy suggested by [44] for zero-shot classification using prompt engineering. For each dataset, we use the names of the classes to form a set of natural sentences such as ‘a photo of the $\{ \mathrm { c l a s s ~ n a m e } \} ^ { \mathrm { , } }$ , ‘a sketch of the {class name}’ and more. These are passed through the text encoder to get a set of text embeddings for that class. This set of text embeddings are $\ell _ { 2 }$ -normalized, averaged, and further $\ell _ { 2 }$ -normalized to obtain a single text embedding for that class. For a given image, the image embedding is obtained as described in $\ S 2$ . The class whose text embedding (as described above) is closest to the test image is taken to be the predicted label. The zero-shot performance of the models is presented in Table 1.
+
+Table 1: Zero-shot TopK classification accuracy $( \% )$ where $\mathsf { K } \in \{ 1 , 3 , 5 \}$   
+
+<table><tr><td rowspan="2"></td><td colspan="3">CIFAR-10</td><td colspan="3">CIFAR-100</td><td colspan="3">ImageNet1K</td></tr><tr><td>Top1</td><td>Top3</td><td>Top5</td><td>Top1</td><td>Top3</td><td>Top5</td><td>Top1</td><td>Top3</td><td>Top5</td></tr><tr><td>CLIP</td><td>46.54</td><td>78.22</td><td>91.16</td><td>18.69</td><td>34.72</td><td>43.97</td><td>20.03</td><td>33.04</td><td>39.35</td></tr><tr><td>CYCLIP</td><td>51.45</td><td>79.57</td><td>91.80</td><td>23.15</td><td>41.46</td><td>50.66</td><td>22.08</td><td>35.98</td><td>42.30</td></tr><tr><td>%GAIN</td><td>+10.6</td><td>+1.7</td><td>+0.7</td><td>+23.9</td><td>+19.4</td><td>+15.2</td><td>+10.2</td><td>+8.9</td><td>+7.5</td></tr></table>
+
+We observe that the CYCLIP outperforms CLIP across all the datasets and on all TopK metrics, with gains in the range of $1 0 \% - 2 4 \%$ for $\mathrm { K } = 1$ . Our results on zero-shot transfer indicate the usefulness of having geometrical consistency for improved downstream performance of CLIP.
+
+# 3.2 Robustness to Natural Distribution Shifts
+
+One of the major successes of CLIP was its state-of-the-art performance on the natural distribution shift benchmarks. These benchmarks include images depicting sketches, cartoons, adversaries generated using attacks on trained ImageNet models. In Table 2, we evaluate the zero-shot classification accuracy of CYCLIP on four natural distribution shift benchmarks for the ImageNet dataset: ImageNetV2 [48], ImageNetSketch [57], ImageNet-A [27], and ImageNet-R [25].
+
+For most of the distribution shift benchmarks, both CLIP and CYCLIP undergo a significant reduction in their zero-shot performance compared to the original ImageNet1K dataset (last three columns in Table 1). However, we observe that CYCLIP outperforms CLIP on all of the datasets considered in this experiment by a significant margin of improvement $( 1 0 - 2 7 \% )$ ). This result indicates that having cyclic consistency in the learned representations preserves the robustness on the traditional datasets.
+
+Table 2: Zeroshot Classification on Natural Distribution Shifts $( \% )$   
+
+<table><tr><td></td><td colspan="3">ImageNetV2</td><td colspan="3">ImageNetSketch</td><td colspan="3">ImageNet-A</td><td colspan="3">ImageNet-R</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>Top1 Top3 Top5 Top1 Top3 Top5 Top1 Top3 Top5 Top1 Top3 Top5</td><td></td><td></td><td></td><td></td></tr><tr><td>CLIP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>16.91 29.28 34.99 10.37 19.15 24.20 4.2311.3516.88 24.32 39.69 47.20</td><td></td><td></td><td></td><td></td></tr><tr><td>CYCLIP 19.22 32.29 38.41 12.26 22.56 28.17 5.3513.53 19.51 26.79 42.31 50.03</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>%GAIN +13.7 +10.3 +9.8 +18.2 +17.8 +16.4 +26.5 +19.2 +15.6 +10.2 +6.6 +6.0</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+# 3.3 Linear Probing
+
+While the primary focus of CLIP and CYCLIP is zero-shot generalization, we can also assess if the benefits of our cyclic consistency constraints in mitigating inconsistency can be recovered with extra in-domain and in-modality supervision i.e., in the presence of in-distribution training samples from in-domain visual datasets. To this end, we conduct an additional experiments on linear probing where we fit a linear classifier on the representations learned by the visual encoder (ResNet-50) of CLIP and CYCLIP on a range of image classification datasets.
+
+Table 3: Transfer CLIP and CYCLIP to 14 downstream visual datasets using linear probing. Our CYCLIP performs marginally better on 9 out of 14 datasets. For training ImageNet1K, we use a random subset of 50K images from its original training dataset.   
+
+<table><tr><td>iroiiir0 eeeee srtprtiois Crreaer CIPAAII1 CEIPRIPI0 DPodppttrt Arrit IorPoon TTSSP OITLS NH∧S 0 CLIP 79.80 78.26 54.85 59.02 28.00 83.50 54.44 69.72 35.93 57.66 53.82 20.00 89.23 47.28|</td></tr></table>
+
+We present our results in Table 3. We find that both CLIP and CYCLIP can recover most of the performance lost due to inconsistency when provided extra in-domain and in-modality supervision, with CYCLIP marginally outperforming the CLIP on 9 out of 14 visual datasets.
+
+# 4 Analysis
+
+Previously, we demonstrated the gains of CYCLIP over CLIP on downstream tasks that involve joint reasoning over the image and text spaces. In the current section, we wish to better understand the relative behavior of the two models on a set of challenging tasks.
+
+# 4.1 Consistency in Image and Text Spaces
+
+We begin by quantitatively measuring the inconsistency problem illustrated in Figure 1. That is, we wish to evaluate to what extent are the predictions in the image-text space (zero-shot) consistent with the ones made purely within the image space, as measured by our consistency metric in Eq. 2.
+
+Table 4 presents our results over standard benchmarks (CIFAR-10, CIFAR-100, ImageNet1K). The consistency score is calculated over 10K, 10K, and 50K testing images of the CIFAR-10, CIFAR-100 and ImageNet dataset respectively. We use 50K samples from the training set of each dataset for $\mathbf { k }$ -Nearest Neighbor prediction. CYCLIP is more consistent than CLIP across all the datasets as we explicitly symmetrize the cross-modal and in-modal distances. Hence, the representations learned by CYCLIP can be better used interchangeably than CLIP.
+
+Table 4: Consistency score $( \% )$ trend for CLIP and CYCLIP across standard benchmarks . Top- $\mathbf { \nabla } \cdot \mathbf { k }$ consistency score implies the fraction of times, the zero-shot predicted label in the text space is identical to the k-Nearest Neighbor predicted label in the image space (using the training dataset).   
+
+<table><tr><td></td><td colspan="4">CIFAR-10</td><td colspan="4">CIFAR-100</td><td colspan="4">ImageNet1K</td></tr><tr><td></td><td></td><td>Top1 Top3Top5Top10 Top1</td><td></td><td></td><td></td><td></td><td></td><td>Top3Top5Top10 Top1 T</td><td></td><td></td><td></td><td>Top3Top5Top10</td></tr><tr><td>CLIP</td><td>44.60</td><td>46.04</td><td>47.06</td><td>48.45</td><td>16.21</td><td>17.28</td><td>18.42</td><td>19.36</td><td>16.34</td><td>17.42</td><td>18.58</td><td>19.78</td></tr><tr><td>CYCLIP</td><td>48.81</td><td>50.89</td><td>52.30</td><td>53.71</td><td>20.43</td><td>21.96</td><td>23.18</td><td>24.31</td><td>19.20</td><td>20.31</td><td></td><td>21.9523.94</td></tr><tr><td>%GAIN</td><td>+8.6</td><td>+9.5</td><td>+10.0</td><td>）+9.8</td><td>+20.7</td><td></td><td></td><td>+21.3 +20.5 +20.4</td><td>+14.9 +14.2 +15.4 +17.4</td><td></td><td></td><td></td></tr></table>
+
+# 4.2 Fine-grained and Coarse-grained Performance
+
+In $\ S 3 . 1$ , we observed that CYCLIP outperforms CLIP on zero-shot transfer across various datasets. We perform an error analysis investigating both models’ coarse and fine-grained classification performance to understand the transfer phenomena better. Given a hierarchical class structure dataset, coarse-grained classification differentiates between high-level (parent) classes, i.e., zeroshot classification into aquatic mammals and fish. The fine-grained classification task focuses on differentiating low-level (child) classes, i.e., zero-shot classification into a dolphin, otter, and seal (subclasses of aquatic mammals). We perform this analysis on the CIFAR-100, ImageNet1K, ImageNetV2, ImageNetSketch, ImageNet-A, and ImageNet-R datasets.
+
+Formally, we consider a test set of $N$ image-subclass-superclass triplets, $\{ I _ { j } , C _ { j } , P _ { j } \} _ { j = 1 } ^ { N }$ , where $I _ { j }$ , $C _ { j }$ , $P _ { j }$ represent the image, the subclass (child) and superclass (parent) respectively. The image embedding $I _ { j } ^ { e } \in \mathbb { R } ^ { d }$ is obtained as described in $\ S 2$ , and the subclass embedding $C _ { j } ^ { e } \in \mathbb { R } ^ { d }$ and superclass embedding ${ P } _ { j } ^ { e } \in \mathbb { R } ^ { d }$ are obtained as described in $\ S 3 . 1$ . Let the total number of superclasses and subclasses in the dataset be $n _ { \mathrm { p } }$ and $n _ { \mathrm { c } }$ , respectively. Further, let $F$ be a unique mapping from a subclass to the superclass, and $\mho$ denote the inverse mapping from a superclass to the set of subclasses i.e. $\forall P \in \{ 1 , \cdot \cdot \cdot , n _ { \mathrm { p } } \}$ , $G ( P ) = \{ C : F ( C ) = P$ and $\mathbf { \bar { \it C } } \in \{ 1 , \dots , n _ { \mathrm { c } } \} \}$ . Under this setup, the fine-grained and coarse-grained accuracies are defined as:
+
+$$
+\begin{array} { r l } & { \mathrm { F i n e - g r a i n e d ~ A c c u r a c y } = \displaystyle \frac { 1 } { N } \sum _ { j = 1 } ^ { N } 1 \left[ \mathrm { a r g m a x } \ \langle I _ { j } ^ { e } , C \rangle = C _ { j } \right] } \\ & { \mathrm { C o a r s e - g r a i n e d ~ A c c u r a c y } = \displaystyle \frac { 1 } { N } \sum _ { j = 1 } ^ { N } 1 \left[ \mathrm { a r g m a x } \ \langle I _ { j } ^ { e } , C \rangle \in G \left( P _ { j } \right) \right] } \end{array}
+$$
+
+In Figure 3 we visualize how CLIP and CYCLIP compare with each other on the above metrics. The difference between the zero-shot performance of CYCLIP and CLIP is much more significant for coarse-grained classification than fine-grained classification across all the datasets. This observation indicates that concept-level knowledge is better captured in CYCLIP compared to CLIP. The drastic difference in the coarse-grained performance of CYCLIP and CLIP may be attributed to the rigid separation that the default cross-entropy loss in CLIP enforces between the positive pairs and negative pairs, which might degrade performance when some pairs in the negative batch belong to a similar entity. However, CYCLIP does not suffer from this problem as much because it poses cycle constraints on the overall geometry of all the data pairs rather than forcing a rigid separation.
+
+# 4.3 Alignment and Uniformity on the Unit Hypersphere
+
+[58] argues that contrastive learning directly optimizes for (a) alignment (closeness) of the representations of the positive pairs and (b) uniformity (coverage) of the representation space on the unit hypersphere. We extend these properties for multimodal contrastive representation learning as:
+
+$$
+\mathrm { A l i g n m e n t } = \frac { 1 } { N } \sum _ { j = 1 } ^ { N } \langle I _ { j } ^ { e } , T _ { j } ^ { e } \rangle \qquad \mathrm { U n i f o r m i t y } = \log \left( \frac { 1 } { N ( N - 1 ) } \sum _ { j = 1 } ^ { N } \sum _ { k = 1 , j \neq k } ^ { N } e ^ { - \langle I _ { j } ^ { e } , T _ { k } ^ { e } \rangle } \right)
+$$
+
+![](images/9aeb14c15ecc9ded2d517c638f8177fda1c5a997e450914e5b9f6646bfb05089.jpg)  
+Figure 3: The gap between the performances of CLIP and CYCLIP is much larger in coarse-grained scenario highlighting better entity-level knowledge representation in CYCLIP.
+
+We desire our models to achieve high alignment and uniformity scores so that the image-text representations are close for the matched pairs and better spread over the unit hypersphere for different categories. We analyze the effect of cross-modal and in-modal consistency on the alignment and uniformity of the shared representations. For this, we train two ablated versions of CYCLIP, 1) C-CYCLIP with only cross-modal consistency component i.e. $\lambda _ { 1 } = 0 , \lambda _ { 2 } = 0 . 5$ , and 2) I-CYCLIP with only in-modal consistency component i.e. $\lambda _ { 1 } = 0 . 5 , \lambda _ { 2 } = 0$ (in Eq. 5). We design proxy captions for classes as discussed in $\ S 3 . 1$ to act as text embeddings. We present the results in Table 5.
+
+Table 5: Alignment and Uniformity values for CLIP and Cyclic CLIP models. We abbreviate Alignment by A, Uniformity by U, and Zero-shot Top1 classification accuracy $( \% )$ by ZS-Top1.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="3">CIFAR-10</td><td colspan="3">CIFAR-100</td><td colspan="3">ImageNet1K</td></tr><tr><td>A</td><td>U</td><td>ZS-Top1</td><td>A</td><td>U</td><td>ZS-Top1</td><td>A</td><td>U</td><td>ZS-Top1</td></tr><tr><td>CLIP</td><td>0.36</td><td>-0.27</td><td>46.54</td><td>0.36</td><td>-0.25</td><td>18.69</td><td>0.39</td><td>-0.18</td><td>20.03</td></tr><tr><td>CYCLIP</td><td>0.36</td><td>-0.34</td><td>51.45</td><td>0.37</td><td>-0.33</td><td>23.15</td><td>0.38</td><td>-0.32</td><td>22.08</td></tr><tr><td>I-CYCLIP</td><td>0.60</td><td>-0.57</td><td>50.97</td><td>0.60</td><td>-0.57</td><td>22.35</td><td>0.61</td><td>-0.55</td><td>21.21</td></tr><tr><td>C-CYCLIP</td><td>0.05</td><td>-0.02</td><td>55.52</td><td>0.06</td><td>-0.02</td><td>25.49</td><td>0.07</td><td>-0.02</td><td>21.73</td></tr></table>
+
+We observe that I-CYCLIP learns representations that are better aligned in the representation space; however, they do not cover the hypersphere uniformly. The representations learned by C-CYCLIP are more uniformly spread but poorly aligned compared to I-CYCLIP. In this light, the components of CYCLIP can be seen to encourage a balance of good alignment and uniformity. Further, we find that CLIP is more uniform than CYCLIP in all datasets, but contrary to prior beliefs, this does not translate to improved downstream performance. C-CYCLIP has the best downstream zero-shot performance for CIFAR-10 and CIFAR-100 despite its poor alignment score. Further, all 3 variants of CYCLIP outperform CLIP on all 3 datasets, with CYCLIP performing the best on ImageNet1K.
+
+# 4.4 Image-Text Retrieval
+
+We evaluate the effectiveness of the proposed method on the cross-modal (image to text and text to image) retrieval downstream task in the zero-shot as well as fine-tuned settings. We consider the standard benchmark datasets: Flickr30K [42] and MSCOCO [8]. We assess our models on the test set of Flickr30K (1K) and MSCOCO (5K) obtained from the well-known Karpathy [30] split. Both the datasets contains 5 paired captions per image that makes text retrieval per image more easier than image retrieval per caption. We confirm the same in our results below. We perform fine-tuning on the Karpathy’s training split with the batch size of 48. We fine-tune on Flick30K for 10 epochs and MSCOCO for 5 epochs. All the other hyperparameters are identical to that of pre-training.
+
+Table 6: Zero-shot and fine-tuned cross-modal image-text retrieval (text-to-image and image-to-text) results of CLIP and CYCLIP on Flick30K and MSCOCO datasets.   
+
+<table><tr><td rowspan="3"></td><td rowspan="3"></td><td colspan="4">Flickr30K (1K)</td><td colspan="4">MSCOCO (5K)</td></tr><tr><td colspan="2">Text Retrieval</td><td colspan="2">Image Retrieval</td><td colspan="2">Text Retrieval</td><td colspan="2">Image I Retrieval</td></tr><tr><td> R@1 R@5 R@10 R@1 R@5 R@10 R@1 R@5 R@10 R@1 R@5 R@10</td><td></td><td colspan="2"></td><td></td><td></td><td></td><td></td></tr><tr><td>Zero-shot</td><td>CLIP CyCLIP</td><td>88.2 93.9 88.1 93.7</td><td>95.8</td><td>29.9 57.2</td><td>68.0</td><td>82.1 85.6</td><td>87.8</td><td>8.4 19.5</td><td>26.6</td></tr><tr><td></td><td></td><td></td><td></td><td>95.9 30.9</td><td>57.8</td><td>69.1 82.1</td><td>85.6</td><td>87.7</td><td>8.6 20.0</td><td>27.0</td></tr><tr><td>Fine-tuned</td><td>CLIP</td><td>91.9</td><td>97.0</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td>98.0 46.3</td><td>74.7</td><td>83.6 83.2</td><td>87.6</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>90.0</td><td>10.6 23.9</td><td>31.3</td></tr><tr><td></td><td>CYCLIP</td><td>92.3</td><td>97.0</td><td>98.4</td><td>47.3</td><td>83.2</td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>76.6</td><td>85.4</td><td>87.8</td><td>90.3</td><td>11.4 25.8</td><td>33.4</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+Table 6 presents our cross-modal image-text retrieval results for CLIP and CYCLIP. In the zero-shot setting, we find that CYCLIP marginally outperforms CLIP on the image retrieval task on both the datasets. The relatively lower performance of both CLIP and CYCLIP in the zero-shot setting may be attributed to the more complicated nature of the two datasets where the models are expected to find similarities between the image and text at multiple resolutions as opposed to image classification where there is mostly single object to be matched with a simpler caption. It is not clear as to what distinctions in the raw input and text space are reflected in the embedding space too. Hence, we perform fine-tuning on both the datasets to better inform our models of the downstream datasets. In the fine-tuning setting, we find that the performance of both the models increases across both the datasets. However, we observe clear benefits of the soft consistent regularization on the image retrieval results for both the datasets.
+
+# 4.5 CYCLIP preserves the Effective Robustness of CLIP
+
+[36] shows that there is a strong correlation between the in-distribution and out-of-distribution generalization of the models trained on ImageNet1K, as illustrated by the linear fit (red) in Figure 4. Ideally, any model that does not undergo distribution shift would fall on the $y = x$ trendline (black). For other models, the deviations of the models from this ideal fit indicate their effective robustness. Previously, [45] showed that the zero-shot CLIP classifier trained on 400M image-text pairs improves effective robustness significantly compared to prior approaches to robustness. Subsequently, [28] demonstrated that CLIP models trained at small scales also exhibit high effective robustness that allows them to be used as a proxy to study the robustness properties of CLIP.
+
+![](images/e98bb549238d19b78531727425dfe31fd161eaa892b75cfe6cd33676963af21f.jpg)  
+Figure 4: Effect of varying the training dataset size on (a) Classification accuracy on ImageNet1K and (b) Effective Robustness on ImageNetV2.
+
+We evaluate the effect of cyclic consistency on effective robustness. We trained 4 CLIP and CYCLIP models, varying the training dataset sizes from 500K to 4M image-text pairs from the $\mathrm { C C 3 M + }$ CC12M datasets. In Figure 4, (a) we observe that for all training data sizes, CYCLIP shows a significant improvement over CLIP, showcasing its effectiveness in a diverse set of data regimes. Further, Figure 4 (b) shows that CYCLIP lies way above the baseline trend and preserves the effective robustness of CLIP.
+
+# 5 Related Work
+
+Our work fits into the broader theme of unsupervised pretraining with multiple modalities and has been successfully applied for learning representations of modalities such as images, text, and speech [2, 15, 1, 59, 43, 34]. Similar to the unimodal setting, two predominant approaches for multimodal pretraining are contrastive and generative, as described below.
+
+Contrastive Representation Learning: Contrastive learning was originally proposed for selfsupervised representation learning in the unimodal context where the embeddings of a sample are brought closer to an augmented version of the sample. In contrast, the embeddings are pushed away for other samples, and their augmentation [11, 51, 39, 55, 21, 7, 16, 40, 66, 23, 18]. [63] and [3] impose additional constraints to remove redundancies and prevent dimensional collapse in the visual representations. Recently, contrastive learning has also been used to learn robust representations of the multimodal data [62, 47]. Many works use additional losses to imbibe extra supervisory multimodal knowledge during the training process [54, 65, 64, 14, 35]. In this work, we focus on having cyclic consistency in addition to the contrastive loss to learn more robust image-text representations.
+
+Contrastive Language-Image Pretraining: CLIP [44], ALIGN [29] and BASIC [41] have enjoyed great success in extending contrastive learning to paired image-text data, with impressive zero-shot classification and robustness performance. These works have been further extended recently to include visual self-supervision [37], additional nearest neighbor supervision [33], and utilization of unpaired data [56]. Our work complements much of this literature as it identifies consistency regularizers that can be augmented to the learning objective of the above works.
+
+Generative Representation Learning: Generative models have been applied for learning representations of multimodal data [60, 53]. In particular, [67, 61, 10] proposed a notion of cyclic consistency for learning from unpaired multimodal data using GANs [17], which was extended later to normalizing flows [20, 19]. While these works focus on regularizing a generative mapping between modalities, our notion of cycle consistency applies to embeddings learned via a contrastive framework.
+
+# 6 Conclusion
+
+We presented CYCLIP, a framework for cycle consistent multimodal representation learning for image and text modality. The main benefits of CYCLIP stem from including cross-modal consistency and in-modal consistency regularizers to prevent inconsistent inference in the image and text spaces. Empirically, we show that CYCLIP performs much better than CLIP on zero-shot classification and is more robust on benchmarks for distributional robustness. We also showed that the representations learned by CYCLIP are more consistent than CLIP and better capture concept-level knowledge, as evidenced by our analysis of fine-grained and coarse-grained accuracies.
+
+We believe this work can motivate further studies on understanding the geometry of the representation spaces learned via the contrastive objective applied to paired multimodal data and, in particular, identify conditions and regularization strategies under which the learned representations are synergistic across the various modalities for downstream applications.
+
+One important future direction and a current limitation is scaling CYCLIP to larger datasets. While we do not possess the resources for this study, it is imperative to study the extent to which the benefits of cycle consistency remain at the scale on which the original CLIP was trained (400M image-text pairs). Finally, for real-world deployment of CLIP and their variants, such as CYCLIP, we need to be cautious about amplifying societal biases as these models are trained on large uncurated datasets scraped from the web [9]. Additionally, it is easy to add malicious data to the web, which poses a severe security threat [5]. Alleviating such harms is an important and active area of research.
+
+# Acknowledgements
+
+This research is supported by an Adobe Data Science Research Award for Aditya Grover. We would like to thank the IDRE’s Research Technology group for the GPU computing resources on the UCLA Hoffman2 Cluster. We also want to thank Tung Duc Nguyen, Satvik Mashkaria, Siddarth Krishnamoorthy, Varuni Sarwal, and Ashima Suvarna for their helpful suggestions.
+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] Section 6.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] Section 6.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes]   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] Due to extremely-compute heavy experiments.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes]
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]   
+(b) Did you mention the license of the assets? [No] All the non-proprietary datasets and code used are public under MIT, BSD or CC licenses.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/dev/I-6yh2-dkyD/I-6yh2-dkyD_content_list.json

@@ -0,0 +1,1601 @@
+[
+    {
+        "type": "text",
+        "text": "CYCLIP: Cyclic Contrastive Language-Image Pretraining ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Shashank Goel∗ UCLA shashankgoel@ucla.edu ",
+        "bbox": [
+            205,
+            226,
+            387,
+            268
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Hritik Bansal∗ UCLA hbansal@ucla.edu ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Sumit Bhatia MDSR Lab, Adobe Systems sumit.bhatia@adobe.com ",
+        "bbox": [
+            602,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ryan A. Rossi Adobe Research ryrossi@adobe.com ",
+        "bbox": [
+            222,
+            289,
+            370,
+            332
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Vishwa Vinay Adobe Research vinay@adobe.com ",
+        "bbox": [
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+            330
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Aditya Grover UCLA adityag@cs.ucla.edu ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recent advances in contrastive representation learning over paired image-text data have led to models such as CLIP [44] that achieve state-of-the-art performance for zero-shot classification and distributional robustness. Such models typically require joint reasoning in the image and text representation spaces for downstream inference tasks. Contrary to prior beliefs, we demonstrate that the image and text representations learned via a standard contrastive objective are not interchangeable and can lead to inconsistent downstream predictions. To mitigate this issue, we formalize consistency and propose CYCLIP, a framework for contrastive representation learning that explicitly optimizes for the learned representations to be geometrically consistent in the image and text space. In particular, we show that consistent representations can be learned by explicitly symmetrizing (a) the similarity between the two mismatched image-text pairs (cross-modal consistency); and (b) the similarity between the image-image pair and the text-text pair (in-modal consistency). Empirically, we show that the improved consistency in CYCLIP translates to significant gains over CLIP, with gains ranging from $1 \\dot { 0 } \\% - 2 4 \\%$ for zero-shot classification accuracy on standard benchmarks (CIFAR-10, CIFAR-100, ImageNet1K) and $1 0 \\% - 2 7 \\%$ for robustness to various natural distribution shifts. The code is available at https://github.com/goel-shashank/CyCLIP. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The ability to learn general-purpose representations from diverse data modalities is a long-standing goal of artificial intelligence (AI) [4, 32]. In this regard, recent instantiations such as CLIP [44], ALIGN [29], and BASIC [41] have scaled up vision-language contrastive pretraining to jointly learn image and text embeddings, by exploiting an enormous amount of paired image-text data on the web. Post pretraining, these embeddings exhibit impressive zero-shot classification performance [13] and robustness to natural distribution shifts [48, 57, 24, 26]. Recently, these embeddings have been extended to text-guided generation of natural images [47, 12, 38, 46] and transferred to modalities such as 3-D shapes [50] by emphasizing the interchangeability of the image and text embeddings. ",
+        "bbox": [
+            174,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In the context of vision-language pretraining, the standard contrastive learning objective aims to maximize the similarity between matched image-text pairs (“positives\") against all the mismatched image-text pairs (“negatives\") [45, 7, 40, 22]. While such an objective aligns the true image-text pairs, it poses no constraints on the overall geometry of all data pairs, including the mismatched pairs and pairs within the same modality. In Figure 1 (a), we illustrate this effect where matched image-text pairs, $( I _ { \\mathrm { d o g } } , T _ { \\mathrm { d o g } } )$ and $( I _ { \\mathrm { c a t } } , T _ { \\mathrm { c a t } } )$ , get close to each other but the overall geometry of pairwise distances can be highly irregular (see e.g., $( I _ { \\mathrm { d o g } } , T _ { \\mathrm { c a t } } )$ and $( I _ { \\mathrm { c a t } } , T _ { \\mathrm { d o g } } ) )$ . If we use such representations for downstream inference, such irregularities can translate into inconsistent reasoning in the image and text spaces. For example, CLIP designs proxy captions for class labels and uses the most similar class caption to perform zero-shot classification for images; using the default captions in Figure 1 (a), this would imply that a test image $I _ { \\mathrm { t e s t } }$ gets classified as a dog in the image space even when a simple nearest neighbor classifier in the text space would correctly infer the label to be a cat. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/113dedcfa7f5e22fc188dc3390117ab472deeaa1e0b6883316da098994bce5ea.jpg",
+        "image_caption": [
+            "Figure 1: An illustration of the planar geometry of the learned representations of image-text pairs by (a) CLIP and (b) CYCLIP. The edges indicate the distance between the representations i.e., $\\dot { d ( e _ { 1 } , e _ { 2 } ) } = 1 - \\langle e _ { 1 } , e _ { 2 } \\rangle$ , where $\\langle \\cdot , \\cdot \\rangle$ is the inner product. CYCLIP is cyclic consistent between image-text pairs as the in-modal distances, $d ( T _ { \\mathrm { c a t } } , T _ { \\mathrm { d o g } } ) \\sim d ( I _ { \\mathrm { c a t } } , I _ { \\mathrm { d o g } } )$ , and the cross-modal distances, $d ( \\bar { T _ { \\mathrm { c a t } } } , \\bar { I _ { \\mathrm { d o g } } } ) ~ \\sim ~ d ( \\bar { I _ { \\mathrm { c a t } } } , \\bar { T _ { \\mathrm { d o g } } } )$ , are similar to each other unlike CLIP. Due to explicit consistency constraints, the test image of a cat is classified as a cat in the image as well as the text space. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To mitigate these challenges, we propose Cyclic Contrastive Language-Image Pretraining (CYCLIP), a framework that imposes additional geometric structure on the learned representations. Specifically, given two image-text pairs, we augment the contrastive learning objective with two symmetrization terms. The first term provides for in-modal consistency by encouraging the distance between the two image embeddings to be close to the distance between the corresponding text embeddings. The second term for the cross-modal consistency that encourages the distance between the image and text embedding from the first and second pairs respectively to be close to the distance between the text and image embeddings from the first and second pairs respectively. As shown in Figure 1 (b), if representations of any two image-text pairs, $( I _ { \\mathrm { d o g } } , T _ { \\mathrm { d o g } } )$ and $( \\dot { I } _ { \\mathrm { c a t } } , T _ { \\mathrm { c a t } } )$ exactly satisfy both forms of cyclic consistency, then we can guarantee that any test image $I _ { \\mathrm { t e s t } }$ respects the ordering of distances in both image and text spaces (i.e., if $d ( I _ { \\mathrm { t e s t } } , I _ { \\mathrm { d o g } } ) > d ( I _ { \\mathrm { t e s t } } , I _ { \\mathrm { c a t } } )$ , then $d ( I _ { \\mathrm { t e s t } } , T _ { \\mathrm { d o g } } ) > d ( I _ { t e s t } , T _ { \\mathrm { c a t } } ) )$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Empirically, we demonstrate that the improved consistency in CYCLIP translates to improvements over CLIP. In all cases, we pre-train our models on the Conceptual Captions 3M dataset[52]. On zero-shot classification, we observe that CYCLIP improves over CLIP by $1 0 . 2 \\%$ on ImageNet1K, $1 0 . 6 \\%$ on CIFAR-10 and $2 3 . 9 \\%$ on CIFAR-100 respectively. Further, CYCLIP outperforms CLIP with an average relative gain of $+ 1 7 \\%$ on ImageNet natural distribution shift benchmarks. We further analyze the improved performance of CYCLIP and find that the additional geometric structure in the representation space better captures the coarse and fine-grained concept hierarchies of datasets. ",
+        "bbox": [
+            174,
+            736,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our contributions are as follows: ",
+        "bbox": [
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+            839,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1. We analyze contrastive learning for representation learning jointly over image and text modalities. We identify a critical shortcoming in the geometry of the learned representation space that can lead to inconsistent predictions in image and text domains. ",
+        "bbox": [
+            214,
+            869,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2. We propose CYCLIP, a simple and effective framework for contrastive representation learning with two additional cycle consistency constraints for mitigating the above issue. 3. We demonstrate that CYCLIP achieves significant empirical improvements over CLIP on zero-shot classification and robustness benchmarks. We further explain these improvements by analyzing the impact of consistency on the hierarchical structure of datasets. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 Cycle Consistent Representation Learning ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 Preliminaries ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            220,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We are interested in using text supervision to learn general-purpose visual representations that can be generalized to downstream predictive tasks. To this end, there have been several recent advances in language-image pretraining concerning model architectures, training objectives, and sources of supervision. Our work is most closely related to Contrastive Language-Image Pretraining (CLIP) [44] which combines many such advances in a highly scalable and generalizable learning framework. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "CLIP is trained on millions of images with their captions scraped from the web. Formally, we consider a dataset $S \\subset \\mathcal { T } \\times \\mathcal { T }$ consisting of pairs $( I _ { j } , T _ { j } )$ where $I _ { j }$ is a raw image and $T _ { j }$ is a text caption. We use $\\mathcal { T }$ and $\\tau$ to denote the domain of images and text, respectively. The CLIP architecture consists of 3 components: (i) an image encoder network, $f _ { I } : \\mathcal { T } \\mapsto \\mathbb { R } ^ { d }$ , to encode the raw image into an embedding vector of dimension $d$ , (ii) a text encoder network, $f _ { T } : T \\mapsto \\mathbb { R } ^ { d }$ , to encode the raw text into an embedding vector of dimension $d$ , (iii) a contrastive objective that pulls the embeddings of paired image-caption pairs together while pushing apart embeddings of unmatched pairs. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Formally, during training, consider a batch of $N$ image-captions pairs, $\\{ I _ { j } , T _ { j } \\} _ { j = 1 } ^ { N }$ , where $I _ { j }$ and $T _ { j }$ represent the raw image and text pair, respectively. The image embedding $\\bar { I } _ { j } ^ { e } \\in \\mathbb { R } ^ { d }$ and text embedding $T _ { j } ^ { e } \\in \\mathbb { R } ^ { d }$ are obtained by passing $I _ { j }$ and $T _ { j }$ through the image encoder $f _ { I }$ and text encoder $f _ { T }$ , respectively; i.e. $I _ { j } ^ { e } = f _ { I } ( I _ { j } )$ and $T _ { j } ^ { e } = f _ { T } ( T _ { j } )$ . Further, we assume they are normalized to have unit $\\ell _ { 2 }$ -norm. The contrastive objective in CLIP aims to align the image and text representations by minimizing the loss function ${ \\mathcal { L } } _ { \\mathrm { C L I P } }$ shown below: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cf6f4aec5eb6a0ccaa36ba52693dc6486e216507f408ac680542f717dfb85160.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { C L I P } } = - \\frac { 1 } { 2 N } \\sum _ { j = 1 } ^ { N } \\log \\left[ \\frac { \\exp \\left( \\langle I _ { j } ^ { e } , T _ { j } ^ { e } \\rangle / \\tau \\right) } { \\displaystyle \\sum _ { k = 1 } ^ { N } \\exp \\left( \\langle I _ { j } ^ { e } , T _ { k } ^ { e } \\rangle / \\tau \\right) } \\right] - \\frac { 1 } { 2 N } \\sum _ { k = 1 } ^ { N } \\log \\left[ \\frac { \\exp \\left( \\langle I _ { k } ^ { e } , T _ { k } ^ { e } \\rangle / \\tau \\right) } { \\displaystyle \\sum _ { j = 1 } ^ { N } \\exp \\left( \\langle I _ { j } ^ { e } , T _ { k } ^ { e } \\rangle / \\tau \\right) } \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\langle \\cdot , \\cdot \\rangle$ represents the inner product, and $\\tau$ is a trainable temperature parameter. CLIP and its variants can be used to perform zero-shot image classification, i.e., classifying test images into categories not seen at training time. We first transform each category into a suitable caption (e.g., the airplane category in CIFAR-10 can be expressed as ‘a photo of an airplane’). Then, the similarity of the test image to each caption is computed (e.g., cosine distance), and the model predicts the category for which the image-caption similarity is the highest. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 Inconsistent Representation Learning in CLIP ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As illustrated in Figure 1 (a), the standard contrastive objective in CLIP can learn image-text representations such that the predicted labels for the test image are different in the image and text spaces. Here, we reason about such inconsistencies more formally in the context of downstream classification. As discussed above, we can predict a label in the text embedding space (zero-shot setting) by selecting the label that is closest to the test image $( P _ { T } )$ . Additionally, for classification in the image embedding space, if we had access to a labeled training set, then one natural way to infer the predicted label $( { \\dot { P } } _ { I } ^ { k } )$ of a test image $I _ { t e s t }$ is by taking a majority vote from the true labels associated with the $\\mathbf { k }$ -nearest training images. Formally, we define a consistency score that measures the synchrony between the predicted labels in the image and text spaces as: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/a1fca37a434834539ae005b54f2227219c3c73d2e5bd566f8fe8abde7bd51ec3.jpg",
+        "image_caption": [
+            "Figure 2: Illustrative overview for CYCLIP $N = 2 \\AA$ ). It consists of 3 major components: (a) cross-modal contrastive alignment, (b) cross-modal consistency, and (c) in-modal consistency. Only (a) is present in CLIP, whereas our proposed regularizers in (b) and (c) mitigate inconsistency. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b1e7cdf27c20d1d0ae9769a52678d00d01d5eeea72ed6b6536ae101f3c7b3255.jpg",
+        "text": "$$\n{ \\mathrm { C o n s i s t e n c y ~ S c o r e } } _ { k } = { \\frac { 1 } { N } } \\sum _ { j = 1 } ^ { N } \\mathbb { 1 } \\left[ P _ { I } ^ { k } ( I _ { j } ) = P _ { T } ( I _ { j } ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $_ \\mathrm { N }$ is the number of test images. In our experiments (discussed in detail in $\\ S 3$ ), we found the CLIP’s consistency score $k = 1$ ) to be $44 \\%$ , $16 \\%$ , and $16 \\%$ on the standard benchmarks CIFAR-10, CIFAR-100, and ImageNet1K, respectively, showing a very high degree of disagreement in the image and text spaces. In the following section, we describe our approach to alleviate the inconsistent inference problem and quantitatively show that our solution improves the consistency score in $\\ S 4 . 1$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.3 Cycle Consistent Representation Learning via CYCLIP ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We showed that the visual representations learned by CLIP could be inconsistent when used for inference in the image and text spaces. To mitigate this problem, we propose CYCLIP, a learning framework that builds upon CLIP by augmenting the contrastive loss in Eq. 1 with additional geometric consistency regularizers. The intuition follows directly from Figure 1 (b), where we showed that inconsistency in the image and text spaces could be eliminated if we symmetrize the similarity between the two mismatched image-text pairs and the similarity between the image-image pair and the text-text pair. We formalize this intuition with two consistency regularizers. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "(1) The cross-modal consistency regularizer reduces the gap in the similarity scores between the embeddings of all the mismatched image-text pairs in a batch, two at a time: ",
+        "bbox": [
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+    },
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+        "type": "equation",
+        "img_path": "images/2b17a618af8bd41c51f1ef8cbc4cadc58d98d90641b6496511856ebc2cf7f1a8.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { C - C y c l i c } } = \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } \\sum _ { k = 1 } ^ { N } \\left( \\langle I _ { j } ^ { e } , T _ { k } ^ { e } \\rangle - \\langle I _ { k } ^ { e } , T _ { j } ^ { e } \\rangle \\right) ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "(2) The in-modal consistency regularizer reduces the gap in the similarity scores between the embeddings of all combinations of image pairs and their corresponding text pairs in a batch: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/43e8dc0add8684d8181508ec2bf09767e861ebdde8341cb7ba4dd1f325da1e54.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { I - C y c l i c } } = \\frac { 1 } { N } \\mathrm {sum _ { \\substack { j = 1 } } ^ { N } } \\mathrm { \\sum _ { \\substack { k = 1 } } ^ { N } } \\left( \\langle I _ { j } ^ { e } , I _ { k } ^ { e } \\rangle - \\langle T _ { k } ^ { e } , T _ { j } ^ { e } \\rangle \\right) ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Hence, our overall loss for CYCLIP is given as: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/79b413a6628715e2321e5330b19c77d94fb274ae8cf8990cb6aaf7b8a7481b79.jpg",
+        "text": "$$\n{ \\mathcal { L } } _ { \\mathrm { C Y C L I P } } = { \\mathcal { L } } _ { \\mathrm { C L I P } } + \\lambda _ { 1 } { \\mathcal { L } } _ { \\mathrm { I - C y c l i c } } + \\lambda _ { 2 } { \\mathcal { L } } _ { \\mathrm { C - C y c l i c } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\lambda _ { 1 } > 0$ and $\\lambda _ { 2 } > 0$ are hyperparameters controlling the importance of the in-modal and cross-modal cyclic consistency regularizers relative to the contrastive loss in CLIP. We can also characterize the effect of the regularizers in terms of symmetrizing the in-modal and cross-modal similarity matrices, as illustrated in Figure 2. Note that the optimal solution to the contrastive loss formulation would push the similarity between the normalized embeddings of the matched pairs towards 1 while forcing all other pairs of similarities to 0, thereby also symmetrizing the cross-modal similarity matrix and minimizing the cross-modal consistency loss. However, this idealized scenario does not occur in practice, and we find that explicit regularization via cycle-consistency in CYCLIP facilitates improved learning, as we show in our experiments. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Setup: We use Conceptual Captions 3M [52] (CC3M) image-caption pairs as the source of multimodal pretraining data for all our models. Note while this dataset is smaller than the custom dataset (400 million pairs) used in the original work on CLIP [44], it is suitable for our available data and compute and has been used for benchmark evaluations in many subsequent works on language-image pretraining [5, 33, 37, 56]. Following prior work [44], our CLIP models use ResNet-50 as the image encoder and a transformer architecture as the text encoder. Further, we train our models from scratch for 64 epochs on 4 V100 GPUs with a batch size of 128 and an initial learning rate of 0.0005 with cosine scheduling and 10000 warmup steps. The dimension of the image and text embeddings is 1024. For CYCLIP, we use $\\lambda _ { 1 } = 0 . 2 5$ and $\\lambda _ { 2 } = 0 . 2 5$ across all our experiments. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.1 Zero-Shot Transfer ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    {
+        "type": "text",
+        "text": "We compare the zero-shot performance of CLIP and CYCLIP on standard image classification datasets: CIFAR-10, CIFAR-100 [31], and ImageNet1K [49]. We follow the evaluation strategy suggested by [44] for zero-shot classification using prompt engineering. For each dataset, we use the names of the classes to form a set of natural sentences such as ‘a photo of the $\\{ \\mathrm { c l a s s ~ n a m e } \\} ^ { \\mathrm { , } }$ , ‘a sketch of the {class name}’ and more. These are passed through the text encoder to get a set of text embeddings for that class. This set of text embeddings are $\\ell _ { 2 }$ -normalized, averaged, and further $\\ell _ { 2 }$ -normalized to obtain a single text embedding for that class. For a given image, the image embedding is obtained as described in $\\ S 2$ . The class whose text embedding (as described above) is closest to the test image is taken to be the predicted label. The zero-shot performance of the models is presented in Table 1. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/4f89e882313a27a95f143127b2f55d6a511d0449227ffec744e360a66ab285b6.jpg",
+        "table_caption": [
+            "Table 1: Zero-shot TopK classification accuracy $( \\% )$ where $\\mathsf { K } \\in \\{ 1 , 3 , 5 \\}$ "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"3\">CIFAR-10</td><td colspan=\"3\">CIFAR-100</td><td colspan=\"3\">ImageNet1K</td></tr><tr><td>Top1</td><td>Top3</td><td>Top5</td><td>Top1</td><td>Top3</td><td>Top5</td><td>Top1</td><td>Top3</td><td>Top5</td></tr><tr><td>CLIP</td><td>46.54</td><td>78.22</td><td>91.16</td><td>18.69</td><td>34.72</td><td>43.97</td><td>20.03</td><td>33.04</td><td>39.35</td></tr><tr><td>CYCLIP</td><td>51.45</td><td>79.57</td><td>91.80</td><td>23.15</td><td>41.46</td><td>50.66</td><td>22.08</td><td>35.98</td><td>42.30</td></tr><tr><td>%GAIN</td><td>+10.6</td><td>+1.7</td><td>+0.7</td><td>+23.9</td><td>+19.4</td><td>+15.2</td><td>+10.2</td><td>+8.9</td><td>+7.5</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We observe that the CYCLIP outperforms CLIP across all the datasets and on all TopK metrics, with gains in the range of $1 0 \\% - 2 4 \\%$ for $\\mathrm { K } = 1$ . Our results on zero-shot transfer indicate the usefulness of having geometrical consistency for improved downstream performance of CLIP. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 Robustness to Natural Distribution Shifts ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            767,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "One of the major successes of CLIP was its state-of-the-art performance on the natural distribution shift benchmarks. These benchmarks include images depicting sketches, cartoons, adversaries generated using attacks on trained ImageNet models. In Table 2, we evaluate the zero-shot classification accuracy of CYCLIP on four natural distribution shift benchmarks for the ImageNet dataset: ImageNetV2 [48], ImageNetSketch [57], ImageNet-A [27], and ImageNet-R [25]. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For most of the distribution shift benchmarks, both CLIP and CYCLIP undergo a significant reduction in their zero-shot performance compared to the original ImageNet1K dataset (last three columns in Table 1). However, we observe that CYCLIP outperforms CLIP on all of the datasets considered in this experiment by a significant margin of improvement $( 1 0 - 2 7 \\% )$ ). This result indicates that having cyclic consistency in the learned representations preserves the robustness on the traditional datasets. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/96413a6198e854f4d42e38e2455c007c6b3515e3ae68403075f7c2042207f8a5.jpg",
+        "table_caption": [
+            "Table 2: Zeroshot Classification on Natural Distribution Shifts $( \\% )$ "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">ImageNetV2</td><td colspan=\"3\">ImageNetSketch</td><td colspan=\"3\">ImageNet-A</td><td colspan=\"3\">ImageNet-R</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>Top1 Top3 Top5 Top1 Top3 Top5 Top1 Top3 Top5 Top1 Top3 Top5</td><td></td><td></td><td></td><td></td></tr><tr><td>CLIP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>16.91 29.28 34.99 10.37 19.15 24.20 4.2311.3516.88 24.32 39.69 47.20</td><td></td><td></td><td></td><td></td></tr><tr><td>CYCLIP 19.22 32.29 38.41 12.26 22.56 28.17 5.3513.53 19.51 26.79 42.31 50.03</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>%GAIN +13.7 +10.3 +9.8 +18.2 +17.8 +16.4 +26.5 +19.2 +15.6 +10.2 +6.6 +6.0</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "3.3 Linear Probing ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "While the primary focus of CLIP and CYCLIP is zero-shot generalization, we can also assess if the benefits of our cyclic consistency constraints in mitigating inconsistency can be recovered with extra in-domain and in-modality supervision i.e., in the presence of in-distribution training samples from in-domain visual datasets. To this end, we conduct an additional experiments on linear probing where we fit a linear classifier on the representations learned by the visual encoder (ResNet-50) of CLIP and CYCLIP on a range of image classification datasets. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/3ff584348c71340e8c7bacca5c35e1a11e6538d32abc30e557f723e23d6b5180.jpg",
+        "table_caption": [
+            "Table 3: Transfer CLIP and CYCLIP to 14 downstream visual datasets using linear probing. Our CYCLIP performs marginally better on 9 out of 14 datasets. For training ImageNet1K, we use a random subset of 50K images from its original training dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>iroiiir0 eeeee srtprtiois Crreaer CIPAAII1 CEIPRIPI0 DPodppttrt Arrit IorPoon TTSSP OITLS NH∧S 0 CLIP 79.80 78.26 54.85 59.02 28.00 83.50 54.44 69.72 35.93 57.66 53.82 20.00 89.23 47.28|</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We present our results in Table 3. We find that both CLIP and CYCLIP can recover most of the performance lost due to inconsistency when provided extra in-domain and in-modality supervision, with CYCLIP marginally outperforming the CLIP on 9 out of 14 visual datasets. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4 Analysis ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Previously, we demonstrated the gains of CYCLIP over CLIP on downstream tasks that involve joint reasoning over the image and text spaces. In the current section, we wish to better understand the relative behavior of the two models on a set of challenging tasks. ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "4.1 Consistency in Image and Text Spaces ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "We begin by quantitatively measuring the inconsistency problem illustrated in Figure 1. That is, we wish to evaluate to what extent are the predictions in the image-text space (zero-shot) consistent with the ones made purely within the image space, as measured by our consistency metric in Eq. 2. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 4 presents our results over standard benchmarks (CIFAR-10, CIFAR-100, ImageNet1K). The consistency score is calculated over 10K, 10K, and 50K testing images of the CIFAR-10, CIFAR-100 and ImageNet dataset respectively. We use 50K samples from the training set of each dataset for $\\mathbf { k }$ -Nearest Neighbor prediction. CYCLIP is more consistent than CLIP across all the datasets as we explicitly symmetrize the cross-modal and in-modal distances. Hence, the representations learned by CYCLIP can be better used interchangeably than CLIP. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/a3796c1bd77c99aa7534bdbc3b45776b80ddeb69d1348f8da0a081d126780de9.jpg",
+        "table_caption": [
+            "Table 4: Consistency score $( \\% )$ trend for CLIP and CYCLIP across standard benchmarks . Top- $\\mathbf { \\nabla } \\cdot \\mathbf { k }$ consistency score implies the fraction of times, the zero-shot predicted label in the text space is identical to the k-Nearest Neighbor predicted label in the image space (using the training dataset). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"4\">CIFAR-10</td><td colspan=\"4\">CIFAR-100</td><td colspan=\"4\">ImageNet1K</td></tr><tr><td></td><td></td><td>Top1 Top3Top5Top10 Top1</td><td></td><td></td><td></td><td></td><td></td><td>Top3Top5Top10 Top1 T</td><td></td><td></td><td></td><td>Top3Top5Top10</td></tr><tr><td>CLIP</td><td>44.60</td><td>46.04</td><td>47.06</td><td>48.45</td><td>16.21</td><td>17.28</td><td>18.42</td><td>19.36</td><td>16.34</td><td>17.42</td><td>18.58</td><td>19.78</td></tr><tr><td>CYCLIP</td><td>48.81</td><td>50.89</td><td>52.30</td><td>53.71</td><td>20.43</td><td>21.96</td><td>23.18</td><td>24.31</td><td>19.20</td><td>20.31</td><td></td><td>21.9523.94</td></tr><tr><td>%GAIN</td><td>+8.6</td><td>+9.5</td><td>+10.0</td><td>）+9.8</td><td>+20.7</td><td></td><td></td><td>+21.3 +20.5 +20.4</td><td>+14.9 +14.2 +15.4 +17.4</td><td></td><td></td><td></td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 Fine-grained and Coarse-grained Performance ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In $\\ S 3 . 1$ , we observed that CYCLIP outperforms CLIP on zero-shot transfer across various datasets. We perform an error analysis investigating both models’ coarse and fine-grained classification performance to understand the transfer phenomena better. Given a hierarchical class structure dataset, coarse-grained classification differentiates between high-level (parent) classes, i.e., zeroshot classification into aquatic mammals and fish. The fine-grained classification task focuses on differentiating low-level (child) classes, i.e., zero-shot classification into a dolphin, otter, and seal (subclasses of aquatic mammals). We perform this analysis on the CIFAR-100, ImageNet1K, ImageNetV2, ImageNetSketch, ImageNet-A, and ImageNet-R datasets. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Formally, we consider a test set of $N$ image-subclass-superclass triplets, $\\{ I _ { j } , C _ { j } , P _ { j } \\} _ { j = 1 } ^ { N }$ , where $I _ { j }$ , $C _ { j }$ , $P _ { j }$ represent the image, the subclass (child) and superclass (parent) respectively. The image embedding $I _ { j } ^ { e } \\in \\mathbb { R } ^ { d }$ is obtained as described in $\\ S 2$ , and the subclass embedding $C _ { j } ^ { e } \\in \\mathbb { R } ^ { d }$ and superclass embedding ${ P } _ { j } ^ { e } \\in \\mathbb { R } ^ { d }$ are obtained as described in $\\ S 3 . 1$ . Let the total number of superclasses and subclasses in the dataset be $n _ { \\mathrm { p } }$ and $n _ { \\mathrm { c } }$ , respectively. Further, let $F$ be a unique mapping from a subclass to the superclass, and $\\mho$ denote the inverse mapping from a superclass to the set of subclasses i.e. $\\forall P \\in \\{ 1 , \\cdot \\cdot \\cdot , n _ { \\mathrm { p } } \\}$ , $G ( P ) = \\{ C : F ( C ) = P$ and $\\mathbf { \\bar { \\it C } } \\in \\{ 1 , \\dots , n _ { \\mathrm { c } } \\} \\}$ . Under this setup, the fine-grained and coarse-grained accuracies are defined as: ",
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/228004ffb05457aab3de42024de429b8ecc75ea444e16d247b612cd603259f27.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathrm { F i n e - g r a i n e d ~ A c c u r a c y } = \\displaystyle \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } 1 \\left[ \\mathrm { a r g m a x } \\ \\langle I _ { j } ^ { e } , C \\rangle = C _ { j } \\right] } \\\\ & { \\mathrm { C o a r s e - g r a i n e d ~ A c c u r a c y } = \\displaystyle \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } 1 \\left[ \\mathrm { a r g m a x } \\ \\langle I _ { j } ^ { e } , C \\rangle \\in G \\left( P _ { j } \\right) \\right] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In Figure 3 we visualize how CLIP and CYCLIP compare with each other on the above metrics. The difference between the zero-shot performance of CYCLIP and CLIP is much more significant for coarse-grained classification than fine-grained classification across all the datasets. This observation indicates that concept-level knowledge is better captured in CYCLIP compared to CLIP. The drastic difference in the coarse-grained performance of CYCLIP and CLIP may be attributed to the rigid separation that the default cross-entropy loss in CLIP enforces between the positive pairs and negative pairs, which might degrade performance when some pairs in the negative batch belong to a similar entity. However, CYCLIP does not suffer from this problem as much because it poses cycle constraints on the overall geometry of all the data pairs rather than forcing a rigid separation. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3 Alignment and Uniformity on the Unit Hypersphere ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "[58] argues that contrastive learning directly optimizes for (a) alignment (closeness) of the representations of the positive pairs and (b) uniformity (coverage) of the representation space on the unit hypersphere. We extend these properties for multimodal contrastive representation learning as: ",
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/36e3dba549bbfcd13f40600a8d08a7085b78799d6928cfff4d4ee41662ae4c42.jpg",
+        "text": "$$\n\\mathrm { A l i g n m e n t } = \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } \\langle I _ { j } ^ { e } , T _ { j } ^ { e } \\rangle \\qquad \\mathrm { U n i f o r m i t y } = \\log \\left( \\frac { 1 } { N ( N - 1 ) } \\sum _ { j = 1 } ^ { N } \\sum _ { k = 1 , j \\neq k } ^ { N } e ^ { - \\langle I _ { j } ^ { e } , T _ { k } ^ { e } \\rangle } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/9aeb14c15ecc9ded2d517c638f8177fda1c5a997e450914e5b9f6646bfb05089.jpg",
+        "image_caption": [
+            "Figure 3: The gap between the performances of CLIP and CYCLIP is much larger in coarse-grained scenario highlighting better entity-level knowledge representation in CYCLIP. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We desire our models to achieve high alignment and uniformity scores so that the image-text representations are close for the matched pairs and better spread over the unit hypersphere for different categories. We analyze the effect of cross-modal and in-modal consistency on the alignment and uniformity of the shared representations. For this, we train two ablated versions of CYCLIP, 1) C-CYCLIP with only cross-modal consistency component i.e. $\\lambda _ { 1 } = 0 , \\lambda _ { 2 } = 0 . 5$ , and 2) I-CYCLIP with only in-modal consistency component i.e. $\\lambda _ { 1 } = 0 . 5 , \\lambda _ { 2 } = 0$ (in Eq. 5). We design proxy captions for classes as discussed in $\\ S 3 . 1$ to act as text embeddings. We present the results in Table 5. ",
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+        "type": "table",
+        "img_path": "images/b5390091bc120b42dba967fbbc26586d165649f5ccccc188b5c89f399f9709b8.jpg",
+        "table_caption": [
+            "Table 5: Alignment and Uniformity values for CLIP and Cyclic CLIP models. We abbreviate Alignment by A, Uniformity by U, and Zero-shot Top1 classification accuracy $( \\% )$ by ZS-Top1. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"3\">CIFAR-10</td><td colspan=\"3\">CIFAR-100</td><td colspan=\"3\">ImageNet1K</td></tr><tr><td>A</td><td>U</td><td>ZS-Top1</td><td>A</td><td>U</td><td>ZS-Top1</td><td>A</td><td>U</td><td>ZS-Top1</td></tr><tr><td>CLIP</td><td>0.36</td><td>-0.27</td><td>46.54</td><td>0.36</td><td>-0.25</td><td>18.69</td><td>0.39</td><td>-0.18</td><td>20.03</td></tr><tr><td>CYCLIP</td><td>0.36</td><td>-0.34</td><td>51.45</td><td>0.37</td><td>-0.33</td><td>23.15</td><td>0.38</td><td>-0.32</td><td>22.08</td></tr><tr><td>I-CYCLIP</td><td>0.60</td><td>-0.57</td><td>50.97</td><td>0.60</td><td>-0.57</td><td>22.35</td><td>0.61</td><td>-0.55</td><td>21.21</td></tr><tr><td>C-CYCLIP</td><td>0.05</td><td>-0.02</td><td>55.52</td><td>0.06</td><td>-0.02</td><td>25.49</td><td>0.07</td><td>-0.02</td><td>21.73</td></tr></table>",
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+        "type": "text",
+        "text": "We observe that I-CYCLIP learns representations that are better aligned in the representation space; however, they do not cover the hypersphere uniformly. The representations learned by C-CYCLIP are more uniformly spread but poorly aligned compared to I-CYCLIP. In this light, the components of CYCLIP can be seen to encourage a balance of good alignment and uniformity. Further, we find that CLIP is more uniform than CYCLIP in all datasets, but contrary to prior beliefs, this does not translate to improved downstream performance. C-CYCLIP has the best downstream zero-shot performance for CIFAR-10 and CIFAR-100 despite its poor alignment score. Further, all 3 variants of CYCLIP outperform CLIP on all 3 datasets, with CYCLIP performing the best on ImageNet1K. ",
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+        "type": "text",
+        "text": "4.4 Image-Text Retrieval ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We evaluate the effectiveness of the proposed method on the cross-modal (image to text and text to image) retrieval downstream task in the zero-shot as well as fine-tuned settings. We consider the standard benchmark datasets: Flickr30K [42] and MSCOCO [8]. We assess our models on the test set of Flickr30K (1K) and MSCOCO (5K) obtained from the well-known Karpathy [30] split. Both the datasets contains 5 paired captions per image that makes text retrieval per image more easier than image retrieval per caption. We confirm the same in our results below. We perform fine-tuning on the Karpathy’s training split with the batch size of 48. We fine-tune on Flick30K for 10 epochs and MSCOCO for 5 epochs. All the other hyperparameters are identical to that of pre-training. ",
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+        "type": "table",
+        "img_path": "images/bc71cea7a9c04b02057e4ecfdd88993a104a03027a36dd2368388bfdad7143d4.jpg",
+        "table_caption": [
+            "Table 6: Zero-shot and fine-tuned cross-modal image-text retrieval (text-to-image and image-to-text) results of CLIP and CYCLIP on Flick30K and MSCOCO datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\"></td><td rowspan=\"3\"></td><td colspan=\"4\">Flickr30K (1K)</td><td colspan=\"4\">MSCOCO (5K)</td></tr><tr><td colspan=\"2\">Text Retrieval</td><td colspan=\"2\">Image Retrieval</td><td colspan=\"2\">Text Retrieval</td><td colspan=\"2\">Image I Retrieval</td></tr><tr><td> R@1 R@5 R@10 R@1 R@5 R@10 R@1 R@5 R@10 R@1 R@5 R@10</td><td></td><td colspan=\"2\"></td><td></td><td></td><td></td><td></td></tr><tr><td>Zero-shot</td><td>CLIP CyCLIP</td><td>88.2 93.9 88.1 93.7</td><td>95.8</td><td>29.9 57.2</td><td>68.0</td><td>82.1 85.6</td><td>87.8</td><td>8.4 19.5</td><td>26.6</td></tr><tr><td></td><td></td><td></td><td></td><td>95.9 30.9</td><td>57.8</td><td>69.1 82.1</td><td>85.6</td><td>87.7</td><td>8.6 20.0</td><td>27.0</td></tr><tr><td>Fine-tuned</td><td>CLIP</td><td>91.9</td><td>97.0</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td>98.0 46.3</td><td>74.7</td><td>83.6 83.2</td><td>87.6</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>90.0</td><td>10.6 23.9</td><td>31.3</td></tr><tr><td></td><td>CYCLIP</td><td>92.3</td><td>97.0</td><td>98.4</td><td>47.3</td><td>83.2</td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>76.6</td><td>85.4</td><td>87.8</td><td>90.3</td><td>11.4 25.8</td><td>33.4</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 6 presents our cross-modal image-text retrieval results for CLIP and CYCLIP. In the zero-shot setting, we find that CYCLIP marginally outperforms CLIP on the image retrieval task on both the datasets. The relatively lower performance of both CLIP and CYCLIP in the zero-shot setting may be attributed to the more complicated nature of the two datasets where the models are expected to find similarities between the image and text at multiple resolutions as opposed to image classification where there is mostly single object to be matched with a simpler caption. It is not clear as to what distinctions in the raw input and text space are reflected in the embedding space too. Hence, we perform fine-tuning on both the datasets to better inform our models of the downstream datasets. In the fine-tuning setting, we find that the performance of both the models increases across both the datasets. However, we observe clear benefits of the soft consistent regularization on the image retrieval results for both the datasets. ",
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+    {
+        "type": "text",
+        "text": "4.5 CYCLIP preserves the Effective Robustness of CLIP ",
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+        "type": "text",
+        "text": "[36] shows that there is a strong correlation between the in-distribution and out-of-distribution generalization of the models trained on ImageNet1K, as illustrated by the linear fit (red) in Figure 4. Ideally, any model that does not undergo distribution shift would fall on the $y = x$ trendline (black). For other models, the deviations of the models from this ideal fit indicate their effective robustness. Previously, [45] showed that the zero-shot CLIP classifier trained on 400M image-text pairs improves effective robustness significantly compared to prior approaches to robustness. Subsequently, [28] demonstrated that CLIP models trained at small scales also exhibit high effective robustness that allows them to be used as a proxy to study the robustness properties of CLIP. ",
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+    {
+        "type": "image",
+        "img_path": "images/e98bb549238d19b78531727425dfe31fd161eaa892b75cfe6cd33676963af21f.jpg",
+        "image_caption": [
+            "Figure 4: Effect of varying the training dataset size on (a) Classification accuracy on ImageNet1K and (b) Effective Robustness on ImageNetV2. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We evaluate the effect of cyclic consistency on effective robustness. We trained 4 CLIP and CYCLIP models, varying the training dataset sizes from 500K to 4M image-text pairs from the $\\mathrm { C C 3 M + }$ CC12M datasets. In Figure 4, (a) we observe that for all training data sizes, CYCLIP shows a significant improvement over CLIP, showcasing its effectiveness in a diverse set of data regimes. Further, Figure 4 (b) shows that CYCLIP lies way above the baseline trend and preserves the effective robustness of CLIP. ",
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+    {
+        "type": "text",
+        "text": "5 Related Work ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our work fits into the broader theme of unsupervised pretraining with multiple modalities and has been successfully applied for learning representations of modalities such as images, text, and speech [2, 15, 1, 59, 43, 34]. Similar to the unimodal setting, two predominant approaches for multimodal pretraining are contrastive and generative, as described below. ",
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+    {
+        "type": "text",
+        "text": "Contrastive Representation Learning: Contrastive learning was originally proposed for selfsupervised representation learning in the unimodal context where the embeddings of a sample are brought closer to an augmented version of the sample. In contrast, the embeddings are pushed away for other samples, and their augmentation [11, 51, 39, 55, 21, 7, 16, 40, 66, 23, 18]. [63] and [3] impose additional constraints to remove redundancies and prevent dimensional collapse in the visual representations. Recently, contrastive learning has also been used to learn robust representations of the multimodal data [62, 47]. Many works use additional losses to imbibe extra supervisory multimodal knowledge during the training process [54, 65, 64, 14, 35]. In this work, we focus on having cyclic consistency in addition to the contrastive loss to learn more robust image-text representations. ",
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+    {
+        "type": "text",
+        "text": "Contrastive Language-Image Pretraining: CLIP [44], ALIGN [29] and BASIC [41] have enjoyed great success in extending contrastive learning to paired image-text data, with impressive zero-shot classification and robustness performance. These works have been further extended recently to include visual self-supervision [37], additional nearest neighbor supervision [33], and utilization of unpaired data [56]. Our work complements much of this literature as it identifies consistency regularizers that can be augmented to the learning objective of the above works. ",
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+    {
+        "type": "text",
+        "text": "Generative Representation Learning: Generative models have been applied for learning representations of multimodal data [60, 53]. In particular, [67, 61, 10] proposed a notion of cyclic consistency for learning from unpaired multimodal data using GANs [17], which was extended later to normalizing flows [20, 19]. While these works focus on regularizing a generative mapping between modalities, our notion of cycle consistency applies to embeddings learned via a contrastive framework. ",
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+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We presented CYCLIP, a framework for cycle consistent multimodal representation learning for image and text modality. The main benefits of CYCLIP stem from including cross-modal consistency and in-modal consistency regularizers to prevent inconsistent inference in the image and text spaces. Empirically, we show that CYCLIP performs much better than CLIP on zero-shot classification and is more robust on benchmarks for distributional robustness. We also showed that the representations learned by CYCLIP are more consistent than CLIP and better capture concept-level knowledge, as evidenced by our analysis of fine-grained and coarse-grained accuracies. ",
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+    {
+        "type": "text",
+        "text": "We believe this work can motivate further studies on understanding the geometry of the representation spaces learned via the contrastive objective applied to paired multimodal data and, in particular, identify conditions and regularization strategies under which the learned representations are synergistic across the various modalities for downstream applications. ",
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+        "type": "text",
+        "text": "One important future direction and a current limitation is scaling CYCLIP to larger datasets. While we do not possess the resources for this study, it is imperative to study the extent to which the benefits of cycle consistency remain at the scale on which the original CLIP was trained (400M image-text pairs). Finally, for real-world deployment of CLIP and their variants, such as CYCLIP, we need to be cautious about amplifying societal biases as these models are trained on large uncurated datasets scraped from the web [9]. Additionally, it is easy to add malicious data to the web, which poses a severe security threat [5]. Alleviating such harms is an important and active area of research. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "This research is supported by an Adobe Data Science Research Award for Aditya Grover. We would like to thank the IDRE’s Research Technology group for the GPU computing resources on the UCLA Hoffman2 Cluster. We also want to thank Tung Duc Nguyen, Satvik Mashkaria, Siddarth Krishnamoorthy, Varuni Sarwal, and Ashima Suvarna for their helpful suggestions. ",
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+    {
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parse/dev/ITw9edRDlD/ITw9edRDlD.md

@@ -0,0 +1,338 @@
+# Are Emergent Abilities of Large Language Models a Mirage?
+
+# Rylan Schaeffer
+
+Brando Miranda Computer Science Stanford University brando9@cs.stanford.edu
+
+Computer Science Stanford University rschaef@cs.stanford.edu
+
+Sanmi Koyejo   
+Computer Science   
+Stanford University   
+sanmi@cs.stanford.edu
+
+# Abstract
+
+Recent work claims that large language models display emergent abilities: abilities not present in smaller-scale models that are present in larger-scale models. What makes emergent abilities intriguing is two-fold: their sharpness, transitioning seemingly instantaneously from not present to present, and their unpredictability, appearing at seemingly unforeseeable model scales. Here, we present an alternative explanation for emergent abilities: for a particular task and model family, when analyzing fixed model outputs, emergent abilities appear due to the researcher’s choice of metric rather than due to fundamental changes in models with scale. Specifically, nonlinear or discontinuous metrics produce seemingly emergent abilities, whereas linear or continuous metrics produce smooth, continuous, predictable changes in model performance. We present our alternative explanation in a simple mathematical model, then test it in three complementary ways: we (1) make, test and confirm three predictions on the effect of metric choice using the InstructGPT/GPT-3 family on tasks with claimed emergent abilities; (2) make, test and confirm two predictions about metric choices in a meta-analysis of emergent abilities on the Beyond the Imitation Game Benchmark (BIG-Bench); and (3) show how to choose metrics to produce never-before-seen seemingly emergent abilities in multiple vision tasks across diverse deep network architectures. Via all three analyses, we provide evidence that emergent abilities disappear with different metrics or with better statistics, and may not be a fundamental property of scaling AI models.
+
+# 1 Introduction
+
+Emergent properties of complex systems have long been studied across disciplines, from physics to biology to mathematics. The idea of emergence was popularized by Nobel Prize-winning physicist P.W. Anderson’s “More Is Different" [1], which argues that as the complexity of a system increases, new properties may materialize that cannot be predicted even from a precise quantitative understanding of the system’s microscopic details. Recently, the idea of emergence gained significant attention in machine learning due to observations that large language models (LLMs) such as GPT [4], PaLM [7] and LaMDA [35] exhibit so-called “emergent abilities" [38, 9, 33, 4] (Fig. 1).
+
+The term “emergent abilities of LLMs" was recently and crisply defined as “abilities that are not present in smaller-scale models but are present in large-scale models; thus they cannot be predicted by simply extrapolating the performance improvements on smaller-scale models" [38]. Such emergent abilities were first discovered in the GPT-3 family [4]. Subsequent work emphasized the discovery, writing that “[although model] performance is predictable at a general level, performance on a specific task can sometimes emerge quite unpredictably and abruptly at scale" [9]. These quotations collectively identify the two defining properties of emergent abilities in LLMs:
+
+1. Sharpness, transitioning seemingly instantaneously from not present to present
+
+![](images/58619c355bf48fd739a8be2b0fcd8ad7e47c401ff272f138604393721c9fb37d.jpg)  
+Figure 1: Emergent abilities of large language models. Model families display sharp and unpredictable increases in performance at specific tasks as scale increases. Source: Fig. 2 from [38].
+
+2. Unpredictability, transitioning at seemingly unforeseeable model scales
+
+These emergent abilities have garnered significant interest, raising questions such as: What controls which abilities will emerge? What controls when abilities will emerge? How can we make desirable abilities emerge faster, and ensure undesirable abilities never emerge? These questions are especially pertinent to AI safety and alignment, as emergent abilities forewarn that larger models might one day, without warning, acquire undesired mastery over dangerous capabilities [34, 12, 19, 20].
+
+In this paper, we call into question the claim that LLMs possess emergent abilities, by which we specifically mean sharp and unpredictable changes in model outputs as a function of model scale on specific tasks. Our doubt stems from the observation that emergent abilities seem to appear only under metrics that nonlinearly or discontinuously scale any model’s per-token error rate. For instance, as we later show, $> 9 2 \%$ of emergent abilities on BIG-Bench tasks [33] (hand-annotated by [37]) appear under either of these two metrics:
+
+$$
+\begin{array} { r l } { \mathrm { M u l t i p l e ~ C h o i c e ~ G r a d e ~ { \stackrel { d e f } { = } } ~ } } & { { } { \left\{ \begin{array} { l l } { 1 } & { { \mathrm { i f ~ h i g h e s t ~ p r o b a b i l i t y ~ m a s s ~ o n ~ c o r r e c t ~ o p t i o n } } } \\ { 0 } & { { \mathrm { o t h e r w i s e } } } \end{array} \right. } } \\ { \mathrm { E x a c t ~ S t r i n g ~ M a t c h ~ { \stackrel { d e f } { = } } ~ } } & { { } { \left\{ \begin{array} { l l } { 1 } & { { \mathrm { i f ~ o u t p u t ~ s t r i n g ~ e x a c t l y ~ m a t c h e s ~ t a r g e t ~ s t r i n g } } } \\ { 0 } & { { \mathrm { o t h e r w i s e } } } \end{array} \right. } } \end{array}
+$$
+
+This raises the possibility of an alternative explanation for the origin of LLMs’ emergent abilities: sharp and unpredictable changes might be induced by the researcher’s choice of measurement, even though the model family’s per-token error rate changes smoothly, continuously and predictably with increasing scale. Specifically, our alternative posits that emergent abilities are a mirage caused primarily by the researcher choosing a metric that nonlinearly or discontinuously deforms per-token error rates, and secondarily by possessing too few test data to accurately estimate the performance of smaller models, thereby causing smaller models to appear wholly unable to perform the task.
+
+To communicate our alternative explanation, we present it as a simple mathematical model and demonstrate how it quantitatively reproduces the evidence offered in support of emergent abilities of LLMs. We then test our alternative explanation in three complementary ways:
+
+![](images/e3f92f76fe9724d253565131e8cb279499fb11c062052e48ec6575ba7683ccb7.jpg)  
+Figure 2: Emergent abilities of large language models are created by the researcher’s chosen metrics, not unpredictable changes in model behavior with scale. (A) Suppose the per-token cross-entropy loss decreases monotonically with model scale, e.g., $\mathcal { L } _ { C E }$ scales as a power law. (B) The per-token probability of selecting the correct token asymptotes towards 1. (C) If the researcher scores models’ outputs using a nonlinear metric such as Accuracy (which requires a sequence of tokens to all be correct), the metric choice nonlinearly scales performance, causing performance to change sharply and unpredictably in a manner that qualitatively matches published emergent abilities (inset). (D) If the researcher instead scores models’ outputs using a discontinuous metric such as Multiple Choice Grade (akin to a step function), the metric choice discontinuously scales performance, again causing performance to change sharply and unpredictably. (E) Changing from a nonlinear metric to a linear metric such as Token Edit Distance, scaling shows smooth, continuous and predictable improvements, ablating the emergent ability. (F) Changing from a discontinuous metric to a continuous metric such as Brier Score again reveals smooth, continuous and predictable improvements in task performance. Consequently, the observation of "emergent abilities" can be explained by the researcher’s choice of metrics, and does not require fundamental changes in model family behavior on specific tasks with scale.
+
+1. We make, test and confirm three predictions based on our alternative hypotheses using the InstructGPT [27] / GPT-3 [4] model family.   
+2. We meta-analyze published benchmarks [33, 38] to reveal that emergent abilities only appear for specific metrics, not for model families on particular tasks, and that changing the metric causes the emergence phenomenon to disappear.   
+3. We induce never-before-seen, seemingly emergent abilities in multiple architectures across various vision tasks by intentionally changing the metrics used for evaluation.
+
+# 2 Alternative Explanation for Emergent Abilities
+
+How might smooth, continuous, predictable changes in model family performance appear sharp and unpredictable? The answer is that the researcher’s choice of a nonlinear or discontinuous metric can distort the model family’s performance to appear sharp and unpredictable.
+
+To expound, suppose that within a model family, the test loss falls smoothly, continuously, and predictably with the number of model parameters. One reason to believe this is the phenomenon known as neural scaling laws: empirical observations that deep networks exhibit power law scaling in the test loss as a function of training dataset size, number of parameters or compute [15, 32, 13, 18,
+
+10, 14, 17, 39, 16, 8, 29]. For concreteness, suppose we have a model family of different numbers of parameters $N > 0$ and assume that each model’s per-token cross entropy falls as a power law with the number of parameters $N$ for constants $c > 0 , \alpha < 0$ (Fig. 2A):
+
+$$
+\mathcal { L } _ { C E } ( N ) = \left( \frac { N } { c } \right) ^ { \alpha }
+$$
+
+To be clear, we do not require this particular functional form to hold; rather, we use it for illustrative purposes. Let $V$ denote the set of possible tokens, $p$ denote the true but unknown probability distribution, and $\hat { p } _ { N }$ denote the $N$ -parameter model’s predicted probability distribution. The pertoken cross entropy as a function of number of parameters $N$ is:
+
+$$
+\mathcal { L } _ { C E } ( N ) \ \stackrel { \mathrm { d e f } } { = } \ - \sum _ { v \in V } p ( v ) \log \hat { p } _ { N } ( v )
+$$
+
+In practice, $p$ is unknown, so we substitute a one-hot distribution of the observed token $v ^ { * }$
+
+$$
+\mathcal { L } _ { C E } ( N ) = - \log \hat { p } _ { N } ( v ^ { * } )
+$$
+
+A model with $N$ parameters then has a per-token probability of selecting the correct token (Fig. 2B):
+
+$$
+p ( \mathrm { s i n g l e ~ t o k e n ~ c o r r e c t } ) = \exp \Big ( - \mathcal { L } _ { C E } ( N ) \Big ) = \exp \Big ( - \big ( N / c \big ) ^ { \alpha } \Big )
+$$
+
+Suppose the researcher then chooses a metric that requires selecting $L$ tokens correctly. For example, our task might be $L$ -digit integer addition, and a model’s output is scored 1 if all $L$ output digits exactly match all target digits with no additions, deletions or substitutions, 0 otherwise. If the probability each token is correct is independent1, the probability of scoring 1 is:
+
+$$
+\operatorname { A c c u r a c y } ( N ) \approx p _ { N } ( { \mathrm { s i n g l e ~ t o k e n ~ c o r r e c t } } ) ^ { \mathrm { n u m . ~ o f ~ t o k e n s } } = \exp { \Big ( } - ( N / c ) ^ { \alpha } { \Big ) } ^ { L }
+$$
+
+This choice of metric nonlinearly scales performance with increasing token sequence length. When plotting performance on a linear-log plot, one sees a sharp, unpredictable emergent ability on longer sequences (Fig. 2C) that closely matches claimed emergent abilities (inset). What happens if the researcher switches from a nonlinear metric like Accuracy, under which the per-token error rate scales geometrically in target length (App. A.3), to an approximately linear metric like Token Edit Distance, under which the per-token error rate scales quasi-linearly in target length (App. A.2)?
+
+$$
+{ \mathrm { T o k e n ~ E d i t ~ D i s t a n c e } } ( N ) \approx L \left( 1 - p _ { N } ( { \mathrm { s i n g l e ~ t o k e n ~ c o r r e c t } } ) \right) = L \left( 1 - \exp { \big ( } - ( N / c ) ^ { \alpha } { \big ) } \right)
+$$
+
+The linear metric reveals smooth, continuous, predictable changes in model performance (Fig. 2E). Similarly, if the researcher uses a discontinuous metric like Multiple Choice Grade, the researcher can find emergent abilities (Fig. 2D), but switching to a continuous metric like Brier Score removes such abilities (Fig. 2F). In summary, sharp and unpredictable changes with increasing scale can be fully explained by three interpretable factors: (1) the researcher choosing a metric that nonlinearly or discontinuously scales the per-token error rate, (2) having insufficient resolution to estimate model performance in the smaller parameter regime, with resolution2 set by 1/test dataset size, and (3) insufficiently sampling the larger parameter regime.
+
+# 3 Analyzing InstructGPT/GPT-3’s Emergent Arithmetic Abilities
+
+Previous papers prominently claimed the GPT [4, 27] family3 displays emergent abilities at integer arithmetic tasks [9, 33, 38] (Fig. 1A). We chose these tasks as they were prominently presented [4, 9, 33, 38], and we focused on the GPT family due to it being publicly queryable. As explained mathematically and visually in Sec. 2, our alternative explanation makes three predictions:
+
+![](images/bbb64d43dfd754116f39b15b55fbaf9e5565772cd7c477a77fef3cabb9c1d7e0.jpg)  
+Figure 3: Claimed emergent abilities evaporate upon changing the metric. Top: When performance is measured by a nonlinear metric (e.g., Accuracy), the InstructGPT/GPT-3 [4, 27] family’s performance appears sharp and unpredictable on longer target lengths. Bottom: When performance is instead measured by a linear metric (e.g., Token Edit Distance), the family exhibits smooth, predictable performance improvements.
+
+![](images/f69b19a9a3642de63e2e396a4a87ed592deb194782e67f78c0e5072ad4656e8b.jpg)  
+Figure 4: Claimed emergent abilities evaporate upon using better statistics. Based on the predictable effect Accuracy has on performance, measuring performance requires high resolution. Generating additional test data increases the resolution and reveals that even on Accuracy, the InstructGPT/GPT-3 family’s [4, 27] performance is above chance and improves in a smooth, continuous, predictable manner that qualitatively matches the mathematical model.
+
+1. Changing the metric from a nonlinear or discontinuous metric (Fig. 2CD) to a linear or continuous metric (Fig. 2 EF) should reveal smooth, continuous, predictable performance improvement with model scale.   
+2. For nonlinear metrics, increasing the resolution of measured model performance by increasing the test dataset size should reveal smooth, continuous, predictable model improvements commensurate with the predictable nonlinear effect of the chosen metric.   
+3. Regardless of metric, increasing the target string length should predictably affect the model’s performance as a function of the length-1 target performance: approximately geometrically for accuracy and approximately quasilinearly for token edit distance.
+
+To test these predictions, we collected outputs from the InstructGPT/GPT-3 family on two tasks: 2-shot multiplication between two 2-digit integers and 2-shot addition between two 4-digit integers.
+
+Prediction: Emergent Abilities Disappear With Different Metrics On both arithmetic tasks, the GPT family displays emergent abilities if the target has 4 or 5 digits and if the metric is Accuracy (Fig. 3, top) [4, 9, 38]. However, if one changes from nonlinear Accuracy to linear Token Edit Distance while keeping the models’ outputs fixed, the family’s performance smoothly, continuously and predictably improves with increasing scale (Fig. 3, bottom). This confirms our first prediction and supports our alternative explanation that the observation of emergent abilities can be explained by the researcher’s choice of metric, not changes in the model family’s outputs. We also observe that under Token Edit Distance, increasing the length of the target string from 1 to 5 predictably decreases the family’s performance in an approximately quasilinear manner, confirming the first half of our third prediction.
+
+Prediction: Emergent Abilities Disappear With Better Statistics We next tested our second prediction: that even on nonlinear metrics such as accuracy, smaller models do not have zero accuracy, but rather have non-zero above-chance accuracy commensurate with choosing to use accuracy as the metric. In order to accurately measure models’ accuracy, we increased the resolution by generating additional test data, and found that on both arithmetic tasks, all models in the InstructGPT/GPT-3 family achieve above-chance accuracy (Fig. 4). This confirms our second prediction. We also observe that as the target string length increases, the accuracy falls approximately geometrically with the length of the target string, confirming the second half of our third prediction. These results additionally demonstrate that the researcher’s choice of metric has the effect that one should predict accuracy to have, i.e., geometric decay with the target length.
+
+# 4 Meta-Analysis of Claimed Emergent Abilities
+
+Analyzing the GPT family is possible because the models are publicly queryable. However, at the time of this analysis, other model families claimed to exhibit emergent abilities are not publicly queryable, nor are their generated outputs publicly available, meaning we are limited to analyzing the published results themselves [9, 38, 37]. Our alternative explanation makes two predictions.
+
+1. At the “population level" of Task-Metric-Model Family triplets, emergent abilities should appear predominantly on specific metrics, not task-model family pairs, and specifically with nonlinear and/or discontinuous metrics. 2. On individual Task-Metric-Model Family triplets that display an emergent ability, changing the metric to a linear and/or continuous metric should remove the emergent ability.
+
+To test these predictions, we used claimed emergent abilities on BIG-Bench [33, 38] due to the benchmark being pertinent and publicly available.
+
+Prediction: Emergent Abilities Should Appear with Metrics, not Task-Model Families If emergent abilities are real, one should expect task-model family pairs to show emergence for all reasonable metrics. However, if our alternative explanation is correct, we should expect emergent abilities to appear only under certain metrics. To test this, we analyzed on which metrics emergent abilities appear. To determine whether a task-metric-model family triplet exhibits a possible emergent ability, we used a metric from previous work [33]. Letting $y _ { i } \in \mathbb { R }$ denote model performance at model scales $x _ { i } \in \mathbb { R }$ , sorted such that $x _ { i } < x _ { i + 1 }$ , the emergence score is:
+
+$$
+{ \begin{array} { r l r l } { \cdot { \operatorname { S c o r e } } { \Big ( } { \Big \{ } ( x _ { n } , y _ { n } ) { \Big \} } _ { n = 1 } ^ { N } { \Big ) } } & { } & { { \stackrel { \mathrm { d e f } } { = } } } & { { \frac { \operatorname { s i g n } ( \operatorname { a r g m a x } _ { i } y _ { i } - \operatorname { a r g m i n } _ { i } y _ { i } ) ( \operatorname* { m a x } _ { i } y _ { i } - \operatorname* { m i n } _ { i } y _ { i } ) } { \sqrt { { \mathsf { M e d i a n } } ( \{ ( y _ { i } - y _ { i - 1 } ) ^ { 2 } \} _ { i } ) } } } } \end{array} }
+$$
+
+We found that most metrics used in BIG-Bench have zero task-model family pairs that exhibit emergent abilities: of the 39 preferred metrics in BIG-Bench, at most 5 display emergence (Fig. 5A). Many of the 5 are nonlinear and/or discontinuous, e.g., Exact String Match, Multiple Choice Grade, ROUGE-L-Sum (App. A.4). Notably, because BIG-Bench often scores models on tasks using multiple metrics, the lack of emergent abilities under other metrics suggests that emergent abilities do not appear when model outputs are scored using other metrics.
+
+![](images/2151594b0144a294047300f651dcce9bfc628b1e2f77cedebc288ed61472039a.jpg)  
+Figure 5: Emergent abilities appear only for specific metrics, not task-model families. (A) Possible emergent abilities appear with at most 5 out of 39 BIG-Bench metrics. (B) Hand-annotated data by [37] reveal emergent abilities appear only under 4 preferred metrics. $\mathrm { ( C ) > 9 2 \% }$ of emergent abilities appear under one of two metrics: Multiple Choice Grade and Exact String Match.
+
+Because emergence score only suggests emergence, we also analyzed hand-annotated task-metricmodel family triplets [37], which revealed emergent abilities appear with $4 / 3 9$ metrics (Fig. 5B), and 2 metrics account for $> 9 2 \%$ of claimed emergent abilities (Fig. 5C): Multiple Choice Grade and Exact String Match. Multiple Choice Grade is discontinuous, and Exact String Match is nonlinear.
+
+Prediction: Changing Metric Removes Emergent Abilities To test our second prediction, we focused on the LaMDA family [35] because its outputs are available through BIG-Bench. We identified tasks on which LaMDA displays emergent abilities with Multiple Choice Grade, then asked whether LaMDA still displays emergent abilities on the same tasks with a different BIG-Bench metric: Brier Score [3]. Brier Score is a strictly proper scoring rule for predictions of mutually exclusive outcomes; for a binary outcome, the Brier Score simplifies to the squared error between 1 and the model’s probability mass on the outcome. LaMDA’s emergent abilities on the discontinuous Multiple Choice Grade disappeared when we changed the metric to the continuous Brier Score (Fig. 6). These results support our alternative explanation that emergent abilities are induced by the chosen metric.
+
+![](images/9f821e06113838668f620fece0360ab9f0a920b262835347b401956be4719242.jpg)  
+Figure 6: Changing the metric when evaluating task-model family pairs causes emergent abilities to disappear. Top: The LaMDA model family displays emergent abilities when measured under the discontinuous Multiple Choice Grade. Bottom: The LaMDA model family’s emergent abilities disappear when measured under a continuous BIG-Bench metric: Brier Score.
+
+# 5 Inducing Emergent Abilities in Networks on Vision Tasks
+
+To demonstrate how emergent abilities can be induced by the researcher’s choice of metric, we show how to produce emergent abilities in deep networks of various architectures: fully connected, convolutional, self-attentional. We focus on vision tasks because abrupt transitions in vision models’ capabilities have not been observed to the best of our knowledge; this is one reason why emergence in large language models is considered so interesting. For the convolutional example, see App. B.
+
+Emergent Reconstruction of CIFAR100 Natural Images by Nonlinear Autoencoders We first induce an emergent ability to reconstruct images in shallow (i.e., single hidden layer) nonlinear autoencoders trained on CIFAR100 natural images [21]. To emphasize that the sharpness of the metric is responsible for emergent abilities, and to show that sharpness extends to metrics beyond Accuracy, we intentionally define a discontinuous metric that measures a network’s ability to reconstruct a dataset as the average number of test data with squared reconstruction error below cutoff $c$ :
+
+$$
+\mathrm { R e c o n s t r u c t i o n } _ { c } \Big ( \{ x _ { n } \} _ { n = 1 } ^ { N } \Big ) \stackrel { \mathrm { \scriptsize ~ d e f } } { = } \frac { 1 } { N } \sum _ { n } \mathbb { I } \Big [ | | x _ { n } - \hat { x } _ { n } | | ^ { 2 } < c \Big ] ,
+$$
+
+where $\mathbb { I } ( \cdot )$ denotes an indicator variable and ${ \hat { x } } _ { n }$ is the autoencoder’s reconstruction of $x _ { n }$ . The autoencoder family displays smoothly decreasing squared reconstruction error as the number of bottleneck units increases (Fig. 7B). Under our newly defined Reconstructionc metric and for particular choices of $c$ , the autoencoder family exhibits a sharp and seemingly unpredictable image reconstruction ability (Fig. 7C) that qualitatively matches published emergent abilities (Fig. 7A).
+
+Emergent Classification of Omniglot Characters by Autoregressive Transformers We next induce emergent abilities in Transformers [36] trained to autoregressively classify Omniglot handwritten characters [22], in a setup inspired by recent work [6]: Omniglot images are embedded by convolutional layers, then sequences of embedded image-image class label pairs are fed into decoder-only transformers. We measure image classification performance on sequences of length $L \in [ 1 , 5 ]$ , again via subset accuracy: 1 if all $L$ images are classified correctly (Fig. 8B), 0 otherwise. Causal transformers display a seemingly emergent ability to correctly classify Omniglot handwritten characters (Fig. 8C) that qualitatively matches published emergent abilities (Fig. 8A).
+
+![](images/aeaa81388b37adea8775476623a02dd5dc3201b4f674b5d84db994bf94e56554.jpg)  
+Figure 7: Induced emergent reconstruction ability in shallow nonlinear autoencoders. (A) A published emergent ability at the BIG-Bench Periodic Elements task [33]. (B) Shallow nonlinear autoencoders trained on CIFAR100 [21] display smoothly decreasing mean squared reconstruction error. (C) Using a newly defined Reconstructionc metric (Eqn. 1) induces an unpredictable change.
+
+![](images/3489b3534ad2521f4b7d830b0635c6120f9030d86dee59d62ef68e143d62e50e.jpg)  
+Figure 8: Induced emergent classification ability in autoregressive Transformers. (A) A published emergent ability on the MMLU benchmark [9]. (B) Autoregressive transformers trained to classify Omniglot images display increasing accuracy with increasing scale. (C) When accuracy is redefined as classifying all images correctly, a seemingly emergent ability appears.
+
+# 6 Limitations
+
+This paper has several limitations. First, nothing in this paper should be interpreted as claiming that large language models cannot display emergent abilities; rather, our message is that some previously claimed emergent abilities appear to be mirages induced by researcher analyses. Second, our experiments and analyses are limited because some LLMs with claimed emergent abilities (e.g., PaLM 1, Gopher, Chinchilla) are private and not queryable at the time of our analysis. Lastly, the best metric(s) arguably depends on human preferences, which may exhibit qualitatively different behavior; we are unaware of studies quantifying whether human judgment is thresholded in an “emergent" way.
+
+# 7 Related Work
+
+Srivastava et al. [33] observed that while accuracy at a particular task can empirically appear sharp and unpredictable, cross-entropy does not appear so; the authors then discussed whether emergent abilities may be partially attributed to the metric. Our paper converts their discussion into precise predictions, then quantitatively tests the predictions to reveal metric choice is possibly responsible for some claimed emergent abilities; well-known and widely-used metrics (including metrics used by [33]) capture graded improvements; emergent abilities do not appear only on tasks involving multiple steps, such as the discontinuous Multiple Choice Grade; metric choice can be used to induce emergent abilities in a novel domain (vision) in diverse architectures and tasks.
+
+Alternative explanations exist for the origin of emergent abilities. Caballero et al. [5] explain emergence by assuming a piece-wise power law functional form; under this view, emergent abilities are real, caused by a “break" (or possibly multiple breaks) in the governing power law. In contrast, our work suggests that emergent abilities can be induced by the researcher under a single power law. Both explanations could be true: some emergent abilities might genuinely be abruptly appearing, whereas some emergent abilities might be attributable to the metric. Michaud et al. [28] posits that language modeling data might be comprised of discrete subtasks (“quanta”) that networks learn; if larger networks have greater capacity, and are thus more capable of learning more of these quanta, then if some downstream task requires a network to learn some combination of quanta, larger networks are more likely to have all the requisite capabilities and thus are capable of performing this downstream task. We think that this is a very interesting hypothesis. Whether language modeling data can or should be understood from this quantization perspective, and whether these quanta indeed are the origin of emergent abilities, are really exciting questions that we think merit more study.
+
+# 8 Discussion
+
+Our paper presents an alternative explanation for the claimed emergent abilities of large language models. For a fixed task and a fixed model family, the researcher can choose a metric to create an emergent ability or choose a metric to ablate an emergent ability. Ergo, emergent abilities may be creations of the researcher’s choices, not a fundamental property of the model family on the specific task.
+
+Our work has several implications. Firstly, a task and a metric are distinct and meaningful choices when constructing a benchmark. Secondly, when choosing metric(s), if the goal is to accurately predict scaling behavior, then one should consider the interplay between cross-entropy, transformations, and resolution-limited evaluations so that one isn’t surprised. As a corollary, continuous/linear metrics are probably better for accurate scaling forecasts, but if discontinuous/nonlinear metrics are preferred, then one may need a lot of data for sufficient resolution to accurately measure performance. The key is thinking through the consequences of one’s choices! Thirdly, when making claims about capabilities of large models, including proper controls is critical. In this particular setting, emergent abilities claims are possibly infected by a failure to control for multiple comparisons. In BIG-Bench alone, there are $\geq 2 2 0$ tasks, $\sim 4 0$ metrics per task, $\sim 1 0$ model families, for a total of $\sim 1 0 ^ { 6 }$ taskmetric-model family triplets, meaning the probability that no task-metric-model family triplet exhibits an emergent ability by random chance might be small. Fourthly, scientific progress can be hampered when models and their outputs are not made available for independent scientific investigation.
+
+# 9 Contributions
+
+RS conceived of the research direction collected data, ran experiments, and analyzed results. SK supervised and guided the project. BM also provided guidance. All authors helped write the manuscript.
+
+# 10 Acknowledgements
+
+This work is partially supported by the National Science Foundation under grants No. 2046795, 1934986, 2205329, NIH 1R01MH116226-01A, NIFA award 2020-67021-32799, the Alfred P. Sloan Foundation, and Google Inc. RS is partially supported by a Stanford Data Science Scholarship and BM is partially supported by a Stanford School of Engineering Fellowship and a Stanford EDGE Scholar Fellowship. We thank our colleagues Professor Tatsunori Hashimoto, Eric Han, Max Lamparth, Mikail Khona, Kateryna Pistunova, Victor Lecomte, and Zane Durante for discussing our findings with us and providing much-appreciated feedback.
+
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+
+# A Approximate Behavior of Metrics on Sequential Data
+
+How do different metrics behave when used to measure autoregressive model outputs? Precisely answering this question is tricky and possibly analytically unsolvable, so we provide an approximate answer here.
+
+Notationally, we consider $N$ test data of length $L$ (here, length is measured in tokens) with targets denoted $t _ { n } \ { \stackrel { \mathrm { d e f } } { = } } \ ( t _ { n 1 } , t _ { n 2 } , . . . t _ { n L } )$ , the autoregressive model has a true-but-unknown per-token error probability of $\epsilon \in [ 0 , 1 ]$ and the model outputs prediction $\boldsymbol { \hat { t } _ { n } } \ { \stackrel { \mathrm { d e f } } { = } } \ ( { \hat { t } _ { n 1 } } , { \hat { t } _ { n 2 } } , . . . { \hat { t } _ { n L } } )$ . This assumes that the model’s per-token error probability is constant, which is empirically false, but modeling the complex dependencies of errors is beyond our scope.
+
+# A.1 Per-Token Error Probability is Resolution-Limited
+
+Note that because we have $N$ test data, each of length $L$ , our resolution for viewing the per-token error probability $\epsilon$ is limited by $1 / N L$ . Here, resolution refers to “the smallest interval measurable by a scientific instrument; the resolving power." To explain what resolution means via an example, suppose one wants to measure a coin’s probability of yielding heads. After a single coin flip, only two outcomes are possible (H, T), so the resolution-limited probability of heads is either 0 or 1. After two coin flips, four outcomes are possible (HH, HT, TH, TT), so the resolution-limited probability of heads is now one of $0 , 0 . 5 , 1$ . After $F$ coin flips, we can only resolve the coin’s probability of yielding heads up to $1 / F$ . Consequently, we introduce a resolution-limited notation:
+
+$a _ { b } \ { \stackrel { \mathrm { d e f } } { = } } \ a$ rounded to the nearest integer multiple of $1 / b$
+
+# A.2 Token Edit Distance
+
+We first consider an adaptation of the Levenshtein (string edit) distance for models that function on tokens rather than characters, an adaptation we term the token edit distance. The token edit distance between two token sequences $t _ { n } , \hat { t _ { n } }$ is defined as the integer number of additions, deletions or substitutions necessary to transform $t _ { n }$ into $\hat { t } _ { n }$ (or vice versa).
+
+$$
+\begin{array} { l } { \mathrm { T o k e n ~ E d i t ~ D i s t a n c e } ( t _ { n } , \hat { t } _ { n } ) \overset { \mathrm { d e f } } { = } \mathrm { N u m ~ S u b s t i u t i o n s } + \mathrm { N u m . ~ A d d i t i o n s } + \mathrm { N u m . ~ D e l e t i o n s } } \\ { \displaystyle = \sum _ { \ell = 1 } ^ { L } \mathbb { I } [ t _ { n \ell } \neq \hat { t } _ { n \ell } ] + \mathrm { N u m . ~ A d d i t i o n s } + \mathrm { N u m . ~ D e l e t i o n s } } \\ { \displaystyle \quad \geq \sum _ { \ell = 1 } ^ { L } \mathbb { I } [ t _ { n \ell } \neq \hat { t } _ { n \ell } ] } \end{array}
+$$
+
+The expected token edit distance is therefore:
+
+$$
+\begin{array} { l } { \displaystyle \mathbb { E } [ \mathrm { T o k e n ~ E d i t ~ D i s t a n c e } ( t _ { n } , \hat { t } _ { n } ) ] \geq \mathbb { E } [ \sum _ { \ell = 1 } ^ { L } \mathbb { I } [ t _ { n \ell } \neq \hat { t } _ { n \ell } ] ] } \\ { \displaystyle = \sum _ { \ell = 1 } ^ { L } p ( t _ { n \ell } \neq \hat { t } _ { n \ell } ) } \\ { \approx L ( 1 - \epsilon ) } \end{array}
+$$
+
+The resolution-limited expected token edit distance is therefore:
+
+$$
+\mathbb { E } [ \mathrm { T o k e n ~ E d i t ~ D i s t a n c e } ( t _ { n } , \hat { t } _ { n } ) ] _ { N L } \ge L \Big ( 1 - \epsilon _ { N L } \Big )
+$$
+
+From this, we see that the expected token edit distance scales approximately linearly with the resolution-limited per-token probability. The real rate is slightly higher than linear because additions
+
+and deletions contribute an additional non-negative cost, but modeling this requires a model of how likely the model is to overproduce or underproduce tokens, which is something we do not currently possess.
+
+# A.3 Accuracy
+
+$$
+\begin{array} { l } { \displaystyle \mathrm { A c c u r a c y } ( t _ { n } , \hat { t } _ { n } ) \stackrel { \mathrm { d e f } } { = } \mathbb { I } [ \mathrm { N o \ a d d i t i o n s } ] \mathbb { I } [ \mathrm { N o \ d e l e t i o n s } ] \prod _ { l = 1 } ^ { L } \mathbb { I } [ t _ { n l } = \hat { t } _ { n l } ] } \\ { \displaystyle \approx \prod _ { l = 1 } ^ { L } \mathbb { I } [ t _ { n l } = \hat { t } _ { n l } ] } \end{array}
+$$
+
+As with the Token Edit Distance (App. A.2), we ignore how likely the language model is to overproduce or underproduce tokens because we do not have a good model of this process. Continuing along,
+
+$$
+\begin{array} { l } { \displaystyle \mathbb { E } [ \log \mathrm { A c c u r a c y } ] = \sum _ { l } \mathbb { E } [ \log \mathbb { I } [ t _ { n l } = \hat { t } _ { n l } ] ] } \\ { \displaystyle \qquad \leq \sum _ { l } \log \mathbb { E } [ \mathbb { I } [ t _ { n l } = \hat { t } _ { n l } ] ] } \\ { \displaystyle \qquad \approx L \log ( 1 - \epsilon ) } \end{array}
+$$
+
+Taking an approximation that would make most mathematicians cry:
+
+$$
+\begin{array} { r } { \mathbb { E } [ \mathrm { A c c u r a c y } ] \approx \exp ( \mathbb { E } [ \mathrm { l o g } \mathrm { A c c u r a c y } ] ) } \\ { = ( 1 - \epsilon ) ^ { L } } \end{array}
+$$
+
+This reveals that accuracy approximately falls geometrically with target token length. The resolutionlimited expected accuracy is therefore:
+
+$$
+\mathbb { E } [ \mathrm { A c c u r a c y } ] _ { N L } = ( 1 - \epsilon ) ^ { L } { } _ { N L }
+$$
+
+From this we can see that choosing a nonlinear metric like Accuracy is affected significantly more than a linear metric by limited resolution because Accuracy forces one to distinguish quantities that decay rapidly.
+
+# A.4 ROUGE-L-Sum
+
+Another BIG-Bench metric [33] is ROUGE-L-Sum [25], a metric based on the longest common subsequence (LCS) between two sequences. Section 3.2 of [25] gives the exact definition, but the key property is that ROUGE-L-Sum measures the “union" LCS, which means “stitching" together LCSs across the candidate and multiple references. As explained in the original paper [25]: if the candidate sequence is $c = w _ { 1 } w _ { 2 } w _ { 3 } w _ { 4 } w _ { 5 }$ , and if there are two reference sequences $r _ { 1 } = w _ { 1 } w _ { 2 } w _ { 6 } w _ { 7 } w _ { 8 }$ and $r _ { 2 } = w _ { 1 } w _ { 3 } w _ { 8 } w _ { 9 } w _ { 5 }$ , then $L C S ( r _ { 1 } , c ) = w _ { 1 } w _ { 2 }$ and $L C S ( r _ { 2 } , c ) = \overline { { w _ { 1 } w _ { 3 } w _ { 5 } } }$ , then the union LCS of $c , r _ { 1 } , r _ { 2 }$ is $w _ { 1 } w _ { 2 } w _ { 3 } w _ { 5 }$ , with length 4. Intuitively, this disproportionately benefits models with smaller error rates because their mistakes can be “stitched" across multiple references; this is confirmed in Monte Carlo simulation (Fig. 9).
+
+# A.5 BLEU
+
+Yet another BIG-Bench metric [33] is BLEU [30], a metric based on shared $\mathbf { n }$ -grams between the generated string and reference strings. BLEU is also a discontinuous and nonlinear metric for several reasons. For an explanation of its discontinuity, consider bleu.compute(predictions $=$ ["hello there general"], references=[["hello there general"]]). At first glance, this might seem like it should also result in a BLEU score of 1.0 since the prediction matches the reference. However, the issue here is the absence of longer n-grams. For the unigrams, bigrams, and trigrams, the precision is 1.0 since they match perfectly. However, for the 4-grams, there are none in both the candidate and the reference. This results in a precision of 0 for the 4-grams because the BLEU score takes the geometric mean of the n-gram precisions, meaning any 0 in the set will make the entire product 0. Hence, despite the match in unigrams, bigrams, and trigrams, the absence of 4-grams results in a BLEU score of 0.0. This behavior of BLEU has been a point of criticism, as short sentences or those with fewer n-grams than the maximum considered (often 4) can yield scores that are counter-intuitive. This is confirmed in Monte Carlo simulations (Fig. 10)
+
+![](images/b232dac27063b9946e0f425e9311d8f3af48091cb9b2c8555faa06ef2edfb1e7.jpg)  
+Figure 9: ROUGE-L-Sum is a sharp metric. Simulations show that as the per-token error probability slightly increases (e.g. from 0.05 to 0.1), the ROUGE-L-Sum metric falls sharply.
+
+![](images/38be85b71e7e7bada1a8b5a5180dbe5be8df06d1287fef6f83fcabce79b76a66.jpg)  
+Figure 10: BLEU is a sharp metric. Simulations show that as the per-token error probability slightly increases (e.g. from 0.05 to 0.1), the ROUGE-L-Sum metric falls sharply.
+
+# B Inducing Emergent Abilities in Networks on Vision Tasks
+
+# B.1 Emergent Classification of MNIST Handwritten Digits by Convolutional Networks
+
+We begin by inducing an emergent classification ability in a LeNet convolutional neural network family [24], trained on the MNIST handwritten digits dataset [23]. This family displays smoothly increasing test accuracy as the number of parameters increases (Fig. 11B). To emulate the accuracy metric used by emergence papers [9, 38, 33], we use subset accuracy: 1 if the network classifies $K$ out of $K$ (independent) test data correctly, 0 otherwise. Under this definition of accuracy, the model family displays an “emergent" ability to correctly classify sets of MNIST digits as $K$ increases from 1 to 5, especially when combined with sparse sampling of model sizes (Fig. 11C). This convolutional family’s emergent classification ability qualitatively matches published emergent abilities, e.g., at the BIG-Bench Grounded Mappings task [38] (Fig. 11A).
+
+![](images/38d1b3571deb00098b83f0272d6e8d14ac421eb06ebecf23348f75954da88ad8.jpg)  
+Figure 11: Induced emergent MNIST classification ability in convolutional networks. (A) A published emergent ability from the BIG-Bench Grounded Mappings task [38]. (B) LeNet trained on MNIST [23] displays a predictable, commonplace sigmoidal increase in test accuracy as model parameters increase. (C) When accuracy is redefined as correctly classifying $K$ out of $K$ independent test data, this newly defined metric induces a seemingly unpredictable change.
+
+# C Relationship Between Emergent Abilities and Grokking
+
+Emergent abilities [4, 9, 33, 38] are sometimes compared with grokking [31, 26, 2, 11], a phenomenon whereby a single model will, over the course of learning, achieve high training accuracy and only much later achieve high test accuracy. There are several differences between grokking and emergent abilities:
+
+1. Grokking is primarily studied within a single model, whereas emergent abilities are studied within a model family (i.e., multiple models).   
+2. Grokking occurs with increasing gradient steps, whereas emergent abilities occur with increasing model scale, typically measured in parameters or effective parameters (although more recently compute).   
+3. Grokking explicitly studies a discrepancy between the model’s train and test behavior, whereas emergent abilities (to the best of our knowledge) do not present separate train & test curves.   
+4. Grokking is primarily studied on toy “algorithmic" tasks in small networks, whereas emergent abilities are often studied on benchmark NLP tasks in large language models.

parse/dev/ITw9edRDlD/ITw9edRDlD_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "Are Emergent Abilities of Large Language Models a Mirage? ",
+        "text_level": 1,
+        "bbox": [
+            303,
+            122,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Rylan Schaeffer ",
+        "text_level": 1,
+        "bbox": [
+            230,
+            227,
+            343,
+            239
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Brando Miranda Computer Science Stanford University brando9@cs.stanford.edu ",
+        "bbox": [
+            406,
+            226,
+            606,
+            281
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Computer Science Stanford University rschaef@cs.stanford.edu ",
+        "bbox": [
+            187,
+            242,
+            385,
+            281
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Sanmi Koyejo   \nComputer Science   \nStanford University   \nsanmi@cs.stanford.edu ",
+        "bbox": [
+            627,
+            227,
+            810,
+            281
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            318,
+            535,
+            334
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recent work claims that large language models display emergent abilities: abilities not present in smaller-scale models that are present in larger-scale models. What makes emergent abilities intriguing is two-fold: their sharpness, transitioning seemingly instantaneously from not present to present, and their unpredictability, appearing at seemingly unforeseeable model scales. Here, we present an alternative explanation for emergent abilities: for a particular task and model family, when analyzing fixed model outputs, emergent abilities appear due to the researcher’s choice of metric rather than due to fundamental changes in models with scale. Specifically, nonlinear or discontinuous metrics produce seemingly emergent abilities, whereas linear or continuous metrics produce smooth, continuous, predictable changes in model performance. We present our alternative explanation in a simple mathematical model, then test it in three complementary ways: we (1) make, test and confirm three predictions on the effect of metric choice using the InstructGPT/GPT-3 family on tasks with claimed emergent abilities; (2) make, test and confirm two predictions about metric choices in a meta-analysis of emergent abilities on the Beyond the Imitation Game Benchmark (BIG-Bench); and (3) show how to choose metrics to produce never-before-seen seemingly emergent abilities in multiple vision tasks across diverse deep network architectures. Via all three analyses, we provide evidence that emergent abilities disappear with different metrics or with better statistics, and may not be a fundamental property of scaling AI models. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            647,
+            310,
+            664
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Emergent properties of complex systems have long been studied across disciplines, from physics to biology to mathematics. The idea of emergence was popularized by Nobel Prize-winning physicist P.W. Anderson’s “More Is Different\" [1], which argues that as the complexity of a system increases, new properties may materialize that cannot be predicted even from a precise quantitative understanding of the system’s microscopic details. Recently, the idea of emergence gained significant attention in machine learning due to observations that large language models (LLMs) such as GPT [4], PaLM [7] and LaMDA [35] exhibit so-called “emergent abilities\" [38, 9, 33, 4] (Fig. 1). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The term “emergent abilities of LLMs\" was recently and crisply defined as “abilities that are not present in smaller-scale models but are present in large-scale models; thus they cannot be predicted by simply extrapolating the performance improvements on smaller-scale models\" [38]. Such emergent abilities were first discovered in the GPT-3 family [4]. Subsequent work emphasized the discovery, writing that “[although model] performance is predictable at a general level, performance on a specific task can sometimes emerge quite unpredictably and abruptly at scale\" [9]. These quotations collectively identify the two defining properties of emergent abilities in LLMs: ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1. Sharpness, transitioning seemingly instantaneously from not present to present ",
+        "bbox": [
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+            902
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/58619c355bf48fd739a8be2b0fcd8ad7e47c401ff272f138604393721c9fb37d.jpg",
+        "image_caption": [
+            "Figure 1: Emergent abilities of large language models. Model families display sharp and unpredictable increases in performance at specific tasks as scale increases. Source: Fig. 2 from [38]. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            89,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2. Unpredictability, transitioning at seemingly unforeseeable model scales ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "These emergent abilities have garnered significant interest, raising questions such as: What controls which abilities will emerge? What controls when abilities will emerge? How can we make desirable abilities emerge faster, and ensure undesirable abilities never emerge? These questions are especially pertinent to AI safety and alignment, as emergent abilities forewarn that larger models might one day, without warning, acquire undesired mastery over dangerous capabilities [34, 12, 19, 20]. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we call into question the claim that LLMs possess emergent abilities, by which we specifically mean sharp and unpredictable changes in model outputs as a function of model scale on specific tasks. Our doubt stems from the observation that emergent abilities seem to appear only under metrics that nonlinearly or discontinuously scale any model’s per-token error rate. For instance, as we later show, $> 9 2 \\%$ of emergent abilities on BIG-Bench tasks [33] (hand-annotated by [37]) appear under either of these two metrics: ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/e40144d99c5950df6256a55196ba929f579447f60232c87be728623c6f571c7d.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\mathrm { M u l t i p l e ~ C h o i c e ~ G r a d e ~ { \\stackrel { d e f } { = } } ~ } } & { { } { \\left\\{ \\begin{array} { l l } { 1 } & { { \\mathrm { i f ~ h i g h e s t ~ p r o b a b i l i t y ~ m a s s ~ o n ~ c o r r e c t ~ o p t i o n } } } \\\\ { 0 } & { { \\mathrm { o t h e r w i s e } } } \\end{array} \\right. } } \\\\ { \\mathrm { E x a c t ~ S t r i n g ~ M a t c h ~ { \\stackrel { d e f } { = } } ~ } } & { { } { \\left\\{ \\begin{array} { l l } { 1 } & { { \\mathrm { i f ~ o u t p u t ~ s t r i n g ~ e x a c t l y ~ m a t c h e s ~ t a r g e t ~ s t r i n g } } } \\\\ { 0 } & { { \\mathrm { o t h e r w i s e } } } \\end{array} \\right. } } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "This raises the possibility of an alternative explanation for the origin of LLMs’ emergent abilities: sharp and unpredictable changes might be induced by the researcher’s choice of measurement, even though the model family’s per-token error rate changes smoothly, continuously and predictably with increasing scale. Specifically, our alternative posits that emergent abilities are a mirage caused primarily by the researcher choosing a metric that nonlinearly or discontinuously deforms per-token error rates, and secondarily by possessing too few test data to accurately estimate the performance of smaller models, thereby causing smaller models to appear wholly unable to perform the task. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To communicate our alternative explanation, we present it as a simple mathematical model and demonstrate how it quantitatively reproduces the evidence offered in support of emergent abilities of LLMs. We then test our alternative explanation in three complementary ways: ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "image",
+        "img_path": "images/e3f92f76fe9724d253565131e8cb279499fb11c062052e48ec6575ba7683ccb7.jpg",
+        "image_caption": [
+            "Figure 2: Emergent abilities of large language models are created by the researcher’s chosen metrics, not unpredictable changes in model behavior with scale. (A) Suppose the per-token cross-entropy loss decreases monotonically with model scale, e.g., $\\mathcal { L } _ { C E }$ scales as a power law. (B) The per-token probability of selecting the correct token asymptotes towards 1. (C) If the researcher scores models’ outputs using a nonlinear metric such as Accuracy (which requires a sequence of tokens to all be correct), the metric choice nonlinearly scales performance, causing performance to change sharply and unpredictably in a manner that qualitatively matches published emergent abilities (inset). (D) If the researcher instead scores models’ outputs using a discontinuous metric such as Multiple Choice Grade (akin to a step function), the metric choice discontinuously scales performance, again causing performance to change sharply and unpredictably. (E) Changing from a nonlinear metric to a linear metric such as Token Edit Distance, scaling shows smooth, continuous and predictable improvements, ablating the emergent ability. (F) Changing from a discontinuous metric to a continuous metric such as Brier Score again reveals smooth, continuous and predictable improvements in task performance. Consequently, the observation of \"emergent abilities\" can be explained by the researcher’s choice of metrics, and does not require fundamental changes in model family behavior on specific tasks with scale. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "1. We make, test and confirm three predictions based on our alternative hypotheses using the InstructGPT [27] / GPT-3 [4] model family.   \n2. We meta-analyze published benchmarks [33, 38] to reveal that emergent abilities only appear for specific metrics, not for model families on particular tasks, and that changing the metric causes the emergence phenomenon to disappear.   \n3. We induce never-before-seen, seemingly emergent abilities in multiple architectures across various vision tasks by intentionally changing the metrics used for evaluation. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 Alternative Explanation for Emergent Abilities ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "How might smooth, continuous, predictable changes in model family performance appear sharp and unpredictable? The answer is that the researcher’s choice of a nonlinear or discontinuous metric can distort the model family’s performance to appear sharp and unpredictable. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To expound, suppose that within a model family, the test loss falls smoothly, continuously, and predictably with the number of model parameters. One reason to believe this is the phenomenon known as neural scaling laws: empirical observations that deep networks exhibit power law scaling in the test loss as a function of training dataset size, number of parameters or compute [15, 32, 13, 18, ",
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+    {
+        "type": "text",
+        "text": "10, 14, 17, 39, 16, 8, 29]. For concreteness, suppose we have a model family of different numbers of parameters $N > 0$ and assume that each model’s per-token cross entropy falls as a power law with the number of parameters $N$ for constants $c > 0 , \\alpha < 0$ (Fig. 2A): ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8af0b76128070114c8b41becd05c431422903df72aa8755799c99fb78638ba59.jpg",
+        "text": "$$\n\\mathcal { L } _ { C E } ( N ) = \\left( \\frac { N } { c } \\right) ^ { \\alpha }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "To be clear, we do not require this particular functional form to hold; rather, we use it for illustrative purposes. Let $V$ denote the set of possible tokens, $p$ denote the true but unknown probability distribution, and $\\hat { p } _ { N }$ denote the $N$ -parameter model’s predicted probability distribution. The pertoken cross entropy as a function of number of parameters $N$ is: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d12d7e0942eee875ba02e247fdac2ce86743247347c9c86b6be84199e00a15af.jpg",
+        "text": "$$\n\\mathcal { L } _ { C E } ( N ) \\ \\stackrel { \\mathrm { d e f } } { = } \\ - \\sum _ { v \\in V } p ( v ) \\log \\hat { p } _ { N } ( v )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            379,
+            256,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In practice, $p$ is unknown, so we substitute a one-hot distribution of the observed token $v ^ { * }$ ",
+        "bbox": [
+            169,
+            297,
+            764,
+            313
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1ee9c99e7812e40c6978800126e5e620c636f1b53224d74a08f05c34d216602c.jpg",
+        "text": "$$\n\\mathcal { L } _ { C E } ( N ) = - \\log \\hat { p } _ { N } ( v ^ { * } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            411,
+            330,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A model with $N$ parameters then has a per-token probability of selecting the correct token (Fig. 2B): ",
+        "bbox": [
+            171,
+            354,
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+            371
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3b5832ac7d413a729a7b0b439ac6e558fb55d2924cc8f9f534d9caea6166a616.jpg",
+        "text": "$$\np ( \\mathrm { s i n g l e ~ t o k e n ~ c o r r e c t } ) = \\exp \\Big ( - \\mathcal { L } _ { C E } ( N ) \\Big ) = \\exp \\Big ( - \\big ( N / c \\big ) ^ { \\alpha } \\Big )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Suppose the researcher then chooses a metric that requires selecting $L$ tokens correctly. For example, our task might be $L$ -digit integer addition, and a model’s output is scored 1 if all $L$ output digits exactly match all target digits with no additions, deletions or substitutions, 0 otherwise. If the probability each token is correct is independent1, the probability of scoring 1 is: ",
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+        "img_path": "images/8b509ff42cbe3fe7c78e89bc130afcf7ca2207b38d561841d48ee943b4ca6f56.jpg",
+        "text": "$$\n\\operatorname { A c c u r a c y } ( N ) \\approx p _ { N } ( { \\mathrm { s i n g l e ~ t o k e n ~ c o r r e c t } } ) ^ { \\mathrm { n u m . ~ o f ~ t o k e n s } } = \\exp { \\Big ( } - ( N / c ) ^ { \\alpha } { \\Big ) } ^ { L }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "This choice of metric nonlinearly scales performance with increasing token sequence length. When plotting performance on a linear-log plot, one sees a sharp, unpredictable emergent ability on longer sequences (Fig. 2C) that closely matches claimed emergent abilities (inset). What happens if the researcher switches from a nonlinear metric like Accuracy, under which the per-token error rate scales geometrically in target length (App. A.3), to an approximately linear metric like Token Edit Distance, under which the per-token error rate scales quasi-linearly in target length (App. A.2)? ",
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+        "img_path": "images/97ad61b112eae0c20313f648501431aa4dc61eed1e7e8b1c4819d7f3e8289df0.jpg",
+        "text": "$$\n{ \\mathrm { T o k e n ~ E d i t ~ D i s t a n c e } } ( N ) \\approx L \\left( 1 - p _ { N } ( { \\mathrm { s i n g l e ~ t o k e n ~ c o r r e c t } } ) \\right) = L \\left( 1 - \\exp { \\big ( } - ( N / c ) ^ { \\alpha } { \\big ) } \\right)\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "The linear metric reveals smooth, continuous, predictable changes in model performance (Fig. 2E). Similarly, if the researcher uses a discontinuous metric like Multiple Choice Grade, the researcher can find emergent abilities (Fig. 2D), but switching to a continuous metric like Brier Score removes such abilities (Fig. 2F). In summary, sharp and unpredictable changes with increasing scale can be fully explained by three interpretable factors: (1) the researcher choosing a metric that nonlinearly or discontinuously scales the per-token error rate, (2) having insufficient resolution to estimate model performance in the smaller parameter regime, with resolution2 set by 1/test dataset size, and (3) insufficiently sampling the larger parameter regime. ",
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+        "type": "text",
+        "text": "3 Analyzing InstructGPT/GPT-3’s Emergent Arithmetic Abilities ",
+        "text_level": 1,
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+        "text": "Previous papers prominently claimed the GPT [4, 27] family3 displays emergent abilities at integer arithmetic tasks [9, 33, 38] (Fig. 1A). We chose these tasks as they were prominently presented [4, 9, 33, 38], and we focused on the GPT family due to it being publicly queryable. As explained mathematically and visually in Sec. 2, our alternative explanation makes three predictions: ",
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+        "image_caption": [
+            "Figure 3: Claimed emergent abilities evaporate upon changing the metric. Top: When performance is measured by a nonlinear metric (e.g., Accuracy), the InstructGPT/GPT-3 [4, 27] family’s performance appears sharp and unpredictable on longer target lengths. Bottom: When performance is instead measured by a linear metric (e.g., Token Edit Distance), the family exhibits smooth, predictable performance improvements. "
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+        "image_caption": [
+            "Figure 4: Claimed emergent abilities evaporate upon using better statistics. Based on the predictable effect Accuracy has on performance, measuring performance requires high resolution. Generating additional test data increases the resolution and reveals that even on Accuracy, the InstructGPT/GPT-3 family’s [4, 27] performance is above chance and improves in a smooth, continuous, predictable manner that qualitatively matches the mathematical model. "
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+        "type": "text",
+        "text": "1. Changing the metric from a nonlinear or discontinuous metric (Fig. 2CD) to a linear or continuous metric (Fig. 2 EF) should reveal smooth, continuous, predictable performance improvement with model scale.   \n2. For nonlinear metrics, increasing the resolution of measured model performance by increasing the test dataset size should reveal smooth, continuous, predictable model improvements commensurate with the predictable nonlinear effect of the chosen metric.   \n3. Regardless of metric, increasing the target string length should predictably affect the model’s performance as a function of the length-1 target performance: approximately geometrically for accuracy and approximately quasilinearly for token edit distance. ",
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+        "type": "text",
+        "text": "To test these predictions, we collected outputs from the InstructGPT/GPT-3 family on two tasks: 2-shot multiplication between two 2-digit integers and 2-shot addition between two 4-digit integers. ",
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+        "text": "Prediction: Emergent Abilities Disappear With Different Metrics On both arithmetic tasks, the GPT family displays emergent abilities if the target has 4 or 5 digits and if the metric is Accuracy (Fig. 3, top) [4, 9, 38]. However, if one changes from nonlinear Accuracy to linear Token Edit Distance while keeping the models’ outputs fixed, the family’s performance smoothly, continuously and predictably improves with increasing scale (Fig. 3, bottom). This confirms our first prediction and supports our alternative explanation that the observation of emergent abilities can be explained by the researcher’s choice of metric, not changes in the model family’s outputs. We also observe that under Token Edit Distance, increasing the length of the target string from 1 to 5 predictably decreases the family’s performance in an approximately quasilinear manner, confirming the first half of our third prediction. ",
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+        "type": "text",
+        "text": "Prediction: Emergent Abilities Disappear With Better Statistics We next tested our second prediction: that even on nonlinear metrics such as accuracy, smaller models do not have zero accuracy, but rather have non-zero above-chance accuracy commensurate with choosing to use accuracy as the metric. In order to accurately measure models’ accuracy, we increased the resolution by generating additional test data, and found that on both arithmetic tasks, all models in the InstructGPT/GPT-3 family achieve above-chance accuracy (Fig. 4). This confirms our second prediction. We also observe that as the target string length increases, the accuracy falls approximately geometrically with the length of the target string, confirming the second half of our third prediction. These results additionally demonstrate that the researcher’s choice of metric has the effect that one should predict accuracy to have, i.e., geometric decay with the target length. ",
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+        "type": "text",
+        "text": "4 Meta-Analysis of Claimed Emergent Abilities ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Analyzing the GPT family is possible because the models are publicly queryable. However, at the time of this analysis, other model families claimed to exhibit emergent abilities are not publicly queryable, nor are their generated outputs publicly available, meaning we are limited to analyzing the published results themselves [9, 38, 37]. Our alternative explanation makes two predictions. ",
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+    {
+        "type": "text",
+        "text": "1. At the “population level\" of Task-Metric-Model Family triplets, emergent abilities should appear predominantly on specific metrics, not task-model family pairs, and specifically with nonlinear and/or discontinuous metrics. 2. On individual Task-Metric-Model Family triplets that display an emergent ability, changing the metric to a linear and/or continuous metric should remove the emergent ability. ",
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+    {
+        "type": "text",
+        "text": "To test these predictions, we used claimed emergent abilities on BIG-Bench [33, 38] due to the benchmark being pertinent and publicly available. ",
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+        "type": "text",
+        "text": "Prediction: Emergent Abilities Should Appear with Metrics, not Task-Model Families If emergent abilities are real, one should expect task-model family pairs to show emergence for all reasonable metrics. However, if our alternative explanation is correct, we should expect emergent abilities to appear only under certain metrics. To test this, we analyzed on which metrics emergent abilities appear. To determine whether a task-metric-model family triplet exhibits a possible emergent ability, we used a metric from previous work [33]. Letting $y _ { i } \\in \\mathbb { R }$ denote model performance at model scales $x _ { i } \\in \\mathbb { R }$ , sorted such that $x _ { i } < x _ { i + 1 }$ , the emergence score is: ",
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+        "img_path": "images/fd58c559a7dbe1dd65712e151478cf16dcab40d1f576cbcec179385137efbd9a.jpg",
+        "text": "$$\n{ \\begin{array} { r l r l } { \\cdot { \\operatorname { S c o r e } } { \\Big ( } { \\Big \\{ } ( x _ { n } , y _ { n } ) { \\Big \\} } _ { n = 1 } ^ { N } { \\Big ) } } & { } & { { \\stackrel { \\mathrm { d e f } } { = } } } & { { \\frac { \\operatorname { s i g n } ( \\operatorname { a r g m a x } _ { i } y _ { i } - \\operatorname { a r g m i n } _ { i } y _ { i } ) ( \\operatorname* { m a x } _ { i } y _ { i } - \\operatorname* { m i n } _ { i } y _ { i } ) } { \\sqrt { { \\mathsf { M e d i a n } } ( \\{ ( y _ { i } - y _ { i - 1 } ) ^ { 2 } \\} _ { i } ) } } } } \\end{array} }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "We found that most metrics used in BIG-Bench have zero task-model family pairs that exhibit emergent abilities: of the 39 preferred metrics in BIG-Bench, at most 5 display emergence (Fig. 5A). Many of the 5 are nonlinear and/or discontinuous, e.g., Exact String Match, Multiple Choice Grade, ROUGE-L-Sum (App. A.4). Notably, because BIG-Bench often scores models on tasks using multiple metrics, the lack of emergent abilities under other metrics suggests that emergent abilities do not appear when model outputs are scored using other metrics. ",
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+        "img_path": "images/2151594b0144a294047300f651dcce9bfc628b1e2f77cedebc288ed61472039a.jpg",
+        "image_caption": [
+            "Figure 5: Emergent abilities appear only for specific metrics, not task-model families. (A) Possible emergent abilities appear with at most 5 out of 39 BIG-Bench metrics. (B) Hand-annotated data by [37] reveal emergent abilities appear only under 4 preferred metrics. $\\mathrm { ( C ) > 9 2 \\% }$ of emergent abilities appear under one of two metrics: Multiple Choice Grade and Exact String Match. "
+        ],
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+    {
+        "type": "text",
+        "text": "Because emergence score only suggests emergence, we also analyzed hand-annotated task-metricmodel family triplets [37], which revealed emergent abilities appear with $4 / 3 9$ metrics (Fig. 5B), and 2 metrics account for $> 9 2 \\%$ of claimed emergent abilities (Fig. 5C): Multiple Choice Grade and Exact String Match. Multiple Choice Grade is discontinuous, and Exact String Match is nonlinear. ",
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+        "text": "Prediction: Changing Metric Removes Emergent Abilities To test our second prediction, we focused on the LaMDA family [35] because its outputs are available through BIG-Bench. We identified tasks on which LaMDA displays emergent abilities with Multiple Choice Grade, then asked whether LaMDA still displays emergent abilities on the same tasks with a different BIG-Bench metric: Brier Score [3]. Brier Score is a strictly proper scoring rule for predictions of mutually exclusive outcomes; for a binary outcome, the Brier Score simplifies to the squared error between 1 and the model’s probability mass on the outcome. LaMDA’s emergent abilities on the discontinuous Multiple Choice Grade disappeared when we changed the metric to the continuous Brier Score (Fig. 6). These results support our alternative explanation that emergent abilities are induced by the chosen metric. ",
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+        "img_path": "images/9f821e06113838668f620fece0360ab9f0a920b262835347b401956be4719242.jpg",
+        "image_caption": [
+            "Figure 6: Changing the metric when evaluating task-model family pairs causes emergent abilities to disappear. Top: The LaMDA model family displays emergent abilities when measured under the discontinuous Multiple Choice Grade. Bottom: The LaMDA model family’s emergent abilities disappear when measured under a continuous BIG-Bench metric: Brier Score. "
+        ],
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+    {
+        "type": "text",
+        "text": "5 Inducing Emergent Abilities in Networks on Vision Tasks ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To demonstrate how emergent abilities can be induced by the researcher’s choice of metric, we show how to produce emergent abilities in deep networks of various architectures: fully connected, convolutional, self-attentional. We focus on vision tasks because abrupt transitions in vision models’ capabilities have not been observed to the best of our knowledge; this is one reason why emergence in large language models is considered so interesting. For the convolutional example, see App. B. ",
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+        "text": "Emergent Reconstruction of CIFAR100 Natural Images by Nonlinear Autoencoders We first induce an emergent ability to reconstruct images in shallow (i.e., single hidden layer) nonlinear autoencoders trained on CIFAR100 natural images [21]. To emphasize that the sharpness of the metric is responsible for emergent abilities, and to show that sharpness extends to metrics beyond Accuracy, we intentionally define a discontinuous metric that measures a network’s ability to reconstruct a dataset as the average number of test data with squared reconstruction error below cutoff $c$ : ",
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+        "img_path": "images/d10024623233c36dfd850f5da05c6889fbd52b7b00b95c5a9183a6c3f28e8126.jpg",
+        "text": "$$\n\\mathrm { R e c o n s t r u c t i o n } _ { c } \\Big ( \\{ x _ { n } \\} _ { n = 1 } ^ { N } \\Big ) \\stackrel { \\mathrm { \\scriptsize ~ d e f } } { = } \\frac { 1 } { N } \\sum _ { n } \\mathbb { I } \\Big [ | | x _ { n } - \\hat { x } _ { n } | | ^ { 2 } < c \\Big ] ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathbb { I } ( \\cdot )$ denotes an indicator variable and ${ \\hat { x } } _ { n }$ is the autoencoder’s reconstruction of $x _ { n }$ . The autoencoder family displays smoothly decreasing squared reconstruction error as the number of bottleneck units increases (Fig. 7B). Under our newly defined Reconstructionc metric and for particular choices of $c$ , the autoencoder family exhibits a sharp and seemingly unpredictable image reconstruction ability (Fig. 7C) that qualitatively matches published emergent abilities (Fig. 7A). ",
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+    {
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+        "text": "Emergent Classification of Omniglot Characters by Autoregressive Transformers We next induce emergent abilities in Transformers [36] trained to autoregressively classify Omniglot handwritten characters [22], in a setup inspired by recent work [6]: Omniglot images are embedded by convolutional layers, then sequences of embedded image-image class label pairs are fed into decoder-only transformers. We measure image classification performance on sequences of length $L \\in [ 1 , 5 ]$ , again via subset accuracy: 1 if all $L$ images are classified correctly (Fig. 8B), 0 otherwise. Causal transformers display a seemingly emergent ability to correctly classify Omniglot handwritten characters (Fig. 8C) that qualitatively matches published emergent abilities (Fig. 8A). ",
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+        "img_path": "images/aeaa81388b37adea8775476623a02dd5dc3201b4f674b5d84db994bf94e56554.jpg",
+        "image_caption": [
+            "Figure 7: Induced emergent reconstruction ability in shallow nonlinear autoencoders. (A) A published emergent ability at the BIG-Bench Periodic Elements task [33]. (B) Shallow nonlinear autoencoders trained on CIFAR100 [21] display smoothly decreasing mean squared reconstruction error. (C) Using a newly defined Reconstructionc metric (Eqn. 1) induces an unpredictable change. "
+        ],
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+        "img_path": "images/3489b3534ad2521f4b7d830b0635c6120f9030d86dee59d62ef68e143d62e50e.jpg",
+        "image_caption": [
+            "Figure 8: Induced emergent classification ability in autoregressive Transformers. (A) A published emergent ability on the MMLU benchmark [9]. (B) Autoregressive transformers trained to classify Omniglot images display increasing accuracy with increasing scale. (C) When accuracy is redefined as classifying all images correctly, a seemingly emergent ability appears. "
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+    {
+        "type": "text",
+        "text": "6 Limitations ",
+        "text_level": 1,
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+        "text": "This paper has several limitations. First, nothing in this paper should be interpreted as claiming that large language models cannot display emergent abilities; rather, our message is that some previously claimed emergent abilities appear to be mirages induced by researcher analyses. Second, our experiments and analyses are limited because some LLMs with claimed emergent abilities (e.g., PaLM 1, Gopher, Chinchilla) are private and not queryable at the time of our analysis. Lastly, the best metric(s) arguably depends on human preferences, which may exhibit qualitatively different behavior; we are unaware of studies quantifying whether human judgment is thresholded in an “emergent\" way. ",
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+        "type": "text",
+        "text": "7 Related Work ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Srivastava et al. [33] observed that while accuracy at a particular task can empirically appear sharp and unpredictable, cross-entropy does not appear so; the authors then discussed whether emergent abilities may be partially attributed to the metric. Our paper converts their discussion into precise predictions, then quantitatively tests the predictions to reveal metric choice is possibly responsible for some claimed emergent abilities; well-known and widely-used metrics (including metrics used by [33]) capture graded improvements; emergent abilities do not appear only on tasks involving multiple steps, such as the discontinuous Multiple Choice Grade; metric choice can be used to induce emergent abilities in a novel domain (vision) in diverse architectures and tasks. ",
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+    {
+        "type": "text",
+        "text": "Alternative explanations exist for the origin of emergent abilities. Caballero et al. [5] explain emergence by assuming a piece-wise power law functional form; under this view, emergent abilities are real, caused by a “break\" (or possibly multiple breaks) in the governing power law. In contrast, our work suggests that emergent abilities can be induced by the researcher under a single power law. Both explanations could be true: some emergent abilities might genuinely be abruptly appearing, whereas some emergent abilities might be attributable to the metric. Michaud et al. [28] posits that language modeling data might be comprised of discrete subtasks (“quanta”) that networks learn; if larger networks have greater capacity, and are thus more capable of learning more of these quanta, then if some downstream task requires a network to learn some combination of quanta, larger networks are more likely to have all the requisite capabilities and thus are capable of performing this downstream task. We think that this is a very interesting hypothesis. Whether language modeling data can or should be understood from this quantization perspective, and whether these quanta indeed are the origin of emergent abilities, are really exciting questions that we think merit more study. ",
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+    {
+        "type": "text",
+        "text": "8 Discussion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our paper presents an alternative explanation for the claimed emergent abilities of large language models. For a fixed task and a fixed model family, the researcher can choose a metric to create an emergent ability or choose a metric to ablate an emergent ability. Ergo, emergent abilities may be creations of the researcher’s choices, not a fundamental property of the model family on the specific task. ",
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+    {
+        "type": "text",
+        "text": "Our work has several implications. Firstly, a task and a metric are distinct and meaningful choices when constructing a benchmark. Secondly, when choosing metric(s), if the goal is to accurately predict scaling behavior, then one should consider the interplay between cross-entropy, transformations, and resolution-limited evaluations so that one isn’t surprised. As a corollary, continuous/linear metrics are probably better for accurate scaling forecasts, but if discontinuous/nonlinear metrics are preferred, then one may need a lot of data for sufficient resolution to accurately measure performance. The key is thinking through the consequences of one’s choices! Thirdly, when making claims about capabilities of large models, including proper controls is critical. In this particular setting, emergent abilities claims are possibly infected by a failure to control for multiple comparisons. In BIG-Bench alone, there are $\\geq 2 2 0$ tasks, $\\sim 4 0$ metrics per task, $\\sim 1 0$ model families, for a total of $\\sim 1 0 ^ { 6 }$ taskmetric-model family triplets, meaning the probability that no task-metric-model family triplet exhibits an emergent ability by random chance might be small. Fourthly, scientific progress can be hampered when models and their outputs are not made available for independent scientific investigation. ",
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+        "type": "text",
+        "text": "9 Contributions ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "RS conceived of the research direction collected data, ran experiments, and analyzed results. SK supervised and guided the project. BM also provided guidance. All authors helped write the manuscript. ",
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+    {
+        "type": "text",
+        "text": "10 Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
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+        "text": "This work is partially supported by the National Science Foundation under grants No. 2046795, 1934986, 2205329, NIH 1R01MH116226-01A, NIFA award 2020-67021-32799, the Alfred P. Sloan Foundation, and Google Inc. RS is partially supported by a Stanford Data Science Scholarship and BM is partially supported by a Stanford School of Engineering Fellowship and a Stanford EDGE Scholar Fellowship. We thank our colleagues Professor Tatsunori Hashimoto, Eric Han, Max Lamparth, Mikail Khona, Kateryna Pistunova, Victor Lecomte, and Zane Durante for discussing our findings with us and providing much-appreciated feedback. ",
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+    {
+        "type": "text",
+        "text": "References   \n[1] Philip W Anderson. More is different: broken symmetry and the nature of the hierarchical structure of science. Science, 177(4047):393–396, 1972.   \n[2] Boaz Barak, Benjamin Edelman, Surbhi Goel, Sham Kakade, Eran Malach, and Cyril Zhang. Hidden progress in deep learning: Sgd learns parities near the computational limit. Advances in Neural Information Processing Systems, 35:21750–21764, 2022.   \n[3] Glenn W Brier et al. Verification of forecasts expressed in terms of probability. Monthly weather review, 78(1):1–3, 1950.   \n[4] Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Language models are few-shot learners. Advances in neural information processing systems, 33:1877–1901, 2020.   \n[5] Ethan Caballero, Kshitij Gupta, Irina Rish, and David Krueger. Broken neural scaling laws. arXiv preprint arXiv:2210.14891, 2022.   \n[6] Stephanie CY Chan, Adam Santoro, Andrew Kyle Lampinen, Jane X Wang, Aaditya K Singh, Pierre Harvey Richemond, James McClelland, and Felix Hill. Data distributional properties drive emergent in-context learning in transformers. In Advances in Neural Information Processing Systems, 2022.   \n[7] Aakanksha Chowdhery, Sharan Narang, Jacob Devlin, Maarten Bosma, Gaurav Mishra, Adam Roberts, Paul Barham, Hyung Won Chung, Charles Sutton, Sebastian Gehrmann, et al. Palm: Scaling language modeling with pathways. arXiv preprint arXiv:2204.02311, 2022.   \n[8] Aidan Clark, Diego De Las Casas, Aurelia Guy, Arthur Mensch, Michela Paganini, Jordan Hoffmann, Bogdan Damoc, Blake Hechtman, Trevor Cai, Sebastian Borgeaud, et al. Unified scaling laws for routed language models. In International Conference on Machine Learning, pages 4057–4086. PMLR, 2022.   \n[9] Deep Ganguli, Danny Hernandez, Liane Lovitt, Amanda Askell, Yuntao Bai, Anna Chen, Tom Conerly, Nova Dassarma, Dawn Drain, Nelson Elhage, et al. Predictability and surprise in large generative models. In 2022 ACM Conference on Fairness, Accountability, and Transparency, pages 1747–1764, 2022.   \n[10] Mitchell A Gordon, Kevin Duh, and Jared Kaplan. Data and parameter scaling laws for neural machine translation. In Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing, pages 5915–5922, 2021.   \n[11] Andrey Gromov. Grokking modular arithmetic. arXiv preprint arXiv:2301.02679, 2023.   \n[12] Dan Hendrycks. Detecting emergent behavior. 2022.   \n[13] Tom Henighan, Jared Kaplan, Mor Katz, Mark Chen, Christopher Hesse, Jacob Jackson, Heewoo Jun, Tom B Brown, Prafulla Dhariwal, Scott Gray, et al. Scaling laws for autoregressive generative modeling. arXiv preprint arXiv:2010.14701, 2020.   \n[14] Danny Hernandez, Jared Kaplan, Tom Henighan, and Sam McCandlish. Scaling laws for transfer. arXiv preprint arXiv:2102.01293, 2021.   \n[15] Joel Hestness, Sharan Narang, Newsha Ardalani, Gregory Diamos, Heewoo Jun, Hassan Kianinejad, Md Patwary, Mostofa Ali, Yang Yang, and Yanqi Zhou. Deep learning scaling is predictable, empirically. arXiv preprint arXiv:1712.00409, 2017.   \n[16] Jordan Hoffmann, Sebastian Borgeaud, Arthur Mensch, Elena Buchatskaya, Trevor Cai, Eliza Rutherford, Diego de Las Casas, Lisa Anne Hendricks, Johannes Welbl, Aidan Clark, et al. Training compute-optimal large language models. arXiv preprint arXiv:2203.15556, 2022.   \n[17] Andy L Jones. Scaling scaling laws with board games. arXiv preprint arXiv:2104.03113, 2021. ",
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+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A Approximate Behavior of Metrics on Sequential Data ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "How do different metrics behave when used to measure autoregressive model outputs? Precisely answering this question is tricky and possibly analytically unsolvable, so we provide an approximate answer here. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Notationally, we consider $N$ test data of length $L$ (here, length is measured in tokens) with targets denoted $t _ { n } \\ { \\stackrel { \\mathrm { d e f } } { = } } \\ ( t _ { n 1 } , t _ { n 2 } , . . . t _ { n L } )$ , the autoregressive model has a true-but-unknown per-token error probability of $\\epsilon \\in [ 0 , 1 ]$ and the model outputs prediction $\\boldsymbol { \\hat { t } _ { n } } \\ { \\stackrel { \\mathrm { d e f } } { = } } \\ ( { \\hat { t } _ { n 1 } } , { \\hat { t } _ { n 2 } } , . . . { \\hat { t } _ { n L } } )$ . This assumes that the model’s per-token error probability is constant, which is empirically false, but modeling the complex dependencies of errors is beyond our scope. ",
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+    {
+        "type": "text",
+        "text": "A.1 Per-Token Error Probability is Resolution-Limited ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Note that because we have $N$ test data, each of length $L$ , our resolution for viewing the per-token error probability $\\epsilon$ is limited by $1 / N L$ . Here, resolution refers to “the smallest interval measurable by a scientific instrument; the resolving power.\" To explain what resolution means via an example, suppose one wants to measure a coin’s probability of yielding heads. After a single coin flip, only two outcomes are possible (H, T), so the resolution-limited probability of heads is either 0 or 1. After two coin flips, four outcomes are possible (HH, HT, TH, TT), so the resolution-limited probability of heads is now one of $0 , 0 . 5 , 1$ . After $F$ coin flips, we can only resolve the coin’s probability of yielding heads up to $1 / F$ . Consequently, we introduce a resolution-limited notation: ",
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+    {
+        "type": "text",
+        "text": "$a _ { b } \\ { \\stackrel { \\mathrm { d e f } } { = } } \\ a$ rounded to the nearest integer multiple of $1 / b$ ",
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+    {
+        "type": "text",
+        "text": "A.2 Token Edit Distance ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We first consider an adaptation of the Levenshtein (string edit) distance for models that function on tokens rather than characters, an adaptation we term the token edit distance. The token edit distance between two token sequences $t _ { n } , \\hat { t _ { n } }$ is defined as the integer number of additions, deletions or substitutions necessary to transform $t _ { n }$ into $\\hat { t } _ { n }$ (or vice versa). ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/25889826914510fd5999570fda3860a6fdf394b61d83134a7bd26d56c641e5c6.jpg",
+        "text": "$$\n\\begin{array} { l } { \\mathrm { T o k e n ~ E d i t ~ D i s t a n c e } ( t _ { n } , \\hat { t } _ { n } ) \\overset { \\mathrm { d e f } } { = } \\mathrm { N u m ~ S u b s t i u t i o n s } + \\mathrm { N u m . ~ A d d i t i o n s } + \\mathrm { N u m . ~ D e l e t i o n s } } \\\\ { \\displaystyle = \\sum _ { \\ell = 1 } ^ { L } \\mathbb { I } [ t _ { n \\ell } \\neq \\hat { t } _ { n \\ell } ] + \\mathrm { N u m . ~ A d d i t i o n s } + \\mathrm { N u m . ~ D e l e t i o n s } } \\\\ { \\displaystyle \\quad \\geq \\sum _ { \\ell = 1 } ^ { L } \\mathbb { I } [ t _ { n \\ell } \\neq \\hat { t } _ { n \\ell } ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The expected token edit distance is therefore: ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e0243250ca9c28b579b35ffb406027eb4dd45b7f519cc9e46cf77b8cc90342d9.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathbb { E } [ \\mathrm { T o k e n ~ E d i t ~ D i s t a n c e } ( t _ { n } , \\hat { t } _ { n } ) ] \\geq \\mathbb { E } [ \\sum _ { \\ell = 1 } ^ { L } \\mathbb { I } [ t _ { n \\ell } \\neq \\hat { t } _ { n \\ell } ] ] } \\\\ { \\displaystyle = \\sum _ { \\ell = 1 } ^ { L } p ( t _ { n \\ell } \\neq \\hat { t } _ { n \\ell } ) } \\\\ { \\approx L ( 1 - \\epsilon ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The resolution-limited expected token edit distance is therefore: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/2e488cd5d0123faca69d3ad78b6b5874fe2c1df04992be3b932ab2f92c00f572.jpg",
+        "text": "$$\n\\mathbb { E } [ \\mathrm { T o k e n ~ E d i t ~ D i s t a n c e } ( t _ { n } , \\hat { t } _ { n } ) ] _ { N L } \\ge L \\Big ( 1 - \\epsilon _ { N L } \\Big )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "From this, we see that the expected token edit distance scales approximately linearly with the resolution-limited per-token probability. The real rate is slightly higher than linear because additions ",
+        "bbox": [
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+        "page_idx": 13
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+    {
+        "type": "text",
+        "text": "and deletions contribute an additional non-negative cost, but modeling this requires a model of how likely the model is to overproduce or underproduce tokens, which is something we do not currently possess. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A.3 Accuracy ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/a17689acc3ce9e552ab8b7a81583f8a3ebe58973b339b735aaa2567308af5ee4.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathrm { A c c u r a c y } ( t _ { n } , \\hat { t } _ { n } ) \\stackrel { \\mathrm { d e f } } { = } \\mathbb { I } [ \\mathrm { N o \\ a d d i t i o n s } ] \\mathbb { I } [ \\mathrm { N o \\ d e l e t i o n s } ] \\prod _ { l = 1 } ^ { L } \\mathbb { I } [ t _ { n l } = \\hat { t } _ { n l } ] } \\\\ { \\displaystyle \\approx \\prod _ { l = 1 } ^ { L } \\mathbb { I } [ t _ { n l } = \\hat { t } _ { n l } ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "As with the Token Edit Distance (App. A.2), we ignore how likely the language model is to overproduce or underproduce tokens because we do not have a good model of this process. Continuing along, ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d16bd2849e775991762208e8b49a2f7a38216d53692d7d4f7b27279d9ad989af.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathbb { E } [ \\log \\mathrm { A c c u r a c y } ] = \\sum _ { l } \\mathbb { E } [ \\log \\mathbb { I } [ t _ { n l } = \\hat { t } _ { n l } ] ] } \\\\ { \\displaystyle \\qquad \\leq \\sum _ { l } \\log \\mathbb { E } [ \\mathbb { I } [ t _ { n l } = \\hat { t } _ { n l } ] ] } \\\\ { \\displaystyle \\qquad \\approx L \\log ( 1 - \\epsilon ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Taking an approximation that would make most mathematicians cry: ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2a0b37bcf22a17705eb78cf5e057a2c33e380e7c33abe875a1f08da0f76248e3.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbb { E } [ \\mathrm { A c c u r a c y } ] \\approx \\exp ( \\mathbb { E } [ \\mathrm { l o g } \\mathrm { A c c u r a c y } ] ) } \\\\ { = ( 1 - \\epsilon ) ^ { L } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "This reveals that accuracy approximately falls geometrically with target token length. The resolutionlimited expected accuracy is therefore: ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6df8e9b5949e4aab27210961a1263e3d0c8f4e152a055e4a5272939a20d94f46.jpg",
+        "text": "$$\n\\mathbb { E } [ \\mathrm { A c c u r a c y } ] _ { N L } = ( 1 - \\epsilon ) ^ { L } { } _ { N L }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            393,
+            604,
+            602,
+            625
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "From this we can see that choosing a nonlinear metric like Accuracy is affected significantly more than a linear metric by limited resolution because Accuracy forces one to distinguish quantities that decay rapidly. ",
+        "bbox": [
+            174,
+            632,
+            825,
+            675
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.4 ROUGE-L-Sum ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            690,
+            330,
+            705
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Another BIG-Bench metric [33] is ROUGE-L-Sum [25], a metric based on the longest common subsequence (LCS) between two sequences. Section 3.2 of [25] gives the exact definition, but the key property is that ROUGE-L-Sum measures the “union\" LCS, which means “stitching\" together LCSs across the candidate and multiple references. As explained in the original paper [25]: if the candidate sequence is $c = w _ { 1 } w _ { 2 } w _ { 3 } w _ { 4 } w _ { 5 }$ , and if there are two reference sequences $r _ { 1 } = w _ { 1 } w _ { 2 } w _ { 6 } w _ { 7 } w _ { 8 }$ and $r _ { 2 } = w _ { 1 } w _ { 3 } w _ { 8 } w _ { 9 } w _ { 5 }$ , then $L C S ( r _ { 1 } , c ) = w _ { 1 } w _ { 2 }$ and $L C S ( r _ { 2 } , c ) = \\overline { { w _ { 1 } w _ { 3 } w _ { 5 } } }$ , then the union LCS of $c , r _ { 1 } , r _ { 2 }$ is $w _ { 1 } w _ { 2 } w _ { 3 } w _ { 5 }$ , with length 4. Intuitively, this disproportionately benefits models with smaller error rates because their mistakes can be “stitched\" across multiple references; this is confirmed in Monte Carlo simulation (Fig. 9). ",
+        "bbox": [
+            173,
+            715,
+            825,
+            842
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.5 BLEU ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            857,
+            261,
+            872
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Yet another BIG-Bench metric [33] is BLEU [30], a metric based on shared $\\mathbf { n }$ -grams between the generated string and reference strings. BLEU is also a discontinuous and nonlinear metric for several reasons. For an explanation of its discontinuity, consider bleu.compute(predictions $=$ [\"hello there general\"], references=[[\"hello there general\"]]). At first glance, this might seem like it should also result in a BLEU score of 1.0 since the prediction matches the reference. However, the issue here is the absence of longer n-grams. For the unigrams, bigrams, and trigrams, the precision is 1.0 since they match perfectly. However, for the 4-grams, there are none in both the candidate and the reference. This results in a precision of 0 for the 4-grams because the BLEU score takes the geometric mean of the n-gram precisions, meaning any 0 in the set will make the entire product 0. Hence, despite the match in unigrams, bigrams, and trigrams, the absence of 4-grams results in a BLEU score of 0.0. This behavior of BLEU has been a point of criticism, as short sentences or those with fewer n-grams than the maximum considered (often 4) can yield scores that are counter-intuitive. This is confirmed in Monte Carlo simulations (Fig. 10) ",
+        "bbox": [
+            174,
+            882,
+            821,
+            911
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/b232dac27063b9946e0f425e9311d8f3af48091cb9b2c8555faa06ef2edfb1e7.jpg",
+        "image_caption": [
+            "Figure 9: ROUGE-L-Sum is a sharp metric. Simulations show that as the per-token error probability slightly increases (e.g. from 0.05 to 0.1), the ROUGE-L-Sum metric falls sharply. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            192,
+            92,
+            802,
+            270
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/38be85b71e7e7bada1a8b5a5180dbe5be8df06d1287fef6f83fcabce79b76a66.jpg",
+        "image_caption": [
+            "Figure 10: BLEU is a sharp metric. Simulations show that as the per-token error probability slightly increases (e.g. from 0.05 to 0.1), the ROUGE-L-Sum metric falls sharply. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            194,
+            332,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            582,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "B Inducing Emergent Abilities in Networks on Vision Tasks ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            755,
+            687,
+            772
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "B.1 Emergent Classification of MNIST Handwritten Digits by Convolutional Networks ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            787,
+            787,
+            803
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We begin by inducing an emergent classification ability in a LeNet convolutional neural network family [24], trained on the MNIST handwritten digits dataset [23]. This family displays smoothly increasing test accuracy as the number of parameters increases (Fig. 11B). To emulate the accuracy metric used by emergence papers [9, 38, 33], we use subset accuracy: 1 if the network classifies $K$ out of $K$ (independent) test data correctly, 0 otherwise. Under this definition of accuracy, the model family displays an “emergent\" ability to correctly classify sets of MNIST digits as $K$ increases from 1 to 5, especially when combined with sparse sampling of model sizes (Fig. 11C). This convolutional family’s emergent classification ability qualitatively matches published emergent abilities, e.g., at the BIG-Bench Grounded Mappings task [38] (Fig. 11A). ",
+        "bbox": [
+            174,
+            814,
+            823,
+            911
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/38d1b3571deb00098b83f0272d6e8d14ac421eb06ebecf23348f75954da88ad8.jpg",
+        "image_caption": [
+            "Figure 11: Induced emergent MNIST classification ability in convolutional networks. (A) A published emergent ability from the BIG-Bench Grounded Mappings task [38]. (B) LeNet trained on MNIST [23] displays a predictable, commonplace sigmoidal increase in test accuracy as model parameters increase. (C) When accuracy is redefined as correctly classifying $K$ out of $K$ independent test data, this newly defined metric induces a seemingly unpredictable change. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            89,
+            821,
+            251
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            358,
+            823,
+            387
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "C Relationship Between Emergent Abilities and Grokking ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            405,
+            676,
+            424
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Emergent abilities [4, 9, 33, 38] are sometimes compared with grokking [31, 26, 2, 11], a phenomenon whereby a single model will, over the course of learning, achieve high training accuracy and only much later achieve high test accuracy. There are several differences between grokking and emergent abilities: ",
+        "bbox": [
+            174,
+            438,
+            826,
+            492
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "1. Grokking is primarily studied within a single model, whereas emergent abilities are studied within a model family (i.e., multiple models).   \n2. Grokking occurs with increasing gradient steps, whereas emergent abilities occur with increasing model scale, typically measured in parameters or effective parameters (although more recently compute).   \n3. Grokking explicitly studies a discrepancy between the model’s train and test behavior, whereas emergent abilities (to the best of our knowledge) do not present separate train & test curves.   \n4. Grokking is primarily studied on toy “algorithmic\" tasks in small networks, whereas emergent abilities are often studied on benchmark NLP tasks in large language models. ",
+        "bbox": [
+            210,
+            505,
+            825,
+            659
+        ],
+        "page_idx": 16
+    }
+]

parse/dev/SlxSY2UZQT/SlxSY2UZQT.md

@@ -0,0 +1,336 @@
+# LABEL-EFFICIENT SEMANTIC SEGMENTATION WITH DIFFUSION MODELS
+
+Dmitry Baranchuk, Ivan Rubachev, Andrey Voynov, Valentin Khrulkov, Artem Babenko
+
+Yandex Research
+
+# ABSTRACT
+
+Denoising diffusion probabilistic models have recently received much research attention since they outperform alternative approaches, such as GANs, and currently provide state-of-the-art generative performance. The superior performance of diffusion models has made them an appealing tool in several applications, including inpainting, super-resolution, and semantic editing. In this paper, we demonstrate that diffusion models can also serve as an instrument for semantic segmentation, especially in the setup when labeled data is scarce. In particular, for several pretrained diffusion models, we investigate the intermediate activations from the networks that perform the Markov step of the reverse diffusion process. We show that these activations effectively capture the semantic information from an input image and appear to be excellent pixel-level representations for the segmentation problem. Based on these observations, we describe a simple segmentation method, which can work even if only a few training images are provided. Our approach significantly outperforms the existing alternatives on several datasets for the same amount of human supervision. The source code of the project is publicly available.
+
+# 1 INTRODUCTION
+
+Denoising diffusion probabilistic models (DDPM) (Sohl-Dickstein et al., 2015; Ho et al., 2020) have recently outperformed alternative approaches to model the distribution of natural images both in the realism of individual samples and their diversity (Dhariwal & Nichol, 2021). These advantages of DDPM are successfully exploited in applications, such as colorization (Song et al., 2021), inpainting (Song et al., 2021), super-resolution (Saharia et al., 2021; Li et al., 2021b), and semantic editing (Meng et al., 2021), where DDPM often achieve more impressive results compared to GANs.
+
+So far, however, DDPM were not exploited as a source of effective image representations for discriminative computer vision problems. While the prior literature has demonstrated that various generative paradigms, such as GANs (Donahue & Simonyan, 2019) or autoregressive models (Chen et al., 2020a), can be used to extract the representations for common vision tasks, it is not clear if DDPM can also serve as representation learners. In this paper, we provide an affirmative answer to this question in the context of semantic segmentation.
+
+In particular, we investigate the intermediate activations from the U-Net network that approximates the Markov step of the reverse diffusion process in DDPM. Intuitively, this network learns to denoise its input, and it is not clear why the intermediate activations should capture semantic information needed for high-level vision problems. Nevertheless, we show that on certain diffusion steps, these activations do capture such information, and therefore, can potentially be used as image representations for downstream tasks. Given these observations, we propose a simple semantic segmentation method, which exploits these representations and works successfully even if only a few labeled images are provided. On several datasets, we show that our DDPM-based segmentation method outperforms the existing baselines for the same amount of supervision.
+
+To sum up, the contributions of our paper are:
+
+1. We investigate the representations learned by the state-of-the-art DDPM and show that they capture high-level semantic information valuable for downstream vision tasks.
+
+2. We design a simple semantic segmentation approach that exploits these representations and outperforms the alternatives in the few-shot operating point.
+
+3. We compare the DDPM-based representations with their GAN-based counterparts on the same datasets and demonstrate the advantages of the former in the context of semantic segmentation.
+
+# 2 RELATED WORK
+
+In this section, we briefly describe the existing lines of research relevant to our work.
+
+Diffusion models (Sohl-Dickstein et al., 2015; Ho et al., 2020) are a class of generative models that approximate the distribution of real images by the endpoint of the Markov chain which originates from a simple parametric distribution, typically a standard Gaussian. Each Markov step is modeled by a deep neural network that effectively learns to invert the diffusion process with a known Gaussian kernel. Ho et al. highlighted the equivalence of diffusion models and score matching (Song & Ermon, 2019; 2020), showing them to be two different perspectives on the gradual conversion of a simple known distribution into a target distribution via the iterative denoising process. Very recent works (Nichol, 2021; Dhariwal & Nichol, 2021) have developed more powerful model architectures as well as different advanced objectives, which led to the “victory” of DDPM over GANs in terms of generative quality and diversity. DDPM have been widely used in several applications, including image colorization (Song et al., 2021), super-resolution (Saharia et al., 2021; Li et al., 2021b), inpainting (Song et al., 2021), and semantic editing (Meng et al., 2021). In our work, we demonstrate that one can also successfully use them for semantic segmentation.
+
+Image segmentation with generative models is an active research direction at the moment, however, existing methods are primarily based on GANs. The first line of works (Voynov & Babenko, 2020; Voynov et al., 2021; Melas-Kyriazi et al., 2021) is based on the evidence that the latent spaces of the state-of-the-art GANs have directions corresponding to effects that influence the foreground/background pixels differently, which allows producing synthetic data to train segmentation models. However, these approaches are currently able to perform binary segmentation only, and it is not clear if they can be used in the general setup of semantic segmentation. The second line of works (Zhang et al., 2021; Tritrong et al., 2021; Xu, 2021; Galeev et al., 2020) is more relevant to our study since they are based on the intermediate representations obtained in GANs. In particular, the method proposed in (Zhang et al., 2021) trains a pixel class prediction model on these representations and confirms their label efficiency. In the experimental section, we compare the method from (Zhang et al., 2021) to our DDPM-based one and demonstrate several distinctive advantages of our solution.
+
+Representations from generative models for discriminative tasks. The usage of generative models, as representation learners, has been widely investigated for global prediction (Donahue & Simonyan, 2019; Chen et al., 2020a), and dense prediction problems (Zhang et al., 2021; Tritrong et al., 2021; Xu, 2021; Xu et al., 2021). While previous works highlighted the practical advantages of these representations, such as out-of-distribution robustness (Li et al., 2021a), generative models as representation learners receive less attention compared to alternative unsupervised methods, e.g., based on contrastive learning (Chen et al., 2020b). The main reason is probably the difficulty of training a high-quality generative model on a complex, diverse dataset. However, given the recent success of DDPM on Imagenet (Deng et al., 2009), one can expect that this direction will attract more attention in the future.
+
+# 3 REPRESENTATIONS FROM DIFFUSION MODELS
+
+In the following section, we investigate the image representations learned by diffusion models. First, we provide a brief overview of the DDPM framework. Then, we describe how to extract features with DDPM and investigate what kind of semantic information these features might capture.
+
+Background. Diffusion models transform noise $x _ { T } { \sim } N ( 0 , I )$ to the sample $x _ { 0 }$ by gradually denoising $x _ { T }$ to less noisy samples $x _ { t }$ . Formally, we are given a forward diffusion process:
+
+$$
+q ( x _ { t } | x _ { t - 1 } ) : = N ( x _ { t } ; \sqrt { 1 - \beta _ { t } } x _ { t - 1 } , \beta _ { t } I ) ,
+$$
+
+for some fixed variance schedule $\beta _ { 1 } , \ldots , \beta _ { t }$ .
+
+![](images/87661b7fbec7eedb49746140ee31bd35cac0e26d89d2b0fe4bb0f41fcdad5e2f.jpg)  
+Figure 1: Overview of the proposed method. (1) $x _ { 0 }  x _ { t }$ by adding noise according to $q ( x _ { t } | x _ { 0 } )$ . (2) Extracting feature maps from a noise predictor $\epsilon _ { \theta } ( x _ { t } , t )$ . (3) Collecting pixel-level representations by upsampling the feature maps to the image resolution and concatenating them. (4) Using the pixel-wise feature vectors to train an ensemble of MLPs to predict a class label for each pixel.
+
+Importantly, a noisy sample $x _ { t }$ can be obtained directly from the data $x _ { 0 }$
+
+$$
+\begin{array} { r } { q ( x _ { t } | x _ { 0 } ) : = \mathcal { N } ( x _ { t } ; \sqrt { \bar { \alpha } _ { t } } x _ { 0 } , ( 1 - \bar { \alpha } _ { t } ) I ) , } \\ { x _ { t } = \sqrt { \bar { \alpha } _ { t } } x _ { 0 } + \sqrt { 1 - \bar { \alpha } _ { t } } \epsilon , \epsilon \sim \mathcal { N } ( 0 , 1 ) , } \end{array}
+$$
+
+where $\begin{array} { r } { \alpha _ { t } : = 1 - \beta _ { t } , \bar { \alpha } _ { t } : = \prod _ { s = 1 } ^ { t } \alpha _ { s } , } \end{array}$
+
+Pretrained DDPM approximates a reverse process:
+
+$$
+p _ { \theta } ( x _ { t - 1 } | x _ { t } ) : = N ( x _ { t - 1 } ; \mu _ { \theta } ( x _ { t } , t ) , \Sigma _ { \theta } ( x _ { t } , t ) ) .
+$$
+
+In practice, rather than predicting the mean of the distribution in Equation (3), the noise predictor network $\epsilon _ { \theta } ( x _ { t } , t )$ predicts the noise component at the step $t$ ; the mean is then a linear combination of this noise component and $x _ { t }$ . The covariance predictor $\Sigma _ { \theta } ( x _ { t } , t )$ can be either a fixed set of scalar covariances or learned as well (the latter was shown to improve the model quality (Nichol, 2021)).
+
+The denoising model $\epsilon _ { \theta } ( x _ { t } , t )$ is typically parameterized by different variants of the UNet architecture (Ronneberger et al., 2015), and in our experiments we investigate the state-of-the-art one proposed in (Dhariwal & Nichol, 2021).
+
+Extracting representations. For a given real image $\boldsymbol { x } _ { 0 } \in \mathbb { R } ^ { H \times W \times 3 }$ , one can compute $T$ sets of activation tensors from the noise predictor network $\epsilon _ { \theta } ( x _ { t } , t )$ . The overall scheme for a timestep $t$ is presented in Figure 1. First, we corrupt $x _ { 0 }$ by adding Gaussian noise according to Equation (2). The noisy $x _ { t }$ is used as an input of $\epsilon _ { \theta } ( x _ { t } , t )$ parameterized by the UNet model. The UNet’s intermediate activations are then upsampled to $H \times W$ with bilinear interpolation. This allows treating them as pixel-level representations of $x _ { 0 }$ .
+
+# 3.1 REPRESENTATION ANALYSIS
+
+We analyze the representations produced by the noise predictor $\epsilon _ { \theta } ( x _ { t } , t )$ for different $t$ . We consider the state-of-the-art DDPM checkpoints trained on the LSUN-Horse and FFHQ-256 datasets1.
+
+The intermediate activations from the noise predictor capture semantic information. For this experiment, we take a few images from the LSUN-Horse and FFHQ datasets and manually assign each pixel to one of the 21 and 34 semantic classes, respectively. Our goal is to understand whether the pixel-level representations produced by DDPM effectively capture the information about semantics. To this end, we train a multi-layer perceptron (MLP) to predict the pixel semantic label from its features produced by one of the 18 UNet decoder blocks on a specific diffusion step $t$ . Note that we consider only the decoder activations because they also aggregate the encoder activations through the skip connections. MLPs are trained on 20 images and evaluated on 20 hold-out ones. The predictive performance is measured in terms of mean IoU.
+
+![](images/6c8aef6a3060d0e4d210d9c6d79e93986f30966a48122d04e3b58bd627cf1468.jpg)  
+Figure 2: The evolution of predictive performance of DDPM-based pixel-wise representations for different UNet decoder blocks and diffusion steps. The blocks are numbered from the deep to shallow ones. The most informative features typically correspond to the later steps of the reverse diffusion process and middle layers of the UNet decoder. The earlier steps correspond to uninformative representations. The plots for other datasets are provided in Appendix A
+
+![](images/77469508dbb11c3b0cb7a0616376e82b590a70e073185e3712b4d67b63ff8fc4.jpg)  
+Figure 3: The evolution of predictive performance of DDPM-based pixel-wise representations on the LSUN-Horse dataset for classes with the smallest (Left) and largest (Right) average areas. The predictive performance for small-sized objects starts growing later in the reverse process. The deeper blocks are more informative for larger objects and the shallower blocks are more informative for smaller objects. A similar evaluation for other datasets is provided in Appendix A.
+
+The evolution of predictive performance across the different blocks and diffusion steps $t$ is presented in Figure 2. The blocks are numbered from the deep to shallow ones. Figure 2 shows that the discriminability of the features produced by the noise predictor $\epsilon _ { \theta } ( x _ { t } , t )$ varies for different blocks and diffusion steps. In particular, the features corresponding to the later steps of the reverse diffusion process typically capture semantic information more effectively. In contrast, the ones corresponding to the early steps are generally uninformative. Across different blocks, the features produced by the layers in the middle of the UNet decoder appear to be the most informative on all diffusion steps.
+
+Also, we separately consider small-sized and large-sized semantic classes based on the average area in the annotated dataset. Then, we evaluate mean IoU for these classes independently across the different UNet blocks and diffusion steps. The results on LSUN-Horse are in Figure 3. As expected, the predictive performance for large-sized objects starts growing earlier in the reverse process. The shallower blocks are more informative for smaller objects, while the deeper blocks are more so for the larger ones. In both cases, the most discriminative features still correspond to the middle blocks.
+
+Figure 2 implies that for certain UNet blocks and diffusion steps, similar DDPM-based representations correspond to the pixels of the same semantics. Figure 4 shows the $\mathbf { k }$ -means clusters $\scriptstyle ( k = 5 )$ ) formed by the features extracted by the FFHQ checkpoint from the blocks $\{ 6 , 8 , 1 0 , 1 2 \}$ on the diffusion steps $\{ 5 0 , 2 0 0 , 4 0 0 , 6 0 0 , 8 0 0 \}$ , and confirms that clusters can span coherent semantic objects and object-parts. In the block $B { = } 6$ , the features correspond to coarse semantic masks. At the other extreme, the features from $B { = } 1 2$ can discriminate between fine-grained face parts but exhibit less semantic meaningness for coarse fragmentation. Across different diffusion steps, the most meaningful features correspond to the later ones. We attribute this behavior to the fact that on the earlier steps of the reverse process, the global structure of a DDPM sample has not yet emerged, therefore, it is hardly possible to predict segmentation masks at this stage. This intuition is qualitatively confirmed by the masks in Figure 4. For $t { = } 8 0 0$ , the masks poorly reflect the content of actual images, while for smaller values of $t$ , the masks and images are semantically coherent.
+
+![](images/8267a2745891e62a04fe53038cf44e31e651e2b80bdf7c4a407cb6be62a8cda6.jpg)  
+Figure 4: Examples of $\mathbf { k }$ -means clusters $\scriptstyle ( k = 5 )$ ) formed by the features extracted from the UNet decoder blocks $\{ 6 , 8 , 1 0 , 1 2 \}$ on the diffusion steps $\{ 5 0 , 2 0 0 , 4 0 0 , 6 0 0 , 8 0 0 \}$ . The clusters from the middle blocks spatially span coherent semantic objects and parts.
+
+# 3.2 DDPM-BASED REPRESENTATIONS FOR FEW-SHOT SEMANTIC SEGMENTATION
+
+The potential effectiveness of the intermediate DDPM activations observed above implies their usage as image representations for dense prediction tasks. Figure 1 schematically presents our overall approach for image segmentation, which exploits the discriminability of these representations. In more detail, we consider a few-shot semi-supervised setup, when a large number of unlabeled images $\{ X _ { 1 } , \ldots , X _ { N } \} \subset \mathbb { R } ^ { H \times W \times 3 }$ from the particular domain are available, and only for $n$ training images $\{ X _ { 1 } , \ldots , X _ { n } \} \subset \mathbb { R } ^ { H \times W \times 3 }$ the groundtruth $K$ -class semantic masks $\{ Y _ { 1 } , . . . , Y _ { n } \} \subset \mathbb { R } ^ { H \times W \times \{ 1 , . . . , K \} }$ are provided.
+
+As a first step, we train a diffusion model on the whole $\{ X _ { 1 } , \ldots , X _ { N } \}$ in an unsupervised manner. Then, this diffusion model is used to extract the pixel-level representations of the labeled images using the subset of the UNet blocks and diffusion steps $t$ . In this work, we use the representations from the middle blocks $B { = } \{ 5 , 6 , 7 , 8 , 1 2 \}$ of the UNet decoder and later steps $t { = } \{ 5 0 , 1 5 0 , 2 5 0 \}$ of the reverse diffusion process. These blocks and time steps are motivated by the insights from Section 3.1 but intentionally not tuned for each dataset.
+
+While the feature extraction at the particular time step is stochastic, we fix the noise for all timesteps $t$ and ablate this in Section 4.1. The extracted representations from all blocks $B$ and steps $t$ are upsampled to the image size and concatenated, forming the feature vectors for all pixels of the training images. The overall dimension of the pixel-level representations is 8448.
+
+Then, following (Zhang et al., 2021), we train an ensemble of independent multi-layer perceptrons (MLPs) on these feature vectors, which aim to predict a semantic label of each pixel available for training images. We adopt the ensemble configuration and training settings from (Zhang et al., 2021) and exploit them across all other methods in our experiments, see Appendix C for details.
+
+To segment a test image, we extract its DDPM-based pixel-wise representations and use them to predict the pixel labels by the ensemble. The final prediction is obtained by majority voting.
+
+# 4 EXPERIMENTS
+
+This section experimentally confirms the advantage of the DDPM-based representations for the semantic segmentation problem. We start from a thorough comparison to the existing alternatives and then dissect the reasons for the DDPM success by additional analysis.
+
+Datasets. In our evaluation, we mainly work with the “bedroom”, “cat” and “horse” categories from LSUN (Yu et al., 2015) and FFHQ-256 (Karras et al., 2019). As a training set for each dataset, we consider several images for which the fine-grained semantic masks are collected following the protocol from (Zhang et al., 2021). For each dataset, a professional assessor was hired to annotate train and test samples. We denote the collected datasets as Bedroom-28, FFHQ-34, Cat-15, Horse21, where the number corresponds to the number of semantic classes.
+
+Table 1: Number of annotated images for each dataset used in our evaluation.   
+
+<table><tr><td>Dataset</td><td>RealTrain</td><td>RealTest</td><td>GAN</td><td>DDPM</td><td>Total</td></tr><tr><td>Bedroom-28</td><td>40</td><td>20</td><td>40</td><td>40</td><td>140</td></tr><tr><td>FFHQ-34</td><td>20</td><td>20</td><td>20</td><td>20</td><td>80</td></tr><tr><td>Cat-15</td><td>30</td><td>20</td><td>30</td><td>30</td><td>110</td></tr><tr><td>Horse-21</td><td>30</td><td>30</td><td>30</td><td>30</td><td>120</td></tr><tr><td>CelebA-19</td><td>20</td><td>500</td><td>一</td><td>一</td><td>520</td></tr><tr><td>ADE-Bedroom-30</td><td>50</td><td>650</td><td>丨</td><td>丨</td><td>700</td></tr></table>
+
+Additionally, we consider two datasets, which, in contrast to others, have publicly available annotations and sizable evaluation sets:
+
+• ADE-Bedroom-30 is a subset of the ADE20K dataset (Zhou et al., 2018), where we extract only images of bedroom scenes with 30 most frequent classes. We resize each image to 256 for the smaller side and then crop them to obtain the $2 5 6 \times 2 5 6$ samples. • CelebA-19 is a subset of the CelebAMask-HQ dataset (Lee et al., 2020), which provides the annotation for 19 facial attributes. All images are resized to 256 resolution.
+
+The number of annotated images for each dataset are in Table 1. Other details are in Appendix E.
+
+Methods. In the evaluation, we compare our method (denoted as DDPM) to several prior approaches which tackle the few-shot semantic segmentation setup. First, we describe the baselines that produce a large set of annotated synthetic images to train a segmentation model:
+
+• DatasetGAN (Zhang et al., 2021) — this method exploits the discriminability of pixel-level features produced by GANs. In more detail, assessors annotate a few GAN-produced images. Then, the latent codes of these images are used to obtain the intermediate generator activations, which are considered as pixel-level representations. Given these representations, a classifier is trained to predict a semantic label for each pixel. This classifier is then used to label new synthetic GAN images, which, for their part, serve as a training set for the DeepLabV3 segmentation model (Chen et al., 2017). For each dataset, we increase the number of synthetic images until the performance on the validation set is not saturated. According to (Zhang et al., 2021), we also remove $1 0 \%$ of synthetic samples with the most uncertain predictions. • DatasetDDPM mirrors the DatasetGAN baseline with the only difference being that GANs are replaced with DDPMs. We include this baseline to compare the GAN-based and DDPM-based representations in the same scenario.
+
+Note that our segmentation method described in Section 3.2 is more straightforward compared to DatasetGAN and DatasetDDPM since it does not require auxiliary steps of the synthetic dataset generation and training the segmentation model on it.
+
+Then, we consider a set of baselines that allow extracting intermediate activations from the real images directly and use them as pixel-level representations similarly to our method. In contrast to DatasetGAN and DatasetDDPM, these methods can potentially be beneficial due to the absence of the domain gap between real and synthetic images.
+
+• MAE (He et al., 2021) — one of the state-of-the-art self-supervised methods, which learns a denoising autoencoder to reconstruct missing patches. We use ViT-Large (Dosovitskiy et al., 2021) as a backbone model and reduce the patch size to $8 \times 8$ to increase the spatial dimensions of the feature maps. We pretrain all models on the same datasets as DDPM using the official code2. The feature extraction for this method is described in Appendix F. • SwAV (Caron et al., 2020) — one more recent self-supervised approach. We consider a twice wider ResNet-50 model for evaluation. All models are pretrained on the same datasets as DDPM also using the official source code3. The input image resolution is 256. GAN Inversion employs the state-of-the-art method (Tov et al., 2021) to obtain the latent codes for real images. We map the annotated real images to the GAN latent space, which allows computing the intermediate generator activations and using them as pixel-level representations.
+
+Table 2: The comparison of the segmentation methods in terms of mean IoU. $( ^ { * } )$ On CelebA-19 and ADE Bedroom-30, we evaluate models trained on FFHQ-256 and LSUN Bedroom, respectively.   
+
+<table><tr><td>Method</td><td>Bedroom-28</td><td>FFHQ-34</td><td>Cat-15</td><td>Horse-21</td><td>CelebA-19*</td><td>ADE Bedroom-30*</td></tr><tr><td>ALAE</td><td>20.0 ± 1.0</td><td>48.1 ± 1.3</td><td></td><td></td><td>49.7 ± 0.7</td><td>15.0 ± 0.5</td></tr><tr><td>VDVAE</td><td></td><td>57.3 ± 1.1</td><td></td><td></td><td>54.1 ± 1.0</td><td></td></tr><tr><td>GAN Inversion</td><td>13.9 ± 0.6</td><td>51.7± 0.8</td><td>21.4 ± 1.7</td><td>17.7 ± 0.4</td><td>51.5± 2.3</td><td>11.1 ± 0.2</td></tr><tr><td>GAN Encoder</td><td>22.4 ± 1.6</td><td>53.9 ± 1.3</td><td>32.0± 1.8</td><td>26.7 ± 0.7</td><td>53.9±0.8</td><td>15.7 ± 0.3</td></tr><tr><td>SwAV</td><td>42.4 ± 1.7</td><td>56.9 ± 1.3</td><td>45.1 ± 2.1</td><td>54.0± 0.9</td><td>52.4± 1.3</td><td>30.6 ± 1.6</td></tr><tr><td>MAE</td><td>45.0 ± 2.0</td><td>58.8 ± 1.1</td><td>52.4 ± 2.3</td><td>63.4± 1.4</td><td>57.8±0.4</td><td>31.7 ± 1.8</td></tr><tr><td>DatasetGAN</td><td>31.3 ± 2.3</td><td>57.0 ± 1.1</td><td>36.5± 2.3</td><td>45.4 ± 1.4</td><td></td><td></td></tr><tr><td>DatasetDDPM (Ours)</td><td>47.9 ± 2.9</td><td>56.0± 0.9</td><td>47.6 ± 1.5</td><td>60.8 ± 1.0</td><td></td><td></td></tr><tr><td>DDPM (Ours)</td><td>49.4 ± 1.9</td><td>59.1 ± 1.4</td><td>53.7± 3.3</td><td>65.0 ± 0.8</td><td>59.9 ± 1.0</td><td>34.6 ± 1.7</td></tr></table>
+
+• GAN Encoder — while GAN Inversion struggles to reconstruct images from LSUN domains, we also consider the activations of the pretrained GAN encoder used for GAN Inversion.
+
+• VDVAE (Child, 2021) — state-of-the-art autoencoder model. The intermediate activations are extracted from both encoder and decoder and concatenated. While there are no pretrained models on the LSUN datasets, we evaluate this model only on the publicly available checkpoint4 on FFHQ-256. Note that VAEs are still significantly inferior to GANs and DDPMs on LSUN.
+
+• ALAE (Pidhorskyi et al., 2020) adopts StyleGANv1 generator and adds an encoder network to the adversarial training. We extract features from the encoder model. In our evaluation, we use publicly available models on LSUN-Bedroom and FFHQ- $1 0 2 4 ^ { 5 }$ .
+
+Generative pretrained models. In our experiments, we use the state-of-the-art StyleGAN2 (Karras et al., 2020) models for the GAN-based baselines and the state-of-the-art pretrained ADMs (Dhariwal & Nichol, 2021) for our DDPM-based method. Since there is not a pretrained model for FFHQ-256, we train it ourselves using the official implementation6. For evaluation on the ADEBedroom-30 dataset, we use the models (including the baselines) pretrained on LSUN-Bedroom. For Celeba-19, we evaluate the models trained on FFHQ-256.
+
+Main results. The comparison of the methods in terms of the mean IoU measure is presented in Table 2. The results are averaged over 5 independent runs for different data splits. We also report per class IoUs in Appendix D. Additionally, we provide several qualitative examples of segmentation with our method in Figure 5. Below we highlight several key observations:
+
+• The proposed method based on the DDPM representations significantly outperforms the alternatives on most datasets.   
+• The MAE baseline is the strongest competitor to the DDPM-based segmentation and demonstrates comparable results on the FFHQ-34 and Cat-15 datasets.   
+• The SwAV baseline underperforms compared to the DDPM-based segmentation. We attribute this behavior to the fact that this baseline is trained in the discriminative fashion and can suppress the details, which are needed for fine-grained semantic segmentation. This result is consistent with the recent findings in (Cole et al., 2021), which shows that the state-of-the-art contrastive methods produce representations, which are suboptimal for fine-grained problems.   
+• DatasetDDPM outperforms its counterpart DatasetGAN against most benchmarks. Note that both these methods use the DeepLabV3 network. We attribute this superiority to the higher quality of DDPM synthetics, therefore, a smaller domain gap between synthetic and real data.   
+• On most datasets, DDPM outperforms the DatasetDDPM competitor. We provide an additional experiment to investigate this in the discussion section below.
+
+Overall, the proposed DDPM-based segmentation outperforms the baselines that exploit alternative generative models and also the baselines trained in the self-supervised fashion. This result highlights the potential of using the state-of-the-art DDPMs as strong unsupervised representation learners.
+
+![](images/9efda8b75a8f1afda524a62c282f6f69bea23026668dc1a1736ed93231888387.jpg)  
+Figure 5: The examples of segmentation masks predicted by our method on the test images along with the groundtruth annotated masks.
+
+Table 3: Performance of DDPM-based segmentation when trained on real and synthetic images. When trained on DDPM-produced data, DDPM demonstrates comparable performance to DatasetDDPM. When trained on GAN-produced data, DDPM still significantly outperforms DatasetGAN, but the gap between them reduces.   
+
+<table><tr><td></td><td colspan="3">Bedroom-28</td><td colspan="3">Cat-15</td><td colspan="3">Horse-21</td></tr><tr><td>Train data</td><td>Real</td><td>DDPM</td><td>GAN</td><td>Real</td><td>DDPM</td><td>GAN</td><td>Real</td><td>DDPM</td><td>GAN</td></tr><tr><td>DatasetGAN</td><td>丨</td><td>一</td><td>31.3 ± 2.3</td><td>丨</td><td>一</td><td>36.5± 2.3</td><td>1</td><td>一</td><td>45.4 ± 1.4</td></tr><tr><td>DatasetDDPM</td><td>一</td><td>47.9 ± 2.9</td><td>丨</td><td>一</td><td>47.6 ± 1.5</td><td>1</td><td>一</td><td>60.8 ±1.0</td><td>1</td></tr><tr><td>DDPM</td><td>49.4 ± 1.9</td><td>48.7±2.6</td><td>43.3±2.9</td><td>53.7± 3.3</td><td>47.9 ± 2.7</td><td>41.1 ± 2.2</td><td>65.0±0.8</td><td>62.4 ± 1.0</td><td>60.0 ±1.0</td></tr></table>
+
+# 4.1 DISCUSSION
+
+The effect of training on real data. The proposed DDPM method is trained on annotated real images, while DatasetDDPM and DatasetGAN are trained on synthetic ones, which are typically less natural, diverse, and can lack objects of particular classes. Moreover, synthetic images are harder for human annotation since they might have some distorted objects that are difficult to assign to a particular class. In the following experiment, we quantify the performance drop caused by training on real or synthetic data. Specifically, Table 3 reports the performance of the DDPM approach trained on real, DDPM-produced and GAN-produced annotated images. As can be seen, training on real images is very beneficial on the domains where the fidelity of generative models is still relatively low, e.g., LSUN-Cat, which indicates that annotated real images are a more reliable source of supervision. Moreover, if the DDPM method is trained on synthetic images, its performance becomes on par with DatasetDDPM. On the other hand, when trained on GAN-produced samples, DDPM significantly outperforms DatasetGAN. We attribute this to the fact that DDPMs provide more semantically-valuable pixel-wise representations compared to GANs.
+
+Sample-efficiency. In this experiment, we evaluate the performance of our method when it utilizes less annotated data. We provide mIoU for four datasets in Table 4. Importantly, DDPM is still able to outperform most baselines in Table 2, using significantly less supervision.
+
+The effect of stochastic feature extraction. Here, we investigate whether our method can benefit from the stochastic feature extraction described in Section 3.2. We consider the deterministic case, when the noise $\epsilon { \sim } N ( 0 , I )$ is sampled once and used in (2) to obtain $x _ { t }$ for all timesteps $t$ during both training and evaluation. Then, we compare it to the following stochastic options:
+
+First, different $\epsilon _ { t }$ are sampled for different timesteps $t$ and shared during the training and evaluation. Second, one samples different noise for all timesteps at each training iteration; during the evaluation the method also uses unseen noise samples.
+
+![](images/12484b8608423757a49d1e7feffe10d612930f88ef18f447c0e4aeb4f9d34758.jpg)  
+Figure 6: mIoU degradation for different image corruption levels on the Bedroom-28 and Horse-21 datasets. DDPM demonstrates higher robustness and preserves its advantage for all distortion levels.
+
+Table 4: Evaluation of the proposed method with a different number of labeled training data. Even using less annotated data, DDPM still outperforms most baselines in Table 2.   
+
+<table><tr><td></td><td colspan="3">Bedroom-28</td><td colspan="3">Cat-15</td><td colspan="3">Horse-21</td></tr><tr><td>Method</td><td>40</td><td>20</td><td>10</td><td>30</td><td>20</td><td>10</td><td>30</td><td>20</td><td>10</td></tr><tr><td>DDPM</td><td></td><td></td><td></td><td>49.4±1.9 46.2 ±3.6 38.2±2.9 53.7 ±3.3 49.2 ±4.2 42.0±4.8 65.0±0.8</td><td></td><td></td><td></td><td>63.8± 0.7</td><td>56.9± 2.4</td></tr></table>
+
+<table><tr><td>ShareTrain/Test</td><td>Share for t</td><td>Bedroom-28</td><td>FFHQ-34</td></tr><tr><td>+</td><td>+</td><td>49.3 ± 1.9</td><td>59.1 ± 1.4</td></tr><tr><td>+</td><td>=</td><td>49.1 ± 2.2</td><td>59.3 ± 1.5</td></tr><tr><td>-</td><td>-</td><td>48.9 ± 1.6</td><td>59.3 ± 1.4</td></tr></table>
+
+Table 5: Performance of the DDPM-based method for different feature extraction variations. All considered stochastic options provide a similar mIoU to the determinstic one.
+
+The results are provided in Table 5. As one can see, the difference in the performance is marginal. We attribute this behavior to the following reasons:
+
+• Our method uses later $t$ of the reverse diffusion process where the noise magnitude is low. • Since we exploit the deep layers of the UNet model, the noise might not affect the activations from these layers significantly.
+
+Robustness to input corruptions. In this experiment, we investigate the robustness of DDPMbased representations. First, we learn pixel classifiers on the clean images using the DDPM, SwAV and MAE representations on the Bedroom-28 and Horse-21 datasets. Then, 18 diverse corruption types, adopted from (Hendrycks & Dietterich, 2019), are applied to test images. Each corruption has five levels of severity. In Figure 6, we provide mean IoUs computed over all corruption types for 1, 3, 5 levels of severity, denoted as “weak”, “medium” and “strong”, respectively.
+
+One can observe that the proposed DDPM-based method demonstrates higher robustness and preserves its advantage over the SwAV and MAE models even for severe image distortions.
+
+# 5 CONCLUSION
+
+This paper demonstrates that DDPMs can serve as representation learners for discriminative computer vision problems. Compared to GANs, diffusion models allow for a straightforward computation of these representations for real images, and one does not need to learn an additional encoder, which maps images to the latent space. This DDPM’s advantage and superior generative quality provide state-of-the-art performance in the few-shot semantic segmentation task. The notable restraint of the DDPM-based segmentation is a requirement of high-quality diffusion models trained on the dataset at hand, which can be challenging for complex domains, like ImageNet or MSCOCO. However, given the rapid research progress on DDPM, we expect they will reach these milestones in the nearest future, thereby extending the range of applicability for the corresponding representations.
+
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+
+# APPENDIX
+
+# A EVOLUTION OF PREDICTIVE PERFORMANCE
+
+![](images/9336d075211187cebcba86e23520979299dff826590458419e8a1314a5dc8b91.jpg)  
+Figure 7: The evolution of predictive performance of DDPM-based pixel-wise representations for different UNet blocks and diffusion steps on LSUN-Cat and LSUN-Bedroom. The blocks are numbered from the deep to shallow ones.
+
+![](images/1b63c382cd2bf56e9b8c04caee70ec3bc0c207c8521a06e0c72a678d7c57555a.jpg)  
+Figure 8: The evolution of predictive performance of DDPM-based pixel-wise representations on the FFHQ-256, LSUN-Cat and LSUN-Bedroom datasets for classes with the smallest (Left) and largest (Right) average areas.
+
+B DATASETDDPM & DATASETGAN SATURATION   
+Table 6: Performance of DatasetDDPM and DatasetGAN for $1 0 K { - } 5 0 K$ synthetic images in the training dataset. Mean IoU of both methods saturates at $3 0 K { - } 5 0 K$ of synthetic data.   
+
+<table><tr><td colspan="7">DatasetDDPM</td><td colspan="4">DatasetGAN</td></tr><tr><td>Dataset</td><td>10k</td><td>20K</td><td>30K</td><td>40K</td><td>50K</td><td>10K</td><td>20K</td><td>30K</td><td>40K</td><td>50K</td></tr><tr><td>Bedroom-2845.1±2.346.2±2.346.1±2.847.8±2.347.9±2.930.6±2.330.4±3.1 30.9±2.430.9±2.431.3±2.7</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>FFHQ-34</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>55.9±0.8 55.8±0.7 55.9±0.7 56.0±0.8 55.9±0.7 56.4±1.0 56.9±1.0 57.0±1.1 57.0±1.2 57.0±1.2</td></tr><tr><td>Cat-15</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>43.6±3.0 46.4±1.7 46.2±1.9 47.4±1.7 47.6±1.5 34.7±2.8 34.8±2.9 36.3±2.335.8±2.5 36.5 ±2.3</td></tr><tr><td>Horse-21</td><td></td><td></td><td>57.0±1.2 59.5±0.5 59.0±2.0 60.4±1.1 60.8±0.9 41.6±2.0 43.1± 1.8 45.4±1.4 44.5±1.2 44.6 ± 1.4</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+# C TRAINING SETUP
+
+The ensemble of MLPs consists of 10 independent models. Each MLP is trained for ${ \sim } 4$ epochs using the Adam optimizer (Kingma & Ba, 2015) with 0.001 learning rate. The batch size is 64. This setting is used for all methods and datasets.
+
+MLP architecture. We adopt the MLP architecture from (Zhang et al., 2021). Specifically, we use MLPs with two hidden layers with ReLU nonlinearity and batch normalization. The sizes of hidden layers are 128 and 32 for datasets with a number of classes less than 30, and 256 and 128 for others.
+
+Also, we evaluate the performance of the proposed method for twice wider / deeper MLPs on the Bedroom-28 and FFHQ-34 datasets and do not observe any noticeable difference, see Table 7.
+
+<table><tr><td>Method</td><td>Bedroom-28</td><td>FFHQ-34</td></tr><tr><td>Original MLP</td><td>49.4</td><td>59.1</td></tr><tr><td>Wider MLP</td><td>49.5</td><td>59.1</td></tr><tr><td>Deeper MLP</td><td>49.3</td><td>58.9</td></tr></table>
+
+Table 7: Performance of the proposed method for twice wider / deeper MLP architecture within the ensemble. More expressive MLPs do not improve the performance.
+
+# D PER CLASS IOUS
+
+![](images/02f4b1e7cd5604421b93ea732e15e6ee885442fcc90f2409dc5fdf67449a18f4.jpg)  
+Figure 9: Per class IoUs for DatasetGAN, DatasetDDPM and DDPM.
+
+![](images/ab730fa2e015904bdb829ac09c57fbd53b4a26673dd3daefc00740d32a680ac7.jpg)  
+Figure 10: Number of instances of each semantic class in the annotated real and synthetic train sets.
+
+# E DATASET DETAILS
+
+# E.1 CLASS NAMES
+
+Bedroom-28: [bed, footboard, headboard, side rail, carpet, ceiling, chandelier, curtain, cushion, floor, table, table top, picture, pillow, lamp column, lamp shade, wall, window, curtain rod, window frame, chair, picture frame, plinth, door, pouf, wardrobe, plant, table staff]
+
+FFHQ-34: [background, head, cheek, chin, ear, helix, lobule, bottom lid, eyelashes, iris, pupil, sclera, tear duct, top lid, eyebrow, forehead, frown, hair, sideburns, jaw, moustache, inferior lip, oral commissure, superior lip, teeth, neck, nose, ala of nose, bridge, nose tip, nostril, philtrum, temple, wrinkles]
+
+Cat-15: [background, back, belly, chest, leg, paw, head, ear, eye, mouth, tongue, tail, nose, whiskers, neck]
+
+Horse-21: [background, person, back, barrel, bridle, chest, ear, eye, forelock, head, hoof, leg, mane, muzzle, neck, nostril, tail, thigh, saddle, shoulder, leg protection]
+
+CelebA-19: [background, cloth, ear r, eye g, hair, hat, l brow, l ear, l eye, l lip, mouth, neck, neck l, nose, r brow, r ear, r eye, skin, u lip]
+
+ADE-Bedroom-30: [wall, bed, floor, table, lamp, ceiling, painting, windowpane, pillow, curtain, cushion, door, chair, cabinet, chest, mirror, rug, armchair, book, sconce, plant, wardrobe, clock, light, flower, vase, fan, box, shelf, television]
+
+# E.2 CLASS STATISTICS
+
+In Figure 10, we report the statistics of classes computed over annotated real images as well as annotated synthetic images produced by GAN and DDPM.
+
+# F EXTRACTING REPRESENTATIONS FROM MAE
+
+To obtain pixelwise representations, we apply the model to a fully observed image (mask ratio $\scriptstyle 1 = 0$ ) of resolution 256 and extract feature maps from the deepest 12 ViT-L blocks . The feature maps from each block have $1 0 2 4 \times 3 2 \times 3 2$ dimensions. Similarly to other methods, we upsample the extracted feature maps to $2 5 6 \times 2 5 6$ and concatenate them. The overall dimension of the pixel representation is 12288.
+
+In addition, we investigated other feature extraction strategies and got the following observations:
+
+1. Including activations from the decoder did not provide any noticeable gains;   
+2. Extracting activations right after self-attention layers caused slightly inferior performance;   
+3. Extracting activations from every second encoder block also provided a bit worse results.

parse/dev/SlxSY2UZQT/SlxSY2UZQT_content_list.json

@@ -0,0 +1,1771 @@
+[
+    {
+        "type": "text",
+        "text": "LABEL-EFFICIENT SEMANTIC SEGMENTATION WITH DIFFUSION MODELS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            821,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Dmitry Baranchuk, Ivan Rubachev, Andrey Voynov, Valentin Khrulkov, Artem Babenko ",
+        "bbox": [
+            184,
+            170,
+            802,
+            185
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yandex Research ",
+        "bbox": [
+            186,
+            191,
+            297,
+            205
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            242,
+            544,
+            257
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Denoising diffusion probabilistic models have recently received much research attention since they outperform alternative approaches, such as GANs, and currently provide state-of-the-art generative performance. The superior performance of diffusion models has made them an appealing tool in several applications, including inpainting, super-resolution, and semantic editing. In this paper, we demonstrate that diffusion models can also serve as an instrument for semantic segmentation, especially in the setup when labeled data is scarce. In particular, for several pretrained diffusion models, we investigate the intermediate activations from the networks that perform the Markov step of the reverse diffusion process. We show that these activations effectively capture the semantic information from an input image and appear to be excellent pixel-level representations for the segmentation problem. Based on these observations, we describe a simple segmentation method, which can work even if only a few training images are provided. Our approach significantly outperforms the existing alternatives on several datasets for the same amount of human supervision. The source code of the project is publicly available. ",
+        "bbox": [
+            233,
+            276,
+            764,
+            484
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            517,
+            336,
+            534
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Denoising diffusion probabilistic models (DDPM) (Sohl-Dickstein et al., 2015; Ho et al., 2020) have recently outperformed alternative approaches to model the distribution of natural images both in the realism of individual samples and their diversity (Dhariwal & Nichol, 2021). These advantages of DDPM are successfully exploited in applications, such as colorization (Song et al., 2021), inpainting (Song et al., 2021), super-resolution (Saharia et al., 2021; Li et al., 2021b), and semantic editing (Meng et al., 2021), where DDPM often achieve more impressive results compared to GANs. ",
+        "bbox": [
+            174,
+            551,
+            825,
+            635
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "So far, however, DDPM were not exploited as a source of effective image representations for discriminative computer vision problems. While the prior literature has demonstrated that various generative paradigms, such as GANs (Donahue & Simonyan, 2019) or autoregressive models (Chen et al., 2020a), can be used to extract the representations for common vision tasks, it is not clear if DDPM can also serve as representation learners. In this paper, we provide an affirmative answer to this question in the context of semantic segmentation. ",
+        "bbox": [
+            174,
+            642,
+            823,
+            726
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In particular, we investigate the intermediate activations from the U-Net network that approximates the Markov step of the reverse diffusion process in DDPM. Intuitively, this network learns to denoise its input, and it is not clear why the intermediate activations should capture semantic information needed for high-level vision problems. Nevertheless, we show that on certain diffusion steps, these activations do capture such information, and therefore, can potentially be used as image representations for downstream tasks. Given these observations, we propose a simple semantic segmentation method, which exploits these representations and works successfully even if only a few labeled images are provided. On several datasets, we show that our DDPM-based segmentation method outperforms the existing baselines for the same amount of supervision. ",
+        "bbox": [
+            174,
+            733,
+            825,
+            858
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To sum up, the contributions of our paper are: ",
+        "bbox": [
+            176,
+            866,
+            472,
+            878
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1. We investigate the representations learned by the state-of-the-art DDPM and show that they capture high-level semantic information valuable for downstream vision tasks. ",
+        "bbox": [
+            183,
+            895,
+            823,
+            922
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "2. We design a simple semantic segmentation approach that exploits these representations and outperforms the alternatives in the few-shot operating point. ",
+        "bbox": [
+            181,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3. We compare the DDPM-based representations with their GAN-based counterparts on the same datasets and demonstrate the advantages of the former in the context of semantic segmentation. ",
+        "bbox": [
+            184,
+            137,
+            823,
+            166
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            188,
+            341,
+            204
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this section, we briefly describe the existing lines of research relevant to our work. ",
+        "bbox": [
+            176,
+            219,
+            727,
+            234
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Diffusion models (Sohl-Dickstein et al., 2015; Ho et al., 2020) are a class of generative models that approximate the distribution of real images by the endpoint of the Markov chain which originates from a simple parametric distribution, typically a standard Gaussian. Each Markov step is modeled by a deep neural network that effectively learns to invert the diffusion process with a known Gaussian kernel. Ho et al. highlighted the equivalence of diffusion models and score matching (Song & Ermon, 2019; 2020), showing them to be two different perspectives on the gradual conversion of a simple known distribution into a target distribution via the iterative denoising process. Very recent works (Nichol, 2021; Dhariwal & Nichol, 2021) have developed more powerful model architectures as well as different advanced objectives, which led to the “victory” of DDPM over GANs in terms of generative quality and diversity. DDPM have been widely used in several applications, including image colorization (Song et al., 2021), super-resolution (Saharia et al., 2021; Li et al., 2021b), inpainting (Song et al., 2021), and semantic editing (Meng et al., 2021). In our work, we demonstrate that one can also successfully use them for semantic segmentation. ",
+        "bbox": [
+            173,
+            241,
+            825,
+            421
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Image segmentation with generative models is an active research direction at the moment, however, existing methods are primarily based on GANs. The first line of works (Voynov & Babenko, 2020; Voynov et al., 2021; Melas-Kyriazi et al., 2021) is based on the evidence that the latent spaces of the state-of-the-art GANs have directions corresponding to effects that influence the foreground/background pixels differently, which allows producing synthetic data to train segmentation models. However, these approaches are currently able to perform binary segmentation only, and it is not clear if they can be used in the general setup of semantic segmentation. The second line of works (Zhang et al., 2021; Tritrong et al., 2021; Xu, 2021; Galeev et al., 2020) is more relevant to our study since they are based on the intermediate representations obtained in GANs. In particular, the method proposed in (Zhang et al., 2021) trains a pixel class prediction model on these representations and confirms their label efficiency. In the experimental section, we compare the method from (Zhang et al., 2021) to our DDPM-based one and demonstrate several distinctive advantages of our solution. ",
+        "bbox": [
+            173,
+            429,
+            825,
+            594
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Representations from generative models for discriminative tasks. The usage of generative models, as representation learners, has been widely investigated for global prediction (Donahue & Simonyan, 2019; Chen et al., 2020a), and dense prediction problems (Zhang et al., 2021; Tritrong et al., 2021; Xu, 2021; Xu et al., 2021). While previous works highlighted the practical advantages of these representations, such as out-of-distribution robustness (Li et al., 2021a), generative models as representation learners receive less attention compared to alternative unsupervised methods, e.g., based on contrastive learning (Chen et al., 2020b). The main reason is probably the difficulty of training a high-quality generative model on a complex, diverse dataset. However, given the recent success of DDPM on Imagenet (Deng et al., 2009), one can expect that this direction will attract more attention in the future. ",
+        "bbox": [
+            173,
+            603,
+            825,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 REPRESENTATIONS FROM DIFFUSION MODELS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            763,
+            586,
+            780
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the following section, we investigate the image representations learned by diffusion models. First, we provide a brief overview of the DDPM framework. Then, we describe how to extract features with DDPM and investigate what kind of semantic information these features might capture. ",
+        "bbox": [
+            174,
+            795,
+            825,
+            838
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Background. Diffusion models transform noise $x _ { T } { \\sim } N ( 0 , I )$ to the sample $x _ { 0 }$ by gradually denoising $x _ { T }$ to less noisy samples $x _ { t }$ . Formally, we are given a forward diffusion process: ",
+        "bbox": [
+            173,
+            844,
+            823,
+            873
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e4c99ee41fba176a1728d5607d7afdf61a72d53d78b5e8ac5b619eca1222854d.jpg",
+        "text": "$$\nq ( x _ { t } | x _ { t - 1 } ) : = N ( x _ { t } ; \\sqrt { 1 - \\beta _ { t } } x _ { t - 1 } , \\beta _ { t } I ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            881,
+            635,
+            901
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "for some fixed variance schedule $\\beta _ { 1 } , \\ldots , \\beta _ { t }$ . ",
+        "bbox": [
+            174,
+            909,
+            467,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/87661b7fbec7eedb49746140ee31bd35cac0e26d89d2b0fe4bb0f41fcdad5e2f.jpg",
+        "image_caption": [
+            "Figure 1: Overview of the proposed method. (1) $x _ { 0 }  x _ { t }$ by adding noise according to $q ( x _ { t } | x _ { 0 } )$ . (2) Extracting feature maps from a noise predictor $\\epsilon _ { \\theta } ( x _ { t } , t )$ . (3) Collecting pixel-level representations by upsampling the feature maps to the image resolution and concatenating them. (4) Using the pixel-wise feature vectors to train an ensemble of MLPs to predict a class label for each pixel. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            79,
+            820,
+            279
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Importantly, a noisy sample $x _ { t }$ can be obtained directly from the data $x _ { 0 }$ ",
+        "bbox": [
+            173,
+            367,
+            651,
+            382
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1ad7b9f5590dff949f9cc140e403f7d78a32e6b37cce5c5f59cb380ba8b0e4bb.jpg",
+        "text": "$$\n\\begin{array} { r } { q ( x _ { t } | x _ { 0 } ) : = \\mathcal { N } ( x _ { t } ; \\sqrt { \\bar { \\alpha } _ { t } } x _ { 0 } , ( 1 - \\bar { \\alpha } _ { t } ) I ) , } \\\\ { x _ { t } = \\sqrt { \\bar { \\alpha } _ { t } } x _ { 0 } + \\sqrt { 1 - \\bar { \\alpha } _ { t } } \\epsilon , \\epsilon \\sim \\mathcal { N } ( 0 , 1 ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            361,
+            397,
+            635,
+            438
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { \\alpha _ { t } : = 1 - \\beta _ { t } , \\bar { \\alpha } _ { t } : = \\prod _ { s = 1 } ^ { t } \\alpha _ { s } , } \\end{array}$ ",
+        "bbox": [
+            174,
+            440,
+            415,
+            458
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Pretrained DDPM approximates a reverse process: ",
+        "bbox": [
+            173,
+            463,
+            506,
+            477
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a647d1fb064d6a9d1b160666857943482854e7ce364759485ef4feabbc483cdb.jpg",
+        "text": "$$\np _ { \\theta } ( x _ { t - 1 } | x _ { t } ) : = N ( x _ { t - 1 } ; \\mu _ { \\theta } ( x _ { t } , t ) , \\Sigma _ { \\theta } ( x _ { t } , t ) ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            481,
+            651,
+            500
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In practice, rather than predicting the mean of the distribution in Equation (3), the noise predictor network $\\epsilon _ { \\theta } ( x _ { t } , t )$ predicts the noise component at the step $t$ ; the mean is then a linear combination of this noise component and $x _ { t }$ . The covariance predictor $\\Sigma _ { \\theta } ( x _ { t } , t )$ can be either a fixed set of scalar covariances or learned as well (the latter was shown to improve the model quality (Nichol, 2021)). ",
+        "bbox": [
+            173,
+            502,
+            823,
+            559
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The denoising model $\\epsilon _ { \\theta } ( x _ { t } , t )$ is typically parameterized by different variants of the UNet architecture (Ronneberger et al., 2015), and in our experiments we investigate the state-of-the-art one proposed in (Dhariwal & Nichol, 2021). ",
+        "bbox": [
+            176,
+            565,
+            823,
+            607
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Extracting representations. For a given real image $\\boldsymbol { x } _ { 0 } \\in \\mathbb { R } ^ { H \\times W \\times 3 }$ , one can compute $T$ sets of activation tensors from the noise predictor network $\\epsilon _ { \\theta } ( x _ { t } , t )$ . The overall scheme for a timestep $t$ is presented in Figure 1. First, we corrupt $x _ { 0 }$ by adding Gaussian noise according to Equation (2). The noisy $x _ { t }$ is used as an input of $\\epsilon _ { \\theta } ( x _ { t } , t )$ parameterized by the UNet model. The UNet’s intermediate activations are then upsampled to $H \\times W$ with bilinear interpolation. This allows treating them as pixel-level representations of $x _ { 0 }$ . ",
+        "bbox": [
+            173,
+            613,
+            825,
+            698
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 REPRESENTATION ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            714,
+            411,
+            728
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We analyze the representations produced by the noise predictor $\\epsilon _ { \\theta } ( x _ { t } , t )$ for different $t$ . We consider the state-of-the-art DDPM checkpoints trained on the LSUN-Horse and FFHQ-256 datasets1. ",
+        "bbox": [
+            171,
+            739,
+            823,
+            768
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The intermediate activations from the noise predictor capture semantic information. For this experiment, we take a few images from the LSUN-Horse and FFHQ datasets and manually assign each pixel to one of the 21 and 34 semantic classes, respectively. Our goal is to understand whether the pixel-level representations produced by DDPM effectively capture the information about semantics. To this end, we train a multi-layer perceptron (MLP) to predict the pixel semantic label from its features produced by one of the 18 UNet decoder blocks on a specific diffusion step $t$ . Note that we consider only the decoder activations because they also aggregate the encoder activations through the skip connections. MLPs are trained on 20 images and evaluated on 20 hold-out ones. The predictive performance is measured in terms of mean IoU. ",
+        "bbox": [
+            173,
+            776,
+            825,
+            901
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/6c8aef6a3060d0e4d210d9c6d79e93986f30966a48122d04e3b58bd627cf1468.jpg",
+        "image_caption": [
+            "Figure 2: The evolution of predictive performance of DDPM-based pixel-wise representations for different UNet decoder blocks and diffusion steps. The blocks are numbered from the deep to shallow ones. The most informative features typically correspond to the later steps of the reverse diffusion process and middle layers of the UNet decoder. The earlier steps correspond to uninformative representations. The plots for other datasets are provided in Appendix A "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            77,
+            823,
+            234
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/77469508dbb11c3b0cb7a0616376e82b590a70e073185e3712b4d67b63ff8fc4.jpg",
+        "image_caption": [
+            "Figure 3: The evolution of predictive performance of DDPM-based pixel-wise representations on the LSUN-Horse dataset for classes with the smallest (Left) and largest (Right) average areas. The predictive performance for small-sized objects starts growing later in the reverse process. The deeper blocks are more informative for larger objects and the shallower blocks are more informative for smaller objects. A similar evaluation for other datasets is provided in Appendix A. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            329,
+            823,
+            452
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The evolution of predictive performance across the different blocks and diffusion steps $t$ is presented in Figure 2. The blocks are numbered from the deep to shallow ones. Figure 2 shows that the discriminability of the features produced by the noise predictor $\\epsilon _ { \\theta } ( x _ { t } , t )$ varies for different blocks and diffusion steps. In particular, the features corresponding to the later steps of the reverse diffusion process typically capture semantic information more effectively. In contrast, the ones corresponding to the early steps are generally uninformative. Across different blocks, the features produced by the layers in the middle of the UNet decoder appear to be the most informative on all diffusion steps. ",
+        "bbox": [
+            173,
+            563,
+            825,
+            660
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Also, we separately consider small-sized and large-sized semantic classes based on the average area in the annotated dataset. Then, we evaluate mean IoU for these classes independently across the different UNet blocks and diffusion steps. The results on LSUN-Horse are in Figure 3. As expected, the predictive performance for large-sized objects starts growing earlier in the reverse process. The shallower blocks are more informative for smaller objects, while the deeper blocks are more so for the larger ones. In both cases, the most discriminative features still correspond to the middle blocks. ",
+        "bbox": [
+            173,
+            666,
+            825,
+            750
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Figure 2 implies that for certain UNet blocks and diffusion steps, similar DDPM-based representations correspond to the pixels of the same semantics. Figure 4 shows the $\\mathbf { k }$ -means clusters $\\scriptstyle ( k = 5 )$ ) formed by the features extracted by the FFHQ checkpoint from the blocks $\\{ 6 , 8 , 1 0 , 1 2 \\}$ on the diffusion steps $\\{ 5 0 , 2 0 0 , 4 0 0 , 6 0 0 , 8 0 0 \\}$ , and confirms that clusters can span coherent semantic objects and object-parts. In the block $B { = } 6$ , the features correspond to coarse semantic masks. At the other extreme, the features from $B { = } 1 2$ can discriminate between fine-grained face parts but exhibit less semantic meaningness for coarse fragmentation. Across different diffusion steps, the most meaningful features correspond to the later ones. We attribute this behavior to the fact that on the earlier steps of the reverse process, the global structure of a DDPM sample has not yet emerged, therefore, it is hardly possible to predict segmentation masks at this stage. This intuition is qualitatively confirmed by the masks in Figure 4. For $t { = } 8 0 0$ , the masks poorly reflect the content of actual images, while for smaller values of $t$ , the masks and images are semantically coherent. ",
+        "bbox": [
+            173,
+            757,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/8267a2745891e62a04fe53038cf44e31e651e2b80bdf7c4a407cb6be62a8cda6.jpg",
+        "image_caption": [
+            "Figure 4: Examples of $\\mathbf { k }$ -means clusters $\\scriptstyle ( k = 5 )$ ) formed by the features extracted from the UNet decoder blocks $\\{ 6 , 8 , 1 0 , 1 2 \\}$ on the diffusion steps $\\{ 5 0 , 2 0 0 , 4 0 0 , 6 0 0 , 8 0 0 \\}$ . The clusters from the middle blocks spatially span coherent semantic objects and parts. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            75,
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+    {
+        "type": "text",
+        "text": "3.2 DDPM-BASED REPRESENTATIONS FOR FEW-SHOT SEMANTIC SEGMENTATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The potential effectiveness of the intermediate DDPM activations observed above implies their usage as image representations for dense prediction tasks. Figure 1 schematically presents our overall approach for image segmentation, which exploits the discriminability of these representations. In more detail, we consider a few-shot semi-supervised setup, when a large number of unlabeled images $\\{ X _ { 1 } , \\ldots , X _ { N } \\} \\subset \\mathbb { R } ^ { H \\times W \\times 3 }$ from the particular domain are available, and only for $n$ training images $\\{ X _ { 1 } , \\ldots , X _ { n } \\} \\subset \\mathbb { R } ^ { H \\times W \\times 3 }$ the groundtruth $K$ -class semantic masks $\\{ Y _ { 1 } , . . . , Y _ { n } \\} \\subset \\mathbb { R } ^ { H \\times W \\times \\{ 1 , . . . , K \\} }$ are provided. ",
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+        "type": "text",
+        "text": "As a first step, we train a diffusion model on the whole $\\{ X _ { 1 } , \\ldots , X _ { N } \\}$ in an unsupervised manner. Then, this diffusion model is used to extract the pixel-level representations of the labeled images using the subset of the UNet blocks and diffusion steps $t$ . In this work, we use the representations from the middle blocks $B { = } \\{ 5 , 6 , 7 , 8 , 1 2 \\}$ of the UNet decoder and later steps $t { = } \\{ 5 0 , 1 5 0 , 2 5 0 \\}$ of the reverse diffusion process. These blocks and time steps are motivated by the insights from Section 3.1 but intentionally not tuned for each dataset. ",
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+        "type": "text",
+        "text": "While the feature extraction at the particular time step is stochastic, we fix the noise for all timesteps $t$ and ablate this in Section 4.1. The extracted representations from all blocks $B$ and steps $t$ are upsampled to the image size and concatenated, forming the feature vectors for all pixels of the training images. The overall dimension of the pixel-level representations is 8448. ",
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+    {
+        "type": "text",
+        "text": "Then, following (Zhang et al., 2021), we train an ensemble of independent multi-layer perceptrons (MLPs) on these feature vectors, which aim to predict a semantic label of each pixel available for training images. We adopt the ensemble configuration and training settings from (Zhang et al., 2021) and exploit them across all other methods in our experiments, see Appendix C for details. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To segment a test image, we extract its DDPM-based pixel-wise representations and use them to predict the pixel labels by the ensemble. The final prediction is obtained by majority voting. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This section experimentally confirms the advantage of the DDPM-based representations for the semantic segmentation problem. We start from a thorough comparison to the existing alternatives and then dissect the reasons for the DDPM success by additional analysis. ",
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+    },
+    {
+        "type": "text",
+        "text": "Datasets. In our evaluation, we mainly work with the “bedroom”, “cat” and “horse” categories from LSUN (Yu et al., 2015) and FFHQ-256 (Karras et al., 2019). As a training set for each dataset, we consider several images for which the fine-grained semantic masks are collected following the protocol from (Zhang et al., 2021). For each dataset, a professional assessor was hired to annotate train and test samples. We denote the collected datasets as Bedroom-28, FFHQ-34, Cat-15, Horse21, where the number corresponds to the number of semantic classes. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/d906d8c5f7b096181f42a08cddac7a21c790f85e022fce01ee34946734a14a9d.jpg",
+        "table_caption": [
+            "Table 1: Number of annotated images for each dataset used in our evaluation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>RealTrain</td><td>RealTest</td><td>GAN</td><td>DDPM</td><td>Total</td></tr><tr><td>Bedroom-28</td><td>40</td><td>20</td><td>40</td><td>40</td><td>140</td></tr><tr><td>FFHQ-34</td><td>20</td><td>20</td><td>20</td><td>20</td><td>80</td></tr><tr><td>Cat-15</td><td>30</td><td>20</td><td>30</td><td>30</td><td>110</td></tr><tr><td>Horse-21</td><td>30</td><td>30</td><td>30</td><td>30</td><td>120</td></tr><tr><td>CelebA-19</td><td>20</td><td>500</td><td>一</td><td>一</td><td>520</td></tr><tr><td>ADE-Bedroom-30</td><td>50</td><td>650</td><td>丨</td><td>丨</td><td>700</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Additionally, we consider two datasets, which, in contrast to others, have publicly available annotations and sizable evaluation sets: ",
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+    {
+        "type": "text",
+        "text": "• ADE-Bedroom-30 is a subset of the ADE20K dataset (Zhou et al., 2018), where we extract only images of bedroom scenes with 30 most frequent classes. We resize each image to 256 for the smaller side and then crop them to obtain the $2 5 6 \\times 2 5 6$ samples. • CelebA-19 is a subset of the CelebAMask-HQ dataset (Lee et al., 2020), which provides the annotation for 19 facial attributes. All images are resized to 256 resolution. ",
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+    {
+        "type": "text",
+        "text": "The number of annotated images for each dataset are in Table 1. Other details are in Appendix E. ",
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+    {
+        "type": "text",
+        "text": "Methods. In the evaluation, we compare our method (denoted as DDPM) to several prior approaches which tackle the few-shot semantic segmentation setup. First, we describe the baselines that produce a large set of annotated synthetic images to train a segmentation model: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• DatasetGAN (Zhang et al., 2021) — this method exploits the discriminability of pixel-level features produced by GANs. In more detail, assessors annotate a few GAN-produced images. Then, the latent codes of these images are used to obtain the intermediate generator activations, which are considered as pixel-level representations. Given these representations, a classifier is trained to predict a semantic label for each pixel. This classifier is then used to label new synthetic GAN images, which, for their part, serve as a training set for the DeepLabV3 segmentation model (Chen et al., 2017). For each dataset, we increase the number of synthetic images until the performance on the validation set is not saturated. According to (Zhang et al., 2021), we also remove $1 0 \\%$ of synthetic samples with the most uncertain predictions. • DatasetDDPM mirrors the DatasetGAN baseline with the only difference being that GANs are replaced with DDPMs. We include this baseline to compare the GAN-based and DDPM-based representations in the same scenario. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Note that our segmentation method described in Section 3.2 is more straightforward compared to DatasetGAN and DatasetDDPM since it does not require auxiliary steps of the synthetic dataset generation and training the segmentation model on it. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Then, we consider a set of baselines that allow extracting intermediate activations from the real images directly and use them as pixel-level representations similarly to our method. In contrast to DatasetGAN and DatasetDDPM, these methods can potentially be beneficial due to the absence of the domain gap between real and synthetic images. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• MAE (He et al., 2021) — one of the state-of-the-art self-supervised methods, which learns a denoising autoencoder to reconstruct missing patches. We use ViT-Large (Dosovitskiy et al., 2021) as a backbone model and reduce the patch size to $8 \\times 8$ to increase the spatial dimensions of the feature maps. We pretrain all models on the same datasets as DDPM using the official code2. The feature extraction for this method is described in Appendix F. • SwAV (Caron et al., 2020) — one more recent self-supervised approach. We consider a twice wider ResNet-50 model for evaluation. All models are pretrained on the same datasets as DDPM also using the official source code3. The input image resolution is 256. GAN Inversion employs the state-of-the-art method (Tov et al., 2021) to obtain the latent codes for real images. We map the annotated real images to the GAN latent space, which allows computing the intermediate generator activations and using them as pixel-level representations. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/c432ee1b6eb005f9870c22b3b36fd00df98792f43d403a76188672deb91c97bc.jpg",
+        "table_caption": [
+            "Table 2: The comparison of the segmentation methods in terms of mean IoU. $( ^ { * } )$ On CelebA-19 and ADE Bedroom-30, we evaluate models trained on FFHQ-256 and LSUN Bedroom, respectively. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Bedroom-28</td><td>FFHQ-34</td><td>Cat-15</td><td>Horse-21</td><td>CelebA-19*</td><td>ADE Bedroom-30*</td></tr><tr><td>ALAE</td><td>20.0 ± 1.0</td><td>48.1 ± 1.3</td><td></td><td></td><td>49.7 ± 0.7</td><td>15.0 ± 0.5</td></tr><tr><td>VDVAE</td><td></td><td>57.3 ± 1.1</td><td></td><td></td><td>54.1 ± 1.0</td><td></td></tr><tr><td>GAN Inversion</td><td>13.9 ± 0.6</td><td>51.7± 0.8</td><td>21.4 ± 1.7</td><td>17.7 ± 0.4</td><td>51.5± 2.3</td><td>11.1 ± 0.2</td></tr><tr><td>GAN Encoder</td><td>22.4 ± 1.6</td><td>53.9 ± 1.3</td><td>32.0± 1.8</td><td>26.7 ± 0.7</td><td>53.9±0.8</td><td>15.7 ± 0.3</td></tr><tr><td>SwAV</td><td>42.4 ± 1.7</td><td>56.9 ± 1.3</td><td>45.1 ± 2.1</td><td>54.0± 0.9</td><td>52.4± 1.3</td><td>30.6 ± 1.6</td></tr><tr><td>MAE</td><td>45.0 ± 2.0</td><td>58.8 ± 1.1</td><td>52.4 ± 2.3</td><td>63.4± 1.4</td><td>57.8±0.4</td><td>31.7 ± 1.8</td></tr><tr><td>DatasetGAN</td><td>31.3 ± 2.3</td><td>57.0 ± 1.1</td><td>36.5± 2.3</td><td>45.4 ± 1.4</td><td></td><td></td></tr><tr><td>DatasetDDPM (Ours)</td><td>47.9 ± 2.9</td><td>56.0± 0.9</td><td>47.6 ± 1.5</td><td>60.8 ± 1.0</td><td></td><td></td></tr><tr><td>DDPM (Ours)</td><td>49.4 ± 1.9</td><td>59.1 ± 1.4</td><td>53.7± 3.3</td><td>65.0 ± 0.8</td><td>59.9 ± 1.0</td><td>34.6 ± 1.7</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "• GAN Encoder — while GAN Inversion struggles to reconstruct images from LSUN domains, we also consider the activations of the pretrained GAN encoder used for GAN Inversion. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "• VDVAE (Child, 2021) — state-of-the-art autoencoder model. The intermediate activations are extracted from both encoder and decoder and concatenated. While there are no pretrained models on the LSUN datasets, we evaluate this model only on the publicly available checkpoint4 on FFHQ-256. Note that VAEs are still significantly inferior to GANs and DDPMs on LSUN. ",
+        "bbox": [
+            192,
+            340,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "• ALAE (Pidhorskyi et al., 2020) adopts StyleGANv1 generator and adds an encoder network to the adversarial training. We extract features from the encoder model. In our evaluation, we use publicly available models on LSUN-Bedroom and FFHQ- $1 0 2 4 ^ { 5 }$ . ",
+        "bbox": [
+            192,
+            400,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Generative pretrained models. In our experiments, we use the state-of-the-art StyleGAN2 (Karras et al., 2020) models for the GAN-based baselines and the state-of-the-art pretrained ADMs (Dhariwal & Nichol, 2021) for our DDPM-based method. Since there is not a pretrained model for FFHQ-256, we train it ourselves using the official implementation6. For evaluation on the ADEBedroom-30 dataset, we use the models (including the baselines) pretrained on LSUN-Bedroom. For Celeba-19, we evaluate the models trained on FFHQ-256. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Main results. The comparison of the methods in terms of the mean IoU measure is presented in Table 2. The results are averaged over 5 independent runs for different data splits. We also report per class IoUs in Appendix D. Additionally, we provide several qualitative examples of segmentation with our method in Figure 5. Below we highlight several key observations: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "• The proposed method based on the DDPM representations significantly outperforms the alternatives on most datasets.   \n• The MAE baseline is the strongest competitor to the DDPM-based segmentation and demonstrates comparable results on the FFHQ-34 and Cat-15 datasets.   \n• The SwAV baseline underperforms compared to the DDPM-based segmentation. We attribute this behavior to the fact that this baseline is trained in the discriminative fashion and can suppress the details, which are needed for fine-grained semantic segmentation. This result is consistent with the recent findings in (Cole et al., 2021), which shows that the state-of-the-art contrastive methods produce representations, which are suboptimal for fine-grained problems.   \n• DatasetDDPM outperforms its counterpart DatasetGAN against most benchmarks. Note that both these methods use the DeepLabV3 network. We attribute this superiority to the higher quality of DDPM synthetics, therefore, a smaller domain gap between synthetic and real data.   \n• On most datasets, DDPM outperforms the DatasetDDPM competitor. We provide an additional experiment to investigate this in the discussion section below. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Overall, the proposed DDPM-based segmentation outperforms the baselines that exploit alternative generative models and also the baselines trained in the self-supervised fashion. This result highlights the potential of using the state-of-the-art DDPMs as strong unsupervised representation learners. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/9efda8b75a8f1afda524a62c282f6f69bea23026668dc1a1736ed93231888387.jpg",
+        "image_caption": [
+            "Figure 5: The examples of segmentation masks predicted by our method on the test images along with the groundtruth annotated masks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            169,
+            79,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/7a225f56c63c26aeb6b4e9fe718739ce31db5918cd9a59dfa483342c3abeb0f8.jpg",
+        "table_caption": [
+            "Table 3: Performance of DDPM-based segmentation when trained on real and synthetic images. When trained on DDPM-produced data, DDPM demonstrates comparable performance to DatasetDDPM. When trained on GAN-produced data, DDPM still significantly outperforms DatasetGAN, but the gap between them reduces. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">Bedroom-28</td><td colspan=\"3\">Cat-15</td><td colspan=\"3\">Horse-21</td></tr><tr><td>Train data</td><td>Real</td><td>DDPM</td><td>GAN</td><td>Real</td><td>DDPM</td><td>GAN</td><td>Real</td><td>DDPM</td><td>GAN</td></tr><tr><td>DatasetGAN</td><td>丨</td><td>一</td><td>31.3 ± 2.3</td><td>丨</td><td>一</td><td>36.5± 2.3</td><td>1</td><td>一</td><td>45.4 ± 1.4</td></tr><tr><td>DatasetDDPM</td><td>一</td><td>47.9 ± 2.9</td><td>丨</td><td>一</td><td>47.6 ± 1.5</td><td>1</td><td>一</td><td>60.8 ±1.0</td><td>1</td></tr><tr><td>DDPM</td><td>49.4 ± 1.9</td><td>48.7±2.6</td><td>43.3±2.9</td><td>53.7± 3.3</td><td>47.9 ± 2.7</td><td>41.1 ± 2.2</td><td>65.0±0.8</td><td>62.4 ± 1.0</td><td>60.0 ±1.0</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.1 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The effect of training on real data. The proposed DDPM method is trained on annotated real images, while DatasetDDPM and DatasetGAN are trained on synthetic ones, which are typically less natural, diverse, and can lack objects of particular classes. Moreover, synthetic images are harder for human annotation since they might have some distorted objects that are difficult to assign to a particular class. In the following experiment, we quantify the performance drop caused by training on real or synthetic data. Specifically, Table 3 reports the performance of the DDPM approach trained on real, DDPM-produced and GAN-produced annotated images. As can be seen, training on real images is very beneficial on the domains where the fidelity of generative models is still relatively low, e.g., LSUN-Cat, which indicates that annotated real images are a more reliable source of supervision. Moreover, if the DDPM method is trained on synthetic images, its performance becomes on par with DatasetDDPM. On the other hand, when trained on GAN-produced samples, DDPM significantly outperforms DatasetGAN. We attribute this to the fact that DDPMs provide more semantically-valuable pixel-wise representations compared to GANs. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Sample-efficiency. In this experiment, we evaluate the performance of our method when it utilizes less annotated data. We provide mIoU for four datasets in Table 4. Importantly, DDPM is still able to outperform most baselines in Table 2, using significantly less supervision. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The effect of stochastic feature extraction. Here, we investigate whether our method can benefit from the stochastic feature extraction described in Section 3.2. We consider the deterministic case, when the noise $\\epsilon { \\sim } N ( 0 , I )$ is sampled once and used in (2) to obtain $x _ { t }$ for all timesteps $t$ during both training and evaluation. Then, we compare it to the following stochastic options: ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "First, different $\\epsilon _ { t }$ are sampled for different timesteps $t$ and shared during the training and evaluation. Second, one samples different noise for all timesteps at each training iteration; during the evaluation the method also uses unseen noise samples. ",
+        "bbox": [
+            176,
+            882,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/12484b8608423757a49d1e7feffe10d612930f88ef18f447c0e4aeb4f9d34758.jpg",
+        "image_caption": [
+            "Figure 6: mIoU degradation for different image corruption levels on the Bedroom-28 and Horse-21 datasets. DDPM demonstrates higher robustness and preserves its advantage for all distortion levels. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            79,
+            818,
+            212
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/cca982869659906946976fdd733248c53f349231c32c1c2f1ac6fa5b0674f3a3.jpg",
+        "table_caption": [
+            "Table 4: Evaluation of the proposed method with a different number of labeled training data. Even using less annotated data, DDPM still outperforms most baselines in Table 2. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">Bedroom-28</td><td colspan=\"3\">Cat-15</td><td colspan=\"3\">Horse-21</td></tr><tr><td>Method</td><td>40</td><td>20</td><td>10</td><td>30</td><td>20</td><td>10</td><td>30</td><td>20</td><td>10</td></tr><tr><td>DDPM</td><td></td><td></td><td></td><td>49.4±1.9 46.2 ±3.6 38.2±2.9 53.7 ±3.3 49.2 ±4.2 42.0±4.8 65.0±0.8</td><td></td><td></td><td></td><td>63.8± 0.7</td><td>56.9± 2.4</td></tr></table>",
+        "bbox": [
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+            265,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/38d1594b95dc2e48f5e2eecb8326badd4dc4008b0f98a7212c937c77bd8ca190.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>ShareTrain/Test</td><td>Share for t</td><td>Bedroom-28</td><td>FFHQ-34</td></tr><tr><td>+</td><td>+</td><td>49.3 ± 1.9</td><td>59.1 ± 1.4</td></tr><tr><td>+</td><td>=</td><td>49.1 ± 2.2</td><td>59.3 ± 1.5</td></tr><tr><td>-</td><td>-</td><td>48.9 ± 1.6</td><td>59.3 ± 1.4</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 5: Performance of the DDPM-based method for different feature extraction variations. All considered stochastic options provide a similar mIoU to the determinstic one. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The results are provided in Table 5. As one can see, the difference in the performance is marginal. We attribute this behavior to the following reasons: ",
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+    {
+        "type": "text",
+        "text": "• Our method uses later $t$ of the reverse diffusion process where the noise magnitude is low. • Since we exploit the deep layers of the UNet model, the noise might not affect the activations from these layers significantly. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Robustness to input corruptions. In this experiment, we investigate the robustness of DDPMbased representations. First, we learn pixel classifiers on the clean images using the DDPM, SwAV and MAE representations on the Bedroom-28 and Horse-21 datasets. Then, 18 diverse corruption types, adopted from (Hendrycks & Dietterich, 2019), are applied to test images. Each corruption has five levels of severity. In Figure 6, we provide mean IoUs computed over all corruption types for 1, 3, 5 levels of severity, denoted as “weak”, “medium” and “strong”, respectively. ",
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+    },
+    {
+        "type": "text",
+        "text": "One can observe that the proposed DDPM-based method demonstrates higher robustness and preserves its advantage over the SwAV and MAE models even for severe image distortions. ",
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+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This paper demonstrates that DDPMs can serve as representation learners for discriminative computer vision problems. Compared to GANs, diffusion models allow for a straightforward computation of these representations for real images, and one does not need to learn an additional encoder, which maps images to the latent space. This DDPM’s advantage and superior generative quality provide state-of-the-art performance in the few-shot semantic segmentation task. The notable restraint of the DDPM-based segmentation is a requirement of high-quality diffusion models trained on the dataset at hand, which can be challenging for complex domains, like ImageNet or MSCOCO. However, given the rapid research progress on DDPM, we expect they will reach these milestones in the nearest future, thereby extending the range of applicability for the corresponding representations. ",
+        "bbox": [
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+    },
+    {
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+        "text": "APPENDIX ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "A EVOLUTION OF PREDICTIVE PERFORMANCE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            145,
+            575,
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+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/9336d075211187cebcba86e23520979299dff826590458419e8a1314a5dc8b91.jpg",
+        "image_caption": [
+            "Figure 7: The evolution of predictive performance of DDPM-based pixel-wise representations for different UNet blocks and diffusion steps on LSUN-Cat and LSUN-Bedroom. The blocks are numbered from the deep to shallow ones. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/1b63c382cd2bf56e9b8c04caee70ec3bc0c207c8521a06e0c72a678d7c57555a.jpg",
+        "image_caption": [
+            "Figure 8: The evolution of predictive performance of DDPM-based pixel-wise representations on the FFHQ-256, LSUN-Cat and LSUN-Bedroom datasets for classes with the smallest (Left) and largest (Right) average areas. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            169,
+            445,
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+            835
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/932ce959597ee947cc0a966d848bea605301dc4af340c4d3bea54d07e34329f6.jpg",
+        "table_caption": [
+            "B DATASETDDPM & DATASETGAN SATURATION ",
+            "Table 6: Performance of DatasetDDPM and DatasetGAN for $1 0 K { - } 5 0 K$ synthetic images in the training dataset. Mean IoU of both methods saturates at $3 0 K { - } 5 0 K$ of synthetic data. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"7\">DatasetDDPM</td><td colspan=\"4\">DatasetGAN</td></tr><tr><td>Dataset</td><td>10k</td><td>20K</td><td>30K</td><td>40K</td><td>50K</td><td>10K</td><td>20K</td><td>30K</td><td>40K</td><td>50K</td></tr><tr><td>Bedroom-2845.1±2.346.2±2.346.1±2.847.8±2.347.9±2.930.6±2.330.4±3.1 30.9±2.430.9±2.431.3±2.7</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>FFHQ-34</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>55.9±0.8 55.8±0.7 55.9±0.7 56.0±0.8 55.9±0.7 56.4±1.0 56.9±1.0 57.0±1.1 57.0±1.2 57.0±1.2</td></tr><tr><td>Cat-15</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>43.6±3.0 46.4±1.7 46.2±1.9 47.4±1.7 47.6±1.5 34.7±2.8 34.8±2.9 36.3±2.335.8±2.5 36.5 ±2.3</td></tr><tr><td>Horse-21</td><td></td><td></td><td>57.0±1.2 59.5±0.5 59.0±2.0 60.4±1.1 60.8±0.9 41.6±2.0 43.1± 1.8 45.4±1.4 44.5±1.2 44.6 ± 1.4</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C TRAINING SETUP ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            315,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The ensemble of MLPs consists of 10 independent models. Each MLP is trained for ${ \\sim } 4$ epochs using the Adam optimizer (Kingma & Ba, 2015) with 0.001 learning rate. The batch size is 64. This setting is used for all methods and datasets. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "MLP architecture. We adopt the MLP architecture from (Zhang et al., 2021). Specifically, we use MLPs with two hidden layers with ReLU nonlinearity and batch normalization. The sizes of hidden layers are 128 and 32 for datasets with a number of classes less than 30, and 256 and 128 for others. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Also, we evaluate the performance of the proposed method for twice wider / deeper MLPs on the Bedroom-28 and FFHQ-34 datasets and do not observe any noticeable difference, see Table 7. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/f548c2d84e40f6d420b44744dfea46b149e0655b595abfa23d68d0f1f3f98ee9.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Bedroom-28</td><td>FFHQ-34</td></tr><tr><td>Original MLP</td><td>49.4</td><td>59.1</td></tr><tr><td>Wider MLP</td><td>49.5</td><td>59.1</td></tr><tr><td>Deeper MLP</td><td>49.3</td><td>58.9</td></tr></table>",
+        "bbox": [
+            375,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 7: Performance of the proposed method for twice wider / deeper MLP architecture within the ensemble. More expressive MLPs do not improve the performance. ",
+        "bbox": [
+            168,
+            577,
+            825,
+            606
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "D PER CLASS IOUS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            633,
+            352,
+            651
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/02f4b1e7cd5604421b93ea732e15e6ee885442fcc90f2409dc5fdf67449a18f4.jpg",
+        "image_caption": [
+            "Figure 9: Per class IoUs for DatasetGAN, DatasetDDPM and DDPM. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            660,
+            821,
+            888
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/ab730fa2e015904bdb829ac09c57fbd53b4a26673dd3daefc00740d32a680ac7.jpg",
+        "image_caption": [
+            "Figure 10: Number of instances of each semantic class in the annotated real and synthetic train sets. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            83,
+            826,
+            460
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "E DATASET DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            517,
+            361,
+            534
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "E.1 CLASS NAMES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            554,
+            316,
+            568
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Bedroom-28: [bed, footboard, headboard, side rail, carpet, ceiling, chandelier, curtain, cushion, floor, table, table top, picture, pillow, lamp column, lamp shade, wall, window, curtain rod, window frame, chair, picture frame, plinth, door, pouf, wardrobe, plant, table staff] ",
+        "bbox": [
+            174,
+            583,
+            825,
+            625
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "FFHQ-34: [background, head, cheek, chin, ear, helix, lobule, bottom lid, eyelashes, iris, pupil, sclera, tear duct, top lid, eyebrow, forehead, frown, hair, sideburns, jaw, moustache, inferior lip, oral commissure, superior lip, teeth, neck, nose, ala of nose, bridge, nose tip, nostril, philtrum, temple, wrinkles] ",
+        "bbox": [
+            174,
+            631,
+            825,
+            688
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Cat-15: [background, back, belly, chest, leg, paw, head, ear, eye, mouth, tongue, tail, nose, whiskers, neck] ",
+        "bbox": [
+            174,
+            695,
+            823,
+            723
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Horse-21: [background, person, back, barrel, bridle, chest, ear, eye, forelock, head, hoof, leg, mane, muzzle, neck, nostril, tail, thigh, saddle, shoulder, leg protection] ",
+        "bbox": [
+            171,
+            729,
+            823,
+            758
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "CelebA-19: [background, cloth, ear r, eye g, hair, hat, l brow, l ear, l eye, l lip, mouth, neck, neck l, nose, r brow, r ear, r eye, skin, u lip] ",
+        "bbox": [
+            171,
+            765,
+            823,
+            795
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "ADE-Bedroom-30: [wall, bed, floor, table, lamp, ceiling, painting, windowpane, pillow, curtain, cushion, door, chair, cabinet, chest, mirror, rug, armchair, book, sconce, plant, wardrobe, clock, light, flower, vase, fan, box, shelf, television] ",
+        "bbox": [
+            174,
+            800,
+            825,
+            843
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "E.2 CLASS STATISTICS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            867,
+            344,
+            881
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In Figure 10, we report the statistics of classes computed over annotated real images as well as annotated synthetic images produced by GAN and DDPM. ",
+        "bbox": [
+            173,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "F EXTRACTING REPRESENTATIONS FROM MAE ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            102,
+            583,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "To obtain pixelwise representations, we apply the model to a fully observed image (mask ratio $\\scriptstyle 1 = 0$ ) of resolution 256 and extract feature maps from the deepest 12 ViT-L blocks . The feature maps from each block have $1 0 2 4 \\times 3 2 \\times 3 2$ dimensions. Similarly to other methods, we upsample the extracted feature maps to $2 5 6 \\times 2 5 6$ and concatenate them. The overall dimension of the pixel representation is 12288. ",
+        "bbox": [
+            174,
+            133,
+            825,
+            203
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In addition, we investigated other feature extraction strategies and got the following observations: ",
+        "bbox": [
+            174,
+            210,
+            810,
+            226
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "1. Including activations from the decoder did not provide any noticeable gains;   \n2. Extracting activations right after self-attention layers caused slightly inferior performance;   \n3. Extracting activations from every second encoder block also provided a bit worse results. ",
+        "bbox": [
+            202,
+            238,
+            825,
+            291
+        ],
+        "page_idx": 14
+    }
+]

parse/dev/fARM4P0gAJV/fARM4P0gAJV.md

@@ -0,0 +1,441 @@
+# Explainable Spatio-Temporal Forecasting with Shape Functions
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 Spatio-temporal modeling and forecasting are challenging due to their complicated   
+2 spatial dependence, temporal dynamics, and scenarios. Many statistical models,   
+3 such as Spatial Auto-regression Model (SAR) and Spatial Dynamic Panel Data   
+4 Model (SDPD), are restricted by a pre-specified spatial weight matrix and thus   
+5 are limited to reflect its flexibility. Graph-based or convolution-based methods   
+6 can learn more flexible representations, but they fail to show the exact interactions   
+7 between locations due to the lack of explainability. This paper proposes a spatial re  
+8 gression model with shape functions to address the limitations of existing methods.   
+9 Our method learns the shape functions by incorporating shape constraints, which   
+10 are able to capture spatial variability or distance-based effects over distance. There  
+11 fore, our approach enjoys a learnable spatial weight matrix with a distance-based   
+12 explanation. We demonstrate our method’s efficiency and forecasting performance   
+13 on synthetic and real data.
+
+# 14 1 Introduction
+
+15 Spatio-temporal data is widely observed in many areas, such as transportation (33; 27), climatology   
+16 (2), and environmental research(19). The popularity of spatio-temporal data brings varieties of tasks   
+17 for researchers, and one of the key tasks is forecasting. Spatio-temporal data has some inherent   
+18 characteristics, namely, spatial dependence and temporal dynamics, which need to be considered for   
+19 modeling and forecasting.   
+20 Spatial dependence means that the observations at different locations are not independent, and   
+21 observations at closer locations often have a stronger correlation. In the statistics community,   
+22 extensive research has been conducted to model spatial dependence, and various spatial models have   
+23 been proposed. For example, in the spatial autoregressive (SAR) models, the spatial dependence is   
+24 modeled by a product of an unknown parameter and a pre-specified spatial weight matrix (4; 1; 11; 12).   
+25 Combined with the panel data, various types of spatial panel data models have been used to analyze   
+26 spatio-temporal data (35; 13; 7; 22). One limitation of the autoregressive models is that the elements   
+27 of the spatial weight matrix are pre-specified, such as an inverse distance. Although these pre  
+28 specified spatial weight matrices are applied to capture decreased distance-based effects, they fail to   
+29 capture complex distance relations in real-world applications.   
+30 Researchers in the computer science community have developed various methods modeling spatio  
+31 temporal data using deep neural networks. Various neural network architectures have been proposed   
+32 and applied to spatio-temporal forecasting, for example, spatio-temporal LSTM (31), fully connected   
+33 gated graph architecture (20), Convolutional LSTM (23) and etc. One advantage of these methods   
+34 is that they can incorporate unstructured data and rely on a high-performance computing platform   
+35 to learn complicated representations for spatio-temporal problems. However, a critical limitation of   
+36 these methods is that they fail to explain how the spatial interaction works explicitly. The lack of   
+37 interpretability restricts its reliability and deep insights into the underlying spatio-temporal process.   
+38 The explanation can be obtained if we can estimate the coefficient matrix that intuitively explains   
+39 spatio-temporal interactions.   
+40 In this paper, we propose an Explainable Spatio-Temporal Forecasting (ESTF) model, which utilizes   
+41 a spatial autoregressive model with shape functions to address the current limitations. Our method   
+42 extends the vector autoregressive (VAR) model (24) by incorporating distance information into the   
+43 temporal coefficient matrix using shape functions (3). The shape constraints are designed to be   
+44 consistent with the common fact that observations from neighbours have stronger spatial dependence   
+45 versus long-distance pairs. It is known as Tobler’s First Law, which is "Everything is related to   
+46 everything else, but near things are more related than distant things"(26; 18). Unlike the pre-specified   
+47 spatial weight matrix, this coefficient matrix is learnable and is thus more flexible in capturing   
+48 real-world complex spatial relations. Moreover, the shape functions are represented as a combination   
+49 of basis functions, and thus a smaller number of parameters needs to be estimated. Finally, ESTF can   
+50 be easily extended to forecasting in non-stationary scenarios using a dynamic spatial weight matrix.   
+51 We conduct experiments on both simulated and real data, and the results demonstrate that our method   
+52 achieves better forecast accuracy and is computationally efficient and more explainable.
+
+# 53 2 Related work
+
+54 Statistical models Several works focus on temporal dynamics when considering spatio-temporal   
+55 forecasting problems. The classical time series models, such as VAR, and ARIMA models, are applied   
+56 to spatio-temporal process modeling(21; 38). Besides, a spatial weight matrix is also introduced to the   
+57 ARIMA model to capture spatial dependence (28). The non-stationarity, particularly unit-root non  
+58 stationarity, is mainly modeled by ARIMA or Co-integration models. In addition, spatial regression   
+59 models or panel data are classical models in econometrics and can also be applied to model spatio  
+60 temporal problems. These models, for example, spatial auto-regression models, take spatial weight   
+61 matrix into consideration and estimate parameters in the framework of regression. However, the   
+62 common characteristics of these models need a pre-specified spatial weight matrix(35; 6). Elements   
+63 in the matrices are generally an inverse distance of corresponding locations. Meanwhile, these   
+64 spatial models focus on statistical inference on the scalar parameters placed before the spatial weight   
+65 matrix(25). Although there are many choices for the spatial weight matrix, such as inverse distance,   
+66 adjacency relationships, and K-nearest neighbors, there is a lack of research on estimating the spatial   
+67 weight matrix. The pre-specified spatial weight matrix restricts models’ application and fails to   
+68 capture more complicated underlying spatial dependence. Some researchers developed a sparse   
+69 spatio-temporal model that can estimate a sparse spatial weight matrix (17). The strict sparse setting   
+70 also restricts the wide application of the spatial weight matrix.   
+71 Graph-based methods Graph-based methods are widely applied for a non-Euclidean domain.   
+72 Some types of spatio-temporal data, for example, traffic flow data or brain network data, can be   
+73 represented as graphs. The graph structures well model the complicated spatial dependence. Thus,   
+74 the definition or pre-specified graphs structure is normally required when developing a graph-based   
+75 model. Related works can be found in (30; 14). The common typical method is GraphCNN, which is   
+76 to apply a convolutional transformation to the neighbors of each node (29; 34). The graph convolution   
+77 can capture patterns and features in the spatial domain. Graph-based methods have been proposed   
+78 and widely applied to lots of real cases. Traffic flow data modeling and forecasting is a popular topic   
+79 in this area (30; 20). Other topics, for example, climate sensor data (16), video (10) and etc, are also   
+80 applied by variant graph-based models. RNN or LSTM combined with graphs, i.e., a sequence of   
+81 graphs, are also considered in spatio-temporal forecasting problems (10).   
+82 CNN-based methods Unlike graph-based methods, CNN-based methods are more suitable for   
+83 modeling spatio-temporal data collected in regular grid locations. It applies filters to find relationships   
+84 between neighboring inputs. Although some works (32) applied convolution neural networks to   
+85 model non-grid traffic data, it is more common to see CNN-based methods process grid structures,   
+86 e.g., images, video rather than a general domain. As some spatio-temporal data are collected from a   
+87 regular grid in the Euclidean space (29), they thus can be viewed as a kind of special image. The CNN   
+88 structure combined with RNN or LSTM has been developed to make forecasting for spatio-temporal   
+89 data, for example, diffusion convolutional RNN (15), Convolutional LSTM networks (23; 36)and etc.
+
+# 90 3 Proposed method
+
+# 3.1 Problem formulation and notation
+
+We use a $n \times 1$ vector $\mathbf { X _ { t } } ~ = ~ \{ \mathbf { x _ { 1 t } } , \mathbf { x _ { 2 t } } , \cdot \cdot \cdot , \mathbf { x _ { n t } } \}$ to denote observations at time $t$ , where $n$ is the number of locations. At each location $i$ , $\bf { S _ { i } } = ( c _ { i } ^ { x } , c _ { i } ^ { y } )$ is the coordinates of the location $i$ . The distance between location $\mathbf { S _ { i } }$ and $\mathbf { S _ { j } }$ is $d _ { i j } = \sqrt { ( d _ { i j } ^ { x } ) ^ { 2 } + ( d _ { i j } ^ { y } ) ^ { 2 } }$ , where $d _ { i j } ^ { x } = | c _ { i } ^ { x } - c _ { j } ^ { x } |$ and $d _ { i j } ^ { y } = | c _ { i } ^ { y } - c _ { j } ^ { y } |$ . Our goal is to make forecasting for spatio-temporal data: given training data set $\mathbf { X _ { 1 } } , \mathbf { X _ { 2 } } , \cdots , \mathbf { X _ { T } }$ , we would like to make forecasting for the next $h$ , $\hat { \mathbf { X } } _ { T + 1 } , \cdot \cdot \cdot , \hat { \mathbf { X } } _ { T + h }$ .
+
+# 3.2 The stationary spatio-temporal model with shape functions
+
+98 We first consider the stationary case. To model the spatio-temporal stationary process, we consider   
+99 the following model
+
+$$
+\mathbf { X _ { t } } = \sum _ { \mathbf { k } = 1 } ^ { \mathbf { p } } \mathbf { W _ { k } } \mathbf { X _ { t - k } } + \epsilon _ { \mathbf { t } } ,
+$$
+
+100 where $\mathbf { W _ { k } }$ is a spatial weight matrix for capturing the spatial dependence at $\log k$ , and $\epsilon _ { \mathbf { t } }$ is white noise. Moreover, we assume the 101 $( i , j )$ th element of $\mathbf { W _ { k } }$ , $w _ { i j } ^ { ( k ) }$ , depends on the distance $d _ { i j }$ . That is, 102 w(k)ij depends on a function $f _ { k } ( d _ { i j } )$ .
+
+103 For spatio-temporal data, the spatial dependence, represented by $w _ { i j } ^ { ( k ) }$ , between locations decreases   
+104 as the distance between two locations increases. In other words, there is a shape constraint for   
+105 the function $f _ { k } ( d )$ , such as a decreasing function. In order to estimate the shape function, we   
+106 model $f _ { k } ( d )$ as a linear combination of basis functions $g _ { i } ( d ) , i = 1 , 2 , \cdots , m$ . More specifically,   
+107 the shape function $f _ { k } ( d )$ is a linear combination of basis functions and coefficients with positive   
+108 value $\bar { f _ { k } } ( d ) = a _ { 1 , k } ^ { 2 } \bar { g _ { 1 } } ( \dot { d } ) + \cdot \cdot \cdot + a _ { m , k } ^ { 2 } g _ { m } ( d )$ , where $a _ { 1 , k } , \cdots , a _ { m , k }$ are parameters to be estimated.   
+109 The constraint of decrease needs parameters non-negative and thus each parameters squared. The   
+110 spatial weight matrix can take the value of decreased shape function directly. The element of $\mathbf { W _ { k } }$   
+111 is $w _ { i j . } ^ { ( k ) } = f _ { k } ( d _ { i j } )$ . The details of the shape function and the corresponding basis functions can be   
+112 found in Section 3.4
+
+The parameters in shape functions can be estimated from the neural network illustrated in Figure 1. The neural network can be trained from the following criterion:
+
+$$
+\operatorname* { m i n } _ { \{ W _ { k } \} _ { k = 1 } ^ { p } } \sum _ { t = 1 } ^ { T } | | \mathbf { X _ { t } } - \hat { \mathbf { X _ { t } } } | | ^ { 2 } = \sum _ { \mathbf { t } = 1 } ^ { \mathbf { T } } | | \mathbf { X _ { t } } - \sum _ { \mathbf { k } = 1 } ^ { \mathbf { p } } \hat { \mathbf { W _ { k } } } \hat { \mathbf { X _ { t - k } } } | | ^ { 2 } .
+$$
+
+# 113 3.3 The non-stationary spatio-temporal model with time-variant shape functions
+
+114 The static spatial weight matrix $\mathbf { W _ { k } }$ can reflect spatial dependence and thus can be applied to   
+115 stationary scenarios. Next, we consider the nonstationary case. Therefore, we extend the stationary   
+116 model to non-stationary cases. The spatial weight matrices only reflect static relationships across time   
+117 lags in the static model. Unlike these settings, we change spatial weight matrices to be time-variant.   
+118 The spatial weight matrices formed by time-variant shape functions can thus capture non-stationary   
+119 dynamic spatial dependence. The non-stationary model has the form below,
+
+$$
+\mathbf { X _ { t } } = \sum _ { \mathbf { k } = 1 } ^ { \mathbf { p } } \mathbf { W _ { t , k } } \mathbf { X _ { t - k } } + \epsilon _ { \mathbf { t } } .
+$$
+
+where $\epsilon _ { \mathbf { t } }$ is white noise, and $\mathbf { W _ { t , k } }$ relies on shape function $f _ { t , k } ( d )$ . Similar with stationary settings, the time-variant shape functions are still represented as a linear combination of basis functions $g _ { i } ( d ) , i = 1 , 2 , \cdots , \bar { m }$ . The coefficients are therefore time-variant. The shape function at time $t$ has the form below $f _ { t , k } ( d ) = a _ { 1 , t , k } ^ { 2 } g _ { 1 } ( d ) + \cdot \cdot \cdot + a _ { m , t , k } ^ { 2 } g _ { m } ( d )$ . Unlike stationary setting, the coefficients of nonstationary setting, $\{ a _ { i , t , k } \} _ { i = 1 } ^ { m }$ , depend on the time $t$ . The non-stationary model can be trained from the criterion by minimizing
+
+$$
+\operatorname* { m i n } _ { \{ W _ { t } , k \} _ { k = 1 } ^ { p } } | | \mathbf { X _ { t } } - \hat { \mathbf { X _ { t } } } | | ^ { 2 } = | | \mathbf { X _ { t } } - \sum _ { \mathbf { k } = 1 } ^ { \mathbf { p } } \hat { \mathbf { W } } _ { \mathbf { t } , \mathbf { k } } \hat { \mathbf { X } } _ { \mathbf { t - k } } | | ^ { 2 } .
+$$
+
+![](images/1b8f28065b5df206375a0dece1e19a6c9d354e44db886268da7cfb359b1d8c90.jpg)  
+Figure 1: The neural network for the stationary spatio-temporal process (left) and non-stationary spatio-temporal process (right).
+
+# 122 3.4 The basis functions for shape functions
+
+123 The shape functions are integrated into our model to obtain distance-based explanations in stationary   
+124 and non-stationary scenarios. The motivation of the proposed shape functions is that as the distance   
+125 between two observations increases, the effects between these two locations decreases. These distance  
+126 based effects can be reflected in spatial weight matrix W and each element in the matrix can measure   
+127 how the corresponding locations interact. The shape function is represented as a linear combination of   
+128 basis functions. The basis functions, satisfying shape constraint, rely on the corresponding definition   
+129 of basis functions.   
+130 Definition of basis functions for various shape constraints. We list the definition of basis functions   
+131 for increased and decreased shape (3). The distance quantile among $\{ d _ { i _ { 1 } , j _ { 1 } } , d _ { i _ { 2 } , j _ { 2 } } , \dots , d _ { i _ { N } , j _ { N } } \}$ at   
+132 quantile level $q _ { 1 } , q _ { 2 } , \cdots , q _ { m }$ is denoted by $\{ d _ { ( 1 ) } , d _ { ( 2 ) } , \cdots , d _ { ( m ) } \}$ , where $0 \leq q _ { 1 } < q _ { 2 } < \cdot \cdot \cdot < q _ { m } \leq$   
+1 and 133 $\{ q _ { 1 } , q _ { 2 } , \cdot \cdot \cdot , q _ { m } \} = \{ { \textstyle \frac { 1 } { m } } , { \textstyle \frac { 2 } { m } } , \cdot \cdot \cdot , 1 \}$ . Here, we can set the number of $m < < n ^ { 2 }$ , and thus, the   
+number of parameters is significantly reduced.   
+135 For the constraint of monotone decreasing function, the basis function is defined as $g _ { i } ( d ) = \mathbf { 1 } _ { \left\{ \mathbf { d } < \mathbf { d } _ { \left( \mathbf { i } \right) } \right\} }$ .   
+136 The basis function for the shape function with the constraint of concave decrease is defined as   
+137 $g _ { i } ( d ) = ( d _ { ( i ) } - d ) \mathbf { 1 } _ { \{ \mathbf { d } _ { ( i ) } \leq \mathbf { d } \} }$ and convex decrease is defined as $g _ { i } ( d ) = ( d _ { ( i ) } - d ) \mathbf { 1 } _ { \{ \mathbf { d } \leq \mathbf { d } _ { ( i ) } \} }$ , for   
+138 $1 \leq i \leq m$ . Figure 2 shows the definition of basis functions for monotone decreased and increased   
+139 shape functions, respectively. We only present four basis functions for each shape and each of them   
+140 is related to four quantile levels. The dashed lines indicate the turning points for each basis function   
+141 and they equal one or zero at the beginning and turn to zero or one at turning points.   
+143 The stationary model requires fixed shape functions and related spatial weight matrix are time  
+144 invariant. Given training data set $\mathbf { X _ { 1 } } , \mathbf { X _ { 2 } } , \cdots , \mathbf { X _ { T } }$ , we can estimate spatial weight matrix   
+145 ${ \hat { W } } _ { 1 } , { \hat { W } } _ { 2 } , \cdot \cdot \cdot , { \hat { W } } _ { p }$ and make forecasting iteratively. That is $\begin{array} { r } { \hat { { \bf X } } _ { T + 1 } = \sum _ { k = 1 } ^ { p } \hat { W } _ { k } { \bf X _ { T + 1 - k } } , \hat { { \bf X } } _ { T + 2 } = } \end{array}$   
+146 $\begin{array} { r } { \hat { W } _ { 1 } \hat { \mathbf { X } } _ { T + 1 } + \sum _ { k = 2 } ^ { \bar { p } } \hat { W } _ { k } \mathbf { X } _ { \mathbf { T } + 2 - \mathbf { k } } , \cdot \cdot \cdot \hat { \mathbf { X } } _ { T + h } = \sum _ { k = 1 } ^ { p } \hat { W } _ { k } \hat { \mathbf { X } } _ { T + h - k } } \end{array}$ .
+
+![](images/8db77b9cc5e93deb1966a1a21c820773c0c8a69b925b2d3a8fdceab9ce3b175b.jpg)  
+Figure 2: The basis functions for decreased shape (left) and for increased shape (right). The arrows indicate domain of each basis functions.
+
+The non-stationary model incorporate time-variant spatial weight matrix ${ \hat { W } } _ { t , \cdot } .$ . Given the training data set $\mathbf { X _ { 1 } } , \mathbf { X _ { 2 } } , \cdots , \mathbf { X _ { T } }$ , we can obtain corresponding shape functions $\hat { f } _ { 1 , \cdot } , \hat { f } _ { 2 , \cdot } , \cdot \cdot \cdot , \hat { f } _ { T , \cdot }$ , where · denotes time lag. For lag $p = 1$ , we can use $\{ \hat { f } _ { t } \} _ { t = 1 } ^ { T }$ to represent time-variant shape functions for convenience. We can make dynamic forecasts for the next $h$ windows. One simple forecasting method is to use $\hat { W } _ { T , k }$ to make forecast for $\hat { \mathbf { X } } _ { T + h }$ , that is
+
+$$
+\hat { \mathbf { X } } _ { T + h } = \sum _ { k = 1 } ^ { p } \hat { W } _ { T , k } \mathbf { X } _ { \mathbf { T } + \mathbf { h } - \mathbf { k } } .
+$$
+
+The alternative method is to retrain the new forecast to obtain the latest shape functions as well as spatial weight matrix. Given long-term forecast window $L$ , we first make short-term forecast for $h$ steps
+
+$$
+\hat { \mathbf { X } } _ { T + h } = \sum _ { k = 1 } ^ { p } \hat { W } _ { T + h , k } \mathbf { X } _ { \mathbf { T } + \mathbf { h } - \mathbf { k } } ,
+$$
+
+where 147 $h = 1 , 2 , \cdots$ and $\hat { W } _ { T + h , k }$ is estimated by training forecast value of $\hat { \mathbf { X } } _ { T + h - k }$ . We repeat the 148 process until $L$ steps in total have been predicted.
+
+49 We summarize the whole process of our model when making spatio-temporal forecasts.
+
+Step 1 Given the observation $\{ \mathbf { X _ { t } } \} _ { \mathbf { t = 1 } } ^ { \mathbf { T } }$ and its coordinates, calculate all distance pairs among all locations, denoted by $\{ d _ { i _ { 1 } , j _ { i } } , \cdot \cdot \cdot , d _ { i _ { N } , j _ { N } } \}$ .
+
+Step 2 Calculate $\textstyle \left\{ { \frac { 1 } { m } } , { \frac { 2 } { m } } , \cdots , 1 \right\}$ quantile levels and obtain corresponding distance quantile value $\{ d _ { ( 1 ) } , d _ { ( 2 ) } , \cdots , \ " { d _ { ( m ) } } \}$ .
+
+Step 3 Determine the shape constraints and construct corresponding basis functions. Specify the time lag $p$ .
+
+Step 4 Train the model according to the illustration of Figure 1.
+
+# 157 4 Experiment
+
+158 In order to assess our model in stationary and non-stationary scenarios, we synthesize data. Then,   
+159 we apply our model to make some comparisons. On the one hand, we need to evaluate how   
+160 the estimated shape functions look and assess their similarity and accuracy. On the other hand,   
+161 our model can make spatio-temporal forecasting after estimating for spatial weight matrix. The   
+162 basic idea for completing the two goals is to set up the expected shape function and compare   
+163 estimated parameters with the real one. Next, we assess the forecasting performance with baseline   
+164 models. Codes and data for replicating our experiments are anonymously published at https:   
+165 //anonymous.4open.science/r/STVAR-F16E/.
+
+# 166 4.1 Simulation for stationary model
+
+Here, we synthesize 100 stationary spatio-temporal data sets. The spatial domain consists of 30 locations and their coordinates can be found at https://anonymous.4open.science/r/ STVAR-F16E/. For each location, we observe 500 values. The observation is generated from the stationary model $\begin{array} { r } { X _ { t } = \sum _ { k = 1 } ^ { p } W _ { k } X _ { t - k } + \epsilon _ { t } } \end{array}$ , where $\epsilon _ { t }$ is randomly generated from the standard normal distribution. The next step is to construct random spatial weight matrices for each synthesized data set. The shape functions are set to be decreasing, and we set them as a logarithmic function:
+
+$$
+\alpha ( - \log ( d + 1 ) + \log ( 1 7 0 ) ) ,
+$$
+
+167 where $\alpha$ is randomly generated from uniform distribution [0.05,0.06] but kept to be fixed for each   
+168 simulated data set. We use $d + 1$ to avoid zero value. This setting can make the real shape function   
+169 decrease and make it equal to zero when $d = 1 6 9$ . The stationary model can iteratively generate the   
+170 $\mathbf { X _ { t } }$ given initial value $\mathbf { X _ { 0 } }$ , where $\mathbf { X _ { 0 } }$ is randomly generated from a uniform distribution with bounds   
+171 [-0.01,0.01]. The time lags are set as $p = 1$ .   
+172 Estimation for shape functions. In Figure 3,   
+173 the estimated shape function is presented in red,   
+174 while the real shape function is presented in blue.   
+175 It can be seen that the estimated shape function   
+176 can capture the trend of the real shape function.   
+177 Training details. The first 300 steps are used as   
+178 training data, saving the last 200 steps for eval  
+179 uation. We train all models for 100 epochs with   
+180 Adam optimizer (5) and a learning rate of 0.01.   
+181 The process involves parallel training across 10   
+182 CPUs. We select 100 quantile levels, and thus   
+183 100 basis functions $g _ { i } ( d )$ were generated as the   
+184 inputs for the model.   
+185 Assessment for forecasting. We assess the fore  
+186 casting performance for the stationary model   
+187 with baseline models. As introduced in the liter  
+188 ature review, the baseline models are selected from the VAR model(21), the spatial panel data(SPE)   
+189 model that applied pre-specified spatial weigh matrix (28), graph-based models (20; 37) and   
+190 convolution-based models (15; 23). The error metrics are mean absolute error and root mean squared   
+191 error defined by 1Nn PNj=1 Pni=1 $\begin{array} { r } { \frac { 1 } { N n } \sum _ { j = 1 } ^ { N } \sum _ { i = 1 } ^ { n } \frac { \sum _ { t = T } ^ { T + h } | \hat { X } _ { i t } ^ { ( j ) } - X _ { i t } ^ { ( j ) } | } { h } } \end{array}$ , $\begin{array} { r } { \frac { 1 } { N n } \sum _ { j = 1 } ^ { N } \sum _ { i = 1 } ^ { n } \sqrt { \frac { 1 } { h } \sum _ { t = T } ^ { T + h } ( X _ { i t } ^ { ( j ) } - \hat { X } _ { i t } ^ { ( j ) } ) ^ { 2 } } } \end{array}$ ,   
+192 respectively. The Table 1 shows the six baseline models with the proposed model. As totally we   
+193 have 100 synthesised data sets, $X _ { i t } ^ { ( j ) }$ and $\hat { X } _ { i t } ^ { ( j ) }$ denote $i - t h$ variable in $j - t h$ data sets. $n = 3 0$ is   
+194 the number of locations and $N = 1 0 0$ is the number of synthesised data. We conducted one-step   
+195 forecasting for the next 200 observations.   
+96 Compared with baseline models, the proposed model performs better under the metric MAE and   
+97 RMSE. The proposed method outperforms the closest competing method, DC-RNN, by $10 \%$ .
+
+![](images/3f90af4b01d3f86e6ea9005e43f224ee9862c72c8e5e1f8c38e688ce1b660d08.jpg)  
+Figure 3: The sample of estimated shape function. Distances are shown every $2 0 ^ { t h }$ quantile.
+
+# 198 4.2 Experiments for non-stationary model
+
+We conduct a simulation for the non-stationary model with time lag $p = 1$ and synthesize 100 data sets using a similar approach to the stationary model simulation. The initial value $\mathbf { X _ { 0 } }$ and $\epsilon _ { t }$ are generated from a uniform and normal distribution respectively. The locations of observations are the same as those in the stationary model simulation. In order to construct $W _ { t }$ , the time-varying shape functions are created under the decreased constraint. The shape function at time $t$ is constructed as
+
+$$
+\alpha _ { t } ( - \log ( d + 1 ) + \log ( 1 7 0 ) ) ,
+$$
+
+99 where $\alpha _ { t }$ controls the level of value at each time $t$ . $\epsilon _ { t }$ is generated from a normal distribution. $\mathbf { X _ { 0 } }$ is   
+00 generated from a uniform distribution with bound [-0.001,0.001].   
+201 Shape functions settings and estimation. The shape functions are set as time-variant, as they can   
+202 simulate the non-stationary process across time. We specified $\alpha _ { 0 }$ at $t = 0$ from uniform distribution   
+203 $[ 1 \times 1 0 ^ { - 4 } , 2 \times 1 0 ^ { - 4 } ]$ and then make an interpolation from $\alpha _ { 0 }$ to $\alpha _ { 5 0 0 }$ . The total length for every   
+204 location is 500 and we set $\alpha _ { 5 0 0 } = 1 0 \times \alpha _ { 0 }$ . For example, generally if $\alpha _ { 0 } ~ = ~ 0 . 0 0 0 1$ , we have   
+205 $\begin{array} { r } { \alpha _ { t } = 0 . 0 0 0 1 ( 1 - \frac { t } { T } ) + 0 . 0 0 1 \frac { t } { T } } \end{array}$ , where $T = 5 0 0$ . This setting guarantee that shape functions vary   
+206 from lower level to higher level. The larger $\alpha _ { t }$ is, the more larger distance-based effects they have.   
+207 Thus, the corresponding spatial weight matrix consists of dynamic shape functions and can reflect the   
+208 non-stationary dependence among each site. We present the estimated shape functions in Figure 4   
+209 and compare them with the real ones.   
+10 Training details. Similar to the stationary simulation, the train-test split is $3 0 0 - 2 0 0$ over the data   
+11 size of 500. However, we train all models for 100 epochs with Adam optimizer (5) at a learning rate   
+12 of 0.001. We train models in parallel across 10 CPUs.
+
+![](images/1bb651c4367ceea65a5aca2eebdb4f0acb26ea67b9ed6301cd50eeef841b3a2f.jpg)  
+Figure 4: The sample of estimated shape function for the 120 testing time steps. Distances are shown every $4 0 ^ { t h }$ quantile.
+
+Forecasting performance. The forecasting performance is assessed by the same metrics used in the previous simulation for the stationary case. We made a one-step forecast by our model. As for the baseline models, we adjusted their published code accordingly. The results show that the proposed model can still capture non-stationary processes compared with baseline models. The proposed method outperforms the other competing methods. The error metric is shown in Table 1.
+
+Table 1: The error metrics with baseline models for simulation.   
+
+<table><tr><td rowspan="2">Methods</td><td colspan="2">Stationary Simulation</td><td colspan="2">Non-stationary Simulation</td></tr><tr><td>MAE</td><td>RMSE</td><td>MAE</td><td>RMSE</td></tr><tr><td>VAR</td><td>2.9611 ± 1.8573</td><td>3.2588 ± 1.8077</td><td>2.4426 ±1.2285</td><td>2.7676 ± 1.2015</td></tr><tr><td>SPM</td><td>1.8850 ± 0.6348</td><td>1.8671 ± 0.6778</td><td>2.1918 ± 0.7350</td><td>2.2161 ± 0.6876</td></tr><tr><td>DC-RNN</td><td>0.8960 ± 0.0370</td><td>1.1168 ± 0.0426</td><td>0.9017 ± 0.0358</td><td>1.1328 ± 0.0463</td></tr><tr><td>FC-GAGA</td><td>2.5425 ± 0.2965</td><td>3.1066 ± 0.3633</td><td>1.0270 ± 0.0080</td><td>1.2939 ± 0.0120</td></tr><tr><td>GMAN</td><td>1.6806 ± 0.1491</td><td>1.9293 ± 0.1483</td><td>1.5714 ± 0.1104</td><td>1.8608 ± 0.1155</td></tr><tr><td>ConvLSTM</td><td>2.9495 ± 0.2980</td><td>3.2509 ± 0.2887</td><td>2.2478 ± 0.2295</td><td>2.5469 ± 0.2324</td></tr><tr><td>ESTF</td><td>0.7997 ± 0.0015</td><td>1.0017 ± 0.0016</td><td>0.8075 ± 0.0016</td><td>1.0112 ± 0.0020</td></tr></table>
+
+# 218 4.3 Real case studies
+
+Air quality data. We apply our model to air quality data, which records air quality in California over 2021 1. The daily mean of $\mathrm { P M } 2 . 5$ is recorded across 172 sites.
+
+We obtain the first 200 steps for training and perform forecasting for the next 165 steps. All models are trained for 100 epochs using Adam optimizer (5), at a learning rate of 0.01 and batch size of 50. We present the estimated time-variant shape functions in supplemental file. The value of shape functions decays to zero at around 5.926, which is $80 \%$ quantile in the sample of distance pairs. In other words, the distance-based effects decay to zero at a distance equal or larger than 5.926. Our
+
+226 model has ideal performance with low time consummation compared with baseline models. We put   
+227 detailed forecasting results of simulation and real cases in a supplemental file.   
+228 The result is shown in Table 2. The ESTF performs best in terms of RMSE, while the DC-RNN   
+229 method performs best in terms of MAE. For the computational time, the ESTF method is significantly   
+230 faster than most machine learning methods, and only takes around $1 / 1 0$ time of DC-RNN. In Figure 5,   
+231 MAE, RMSE, and time are presented with different numbers of $m$ . As $m$ increases, the computational   
+232 time increases while both MAE and RMSE decrease. There is a significant increase in the forecasting   
+233 performance when $m$ increases from 10 to 50. For $m > 5 0$ , the forecasting performance does not   
+234 increase much as $m$ increases.   
+235 One key advantage of the ESTF method is that we can make an explicit distance-based explanation   
+236 for our dataset. Figure 6 shows the distance-based effects at time $t = 9$ . We only present the effects   
+237 using a threshold to obtain a more concise visualization. The estimated shape function $\hat { f } _ { 9 }$ ranges   
+238 from 0 to 9.8 and we set 5 as the threshold. The red line indicates the value of the shape function   
+239 larger than 7, while the gray line indicates the value between 5 and 7. Figure 6 shows how any two   
+240 locations interact and measure the distance-based effects quantitatively. For example, air quality   
+241 monitoring sites around the Greater Los Angeles(red circle in Figure 6) area have a strong spatial   
+242 interaction with each other, such as node 7 and node 8.
+
+![](images/dd4d473906519ef5ba947df7fea59f24905c50959bed2dd219ffc764876dfad4.jpg)  
+Figure 6: The significant distance-based effect Figure 5: Comparing efficiency vs. performanceamong all 30 locations. trade-off at different quantile values.
+
+![](images/46cdb021a33e3b06570a98c9f9aad44cf364e116f094694b1e15aa7d5b312e72.jpg)
+
+# 4.4 $S O _ { 2 }$ data
+
+Texas is the second largest manufacturing state in the USA and prediction for $S O _ { 2 }$ is critical task for researchers. The data 2 records daily $S O _ { 2 }$ at 31 locations in 2021. More detailed spatial information can be found in the supplemental file. The numeric result is listed in Table 2.
+
+Table 2: The error metrics with baseline models for real case study. Clock time (in seconds) for real case study is recorded when training each model for 100 epochs on a single CPU.   
+
+<table><tr><td rowspan="2">Methods</td><td colspan="6">Air quality data</td><td rowspan="2">SO2 data Training Time (s)</td><td rowspan="2">Inference Time (s)</td></tr><tr><td>MAE</td><td>RMSE</td><td>Training time (s)</td><td>Inference Time (s)</td><td>MAE</td><td>RMSE</td></tr><tr><td>VAR</td><td>16.9844</td><td>22.3410</td><td>3.56</td><td>0.04</td><td>6.2705</td><td>9.1388</td><td>3.330</td><td>0.016</td></tr><tr><td>SPM</td><td>8.4547</td><td>13.8262</td><td>0.31</td><td>0.03</td><td>7.1453</td><td>9.1086</td><td>0.143</td><td>0.027</td></tr><tr><td>DC-RNN</td><td>4.7157</td><td>9.3873</td><td>203</td><td>1.211</td><td>3.5094</td><td>6.8681</td><td>264.215</td><td>1.366</td></tr><tr><td>FC-GAGA</td><td>7.8671</td><td>18.1870</td><td>181</td><td>2.759</td><td>4.5976</td><td>7.7528</td><td>169.425</td><td>2.889</td></tr><tr><td>GMAN</td><td>12.5268</td><td>17.3817</td><td>140</td><td>1.823</td><td>4.1099</td><td>7.4806</td><td>172.016</td><td>1.581</td></tr><tr><td>ConvLSTM</td><td>12.6292</td><td>17.9149</td><td>53</td><td>1.940</td><td>4.1445</td><td>8.0688</td><td>96.233</td><td>1.656</td></tr><tr><td>ESTF</td><td>5.2237</td><td>9.2169</td><td>22</td><td>1.625</td><td>4.2966</td><td>6.8307</td><td>31.050</td><td>1.868</td></tr></table>
+
+247 Similar conclusions can be drawn in $S O _ { 2 }$ data as that of air quality data. The ESTF model performs   
+248 best under the RMSE metric, while DC-RNN is best in the MAE metric. In terms of training time,   
+251 The efficiency analysis and performance at different quantiles are shown in Figure 7. Together with   
+252 Figure 5, we can see that the increasing number of basis functions does not have much improvement   
+253 when the number of basis functions is larger than 50, while the training time increases as the number   
+254 of basis functions increases. The spatial distribution at time $t = 9 0$ is presented in Figure 8 where   
+255 coordinates are denoted by latitude and longitude. Two significant clusters, representing Houston   
+256 and Dallas respectively, have the strongest distance-based effect. It quantitatively shows how these   
+257 neighbors affect each other. Counties around Dallas-Fort Worth metropolitan area show strong   
+258 interaction, which should be noted by environmental policy-makers. More detailed results are   
+259 presented in the supplemental file.
+
+![](images/99b92a32e5602d31d5e3d0bd9d9e3978f4094cd4390f92648068ad7dc59e4c58.jpg)  
+Figure 8: The significant distance-based effect Figure 7: Comparing efficiency vs. performanceamong all 31 locations at $t = 9 0$ trade-off at different quantile values.
+
+![](images/235426f14840fbb29557b205ac1dba42bc7e460606884504bc3963cd61bc31a0.jpg)
+
+# 260 5 Discussion
+
+This paper applies learnable shape functions to capture distance-based effects. It can model dynamic spatial dependence for stationary and non-stationary spatio-temporal data based on their distance. The model does not have the limitations of classical statistical spatial models and provides a more explanatory model than usual deep learning methods. Furthermore, some spatio-temporal data, such as temperature for sea surface and air quality monitoring data, usually viewed as collected from the continuous field, are more suitable for the proposed models since these kinds of data follow the basic rule that variability between two locations is significantly affected by their distance. However, some spatio-temporal data, such as traffic flow or some biology data, do not follow the rule. As a result, the spatial dependence may rely on road structure or biological mechanisms instead of distance. It is worth researching such data by considering graph structure when estimating spatial weight matrix. In addition, we can develop spatio-temporal causal inference based on the ESTF model. Grander causal analysis can be done by fitting the first-order VAR model (24). The estimation of the coefficients matrix of the VAR model attracts researchers’ interest as it can be treated as a causal transition matrix. In the causal inference community, lots of work have been conducted on the VAR model (8; 9). However, there is a lack of research on causal inference under the spatio-temporal process. The quantitative distance-based effects in ESTF can be further researched and extended to develop a spatio-temporal causal model.
+
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+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] We describe limitations in discussion section   
+(c) Did you discuss any potential negative societal impacts of your work? [No]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
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+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We upload data and code to github   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] Please check training details in simulation and real case section   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] They are included in training details
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
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+(a) If your work uses existing assets, did you cite the creators? [Yes]   
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+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
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parse/dev/fARM4P0gAJV/fARM4P0gAJV_content_list.json

@@ -0,0 +1,1161 @@
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+        "text": "Explainable Spatio-Temporal Forecasting with Shape Functions ",
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+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
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+        "text": "Abstract ",
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+        "text": "1 Spatio-temporal modeling and forecasting are challenging due to their complicated   \n2 spatial dependence, temporal dynamics, and scenarios. Many statistical models,   \n3 such as Spatial Auto-regression Model (SAR) and Spatial Dynamic Panel Data   \n4 Model (SDPD), are restricted by a pre-specified spatial weight matrix and thus   \n5 are limited to reflect its flexibility. Graph-based or convolution-based methods   \n6 can learn more flexible representations, but they fail to show the exact interactions   \n7 between locations due to the lack of explainability. This paper proposes a spatial re  \n8 gression model with shape functions to address the limitations of existing methods.   \n9 Our method learns the shape functions by incorporating shape constraints, which   \n10 are able to capture spatial variability or distance-based effects over distance. There  \n11 fore, our approach enjoys a learnable spatial weight matrix with a distance-based   \n12 explanation. We demonstrate our method’s efficiency and forecasting performance   \n13 on synthetic and real data. ",
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+        "type": "text",
+        "text": "14 1 Introduction ",
+        "text_level": 1,
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+        "text": "15 Spatio-temporal data is widely observed in many areas, such as transportation (33; 27), climatology   \n16 (2), and environmental research(19). The popularity of spatio-temporal data brings varieties of tasks   \n17 for researchers, and one of the key tasks is forecasting. Spatio-temporal data has some inherent   \n18 characteristics, namely, spatial dependence and temporal dynamics, which need to be considered for   \n19 modeling and forecasting.   \n20 Spatial dependence means that the observations at different locations are not independent, and   \n21 observations at closer locations often have a stronger correlation. In the statistics community,   \n22 extensive research has been conducted to model spatial dependence, and various spatial models have   \n23 been proposed. For example, in the spatial autoregressive (SAR) models, the spatial dependence is   \n24 modeled by a product of an unknown parameter and a pre-specified spatial weight matrix (4; 1; 11; 12).   \n25 Combined with the panel data, various types of spatial panel data models have been used to analyze   \n26 spatio-temporal data (35; 13; 7; 22). One limitation of the autoregressive models is that the elements   \n27 of the spatial weight matrix are pre-specified, such as an inverse distance. Although these pre  \n28 specified spatial weight matrices are applied to capture decreased distance-based effects, they fail to   \n29 capture complex distance relations in real-world applications.   \n30 Researchers in the computer science community have developed various methods modeling spatio  \n31 temporal data using deep neural networks. Various neural network architectures have been proposed   \n32 and applied to spatio-temporal forecasting, for example, spatio-temporal LSTM (31), fully connected   \n33 gated graph architecture (20), Convolutional LSTM (23) and etc. One advantage of these methods   \n34 is that they can incorporate unstructured data and rely on a high-performance computing platform   \n35 to learn complicated representations for spatio-temporal problems. However, a critical limitation of   \n36 these methods is that they fail to explain how the spatial interaction works explicitly. The lack of   \n37 interpretability restricts its reliability and deep insights into the underlying spatio-temporal process.   \n38 The explanation can be obtained if we can estimate the coefficient matrix that intuitively explains   \n39 spatio-temporal interactions.   \n40 In this paper, we propose an Explainable Spatio-Temporal Forecasting (ESTF) model, which utilizes   \n41 a spatial autoregressive model with shape functions to address the current limitations. Our method   \n42 extends the vector autoregressive (VAR) model (24) by incorporating distance information into the   \n43 temporal coefficient matrix using shape functions (3). The shape constraints are designed to be   \n44 consistent with the common fact that observations from neighbours have stronger spatial dependence   \n45 versus long-distance pairs. It is known as Tobler’s First Law, which is \"Everything is related to   \n46 everything else, but near things are more related than distant things\"(26; 18). Unlike the pre-specified   \n47 spatial weight matrix, this coefficient matrix is learnable and is thus more flexible in capturing   \n48 real-world complex spatial relations. Moreover, the shape functions are represented as a combination   \n49 of basis functions, and thus a smaller number of parameters needs to be estimated. Finally, ESTF can   \n50 be easily extended to forecasting in non-stationary scenarios using a dynamic spatial weight matrix.   \n51 We conduct experiments on both simulated and real data, and the results demonstrate that our method   \n52 achieves better forecast accuracy and is computationally efficient and more explainable. ",
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+        "text": "53 2 Related work ",
+        "text_level": 1,
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+        "text": "54 Statistical models Several works focus on temporal dynamics when considering spatio-temporal   \n55 forecasting problems. The classical time series models, such as VAR, and ARIMA models, are applied   \n56 to spatio-temporal process modeling(21; 38). Besides, a spatial weight matrix is also introduced to the   \n57 ARIMA model to capture spatial dependence (28). The non-stationarity, particularly unit-root non  \n58 stationarity, is mainly modeled by ARIMA or Co-integration models. In addition, spatial regression   \n59 models or panel data are classical models in econometrics and can also be applied to model spatio  \n60 temporal problems. These models, for example, spatial auto-regression models, take spatial weight   \n61 matrix into consideration and estimate parameters in the framework of regression. However, the   \n62 common characteristics of these models need a pre-specified spatial weight matrix(35; 6). Elements   \n63 in the matrices are generally an inverse distance of corresponding locations. Meanwhile, these   \n64 spatial models focus on statistical inference on the scalar parameters placed before the spatial weight   \n65 matrix(25). Although there are many choices for the spatial weight matrix, such as inverse distance,   \n66 adjacency relationships, and K-nearest neighbors, there is a lack of research on estimating the spatial   \n67 weight matrix. The pre-specified spatial weight matrix restricts models’ application and fails to   \n68 capture more complicated underlying spatial dependence. Some researchers developed a sparse   \n69 spatio-temporal model that can estimate a sparse spatial weight matrix (17). The strict sparse setting   \n70 also restricts the wide application of the spatial weight matrix.   \n71 Graph-based methods Graph-based methods are widely applied for a non-Euclidean domain.   \n72 Some types of spatio-temporal data, for example, traffic flow data or brain network data, can be   \n73 represented as graphs. The graph structures well model the complicated spatial dependence. Thus,   \n74 the definition or pre-specified graphs structure is normally required when developing a graph-based   \n75 model. Related works can be found in (30; 14). The common typical method is GraphCNN, which is   \n76 to apply a convolutional transformation to the neighbors of each node (29; 34). The graph convolution   \n77 can capture patterns and features in the spatial domain. Graph-based methods have been proposed   \n78 and widely applied to lots of real cases. Traffic flow data modeling and forecasting is a popular topic   \n79 in this area (30; 20). Other topics, for example, climate sensor data (16), video (10) and etc, are also   \n80 applied by variant graph-based models. RNN or LSTM combined with graphs, i.e., a sequence of   \n81 graphs, are also considered in spatio-temporal forecasting problems (10).   \n82 CNN-based methods Unlike graph-based methods, CNN-based methods are more suitable for   \n83 modeling spatio-temporal data collected in regular grid locations. It applies filters to find relationships   \n84 between neighboring inputs. Although some works (32) applied convolution neural networks to   \n85 model non-grid traffic data, it is more common to see CNN-based methods process grid structures,   \n86 e.g., images, video rather than a general domain. As some spatio-temporal data are collected from a   \n87 regular grid in the Euclidean space (29), they thus can be viewed as a kind of special image. The CNN   \n88 structure combined with RNN or LSTM has been developed to make forecasting for spatio-temporal   \n89 data, for example, diffusion convolutional RNN (15), Convolutional LSTM networks (23; 36)and etc. ",
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+        "text": "90 3 Proposed method ",
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+        "text": "3.1 Problem formulation and notation ",
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+        "text": "We use a $n \\times 1$ vector $\\mathbf { X _ { t } } ~ = ~ \\{ \\mathbf { x _ { 1 t } } , \\mathbf { x _ { 2 t } } , \\cdot \\cdot \\cdot , \\mathbf { x _ { n t } } \\}$ to denote observations at time $t$ , where $n$ is the number of locations. At each location $i$ , $\\bf { S _ { i } } = ( c _ { i } ^ { x } , c _ { i } ^ { y } )$ is the coordinates of the location $i$ . The distance between location $\\mathbf { S _ { i } }$ and $\\mathbf { S _ { j } }$ is $d _ { i j } = \\sqrt { ( d _ { i j } ^ { x } ) ^ { 2 } + ( d _ { i j } ^ { y } ) ^ { 2 } }$ , where $d _ { i j } ^ { x } = | c _ { i } ^ { x } - c _ { j } ^ { x } |$ and $d _ { i j } ^ { y } = | c _ { i } ^ { y } - c _ { j } ^ { y } |$ . Our goal is to make forecasting for spatio-temporal data: given training data set $\\mathbf { X _ { 1 } } , \\mathbf { X _ { 2 } } , \\cdots , \\mathbf { X _ { T } }$ , we would like to make forecasting for the next $h$ , $\\hat { \\mathbf { X } } _ { T + 1 } , \\cdot \\cdot \\cdot , \\hat { \\mathbf { X } } _ { T + h }$ . ",
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+        "text": "3.2 The stationary spatio-temporal model with shape functions ",
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+        "text": "98 We first consider the stationary case. To model the spatio-temporal stationary process, we consider   \n99 the following model ",
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+    {
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+        "img_path": "images/c4ae415f4c4435398d331df745db97e8ef13a9695838b5efdb482d2fd41396d3.jpg",
+        "text": "$$\n\\mathbf { X _ { t } } = \\sum _ { \\mathbf { k } = 1 } ^ { \\mathbf { p } } \\mathbf { W _ { k } } \\mathbf { X _ { t - k } } + \\epsilon _ { \\mathbf { t } } ,\n$$",
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+        "text": "100 where $\\mathbf { W _ { k } }$ is a spatial weight matrix for capturing the spatial dependence at $\\log k$ , and $\\epsilon _ { \\mathbf { t } }$ is white noise. Moreover, we assume the 101 $( i , j )$ th element of $\\mathbf { W _ { k } }$ , $w _ { i j } ^ { ( k ) }$ , depends on the distance $d _ { i j }$ . That is, 102 w(k)ij depends on a function $f _ { k } ( d _ { i j } )$ . ",
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+        "text": "103 For spatio-temporal data, the spatial dependence, represented by $w _ { i j } ^ { ( k ) }$ , between locations decreases   \n104 as the distance between two locations increases. In other words, there is a shape constraint for   \n105 the function $f _ { k } ( d )$ , such as a decreasing function. In order to estimate the shape function, we   \n106 model $f _ { k } ( d )$ as a linear combination of basis functions $g _ { i } ( d ) , i = 1 , 2 , \\cdots , m$ . More specifically,   \n107 the shape function $f _ { k } ( d )$ is a linear combination of basis functions and coefficients with positive   \n108 value $\\bar { f _ { k } } ( d ) = a _ { 1 , k } ^ { 2 } \\bar { g _ { 1 } } ( \\dot { d } ) + \\cdot \\cdot \\cdot + a _ { m , k } ^ { 2 } g _ { m } ( d )$ , where $a _ { 1 , k } , \\cdots , a _ { m , k }$ are parameters to be estimated.   \n109 The constraint of decrease needs parameters non-negative and thus each parameters squared. The   \n110 spatial weight matrix can take the value of decreased shape function directly. The element of $\\mathbf { W _ { k } }$   \n111 is $w _ { i j . } ^ { ( k ) } = f _ { k } ( d _ { i j } )$ . The details of the shape function and the corresponding basis functions can be   \n112 found in Section 3.4 ",
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+        "text": "The parameters in shape functions can be estimated from the neural network illustrated in Figure 1. The neural network can be trained from the following criterion: ",
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+        "text": "$$\n\\operatorname* { m i n } _ { \\{ W _ { k } \\} _ { k = 1 } ^ { p } } \\sum _ { t = 1 } ^ { T } | | \\mathbf { X _ { t } } - \\hat { \\mathbf { X _ { t } } } | | ^ { 2 } = \\sum _ { \\mathbf { t } = 1 } ^ { \\mathbf { T } } | | \\mathbf { X _ { t } } - \\sum _ { \\mathbf { k } = 1 } ^ { \\mathbf { p } } \\hat { \\mathbf { W _ { k } } } \\hat { \\mathbf { X _ { t - k } } } | | ^ { 2 } .\n$$",
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+        "text": "113 3.3 The non-stationary spatio-temporal model with time-variant shape functions ",
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+        "text": "114 The static spatial weight matrix $\\mathbf { W _ { k } }$ can reflect spatial dependence and thus can be applied to   \n115 stationary scenarios. Next, we consider the nonstationary case. Therefore, we extend the stationary   \n116 model to non-stationary cases. The spatial weight matrices only reflect static relationships across time   \n117 lags in the static model. Unlike these settings, we change spatial weight matrices to be time-variant.   \n118 The spatial weight matrices formed by time-variant shape functions can thus capture non-stationary   \n119 dynamic spatial dependence. The non-stationary model has the form below, ",
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+        "text": "$$\n\\mathbf { X _ { t } } = \\sum _ { \\mathbf { k } = 1 } ^ { \\mathbf { p } } \\mathbf { W _ { t , k } } \\mathbf { X _ { t - k } } + \\epsilon _ { \\mathbf { t } } .\n$$",
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+        "text": "where $\\epsilon _ { \\mathbf { t } }$ is white noise, and $\\mathbf { W _ { t , k } }$ relies on shape function $f _ { t , k } ( d )$ . Similar with stationary settings, the time-variant shape functions are still represented as a linear combination of basis functions $g _ { i } ( d ) , i = 1 , 2 , \\cdots , \\bar { m }$ . The coefficients are therefore time-variant. The shape function at time $t$ has the form below $f _ { t , k } ( d ) = a _ { 1 , t , k } ^ { 2 } g _ { 1 } ( d ) + \\cdot \\cdot \\cdot + a _ { m , t , k } ^ { 2 } g _ { m } ( d )$ . Unlike stationary setting, the coefficients of nonstationary setting, $\\{ a _ { i , t , k } \\} _ { i = 1 } ^ { m }$ , depend on the time $t$ . The non-stationary model can be trained from the criterion by minimizing ",
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+        "text": "$$\n\\operatorname* { m i n } _ { \\{ W _ { t } , k \\} _ { k = 1 } ^ { p } } | | \\mathbf { X _ { t } } - \\hat { \\mathbf { X _ { t } } } | | ^ { 2 } = | | \\mathbf { X _ { t } } - \\sum _ { \\mathbf { k } = 1 } ^ { \\mathbf { p } } \\hat { \\mathbf { W } } _ { \\mathbf { t } , \\mathbf { k } } \\hat { \\mathbf { X } } _ { \\mathbf { t - k } } | | ^ { 2 } .\n$$",
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+        "image_caption": [
+            "Figure 1: The neural network for the stationary spatio-temporal process (left) and non-stationary spatio-temporal process (right). "
+        ],
+        "image_footnote": [],
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+        "text": "122 3.4 The basis functions for shape functions ",
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+        "text": "123 The shape functions are integrated into our model to obtain distance-based explanations in stationary   \n124 and non-stationary scenarios. The motivation of the proposed shape functions is that as the distance   \n125 between two observations increases, the effects between these two locations decreases. These distance  \n126 based effects can be reflected in spatial weight matrix W and each element in the matrix can measure   \n127 how the corresponding locations interact. The shape function is represented as a linear combination of   \n128 basis functions. The basis functions, satisfying shape constraint, rely on the corresponding definition   \n129 of basis functions.   \n130 Definition of basis functions for various shape constraints. We list the definition of basis functions   \n131 for increased and decreased shape (3). The distance quantile among $\\{ d _ { i _ { 1 } , j _ { 1 } } , d _ { i _ { 2 } , j _ { 2 } } , \\dots , d _ { i _ { N } , j _ { N } } \\}$ at   \n132 quantile level $q _ { 1 } , q _ { 2 } , \\cdots , q _ { m }$ is denoted by $\\{ d _ { ( 1 ) } , d _ { ( 2 ) } , \\cdots , d _ { ( m ) } \\}$ , where $0 \\leq q _ { 1 } < q _ { 2 } < \\cdot \\cdot \\cdot < q _ { m } \\leq$   \n1 and 133 $\\{ q _ { 1 } , q _ { 2 } , \\cdot \\cdot \\cdot , q _ { m } \\} = \\{ { \\textstyle \\frac { 1 } { m } } , { \\textstyle \\frac { 2 } { m } } , \\cdot \\cdot \\cdot , 1 \\}$ . Here, we can set the number of $m < < n ^ { 2 }$ , and thus, the   \nnumber of parameters is significantly reduced.   \n135 For the constraint of monotone decreasing function, the basis function is defined as $g _ { i } ( d ) = \\mathbf { 1 } _ { \\left\\{ \\mathbf { d } < \\mathbf { d } _ { \\left( \\mathbf { i } \\right) } \\right\\} }$ .   \n136 The basis function for the shape function with the constraint of concave decrease is defined as   \n137 $g _ { i } ( d ) = ( d _ { ( i ) } - d ) \\mathbf { 1 } _ { \\{ \\mathbf { d } _ { ( i ) } \\leq \\mathbf { d } \\} }$ and convex decrease is defined as $g _ { i } ( d ) = ( d _ { ( i ) } - d ) \\mathbf { 1 } _ { \\{ \\mathbf { d } \\leq \\mathbf { d } _ { ( i ) } \\} }$ , for   \n138 $1 \\leq i \\leq m$ . Figure 2 shows the definition of basis functions for monotone decreased and increased   \n139 shape functions, respectively. We only present four basis functions for each shape and each of them   \n140 is related to four quantile levels. The dashed lines indicate the turning points for each basis function   \n141 and they equal one or zero at the beginning and turn to zero or one at turning points.   \n143 The stationary model requires fixed shape functions and related spatial weight matrix are time  \n144 invariant. Given training data set $\\mathbf { X _ { 1 } } , \\mathbf { X _ { 2 } } , \\cdots , \\mathbf { X _ { T } }$ , we can estimate spatial weight matrix   \n145 ${ \\hat { W } } _ { 1 } , { \\hat { W } } _ { 2 } , \\cdot \\cdot \\cdot , { \\hat { W } } _ { p }$ and make forecasting iteratively. That is $\\begin{array} { r } { \\hat { { \\bf X } } _ { T + 1 } = \\sum _ { k = 1 } ^ { p } \\hat { W } _ { k } { \\bf X _ { T + 1 - k } } , \\hat { { \\bf X } } _ { T + 2 } = } \\end{array}$   \n146 $\\begin{array} { r } { \\hat { W } _ { 1 } \\hat { \\mathbf { X } } _ { T + 1 } + \\sum _ { k = 2 } ^ { \\bar { p } } \\hat { W } _ { k } \\mathbf { X } _ { \\mathbf { T } + 2 - \\mathbf { k } } , \\cdot \\cdot \\cdot \\hat { \\mathbf { X } } _ { T + h } = \\sum _ { k = 1 } ^ { p } \\hat { W } _ { k } \\hat { \\mathbf { X } } _ { T + h - k } } \\end{array}$ . ",
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+            "Figure 2: The basis functions for decreased shape (left) and for increased shape (right). The arrows indicate domain of each basis functions. "
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+        "text": "The non-stationary model incorporate time-variant spatial weight matrix ${ \\hat { W } } _ { t , \\cdot } .$ . Given the training data set $\\mathbf { X _ { 1 } } , \\mathbf { X _ { 2 } } , \\cdots , \\mathbf { X _ { T } }$ , we can obtain corresponding shape functions $\\hat { f } _ { 1 , \\cdot } , \\hat { f } _ { 2 , \\cdot } , \\cdot \\cdot \\cdot , \\hat { f } _ { T , \\cdot }$ , where · denotes time lag. For lag $p = 1$ , we can use $\\{ \\hat { f } _ { t } \\} _ { t = 1 } ^ { T }$ to represent time-variant shape functions for convenience. We can make dynamic forecasts for the next $h$ windows. One simple forecasting method is to use $\\hat { W } _ { T , k }$ to make forecast for $\\hat { \\mathbf { X } } _ { T + h }$ , that is ",
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+        "text": "$$\n\\hat { \\mathbf { X } } _ { T + h } = \\sum _ { k = 1 } ^ { p } \\hat { W } _ { T , k } \\mathbf { X } _ { \\mathbf { T } + \\mathbf { h } - \\mathbf { k } } .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "The alternative method is to retrain the new forecast to obtain the latest shape functions as well as spatial weight matrix. Given long-term forecast window $L$ , we first make short-term forecast for $h$ steps ",
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+        "text": "$$\n\\hat { \\mathbf { X } } _ { T + h } = \\sum _ { k = 1 } ^ { p } \\hat { W } _ { T + h , k } \\mathbf { X } _ { \\mathbf { T } + \\mathbf { h } - \\mathbf { k } } ,\n$$",
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+        "type": "text",
+        "text": "where 147 $h = 1 , 2 , \\cdots$ and $\\hat { W } _ { T + h , k }$ is estimated by training forecast value of $\\hat { \\mathbf { X } } _ { T + h - k }$ . We repeat the 148 process until $L$ steps in total have been predicted. ",
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+        "type": "text",
+        "text": "49 We summarize the whole process of our model when making spatio-temporal forecasts. ",
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+        "text": "Step 1 Given the observation $\\{ \\mathbf { X _ { t } } \\} _ { \\mathbf { t = 1 } } ^ { \\mathbf { T } }$ and its coordinates, calculate all distance pairs among all locations, denoted by $\\{ d _ { i _ { 1 } , j _ { i } } , \\cdot \\cdot \\cdot , d _ { i _ { N } , j _ { N } } \\}$ . ",
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+        "text": "Step 2 Calculate $\\textstyle \\left\\{ { \\frac { 1 } { m } } , { \\frac { 2 } { m } } , \\cdots , 1 \\right\\}$ quantile levels and obtain corresponding distance quantile value $\\{ d _ { ( 1 ) } , d _ { ( 2 ) } , \\cdots , \\ \" { d _ { ( m ) } } \\}$ . ",
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+        "text": "Step 3 Determine the shape constraints and construct corresponding basis functions. Specify the time lag $p$ . ",
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+        "text": "Step 4 Train the model according to the illustration of Figure 1. ",
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+        "type": "text",
+        "text": "157 4 Experiment ",
+        "text_level": 1,
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+        "text": "158 In order to assess our model in stationary and non-stationary scenarios, we synthesize data. Then,   \n159 we apply our model to make some comparisons. On the one hand, we need to evaluate how   \n160 the estimated shape functions look and assess their similarity and accuracy. On the other hand,   \n161 our model can make spatio-temporal forecasting after estimating for spatial weight matrix. The   \n162 basic idea for completing the two goals is to set up the expected shape function and compare   \n163 estimated parameters with the real one. Next, we assess the forecasting performance with baseline   \n164 models. Codes and data for replicating our experiments are anonymously published at https:   \n165 //anonymous.4open.science/r/STVAR-F16E/. ",
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+        "type": "text",
+        "text": "166 4.1 Simulation for stationary model ",
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+        "type": "text",
+        "text": "Here, we synthesize 100 stationary spatio-temporal data sets. The spatial domain consists of 30 locations and their coordinates can be found at https://anonymous.4open.science/r/ STVAR-F16E/. For each location, we observe 500 values. The observation is generated from the stationary model $\\begin{array} { r } { X _ { t } = \\sum _ { k = 1 } ^ { p } W _ { k } X _ { t - k } + \\epsilon _ { t } } \\end{array}$ , where $\\epsilon _ { t }$ is randomly generated from the standard normal distribution. The next step is to construct random spatial weight matrices for each synthesized data set. The shape functions are set to be decreasing, and we set them as a logarithmic function: ",
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+        "img_path": "images/c6caf52b3c8cac794927274343b311ebd19e8794b9e513b9db5dcfd1099c6254.jpg",
+        "text": "$$\n\\alpha ( - \\log ( d + 1 ) + \\log ( 1 7 0 ) ) ,\n$$",
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+        "type": "text",
+        "text": "167 where $\\alpha$ is randomly generated from uniform distribution [0.05,0.06] but kept to be fixed for each   \n168 simulated data set. We use $d + 1$ to avoid zero value. This setting can make the real shape function   \n169 decrease and make it equal to zero when $d = 1 6 9$ . The stationary model can iteratively generate the   \n170 $\\mathbf { X _ { t } }$ given initial value $\\mathbf { X _ { 0 } }$ , where $\\mathbf { X _ { 0 } }$ is randomly generated from a uniform distribution with bounds   \n171 [-0.01,0.01]. The time lags are set as $p = 1$ .   \n172 Estimation for shape functions. In Figure 3,   \n173 the estimated shape function is presented in red,   \n174 while the real shape function is presented in blue.   \n175 It can be seen that the estimated shape function   \n176 can capture the trend of the real shape function.   \n177 Training details. The first 300 steps are used as   \n178 training data, saving the last 200 steps for eval  \n179 uation. We train all models for 100 epochs with   \n180 Adam optimizer (5) and a learning rate of 0.01.   \n181 The process involves parallel training across 10   \n182 CPUs. We select 100 quantile levels, and thus   \n183 100 basis functions $g _ { i } ( d )$ were generated as the   \n184 inputs for the model.   \n185 Assessment for forecasting. We assess the fore  \n186 casting performance for the stationary model   \n187 with baseline models. As introduced in the liter  \n188 ature review, the baseline models are selected from the VAR model(21), the spatial panel data(SPE)   \n189 model that applied pre-specified spatial weigh matrix (28), graph-based models (20; 37) and   \n190 convolution-based models (15; 23). The error metrics are mean absolute error and root mean squared   \n191 error defined by 1Nn PNj=1 Pni=1 $\\begin{array} { r } { \\frac { 1 } { N n } \\sum _ { j = 1 } ^ { N } \\sum _ { i = 1 } ^ { n } \\frac { \\sum _ { t = T } ^ { T + h } | \\hat { X } _ { i t } ^ { ( j ) } - X _ { i t } ^ { ( j ) } | } { h } } \\end{array}$ , $\\begin{array} { r } { \\frac { 1 } { N n } \\sum _ { j = 1 } ^ { N } \\sum _ { i = 1 } ^ { n } \\sqrt { \\frac { 1 } { h } \\sum _ { t = T } ^ { T + h } ( X _ { i t } ^ { ( j ) } - \\hat { X } _ { i t } ^ { ( j ) } ) ^ { 2 } } } \\end{array}$ ,   \n192 respectively. The Table 1 shows the six baseline models with the proposed model. As totally we   \n193 have 100 synthesised data sets, $X _ { i t } ^ { ( j ) }$ and $\\hat { X } _ { i t } ^ { ( j ) }$ denote $i - t h$ variable in $j - t h$ data sets. $n = 3 0$ is   \n194 the number of locations and $N = 1 0 0$ is the number of synthesised data. We conducted one-step   \n195 forecasting for the next 200 observations.   \n96 Compared with baseline models, the proposed model performs better under the metric MAE and   \n97 RMSE. The proposed method outperforms the closest competing method, DC-RNN, by $10 \\%$ . ",
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+        "image_caption": [
+            "Figure 3: The sample of estimated shape function. Distances are shown every $2 0 ^ { t h }$ quantile. "
+        ],
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+        "type": "text",
+        "text": "198 4.2 Experiments for non-stationary model ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We conduct a simulation for the non-stationary model with time lag $p = 1$ and synthesize 100 data sets using a similar approach to the stationary model simulation. The initial value $\\mathbf { X _ { 0 } }$ and $\\epsilon _ { t }$ are generated from a uniform and normal distribution respectively. The locations of observations are the same as those in the stationary model simulation. In order to construct $W _ { t }$ , the time-varying shape functions are created under the decreased constraint. The shape function at time $t$ is constructed as ",
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+    {
+        "type": "equation",
+        "img_path": "images/8a7723e14b023ebff7f6faa2ed7d026ae7d27c04bb56bd1e5af2b100650cea61.jpg",
+        "text": "$$\n\\alpha _ { t } ( - \\log ( d + 1 ) + \\log ( 1 7 0 ) ) ,\n$$",
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+        "type": "text",
+        "text": "99 where $\\alpha _ { t }$ controls the level of value at each time $t$ . $\\epsilon _ { t }$ is generated from a normal distribution. $\\mathbf { X _ { 0 } }$ is   \n00 generated from a uniform distribution with bound [-0.001,0.001].   \n201 Shape functions settings and estimation. The shape functions are set as time-variant, as they can   \n202 simulate the non-stationary process across time. We specified $\\alpha _ { 0 }$ at $t = 0$ from uniform distribution   \n203 $[ 1 \\times 1 0 ^ { - 4 } , 2 \\times 1 0 ^ { - 4 } ]$ and then make an interpolation from $\\alpha _ { 0 }$ to $\\alpha _ { 5 0 0 }$ . The total length for every   \n204 location is 500 and we set $\\alpha _ { 5 0 0 } = 1 0 \\times \\alpha _ { 0 }$ . For example, generally if $\\alpha _ { 0 } ~ = ~ 0 . 0 0 0 1$ , we have   \n205 $\\begin{array} { r } { \\alpha _ { t } = 0 . 0 0 0 1 ( 1 - \\frac { t } { T } ) + 0 . 0 0 1 \\frac { t } { T } } \\end{array}$ , where $T = 5 0 0$ . This setting guarantee that shape functions vary   \n206 from lower level to higher level. The larger $\\alpha _ { t }$ is, the more larger distance-based effects they have.   \n207 Thus, the corresponding spatial weight matrix consists of dynamic shape functions and can reflect the   \n208 non-stationary dependence among each site. We present the estimated shape functions in Figure 4   \n209 and compare them with the real ones.   \n10 Training details. Similar to the stationary simulation, the train-test split is $3 0 0 - 2 0 0$ over the data   \n11 size of 500. However, we train all models for 100 epochs with Adam optimizer (5) at a learning rate   \n12 of 0.001. We train models in parallel across 10 CPUs. ",
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+        "image_caption": [
+            "Figure 4: The sample of estimated shape function for the 120 testing time steps. Distances are shown every $4 0 ^ { t h }$ quantile. "
+        ],
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+        "type": "text",
+        "text": "Forecasting performance. The forecasting performance is assessed by the same metrics used in the previous simulation for the stationary case. We made a one-step forecast by our model. As for the baseline models, we adjusted their published code accordingly. The results show that the proposed model can still capture non-stationary processes compared with baseline models. The proposed method outperforms the other competing methods. The error metric is shown in Table 1. ",
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+    {
+        "type": "table",
+        "img_path": "images/d5c85dc2380203f3845b090f82221f7f8063b4c461d3066f95eb95545ba74c87.jpg",
+        "table_caption": [
+            "Table 1: The error metrics with baseline models for simulation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Methods</td><td colspan=\"2\">Stationary Simulation</td><td colspan=\"2\">Non-stationary Simulation</td></tr><tr><td>MAE</td><td>RMSE</td><td>MAE</td><td>RMSE</td></tr><tr><td>VAR</td><td>2.9611 ± 1.8573</td><td>3.2588 ± 1.8077</td><td>2.4426 ±1.2285</td><td>2.7676 ± 1.2015</td></tr><tr><td>SPM</td><td>1.8850 ± 0.6348</td><td>1.8671 ± 0.6778</td><td>2.1918 ± 0.7350</td><td>2.2161 ± 0.6876</td></tr><tr><td>DC-RNN</td><td>0.8960 ± 0.0370</td><td>1.1168 ± 0.0426</td><td>0.9017 ± 0.0358</td><td>1.1328 ± 0.0463</td></tr><tr><td>FC-GAGA</td><td>2.5425 ± 0.2965</td><td>3.1066 ± 0.3633</td><td>1.0270 ± 0.0080</td><td>1.2939 ± 0.0120</td></tr><tr><td>GMAN</td><td>1.6806 ± 0.1491</td><td>1.9293 ± 0.1483</td><td>1.5714 ± 0.1104</td><td>1.8608 ± 0.1155</td></tr><tr><td>ConvLSTM</td><td>2.9495 ± 0.2980</td><td>3.2509 ± 0.2887</td><td>2.2478 ± 0.2295</td><td>2.5469 ± 0.2324</td></tr><tr><td>ESTF</td><td>0.7997 ± 0.0015</td><td>1.0017 ± 0.0016</td><td>0.8075 ± 0.0016</td><td>1.0112 ± 0.0020</td></tr></table>",
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+        "type": "text",
+        "text": "218 4.3 Real case studies ",
+        "text_level": 1,
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+        "text": "Air quality data. We apply our model to air quality data, which records air quality in California over 2021 1. The daily mean of $\\mathrm { P M } 2 . 5$ is recorded across 172 sites. ",
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+    {
+        "type": "text",
+        "text": "We obtain the first 200 steps for training and perform forecasting for the next 165 steps. All models are trained for 100 epochs using Adam optimizer (5), at a learning rate of 0.01 and batch size of 50. We present the estimated time-variant shape functions in supplemental file. The value of shape functions decays to zero at around 5.926, which is $80 \\%$ quantile in the sample of distance pairs. In other words, the distance-based effects decay to zero at a distance equal or larger than 5.926. Our ",
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+        "text": "226 model has ideal performance with low time consummation compared with baseline models. We put   \n227 detailed forecasting results of simulation and real cases in a supplemental file.   \n228 The result is shown in Table 2. The ESTF performs best in terms of RMSE, while the DC-RNN   \n229 method performs best in terms of MAE. For the computational time, the ESTF method is significantly   \n230 faster than most machine learning methods, and only takes around $1 / 1 0$ time of DC-RNN. In Figure 5,   \n231 MAE, RMSE, and time are presented with different numbers of $m$ . As $m$ increases, the computational   \n232 time increases while both MAE and RMSE decrease. There is a significant increase in the forecasting   \n233 performance when $m$ increases from 10 to 50. For $m > 5 0$ , the forecasting performance does not   \n234 increase much as $m$ increases.   \n235 One key advantage of the ESTF method is that we can make an explicit distance-based explanation   \n236 for our dataset. Figure 6 shows the distance-based effects at time $t = 9$ . We only present the effects   \n237 using a threshold to obtain a more concise visualization. The estimated shape function $\\hat { f } _ { 9 }$ ranges   \n238 from 0 to 9.8 and we set 5 as the threshold. The red line indicates the value of the shape function   \n239 larger than 7, while the gray line indicates the value between 5 and 7. Figure 6 shows how any two   \n240 locations interact and measure the distance-based effects quantitatively. For example, air quality   \n241 monitoring sites around the Greater Los Angeles(red circle in Figure 6) area have a strong spatial   \n242 interaction with each other, such as node 7 and node 8. ",
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+        "image_caption": [
+            "Figure 6: The significant distance-based effect Figure 5: Comparing efficiency vs. performanceamong all 30 locations. trade-off at different quantile values. "
+        ],
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+        "type": "text",
+        "text": "4.4 $S O _ { 2 }$ data ",
+        "text_level": 1,
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+        "text": "Texas is the second largest manufacturing state in the USA and prediction for $S O _ { 2 }$ is critical task for researchers. The data 2 records daily $S O _ { 2 }$ at 31 locations in 2021. More detailed spatial information can be found in the supplemental file. The numeric result is listed in Table 2. ",
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+        "img_path": "images/df1a6660577b2bd246541ce2256798752f4a9b22efcef7d70163c4e0b2d2545c.jpg",
+        "table_caption": [
+            "Table 2: The error metrics with baseline models for real case study. Clock time (in seconds) for real case study is recorded when training each model for 100 epochs on a single CPU. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Methods</td><td colspan=\"6\">Air quality data</td><td rowspan=\"2\">SO2 data Training Time (s)</td><td rowspan=\"2\">Inference Time (s)</td></tr><tr><td>MAE</td><td>RMSE</td><td>Training time (s)</td><td>Inference Time (s)</td><td>MAE</td><td>RMSE</td></tr><tr><td>VAR</td><td>16.9844</td><td>22.3410</td><td>3.56</td><td>0.04</td><td>6.2705</td><td>9.1388</td><td>3.330</td><td>0.016</td></tr><tr><td>SPM</td><td>8.4547</td><td>13.8262</td><td>0.31</td><td>0.03</td><td>7.1453</td><td>9.1086</td><td>0.143</td><td>0.027</td></tr><tr><td>DC-RNN</td><td>4.7157</td><td>9.3873</td><td>203</td><td>1.211</td><td>3.5094</td><td>6.8681</td><td>264.215</td><td>1.366</td></tr><tr><td>FC-GAGA</td><td>7.8671</td><td>18.1870</td><td>181</td><td>2.759</td><td>4.5976</td><td>7.7528</td><td>169.425</td><td>2.889</td></tr><tr><td>GMAN</td><td>12.5268</td><td>17.3817</td><td>140</td><td>1.823</td><td>4.1099</td><td>7.4806</td><td>172.016</td><td>1.581</td></tr><tr><td>ConvLSTM</td><td>12.6292</td><td>17.9149</td><td>53</td><td>1.940</td><td>4.1445</td><td>8.0688</td><td>96.233</td><td>1.656</td></tr><tr><td>ESTF</td><td>5.2237</td><td>9.2169</td><td>22</td><td>1.625</td><td>4.2966</td><td>6.8307</td><td>31.050</td><td>1.868</td></tr></table>",
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+        "text": "247 Similar conclusions can be drawn in $S O _ { 2 }$ data as that of air quality data. The ESTF model performs   \n248 best under the RMSE metric, while DC-RNN is best in the MAE metric. In terms of training time,   \n251 The efficiency analysis and performance at different quantiles are shown in Figure 7. Together with   \n252 Figure 5, we can see that the increasing number of basis functions does not have much improvement   \n253 when the number of basis functions is larger than 50, while the training time increases as the number   \n254 of basis functions increases. The spatial distribution at time $t = 9 0$ is presented in Figure 8 where   \n255 coordinates are denoted by latitude and longitude. Two significant clusters, representing Houston   \n256 and Dallas respectively, have the strongest distance-based effect. It quantitatively shows how these   \n257 neighbors affect each other. Counties around Dallas-Fort Worth metropolitan area show strong   \n258 interaction, which should be noted by environmental policy-makers. More detailed results are   \n259 presented in the supplemental file. ",
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+        "image_caption": [
+            "Figure 8: The significant distance-based effect Figure 7: Comparing efficiency vs. performanceamong all 31 locations at $t = 9 0$ trade-off at different quantile values. "
+        ],
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "260 5 Discussion ",
+        "text_level": 1,
+        "bbox": [
+            150,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "This paper applies learnable shape functions to capture distance-based effects. It can model dynamic spatial dependence for stationary and non-stationary spatio-temporal data based on their distance. The model does not have the limitations of classical statistical spatial models and provides a more explanatory model than usual deep learning methods. Furthermore, some spatio-temporal data, such as temperature for sea surface and air quality monitoring data, usually viewed as collected from the continuous field, are more suitable for the proposed models since these kinds of data follow the basic rule that variability between two locations is significantly affected by their distance. However, some spatio-temporal data, such as traffic flow or some biology data, do not follow the rule. As a result, the spatial dependence may rely on road structure or biological mechanisms instead of distance. It is worth researching such data by considering graph structure when estimating spatial weight matrix. In addition, we can develop spatio-temporal causal inference based on the ESTF model. Grander causal analysis can be done by fitting the first-order VAR model (24). The estimation of the coefficients matrix of the VAR model attracts researchers’ interest as it can be treated as a causal transition matrix. In the causal inference community, lots of work have been conducted on the VAR model (8; 9). However, there is a lack of research on causal inference under the spatio-temporal process. The quantitative distance-based effects in ESTF can be further researched and extended to develop a spatio-temporal causal model. ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "References ",
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+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
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parse/dev/v54eUIayFh/v54eUIayFh.md

@@ -0,0 +1,520 @@
+# UniControl: A Unified Diffusion Model for Controllable Visual Generation In the Wild
+
+Can $\mathrm { Q i n } ^ { \dag \star }$ , Shu Zhang†, Ning $\mathrm { Y u ^ { \dag } }$ , Yihao Feng†, Xinyi Yang†, Yingbo Zhou†, Huan Wang†, Juan Carlos Niebles†, Caiming Xiong†, Silvio Savarese†, Stefano Ermon‡, Yun $\operatorname { F u } ^ { \star }$ , and Ran $\mathrm { { X u ^ { \dag } } }$
+
+†Salesforce AI Research, ⋆Northeastern University, ‡Stanford Univeristy, qin.ca@northeastern.edu, ermon@cs.stanford.edu, yunfu@ece.neu.edu, {shu.zhang, ning.yu, yihaof, x.yang, yingbo.zhou, huan.wang, jniebles, cxiong, ssavarese, ran.xu}@salesforce.com
+
+# Abstract
+
+Achieving machine autonomy and human control often represent divergent objectives in the design of interactive AI systems. Visual generative foundation models such as Stable Diffusion show promise in navigating these goals, especially when prompted with arbitrary languages. However, they often fall short in generating images with spatial, structural, or geometric controls. The integration of such controls, which can accommodate various visual conditions in a single unified model, remains an unaddressed challenge. In response, we introduce UniControl , a new generative foundation model that consolidates a wide array of controllable condition-to-image (C2I) tasks within a singular framework, while still allowing for arbitrary language prompts. UniControl enables pixel-level-precise image generation, where visual conditions primarily influence the generated structures and language prompts guide the style and context. To equip UniControl with the capacity to handle diverse visual conditions, we augment pretrained text-to-image diffusion models and introduce a task-aware HyperNet to modulate the diffusion models, enabling the adaptation to different C2I tasks simultaneously. Trained on nine unique C2I tasks, UniControl demonstrates impressive zero-shot generation abilities with unseen visual conditions. Experimental results show that UniControl often surpasses the performance of single-task-controlled methods of comparable model sizes. This control versatility positions UniControl as a significant advancement in the realm of controllable visual generation. 1
+
+# 1 Introduction
+
+Generative foundation models are revolutionizing the ways that humans and AI interact in natural language processing (NLP) [1–6], computer vision (CV) [7–10], audio processing (AP) [11, 12], and robotic controls [13–15], to name a few. In NLP, generative foundation models such as InstructGPT or GPT-4, achieve excellent performance on a wide range of tasks, e.g., question answering, summarization, text generation, or machine translation within a single-unified model. Such multi-tasking ability is one of the most appealing characteristics of generative foundation models. Furthermore, generative foundation models can also perform zero-shot or few-shot learning on unseen tasks [3, 16, 17].
+
+For generative models in vision domains [9, 18–20], such multi-tasking ability is less clear. Stable Diffusion Model (SDM) [9] has established itself as the major cornerstone for text-conditioned image generation. However, while text descriptions provide a very flexible way to control the generated images, their ability to provide pixel-level precision for spatial, structural, or geometric controls is often inadequate. A recent work, ControlNet [21], was proposed to augment SDM to enable visual conditions (e.g., edge maps, depth maps). With the additional visual conditions, ControlNet can achieve explicit spatial, structural, or geometric control over generated structures, without losing the semantic control from textual captions. Unfortunately, unlike language prompts that a unified module such as CLIP [22] can handle, each ControlNet model can only handle a specific control modality that it was trained on (e.g., edge map). Retraining a separate model is necessary to handle a different modality of visual conditions, incurring non-trivial time and spatial complexity costs.
+
+![](images/e2f7a1a82640708ece2e219ec1c356d11a92c8831241ba8bcd8afc8f321db7de.jpg)  
+Figure 1: UniControl is trained with multiple tasks with a unified model, and it further demonstrates promising capability in zero-shot tasks generalization with visual example results shown above.
+
+To overcome the limitation of previous works, we present UniControl, a unified diffusion model for controllable visual generation in the wild, which is capable of simultaneously handling both language and various visual conditions. Naturally, UniControl can perform multi-tasking and can encode visual conditions from different tasks into a universal representation space, seeking a common representation structure among tasks. The unified design of UniControl allows us to enjoy the advantages of improved training and inference efficiency, as well as enhanced controllable generation. On the one hand, the model size of UniControl does not significantly increase as the number of tasks scales up. On the other hand, UniControl derives advantages from the inherent connections between different visual conditions [e.g., 23–25]. These relationships, such as depth and segmentation mapping, leverage shared geometric information to enhance the controllable generation quality.
+
+The unified controllable generation ability of UniControl relies on two novel designed modules, a mixture of expert (MOE)-style adapter and a task-aware HyperNet [26, 27]. The MOE-style adapter can learn necessary low-level feature maps from various visual conditions, allowing UniControl to capture unique information from different visual conditions. The task-aware HyperNet, which takes the task instruction as natural language prompt inputs, and outputs a task-aware embedding. The output embeddings can be incorporated to modulate ControlNet [21] for task-aware visual condition controls, where each task corresponds to a particular format of visual condition. As a result, the task-aware HyperNet allows UniControl to learn meta-knowledge across various tasks, and obtain abilities to generalize to unseen tasks. As Tab. 1, UniControl has significantly compressed the model size compared with its direct baseline, i.e., Multi-ControlNet, by unifying nine tasks into ONE model.
+
+Table 1: Architecture and Model Size (#Params): UniControl vs. Multi-ControlNet   
+
+<table><tr><td></td><td>Stable Diffusion</td><td>ControlNet</td><td>MoE-Adapter</td><td>TaskHyperNet</td><td>Total</td></tr><tr><td>UniControl</td><td>1065.7M</td><td>361M</td><td>0.06M</td><td>12.7M</td><td>1.44B</td></tr><tr><td>Multi-ControlNet</td><td>1065.7M</td><td>361M×9</td><td>-</td><td>-</td><td>4.32B</td></tr></table>
+
+To obtain multi-tasking and zero-shot learning abilities, we pre-train UniControl on nine distinct tasks across five categories: 1) edges (Canny, HED, User Sketch); 2) region-wise maps (Segmentation Maps, Bounding Boxes); 3) skeletons (Human Pose Skeletons); 4) geometric maps Depth, Surface Normal); 5) editing (Image Outpainting). We build MultiGen-20M dataset, comprising over 20 million high-quality triplets of original images, language prompts, and visual conditions for all the tasks. Then UniControl is trained for over 5,000 GPU hours on NVIDIA A100-40G hardware that is comparable with the overall training cost of different ControlNets. Moreover, UniControl exhibits a remarkable capacity for zero-shot adaptation to new tasks, highlighting its potential for deployment in real-world applications. Our contributions are summarized below:
+
+• We present UniControl, a unified model capable of handling various visual conditions for the controllable visual generation.
+
+• We collect a new dataset for multi-condition visual generation with more than 20 million imagetext-condition triplets over nine distinct tasks across five categories.
+
+• We conduct extensive experiments to demonstrate that the unified model UniControl outperforms each single-task controlled image generation, thanks to learning the intrinsic relationships between different visual conditions.
+
+• UniControl shows the ability to adapt to unseen tasks in a zero-shot manner, highlighting its versatility and potential for widespread adoption in the wild.
+
+# 2 Related Works
+
+Diffusion-based Generative Models. Diffusion models were initially introduced in [28] that yield favorable outcomes for generating images [18, 21]. Improvements have been made through various training and sampling techniques such as score-based diffusion [29, 30], Denoising Diffusion Probabilistic Model (DDPM) [31], and Denoising Diffusion Implicit Model (DDIM) [32], When training U-Net denoisers [33] with high-resolution images, researchers involve speed-up techniques including pyramids [34], multiple stages [20], or latent representations [9]. In particular, UniControl leverages Stable Diffusion Models (SDM) [9] as the base model to perform multi-tasking.
+
+Text-to-Image Diffusion. Diffusion models emerge to set up a cutting-edge performance in text-to-image generation tasks [20, 19], by cross-attending U-Net denoiser in diffusion generators with CLIP [22] or T5-pretrained [2] text embeddings. GLIDE [35] is another example of a textguided diffusion model that supports image generation and editing. UniControl and closely related
+
+ControlNet [21] are both built upon previous works on diffusion-based text-to-image generation [9].   
+[36] introduces the compositional conditions to guide visual generation.
+
+Image-to-Image Translation. Image-to-image (I2I) translation task was initially proposed in Pix2Pix [37], focusing on learning a mapping between images in different domains. Recently, diffusion-based approaches [38, 39, 21] set up the new state of the art results. Recent diffusionbased image editing methods show outstanding performances without requiring paired data, e.g., SDEdit [40], prompt-to-prompt [41], Edict [42]. Other image editing examples include various diffusion bridges and flows [43–47], classifier guidance [30] based methods for colorization, superresolution [34], inpainting [48], and etc. ControlNet [21] takes both visual and text conditions and achieves new state-of-the-art controllable image generation. Our proposed UniControl unifies various visual conditions of ControlNet, and is capable of performing zero-shot learning on newly unseen tasks. Concurrently, Prompt Diffusion [49] introduces visual prompt [50] from image inpainting to controllable diffusion models, which requires two additional image pairs as the in-context example for both training and inference. By contrast, UniControl takes only a single visual condition while still capable of both multi-tasking and zero-shot learning.
+
+# 3 UniControl
+
+In this section, we describe the training and the model design of our unified controllable diffusion model UniControl. Specifically, we first provide the problem setup and training objectives in Sec. 3.1, and then show the novel network design of UniControl in Sec. 3.2. Finally, we explain how to perform zero-shot image generation with the trained UniControl in Sec. 3.3.
+
+# 3.1 Training Setup
+
+Different from the previous generative models such as Stable Diffusion Models (SDM) [9] or ControlNet [21], where the image generation conditions are single language prompt, or single type of visual condition such as canny, UniControl is required to take a wide range of visual conditions from different tasks, as well as the language prompt.
+
+To achieve this, we reformulate the training conditions and target pairs for UniControl. Specifically, suppose we have a dataset consisting of $K$ tasks : $\mathcal { D } : = \{ { \mathcal { D } } _ { 1 } \cup \cdot \cdot \cdot \cup { \mathcal { D } } _ { K } \}$ , and for each task training set $\mathcal { D } _ { k }$ , denote the training pairs by $( [ c _ { \mathrm { t e x t } } , c _ { \mathrm { t a s k } } ] ,  { \mathcal { T } } _ { c } , \pmb { x } )$ , with $c _ { \mathrm { t a s k } }$ being the task instruction that indicates the task type, $c _ { \mathrm { t e x t } }$ being the language prompt describing the target image, $\mathcal { T } _ { c }$ being the visual conditions, and $_ { \pmb { x } }$ being the target image. With the additional task instruction, UniControl can differentiate visual conditions from different tasks. A concrete training example pair is the following:
+
+# Task-Aware Vision-Language Condition
+
+![](images/fcb8b0ba4b724590545b882ca6e410285660539ed63b095b991438633dd6c62a.jpg)  
+Visual Condition $\mathcal { T } _ { c }$
+
+Language Prompt :   
+“Camp on a mountain top: Birthday Presents,   
+Adventure, Outdoor, Mountain Camps, Great   
+View, Places, Hiking, Mornings Lights,   
+Himalayan Sunri”
+
+![](images/38e8fa6599c74af649d251c2351c779c38e3f2b94a8f40459a98969d57a74df5.jpg)  
+Target output
+
+Task Instruction $c _ { \mathrm { t a s k } }$ : “Canny Edge to Image”
+
+where the task is to translate the canny edge to real images following language prompt. With the induced training pairs $( \pmb { x } , [ c _ { \mathrm { t a s k } } , c _ { \mathrm { t e x t } } ] , \pmb { \mathcal { T } } _ { c } )$ , we define the training loss for task $k$ following LDM [9]:
+
+$\ell ^ { k } ( \theta ) : = \mathbb { E } _ { z , \varepsilon , t , c _ { \mathrm { t a s k } } , c _ { \mathrm { t e x t } } , { T _ { c } } } \left[ \lVert \varepsilon - \varepsilon _ { \theta } ( z _ { t } , t , c _ { \mathrm { t a s k } } , c _ { \mathrm { t e x t } } , \mathcal { T } _ { c } ) \rVert _ { 2 } ^ { 2 } \right]$ , with $( [ c _ { \mathrm { t a s k } } , c _ { \mathrm { t e x t } } ] , \mathcal { T } _ { c } , \pmb { x } ) \sim \mathcal { D } _ { k } .$ , where $t$ represents the time step, $z _ { t }$ is the noise-corrupted latent tensor at time step $t$ , $z _ { 0 } = E ( \pmb { x } )$ , and $\theta$ is the trainable parameters of UniControl . We also apply classifier-free guidance [51] to randomly drop $30 \%$ text prompts to enhance the controllability of input visual conditions. We train UniControl uniformly on the $K$ tasks. To be more specific, we first randomly select a task $k$ and sample a mini-match from $\mathcal { D } _ { k }$ , and optimize $\theta$ with the calculated loss $\ell ^ { k } ( \theta )$ .
+
+# 3.2 Model Design
+
+Since our unified model UniControl needs to achieve superior performance on a set of diverse tasks, it is necessary to ensure the network design enjoys the following properties: 1) The model can overcome the misalignment of low-level features from different tasks; 2) The model can learn meta-knowledge across tasks, and adapt to each task effectively.
+
+![](images/567eb74edf244be03efaf0e3179e4c11be587c12bc7456afce228a991e56a28a.jpg)  
+Figure 2: This figure shows our proposed UniControl method. To accommodate diverse tasks, we’ve designed a Mixture of Experts (MOE) Adapter, containing roughly $\mathord { \sim } 7 0 \mathrm { K }$ $\#$ params for each task, and a Task-aware HyperNet $\mathrm { \sim } 1 2 \mathrm { M }$ #params) to modulate $N$ (i.e., 7) zero-conv layers. This structure allows for multi-task functionality within a singular model, significantly reducing the model size compared to an equivalent stack of single-task models, each with around 1.4B #params.
+
+The first property can ensure that UniControl can learn necessary and unique information from all tasks. For instance, if UniControl takes the segmentation map as the visual condition, the model might ignore the 3D information. As a result, the feature map learned may not be suitable for the task that takes the depth map images as visual condition. The second property would allow the model to learn the shared knowledge across tasks, as well as the differences among them.
+
+We introduce two novel designed modules, MOE-style adapter and task-aware HyperNet, that allows UniControl enjoys the above two properties. An overview of the model design for UniControl is in Fig. 2. We describe the detailed designs of these modules below.
+
+MOE-Style Adapter. Inspired by the design of Mixture-of-Experts (MOEs) [52], we devise a group of convolution modules to serve as the adapter for UniControl to capture features of various low-level visual conditions. Precisely, the designed adapter module can be expressed as
+
+$$
+\mathcal { F } _ { \mathrm { A d a p t e r } } ( \mathcal { Z } _ { c } ^ { k } ) : = \sum _ { i = 1 } ^ { K } \mathbb { 1 } ( i = = k ) \cdot \mathcal { F } _ { \mathrm { C o v 1 } } ^ { ( i ) } \circ \mathcal { F } _ { \mathrm { C o v 2 } } ^ { ( i ) } ( \mathcal { Z } _ { c } ^ { k } ) ,
+$$
+
+where $\mathbb { 1 } ( \cdot )$ is the indicator func on, $\mathcal { T } _ { c } ^ { k }$ is the conditioned image from task $k$ , and $\mathcal { F } _ { \mathrm { { C o v 1 } } } ^ { ( i ) } , \mathcal { F } _ { \mathrm { { C o v 2 } } } ^ { ( i ) }$ are the convolution layers of the $i$ -th module of the adapter. We remove the weights of the original MOEs since our designed adapter is required to differentiate various visual conditions. Meanwhile, naive MOE modules can not explicitly distinguish different visual conditions when the weights are learnable. Moreover, such task-specific MOE adapters facilitate the zero-shot tasks with explicit retrieval of the adapters of highly related pre-training tasks. Besides, the number of parameters for each convolution module is approximately 70K, which is computationally efficient.
+
+![](images/ddfdf7bb249088e3f366e8b8ba221b8f472a1b46e62e984081e2fc92dd49c629.jpg)  
+Figure 3: Illustration of MOE’s behaviors under zero-shot scenarios. The left part shows the capacity of the MOE to generalize to hybrid task conditions, achieved through the integration of outputs from two pertinent convolution layers. The right part illustrates the ability of the MOE-style adapter to generalize to unseen tasks, facilitated by the aggregation of pre-trained tasks using estimated weights.
+
+Task-Aware HyperNet. The task-aware HyperNet modulates the zero-convolution modules of ControlNet [21] with the task instruction condition $c _ { \mathrm { t a s k } }$ . As shown in Figure 2, our hyperNet first projects the task instruction $c _ { \mathrm { t a s k } }$ into task embedding with the help of CLIPText encoder. Then similar in spirit of style modulation in StyleGAN2 [53], we inject the task embedding into the trainable copy of ControlNet, by multiplying the task embedding to each zero-conv layer. In specific, the length of the embedding is the same as the number of input channels of the zero-conv layer, and each element scalar in the embedding is multiplied to the convolution kernel per input channel. We also show that our newly designed task-aware HyperNet can also efficiently learn from training instances and task supervision following a similar analysis as in ControlNet [21].
+
+# 3.3 Task Generalization Ability
+
+With the comprehensive pretraining on the MultiGen-20M dataset, UniControl exhibits zero-shot capabilities on tasks that were not encountered during its training, suggesting that Unicontrol possesses the ability to transcend in-domain distributions for broader generalization. We demonstrate the zeroshot ability of UniControl in the following two scenarios:
+
+Hybrid Tasks Generalization. As shown in the left side of Fig. 3, We consider two different visual conditions as the input of UniControl, a hybrid combination of segmentation maps and human skeletons, and augment specific keywords "background" and "foreground" into the text prompts. Besides, we rewrite the hybrid task instruction as a blend of instructions of the combined two tasks such as "segmentation map and human skeleton to image".
+
+Zero-Shot New Tasks Generalization. As shown in the right side of Fig. 3, UniControl needs to generate controllable images on a newly unseen visual condition. To achieve this, estimating the task weights based on the relationship between unseen and seen pre-trained tasks is essential. The task weights can be estimated by either manual assignment or calculating the similarity score of task instructions in the embedding space. The example result in Fig. 5 (d) is generated by our manually assigned MOE weights as “depth: 0.6, seg: 0.3, canny: 0.1” for colorization. The MOE-style adapter can be linearly assembled with the estimated task weights to extract shallow features from the newly unseen visual condition.
+
+# 4 Experiments
+
+We empirically evaluate the effectiveness and robustness of UniControl. We conduct a series of comprehensive experiments across various conditions and tasks, utilizing diverse datasets to challenge the model’s adaptability and versatility. Experimental setup, methodologies, and results analysis are provided in the subsequent sections.
+
+# 4.1 Experiment Setup
+
+Implementation. The UniControl is illustrated as Fig. 2 with Stable Diffusion, ControlNet, MOE Adapter, and Task-aware HyperNet consisting ${ \sim } 1 . 5 \mathrm { B }$ parameters. MOE Adapter consists of parallel convolutional modules, each of which corresponds to one task. The task-aware HyperNet inputs the CLIP text embedding [22] of task instructions and outputs the task embeddings to modulate the weights of zero-conv kernels. We implement our model upon the ControlNet . We take the AdamW [54] as the optimizer based on PyTorch Lightning [55]. The learning rate is assigned as $1 \times 1 0 ^ { - 5 }$ . Our full-version UniControl model is trained on 16 Nvidia-A100 GPUs with the batch size of 4, requiring $\sim 5$ , 000 GPU hours. We have also applied Safety-Checker as safeguards of results.
+
+![](images/3f78c1bb95cc3e20339cf41b1c27cd81750dd296c828198bfdec2efd35181699.jpg)  
+Figure 4: Visual comparison between official or re-implemented task-specific ControlNet and our proposed model. The example data is collected from our testing set sampled from COCO and Laion.
+
+Data Collection. Since the training set of ControlNet is currently unavailable, we initiate our own data collection process from scratch and name it as MultiGen-20M. We use a subset of LaionAesthetics-V2 [56] with aesthetics ratings over six, excluding low-resolution images smaller than 512. This yields approximately 2.8 million image-text pairs. Subsequently, we process this dataset for nine distinct tasks across five categories (edges, regions, skeletons, geometric maps, real images):
+
+• Canny (2.8M): Utilize the Canny edge detector [57] with randomized thresholds.
+
+• HED (2.8M): Deploy the Holistically-nested edge detection [58] for robust boundary determination.
+
+• Depth (2.8M): Employ the Midas [59] for monocular depth estimation.
+
+• Normal (2.8M): Use the depth estimation results from the depth task to estimate scene or object surface normals.
+
+• Segmentation (2.8M): Implement the Uniformer [60] model, pre-trained on the ADE20K [61] dataset, to generate segmentation maps across 150 classes.
+
+• Object Bounding Box (874K): Utilize YOLO V4 [62] pre-trained on the COCO [63] dataset for bounding box labelling across 80 object classes.
+
+• Human Skeleton (1.3M): Employ the pre-trained Openpose [64] model to generate human skeleton labels from source images.
+
+• Image Outpainting (2.8M): Create boundary masks for source images with random masking percentages from $20 \%$ to $80 \%$ .
+
+Further processings are carried out on HED maps using Gaussian filtering and binary thresholding to simulate user sketching. Overall, we amass over 20 million image-prompt-condition triplets. Task instructions were naturally derived from the respective conditions, with each task corresponding to a specific instruction, such as "canny edge to image" for the canny task. We maintain a one-toone correspondence between tasks and instructions without introducing variance to ensure stability during training. We have additionally collected a testing dataset for evaluation with 100-300 imagecondition-prompt triplets for each task. The source data is collected from Laion and COCO. We will open-source our training and testing data to contribute to the community.
+
+Benchmark Models. The most straightforward comparison for UniControl comes from task-specific ControlNet models. Six tasks overlap with those presented in ControlNet, so their official models are chosen as baselines for these tasks. For fair comparison, we re-implement the ControlNet model (single task) using our collected data. Our unified multi-task UniControl is compared against these task-aware models for each task. We apply default sampler as DDIM [32] with guidance weight 9 and steps 50. All single-task models used for comparison are trained by 100K iterations and our multi-task model is trained around 900K with similar iterations for each task to ensure fairness. The efficiency and compact design of our proposed model are evident in its construction. The total size of UniControl is around 1.5B #params and a single task ControlNet $^ +$ SDM takes 1.4B. In order to achieve the same nine-task functionality, a single-task strategy would require the ensemble of a SDM with nine task-specific ControlNet models, amounting to approximately 4.3B #params in total.
+
+![](images/6c0ea7ba2a2a2c99b0745134a484112af74d27e5d6570f3bad24e796d908e009.jpg)  
+Figure 5: (a)-(b): Example results of UniControl over hybrid (unseen combination) conditions with key words "background" and "foreground" attached in prompts. (c)-(e): Example results of UniControl on three unseen tasks (deblurring, colorization, inpainting).
+
+![](images/3fe5b43a671f7a865acfe456334ba3770564c1cb59a286ee1b3354c2ef64d584.jpg)  
+Figure 6: User study between our method and official ControlNet checkpoints on six tasks. Our method outperforms ControlNet on all tasks.
+
+# 4.2 Visual Comparison
+
+We visually compare different tasks (Canny, HED, Depth, Normal, Segmentation, Openpose, Bounding Box, and Outpainting) in Fig. 4. Our method consistently outperforms the baseline ControlNet model. This superiority is in terms of both visual quality and alignment with conditions or prompts.
+
+For the Canny task, the results generated by our model exhibit a higher degree of detail preservation and visual consistency. The outputs of UniControl maintain a faithful reproduction of the edge information (i.e., round table) compared to ControlNet. In the HED task, our model effectively captures the robust boundaries, leading to visually appealing images with clear and sharp edge transitions, whereas ControlNet results appear to be non-factual. Moreover, our model demonstrate a more subtle understanding of 3D geometrical guidance of depth maps and surface normals than ControlNet. The depth map conditions produce visibly more accurate outputs. In the Normal task, our model faithfully reproduces the normal surface information (i.e., ski pole), leading to more realistic and visually superior outputs. During the Segmentation, Openpose, and Object Bounding Box tasks, the produced images generated by our model are better aligned with the given conditions than that by ControlNet, ensuring a higher fidelity to the input prompts. For example, the re-implemented ControlNet-BBox misunderstands “a woman near a statue”, whereas our outputs exhibit a high degree of accuracy and detail. In the Outpainting task, our model demonstrates its superiority by generating reasonable images with smooth transitions and natural-looking textures. It outperforms the ControlNet model, which produces less coherent results - “a bear missing one leg”. This visual comparison underscores the strength and versatility of our approach across a diverse set of tasks.
+
+![](images/a44d12886fd18a351b70182433054e16acefca7e6bf3e28199ae43767333bc04.jpg)  
+Figure 7: User study between our multi-task model (Ours-multi) and single task model (Ours-single) on eight tasks. Our method outperforms baselines on most of tasks, and achieves big performance gains on tasks of seg-to-image and outpainting-to-image. Moreover, the p-value of voting Ours-multi in all cases is computed as 0.0028 that is statistically significant according to the criteria of $< 0 . 0 5$ .
+
+# 4.3 Quantitative Evaluation
+
+User Study. We compare the performance of our method with both the released ControlNet model and the re-implemented single-task ControlNet on our training set. As shown in Fig. 6, our approach consistently outperforms the alternatives in all cases. In the HED-to-image generation task, our method significantly surpasses ControlNet. This superiority is even more pronounced in the depth and normal surface to image generation tasks, where users overwhelmingly favor our method, demonstrating its ability to handle complex geometric interpretations. When compared to the re-implemented single-task model, Fig. 7 reveals that our approach maintains a smaller advantage, yet it still demonstrates its benefits by effectively discerning image regions to guide content generation. Even in the challenging outpainting task, our model outperforms the baseline, highlighting its robustness and capacity to generalize.
+
+Table 2: Image Perceptual Distance   
+
+<table><tr><td></td><td>Canny↓</td><td>HED↓</td><td>Normal↓</td><td>Depth↓</td><td>Pose↓</td><td>Segmentation ↓</td></tr><tr><td>UniControl</td><td>0.546</td><td>0.466</td><td>0.623</td><td>0.654</td><td>0.741</td><td>0.693</td></tr><tr><td>ControlNet</td><td>0.577</td><td>0.582</td><td>0.778</td><td>0.700</td><td>0.747</td><td>0.693</td></tr></table>
+
+Image Perceptual Metric. We evaluate the distance between our output and the ground truth image. As we aim to obtain
+
+a structural similar image to the ground truth image, we adopt the perceptual metric in [65], where a lower value indicates more similar images. As shown in Tab. 2, UniControl outperforms ControlNet on five tasks, and obtains the same image distance to ControlNet on Segmentation.
+
+Fréchet Inception Distance (FID). We’ve further conducted quantitative analysis with FID [66] to include more classic single-task-controlled methods such as GLIGEN [67] and T2I-adapter [68]. With a collection of over 2,000 test samples sourced from Laion and COCO, we’ve assessed a wide range of tasks covering edges (Canny, HED), regions (Seg), skeletons (Pose), and geometric maps (Depth, Normal). The Tab. 3 demonstrates that our UniControl consistently surpasses the baseline methods across the majority of tasks. Notably, UniControl achieves this while maintaining a more compact and efficient architecture than its counterparts.
+
+Ablation Study. We’ve conducted an ablation study, specifically focusing on the MoE-Style Adapter and TaskHyperNet in Tab. 4 with FID scores reported as the previous part. It is noticeable that the full-version UniControl (MoE-Style Adapter $^ +$ TaskHyperNet) significantly outperforms the ablations which demonstrates the superiority of proposed MoE-Style Adapter and TaskHyperNet.
+
+Table 3: Quantitative Comparison (FID)   
+
+<table><tr><td></td><td>Canny↓</td><td>HED↓</td><td>Depth ↓</td><td>Normal↓</td><td>Seg↓</td><td>Pose↓</td></tr><tr><td>GLIGEN [67]</td><td>24.9</td><td>27.8</td><td>25.8</td><td>27.7</td><td>-</td><td>=</td></tr><tr><td>T2I-Adapter [68]</td><td>23.6</td><td>1</td><td>25.4</td><td>-</td><td>27.1</td><td>28.9</td></tr><tr><td>ControlNet [21]</td><td>22.7</td><td>25.1</td><td>25.5</td><td>28.4</td><td>26.7</td><td>28.8</td></tr><tr><td>UniControl</td><td>22.9</td><td>23.6</td><td>21.3</td><td>23.4</td><td>25.5</td><td>27.4</td></tr></table>
+
+Table 4: Ablation Study (FID)   
+
+<table><tr><td>MoE-Adapter</td><td>TaskHyperNet</td><td>Canny↓</td><td>HED↓</td><td>Depth ↓</td><td>Normal↓</td><td>Seg↓</td><td>Pose↓</td><td>Avg</td></tr><tr><td></td><td>X</td><td>27.2</td><td>29.0</td><td>27.6</td><td>28.8</td><td>29.1</td><td>30.2</td><td>28.7</td></tr><tr><td></td><td></td><td>24.5</td><td>26.1</td><td>23.7</td><td>24.8</td><td>26.9</td><td>28.3</td><td>25.7</td></tr><tr><td>x&lt;&gt;</td><td>X</td><td>22.9</td><td>23.6</td><td>21.3</td><td>23.4</td><td>25.5</td><td>27.4</td><td>24.0</td></tr></table>
+
+# 4.4 Zero-shot Generalization
+
+We further showcase the surprising capabilities of our method to undertake the zero-shot challenge of hybrid conditions combination and unseen tasks generalization.
+
+Hybrid Tasks Combination. This involves generating results from two distinct conditions simultaneously. Our model’s zero-shot ability is tested with combinations such as depth and human skeleton or segmentation map and human skeleton. The results are shown in Fig. 5 (a)-(b). When the background is conditioned on a depth map, the model effectively portrays the intricate 3D structure of the scene, while maintaining the skeletal structure of the human subject. Similarly, when the model is presented with a combination of a segmentation map and human skeleton, the output skillfully retains the structural details of the subject, while adhering to the segmentation boundaries. These examples illustrate our model’s adaptability and robustness, highlighting its ability to handle complex hybrid tasks without any prior explicit training.
+
+Unseen Tasks Generalization. To evaluate the zero-shot ability to generalize to unseen tasks such as gray image colorization, image deblurring, and image inpainting, we conduct the case analysis in Fig. 5 (c)-(e). The model skillfully handles the unseen tasks, producing compelling results. This capability is deeply rooted in the shared attributes and implicit correlations among pre-training and new tasks, allowing our model to adapt seamlessly. For instance, the colorization task leverages the model’s understanding of image structures from the segmentation task and depth estimation task, while deblurring and inpainting tasks benefit from the model’s familiarity with edge detection and outpainting ones.
+
+# 5 Conclusion and Discussion
+
+We introduce UniControl , a novel unified model for incorporating a wide range of conditions into the generation process of diffusion models. UniControl has been designed to be adaptable to various tasks through the employment of two key components: a Mixture-of-Experts (MOE) style adapter and a task-aware HyperNet. The experimental results have showcased the model’s robust performance and adaptability across different tasks and conditions, demonstrating its potential for handling complex text-to-image generation tasks.
+
+Limitation and Broader Impact. While UniControl demonstrates impressive performance, it still inherits the limitation of diffusion-based image generation models. Specifically, it is limited by our training data, which is obtained from a subset of the Laion-Aesthetics datasets. We observe that there is a data bias in this dataset. Although we have performed keywords and image based data filtering methods, we are aware that the model may generate biased or low-fidelity output. Our model is also limited when high-quality human output is desired. UniControl could be improved if better open-source datasets are available to block the creation of biased, toxic, sexualized, or other harmful content. We hope our work can motivate researchers to develop visual generative foundation models.
+
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+
+# Appendix
+
+# A Details of Implementation
+
+# A.1 MOE-Style Adapter
+
+The MOE adapter is implemented as a set of parallel ConvNets composed of three consecutive convolution and non-linear activation layers. The entire model is comprised of nine individual MOE adapters, each of which consumes 70K parameters. Task keys are designated to each adapter, ensuring that they align with the corresponding visual conditions. Once the MOE adapter processes the input, the remaining model parameters become shared across all tasks. This architecture facilitates task adaptability while promoting parameter efficiency.
+
+# A.2 Task-aware HyperNet
+
+The task-aware hypernet is applied to modulate the parameters of zero-conv layers in the ControlNet. Since the ControlNet can be considered as the hypernet of Stable Diffusion (fixed copy). Our idea can be concluded as the control over control or meta-control to let the task-aware hypernet learn the universe representation that is generalizable across different tasks. To implement it, we firstly map the task keys to instruction with a mapping function as: {"hed": "hed edge to image", "canny": "canny edge to image", "seg": "segmentation map to image", "depth": "depth map to image", "normal": "normal surface map to image", "pose": "human pose skeleton to image", "hedsketch": "sketch to image", "bbox": "bounding box to image", "outpainting": "image outpainting"}. Then, such instructions will be projected as text embeddings with the help of a language model (we adopt CLIPText in our implementation). The Task-aware HyperNet takes these task instruction embeddings, and projects them into different shapes to match the size of different zero-conv kernels, which will be modulated by these task embeddings accordingly. We would fix the parameters of task-aware hyperNet in the later stage of model training to ensure the stability of dynamics.
+
+# A.3 Data Collection
+
+We have collected a large amount of training set (MultiGen-20M) including over 20M conditionimage-prompt triplets across nine different tasks. We firstly download 3/4 of Laion-Aesthetics-V2 with score over six and filter out low-resolution $( < 5 1 2 )$ images. As a result, $2 . 8 \mathbf { M }$ images are selected as source images. Then we apply the visual condition extractors as described in the main paper to collect Canny, HED, Sketch, Depth, Normal Surface, Seg Map, Object Bounding Box, Human Skeleton and Outpainting.
+
+# B Numerical Analysis of Task-Aware Modulated ControlNet
+
+We show that our proposed task-aware modulated ControlNet preserves the properties of the original ControlNet structure. Specifically, we show 1) The new task-aware modulated ControlNet preserves the zero-initialization property of ControlNet; 2) The parameters of the task-aware modulated Controlnet can be updated once we start to train the model.
+
+Denote the input feature map by , the frozen SD Block in Fig. 2 by $\mathcal { F } _ { \mathrm { S D } }$ , the extra condition by $c$ , two zero convolution operators by $\mathcal { Z } _ { \theta _ { 1 } } ^ { 1 } ( \cdot )$ and $\mathcal { Z } _ { \theta _ { 2 } } ^ { 2 } ( \cdot )$ , the trainable copy of SD Block by $\mathcal { G } _ { \theta _ { \mathrm { s } } } ^ { \mathrm { S D } } ( \cdot )$ , the task instruction by $c _ { \mathrm { t a s k } }$ , and the task-aware hyperNet by $\mathcal { H } _ { \boldsymbol { \theta } _ { \mathcal { H } } } ( \cdot )$ . Then the output of the new task-aware modulated Controlnet can be expressed as
+
+$$
+{ \pmb y } _ { c } = \mathcal { F } _ { \mathrm { S D } } ( { \pmb x } ) + \mathcal { Z } _ { \theta _ { 1 } } ^ { 1 } ( \mathcal { G } _ { \theta _ { \mathrm { s } } } ^ { \mathrm { S D } } ( { \pmb x } + \mathcal { Z } _ { \theta _ { 2 } } ^ { 2 } ( c ) \cdot \mathcal { H } _ { \theta _ { \mathcal { H } } } ( c _ { \mathrm { t a s k } } ) ) ) \cdot \mathcal { H } _ { \theta _ { \mathcal { H } } } ( c _ { \mathrm { t a s k } } ) .
+$$
+
+Property of Zero Initialization. Similar to ControlNet [21], the weights and biases of the convolution layers are initialized as zeros. As a result, we have $\mathcal { Z } _ { \theta _ { 1 } } ^ { 1 } ( \cdot ) \equiv 0$ and $\pmb { y } _ { c } = \mathcal { F } _ { \mathrm { S D } } ( \pmb { x } )$ , regardless of the initialization of $\mathcal { H } _ { \theta _ { \mathcal { H } } } ( \cdot )$ .
+
+Gradient Analysis. We analyze the gradient of the modulated part
+
+$$
+\nabla _ { \theta } \left( Z _ { \theta _ { 1 } } ^ { 1 } ( I ) \cdot \mathcal { H } _ { \theta _ { \mathcal { H } } } ( c _ { \mathrm { t a s k } } ) \right) = \mathcal { H } _ { \theta _ { \mathcal { H } } } ( c _ { \mathrm { t a s k } } ) \cdot \nabla _ { \theta } Z _ { \theta _ { 1 } } ^ { 1 } ( I ) + Z _ { \theta _ { 1 } } ^ { 1 } ( I ) \cdot \nabla _ { \theta _ { \mathcal { H } } } \mathcal { H } _ { \theta _ { \mathcal { H } } } ( c _ { \mathrm { t a s k } } ) ,
+$$
+
+where $I$ is the input of the zero convolution layer.
+
+When we start to train the network, the first part of the RHS of (2) follows similar analysis of ControlNet [21] since $\mathcal { H } _ { \boldsymbol { \theta } _ { \mathcal { H } } } \left( c _ { \mathrm { t a s k } } \right)$ is constant when we analyze the gradient $\nabla _ { \theta } Z _ { \theta _ { 1 } } ^ { 1 } ( I )$ . Since the parameters of $\mathcal { H } _ { \theta _ { \mathcal { H } } } ( c _ { \mathrm { t a s k } } )$ are not initialized to zero, it is known that $\mathcal { H } _ { \theta _ { \mathcal { H } } } ( c _ { \mathrm { t a s k } } ) \neq 0$ . So the gradient dynamic follows the analysis of ControlNet. Therefore, we conclude that $Z _ { \theta _ { \bot } } ^ { 1 } ( I ) \neq 0$ after the first gradient update, and that the network can start to learn and update the following standard dynamics of stochastic gradient descent.
+
+As for the second part of the RHS of (2), $Z _ { \theta _ { 1 } } ^ { 1 } ( I ) \equiv 0$ before the first gradient update, so the gradient is zero for $\theta _ { \mathcal { H } }$ . However, after the first gradient update of $\theta _ { 1 }$ , we know $Z _ { \theta _ { 1 } } ^ { 1 } ( I ) \neq 0$ , and $\theta _ { \mathcal { H } }$ can be updated with non-zero gradients.
+
+To conclude, the new task-aware Modulated ControlNet can still be efficiently updated and learned even if the convolution layers are initialized to zero.
+
+# C Zero-shot-task Results and Analysis
+
+We show more zero-shot-task results in this section, where the tasks have not been trained on. In Fig. 13, we show zero-shot deblurring results guided by the keywords. Our deblurred images can successfully recover the fine-grained details of the images without training on such data. We note that some details are still missing, e.g., the details in the painting in the first row are still not clear enough. In Fig. 14, we illustrate two zero-shot image colorization results. We believe that most parts of the generated images are acceptable, though the clothes of the second woman do not look the same to the input blurred image. In Fig. 15, we observe impressive zero-shot inpainting results. In the first row, the duck that is inputted in the text has been successfully generated in the inpainted image. The second row obtains acceptable results as well, though the faces do not look perfect. The overall zero-shot quality of UniControl is remarkable.
+
+While inpainting and outpainting might appear related, they are fundamentally distinct. Inpainting heavily leverages the contextual information from unmasked regions, necessitating a precise match. Conversely, outpainting has more freedom, with the generative model prioritizing prompts to envision new content. As shown in Fig. 8, directly using outpainting model for inpainting tasks can be challenging since the model tends to leave a sharp change over the mask boundaries. Our pretrained UniControl, thanks to intensive training across multiple tasks, has learned edge and region-to-image mappings, which assists in preserving contextual information.
+
+Our model also demonstrates a promising capacity to generalize under scribble conditions, showing parallels to the ControlNet’s ability, even though UniControl hasn’t been directly trained using scribble data. Fig. 9 provides results illustrating the scribble-to-image generation.
+
+# D Details of User Study
+
+In the evaluation steps, we use Amazon Mechanical Turk (Mturk) 2 to perform user study. Specifically, we ask three Mturk master workers to select the best output result for each input condition. As shown in Fig. 10, we provide instructions on guidelines to select the best generated image. The annotators are provided the condition map and the text that describes the image, and are required to select the better output between the two generated images. Considering that images can both in good or bad qualities, we provide the tie option as well. We use the majority vote to determine the result of each image, which means that an image is considered as a better image if two or more annotators vote for it. We use 294 images for the tasks of Canny, HED, Surface Normal, Depth, Segmentation, User Sketch, and Outpainting. We adopt 100 images for the task of Human Skeleton and 187 images for the task of Bounding Box. In summary, we totally obtain 7,035 voting results for all nine tasks. 2/3 of source images in testing set are collected from MSCOCO with the remaining 1/3 from Laion. And it includes a very diverse range of topics including indoor scene, outdoor scene, oil painting, portrait, pencil sketch, animation, cartoon, etc.
+
+![](images/7796cc26a834c8384afb9bcfcd3395271440f7b33f59f84f756afb5088431095.jpg)  
+“Contemporary Bedroom Designs 2015 modern bedroom designs intended design”
+
+![](images/2d355454522171fa2e1b083602cdc59ebab9a153e42804ce92e584efa68ebab9.jpg)  
+Figure 8: Visual comparison of Ours-single-outpainting and UniControl on the inpainting task. The single outpainting model cannot well address the zero-shot inpainting task whereas UniControl demonstrates promising capacity.   
+Figure 9: Visual comparison of ControlNet-Scribble and UniControl on the scribble data. ControlNetScribble is trained by the scribble data which, however, are unseen for UniControl.
+
+![](images/16698d49a4b5517d756c624b1810c4b1688ca335a2f7854252080f45c1e5e283.jpg)  
+Figure 10: Mturk interface to select the better generated image.
+
+![](images/533f5d7e0c6cc374e12d4d3c734e3d41fc6ad14014462c57fa1def5de3219eb5.jpg)  
+Figure 11: User study results of User Sketch to image generation.
+
+# E Failure Cases
+
+We illustrate some failure cases in Fig. 12. In the first row, although our generated image successfully aligns the Bounding Box condition, the generated human has a distorted body. In the second row, our generated image looks similar to the ground truth; however, the human faces are blurred. We think that the reason is that UniControl inherits the data and model bias of Stable Diffusion, where the generated human commonly have issues. In the third row, the generated image does not look realistic. We believe that the training data can be improved both quantitatively and qualitatively.
+
+# F Additional Results
+
+We illustrate more visualized results in this section on tasks Canny (Fig. 16), HED (Fig. 17), Depth (Fig. 18), Surface Normal (Fig. 19), Human Skeleton (Fig. 20), Bounding Box (Fig. 21), Segmentation (Fig. 22) and Outpainting (Fig. 23). These results further demonstrate the effectiveness of our proposed method. Moreover, due to the space limitation in the main paper, we report results of the last task, User Sketch. Given a sketched image, UniControl is able to achieve promising realistic images. The visualized results are in Fig. 24. The user study result can be found in Fig. 11, where it is observed that UniControl obtains significantly more votes than the single task model.
+
+![](images/864ebf88f6ec3183b5f05edf0269ae982cf62ec77abde0f6f63e4d7710948da3.jpg)
+
+![](images/05a4ebc5894b0db0cea1eaa2b852f1eb3fba973b98e17810371f725e3b5e3c51.jpg)  
+“La tricoteuse Realism William Adolphe Bouguereau Oil Paintings”
+
+![](images/08c58339e14946f66088b5521c45f1e8932b94d016071608b8865306d9348e5f.jpg)
+
+“A man and woman in ski gear standing in front of a mountain. “ “The Taj Mahal mirrored by a water fountain's reflection. - Agra, Uttar Pradesh, India - Daily Travel Photos”
+
+![](images/ed2f7c68e031fb8b210652223f09edaaa385153ed362bc6c82925dc718153393.jpg)  
+Figure 12: Failure Cases: distorted body (row one); blurred faces (row two); incorrect creation (row three).
+
+![](images/ce300439a093037af0e22018b9d26d71a58799df23ce9342b111dfd41d49307c.jpg)
+
+“Christa McAuliffe (right, sat with her backup crew member Barbara Morgan) was a social studies teacher who had won NASA's Teacher in Space contest and earned herself a spot on the mission”
+
+![](images/406de263650c6bfb5fe8a6742a82c576f8bdf7771416a4ff2faecafc56ca8408.jpg)  
+Figure 13: More zero-shot-task deblurring results.
+
+Gray Image
+
+![](images/a3270977f741084bdf043f1e398e79f8a4eda204077b26f6b699967ba6bf420a.jpg)
+
+Our Result
+
+![](images/1fe38f8b80c193320b1b25b181e91c2293b6f9e87787b2cf78ff1bc057b56e01.jpg)
+
+# “Long White Casual Wedding Dress”
+
+![](images/0ff793313409af7320cb2ee71b0336a463c8f08a5e0364bf684fad7d31969317.jpg)
+
+![](images/e4a211c18f21ce16dc9b45fa2d06a8fb02cb21839e8c77d31f5659b5a8a2c317.jpg)
+
+“Pixie Cropped Short Layered Synthetic Wig for Women-KAMI WIGS”
+
+![](images/0164e828eb13380270f067c363b0186d1bbcb1960acfff09d450029d44bfae3b.jpg)  
+Figure 14: More zero-shot-task gray-to-RGB colorization results.
+
+![](images/d8ab528fc162e3fedd70130a5e90757521ec1ccba11b58ef899b5eafafedecf2.jpg)
+
+“Early morning view over the town of Tinerhir, south of the Todra Gorge, Morocco, North Africa, Africa”
+
+![](images/f135750c3bf538a1ff4c9800056787e401a4ba3c39c05f42b7e5365257228845.jpg)  
+“A lone duck basks in the calm lake's mirror reflection of the Chugach mountain valley”
+
+![](images/1a04600ceab9a09733a2fa4974eaef5863da84806ec06ea5229185e5b32c546c.jpg)  
+“Chancellor of the Exchequer Rishi Sunak was the most high-profile, and unexpected, appointment of the day”
+
+![](images/14e02bdaff82cfd7812f3eda8815ab30625440fa03efb8d9ab27d8ab1aec96cf.jpg)  
+“Contemporary Bedroom Designs 2015 modern bedroom designs intended design “   
+Figure 15: More zero-shot-task image in-painting results. The in-painting MOE adapter weights are directly inherited from outpainting.
+
+![](images/8e4c74d2cca91527f6427f7df6ce14f59247ff3cf6638d066c185853ca69c6df.jpg)
+
+![](images/9ecf91e675f8d8d800075a4ae88a33aef4033b041c54d7252883d8413fccc53c.jpg)  
+Figure 17: HED to Image Generation
+
+(d) “A young girl who is brushing her teeth with a toothbrush.”
+
+![](images/608c7b4897f463da978e27197fc39ed8b66a83e7604f69f386112f6b0918d592.jpg)  
+Figure 18: Depth to Image Generation
+
+![](images/ca0399c20aef42ff8505a70b62dc5bea44a6c393767e01717313e124b3666f8c.jpg)
+
+![](images/307da515b66bcf3263520796363500a159592a4beed485f9abf975cb00a1a13e.jpg)
+
+![](images/edc65b5f77ef0dd345e936158b7a838ee8c6cf8d0c81b2d841c7c90e82621044.jpg)  
+Figure 19: Surface Normal to Image Generation
+
+![](images/e2c5a982dc82ee252be359b3af2b07dde3d18070a621e58b6df7d3118d6b8b1a.jpg)
+
+![](images/9e18dba2bbc4da954773e50475189a163a3b565f1883a8a74f58c087783cd511.jpg)
+
+(a) “Photo of handsome man in black leather jacket”
+
+![](images/fa35fbc7943b8c85b6b5ea26f058940226239c3b9410569a34218ac3e9c58baa.jpg)  
+Input Image<Captio
+
+![](images/fd9e4e671d6ef27027d3e893624e861f319dbfe87dd74710fc10fc8abb2232a6.jpg)  
+Our Method Outputhe snow.
+
+(b) “A woman is sitting near a prominent landmark”
+
+![](images/bd8a798fcfbbf0d3b80669a48d7e2732efead276d6aa020f703b3c124775fff6.jpg)  
+Our Method Outputdesk
+
+(c) “A man that has ski’s and is standing in the snow.”
+
+![](images/47cd8d08dfba650e8d11972cc7c3f075e2af3fdcaee7764d368b5a4950bb88b7.jpg)  
+Input Image<   
+Figure 20: Human Pose Skeleton to Image Generation
+
+(d) “A woman is sitting in front of a desk”
+
+![](images/dc7804f648f95a2a22cf80a3473e45e1d7e39a8f68d1387d94119c16ed8279e3.jpg)
+
+![](images/8b48892dcdf5997bd253d939ad73025ccc80eec99a968be71ac95455d87347cc.jpg)  
+Input Image
+
+![](images/ac0d32d51c3a9f5f9e32428d2797560fa0131b431c76607efbdf0a8568b787bc.jpg)  
+Our Method Output
+
+(a) “A bench at the beach next to the sea”ion>: Water traffic along the Thames by Big
+
+![](images/b9bc5fe441a84ca5ed0902d7991aa92463777b428224fad431090fe7457e7254.jpg)
+
+![](images/315d4ce3f1b36966270e3abd883743371420489d36f7a040fadca6242fd11d55.jpg)  
+Our Method Output
+
+(b) “Water traffic along the Thames by Big Ben”
+
+![](images/169a4907042a8a0f146ea427fe8b9df037cad21d932aaa9d2b192489c82562b1.jpg)
+
+(c) “A well-lit and well-decorated living room shows a glimpse of a glass front door through the corridor.”
+
+![](images/9c7333bbce3405b4086a56ad84a32c1c3e9b3432694e06f759f5790b04f6ad72.jpg)  
+Figure 22: Segmentation Map (by Uniformer-ADE20K) to Image Generation
+
+![](images/853bb3400ea0e0bc5aa6b87da79c91ab47a4714f6eef163977f142964fd5aed2.jpg)  
+Figure 23: Image Outpainting
+
+(d) “Beautiful kitchen grand scale living pinterest for Kitchen cabinets lowes with old world metal wall art”
+
+![](images/c113ff5911f48b8ab6744e8638c79653ea69d2209d8794f4319072c9f2846759.jpg)
+
+![](images/0f48ffb5077bb8a7f0ec71d24de98c629e02d50df7b51f1664154ac9a5c945f5.jpg)  
+(a) “A Limited Edition, Fine Art photograph of a beautiful sunrise at Lake Jackson in Sebring, Florida. Available as a Fine Art print”   
+Figure 24: User Sketch to Image Generation
+
+(d) “Superhero watching over city. No transparency used. Basic (linear) gradients. A4 proportions.”