Datasets:

zai-org
/

RPC-Bench

+# MODULAR ACTION CONCEPT GROUNDINGIN SEMANTIC VIDEO PREDICTION
+Anonymous authors Paper under double-blind review
+# ABSTRACT
+Recent works in video prediction have mainly focused on passive forecasting and low-level action-conditional prediction, which sidesteps the learning of interaction between agents and objects. We introduce the task of semantic actionconditional video prediction, which uses semantic action labels to describe those interactions and can be regarded as an inverse problem of action recognition. The challenge of this new task primarily lies in how to effectively inform the model of semantic action information. Inspired by the idea of Mixture of Experts, we embody each abstract label by a structured combination of various visual concept learners and propose a novel video prediction model, Modular Action Concept Network (MAC). Our method is evaluated on two newly designed synthetic datasets, CLEVR-Building-Blocks and Sapien-Kitchen, and one real-world dataset called Tower-Creation. Extensive experiments demonstrate that MAC can correctly condition on given instructions and generate corresponding future frames without need of bounding boxes. We further show that the trained model can make out-of-distribution generalization, be quickly adapted to new object categories and exploit its learnt features for object detection, showing the progression towards higher-level cognitive abilities. More visualizations can be found at https://iclr-mac.github.io/MAC/.
+# 1 INTRODUCTION
+Recently, video prediction has drawn a lot of attention due to its ability to capture meaningful representations through self-supervision (Wang et al. (2018b); Yu et al. (2019)). Although modern video prediction methods have made significant progress in improving predictive accuracy, most of their applications are limited in the scenarios of passive forecasting (Villegas et al. (2017); Wang et al. (2018a); Byeon et al. (2018); Jin et al. (2020)), meaning models can only passively observe a short period of dynamics and accordingly make a short-term extrapolation. Such settings neglect the fact that the observer can also become an active participant in the environment.
+To model the movements of active manipulators, several low-level action-conditional video prediction models have been proposed in the community (Oh et al. (2015); Mathieu et al. (2015); Babaeizadeh et al. (2017); Ebert et al. (2017)). In this work, we go one step further by introducing the task of semantic action-conditional video prediction which emphasizes the modeling of interactions between agents and environment. Instead of using low-level single-entity actions such as action vectors of robot arms as done in prior works (Finn et al. (2016); Kurutach et al. (2018)), our new task provides semantic descriptions of interactive actions, e.g. "Open the door", and asks the model to imagine "What if I open the door" in the form of future frames. This task requires the model to recognize the object identity, assign correct affordances to objects and envision the long-term expectation by planning a reasonable trajectory toward the goal, which resembles how humans might imagine conditional futures. The ability to predict correct and semantically consistent future perceptual information is indicative of conceptual grounding of actions, in a manner similar to object grounding in image-based detection and generation tasks.
+The challenge of action-conditional video prediction primarily lies in how to correctly inform the model of more abstract semantic action information. Existing low-level counterparts usually achieve this by employing a naive concatenation (Finn et al. (2016); Babaeizadeh et al. (2017)) with action vector of each timestep. While this implementation might enable model to move the desired objects, it fails to produce consistent long-term predictions toward target locations in the multi-entity settings because it was originally designed to only encode the motion information of a single entity. If we take "put A on $B "$ as an example, it turns out to be difficult to make the model learn what and where $B$ is, because the main self-supervisory signals in the framework of video prediction are pixel changes and $B$ is not moving in this case. In order to distinguish and locate instances in the scene, other related works heavily rely on pre-trained object detectors or ground-truth bounding boxes (Bar et al. (2020); Ji et al. (2020); Huang et al. (2018); Wu et al. (2020)). However, we argue that utilizing a pre-trained detector actually simplifies the task since such a detector already solves the major difficulty by mapping high-dimension inputs to low-dimension groundings. Furthermore, bounding boxes cannot effectively describe complex visual changes including rotations and occlusions. Thus, a more flexible way of representing objects and actions is required.
+![](images/18a22a9ed4bce1235e535341f86718ebefdcca7ae56cf92f60f98c955783bfeb.jpg)
+Figure 1: Concept Grounding in Semantic Video Prediction. After observing the scene, an agent predicts future frames conditioned on a series of semantic actions describing agent-object interactions. Neither bounding boxes nor key points are provided. Conditioning on different action labels leads to Counterfactual generations.
+We present a new video prediction model, MAC, short for Modular Action Concept Network. Inspired by the idea of Mixture of Experts, MAC embodies each semantic label by a structured combination of various concept slots, each of which encodes the spatial representation of a specific concept. Such design allows MAC to reuse and integrate the knowledge learnt from different scenarios so that it can perceive the locations of motionless objects and extrapolate to unseen cases, showing the progression towards higher-level cognitive abilities. The contributions of this work are summarized as follows:
+1. We introduce a new task, semantic action-conditional video prediction as illustrated in Fig 1, which can be viewed as an inverse problem of action recognition.
+2. We create two new synthetic video datasets, CLEVR-Building-blocks and Sapien-Kitchen, and label one real-world dataset called Tower-Creation for evaluation.
+3. We propose a novel video prediction model, Modular Action Concept Network, in which routing of visual concept slots is directly controlled by action labels. We show that MAC can successfully depict the long-term counterfactual evolution without need of bounding boxes.
+4. We demonstrate that the trained MAC can make out-of-distribution generalization, be adapted for new object categories with a small number of samples and exploit its learnt features for detection.
+# 2 APPROACH
+We begin with defining the task of semantic action-conditional video prediction. Given an initial frame $x _ { 0 }$ and a sequence of action labels $a _ { 1 : T }$ , the model is required to predict the corresponding future frames $x _ { 1 : T }$ . Each action label is a pre-defined semantic description of a spatiotemporal movement that involves multiple objects in a scene and spans over multiple frames such as "take the yellow cup on the table" from $t = 0$ to $t = 1 0$ . So technically, one can regard this task as an inverse problem of action recognition. It should also be pointed out that our semantic task is different from common dense video prediction and generation tasks in the sense that it focuses on predicting time-agnostic events. Hence, we design the corresponding datasets as videos capturing sufficient key frames of entire actions. In future practices, we can further apply video interpolation methods in CV or motion planner algorithms in RL to make up the intermediate process if needed.
+![](images/dfc5e76ab0af39d06f4ca0181dfcd70b8e217e161695cec07bf20e324f1972f5.jpg)
+Figure 2: The pipeline of MAC in which the computation of concept slot module is elaborated (Better viewed in color). Feature maps extracted by encoder are mapped into the concept slot tensors. Concept slot module receives an action label that controls the collection of concept slot tensors and outputs representations encapsulating this action. A recurrent predictor updates representations before sending them to decoder to predict the next frame.
+# 2.1 MOTIVATION
+The design of our new task is necessary for studying compositional generalization as it detaches the definition of object from its specific location. However, it also requires a successful model to figure out where the desired object is through leveraging abstract labels. Our main idea is that we create a large number of small specialized learners called concept slots for each word in the dictionary of action labels to capture their corresponding spatial representations from observations. During training, action labels will be translated as constituency trees to control the activations of all related concept slots and to assemble the representations of given actions for next-frame prediction. As a result, this language-guided gating mechanism embeds the syntactic structures into the learning system and enables the proposed model to dynamically recombine its learnt concepts so that it can understand the combinatorial complexity of the world. In this paper, we demonstrate that our method possesses many key characteristics of system-2 learning (Goyal et al. (2019); Goyal & Bengio (2020)), including concept grounding, sample efficiency, counterfactual generations, out-of-distribution generalization and fast transfer.
+# 2.2 MODULAR ACTION CONCEPT NETWORK
+The MAC model is composed of 4 modules including encoder $\mathcal { E }$ , decoder $\mathcal { D }$ , concept slot module $\mathcal { C }$ and recurrent predictor $\mathcal { P }$ . The goal of our model is to learn the following mapping:
+$$
+\hat { x } _ { t } = \mathcal { D } ( \mathcal { P } ( \mathcal { C } ( \mathcal { E } _ { t - 1 } ) | a _ { t } ) | h _ { t - 1 } ) )
+$$
+where $x _ { t } , a _ { t }$ and $h _ { t }$ are video frame, action labels and hidden states at time $t$ . The overall architecture of our method is illustrated in Fig 2. In the case of stochastic video generation, another two modules, prior $p ( z )$ and posterior $q ( z )$ , will be added to help estimate the latent distribution of trajectories.
+Encoder and Decoder: At each timestep $t - 1$ , the encoder $\mathcal { E }$ receives visual input $x _ { t - 1 }$ and extracts a set of multi-scale feature maps. In the deterministic setting, we employ a convolutional neural network with an architecture similar to VGG16 (Simonyan $\&$ Zisserman (2014)). The matching decoder $\mathcal { D }$ is a mirrored version of the encoder with down-sampling operations replaced with spatial up-sampling and additional sigmoid output layer. It aggregates the updated latent representations produced by predictor and multi-scale feature maps from encoder to predict the next frame $\hat { x } _ { t }$ .
+In the stochastic setting, we use invertible autoencoder introduced in CrevNet (Yu et al. (2019)) instead as we find this information-preserving architecture can better preserve the attributes of randomly moving objects. The corresponding decoder is the backward pass, i.e. inverse computation, of the same network of the encoder. Readers can find more details about invertible autoencoder and coupling layer in Appendix B.
+Concept Slot Module: The concept slot module $\mathcal { C }$ is the core module of MAC. It resembles the mixture of experts as each slot focuses on only one concept in the space of action labels and will be activated and assembled to represent the given actions through the language-guided gating functions.
+Each action label will first be decomposed into several constituents of sentence. A constituent is a verb or object phrase, like “pick” or "large red bowl". Since we are mostly dealing with manipulation videos, actions are usually divided into 3 constituents, verb, object1, object2. Each constituent will have its own dictionary recording all pre-defined words or concepts and gating functions can be derived based on these dictionaries to establish bottom-up connections from concept slots. The computation of concept slot module is given as follows:
+$$
+\mathbf { w } ^ { i } = \Psi ^ { i } ( \mathbf { f } ) , \qquad \mathbf { c } ^ { j } = \Phi ^ { j } ( \mathrm { C o n c a t } ( \{ \mathbf { w } ^ { i } | \forall i , \delta ^ { j } ( i ) = 1 \} ) )
+$$
+where w and c are concept and constituent representations and $\delta ^ { j }$ is the indicator function for gating function of $j _ { \mathrm { t h } }$ constituent. More specifically, after the feature maps f are extracted from the input image, they are fed into $\kappa$ convolutional units $\Psi ^ { i }$ , i.e. the concept slot layer, to create $\kappa$ concept slot tensors of dimension $N _ { d }$ . Here, $\kappa$ is the total number of possible concepts we pre-defined in the dictionary of action labels. Since verbs can be interpreted as spatiotemporal changes of relationships between objects, not only slots for objects but also slots for verbs, like ’take’ or ’put on’, are computed from the extracted feature maps.
+Next, a gating function will collect all involved concept slot tensors and create an ensemble as input for each constituent. This assembly process simulates the formation of simplified constituency parse trees. Constituent slot layer $\Phi ^ { j }$ can either be resolution-preserving or upsampling operators as spatial information is important for our new task. Finally, outputs of all constituent slots are concatenated pixel-wisely to obtain the representation of actions before sending them to predictor. It is worth noticing that MAC is allowed to have multiple concurrent actions in a scene at inference time. In this case, we copy additional groups of trained constituent slots to represent other actions.
+Learned Prior: We leverage a technique called learned prior from SVG (Denton & Fergus (2018)) to model the stochastic movements in videos. In particular, we build two additional recurrent inference networks, prior and posterior respectively, to capture the randomness of motions. During training, the posterior inference network $q ( z )$ can access to the representations of target frames to estimate a true distribution of trajectory that we expect its prior counterpart $p ( z )$ to mimic at test time. Codes of motions $z _ { t }$ estimated by posterior during training (or by prior during testing) will then be concatenated with latent representations before sent to predictor.
+Predictor: The recurrent predictor $\mathcal { P }$ , implemented as a stack of residual ConvLSTM layers (Shi et al. (2015)), calculates the spatiotemporal evolution for each action label respectively. The memory mechanism of ConvLSTM is essential for MAC to remember its previous actions and to recover the occluded objects. To prevent interference between concurrent actions, hidden states are not shared between actions. The outputs of predictor for all action labels are added point-wisely.
+Training: In the deterministic setting, we train our model by minimizing the mean squared error the between the target frames and the predictions. In the stochastic setting, we optimize the following variational lower bound (ELBO) using re-parameterization trick (Kingma & Welling (2013)):
+$$
+\mathcal { L } _ { \theta , \phi , \psi } ( x _ { 1 : T } ) = \sum _ { t = 1 } ^ { T } [ \mathbb { E } _ { q _ { \phi } ( z _ { 1 : t } | x _ { 1 : t } ) } \log p _ { \theta } ( x _ { t } | z _ { 1 : t } , x _ { 1 : t - 1 } ) - \beta D _ { K L } ( q _ { \phi } ( z _ { t } | x _ { 1 : t } ) | | p _ { \psi } ( z _ { t } | x _ { 1 : t - 1 } ) ]
+$$
+where $p _ { \theta }$ is the future frame generator, $z _ { t }$ represents the latent codes of motion, $p _ { \psi } \big ( z _ { t } | x _ { 1 : t - 1 } \big )$ is the prior distribution, $q _ { \phi } \big ( { z } _ { t } | { x } _ { 1 : t } \big )$ is the posterior distribution and $D _ { K L }$ denotes the Kullback–Leibler (KL) divergence which forces the posterior to approximate the prior distribution. Since $p _ { \theta }$ is modeled by conditional Gaussian, the likelihood term reduces to MSE measure between the ground truth frames and the predictions. The full derivation of ELBO is provided in the Appendix A.
+At the inference phase, the model will use its previous predictions as visual inputs instead except for the first pass. Hence, a training strategy called scheduled sampling (Bengio et al. (2015)) is adopted to alleviate the discrepancy between training and inference.
+# 3 DATASETS
+In this study, we create two new synthetic datesets, CLEVR-Building-blocks and Sapien-Kitchen, and label one real-world dataset called Tower-Creation from Roboturk (Mandlekar et al. (2018))
+for evaluation. This is because most existing video datasets either don’t come with semantic action labels (Babaeizadeh et al. (2017)) or fail to provide necessary visual information in their first frames due to egomotions and occlusions (Hundt et al. (2018)). Although there are several candidate datasets like Penn Action (Zhang et al. (2013)), BAIR (Finn et al. (2016)) and KTH (Schuldt et al. (2004)) for multi-modal learning, they all adopt the same single-entity setting which actually indicates they can be solved by a much simpler model. To tackle the above issues, we design each video in our datatsets as a depiction of certain atomic action performed by an agent with objects which are observable in the starting frame. Furthermore, we add functions to generate bounding boxes of all objects for both synthetic datasets in order to train AG2Vid. It is worth noting that all three of these domains exhibit a key property named combinatorial explosion, resulting in factorial complexity growth in both spatial and temporal dimensions even with a small object set. For instance, a sequence with 6 (out of 32) objects and 6 actions can have 333,396,000 possibilities without considering any continuous factor. Hence, our model only sees a small fraction of these potential scenarios during training.
+# 3.1 CLEVR-BUILDING-BLOCKS DATASET
+CLEVR-Building-blocks dataset is built upon CLEVR environment (Johnson et al. (2017)). For each video, the data generator initializes the scene with 4 - 6 randomly positioned and visually different objects. There are totally 32 combinations of shapes, colors and materials of objects and at most one instance of each combination is allowed to appear in a video sequence. The agent can perform one of the following 8 actions on objects ${ \mathcal { O } } _ { A }$ and $\mathcal { O } _ { B }$ : Pick ${ \mathcal { O } } _ { A }$ , Pick and Rotate ${ \mathcal { O } } _ { A }$ transversely / longitudinally, Put ${ \mathcal { O } } _ { A }$ on $\mathcal { O } _ { B }$ , Put ${ \mathcal { O } } _ { A }$ on the left / right side of $\mathcal { O } _ { B }$ , Put ${ \mathcal { O } } _ { A }$ in the front of / behind $\mathcal { O } _ { B }$ . Each training sample contains a video of three consecutive Pick- and $P u t -$ action pairs and a sequence of semantic action labels of every frame.
+# 3.2 SAPIEN-KITCHEN DATASET
+Sapien-Kitchen Dataset describes a more complicated environment in the sense that: (a). It contains deformable actions like "open" and "close"; (b). The structures of different objects in the same category are highly diverse; (c). Objects can be initialized with randomly assigned relative positions like "along the wall" and "on the dishwasher". We collect totally 21 types of small movable objects in 3 categories, bottle, kettle and kitchen pot, and 19 types of large openable appliances in another 3 categories, oven, refrigerator and dishwasher, from Sapien engine (Xiang et al. (2020)). The agent can perform one of the following 6 atomic actions on small object $\mathcal { O } _ { s }$ and large appliance $\mathcal { O } _ { l }$ : Take $\mathcal { O } _ { s }$ on $\mathcal { O } _ { l }$ , Take $\mathcal { O } _ { s }$ in $\mathcal { O } _ { l }$ , Put $\mathcal { O } _ { s }$ on $\mathcal { O } _ { l }$ , Put $\mathcal { O } _ { s }$ in $\mathcal { O } _ { l }$ , Open $\mathcal { O } _ { l }$ and Close $\mathcal { O } _ { l }$ . Composite action sequences are defined as follows: "Take_on–Put_on", "Take_on–Open–Put_in–Close", "Open–Take_in–Close".
+# 3.3 TOWER-CREATION DATASET
+Each video in Tower-Creation Dataset depicts a robotic arm building a tower with flatware present on the table. We have labeled 524 videos in total since semantic descriptions are not provided and prodce 1867 samples consists of two actions: Pick ${ \mathcal { O } } _ { A }$ and Put ${ \mathcal { O } } _ { A }$ on $\mathcal { O } _ { B }$ . We use 1536 video clips for training and 331 for evaluation. It should be pointed out that the size of Tower-Creation dataset is small compared with commonly used datasets such as BAIR (Finn et al. (2016)) which has $5 9 \mathrm { k }$ videos in total. Thus, our experiments can also tell whether evaluated methods are data efficient.
+# 4 EXPERIMENTAL EVALUATION
+# 4.1 ACTION-CONDITONAL VIDEO PREDICTION
+Baselines and setup: We evaluate the proposed model on CLEVR-Building-blocks and SapienKitchen Datasets. AG2Vid (Bar et al. (2020)) is re-implemented as the baseline model because it is the most related work. Unlike our method which only needs visual input and action sequence, AG2Vid also requires bounding boxes of all objects and progress meters of actions, i.e. clock edge, for training and testing. Furthermore, we conduct an ablation study by replacing concept slot module with the concatenation of features and tiled action vector, which is commonly used in low-level action-conditional video prediction (Finn et al. (2016)), to show the effectiveness of our module.
+Metrics: To estimate the fidelity of action-conditional video prediction, MSE, SSIM (Wang et al. (2004)), PSNR and LPIPS (Zhang et al. (2018)) are calculated between the predictions and groundtruths. However, these metrics may not effectively tell if actions are successfully completed due to the small sizes of the moving objects. Hence, we also perform a human study to assess the accuracy of performing the correct action in generated videos for each model. The human judges annotate whether the model can identify the desired objects, perform actions specified by action labels and maintain the consistent visual appearances of all objects in its generations and only videos meeting all three criterions are scored as correct.
+![](images/3038eadaf59d6f016ac8de99ed21432c8034e5f2bfb1414e7818098f737fcf0c.jpg)
+Figure 3: The qualitative comparison on CLEVR-Building-blocks and Sapien-Kitchen. The first row of each figure is the groundtruth sequence. The red, blue and green boxes highlight the quality of predictions by each method. In contrast to the success of MAC, concatenation-based method fails to find the correct destinations or to preserve attributes of moving objects. Also, bounding boxes used in AG2Vid cannot portray visual changes like rotations correctly.
+<table><tr><td rowspan="2">Model</td><td colspan="4">CLEVR-Building-blocks</td><td colspan="4">Sapien-Kitchen</td></tr><tr><td>SSIM↑</td><td>MSE↓</td><td>LPIPS↓</td><td>Accuracy↑</td><td>SSIM↑</td><td>MSE↓</td><td>LPIPS↓</td><td>Accuracy↑</td></tr><tr><td>Copy-First-Frame</td><td>0.962</td><td>251.38</td><td>0.1320</td><td>-</td><td>0.951</td><td>152.87</td><td>0.0393</td><td>1</td></tr><tr><td>Concatenation Baseline</td><td>0.961</td><td>226.53</td><td>0.1301</td><td>50.8%</td><td>0.962</td><td>23.13</td><td>0.0232</td><td>52.4%</td></tr><tr><td>AG2Vid</td><td>0.956</td><td>58.67</td><td>0.0399</td><td>78.8%</td><td>0.947</td><td>270.87</td><td>0.0684</td><td>5.2%</td></tr><tr><td>MAC</td><td>0.983</td><td>43.52</td><td>0.0303</td><td>95.2%</td><td>0.971</td><td>11.16</td><td>0.0178</td><td>86.4%</td></tr></table>
+Table 1: Quantitative evaluation on CLEVR-Building-blocks and Sapien-Kitchen. All metrics are averaged frame-wisely except for accuracy.
+Results: The quantitative comparisons of all methods are summarized in Table 1. The MAC achieves the best scores on all metrics without access to additional information like bounding boxes, showing the superior performance of our concept slot module. The qualitative analysis in Fig 3 further reveals the drawbacks of other baselines. For CLEVR-Building-blocks, the concatenation-based variant fails to recognize the right objects due to its limited inductive bias. Although AG2Vid has no difficulty in identifying the desired objects, assumptions made by flow warping are too strong to handle rotation and occlusion. Consequently, the adversarial loss enforces AG2Vid to fix these errors by converting them to wrong poses or colors. These limitations of AG2Vid will be further amplified in a more complicated environment, i.e. Sapien-Kitchen. The same architecture used for CLEVR can only learn to remove the moving objects from their starting positions in Sapien-Kitchen because rotation and occlusion occur more often. The concatenation baseline performs better by showing correct generation of open and close actions on large appliance. Yet, it still fails to produce long-term consistent predictions as the visual appearances of moving objects are altered. On the contrary, MAC can authentically depict the correct actions specified by action labels on both datasets.
+![](images/107061b0061d8ec09a04d6e8033e1a292090fa7f6e39534a41674c392584f842.jpg)
+Figure 4: Counterfactual video generation: Conditioning on the same initial frame and different action labels, MAC can produce high-quality imaginations of counterfactual futures. Various visual outcomes present in the final frames are highlighted with red boxes and enlarged in the final column. Top: Generative results on CLEVR-Building-blocks. 34 frames are generated. Bottom: Generative results on Sapien-Kitchen dataset. 35 frames are generated.
+![](images/ab95b63a4716e350a9df25cea955435fa49fa1d2f688651f05b69f93a086d55a.jpg)
+Figure 5: Left: Visual comparison between sMAC and SVG-LP on Tower-Creation. The supposed completions of Pick and Put in the final frames are highlighted by red and yellow boxes while incorrect completions in SVG-LP generations are labelled by grey boxes. The last two rows are counterfactual generations in which models are given different action labels. Right: Quantitative comparison per-frame. Higher SSIM and PSNR indicate better performance.
+# 4.2 COUNTERFACTUAL GENERATION
+Counterfactual generation: The most intriguing application of MAC is counterfactual generation. More specifically, counterfactual generation means that our model will observe the same starting frame but receive different valid action labels to produce the corresponding future frames.
+![](images/0466372b18118f224d4c7199030b2f35ad76a99ed819f8b9789e8c8b8973b942.jpg)
+Figure 6: Compositional generalization and feature reuse.Top: Unobserved scenarios. All red cubes are removed from the tranining data, but the trained model can still manipulate red cube at test time. Middle: Concurrent actions. Inputting two action sequences at the same time. Both actions are depicted correctly. Bottom Left: New-object adaptation. Even with a few training samples, MAC can be fast adapted for generation of new objects. Red arrows point to new objects present in images. Bottom Right: Object detection.
+Results: The visual results of counterfactual generations on each dataset are displayed in Fig 4. As we can see, our model successfully identifies the desired objects, plans correct trajectories toward the target places and generates high-quality imaginations of counterfactual futures. It is also worth noticing that all displayed generations are long-term generations , i.e. more than 30 frames are predicted for each sequence. Our recurrent predictor plays an very important role in sustaining the spatiotemporal consistency and in reconstructing the fully-occluded objects.
+# 4.3 STOCHASTIC VIDEO GENERATION
+Baselines and setup: We continue to evaluate the stochastic version of MAC (sMAC) on TowerCreation dataset. SVG-LP was extended to action-conditional version in (Villegas et al. (2019)) so that we can adopt it as the baseline model to demonstrate the effectiveness of concept slot module.
+Results: The qualitative and quantitative comparison between sMAC and action-conditional SVG-LP is provided in Fig 5. Although SVG-LP can partially understand the given action labels, it often fails to locate and manipulate the desired objects. Consequently, it will generate the moving object out of nowhere and often place it on a wrong target object. In contrast, sMAC can successfully simulate the trajectory of robotic arms and correctly animate the "Pick" and "Put" actions thanks to the concept slot module. Row 3 and 5 in Fig 5 show that sMAC is also capable of producing diverse future frames and predicting counterfactual results following different action instructions.
+# 4.4 COMPOSITIONAL GENERALIZATION
+We further explore other interesting features of our MAC. We first demonstrate that MAC is capable of making out-of-distribution generalization by designing two experiments. We evaluate how quickly our model can be adapted to new objects. It turns out for each new object, the trained MAC only requires a few training video examples to generate decent results. Finally, to verify that our model encodes the spatial information, we add SSD (Liu et al. (2016)) head after the frozen encoder and concept slot layer to conduct object detection.
+Unobserved scenarios: We design an interesting experiment where only a subset of CLEVRBuilding-blocks data are used for training and check what will happen if we input the unobserved action labels to the trained model. More precisely, we exclude all videos manipulating red cubes in the training sets and send the instructions involving red cubes at test time. The visualization of this experiment can be found in Fig 6. As we can see, MAC can still identify and manipulate red cubes correctly, showing its ability to recombine the learnt concept to comprehend new objects.
+Concurrent actions: Concurrent actions means multiple action inputs at the same time. It can be considered as out-of-distribution generalization because our model only observes single-action videos during training. Generating concurrent-action videos needs to employ copied constituent slots and parallel hidden states. As illustrated in Fig 6, MAC can linearly integrate the action information in the latent space and correctly portray 2 concurrent actions in the same scene.
+Adaptation: We add a new openable category "safe" and a new movable category "dispenser" into Sapien-Kitchen and generate 100 video sequences for each new object showing its interaction with other objects. Approximately, there are about 5 new sequences created for each new action pair between 2 objects. Blank concept slots for new categories are attached to trained MAC and we finetune it on this small new training set. Visualization in Fig 6 shows that even with a few training samples, MAC is accurately adapted for video generation of new objects. This is because, with the help of concept slots, MAC can disentangle actions into relatively independent grounded concepts. When it learns new concepts, MAC reuses and integrates prior knowledge learnt from different cases.
+Object detection: The quantitative results of object detection and more visualizations can be found in Appendix D. We observe that the features learnt by MAC can be easily transferred for detection as our video prediction task is highly location-dependent. This result indicates that utilizing bounding boxes might be a little redundant for some video tasks because videos already provide rich motion information that can be used for salient object detection.
+# 5 RELATED WORK
+Video prediction: ConvLSTM (Shi et al. (2015)) was the first deep learning model that employed a hybrid of convolutional and recurrent units for passive video prediction. This architectural design was soon followed by studies looking at a similar problem (Kalchbrenner et al. (2017); Mathieu et al. (2015); Wang et al. (2017); Yu et al. (2019); Wang et al. (2018b)). However, the capability of passive video prediction framework is very limited as models usually don’t have sufficient information to predict the long-term future due to partial observation, egomotion and randomness. More importantly, this setting prevents models from interacting with environment. On the other hand, the low-level action-conditional video prediction task provides an action vector at each timestep as additional input to guide the prediction (Oh et al. (2015); Chiappa et al. (2017); Babaeizadeh et al. (2017); Wu et al. (2021)). CDNA (Finn et al. (2016)) is a representative of such models. In CDNA, the states and action vectors of the robotic manipulator are first spatially tiled and integrated into the model through concatenation. SVG (Denton & Fergus (2018)) was initially proposed for stochastic video generation but later was extended to action-conditional version in (Villegas et al. (2019)). It is worth noticing that SVG also used concatenation to incorporate action information. Such implementations are prevalent in low-level action-conditional video prediction because the action vector only encodes the spatial information of a single entity, usually a robotic manipulator (Finn et al. (2016)) or a human hand. A common failure case for such models is the presence of multiple affordable entities (Kim et al. (2019)), a scenario that our task definition and datasets focus on.
+Modularity: Mixture of Experts refers to a classical machine learning technique where various learners are employed, each of which specializes in one particular function, and their output are aggregated through a gating function. This modular design makes each submodule relatively independent and thus leads to better generalization and robustness to compositional changes, which has been studied in several works (Goyal et al. (2019); Afshar et al. (2021); Sabour et al. (2017); Henaff et al. (2016)). In this work, we hypothesis that the underlying syntactic structures of semantic labels can tell how to aggregate the representations of individual concept learners. By translating labels into constituency trees, action graphs are embedded into the learning system to get the entire perspective of ongoing activities while each concept learner can focus on its specific subtask.
+# 6 CONCLUSION
+In this work, we propose the new task of semantic action-conditional video prediction and introduce 3 new datasets that are meant to bridge the gap towards a robust solution to this task in complex interactive scenarios. MAC, a novel video prediction model, was also designed by utilizing the idea of Mixture of Experts to ground action concept for video generation. Our proposed model can generate alternative futures without requiring additional auxiliary data such as bounding boxes, and is shown to be both quickly extendible and adaptable to novel scenarios and entities. It is our hope that our contributions will advance progress and understanding within this new task space, and that a model robust enough for real-world applications (i.e. in robotic systems) in perception and control will be eventually proposed as a descendant of this work.
+# REFERENCES
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+Yunbo Wang, Zhifeng Gao, Mingsheng Long, Jianmin Wang, and Philip S Yu. Predrnn $^ { + + }$ : Towards a resolution of the deep-in-time dilemma in spatiotemporal predictive learning. arXiv preprint arXiv:1804.06300, 2018a.
+Yunbo Wang, Lu Jiang, Ming-Hsuan Yang, Li-Jia Li, Mingsheng Long, and Li Fei-Fei. Eidetic 3d lstm: A model for video prediction and beyond. In International Conference on Learning Representations, 2018b.
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+Bohan Wu, Suraj Nair, Roberto Martin-Martin, Li Fei-Fei, and Chelsea Finn. Greedy hierarchical variational autoencoders for large-scale video prediction. arXiv preprint arXiv:2103.04174, 2021.
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+# A VARIATIONAL LOWER BOUND DERIVATION
+The original variational lower bound was derived in (Kingma & Welling (2013)).
+$$
+\begin{array} { r l } & { \quad \log \gamma _ { \mathbb { P } } ( \mathbf { x } ) = \log \int _ { \mathcal { R } } \log \left( \mathbf { x } \mid \mathbf { x } _ { \mathcal { P } } \rho ( \mathbf { x } ) \mid \mathbf { x } _ { \mathcal { P } } \rho ( \mathbf { x } ) \right. } \\ & { = \log \int _ { \mathcal { R } } p ( \mathbf { x } ) \mid p ( \mathbf { x } ) \mid p ( \mathbf { x } ) \mid \frac { \log ( \mathbf { x } ) \mid \exp ( \mathbf { x } ) } { \exp ( \mathbf { x } ) } } \\ & { = \log \mathbb { E } _ { \rho _ { \xi } \in \{ \mathbf { x } , \mathbf { x } \} } \frac { p ( \mathbf { x } ) \exp ( \mathbf { x } ) \mid \exp ( \mathbf { x } ) } { \exp ( \mathbf { x } ) } } \\ & { \geq \mathbb { E } _ { \mathbf { x } _ { \mathbb { P } } ( \mathbf { x } ) = \mathbf { b } } \frac { p ( \mathbf { x } ) \mid p ( \mathbf { x } ) \mid p ( \mathbf { x } ) } { \exp ( \mathbf { x } ) } } \\ & { = \mathbb { E } _ { \mathbf { x } _ { \mathbb { P } } ( \mathbf { x } ) = \mathbf { b } } \frac { \log ( \mathbf { x } ) \mid \exp ( \mathbf { x } ) } { \exp ( \mathbf { x } ) } } \\ & { = \mathbb { E } _ { \mathbf { x } _ { \mathbb { P } } ( \mathbf { x } ) = \mathbf { b } } \frac { \log ( \mathbf { x } ) \mid \exp ( \mathbf { x } ) - \mathbf { E } _ { \mathbf { x } _ { \mathbb { P } } ( \mathbf { x } ) } \mid \log \frac { \log ( \mathbf { x } \mid \mathbf { x } ) } { p ( \mathbf { x } ) } } { \exp ( \mathbf { x } ) } } \\ &  = \mathbb { E } _ { \mathbf { x } _ { \mathbb { P } } ( \mathbf { x } ) \mid \mathbf { x } _ { \mathcal { P } } \rho ( \mathbf { x } ) \mid \exp ( \mathbf { x } ) - \mathcal { P } _ { \mathbf { x } _ { \mathbb { P } } ( \mathbf { x } _ { \mathbb { P } } ( \mathbf { x } ) \mid \rho ( \mathbf { x } ) \mid \exp ( \mathbf { x } ) } { \exp ( \mathbf { x } ) \mid \exp ( \mathbf { x } \mid \mathbf { x } _ { \mathbb { P } } \rho ( \mathbf { x } ) \mid \mathbf { x } ) } } \\ &  = \sum _  \mathbf { x } \in \{ \mathbf { x } , \mathbf { x } \mid \mathbf { x } _ { \mathbb { P } } \rho ( \mathbf  x \end{array}
+$$
+The final step (Denton & Fergus (2018)) is obtained through the factorization of reconstruction and KL-Divergence term into individual time steps due to the independence across time.
+# B INVERTIBLE ARCHITECTURE AND COUPLING LAYER
+The additive coupling layer was first introduced in (Dinh et al. (2014)). Following (Yu et al. (2019)), we use it as the building block to construct the invertible autoencoder. More specifically, the reshaped input $x$ is divided into two groups, denoted as $x ^ { 1 }$ and $x ^ { 2 }$ , channel-wisely. In its forward pass, one group, e.g. $x ^ { 1 }$ , passes through several convolutional layers and updates the other group, $x ^ { 2 }$ , through addition.
+$$
+\begin{array} { r } { \hat { x } ^ { 2 } = x ^ { 2 } + \mathcal { F } _ { 1 } ( x ^ { 1 } ) } \\ { \hat { x } ^ { 1 } = x ^ { 1 } + \mathcal { F } _ { 2 } ( \hat { x } ^ { 2 } ) } \end{array}
+$$
+where $\mathcal { F }$ is a composite non-linear transformation consisting of convolutions and activations, and ${ \hat { x } } ^ { 1 }$ and ${ \hat { x } } ^ { 2 }$ are the updated $x ^ { 1 }$ and $x ^ { 2 }$ . In its backward pass, we can retrieve $x ^ { 1 }$ and $x ^ { 2 }$ from ${ \hat { x } } ^ { 2 }$ and ${ \hat { x } } ^ { 1 }$ by the following inverse computation:
+$$
+\begin{array} { r } { x ^ { 1 } = \hat { x } ^ { 1 } - \mathcal { F } _ { 2 } ( \hat { x } ^ { 2 } ) } \\ { x ^ { 2 } = \hat { x } ^ { 2 } - \mathcal { F } _ { 1 } ( x ^ { 1 } ) } \end{array}
+$$
+Pixel shuffle layer (Shi et al. (2016)), a bijective downsampling, is also employed to change the shape of feature from $( w , h , c )$ to $( w / n , h / \bar { n } , c \times n ^ { 2 } )$ to enable the invertibility of the entire network. Stacking these building blocks and downsampling in an alternating fashion between two groups, we will obtain a two-way autoencoder. The property of invertibility ensures no information loss during feature extraction, which is better at preserving the attributes of moving objects. The same network can serve as both the encoder and the decoder by using its forward and backward pass respectively.
+# C TRAINING SETUP
+In the deterministic setting, MAC adopts VGG16 (Simonyan & Zisserman (2014)) and a mirrored network as encoder and decoder and 2 layers of residual ConvLSTM as predictor. In the stochastic setting, sMAC replaces its encoder with 24-layer invertible autoencoder and use its backward pass as decoder. Additionally, it also deploys two inference networks composed of 2 layers of ConvLSTM, named prior and posterior, to model conditionally Gaussian distribution of trajectories .
+We use the Adam optimizer (Kingma & Ba (2014)) with a starting learning rate of $2 \times 1 0 ^ { - 4 }$ to optimize the MAC and sMAC. The training process is stopped after $2 0 0 , 0 0 0$ iterations with the batch size of 4. 20,000 video clips of CLEVR-Building-Blocks and 30,000 of Sapien-Kitchen are generated for model training and additionally 5,000 videos are generated for each dataset for evaluation. Considering the size of Tower-Creation dataset, various traditional data augmentation methods are used and we also implement a new trick in which the neighbouring frames of key frames are sampled from Gaussian distributions to serve as small temporal variations. This trick can significantly improve the visual quality and diversity of stochastic generation for both sMAC and SVG-LP.
+Table 2: Quantitative measures of object detection on Sapien-Kitchen in terms of average precision.
+<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Oven</td><td rowspan=1 colspan=1>Fridge</td><td rowspan=1 colspan=2>DishwashetBottle</td><td rowspan=1 colspan=1>Kettle</td><td rowspan=1 colspan=1>Kitchenpot</td><td rowspan=1 colspan=1>mAP</td></tr><tr><td rowspan=1 colspan=1>MACencoder+SSD</td><td rowspan=1 colspan=1>92.75</td><td rowspan=1 colspan=1>94.56</td><td rowspan=1 colspan=1>90.89</td><td rowspan=1 colspan=1>83.25</td><td rowspan=1 colspan=1>77.18</td><td rowspan=1 colspan=1>81.32</td><td rowspan=1 colspan=1>86.66</td></tr></table>
+![](images/d097e23d4c7e9f3da44e7b0d74273c8fdef42dca11cbfe59335262791a00725d.jpg)
+Figure 7: Visualization of 2D Object Detection on Sapien-Kitchen.
+# D OBJECT DETECTION
+The quantitative results and visualization of object detection is provided in the Table 2 and Fig 7. SSD head was optimized following its protocol while the MAC encoder was frozen to demonstrate that features learnt through self-supervision can be directly transferred for detection because our video prediction task is highly location-dependent.

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+# NASVIT: NEURAL ARCHITECTURE SEARCH FOR EF-FICIENT VISION TRANSFORMERS WITH GRADIENTCONFLICT-AWARE SUPERNET TRAINING
+Chengyue $\mathbf { G o n g } ^ { 2 * }$ , Dilin Wang1, Meng $\mathbf { L i ^ { 1 } }$ , Xinlei Chen1, Zhicheng $\mathbf { Y a n } ^ { 1 }$ ,
+Yuandong Tian1, Qiang $\mathbf { L i u } ^ { 2 }$ , Vikas Chandra1
+1 Meta Reality Labs 2 University of Texas at Austin
+{wdilin, meng.li, xinleic, zyan3, yuandong, vchandra} $@$ fb.com, {cygong, lqiang}@cs.utexas.edu
+# ABSTRACT
+Designing accurate and efficient vision transformers (ViTs) is an important but challenging task. Supernet-based one-shot neural architecture search (NAS) enables fast architecture optimization and has achieved state-of-the-art results on convolutional neural networks (CNNs). However, directly applying the supernetbased NAS to optimize ViTs leads to poor performance - even worse compared to training single ViTs. In this work, we observe that the poor performance is due to a gradient conflict issue: the gradients of different sub-networks conflict with that of the supernet more severely in ViTs than CNNs, which leads to early saturation in training and inferior convergence. To alleviate this issue, we propose a series of techniques, including a gradient projection algorithm, a switchable layer scaling design, and a simplified data augmentation and regularization training recipe. The proposed techniques significantly improve the convergence and the performance of all sub-networks. Our discovered hybrid ViT model family, dubbed NASViT, achieves top-1 accuracy from $7 8 . 2 \%$ to $8 1 . 8 \%$ on ImageNet from 200M to 800M FLOPs, and outperforms all the prior art CNNs and ViTs, including AlphaNet and LeViT. When transferred to semantic segmentation tasks, NASViTs also outperform previous backbones on both Cityscape and ADE20K datasets, achieving $7 3 . 2 \%$ and $3 7 . 9 \%$ mIoU with only 5G FLOPs, respectively. Code is available at https://github.com/facebookresearch/NASViT.
+# 1 INTRODUCTION
+Transformers have recently been applied to various vision tasks, including image classification (Liu et al., 2021; Dong et al., 2021), object detection (Carion et al., 2020; Zhu et al., 2020), semantic segmentation (Xie et al., 2021; Cheng et al., 2021), video understanding (Bertasius et al., 2021; Fan et al., 2021), etc. Vision transformers (ViTs) benefit from high model capacity, large receptive field, and grouping effect, etc (Dosovitskiy et al., 2020), and demonstrate superior performance compared to convolutional neural networks (CNNs) especially with the scaling of the model size and training data size. For example, CoAtNet (Dai et al., 2021) achieves $9 0 . 8 8 \%$ top-1 accuracy on Imagenet by scaling the model to 2586G FLOPs and pre-training the model on JFT-3B dataset (Sun et al., 2017).
+Though promising in the high computation budget regime, the performance of ViTs is still inferior to that of the CNN counterparts on small- or medium-sized architectures, especially compared to CNN architectures that are highly optimized by neural architecture search (NAS), e.g., AlphaNet (Wang et al., 2021a), FBNetV3 (Dai et al., 2020), etc. For example, the initial DeiT-Tiny (Touvron et al., 2020) only achieves $7 2 . 2 \%$ top-1 accuracy with 1.2G FLOPs. The recently proposed LeViT (Graham et al., 2021) makes significant progress to achieve $7 6 . 6 \%$ top-1 accuracy with 305M FLOPs with convolution/transformer hybrid architectures and a $3 \mathbf { x }$ longer training schedule. In contrast, AlphaNet (Wang et al., 2021a) achieves $7 7 . 8 \%$ top-1 accuracy with only 203M FLOPs. The large accuracy gap illustrated above raises a natural question: are transformer blocks that build large and dynamic receptive fields beneficial for small models?
+To answer the question above, in this work, we target at developing a family of efficient ViTs with FLOPs ranging from 200M to 800M . A natural approach is to leverage NAS, which has achieved state-of-the-art (SOTA) accuracy-efficiency trade-off for CNNs (Wang et al., 2021a; Dai et al., 2020; Cai et al., 2019). The recently proposed supernet-based NAS, e.g., BigNAS (Yu et al., 2020a) and AlphaNet (Wang et al., 2021a), builds a weight-sharing graph including all the sub-networks in the architecture search space. A sandwich sampling rule with inplace knowledge distillation (KD) (Yu et al., 2018) is leveraged to simultaneously optimize the supernet and sub-networks for each mini-batch, which stabilizes the training and improves the training convergence.
+To leverage the supernet-based NAS, we first modify the LeViT model to build the architecture search space for ViTs and then jointly optimize the model architectures and parameters following AlphaNet. However, we find that directly applying AlphaNet achieves poor performance on the ViT search space, even worse compared to training single ViTs. To understand the root cause of the poor performance, we examine the supernet training procedure and observe that the gradients of the supernet and the different sub-networks conflict with each other during the sandwich sampling, which makes the training loss saturates much more quickly for ViTs, thus leading to slow convergence.
+To alleviate the issue of conflicting gradients, we propose three different techniques to improve the supernet training. Firstly, instead of directly adding the gradients from different sub-networks together, we find it beneficial to prioritize the training of the sub-networks over the supernet, as our main purpose is to build efficient sub-networks. We achieve this with a projection gradient algorithm which removes the component of the supernet gradient that is conflict with the sub-network gradient. Secondly, to alleviate the gradient conflicts among different sub-networks, we propose to augment each transformer layer with switchable channel-wise scaling layers. The weights of different scaling layers are not shared among different transformer blocks to reduce gradient conflicts. Thirdly, we propose to use a weak data augmentation scheme and reduce the regularization in training to decrease the optimization difficulty and hence reduce gradient conflicts.
+Our proposed techniques significantly alleviate the gradient conflict issue and empirically improve the convergence of supernet training. Compared to the baseline supernet training algorithm in AlphaNet, we can improve the top-1 accuracy to $7 8 . 2 \%$ for the small model with 205M FLOPs and achieve $8 1 . 8 \%$ for the large model with 757M FLOPs. Meanwhile, the resulting model family, NASViT, outperforms all the SOTA CNN and ViT models across a wide range of computation constraints. NASViT also demonstrates good performance on downstream tasks. When transferring to semantic segmentation tasks, NASViT backbones outperform previous CNN and ViT backbones on both Cityscape and ADE20K datasets, achieving $7 3 . 2 \%$ and $3 7 . 9 \%$ mIoU with 5G FLOPs, respectively.
+Related Works Recently, researchers have used supernet-based NAS to optimize the architecture for transformers. For example, HAT (Han et al., 2021) uses supernet for hardware-aware transformer optimization. HAT mainly focuses on NLP tasks and features a design space with heterogeneous transformer layers. AutoFormer (Chen et al., 2021a) and ViTAS (Su et al., 2021) leverages supernetbased NAS to optimize the ViT architecture. By searching the width, depth, K/Q/V dimension, MLP ratio, etc, better accuracy is achieved compared to the baseline DeiT models (Chen et al., 2021a). However, these works focus on large ViT models with more than 1G FLOPs and their accuracy is still inferior to the CNN backbones with similar compute, e.g., EfficientNet (Tan & Le, 2019). We refer readers to appendix for more discussions about related works.
+# 2 NAS FOR EFFICIENT TRANSFORMERS
+Our goal is to design efficient small- and medium-sized ViTs in the FLOPs regime from 200M to 800M. We build our search space inspired by the recently proposed LeViT (Graham et al., 2021). LeViT is a family of efficient models leveraging a hybrid architecture of convolutions and transformers. In LeViT, the convolutions are introduced to handle high resolution inputs thanks to their efficiency from local computation while the transformers are leveraged for lower resolution features to extract global information. We closely follow LeViT to build our search space; see Figure 1 for an overview.
+Search Space We summarize the detailed search dimensions of our search space in Table 1. For each CNN block, we directly follow the design in AlphaNet (Wang et al., 2021a;b) and search for the optimal channel widths, block depths, expansion ratios and kernel sizes; for each transformer
+![](images/da1a4f953ba5b77a13ec2de1120b220d0b596e2e0020a1f8d55db5b3ee9b11fd.jpg)
+Figure 1: An illustration of our ViT search space. MBConv refers to inverted residual blocks (Sandler et al., 2018). All CNN and transformer blocks contain a stack of dynamic layers with searchable architecture configurations. Additionally, we also search for the input resolutions.
+<table><tr><td>Block</td><td>Width</td><td>Depth</td><td></td><td></td><td></td><td></td><td>Kernel size Expansion ratio SE Stride NumberofWindows</td></tr><tr><td>Conv</td><td>{16,24}</td><td>-</td><td>3</td><td>=</td><td>-</td><td>2</td><td></td></tr><tr><td>MBConv-1</td><td>{16,24}</td><td>{1,2}</td><td>{3,5}</td><td>1</td><td>N</td><td>1</td><td></td></tr><tr><td>MBConv-2</td><td>{24,32}</td><td>{3,4,5}</td><td>{3,5}</td><td>{4,5,6}</td><td>N</td><td>2</td><td></td></tr><tr><td>MBConv-3</td><td>{32,40}</td><td>{3,4,5,6}</td><td>{3,5}</td><td>{4,5,6}</td><td>Y</td><td>2</td><td></td></tr><tr><td>Transformer-4</td><td>{64,72}</td><td>{3,4,5,6}</td><td>-</td><td>{1,2}</td><td>-</td><td>2</td><td>1</td></tr><tr><td>Transformer-5</td><td>{112,120,128}</td><td>{3,4,5,6,7,8}</td><td>=</td><td>{1,2}</td><td></td><td>2</td><td>1</td></tr><tr><td>Transformer-6</td><td>{160,168,176,184}</td><td>{3,4,5,6,7,8}</td><td></td><td>{1,2}</td><td></td><td>1</td><td>1</td></tr><tr><td>Transformer-7</td><td>{208,216,224}</td><td>{3,4,5,6}</td><td></td><td>{1,2}</td><td></td><td>2</td><td></td></tr><tr><td>MBPool</td><td>{1792,1984}</td><td>=</td><td>1</td><td>6</td><td></td><td></td><td></td></tr><tr><td>Input resolution</td><td colspan="7">{192,224,256,288}</td></tr></table>
+Table 1: An illustration of our search space. MBConv refers to the inverted residual block (Sandler et al., 2018). MBPool denotes the efficient last stage (Howard et al., 2019). SE represents the squeeze and excite layer (Hu et al., 2018). Transformer stands for the transformer blocks (Vaswani et al., 2017). For MBConv blocks, the expansion ratio refers to the expansion ratio of the depth-wise convolution layer. For transformer layers, it refers to the MLP expansion ratio. For each transformer block, we use $3 \times 3$ depth-wise convolution with stride 2 for down-sampling and the down-sampling layer is placed as the first layer for that block.
+block, we search for the best number of windows, hidden feature dimensions (denoted as Width in Table 1)1, depths and MLP expansion ratios. Compared to CNN blocks, one special search dimension for transformer blocks is the number of windows $k$ . When the number of windows $k$ is greater than 1, we follow Swin transformer (Liu et al., 2021) and partition the input tokens into $k$ groups. We then compute the self-attention weights for each group separately to reduce computational cost. Standard global self-attention is a special case of $k = 1$ . In this work, we only search the number of windows for the first transformer block, as the input resolutions to the other transformer blocks are already small after 4 times of down-sampling. Similar to the search range of AlphaNet, the smallest sub-network in our search space has 190M FLOPs and the largest sub-network has FLOPs of 1,881M. we refer the reader to Appendix B for more description of our search space.
+Naive supernet-based NAS fails to find accurate ViTs We first closely follow the previous best practices in AlphaNet (Wang et al., 2021a) for the supernet training. We train the supernet for 360 epochs on ImageNet (Deng et al., 2009). At each training step, we adopt the sandwich sampling rule (Yu et al., 2018) and sample four sub-networks: the smallest sub-network, the supernet (a.k.a. the largest sub-network), and two random sub-networks. All small sub-networks are supervised by the supernet with $\alpha$ -divergence-based KD; see Algorithm 1 in Appendix C.1 for an overview of the supernet training procedure. Additionally, as our candidate networks contain transformer blocks, we further incorporate the best training recipe from LeViT (Graham et al., 2021) by replacing the SGD optimizer with Adam (Kingma & Ba, 2014) and leveraging an external pre-trained teacher model for the best accuracy. Specifically, we use the pre-trained teacher to supervise the supernet and still constrain all other small sub-networks to learn from the supernet. In this work, we always use an EfficientNet-B5 (Tan & Le, 2019) with $8 3 . 3 \%$ top-1 accuracy on ImageNet as the teacher to train our ViT supernet unless otherwise specified.
+We plot the training curves of the smallest sub-network and the largest sub-network in Figure 2. We find both the smallest sub-network and the largest sub-network from our search space converge poorly compared to the CNN baseline. Specifically, the validation accuracy of both the smallest and the largest sub-network is saturated at around the 250-th epoch, and the final accuracy is much worse than the CNN baselines. To understand the inferior model performance, we investigate the potential issues of our ViT supernet training from the following three directions.
+![](images/d583c7b7b70b8e1371eac7ee76bea6ab3d53afe7a4946fa797a8428054d7f638.jpg)
+Figure 2: (a-b) show the training curves of the smallest sub-network and the largest sub-network (i.e., the supernet), respectively. Note that AlphaNet is trained without external teacher models.
+Table 2: ImageNet top-1 accuracy from subnetworks trained from scratch vs. results from sub-networks sampled from the supernet.
+<table><tr><td>FLOPs (M)</td><td>190</td><td>208</td><td>309</td><td>591</td></tr><tr><td>Scratch</td><td>77.2</td><td>77.5</td><td>79.1</td><td>80.4</td></tr><tr><td>Supernet</td><td>76.4</td><td>76.6</td><td>78.5</td><td>80.6</td></tr></table>
+Table 3: ImageNet Top-1 accuracy from the smallest and the largest sub-network by using different training recipes.
+<table><tr><td></td><td>AlphaNet</td><td>DeiT</td><td>LeViT</td></tr><tr><td>Smallest</td><td>77.0</td><td>76.6</td><td>76.8</td></tr><tr><td>Largest</td><td>82.4</td><td>82.2</td><td>82.2</td></tr></table>
+Investigation 1: Is our search space designed badly? We seek to understand if the performance gap is caused by a bad search space design. To verify, we randomly pick four sub-networks from the search space with computation cost ranging from 190M to 591M FLOPs. Then, we train these networks from scratch with the same data augmentation and regularization. As we can see from Table 2, the sub-networks trained from scratch outperform the sub-networks sampled from the supernet. Note that from previous works (e.g. Yu et al., 2020a), supernet often learns more accurate sub-networks compared to the training from scratch performance, by taking advantage of inplace knowledge KD and weight-sharing. Our observations in Table 2 indicate that the poor performance does not come from the search space but from the interference with the training of the supernet.
+Investigation 2: Are the training settings suitable for ViTs? Our default training settings from AlphaNet are originally optimized for CNNs only. Compared with AlphaNet, recent ViT methods, e.g., DeiT and LeViT, suggest to use stronger data augmentation schemes (e.g., a combination of CutMix (Yun et al., 2019), Mixup (Zhang et al., 2017), randaugment (Cubuk et al., 2020), random erasing (Zhong et al., 2020), and repeated augmentation) and stronger regularization (e.g., large weight decay, large drop path probability) for training. We evaluate the effectiveness of these ViT specific training recipes and summarize our findings in Table 3. As we can see from Table 3, DeiT- or LeViT-based training recipe produces even worse accuracy compared to the results from AlphaNet-based training.
+Investigation 3: Saturated supernet training due to gradient conflicts? Compared to the standard single network training, a major difference of supernet training is that multiple networks are sampled and trained at each step. We hypothesize that the training loss from the supernet and that from the sub-networks may yield conflicting gradients due to the heterogeneous and complex structures of networks, and the conflict gradients may consequently lead to slow convergence and undesirable performance.
+To verify this hypothesis, we compute the cosine similarity between the gradients from the supernet and the averaged gradients from the sub-networks. A negative cosine similarity indicates the supernet and sub-networks produce conflict gradients and tend to update model parameters in opposite directions. To quantitatively examine the gradient conflict issue, we go through the entire ImageNet training set and calculate the percentage of negative cosine similarity between the gradients of supernet and sub-networks among all training images at a per layer granularity. The gradients are computed under the same data augmentation and regularization as the supernet training stage. For AlphaNet, we train the model using its official code 2. As shown in Table 4, our ViT supernet suffers from more severe gradient conflicts compared to the CNN baseline. According to existing works in multi-task learning, large gradient conflict ratios may result in significant accuracy drop even for binary classification problems (see Figure 3 in Du et al. (2018) and Figure 4(b) in Yu et al. (2020b)). We hypothesize that the inferior performance of our ViT supernet is mainly caused by the large percentage of disagreements between the supernet gradients and the subnetworks gradients.
+Table 4: An estimation of negative cosine similarity ratio (gradient conflict ratio) between the supernet gradient and the averaged gradient of the sub-networks.
+<table><tr><td>Epoch</td><td>1st</td><td>90th</td><td>180th</td><td>270th</td><td>360th</td></tr><tr><td>AlphaNet</td><td>27%</td><td>20%</td><td>21%</td><td>24%</td><td>28%</td></tr><tr><td>ViT</td><td>36%</td><td>27%</td><td>27%</td><td>32%</td><td>34%</td></tr></table>
+# 3 GRADIENT CONFLICT AWARE SUPERNET TRAINING
+We propose to improve the ViT supernet training by addressing the gradient conflict issue between the supernet and the sub-networks from three aspects: 1) manually resolving the gradient conflict by projecting the supernet gradients to the normal vector of the sub-networks gradients; 2) introducing switchable scaling layers to the search space to give more optimization freedom for sub-networks; 3) reducing data augmentation and regularization to provide easier training signals.
+Gradient projection to prioritize sub-networks update Our first idea is to focus on training the sub-networks whenever gradients from the supernet and the gradients from the sub-networks conflicted with each other. As we are interested in the sub-networks from the 200M to 800M FLOPs range, we propose to prioritize the optimization of the sub-networks over the supernet when gradient conflicts are observed. Let $\nabla \ell _ { s u p }$ and $\nabla \ell _ { s u b }$ denote the gradients of the supernet and the sub-networks, respectively. To prioritize sub-networks training, we always project $\nabla \ell _ { s u p }$ to the normal vector of $\nabla \ell _ { s u b }$ to avoid gradient conflicts when the cosine similarity between $\nabla \ell _ { s u p }$ and $\nabla \ell _ { s u b }$ is negative. The overall accumulated gradient at each training iteration with projection can be written as follows,
+![](images/6e1beb18a66bc7af0d7217a75f272d4ba9d9948a9673c971ed1bbba09287b7f6.jpg)
+Figure 3: A basic transformer layer with scaling. Activated components are the neurons selected in the forward path for one sub-network. ‘c1’ and $\cdot _ { \mathrm { c 2 } } ,$ represent the number of channels activated in a selfattention layer and MLP, respectively.
+$$
+\begin{array} { r l } & { g = \nabla \ell _ { s u b } + \mathrm { p r o j } ( \nabla \ell _ { s u p } ) \mathrm { w i t h } } \\ & { \mathrm { p r o j } ( \nabla \ell _ { s u p } ) = \left\{ \begin{array} { l l } { \nabla \ell _ { s u p } } & { \mathrm { i f ~ } \mathrm { c o s } ( \nabla \ell _ { s u p } , \nabla \ell _ { s u b } ) \geq 0 , } \\ { \nabla \ell _ { s u p } - \frac { \nabla \ell _ { s u p } ^ { \top } \nabla \ell _ { s u b } } { \| \nabla \ell _ { s u b } \| ^ { 2 } } \nabla \ell _ { s u b } } & { \mathrm { o t h e r w i s e } . } \end{array} \right. } \end{array}
+$$
+Note that $\cos ( \nabla \ell _ { s u b } , \ \mathrm { p r o j } ( \nabla \ell _ { s u p } ) ) = 0$ if $\cos ( \nabla \ell _ { s u p } , \nabla \ell _ { s u b } ) < 0$ , which ensures the gradient cosine similarity is non-negative. In sandwich sampling, since several sub-networks are sampled in each iteration, $\ell _ { s u b }$ is computed as the summation of the gradients from all sub-networks. Note similar ideas of gradient projection have also been explored in multi-tasks learning, see e.g., Yu et al. (2020b); Du et al. (2018); Real et al. (2019); Dery et al. (2021).
+While the gradient projection in Eqn. (1) eliminates the gradient conflicts, it may lead to slow convergence as the resulting gradients are biased. Hence, we also propose the follow two techniques to reduce the gradient conflicts from a search space design and training strategy refinement perspective.
+Switchable scaling layer Motivated by Slimmable NN (Yu et al., 2018), we introduce additional switchable scaling layers to allow sub-networks with different layer widths and depths to re-scale their features in a privatized way. Specifically, for each transformer layer, a switchable scaling layer is introduced at the output of the self-attention (SA) and the MLP, respectively, as shown in Figure 3. Assume $\pmb { x } _ { [ c , d ] } \in \mathbb { R } ^ { c }$ is a input feature of a scaling layer, with $c$ the feature dimension (i.e. the number of selected channels in the forward path) and $d$ the index of this layer in a transformer search block. The scaling layer transforms ${ \pmb x } _ { [ c , d ] }$ as ${ \pmb w } _ { [ c , d ] } \odot { \pmb x } _ { [ c , d ] }$ . Here $\pmb { w } _ { [ c , d ] } \in \mathbb { R } ^ { c }$ are learnable parameters and $\odot$ denotes element-wise multiplication. For each transformer block (see Table 1), each different configuration of $[ c , d ]$ will specify a set of independent switchable scaling layers. Following CaiT (Touvron et al., 2021), we initialize all scaling factors $\pmb { w }$ to a small value (e.g. $1 { \dot { 0 } } ^ { - 4 }$ ) for fast convergence and stable training. Intuitively, the switchable scaling layers effectively increase the model capacity of sub-networks and give the sub-networks more optimization flexibility.
+Reduced data augmentation and regularization Furthermore, we observe that the supernet and the sub-networks are more likely to conflict with each other in the presence of stronger data augmentations and stronger regularization, e.g., large weight decay, large DropConnect (Wan et al., 2013). Hence, we simplify the AlphaNet training recipe and use a weaker data augmentation scheme - RandAugment (Cubuk et al., 2019) with both the number of augmentation transformations and the magnitude set to 1, and remove the regularization, e.g. DropConnect (Wan et al., 2013), dropout and weight decay, from the training; see Table 5 for a comparison.
+Table 5: An illustration of our simplified training settings, where $n$ is the number of augmentation transformations and $m$ the number of magnitudes in RandAugment. A typical setting of RandAugment is $n { = } 2$ and $\scriptstyle m = 9$ for training a single network; see Cubuk et al. (2020); Liu et al. (2021).
+<table><tr><td>Method</td><td>Dataaugmentation</td><td>Weight decay</td><td>DropConnect</td><td>Dropout</td></tr><tr><td>AlphaNet</td><td>AutoAugment</td><td>10-5</td><td>0.2</td><td>0.2</td></tr><tr><td>Ours</td><td>RandAugment (n = 1,m = 1)</td><td>0</td><td>0</td><td>0</td></tr></table>
+# 4 EXPERIMENTS
+We first retrain our ViT supernet with our proposed gradient conflict reduction techniques on ImageNet (Deng et al., 2009); we then conduct an evolutionary search on a subset of the ImageNet training dataset to search the accuracy vs. FLOPs Pareto following (Wang et al., 2021b). We refer the reader to Appendix C.1 for more details. Note all the models are directly sampled from the supernet without retraining or finetuning.
+![](images/b33ff03bfeab5dd2ef33dfa988ce54ce684b202c094e467af4bd14fb8446902f.jpg)
+Figure 4: Comparison with prior-art CNNs and ViTs on ImageNet. Here $" + T "$ indicates methods that are trained with external teacher models. Note that Mobile-Former (Chen et al., 2021c) and MNasNet (Tan et al., 2019) are trained without additional teacher models.
+We call our discovered as NASViT models and compare with state-of-the-art efficient CNNs and ViTs, including FBNetV3 (Dai et al., 2020), AlphaNet (Wang et al., 2021a), LeViT (Graham et al., 2021) and Segformer (Xie et al., 2021), on both image classification (e.g., ImageNet) and semantic segmentation benchmarks (e.g., Cityscapes and ADE20K).
+# 4.1 IMAGENET
+We compare our NASViT models with state-of-the-art NAS-based CNNs, including AlphaNet (Wang et al., 2021a) and FBNetV3 (Dai et al., 2020), and recently-proposed efficient ViTs, e.g., LeViTs.
+Settings Note that our ViT supernet is trained with a pretrained Efficient-B5 teacher model $( 8 3 . 3 \%$ top-1) model. For fair comparison, we retrain AlphaNet with the same teacher. For FBNet-V3 and
+Table 6: Comparison with prior art efficient CNNs and ViTs on ImageNet. The reported AlphaNet models are trained with an external teacher model. The “\*” indicates that the ViTs are trained without external teacher models.
+<table><tr><td>Group</td><td>Method</td><td>M FLOPs</td><td>Top-1 accuracy (%)</td></tr><tr><td rowspan="2">200-300 (M)</td><td>AlphaNet-A0</td><td>203</td><td>77.9</td></tr><tr><td>NASViT-A0 (ours)</td><td>208</td><td>78.2</td></tr><tr><td rowspan="4">300-400 (M)</td><td>LeViT (Graham et al.,2021)</td><td>300</td><td>76.6</td></tr><tr><td>NASViT-A1 (ours)</td><td>309</td><td>79.7</td></tr><tr><td>AlphaNet-A2</td><td>317</td><td>79.4</td></tr><tr><td>FBNetV3 (Dai et al.,2020)</td><td>357</td><td>79.6</td></tr><tr><td rowspan="3">400-500 (M)</td><td>LeViT</td><td>406</td><td>78.6</td></tr><tr><td>NASViT-A2 (ours)</td><td>421</td><td>80.5</td></tr><tr><td>AlphaNet-A4</td><td>444</td><td>80.4</td></tr><tr><td rowspan="4">500-600 (M)</td><td>NASViT-A3 (ours) FBNetV3</td><td>528</td><td>81.0</td></tr><tr><td>NASViT-A4 (ours)</td><td>557</td><td>80.8</td></tr><tr><td></td><td>591</td><td>81.4</td></tr><tr><td>AlphaNet</td><td>596</td><td>81.1</td></tr><tr><td rowspan="3">600 - 1000 (M)</td><td>LeViT NASViT-A5 (ours)</td><td>658</td><td>80.0</td></tr><tr><td>FBNetV3</td><td>757</td><td>81.8</td></tr><tr><td>AutoFormer* (Chen et al.,2021a)</td><td>762</td><td>81.5</td></tr><tr><td rowspan="8">&gt;1000 (M)</td><td>PiT-XS (Heo et al.,2021)</td><td>1,300 1,400</td><td>74.7</td></tr><tr><td>ViTAS-D* (Su et al., 2021)</td><td></td><td>79.1</td></tr><tr><td></td><td>1,600</td><td>76.2</td></tr><tr><td>NASViT (supernet) (ours)</td><td>1,881</td><td>82.9</td></tr><tr><td>CVT-13-NAS*(Wu et al., 2021)</td><td>4,100</td><td>82.2</td></tr><tr><td>Swin-Tiny* (Liu et al., 2021)</td><td>4,500</td><td>81.3</td></tr><tr><td>CVT-13* (Wu et al., 2021)</td><td>4,500</td><td>81.6</td></tr><tr><td>T2T-ViT-14* (Yuan et al., 2021a)</td><td>5,200</td><td>81.5</td></tr><tr><td></td><td>Deep ViT (Zhou et al.,2021)</td><td>6,200</td><td>82.3</td></tr></table>
+LeViT models, these models already use teachers with better performance than Efficient-B5 for training, and therefore we directly report their results following their papers. Specifically, FBNet-V3 use a RegNetY-32G with $8 4 . 5 \%$ top-1 and LeViT use a RegNetY-16G with $8 3 . 6 \%$ top-1 as the teacher model, respectively.
+Results We summarize our results in both Table 6 and Figure 4. Our discovered NASViT models outperform all evaluated CNN and ViT baselines. Our models are the first models with transformers blocks that outperform state-of-the-art efficient CNNs with similar FLOPs on ImageNet. For example, with $< 6 0 0 \mathrm { M }$ FLOPs, our NASViT-A4 achieves $8 1 . 4 \%$ top-1 accuracy on ImageNet. As a reference point, a ResNet-50 model (4G FLOPs) achieves $8 1 . 5 \%$ top-1 accuracy by distilling from a BiT (Kolesnikov et al., 2020) teacher $( 8 7 . 5 \%$ top-1 accuracy) with 1200 epochs of training (Beyer et al., 2021).
+![](images/90ef698b21efb7472680c03fb25c20841793f890a160dfbeab03dde01b671d48.jpg)
+Figure 5: Results of our method and baselines on semantic segmentation. (a-b) show the results on the Cityscapes and ADE20K validation set, respectively.
+# 4.2 SEMANTIC SEGMENTATION
+We evaluate the transfer learning performance of our discovered NASViT models by fine-tuning them on downstream semantic segmentation tasks. In particular, we fine-tune NASViT-A1 to NASViT-A4 as backbones and we show that our NASViT models yield the best segmentation performance compared to the results from the previous efficient CNN backbones, e.g., AlphaNet and MobileNetV3 (Howard et al., 2019), as well as the recently proposed transformer-based Segformer (Xie et al., 2021).
+Settings We evaluate on two benchmark datasets, Cityscapes (Cordts et al., 2016) and ADE20K (Zhou et al., 2017). To handle large input resolutions efficiently, for each transformer block, we set the number of windows to be the input feature map size divided by 7 instead of using our searched settings on ImageNet. We use the recent proposed light-weight Segformer head (Xie et al., 2021) as the decoder head for all the backbones, to achieve a better accuracy efficiency trade-off.
+Results As shown in Figure 5, our models yield the best FLOPs vs. mIoU trade-offs. For example, our model achieves $7 6 . 1 \%$ mIoU and $4 1 . 4 \%$ mIOU with less than 30G FLOPs on the Cityscapes and the ADE20K validation set, respectively.
+# 4.3 ABLATION STUDIES ON GRADIENT CONFLICT AWARE TRAINING
+We conduct ablation studies on ImageNet to have a better understanding on the effectiveness of our proposed methods. We mainly study 1) how our techniques can mitigate the gradient conflict issues and improve the performance 2) whether CNN supernets can also benefit from our techniques. All the comparisons in this section are conducted on ImageNet.
+On the effectiveness of our gradient conflict aware training techniques As demonstrated in Table 7, both weak data augmentation and regularization and switchable scaling layer and can significantly reduce the gradient conflict ratios and in the meantime, improve the top-1 accuracy of both the smallest sub-network and the supernet. By further applying gradient projection to prioritize the sub-networks update (denoted by Prioritize (sub)), the performance of both the smallest and largest sub-network is boosted by around $0 . 3 \%$ on top-1 accuracy.
+Prioritizing the supernet update Instead of focusing on training the sub-networks, we retrain our ViT supernet and prioritize the supernet update by moving the proj(·) term in Eqn. (1) to $\nabla \ell _ { s u b }$ . As demonstrated in the last column of Table 7, this training strategy (denoted by Prioritize (sup)) leads to a slightly improved supernet while resulting in less competitive performance on the smallest sub-network.
+Table 7: Ablation study results on ImageNet. We show the top-1 validation accuracy of the smallest and largest sub-network, and the negative cosine similarity ratio for each case. Note that switchable scaling layer is applied on top of Weak $D A$ & Reg; and Prioritize (sub) is applied on top of both Weak $D A$ & reg and switchable scaling layer.
+<table><tr><td></td><td>Baseline</td><td>Weak DA &amp;Reg</td><td>Switchable scaling</td><td>Prioritize (sub)</td><td>Prioritize (sup)</td></tr><tr><td>Top-1 (smallest)</td><td>76.6</td><td>77.4</td><td>77.6</td><td>78.1</td><td>77.9</td></tr><tr><td>Top-1 (supernet)</td><td>82.2</td><td>82.5</td><td>82.6</td><td>82.9</td><td>83.0</td></tr><tr><td>Negative Cosine Similarity Ratio</td><td>34%</td><td>30%</td><td>29%</td><td>0%</td><td>0%</td></tr></table>
+Improving CNN-based supernets We verify the generalizability of our three techniques to the CNN supernets. In this setting, we applied all three techniques together to improve CNN-based supernets. We show in Table 8 that our method is especially helpful for AlphaNet trained with KL based KD (denoted by AlphaNet $( w / K L ) ,$ ). To further understand the large improvements on AlphaNet $( w / K L )$ , we follow ours studies in Table 4 and compute the gradient conflict ratio for AlphaNet $( w /$ $K L )$ at epoch 1st, 90th, 180th, 270th, and 360th, and the corresponding gradient conflicts ratio is $2 5 \%$ , $18 \%$ , $24 \%$ , $28 \%$ and $31 \%$ , respectively. The gradient conflict issue is more severe for AlphaNet $^ Ḋ r Ḍ w Ḍ$ $\scriptstyle \alpha - K L ,$ compared with AlphaNet trained with $\alpha$ -divergence based KD (AlphaNet $( w / \alpha$ -div)). Our findings indicate that our techniques are not restricted to the ViT supernet training and might be beneficial for all supernets in which a large ratio of gradient conflicts presents.
+# 4.4 ABLATION STUDIES ON SEARCH SPACE
+In this part, we provide additional ablation studies to support some key design choices of our ViT search space. For all the studies, we use a 250M sub-network that is randomly sub-sampled from our search space for consistency.
+Table 8: Improving CNN-based supernets on ImageNet. AlphaNet (w/ KL) and AlphaNet (w/ $\alpha$ -div) denote AlphaNets trained with KL and $\alpha$ -divergence based knowledge distillation, respectively. A0 to A6 are the architectures reported in AlphaNet (Wang et al., 2021a). Note that the AlphaNet supernets here are trained without external teacher models.
+<table><tr><td>Architectures (M FLOPs)</td><td colspan="7">A0 (203M) A1(279M)A2(317M) A3(357M) A4(444M) A5 (491M) A6 (709M)</td></tr><tr><td>AlphaNet (w/ KL)</td><td>77.0</td><td>78.2</td><td>78.5</td><td>78.8</td><td>79.3</td><td>79.6</td><td>80.1</td></tr><tr><td>AlphaNet (w/ KL) + Ours</td><td>77.5</td><td>78.6</td><td>78.9</td><td>79.2</td><td>79.8</td><td>80.1</td><td>80.7</td></tr><tr><td>AlphaNet (w/ α-div)</td><td>77.8</td><td>78.9</td><td>79.2</td><td>79.4</td><td>80.0</td><td>80.3</td><td>80.8</td></tr><tr><td>AlphaNet (w/ α-div) + Ours</td><td>77.8</td><td>78.9</td><td>79.2</td><td>79.4</td><td>80.0</td><td>80.4</td><td>80.9</td></tr></table>
+Global attention vs. local attention In our search space (Table 1), we mainly use the global self-attention for the best representation learning capacity. However, with the more computationally efficient local and linear self-attention schemes, we would be able to use a slightly bigger model under similar FLOPs constraints with a sacrifice of global context modeling. To test this trade-off, we train the aforementioned model from scratch with different types of self-attention strategies. Specifically, in addition to the global self-attention, we further evaluate a number of local and linear self-attention mechanisms, including Swin (Liu et al., 2021), CSwin (Dong et al., 2021), VOLO (Yuan et al., 2021b) and LSH (Kitaev et al., 2020). We uniformly scale the width of the transformer blocks to ensure all models have similar compute FLOPs. In Table 9, we show that the standard global attention achieves the best accuracy compared to other faster local and linear self-attention methods. Additionally, from our evolutionary search results, we also notice that the sub-networks with all standard global self-attention layers often yield the best accuracy vs. FLOPs trade-offs.
+Table 9: ImageNet top-1 accuracy with different types of self-attention mechanisms.
+<table><tr><td>Global</td><td>Swin</td><td>CSwin</td><td>VOLO</td><td>LSH</td></tr><tr><td>78.5</td><td>78.0</td><td>77.9</td><td>78.1</td><td>78.0</td></tr></table>
+The placement of transformer blocks Our ViT supernet has a convolution stem with 3 downsampling strides. We further test the optimal choice of where to switch to transformer blocks. We uniformly scale the width of transformer layers of our baseline model to ensure similar FLOPs for different architecture designs. As shown in Table 10, we notice that our current design, a convolution stem with 3 strides, yields the best performance.
+Table 10: Ablation studies on where to switch to transformer blocks.
+<table><tr><td>#strides</td><td>2</td><td>3</td><td>4</td></tr><tr><td>Top-1 Accuracy</td><td>78.1</td><td>78.5</td><td>78.4</td></tr></table>
+Table 11: Ablation studies on the impact of head dimension.
+<table><tr><td>HeadDimension</td><td>8</td><td>16</td><td>32</td></tr><tr><td>Top-1 Accuracy</td><td>78.5</td><td>78.3</td><td>78.2</td></tr></table>
+Head Dimension In previous ViT works (e.g Touvron et al., 2020; 2021; Zhou et al., 2021; Wu et al., 2021; Liu et al., 2021), the feature dimension of each self-attention head is usually set to be 32, 64, or larger. In this work, as shown in Table 11, we found a smaller head feature dimension (e.g., 8) yields better performance.
+# 5 CONCLUSION
+In this work, we identify one key issue of ViT supernet training that the supernet gradients and the sub-network gradients are likely to disagree with each other, and consequently leading to inferior NAS performance. We fix this gradient conflict issue by introducing a gradient projection method to prioritize the sub-networks update, designing switchable scaling layers to increase the model capacities of sub-networks and simplifying the training recipe to provide easier training signals. With our improved ViT supernet training techniques, our method finds a family of efficient models, called NASViT, that establishes a new state-of-the-art top-1 accuracy vs. FLOPs trade-offs on ImageNet. Our NASViT models are the first ViT variants that outperform prior-art efficient CNNs on the mobile FLOPs regime.
+Acknowledgements Chengyue Gong and Qiang Liu are supported in part by CAREER-1846421, SenSE-2037267, EAGER-2041327, and Office of Navy Research, and NSF AI Institute for Foundations of Machine Learning (IFML). We would like to thank the anonymous reviewers and the area chair for their thoughtful comments and efforts towards improving our manuscript.
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+# A RELATED WORK
+ViTs ViT (Dosovitskiy et al., 2020) and its follow-ups (Wu et al., 2021; Liu et al., 2021; Zhou et al., 2021; Touvron et al., 2021) have been demonstrated to be an alternative choice to CNNs for challenging vision tasks, especially for image classification. In (Dosovitskiy et al., 2020), a large-scale ViT-Large model is trained on JFT-300 to obtain good performance. The follow-ups mainly focus on making the data size and model size smaller without loss of accuracy. A line of works introduce inductive bias or CNN layers to keep the good performance of ViTs while reducing the data and model sizes (e.g. Liu et al., 2021; Dong et al., 2021; Yuan et al., 2021b; Wu et al., 2021). For smaller data sizes, researchers successfully achieve good performance using ViTs without extra data. For example, VOLO (Yuan et al., 2021b) achieves $8 7 . 3 \%$ top-1 accuracy on ImageNet without extra data.
+For smaller model sizes, however, ViTs have not achieved comparable results to efficient CNNS smaller than 1G FLOPs, even if additional CNN layers are introduced (Graham et al., 2021). Dynamic ViTs (Rao et al., 2021; Chen et al., 2021b) propose to dynamically filter the tokens to reduce the computation cost, and the efficiency is not comparable to efficient CNNs. LeViT (Graham et al., 2021), Xiao et al. (2021) and PiT (Heo et al., 2021) processes the high resolution inputs with early convolution layers or spatial-aware layers and also adopt more efficient self-attention designs. Mobileformer (Chen et al., 2021c) proposes a two-branch neural network: one is efficient CNN layers and the other is transformer layers with a small number of tokens (e.g. 6, 8).
+NAS NAS is a powerful tool for automating efficient neural architecture design. It often targets at searching for the best model in a search space under given efficiency-related constraints. Earlier NAS solutions often build on reinforcement learning (e.g. Zoph & Le, 2016; Zoph et al., 2018; Howard et al., 2019) and evolutionary algorithms (e.g. Real et al., 2019; 2017; Wan et al., 2020). More NAS practices have made the search more efficient through weight-sharing and search architectures with gradient-based methods (e.g. Liu et al., 2018; Pham et al., 2018; Stamoulis et al., 2019). This helps alleviate the heavy computational burden of training all candidate networks from scratch and accelerates the NAS process significantly. and researchers work on how to rank the model performance more accurately (Dong & Yang, 2020). Recently, training a large supernet without retraining candidate sub-networks with inplace KD is shown to be an effective mechanism that significantly improves the supernet performance (e.g. Yu et al., 2020a; Wang et al., 2021b;a). In addition to inplace KD, various of KD variants have also been investigated in the literature. For example, Peng et al. (2020) proposes to search a prioritized path as the teacher; Li et al. (2020) proposes to distill the feature level knowledge from an additional teacher model to improve the NAS performance.
+NAS for ViT Most recently, several related works, e.g., AutoFormer (Chen et al., 2021a), and ViTAS (Su et al., 2021), have been proposed to search for ViTs. AutoFormer is the first paper that leverages NAS for ViT optimization. A comprehensive search space for the ViT architecture is proposed and a weight-entanglement training strategy is developed to improve the NAS efficiency. ViTAS leverages a similar supernet-based NAS method compared to AutoFormer and introduces private class token and self-attention maps to cater for the variance of distinct ViT architectures. Both work demonstrate promising accuracy improvement compared to the baseline DeiT models for large models with more than 1G FLOPs.
+Gradient cosine similarity in multi-mask learning A line of prior approaches have observed that the difficult training with multiple objectives can be improved by using the cosine similarity between gradients (e.g. Du et al., 2018; Yu et al., 2020b; Real et al., 2019). The cosine similarity is used to as a regularization or an indicator. Real et al. (2019) adds a regularization term to force the cosine similarity between two different losses to be larger than zero. Du et al. (2018) and Dery et al. (2021) propose to use gradient cosine similarity to identify whether auxiliary tasks can benefit the main task. In Du et al. (2018), once the cosine similarity is negative (gradient conflict), the weight of the auxiliary task is set to be zero. Yu et al. (2020b) is most related to our projection method, which projects the gradient of every loss to achieve orthogonal gradients. To avoid negative gradient cosine similarity, we project the gradient of the supernet to prioritize the training of sub-networks, which have similar intuition as auxiliary losses. In continual learning, many works uses orthogonal gradient descent to restrict the direction of gradient updates of new tasks in order to avoid catastrophic forgetting (e.g. Farajtabar et al., 2020; Bennani et al., 2020; Saha et al., 2021).
+# B SEARCH SPACE
+![](images/67202428f8dd91712368b350a25205c8163436478dcb77286b52fb694ac0a454.jpg)
+Figure 6: A demonstration of our self-attention module. ‘RPE’, ’Dw Conv, ‘Talking Head’, ‘Proj’ and ‘MLP’ refer to relative positional embedding, depth-wise convolutional layer, talking head attention, projection layer and MLP layer, respectively.
+# B.1 EFFICIENT TRANSFORMER BUILDING LAYER
+In this section, we give a detailed introduction about our transformer building layer with self-attention. In the literature, researchers have develop many variants of the standard self-attention with different focuses (e.g. efficiency, convergence, lone-term dependency, etc.). Motivated recent works, e.g., LeVit, SWIN-tranformer and VOLO (Graham et al., 2021; Liu et al., 2021; Yuan et al., 2021b), we develop a transformer layer for the purpose of efficiency and effectiveness in vision tasks. A demonstration of our transformer layer is shown in Figure 6. To enhance the learning capacity of our ViT models, we incorporate talking head (Shazeer et al., 2020) layers and depth-wise convolution layer in the self-attention module. Additionally, following LeViT (Graham et al., 2021), we expand the dimension of V matrix by expansion ratio 4 and introduce activation function after the projection matrix. Following Swin Transformer (Liu et al., 2021), we use relative positional embedding for the attention scores. For efficiency, we reduce the MLP expansion ratio to $\{ 1 , 2 \}$ and add one additional MLP layer to keep the model complexity following MacaronNet (Lu et al., 2019).
+Positional information The positional embedding in transformer architectures is location-dependent trainable parameters. Recent works propose absolute positional embedding, relative positional embedding or additional depth-wise convolution layers (Dong et al., 2021) to enhance the local information. We introduce two additional depth-wise convolution layers into a MHSA with relative positional embedding. For relative positional embedding, we directly follow the implementation in NLP (Shaw et al., 2018). For depth-wise convolution, we add one depth-wise convolution layer in the MLP layer and another depth-wise convolution layer after the linear transformations of V matrix.
+Expansion Ratio In the self-attention design space, researchers have explored whether expanding the channels can have good performance. LeViT proposes to expand the dimension of V. We follow LeViT’s design which expand the dimension of $\mathrm { v }$ by an expansion ratio 4. researchers have explored how many layers (Lu et al., 2019) should we use for MLP in a self-attention block. We follow the strategy developed by (Lu et al., 2019) which adds one more MLP layer for each self-attention block, but reduce the MLP expansion ratio to $\{ 1 , 2 \}$ for efficiency. As displayed in Figure 6, we place an additional MLP layer after the first MLP layer.
+Normalization Layers and activation Many recent works apply additional batch normalization layers, layer normalization layers or activation functions to the network. Taking the computation cost of layer normalization layers into consideration, we do not introduce any new normalization layers to the basic self-attention layers.
+Talking-head attention and number of heads Most of the existing ViTs set the dimension of each head to be 24/32 (e.g Touvron et al., 2020; 2021; Zhou et al., 2021; Wu et al., 2021; Liu et al., 2021). However, for a model with few channels, a large head dimension leads to few number of heads. We set a smaller head dimension (e.g. 8, 16) to make the number of heads to be large, and further introduce the talking-head attention to improve the capacity of different heads. Talking-head attention (Shazeer et al., 2020) introduce two additional linear transformation between all the heads, one is before softmax and another is after softmax.
+Classification head Due to the use of depth-wise convolution layers and down-sampling, we remove the classification token for simplicity. While LeViT and DeiT use two heads for the teacher knowledge distillation and supervised labels, we use one head for all the training settings and replace the onelayer fully-connected layer head with MobilenetV3 (Howard et al., 2019) head so as to reducing computation cost.
+Scaling Factor To train very deep transformer models, Touvron et al. (2021) introduces additional learnable channel-wise scaling factors initialized with $1 0 ^ { - 4 }$ into the models. The channel-wise scaling factors are introduced to the output of each MLP and multi-head attention (MHA) layer in the model. Notice that many of sub-networks in our search space are very deep, we introduce architecture-dependent switchable scaling factors into the supernet (see Section 3).
+# C IMPLEMENTATION DETAILS ON TRAINING AND SEARCHING
+# C.1 TRAINING AND SEARCH ALGORITHM
+Training Consider a supernet with trainable parameter $\theta$ and the candidate sub-networks set $\mathcal { A }$ The goal of training a supernet is to learn model parameter $\theta$ target at optimizing all the sub-networks in $\mathcal { A }$ and simultaneously achieving good accuracy. Let $s \sim A$ , $p ( x ; \theta )$ and $q ( x ; \theta _ { s } )$ denote the output probability of the supernet and the sub-network $s$ , we have the loss
+$$
+\mathcal { L } = \mathcal { \ell } ( \boldsymbol { \theta } ) + \sum _ { i = 1 } ^ { k } \ell _ { \mathrm { K D } } ( \theta _ { s _ { i } } , \theta _ { d e t a c h } ) , \ w h e r e \ \ell _ { \mathrm { K D } } ( \theta _ { s _ { i } } , \theta _ { d e t a c h } ) = \mathbb { D } \bigg ( p ( \boldsymbol { x } ; \theta _ { d e t a c h } ) \mid | \mathbf { \ell } q ( \boldsymbol { x } ; \theta _ { s _ { i } } ) \bigg ) .
+$$
+Here, $\ell ( \theta )$ denotes the loss of the supernet, $\theta _ { s _ { i } }$ represents the parameters for a sampled sub-network $s _ { i }$ , and ${ \dot { \mathbb { D } } } ( p | | q )$ is a divergence that measures the difference between $p$ and $q$ . The $\theta _ { d e t a c h }$ denotes a copy of $\theta$ whose gradient is stopped during back-propagation.
+# Algorithm 1 Algorithm: Supernet based NAS training
+<table><tr><td colspan="2">while not converged do</td></tr><tr><td>1: 2:</td><td>Sample a mini-batch data from dataset</td></tr><tr><td>3:</td><td>Sample the supernet (i.e., the biggest sub-network) from the search space and train the</td></tr><tr><td>4:</td><td>supernet with with ground truth labels (or with KD from an external teacher model) Sample k random sub-networks from the search space and train them with KD by using the supernet network as the teacher model</td></tr></table>
+5: end while
+Search After training, a random forest based neural predictor is trained to fit the map from the architecture hyper-parameters to the model performance (e.g., accuracy). A number of sub-networks are sampled from the trained supernet to train the neural predictor.
+We then follow the strategy in previous works (e.g. Cai et al., 2019; Wang et al., 2021a) to do evolutionary search: 1) we randomly sample 1024 sub-networks from the supernet and estimate their accuracy on a sub-sampled subset of the ImageNet training set, which is never used during the supernet training; 2) we apply crossover and random mutation (see Zhou et al. (2011) for more details about evolutionary algorithms) on the best performing 128 sub-networks. We fix both the crossover size and mutation size to be 128, yielding 256 new sub-networks. We then evaluate the performance of these sub-networks; 3) We repeat the second step 40 times. The total number of sub-networks thus evaluated is around 10K.
+# 1: Input: a pretrained supernet with fixed weights
+2: Randomly sample 1024 sub-networks and evaluate their performance on a withhold training set (which is not used during training).
+3: Partition 1024 sub-networks into training and validation subset with equal size. Train a random forest regressor to predict sub-netowrk’s accuracy given the sub-network architecture hyperparameters as the input.
+4: Run evolution algorithm to search the Pareto of sub-networks. The sub-network accuracy is given by the random forest based predictor.
+# C.2 ABLATION STUDIES
+Latency-aware Searching Note that the networks in the paper were optimized for the best FLOPs vs. accuracy trade-off. However, it is expected that the networks that achieve the best FLOPs vs. accuracy trade-off don’t necessary yield the best latency vs. accuracy trade-off in the same time. Therefore, to achieve the best latency vs. accuracy trade-off, we re-search three NASViTs (B0/B1/B2) that form better latency vs. accuracy trade-off compared to the results from NASViTs (A0/A1/A2). For latency comparison, we evaluate the latency of NASViTs-A0/A1/A2 and AlphaNet-A0/A2/A4 on Intel(R) Xeon CPUs with a batch size of 1. See the result in the table below. Here, we do not remove BN or LN layers when estimating latency.
+Algorithm 2 Algorithm: Supernet based NAS searching
+Table 12: NASViT models searched with for better latency vs. accuracy trade-off.
+<table><tr><td>Model</td><td>Accuracy (%)</td><td>CPU latency (ms)</td></tr><tr><td>NASViT-B0</td><td>78.2</td><td>21.0±0.4</td></tr><tr><td>NASViT-A0</td><td>78.2</td><td>21.6±0.5</td></tr><tr><td>AlphaNet-A0</td><td>77.9</td><td>21.4±0.5</td></tr><tr><td>NASViT-B1</td><td>79.6</td><td>26.6±0.6</td></tr><tr><td>NASViT-A1</td><td>79.7</td><td>27.2±0.6</td></tr><tr><td>AlphaNet-A2</td><td>79.4</td><td>27.4±0.5</td></tr><tr><td>NASViT-B2</td><td>80.6</td><td>29.1±0.6</td></tr><tr><td>NASViT-A2</td><td>80.5</td><td>29.8±0.5</td></tr><tr><td>AlphaNet-A4</td><td>80.4</td><td>30.4±0.5</td></tr></table>
+# D ARCHITECTURE VISUALIZATION OF NASVIT MODELS
+Table 13: Here, ‘c’ denotes the number of output channels, ‘d’ denotes number of layers, ‘ks’ denotes kernel size, ‘e’ denotes expansion ratio, $\mathbf { \cdot } _ { \mathbf { k } } ,$ denotes number of windows, ‘s’ denotes stride.
+<table><tr><td></td><td>NASViT-A1</td><td>NASViT-A2</td><td>NASViT-A3</td><td>NASViT-A4</td></tr><tr><td rowspan="4">Conv</td><td>c: 16</td><td>c: 16</td><td>c: 16</td><td>c:16</td></tr><tr><td>d:1</td><td>d:1</td><td>d:1</td><td>d:1</td></tr><tr><td>ks: 3</td><td>ks: 3</td><td>ks: 3</td><td>ks: 3</td></tr><tr><td>s: 2</td><td>s: 2</td><td>s: 2</td><td>s:2</td></tr><tr><td rowspan="5">MBConv-1</td><td>c: 16</td><td>c: 16</td><td>c: 16</td><td>c: 16</td></tr><tr><td>d:1</td><td>d:1</td><td>d:1</td><td>d:1</td></tr><tr><td>ks: 3</td><td>ks: 3</td><td>ks: 3</td><td>ks: 3</td></tr><tr><td>e:3</td><td>e:3</td><td>e:3</td><td>e:3</td></tr><tr><td>s:1</td><td>s:1</td><td>s:1</td><td>s:1</td></tr><tr><td rowspan="5">MBConv-2</td><td>c: 24</td><td>c: 24</td><td>c: 24</td><td>c: 24</td></tr><tr><td>d:3</td><td>d:3</td><td>d:3</td><td>d:3</td></tr><tr><td>ks: 3</td><td>ks: 3</td><td>ks:3</td><td>ks: 3</td></tr><tr><td>e:4</td><td>e: 4</td><td>e:5</td><td>e:4</td></tr><tr><td>s:2</td><td>s:2</td><td>s:2</td><td>s:2</td></tr><tr><td rowspan="5">MBConv-3</td><td>c: 32</td><td>c: 32</td><td>c: 32</td><td>c: 32</td></tr><tr><td>d: 3</td><td>d:3</td><td>d: 3</td><td>d: 3</td></tr><tr><td>ks: 3</td><td>ks: 3</td><td>ks: 3</td><td>ks: 3</td></tr><tr><td>e:4</td><td>e: 6</td><td>e:5</td><td>e:6</td></tr><tr><td>s: 2</td><td>s: 2</td><td>s: 2</td><td>s: 2</td></tr><tr><td rowspan="5">Transformer-4</td><td>c: 64</td><td>c: 64</td><td>c: 64</td><td>c: 64</td></tr><tr><td>d: 4</td><td>d: 4</td><td>d: 4</td><td>d: 4</td></tr><tr><td>k: 1</td><td>k:1</td><td>k:1</td><td>k: 1</td></tr><tr><td>e:1</td><td>e:1</td><td>e:1</td><td>e:1</td></tr><tr><td>s:2</td><td>s: 2</td><td>s: 2</td><td>s: 2</td></tr><tr><td rowspan="4">Transformer-5</td><td>c: 112</td><td>c: 112</td><td>c: 112</td><td>c: 120</td></tr><tr><td>d: 3</td><td>d: 3</td><td>d: 4</td><td>d: 3</td></tr><tr><td>e:1</td><td>e:1</td><td>e:1</td><td>e:1</td></tr><tr><td>s: 2</td><td>s: 2</td><td>s: 2</td><td>s: 2</td></tr><tr><td rowspan="4">Transformer-6</td><td>c: 160</td><td>c: 160</td><td>c: 160</td><td>c: 160</td></tr><tr><td>d: 3</td><td>d: 5</td><td>d: 7</td><td>d: 6</td></tr><tr><td>e:1</td><td>e:1</td><td>e: 1</td><td>e:1</td></tr><tr><td>s: 1</td><td>s:1</td><td>s: 1</td><td>s:1</td></tr><tr><td rowspan="4">Transformer-7</td><td>c: 216</td><td>c: 208</td><td>c: 216</td><td>c: 216</td></tr><tr><td>d: 3</td><td>d: 4</td><td>d: 5</td><td>d: 6</td></tr><tr><td>e:1</td><td>e:1</td><td>e:1</td><td>e: 1</td></tr><tr><td>s:2</td><td>s: 2</td><td>s: 2</td><td>s:2</td></tr><tr><td>MBPool</td><td>c: 1792</td><td>c: 1792</td><td>c: 1984</td><td>c: 1984</td></tr><tr><td>Resolution</td><td>192</td><td>224</td><td>256</td><td>288</td></tr></table>

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+# The medial axis of closed bounded sets is Lipschitz stable with respect to the Hausdorff distance under ambient diffeomorphisms
+Anonymous Author(s)
+Affiliation
+Address
+email
+# Abstract
+1 We prove that the medial axis of closed sets is Hausdorff stable in the following
+2 sense: Let $S \subseteq \mathbb { R } ^ { d }$ be a fixed closed set that contains a bounding sphere. Consider
+3 the space of $C ^ { 1 , 1 }$ diffeomorphisms of $\mathbb { R } ^ { d }$ to itself, which keep the bounding
+4 sphere invariant. The map from this space of diffeomorphisms (endowed with a
+5 Banach norm) to the space of closed subsets of $\mathbb { R } ^ { d }$ (endowed with the Hausdorff
+6 distance), mapping a diffeomorphism $F$ to the closure of the medial axis of $F ( S )$ ,
+7 is Lipschitz.
+This extends a previous stability result of Chazal and Soufflet on the stability of the medial axis of $C ^ { 2 }$ manifolds under $C ^ { 2 }$ ambient diffeomorphisms.
+# 10 1 Introduction
+11 In [19], Federer introduced the reach of a (closed) set $S \subset \mathbb { R } ^ { d }$ as the infimum over all points in $s$ of
+12 the distance from these points to the medial axis $\operatorname { a x } ( S )$ , the set of points in $\mathbb { R } ^ { d }$ for which the closest
+13 point in $s$ is not unique. Federer also introduced the reach at a point $p \in S$ to be the distance from $p$
+14 to the medial axis of $s$ . We now call this quantity the local feature size [3] and denote it by $\operatorname { l f s } ( p )$ .
+15 Federer proved that the reach is stable under $C ^ { 1 , 1 }$ diffeomorphisms of the ambient space. Here, a
+16 $C ^ { 1 , 1 }$ map is a $C ^ { 1 }$ map whose derivative is Lipschitz, and a $C ^ { \hat { 1 } , 1 }$ diffeomorphism is a $\mathbf { \bar { \it C } } ^ { 1 , 1 }$ bijective
+17 map whose inverse is also $C ^ { 1 , 1 }$ . Chazal and Soufflet [13] proved that the medial axis is stable with
+18 respect to the Hausdorff distance under ambient diffeomorphisms, but under stronger assumptions
+19 than the work of Federer, namely assuming that $s$ is a $C ^ { 2 }$ manifold and the distortion is a $C ^ { 2 }$ diffeo
+20 morphism of the ambient space. Chazal and Soufflet based their work on earlier results by Blaschke
+21 [9], which were not as strong as Federer’s.
+22 In this paper we extend the stability result of the medial axis. More concretely, we generalize the
+23 result of Chazal and Soufflet [13] to arbitrary closed sets and $C ^ { 1 , 1 }$ diffeomorphisms of the ambient
+24 space; we show that the Hausdorff distance between the medial axes of the closed set and its image is
+25 bounded in terms of Lipschitz constants stemming from the diffeomorphism of the ambient space.
+26 Our result follows from the work of Federer [19] and in fact shortens the proof in [13] significantly.
+Our bounds on the Hausdorff distance say nothing about the topology of the medial axis, which is known to be highly unstable (see e.g. [5]), although it preserves the homotopy type (see [28]).
+Contribution and related work Our work differs from the majority of the literature in three essential ways:
+Firstly, we make no assumptions on the set we consider apart from that it is closed. The stability of the medial axis of (piecewise) smooth manifolds has been the object of intense study, see for example
+33 [13, 15–17, 24, 30, 37–40]. However, the manifold assumption is impossible to achieve in many
+34 applications — such as in the context of astrophysics, one of the main motivations of this paper.
+35 Secondly, we achieve stability without pruning the medial axis. This contrasts with a large body of
+36 work, such as [6, 12, 16, 29]. Not having to prune the medial axis is a significant advantage. On the
+37 downside, we limit the changes of the considered set to those induced by ambient diffeomorphisms.
+38 Nevertheless, given the standard examples of the instability of the medial axis — see for example [5]
+39 — we believe these limitations are near to the weakest assumptions necessary for Hausdorff stability.
+40 Within the context of ambient homeomorphisms, the results we obtain are close to optimal, as we
+41 specify in Remark 4.2.
+Thirdly, our results hold for sets in arbitrary dimensions and are not sensitive to the dimension of the set itself. A large part of the related work only investigates sets of low dimensions or codimension one manifolds, although there are some notable exceptions such as [39], see also [17], and [12, 29].
+45 Motivation The medial axis has many real world applications — among others, in robot motion
+46 planning [27], triangulation algorithms [4], graphics [35], and shape recognition, segmentation, and
+47 learning [10, 18, 25, 33, 41]. See also the overviews [32, 35]. The reach — the distance between a
+48 set and its medial axis — is a central concept in manifold learning [1, 2, 20–22, 34].
+49 The motivation of this paper is twofold: Firstly, we tackle the following challenge from the processing
+50 of images collected with optical devices which use lenses — such as cameras or telescopes. A shape
+51 extracted from such an image may be imprecise due to the imperfection of the lenses. Our result
+52 implies that the medial axis of such a shape is stable under these imperfections. As a consequence,
+53 the outcome of any shape recognition or shape segmentation algorithm based on the medial axis will
+54 be stable.
+55 In addition to the disciplines listed above, such stability is sought after in astrophysics, in particular
+56 for shape analysis and automated shape identification in observational astronomy. Observational
+57 astronomers are interested in reconstructing objects like stars or galaxies, and their place in the
+58 universe from data gathered by telescopes. They can deduce the distance from the object to the
+59 observer thanks to so-called standard candles or red shift [14, 23, 31]. However, the image gets
+60 distorted due to optical effects — either through gravitational lensing ([7]) or lensing inside the
+61 telescope itself ([36]).
+62 Such a distortion can be modeled as a diffeomorphism of the ambient space. At the same time, this
+63 problem cannot be tackled using the result by Chazal and Soufflet [13], since the observed objects
+64 might not be smooth — for example due to interactions with shock waves or jets. In addition, with
+65 our method astrophysicists can not only reconstruct objects in space (3D), but also in spacetime (4D).
+66 The second motivation is more formal in nature: The stability of the medial axis is instrumental in
+67 establishing its computability. Indeed, when proving properties of algorithms based on the medial
+68 axis, authors generally assume the real RAM model.1 However, as was recently argued in [29], the
+69 medial axis needs to be stable in order to be computable in more realistic models of computation.
+70 There is a more practical component to this formal question: It is not a priori clear if using possibly
+71 noisy real world data or the output of other computer programs as input for these algorithms yields
+72 answers that are close to the ground truth. To be able to prove that the output is correct, we need
+73 (numerical) stability of the medial axis.
+74 Outline After revisiting preliminaries and known results in Section 2, we state the main stability
+75 result in Section 3. In Section 4 we reformulate this result in terms of norms on Banach spaces. This
+76 also exhibits the fact that the stability of the medial axis is Lipschitz in the following sense: We think
+77 of the set $s$ as fixed and consider the map from the space of diffeomorphisms (endowed with a norm
+78 which makes it a Banach space) to the space of closed subsets of $\mathbb { R } ^ { d }$ (endowed with the Hausdorff
+79 distance), mapping each diffeomorphism $F : \mathbb { R } ^ { d }  \mathbb { R } ^ { d }$ to the closure of the medial axis of $F ( S )$
+80 The Lipschitz constant then only depends on the diameter of the bounding sphere of the set $s$ .
+81 We only include proof sketches of the two main theorems in this article. The full proofs of the
+82 theorems and of the supporting lemmas, can be found in the supplementary material.
+# 83 2 Preliminaries: Sets of positive reach and the closest point projection
+84 In this section we recall some definitions and results concerning the medial axis and sets of positive
+85 reach. Essentially, we need three ingredients from the literature to prove our main theorem: the
+86 notions related to the closest point projection, the properties of the generalized normal and tangent
+87 spaces, and Federer’s result on the stability of the reach under ambient diffeomorphisms.
+88 We write $d ( \cdot , \cdot )$ for the Euclidean distance between two points, and the distance between a point and
+89 a set. That is, for any closed set $s$ and point $p$ ,
+$$
+d ( p , S ) = \operatorname* { i n f } _ { q \in S } d ( p , q ) .
+$$
+We denote the Hausdorff distance between two sets 90 $A , B \subseteq \mathbb { R } ^ { d }$ by $d _ { H } ( A , B )$ :
+$$
+d _ { H } ( A , B ) = \operatorname* { m a x } \left\{ \operatorname* { s u p } _ { a \in A } d ( a , B ) , \operatorname* { s u p } _ { b \in B } d ( b , A ) \right\} .
+$$
+91 We write $B ( c , r )$ , resp. $S ( c , r )$ , to denote balls, resp. spheres, with centre $c$ and radius $r$ . Lastly, $\left. \cdot \right.$
+92 denotes the Euclidean norm, and $\lVert \cdot \rVert$ an operator norm.
+93 The closest point projection and related notions The projection of points in the ambient space
+94 $\mathbb { R } ^ { d }$ to the (set of) closest point(s) of the set $S \subseteq \mathbb { R } ^ { d }$ is denoted by $\pi _ { \boldsymbol { S } }$ , and illustrated in Figure 1.
+![](images/3f0be235ed8367b4a9102442acbd51f72ab9bd9e67bf655c834fcf17a00c472b.jpg)
+Figure 1: The closest point projection to the set $s$ of four points in $\mathbb { R } ^ { 2 }$ . When a point lies on the medial axis $\operatorname { a x } ( S )$ , the closest point projection consists of more points.
+The medial axis of 95 $s$ is the set of all points $p \in \mathbb { R } ^ { d }$ where the set $\pi _ { S } ( p )$ consists of more than one 96 point:
+$$
+\operatorname { a x } ( \mathcal { S } ) = \left\{ p \in \mathbb { R } ^ { d } \mid \# \pi _ { \mathcal { S } } ( p ) > 1 \right\} .
+$$
+97 Here, $\# \pi _ { S } ( p )$ denotes the cardinality of the set $\pi _ { S } ( p )$ .
+For a point $p \in { \mathcal { S } }$ , the local feature size of $p$ is the distance from $p$ to the medial axis of the set $s$
+$$
+\begin{array} { r } { \mathrm { l f s } ( p ) = d ( p , \mathrm { a x } ( S ) ) . } \end{array}
+$$
+99 Finally, the reach of the set $s$ is the infimum of the local feature size over all its points:
+$$
+\operatorname { r c h } ( S ) = \operatorname* { i n f } _ { p \in S } \operatorname { l f s } ( p ) = \operatorname* { i n f } _ { p \in S } d ( p , \operatorname { a x } ( S ) ) .
+$$
+100 Throughout this paper we assume that $S \subseteq \mathbb { R } ^ { d }$ is a closed set. We shall further assume that the set
+101 $s$ as well as its medial axis are bounded, and that the bounding sphere of $s$ is contained in $s$ itself.
+102 More specifically, we assume that there exists a closed ball $B$ of positive radius such that ${ \mathcal { S } } \subseteq B$ ,
+103 and $\partial B \subseteq S$ . We call $\partial B$ the bounding sphere of $s$ .
+104 The addition of the bounding sphere $\partial B$ to the set $s$ is necessary to obtain the desired bound on
+105 the Hausdorff distance between the two medial axes of the set $s$ and its image under the ambient
+106 diffeomorphism. Indeed, consider the following example, illustrated in Figure 2.
+107 Let the set $s$ consist of two points in the plane, $S = \{ p , q \} \subseteq \mathbb { R } ^ { 2 }$ . The medial axis of $s$ is then the
+108 bisector line of $p$ and $q$ . After a generic perturbation $F$ of $p$ and $q$ — that is, not a translation and not a
+109 perturbation in the direction $\pm ( p - q )$ — the bisector line $\operatorname { a x } ( F ( S ) )$ of the perturbed points intersects
+110 the bisector $\operatorname { a x } ( S )$ of the original pair. The Hausdorff distance between these two non-parallel lines
+111 is infinite, and thus unboundable.
+112 At the same time, the addition of the bounding sphere $\partial B$ to the considered set $s$ is not a restriction.
+113 Indeed,
+14 Remark 2.1 The medial axes of $s$ and $s \backslash \partial B$ coincide in the interior of the ball $B$ sufficiently far
+15 away from its boundary $\partial B$ . More precisely:
+![](images/984a3465b0e7ed98c686f557e68f88b725441720c1b7bfd4e786358725ab5dfb.jpg)
+Figure 2: In black the set $s$ and its medial axis, in light blue the perturbed set and its medial axis. The Hausdorff distance between $\operatorname { a x } ( S )$ and $\operatorname { a x } ( F ( S ) )$ is infinite.
+• Any point $x \in \operatorname { a x } ( S )$ , such that $\pi _ { S } ( x ) \cap \partial B = \emptyset .$ , lies on the medial axis $\operatorname { a x } ( S \setminus \partial B )$ . • Conversely, if a point $x$ lies on the medial axis $\operatorname { a x } ( S \setminus \partial B )$ , and any (and thus every) point $q \in \pi _ { S \setminus { \partial B } } ( x )$ satisfies $d ( x , q ) < d ( x , \partial B )$ , then $x \in \operatorname { a x } ( S )$ .
+Thus, the medial axis is locally stable if the ambient diffeomorphism is close to the identity.219
+20 A recurring strategy in this article is to start at a point $p$ on the set $s$ , move away from this point in a
+21 ‘normal’ direction, and see if by projecting using the closest point projection $\pi _ { \boldsymbol { S } }$ we get back to $p$ . To
+22 this end, we define the projection range.
+Definition 2.2 (Projection range) Let $p \in S$ be a point and $v \in \mathbb { R } ^ { d } a$ vector. The projection range $d ( p , v , \pi _ { S } )$ in direction v is the maximal distance one can travel from $p$ along $v$ such that the closest point projection yields only the point $p$ :
+$$
+d ( p , v , \pi _ { \mathcal { S } } ) = \operatorname* { s u p } \{ \lambda \in \mathbb { R } \mid \pi _ { \mathcal { S } } ( p + \lambda v ) = \{ p \} \} .
+$$
+Since $\pi _ { S } ( p ) = \{ p \}$ , the projection range is canonically non-negative. Furthermore, the directions for which the range is positive are key to our study, because of the following property:
+Lemma 2.3 (Theorem 4.8 (6) of [19]) Consider a point $p \in S$ and a vector $v \in \mathbb { R } ^ { d }$ . I f
+$$
+0 < d ( p , v , \pi _ { S } ) < \infty ,
+$$
+then $p + d ( p , v , \pi _ { S } ) \cdot v \in \overline { { \operatorname { a x } ( S ) } }$ .
+We call these special directions $v$ back projection vectors:
+Definition 2.4 (Unit back projection vectors) For a point $p \in { \mathcal { S } }$ , $\mathrm { U B P } ( p , S )$ is the set of unit vectors with a positive projection range:
+$$
+\mathrm { U B P } ( p , S ) = \left\{ u \in \mathbb { R } ^ { d } \mid | u | = 1 a n d 0 < d ( p , u , \pi _ { S } ) < \infty \right\} .
+$$
+133 We further define
+$$
+\begin{array} { r l } & { \mathrm { U B P } ( S ) = \left\{ ( p , u ) \in S \times \mathbb { R } ^ { d } \ : \middle | \ : u \in \mathrm { U B P } ( p , S ) \right\} , } \\ & { \quad \mathrm { B P } ( S ) = \left\{ ( p , \lambda u ) \in S \times \mathbb { R } ^ { d } \ : \middle | \ : ( p , u ) \in \mathrm { U B P } ( S ) , \lambda \geq 0 \right\} . } \end{array}
+$$
+2The bounding sphere does allow one to give a relatively clean mathematical statement, see Section 4.
+134 Thanks to Lemma 2.3, the following map is well-defined:
+$$
+\pi _ { \mathrm { a x } , { \mathcal { S } } } : \operatorname { U B P } ( { \mathcal { S } } ) \to \operatorname { \overline { { a x } } } ( { \mathcal { S } } ) , \qquad ( p , u ) \mapsto p + d ( p , u , \pi _ { \mathcal { S } } ) u .
+$$
+135 The generalized tangent and normal space Back projection vectors are intricately related to the
+136 generalized tangent and normal spaces.
+137 Definition 2.5 (Definitions 4.3 and 4.4 of [19]) Let $\boldsymbol { p } \in \textit { s }$ . The generalized tangent space
+138 $\mathrm { T a n } ( p , S )$ is the set of vectors $u \in \mathbb { R } ^ { d }$ , such that either $u = 0$ or, for every $\varepsilon > 0$ there exists
+139 a point $q \in S$ with
+$$
+0 < | q - p | < \varepsilon \qquad \mathit { a n d } \qquad \left| \frac { q - p } { | q - p | } - \frac { u } { | u | } \right| < \varepsilon .
+$$
+140 The generalized normal space $\operatorname { N o r } ( p , S )$ consists of vectors $v \in \mathbb { R } ^ { d }$ such that $\langle v , u \rangle \leq 0$ for all
+141 $u \in { \mathrm { T a n } } ( p , S )$ . Vectors contained in the generalized tangent, resp. normal, space are called tangent,
+142 resp. normal, to $s$ at $p$ .
+143 The generalized tangent and normal spaces are illustrated in Figure 3.
+![](images/ac397c02bd9997e5e3dd538710186040d2f69640915cce96e4c712d3c33d75c2.jpg)
+Figure 3: The (affine) generalized tangent and normal spaces of four points in the set $\mathcal { S } \subset \mathbb { R } ^ { 2 }$ , in light blue and violet, respectively.
+144 Stability of the reach under ambient diffeomorphisms Our last ingredient is the following result
+145 by Federer.
+146 Theorem 2.6 (Stability of the reach under ambient diffeomorphisms, Theorem 4.19 of [19])
+147 Pick two constants $0 < t < \operatorname { r c h } ( S )$ and $s > 0$ . If the map
+$$
+F : \{ x \in \mathbb { R } ^ { d } \mid d ( x , S ) < s \}  \mathbb { R } ^ { n }
+$$
+is injective and continuously differentiable, and the maps 148 $F$ , $F ^ { - 1 }$ , and $D F$ are Lipschitz continuous 149 with Lipschitz constants $\operatorname { L i p } ( F ) , \operatorname { L i p } ( F ^ { - 1 } ) , \operatorname { L i p } ( D F )$ , respectively, then the reach $\operatorname { r c h } ( F ( S ) )$ of the 150 image of the set $s$ under the map $F$ is lower-bounded by
+$$
+\operatorname { r c h } ( F ( S ) ) \geq \operatorname* { m i n } \left\{ { \frac { s } { \mathrm { L i p } ( F ^ { - 1 } ) } } , { \frac { 1 } { \left( { \frac { \mathrm { L i p } ( F ) } { t } } + \mathrm { L i p } ( D F ) \right) \left( \mathrm { L i p } ( F ^ { - 1 } ) \right) ^ { 2 } } } \right\} .
+$$
+# 151 3 Stability of the medial axis under ambient diffeomorphisms
+152 In this section we present the main result of this paper, Theorem 3.9. This theorem extends earlier
+153 work by Chazal and Soufflet [13]. Its proof relies on Federer’s result on the stability of the reach,
+154 Theorem 2.6. To give a more geometrical interpretation we introduce the concept of a weakly tangent
+155 sphere and ball, and a maximal empty weakly tangent ball.
+156 Definition 3.1 (Weakly tangent sphere and ball) Let $p \in S$ . A sphere is called weakly tangent
+157 to $s$ at $p$ if it contains the point $p$ and its centre lies in the (translated) generalized normal space
+158 $\mathrm { N o r } ( p , S ) + p$ . In other words, spheres weakly tangent to $s$ at $p$ are spheres with centres $p + v$ and
+159 radii $| v |$ , for a vector $v \in \operatorname { N o r } ( p , S )$ .
+A ball is called weakly tangent to $s$ at $p$ if its boundary sphere is weakly tangent to $s$ at $p .$
+161 Remark 3.2 Using the definition of $\operatorname { N o r } ( p , S )$ , a weakly tangent ball can also be defined as follows:
+162 A ball $B ( c , r )$ is weakly tangent at $p$ if and only if its centre c and radius $r$ satisfy
+$$
+( p + \operatorname { T a n } ( S , p ) ) \cap B ( c , r ) = \{ p \} .
+$$
+163 We remark:
+164 Lemma 3.3 Let $p \in S$ and $v \in \mathbb { R } ^ { d }$ , and suppose that for some $\lambda > 0$ we have $\pi _ { S } ( p + \lambda v ) \neq \{ p \}$ .
+Then, for all 165 $\lambda ^ { \prime } \geq \lambda$ , we have $\pi s ( p + \lambda ^ { \prime } v ) \neq \{ p \}$ and for all $\lambda ^ { \prime } > \lambda$ , that $p \notin \pi s ( p + \lambda ^ { \prime } v )$ .
+![](images/9a5344e6a0e1629ea3d6abbde82149aec404769a759a00d0533a4dd0a1aaf5c4.jpg)
+Figure 4: Two families of balls weakly tangent to the set $\mathcal { S } \subset \mathbb { R } ^ { 2 }$ (in blue). Each family contains a unique maximal empty ball (in purple). Notice that the centre of the maximal empty ball weakly tangent at the point $p _ { 1 }$ lies at the medial axis $\operatorname { a x } ( S )$ , while the centre of the maximal empty ball weakly tangent at the point $p _ { 2 }$ only lies at its closure, $\overline { { \operatorname { a x } ( \mathcal { S } ) } }$ .
+Lemma 3.3 essentially tells us that a family of weakly tangent balls 166 $\{ B ( p + \lambda v , \lambda | v | ) \} _ { \lambda \geq 0 }$ contains 167 at most one which is maximal with respect to inclusion among those whose interior is disjoint from 168 the set $s$ . Two such families are illustrated in Figure 4.
+169 We call such balls maximal empty. For the purpose of this article, we define maximal empty balls in
+170 terms of unit back projection vectors (Definition 2.4). To see that each maximal empty ball is indeed
+171 weakly tangent, we emphasise:
+72 Lemma 3.4 If $( p , v ) \in \operatorname { B P } ( S )$ , then $( p , v ) \in \operatorname { N o r } ( S )$ . That is, $\mathrm { B P } ( S ) \subseteq \mathrm { N o r } ( S )$ . In particular, for
+3 any pair $( p , u ) \in \operatorname { U B P } ( S )$ and radius $\lambda \geq 0$ , the ball $B ( p + \lambda u , \lambda )$ is weakly tangent to $s$ .
+174 Remark 3.5 For general closed sets, the converse of Lemma 3.4, that is, $\mathrm { N o r } ( S ) \subseteq \mathrm { B P } ( S )$ , is not
+175 true. One counter-example is the graph of the function $x \mapsto | x | ^ { 3 / 2 }$ at the origin. However, the
+176 inclusion $\operatorname { N o r } ( S ) \subseteq \operatorname { B P } ( S )$ holds for sets of positive reach, thanks to Theorem 4.8 (12) of [19]
+177 (recalled in the supplementary material).
+178 Definition 3.6 (Maximal empty weakly tangent ball) Let $( p , u ) \in \operatorname { U B P } ( S )$ . A weakly tangent
+179 ball $B ( p + \lambda u , \lambda )$ is called maximal empty to $\begin{array} { r } { S i f \lambda = d ( p , u , \pi _ { S } ) } \end{array}$ , or, equivalently, if $\pi _ { \mathrm { a x } , S } ( p , u ) =$
+180 $p + \lambda u$ .
+181 (Maximal empty) weakly tangent balls satisfy the following properties. Let $( p , u ) \in \operatorname { U B P } ( S )$
+• For any radius $0 < \lambda \leq d ( p , u , \pi _ { S } )$ , the interior of the ball $B ( p + \lambda u , \lambda )$ is disjoint from the set $s$ . This follows directly from Definition 3.6 and Lemma 3.3.
+• The centres of maximal empty weakly tangent balls lie on the closure of the medial axis of $s$ . This is due to Lemma 2.3 and the definition of the map $\pi _ { \mathrm { a x } , S }$ (equation (1)).
+186 The following lemma moreover tells us, that each point on the medial axis is a centre of a maximal
+187 empty weakly tangent ball.
+188 Lemma 3.7 (Surjectivity on $\mathrm { a x } ( S ) )$ ) For any point $x \in \operatorname { a x } ( S )$ and $p \in \pi _ { S } ( x )$ , there exists a vector
+189 $u \in \mathrm { U B P } ( p , S )$ such that $\pi _ { \mathrm { a x } , S } ( p , u ) = x$ . In other words, $B ( x , | x - p | )$ is a maximally empty
+190 weakly tangent ball. Moreover, we have that
+$$
+\operatorname { a x } ( S ) \subseteq \pi _ { \mathrm { a x } , S } \left( \operatorname { U B P } ( S ) \right) \subseteq { \overline { { \operatorname { a x } ( S ) } } } .
+$$
+191 We are now almost ready to state our main theorem. Before phrasing the result, we walk the reader through the assumptions and fix the notation on the way. The assumptions are illustrated in Figure 5.
+![](images/2055983f7c713af756043cf2118c67cee8e32cd1c949a32f02c78f5a9f80220c.jpg)
+Figure 5: The setting of Theorem 3.9.
+# 193 Assumption 3.8
+• We assume that the set $s$ has a bounding sphere of radius $r$ , which we denote by $S ( r )$ .
+• We consider a $C ^ { 1 }$ diffeomorphism $F : \mathbb { R } ^ { d }  \mathbb { R } ^ { d }$ such that the Lipschitz constants of $F$ and $F ^ { - 1 }$ are bounded by $L _ { F }$ , and the Lipschitz constants of the differentials $D F$ and $D F ^ { - 1 }$ are bounded by $L _ { D F }$ . We call such a diffeomorphism a $\dot { C } ^ { 1 , 1 }$ diffeomorphism.
+• We further assume that the map $F$ leaves the bounding sphere $S ( r )$ invariant, that is, $F ( \overbar { S } ( r ) ) = S ( r )$ .
+• We pick a point $c \in \operatorname { a x } ( S )$ , a point $p \in \pi _ { S } ( c )$ , and write $\rho = | c - p |$ . Observe that since $S \cap \operatorname { a x } ( S ) = \emptyset$ , $\rho$ is positive. By Lemma 3.7, the ball $B ( c , \rho )$ is a maximal empty weakly tangent ball to $s$ at $p$ . Moreover, we define $\begin{array} { r } { u = \frac { c - p } { | c - p | } } \end{array}$ and note that $u \in \mathrm { U B P } ( p , S )$ .
+• We denote the tangent hyperplane to the boundary sphere of $B ( c , \rho )$ at $p$ by $p + T$ . The hyperplane $T$ is the orthocomplement of the vector $u$ : $T = u ^ { \perp }$ .
+• We work with the unit vector at $F ( p )$ that points inside the image of the ball $B ( c , \rho )$ and is orthogonal to the hyperplane $D _ { p } F ( T )$ . We denote this vector by $u ^ { \prime }$ .
+207 Theorem 3.9 Under the above assumptions, there exists a maximal empty weakly tangent ball
+208 $B ( c ^ { \prime } , \rho ^ { \prime } )$ to the set $F ( S )$ , whose boundary sphere has an internal normal $u ^ { \prime }$ . In particular, the
+209 ball $B ( c ^ { \prime } , \rho ^ { \prime } )$ is tangent to the affine hyperplane $F ( p ) + D _ { p } F ( T )$ . Its radius $\rho ^ { \prime }$ is bounded by
+210 $\begin{array} { r } { \rho ^ { \prime } \in \bigg [ \frac { \rho } { ( L _ { F } ) ^ { 3 } + \rho L _ { D F } ( L _ { F } ) ^ { 2 } } , \frac { ( L _ { F } ) ^ { 3 } \rho } { 1 - \rho L _ { D F } ( L _ { F } ) ^ { 2 } } } \end{array}$ . Assume, moreover, that the distortions of both $F$ and $D F$
+211 are bounded, that is, for all $x \in \mathbb { R } ^ { d }$ ,
+$$
+| F ( x ) - x | \leq \varepsilon _ { 1 } , \qquad \| D F _ { x } - \mathrm { I d } \| \leq \varepsilon _ { 2 } < 1 ,
+$$
+and 212 $r \cdot L _ { D F } ( L _ { F } ) ^ { 2 } \leq 1 / 2$ . Define
+$$
+\begin{array} { l } { { C _ { L } ( r , L _ { F } , L _ { D F } , \varepsilon _ { 1 } , \varepsilon _ { 2 } ) = } } \\ { { 2 r \sqrt { 1 + ( L _ { F } ) ^ { 6 } \left( 1 + 4 r L _ { D F } ( L _ { F } ) ^ { 2 } \right) ^ { 2 } - 2 ( L _ { F } ) ^ { 3 } \left( 1 + 4 r L _ { D F } ( L _ { F } ) ^ { 2 } \right) \sqrt { 1 - ( \varepsilon _ { 2 } ) ^ { 2 } } } + \varepsilon _ { 1 } } } \end{array}
+$$
+213 then the map $\pi _ { \mathrm { a x } , S }$ satisfies
+$$
+| \pi _ { \mathrm { a x } , \mathcal { S } } ( p , u ) - \pi _ { \mathrm { a x } , F ( \mathcal { S } ) } ( F ( p ) , u ^ { \prime } ) | \leq C _ { L } ( r , L _ { F } , L _ { D F } , \varepsilon _ { 1 } , \varepsilon _ { 2 } ) .
+$$
+$$
+d _ { H } ( \mathrm { a x } ( S ) , \mathrm { a x } ( F ( S ) ) ) \leq C _ { L } ( r , L _ { F } , L _ { D F } , \varepsilon _ { 1 } , \varepsilon _ { 2 } ) .
+$$
+215 The bound $| F ( x ) - x | \leq \varepsilon _ { 1 }$ is really necessary, because we want our theorem to accommodate for
+216 rotations and translations, which rotate and translate the medial axis without changing distances
+217 and hence have Lipschitz constant 1. We further stress that if the diffeomorphism $F$ is close to the
+218 identity, its Lipschitz constant satisfies $L _ { F } \geq 1$ , because by assumption $F$ leaves the bounding sphere
+219 $S ( r )$ invariant, and $L _ { D F }$ is close to zero.
+220 Sketch of the proof of Theorem 3.9 The idea of the proof is depicted in Figure 5. Thanks to Federer’s
+221 result (Theorem 2.6), we know that the reach of the maximal empty weakly tangent ball $B ( c , \rho )$
+222 does not change too much under the ambient diffeomorphism $F$ . This gives a lower bound on the
+223 radius of every maximal empty weakly tangent ball of the image of this ball — the set $F ( B ( c , \rho ) )$ .
+224 We show that in the interior of $F ( B ( { \dot { c } } , \rho ) )$ , the radii of the maximal empty weakly tangent balls
+225 of $F ( B ( c , \rho ) )$ are close to $\rho$ . One of these balls is also empty weakly tangent to $F ( S )$ at $F ( p )$ ,
+226 though not necessarily maximal. We denote its centre by $c ^ { \prime }$ . Since we can apply the same argument
+227 for the map $F ^ { - 1 }$ , we find an upper and lower bound on the radius of the maximal weakly tangent
+228 ball $B ( c ^ { \prime } , \rho ^ { \prime } )$ of $F ( S )$ at $F ( p )$ that is also weakly tangent to $F ( B ( c , \rho ) )$ , or equivalently tangent to
+229 $D _ { p } F ( T )$ .
+230 While this bound on the difference of the radii is essentially a bound on the distance $\left| \left| c - p \right| - \left| c ^ { \prime } - \right| \right|$
+231 $F ( p ) | |$ between the points $c - p$ and $c ^ { \prime } - F ( p )$ , the bound $\varepsilon _ { 2 }$ on $\| D F - 1 \|$ allows one to bound the
+232 angle between the vectors $c - p$ and $c ^ { \prime } - F ( p )$ . With the assumption (2) we can then derive a bound
+233 the distance between the points $c$ and $c ^ { \prime }$ . Finally, thanks to [19, Theorem 4.8 (6)] (Lemma 2.3) this
+234 induces a bound on the Hausdorff distance between the (closure of the) two medial axes $\operatorname { a x } ( S )$ and
+235 $\operatorname { a x } ( F ( S ) )$ . □
+It was a surprise to the authors that no assumption on the set (apart from closedness) needed to be made, and that the techniques used were that simple and well established; they go back to Federer [19]. In fact, the authors at first envisioned a far more elaborate argument assuming the set had positive $\mu$ -reach [11].
+# 4 Quantifying 240 $C ^ { 1 , 1 }$ diffeomorphisms as deviations from identity
+241 In this section we reformulate the main result in terms of norms on Banach spaces. This reformulation
+242 offers a more theoretical insight, and we believe the reformulated bounds are easier to work with in
+243 certain applications. Indeed, in the context of practical numerical computations, a bound on the
+244 Lipschitz constant of an operator — or, at least, a modulus of continuity — allows to control the
+245 condition number. This control is particularly useful when we calculate with objects such as the
+246 medial axis,whose (numerical) stability is often problematic in practice.
+As we will see below, for this reformulation we somewhat strengthen our assumptions.
+248 We decompose a diffeomorphism $F$ into the identity map $\mathbb { 1 } _ { \mathbb { R } ^ { d } }$ on $\mathbb { R } ^ { d }$ , and a displacement field $\varphi$
+249 $F = \mathbb { 1 } _ { \mathbb { R } ^ { d } } + \varphi$ . For the choice of the displacement field, we restrict ourselves to the vector space $\mathcal { U }$ of
+250 all $C ^ { 1 , 1 }$ maps $\varphi$ from $\mathbb { R } ^ { d }$ to $\mathbb { R } ^ { d }$ whose restriction to the exterior $\mathbb { R } ^ { d } \backslash B ( r )$ of a certain bounding ball
+251 $B ( r )$ equals 0.3
+52 A natural norm associated to $\mathcal { U }$ is one that makes it a Banach space. A typical choice, inherited from general Banach spaces of 53 $C ^ { 1 , 1 }$ functions, would be for example, for $\varphi \in { \mathcal { U } }$ ,
+$$
+\| \varphi \| _ { C ^ { 1 , 1 } } = \operatorname* { m a x } \left( \| \varphi \| _ { \infty } , \| D \varphi \| _ { \infty } , \operatorname { L i p } ( D \varphi ) \right) .
+$$
+54 Here we used the following notation:
+• $\begin{array} { r } { \| \varphi \| _ { \infty } = \operatorname* { s u p } _ { x \in \mathbb { R } ^ { d } } | \varphi ( x ) | } \end{array}$ denotes the sup norm on $x \mapsto | \varphi ( x ) |$ , where $| \cdot |$ is the Euclidean norm in $\mathbb { R } ^ { d }$ ,
+• $\| D \varphi \| _ { \infty } = \operatorname* { s u p } _ { x \in \mathbb { R } ^ { d } } \| D \varphi ( x ) \|$ denotes the sup norm on $x \mapsto \| D \varphi ( x ) \|$ , where $\| D \varphi ( x ) \|$ is the operator norm induced by the Euclidean norm on $\mathbb { R } ^ { d }$ .
+• We write $\mathrm { L i p } ( D \varphi )$ for the Lipschitz semi-norm of $D \varphi$ . The Lipschitz semi-norms of $\varphi$ and $D \varphi$ are defined as
+and
+$$
+\begin{array} { c } { \displaystyle \mathrm { L i p } ( \varphi ) = \displaystyle \operatorname* { s u p } _ { \substack { x , y \in \mathbb { R } ^ { d } , x \neq y } } \frac { \vert \varphi ( y ) - \varphi ( x ) \vert } { \vert y - x \vert } , } \\ { \displaystyle \mathrm { L i p } ( D \varphi ) = \displaystyle \operatorname* { s u p } _ { \substack { x , y \in \mathbb { R } ^ { d } , x \neq y } } \frac { \Vert D \varphi ( y ) - D \varphi ( x ) \Vert } { \vert y - x \vert } . } \end{array}
+$$
+262 The norm defined in (4) makes $\mathcal { U }$ into a Banach space, since every Cauchy sequence in $\mathcal { U }$ has a limit
+263 in $\mathcal { U }$ . In addition, any function $\varphi \in { \mathcal { U } }$ satisfies:
+$$
+\begin{array} { r l } & { \mathrm { L i p } ( \varphi ) = \| D \varphi \| _ { \infty } , } \\ & { \| D \varphi \| _ { \infty } \leq r \mathrm { L i p } ( D \varphi ) , } \\ & { \| \varphi \| _ { \infty } \leq r \mathrm { L i p } ( \varphi ) \leq r ^ { 2 } \mathrm { L i p } ( D \varphi ) , } \end{array}
+$$
+since the restriction of $\varphi$ to $\mathbb { R } ^ { d } \setminus B ( r )$ is 0. This in turn yields that $\mathrm { L i p } ( D \varphi ) \ \leq \ \| \varphi \| _ { C ^ { 1 , 1 } } \ \leq$ $\operatorname* { m a x } ( 1 , r , r ^ { 2 } ) \operatorname { L i p } ( D \varphi )$ . Thus, in $\mathcal { U }$ , the norm $\varphi \mapsto \operatorname { L i p } ( D \varphi )$ is equivalent to the norm $\varphi \mapsto \| \varphi \| _ { C ^ { 1 , 1 } }$ .
+We can now state slightly less general version of Theorem 3.9 in terms of the Banach space $( \mathcal { U } , \varphi \mapsto$ $\mathrm { L i p } ( D \varphi )$ ) .
+Theorem 4.1 Let $S \subseteq \mathbb { R } ^ { d }$ be bounded by the ball $B ( r )$ of radius $r > 0$ , such that $S ( \boldsymbol { r } ) = \partial B ( \boldsymbol { r } ) \subseteq$ $\mathcal { S } _ { \mathbf { \Omega } }$ . Let further $F$ be a $C ^ { 1 , 1 }$ diffeomorphism from $\mathbb { R } ^ { d }$ to itself that leaves the set $\mathbb { R } ^ { d } \backslash B ( r )$ invariant, and define two displacement fields $\varphi , \tilde { \varphi } \in \mathcal { U }$ such that $F = \mathbb { 1 } _ { \mathbb { R } ^ { d } } + \varphi$ and
+$$
+\left( \mathbb { 1 } _ { \mathbb { R } ^ { d } } + \tilde { \varphi } \right) \circ \left( \mathbb { 1 } _ { \mathbb { R } ^ { d } } + \varphi \right) = \mathbb { 1 } _ { \mathbb { R } ^ { d } } .
+$$
+271 Define $\varepsilon = \operatorname* { m a x } \left( \mathrm { L i p } ( D \varphi ) , \mathrm { L i p } ( D \tilde { \varphi } ) \right)$ .
+272 If $r \varepsilon ~ \leq ~ 1 / 4$ , the Hausdorff distance between the medial axes of the set √ $s$ and its image 273 $F ( S )$ is bounded by $d _ { H } ( \operatorname { a x } ( S ) , \operatorname { a x } ( F ( S ) ) ) ~ \leq ~ \left( 1 + { \sqrt { 5 0 } } \right) r ^ { 2 } \varepsilon + { \mathcal { O } } \left( r ^ { 3 } \varepsilon ^ { 2 } \right)$ . In particular, 274 $d _ { H } ( \operatorname { a x } ( S ) , \operatorname { a x } ( F ( S ) ) ) = { \mathcal { O } } \left( r ^ { 2 } \varepsilon \right)$ .
+275 Sketch of the proof Essentially, the proof consists of rewriting Theorem 3.9 in terms of the language
+76 developed in this section. □
+277 Remark 4.2 Observe that the bound $\mathcal { O } \left( r ^ { 2 } \varepsilon \right)$ is consistent with a scaling by factor $\lambda \colon S \mapsto \lambda S$
+278 $F ( \cdot ) \mapsto \lambda F ( \cdot / \lambda )$ . Under such a scaling, the radius $r$ is multiplied by $\lambda$ , while the Lipschitz
+279 constant $\mathrm { L i p } ( D \varphi )$ — and therefore $\varepsilon$ — is divided by $\lambda$ . Furthermore, the Hausdorff distance
+280 $d _ { H } ( \operatorname { a x } ( S ) , \operatorname { a x } ( F ( S ) ) )$ increases by a factor $\lambda .$ . By considering a diffeomorphism that translates the
+281 set $ { \boldsymbol { S } } \setminus  { \boldsymbol { S } } (  { \boldsymbol { r } } )$ while keeping the bounding sphere $S ( r )$ fixed, we see that this bound is asymptotically
+282 optimal.
+# 5 Conclusion and future work
+We proved the Hausdorff stability of the medial axis of a closed set without any further assumption on it (as explained in Remark 2.1, the existence of the bounding sphere serves to formulate the main result in a clean way).
+With regard to applications, our result is the first step towards providing a provably correct image recognition in particular in the context of astrophysics. The next step is to produce physics-informed models for the medial axis as occurring in astronomical data.
+On the mathematical side, we conclude with a conjecture generalizing our result. We believe that our result generalizes to compact Riemannian manifolds with bounded curvature.
+Conjecture 5.1 Let $\mathcal { M }$ be a compact Riemannian manifold with bounded sectional curvature4 and S a closed subset of M. Then the medial axis (also called cut locus [26]) of $s$ in $\mathcal { M }$ is Lipschitz stable under diffeomorphisms of $\mathcal { M }$ .
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+82 [38] Martijn van Manen. Maxwell strata and caustics. In Singularities In Geometry And Topology, pages 787–824. World Scientific, 2007.
+384 [39] C. T. C. Wall. Geometric properties of generic differentiable manifolds. In Jacob Palis and
+385 Manfredo do Carmo, editors, Geometry and Topology, pages 707–774, Berlin, Heidelberg, 1977.
+386 Springer Berlin Heidelberg.
+387 [40] Franz-Erich Wolter. Cut locus and medial axis in global shape interrogation and representation.
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+390 medial axes of 3d shapes. ACM Transactions on Graphics, 35(4):38:1–38:12, July 2016.

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+    {
+        "type": "text",
+        "text": "The medial axis of closed bounded sets is Lipschitz stable with respect to the Hausdorff distance under ambient diffeomorphisms ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "1 We prove that the medial axis of closed sets is Hausdorff stable in the following   \n2 sense: Let $S \\subseteq \\mathbb { R } ^ { d }$ be a fixed closed set that contains a bounding sphere. Consider   \n3 the space of $C ^ { 1 , 1 }$ diffeomorphisms of $\\mathbb { R } ^ { d }$ to itself, which keep the bounding   \n4 sphere invariant. The map from this space of diffeomorphisms (endowed with a   \n5 Banach norm) to the space of closed subsets of $\\mathbb { R } ^ { d }$ (endowed with the Hausdorff   \n6 distance), mapping a diffeomorphism $F$ to the closure of the medial axis of $F ( S )$ ,   \n7 is Lipschitz. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This extends a previous stability result of Chazal and Soufflet on the stability of the medial axis of $C ^ { 2 }$ manifolds under $C ^ { 2 }$ ambient diffeomorphisms. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "10 1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+            525,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "11 In [19], Federer introduced the reach of a (closed) set $S \\subset \\mathbb { R } ^ { d }$ as the infimum over all points in $s$ of   \n12 the distance from these points to the medial axis $\\operatorname { a x } ( S )$ , the set of points in $\\mathbb { R } ^ { d }$ for which the closest   \n13 point in $s$ is not unique. Federer also introduced the reach at a point $p \\in S$ to be the distance from $p$   \n14 to the medial axis of $s$ . We now call this quantity the local feature size [3] and denote it by $\\operatorname { l f s } ( p )$ .   \n15 Federer proved that the reach is stable under $C ^ { 1 , 1 }$ diffeomorphisms of the ambient space. Here, a   \n16 $C ^ { 1 , 1 }$ map is a $C ^ { 1 }$ map whose derivative is Lipschitz, and a $C ^ { \\hat { 1 } , 1 }$ diffeomorphism is a $\\mathbf { \\bar { \\it C } } ^ { 1 , 1 }$ bijective   \n17 map whose inverse is also $C ^ { 1 , 1 }$ . Chazal and Soufflet [13] proved that the medial axis is stable with   \n18 respect to the Hausdorff distance under ambient diffeomorphisms, but under stronger assumptions   \n19 than the work of Federer, namely assuming that $s$ is a $C ^ { 2 }$ manifold and the distortion is a $C ^ { 2 }$ diffeo  \n20 morphism of the ambient space. Chazal and Soufflet based their work on earlier results by Blaschke   \n21 [9], which were not as strong as Federer’s.   \n22 In this paper we extend the stability result of the medial axis. More concretely, we generalize the   \n23 result of Chazal and Soufflet [13] to arbitrary closed sets and $C ^ { 1 , 1 }$ diffeomorphisms of the ambient   \n24 space; we show that the Hausdorff distance between the medial axes of the closed set and its image is   \n25 bounded in terms of Lipschitz constants stemming from the diffeomorphism of the ambient space.   \n26 Our result follows from the work of Federer [19] and in fact shortens the proof in [13] significantly. ",
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+    {
+        "type": "text",
+        "text": "Our bounds on the Hausdorff distance say nothing about the topology of the medial axis, which is known to be highly unstable (see e.g. [5]), although it preserves the homotopy type (see [28]). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Contribution and related work Our work differs from the majority of the literature in three essential ways: ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Firstly, we make no assumptions on the set we consider apart from that it is closed. The stability of the medial axis of (piecewise) smooth manifolds has been the object of intense study, see for example ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "33 [13, 15–17, 24, 30, 37–40]. However, the manifold assumption is impossible to achieve in many   \n34 applications — such as in the context of astrophysics, one of the main motivations of this paper.   \n35 Secondly, we achieve stability without pruning the medial axis. This contrasts with a large body of   \n36 work, such as [6, 12, 16, 29]. Not having to prune the medial axis is a significant advantage. On the   \n37 downside, we limit the changes of the considered set to those induced by ambient diffeomorphisms.   \n38 Nevertheless, given the standard examples of the instability of the medial axis — see for example [5]   \n39 — we believe these limitations are near to the weakest assumptions necessary for Hausdorff stability.   \n40 Within the context of ambient homeomorphisms, the results we obtain are close to optimal, as we   \n41 specify in Remark 4.2. ",
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+    {
+        "type": "text",
+        "text": "Thirdly, our results hold for sets in arbitrary dimensions and are not sensitive to the dimension of the set itself. A large part of the related work only investigates sets of low dimensions or codimension one manifolds, although there are some notable exceptions such as [39], see also [17], and [12, 29]. ",
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+        "text": "45 Motivation The medial axis has many real world applications — among others, in robot motion   \n46 planning [27], triangulation algorithms [4], graphics [35], and shape recognition, segmentation, and   \n47 learning [10, 18, 25, 33, 41]. See also the overviews [32, 35]. The reach — the distance between a   \n48 set and its medial axis — is a central concept in manifold learning [1, 2, 20–22, 34].   \n49 The motivation of this paper is twofold: Firstly, we tackle the following challenge from the processing   \n50 of images collected with optical devices which use lenses — such as cameras or telescopes. A shape   \n51 extracted from such an image may be imprecise due to the imperfection of the lenses. Our result   \n52 implies that the medial axis of such a shape is stable under these imperfections. As a consequence,   \n53 the outcome of any shape recognition or shape segmentation algorithm based on the medial axis will   \n54 be stable.   \n55 In addition to the disciplines listed above, such stability is sought after in astrophysics, in particular   \n56 for shape analysis and automated shape identification in observational astronomy. Observational   \n57 astronomers are interested in reconstructing objects like stars or galaxies, and their place in the   \n58 universe from data gathered by telescopes. They can deduce the distance from the object to the   \n59 observer thanks to so-called standard candles or red shift [14, 23, 31]. However, the image gets   \n60 distorted due to optical effects — either through gravitational lensing ([7]) or lensing inside the   \n61 telescope itself ([36]).   \n62 Such a distortion can be modeled as a diffeomorphism of the ambient space. At the same time, this   \n63 problem cannot be tackled using the result by Chazal and Soufflet [13], since the observed objects   \n64 might not be smooth — for example due to interactions with shock waves or jets. In addition, with   \n65 our method astrophysicists can not only reconstruct objects in space (3D), but also in spacetime (4D).   \n66 The second motivation is more formal in nature: The stability of the medial axis is instrumental in   \n67 establishing its computability. Indeed, when proving properties of algorithms based on the medial   \n68 axis, authors generally assume the real RAM model.1 However, as was recently argued in [29], the   \n69 medial axis needs to be stable in order to be computable in more realistic models of computation.   \n70 There is a more practical component to this formal question: It is not a priori clear if using possibly   \n71 noisy real world data or the output of other computer programs as input for these algorithms yields   \n72 answers that are close to the ground truth. To be able to prove that the output is correct, we need   \n73 (numerical) stability of the medial axis.   \n74 Outline After revisiting preliminaries and known results in Section 2, we state the main stability   \n75 result in Section 3. In Section 4 we reformulate this result in terms of norms on Banach spaces. This   \n76 also exhibits the fact that the stability of the medial axis is Lipschitz in the following sense: We think   \n77 of the set $s$ as fixed and consider the map from the space of diffeomorphisms (endowed with a norm   \n78 which makes it a Banach space) to the space of closed subsets of $\\mathbb { R } ^ { d }$ (endowed with the Hausdorff   \n79 distance), mapping each diffeomorphism $F : \\mathbb { R } ^ { d }  \\mathbb { R } ^ { d }$ to the closure of the medial axis of $F ( S )$   \n80 The Lipschitz constant then only depends on the diameter of the bounding sphere of the set $s$ .   \n81 We only include proof sketches of the two main theorems in this article. The full proofs of the   \n82 theorems and of the supporting lemmas, can be found in the supplementary material. ",
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+    {
+        "type": "text",
+        "text": "83 2 Preliminaries: Sets of positive reach and the closest point projection ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "84 In this section we recall some definitions and results concerning the medial axis and sets of positive   \n85 reach. Essentially, we need three ingredients from the literature to prove our main theorem: the   \n86 notions related to the closest point projection, the properties of the generalized normal and tangent   \n87 spaces, and Federer’s result on the stability of the reach under ambient diffeomorphisms.   \n88 We write $d ( \\cdot , \\cdot )$ for the Euclidean distance between two points, and the distance between a point and   \n89 a set. That is, for any closed set $s$ and point $p$ , ",
+        "bbox": [
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+        "img_path": "images/4bbcbdbf5c2c5b97a3db260b1b8832178a506a7a9aca06b59aea496b2c2d38b0.jpg",
+        "text": "$$\nd ( p , S ) = \\operatorname* { i n f } _ { q \\in S } d ( p , q ) .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "We denote the Hausdorff distance between two sets 90 $A , B \\subseteq \\mathbb { R } ^ { d }$ by $d _ { H } ( A , B )$ : ",
+        "bbox": [
+            145,
+            295,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/61310b9413c347bcc66560ee6b17466c8ffe10009f2dc1b44ec7209d8bad274f.jpg",
+        "text": "$$\nd _ { H } ( A , B ) = \\operatorname* { m a x } \\left\\{ \\operatorname* { s u p } _ { a \\in A } d ( a , B ) , \\operatorname* { s u p } _ { b \\in B } d ( b , A ) \\right\\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "91 We write $B ( c , r )$ , resp. $S ( c , r )$ , to denote balls, resp. spheres, with centre $c$ and radius $r$ . Lastly, $\\left. \\cdot \\right.$   \n92 denotes the Euclidean norm, and $\\lVert \\cdot \\rVert$ an operator norm.   \n93 The closest point projection and related notions The projection of points in the ambient space   \n94 $\\mathbb { R } ^ { d }$ to the (set of) closest point(s) of the set $S \\subseteq \\mathbb { R } ^ { d }$ is denoted by $\\pi _ { \\boldsymbol { S } }$ , and illustrated in Figure 1. ",
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+        "img_path": "images/3f0be235ed8367b4a9102442acbd51f72ab9bd9e67bf655c834fcf17a00c472b.jpg",
+        "image_caption": [
+            "Figure 1: The closest point projection to the set $s$ of four points in $\\mathbb { R } ^ { 2 }$ . When a point lies on the medial axis $\\operatorname { a x } ( S )$ , the closest point projection consists of more points. "
+        ],
+        "image_footnote": [],
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+    },
+    {
+        "type": "text",
+        "text": "The medial axis of 95 $s$ is the set of all points $p \\in \\mathbb { R } ^ { d }$ where the set $\\pi _ { S } ( p )$ consists of more than one 96 point: ",
+        "bbox": [
+            151,
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+    },
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+        "type": "equation",
+        "img_path": "images/001b3cfc96e2da7772f0e3b41db3600208c56b245698f4a16f228dbf46a10649.jpg",
+        "text": "$$\n\\operatorname { a x } ( \\mathcal { S } ) = \\left\\{ p \\in \\mathbb { R } ^ { d } \\mid \\# \\pi _ { \\mathcal { S } } ( p ) > 1 \\right\\} .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "97 Here, $\\# \\pi _ { S } ( p )$ denotes the cardinality of the set $\\pi _ { S } ( p )$ . ",
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+    },
+    {
+        "type": "text",
+        "text": "For a point $p \\in { \\mathcal { S } }$ , the local feature size of $p$ is the distance from $p$ to the medial axis of the set $s$ ",
+        "bbox": [
+            160,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d1575bdd76a887eff443e5e00fa9f70c75acf36b6a7cd4010fd2496544bd0db6.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathrm { l f s } ( p ) = d ( p , \\mathrm { a x } ( S ) ) . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "99 Finally, the reach of the set $s$ is the infimum of the local feature size over all its points: ",
+        "bbox": [
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+        "img_path": "images/4a284227f1e7b8b10a3d36cffcb59483cacfc0b1c2c716ba2819bc5b8c0bad54.jpg",
+        "text": "$$\n\\operatorname { r c h } ( S ) = \\operatorname* { i n f } _ { p \\in S } \\operatorname { l f s } ( p ) = \\operatorname* { i n f } _ { p \\in S } d ( p , \\operatorname { a x } ( S ) ) .\n$$",
+        "text_format": "latex",
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+        "text": "100 Throughout this paper we assume that $S \\subseteq \\mathbb { R } ^ { d }$ is a closed set. We shall further assume that the set   \n101 $s$ as well as its medial axis are bounded, and that the bounding sphere of $s$ is contained in $s$ itself.   \n102 More specifically, we assume that there exists a closed ball $B$ of positive radius such that ${ \\mathcal { S } } \\subseteq B$ ,   \n103 and $\\partial B \\subseteq S$ . We call $\\partial B$ the bounding sphere of $s$ .   \n104 The addition of the bounding sphere $\\partial B$ to the set $s$ is necessary to obtain the desired bound on   \n105 the Hausdorff distance between the two medial axes of the set $s$ and its image under the ambient   \n106 diffeomorphism. Indeed, consider the following example, illustrated in Figure 2.   \n107 Let the set $s$ consist of two points in the plane, $S = \\{ p , q \\} \\subseteq \\mathbb { R } ^ { 2 }$ . The medial axis of $s$ is then the   \n108 bisector line of $p$ and $q$ . After a generic perturbation $F$ of $p$ and $q$ — that is, not a translation and not a   \n109 perturbation in the direction $\\pm ( p - q )$ — the bisector line $\\operatorname { a x } ( F ( S ) )$ of the perturbed points intersects   \n110 the bisector $\\operatorname { a x } ( S )$ of the original pair. The Hausdorff distance between these two non-parallel lines   \n111 is infinite, and thus unboundable.   \n112 At the same time, the addition of the bounding sphere $\\partial B$ to the considered set $s$ is not a restriction.   \n113 Indeed,   \n14 Remark 2.1 The medial axes of $s$ and $s \\backslash \\partial B$ coincide in the interior of the ball $B$ sufficiently far   \n15 away from its boundary $\\partial B$ . More precisely: ",
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+        "image_caption": [
+            "Figure 2: In black the set $s$ and its medial axis, in light blue the perturbed set and its medial axis. The Hausdorff distance between $\\operatorname { a x } ( S )$ and $\\operatorname { a x } ( F ( S ) )$ is infinite. "
+        ],
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+        "text": "• Any point $x \\in \\operatorname { a x } ( S )$ , such that $\\pi _ { S } ( x ) \\cap \\partial B = \\emptyset .$ , lies on the medial axis $\\operatorname { a x } ( S \\setminus \\partial B )$ . • Conversely, if a point $x$ lies on the medial axis $\\operatorname { a x } ( S \\setminus \\partial B )$ , and any (and thus every) point $q \\in \\pi _ { S \\setminus { \\partial B } } ( x )$ satisfies $d ( x , q ) < d ( x , \\partial B )$ , then $x \\in \\operatorname { a x } ( S )$ . ",
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+    {
+        "type": "text",
+        "text": "Thus, the medial axis is locally stable if the ambient diffeomorphism is close to the identity.219 ",
+        "bbox": [
+            151,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "20 A recurring strategy in this article is to start at a point $p$ on the set $s$ , move away from this point in a   \n21 ‘normal’ direction, and see if by projecting using the closest point projection $\\pi _ { \\boldsymbol { S } }$ we get back to $p$ . To   \n22 this end, we define the projection range. ",
+        "bbox": [
+            156,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Definition 2.2 (Projection range) Let $p \\in S$ be a point and $v \\in \\mathbb { R } ^ { d } a$ vector. The projection range $d ( p , v , \\pi _ { S } )$ in direction v is the maximal distance one can travel from $p$ along $v$ such that the closest point projection yields only the point $p$ : ",
+        "bbox": [
+            165,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/77284a9f9920189e6dee210b3465c601bdf9194bc2a8ccc4221a343978f4e701.jpg",
+        "text": "$$\nd ( p , v , \\pi _ { \\mathcal { S } } ) = \\operatorname* { s u p } \\{ \\lambda \\in \\mathbb { R } \\mid \\pi _ { \\mathcal { S } } ( p + \\lambda v ) = \\{ p \\} \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            627,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Since $\\pi _ { S } ( p ) = \\{ p \\}$ , the projection range is canonically non-negative. Furthermore, the directions for which the range is positive are key to our study, because of the following property: ",
+        "bbox": [
+            160,
+            654,
+            825,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Lemma 2.3 (Theorem 4.8 (6) of [19]) Consider a point $p \\in S$ and a vector $v \\in \\mathbb { R } ^ { d }$ . I f ",
+        "bbox": [
+            169,
+            691,
+            750,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e3675cc67a239b9263602ba6d2d2df1c35b494fe1f1a156046d6f2d4bbc161ca.jpg",
+        "text": "$$\n0 < d ( p , v , \\pi _ { S } ) < \\infty ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            424,
+            710,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "then $p + d ( p , v , \\pi _ { S } ) \\cdot v \\in \\overline { { \\operatorname { a x } ( S ) } }$ . ",
+        "bbox": [
+            176,
+            732,
+            392,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We call these special directions $v$ back projection vectors: ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Definition 2.4 (Unit back projection vectors) For a point $p \\in { \\mathcal { S } }$ , $\\mathrm { U B P } ( p , S )$ is the set of unit vectors with a positive projection range: ",
+        "bbox": [
+            166,
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/43399e16677a50b5b7f6c789225895b76863f727ed52cb3f8b4f35c2750e7b3e.jpg",
+        "text": "$$\n\\mathrm { U B P } ( p , S ) = \\left\\{ u \\in \\mathbb { R } ^ { d } \\mid | u | = 1 a n d 0 < d ( p , u , \\pi _ { S } ) < \\infty \\right\\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            294,
+            814,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "133 We further define ",
+        "bbox": [
+            147,
+            837,
+            287,
+            851
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/83764343e8fba12b06e01e660cb5c1a47dfcbd75ec8a6bd09e7c53f4f8e26eb8.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathrm { U B P } ( S ) = \\left\\{ ( p , u ) \\in S \\times \\mathbb { R } ^ { d } \\ : \\middle | \\ : u \\in \\mathrm { U B P } ( p , S ) \\right\\} , } \\\\ & { \\quad \\mathrm { B P } ( S ) = \\left\\{ ( p , \\lambda u ) \\in S \\times \\mathbb { R } ^ { d } \\ : \\middle | \\ : ( p , u ) \\in \\mathrm { U B P } ( S ) , \\lambda \\geq 0 \\right\\} . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2The bounding sphere does allow one to give a relatively clean mathematical statement, see Section 4. ",
+        "bbox": [
+            186,
+            897,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "134 Thanks to Lemma 2.3, the following map is well-defined: ",
+        "bbox": [
+            142,
+            90,
+            552,
+            106
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a7b203475ad037e0d99cce0584d32b616c6977337f9617e49bc980705b1a89e3.jpg",
+        "text": "$$\n\\pi _ { \\mathrm { a x } , { \\mathcal { S } } } : \\operatorname { U B P } ( { \\mathcal { S } } ) \\to \\operatorname { \\overline { { a x } } } ( { \\mathcal { S } } ) , \\qquad ( p , u ) \\mapsto p + d ( p , u , \\pi _ { \\mathcal { S } } ) u .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            302,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "135 The generalized tangent and normal space Back projection vectors are intricately related to the   \n136 generalized tangent and normal spaces.   \n137 Definition 2.5 (Definitions 4.3 and 4.4 of [19]) Let $\\boldsymbol { p } \\in \\textit { s }$ . The generalized tangent space   \n138 $\\mathrm { T a n } ( p , S )$ is the set of vectors $u \\in \\mathbb { R } ^ { d }$ , such that either $u = 0$ or, for every $\\varepsilon > 0$ there exists   \n139 a point $q \\in S$ with ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 4
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+    {
+        "type": "equation",
+        "img_path": "images/847a8d8872ac407475e04b5f483cc5898542c9943e9d23b8de7ffeaa78c7a3bd.jpg",
+        "text": "$$\n0 < | q - p | < \\varepsilon \\qquad \\mathit { a n d } \\qquad \\left| \\frac { q - p } { | q - p | } - \\frac { u } { | u | } \\right| < \\varepsilon .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "140 The generalized normal space $\\operatorname { N o r } ( p , S )$ consists of vectors $v \\in \\mathbb { R } ^ { d }$ such that $\\langle v , u \\rangle \\leq 0$ for all   \n141 $u \\in { \\mathrm { T a n } } ( p , S )$ . Vectors contained in the generalized tangent, resp. normal, space are called tangent,   \n142 resp. normal, to $s$ at $p$ . ",
+        "bbox": [
+            142,
+            280,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "143 The generalized tangent and normal spaces are illustrated in Figure 3. ",
+        "bbox": [
+            143,
+            333,
+            629,
+            349
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/ac397c02bd9997e5e3dd538710186040d2f69640915cce96e4c712d3c33d75c2.jpg",
+        "image_caption": [
+            "Figure 3: The (affine) generalized tangent and normal spaces of four points in the set $\\mathcal { S } \\subset \\mathbb { R } ^ { 2 }$ , in light blue and violet, respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            316,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "144 Stability of the reach under ambient diffeomorphisms Our last ingredient is the following result   \n145 by Federer.   \n146 Theorem 2.6 (Stability of the reach under ambient diffeomorphisms, Theorem 4.19 of [19])   \n147 Pick two constants $0 < t < \\operatorname { r c h } ( S )$ and $s > 0$ . If the map ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            148,
+            643,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/41a8f6f3eb8c038e35075815cb985daf93ac406ae63b41a79e89b6a3bcd69a2d.jpg",
+        "text": "$$\nF : \\{ x \\in \\mathbb { R } ^ { d } \\mid d ( x , S ) < s \\}  \\mathbb { R } ^ { n }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            382,
+            679,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "is injective and continuously differentiable, and the maps 148 $F$ , $F ^ { - 1 }$ , and $D F$ are Lipschitz continuous 149 with Lipschitz constants $\\operatorname { L i p } ( F ) , \\operatorname { L i p } ( F ^ { - 1 } ) , \\operatorname { L i p } ( D F )$ , respectively, then the reach $\\operatorname { r c h } ( F ( S ) )$ of the 150 image of the set $s$ under the map $F$ is lower-bounded by ",
+        "bbox": [
+            142,
+            705,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4ed7bfa4cb0d6e59150890e697f4d1f6f9ba16e91fcac0bbccc45cd6870665fa.jpg",
+        "text": "$$\n\\operatorname { r c h } ( F ( S ) ) \\geq \\operatorname* { m i n } \\left\\{ { \\frac { s } { \\mathrm { L i p } ( F ^ { - 1 } ) } } , { \\frac { 1 } { \\left( { \\frac { \\mathrm { L i p } ( F ) } { t } } + \\mathrm { L i p } ( D F ) \\right) \\left( \\mathrm { L i p } ( F ^ { - 1 } ) \\right) ^ { 2 } } } \\right\\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            258,
+            756,
+            738,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "151 3 Stability of the medial axis under ambient diffeomorphisms ",
+        "text_level": 1,
+        "bbox": [
+            143,
+            823,
+            702,
+            842
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "152 In this section we present the main result of this paper, Theorem 3.9. This theorem extends earlier   \n153 work by Chazal and Soufflet [13]. Its proof relies on Federer’s result on the stability of the reach,   \n154 Theorem 2.6. To give a more geometrical interpretation we introduce the concept of a weakly tangent   \n155 sphere and ball, and a maximal empty weakly tangent ball.   \n156 Definition 3.1 (Weakly tangent sphere and ball) Let $p \\in S$ . A sphere is called weakly tangent   \n157 to $s$ at $p$ if it contains the point $p$ and its centre lies in the (translated) generalized normal space   \n158 $\\mathrm { N o r } ( p , S ) + p$ . In other words, spheres weakly tangent to $s$ at $p$ are spheres with centres $p + v$ and   \n159 radii $| v |$ , for a vector $v \\in \\operatorname { N o r } ( p , S )$ . ",
+        "bbox": [
+            142,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            143,
+            90,
+            825,
+            148
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "A ball is called weakly tangent to $s$ at $p$ if its boundary sphere is weakly tangent to $s$ at $p .$ ",
+        "bbox": [
+            160,
+            152,
+            764,
+            169
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "161 Remark 3.2 Using the definition of $\\operatorname { N o r } ( p , S )$ , a weakly tangent ball can also be defined as follows:   \n162 A ball $B ( c , r )$ is weakly tangent at $p$ if and only if its centre c and radius $r$ satisfy ",
+        "bbox": [
+            151,
+            180,
+            825,
+            210
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/07c51f7db4499bcf365f693c001000fab460255710adeddeae100cd86a6e0cca.jpg",
+        "text": "$$\n( p + \\operatorname { T a n } ( S , p ) ) \\cap B ( c , r ) = \\{ p \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            383,
+            217,
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+            234
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "163 We remark: ",
+        "bbox": [
+            143,
+            250,
+            251,
+            263
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "164 Lemma 3.3 Let $p \\in S$ and $v \\in \\mathbb { R } ^ { d }$ , and suppose that for some $\\lambda > 0$ we have $\\pi _ { S } ( p + \\lambda v ) \\neq \\{ p \\}$ .   \nThen, for all 165 $\\lambda ^ { \\prime } \\geq \\lambda$ , we have $\\pi s ( p + \\lambda ^ { \\prime } v ) \\neq \\{ p \\}$ and for all $\\lambda ^ { \\prime } > \\lambda$ , that $p \\notin \\pi s ( p + \\lambda ^ { \\prime } v )$ . ",
+        "bbox": [
+            143,
+            273,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/9a5344e6a0e1629ea3d6abbde82149aec404769a759a00d0533a4dd0a1aaf5c4.jpg",
+        "image_caption": [
+            "Figure 4: Two families of balls weakly tangent to the set $\\mathcal { S } \\subset \\mathbb { R } ^ { 2 }$ (in blue). Each family contains a unique maximal empty ball (in purple). Notice that the centre of the maximal empty ball weakly tangent at the point $p _ { 1 }$ lies at the medial axis $\\operatorname { a x } ( S )$ , while the centre of the maximal empty ball weakly tangent at the point $p _ { 2 }$ only lies at its closure, $\\overline { { \\operatorname { a x } ( \\mathcal { S } ) } }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            351,
+            323,
+            647,
+            470
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Lemma 3.3 essentially tells us that a family of weakly tangent balls 166 $\\{ B ( p + \\lambda v , \\lambda | v | ) \\} _ { \\lambda \\geq 0 }$ contains 167 at most one which is maximal with respect to inclusion among those whose interior is disjoint from 168 the set $s$ . Two such families are illustrated in Figure 4. ",
+        "bbox": [
+            145,
+            549,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "169 We call such balls maximal empty. For the purpose of this article, we define maximal empty balls in   \n170 terms of unit back projection vectors (Definition 2.4). To see that each maximal empty ball is indeed   \n171 weakly tangent, we emphasise:   \n72 Lemma 3.4 If $( p , v ) \\in \\operatorname { B P } ( S )$ , then $( p , v ) \\in \\operatorname { N o r } ( S )$ . That is, $\\mathrm { B P } ( S ) \\subseteq \\mathrm { N o r } ( S )$ . In particular, for   \n3 any pair $( p , u ) \\in \\operatorname { U B P } ( S )$ and radius $\\lambda \\geq 0$ , the ball $B ( p + \\lambda u , \\lambda )$ is weakly tangent to $s$ .   \n174 Remark 3.5 For general closed sets, the converse of Lemma 3.4, that is, $\\mathrm { N o r } ( S ) \\subseteq \\mathrm { B P } ( S )$ , is not   \n175 true. One counter-example is the graph of the function $x \\mapsto | x | ^ { 3 / 2 }$ at the origin. However, the   \n176 inclusion $\\operatorname { N o r } ( S ) \\subseteq \\operatorname { B P } ( S )$ holds for sets of positive reach, thanks to Theorem 4.8 (12) of [19]   \n177 (recalled in the supplementary material).   \n178 Definition 3.6 (Maximal empty weakly tangent ball) Let $( p , u ) \\in \\operatorname { U B P } ( S )$ . A weakly tangent   \n179 ball $B ( p + \\lambda u , \\lambda )$ is called maximal empty to $\\begin{array} { r } { S i f \\lambda = d ( p , u , \\pi _ { S } ) } \\end{array}$ , or, equivalently, if $\\pi _ { \\mathrm { a x } , S } ( p , u ) =$   \n180 $p + \\lambda u$ . ",
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            143,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "181 (Maximal empty) weakly tangent balls satisfy the following properties. Let $( p , u ) \\in \\operatorname { U B P } ( S )$ ",
+        "bbox": [
+            150,
+            821,
+            789,
+            839
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• For any radius $0 < \\lambda \\leq d ( p , u , \\pi _ { S } )$ , the interior of the ball $B ( p + \\lambda u , \\lambda )$ is disjoint from the set $s$ . This follows directly from Definition 3.6 and Lemma 3.3. ",
+        "bbox": [
+            217,
+            848,
+            826,
+            878
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• The centres of maximal empty weakly tangent balls lie on the closure of the medial axis of $s$ . This is due to Lemma 2.3 and the definition of the map $\\pi _ { \\mathrm { a x } , S }$ (equation (1)). ",
+        "bbox": [
+            217,
+            882,
+            826,
+            912
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "186 The following lemma moreover tells us, that each point on the medial axis is a centre of a maximal   \n187 empty weakly tangent ball.   \n188 Lemma 3.7 (Surjectivity on $\\mathrm { a x } ( S ) )$ ) For any point $x \\in \\operatorname { a x } ( S )$ and $p \\in \\pi _ { S } ( x )$ , there exists a vector   \n189 $u \\in \\mathrm { U B P } ( p , S )$ such that $\\pi _ { \\mathrm { a x } , S } ( p , u ) = x$ . In other words, $B ( x , | x - p | )$ is a maximally empty   \n190 weakly tangent ball. Moreover, we have that ",
+        "bbox": [
+            147,
+            90,
+            825,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            151,
+            128,
+            826,
+            171
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dfccea87b44f55a4d635716aa1a6132b365efa30aba33459366ed299824e5ff4.jpg",
+        "text": "$$\n\\operatorname { a x } ( S ) \\subseteq \\pi _ { \\mathrm { a x } , S } \\left( \\operatorname { U B P } ( S ) \\right) \\subseteq { \\overline { { \\operatorname { a x } ( S ) } } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            176,
+            620,
+            195
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "191 We are now almost ready to state our main theorem. Before phrasing the result, we walk the reader through the assumptions and fix the notation on the way. The assumptions are illustrated in Figure 5. ",
+        "bbox": [
+            137,
+            205,
+            828,
+            236
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/2055983f7c713af756043cf2118c67cee8e32cd1c949a32f02c78f5a9f80220c.jpg",
+        "image_caption": [
+            "Figure 5: The setting of Theorem 3.9. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            223,
+            247,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "193 Assumption 3.8 ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "• We assume that the set $s$ has a bounding sphere of radius $r$ , which we denote by $S ( r )$ .   \n• We consider a $C ^ { 1 }$ diffeomorphism $F : \\mathbb { R } ^ { d }  \\mathbb { R } ^ { d }$ such that the Lipschitz constants of $F$ and $F ^ { - 1 }$ are bounded by $L _ { F }$ , and the Lipschitz constants of the differentials $D F$ and $D F ^ { - 1 }$ are bounded by $L _ { D F }$ . We call such a diffeomorphism a $\\dot { C } ^ { 1 , 1 }$ diffeomorphism.   \n• We further assume that the map $F$ leaves the bounding sphere $S ( r )$ invariant, that is, $F ( \\overbar { S } ( r ) ) = S ( r )$ .   \n• We pick a point $c \\in \\operatorname { a x } ( S )$ , a point $p \\in \\pi _ { S } ( c )$ , and write $\\rho = | c - p |$ . Observe that since $S \\cap \\operatorname { a x } ( S ) = \\emptyset$ , $\\rho$ is positive. By Lemma 3.7, the ball $B ( c , \\rho )$ is a maximal empty weakly tangent ball to $s$ at $p$ . Moreover, we define $\\begin{array} { r } { u = \\frac { c - p } { | c - p | } } \\end{array}$ and note that $u \\in \\mathrm { U B P } ( p , S )$ .   \n• We denote the tangent hyperplane to the boundary sphere of $B ( c , \\rho )$ at $p$ by $p + T$ . The hyperplane $T$ is the orthocomplement of the vector $u$ : $T = u ^ { \\perp }$ .   \n• We work with the unit vector at $F ( p )$ that points inside the image of the ball $B ( c , \\rho )$ and is orthogonal to the hyperplane $D _ { p } F ( T )$ . We denote this vector by $u ^ { \\prime }$ . ",
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+        "type": "text",
+        "text": "207 Theorem 3.9 Under the above assumptions, there exists a maximal empty weakly tangent ball   \n208 $B ( c ^ { \\prime } , \\rho ^ { \\prime } )$ to the set $F ( S )$ , whose boundary sphere has an internal normal $u ^ { \\prime }$ . In particular, the   \n209 ball $B ( c ^ { \\prime } , \\rho ^ { \\prime } )$ is tangent to the affine hyperplane $F ( p ) + D _ { p } F ( T )$ . Its radius $\\rho ^ { \\prime }$ is bounded by   \n210 $\\begin{array} { r } { \\rho ^ { \\prime } \\in \\bigg [ \\frac { \\rho } { ( L _ { F } ) ^ { 3 } + \\rho L _ { D F } ( L _ { F } ) ^ { 2 } } , \\frac { ( L _ { F } ) ^ { 3 } \\rho } { 1 - \\rho L _ { D F } ( L _ { F } ) ^ { 2 } } } \\end{array}$ . Assume, moreover, that the distortions of both $F$ and $D F$   \n211 are bounded, that is, for all $x \\in \\mathbb { R } ^ { d }$ , ",
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+        "text": "$$\n| F ( x ) - x | \\leq \\varepsilon _ { 1 } , \\qquad \\| D F _ { x } - \\mathrm { I d } \\| \\leq \\varepsilon _ { 2 } < 1 ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "and 212 $r \\cdot L _ { D F } ( L _ { F } ) ^ { 2 } \\leq 1 / 2$ . Define ",
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+        "text": "$$\n\\begin{array} { l } { { C _ { L } ( r , L _ { F } , L _ { D F } , \\varepsilon _ { 1 } , \\varepsilon _ { 2 } ) = } } \\\\ { { 2 r \\sqrt { 1 + ( L _ { F } ) ^ { 6 } \\left( 1 + 4 r L _ { D F } ( L _ { F } ) ^ { 2 } \\right) ^ { 2 } - 2 ( L _ { F } ) ^ { 3 } \\left( 1 + 4 r L _ { D F } ( L _ { F } ) ^ { 2 } \\right) \\sqrt { 1 - ( \\varepsilon _ { 2 } ) ^ { 2 } } } + \\varepsilon _ { 1 } } } \\end{array}\n$$",
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+        "type": "text",
+        "text": "213 then the map $\\pi _ { \\mathrm { a x } , S }$ satisfies ",
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+        "type": "equation",
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+        "text": "$$\n| \\pi _ { \\mathrm { a x } , \\mathcal { S } } ( p , u ) - \\pi _ { \\mathrm { a x } , F ( \\mathcal { S } ) } ( F ( p ) , u ^ { \\prime } ) | \\leq C _ { L } ( r , L _ { F } , L _ { D F } , \\varepsilon _ { 1 } , \\varepsilon _ { 2 } ) .\n$$",
+        "text_format": "latex",
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+        "img_path": "images/f21e2528edf51f82a7b039276b98e1e9e396cc51024ac51fe4e47b29ed606bab.jpg",
+        "text": "$$\nd _ { H } ( \\mathrm { a x } ( S ) , \\mathrm { a x } ( F ( S ) ) ) \\leq C _ { L } ( r , L _ { F } , L _ { D F } , \\varepsilon _ { 1 } , \\varepsilon _ { 2 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "215 The bound $| F ( x ) - x | \\leq \\varepsilon _ { 1 }$ is really necessary, because we want our theorem to accommodate for   \n216 rotations and translations, which rotate and translate the medial axis without changing distances   \n217 and hence have Lipschitz constant 1. We further stress that if the diffeomorphism $F$ is close to the   \n218 identity, its Lipschitz constant satisfies $L _ { F } \\geq 1$ , because by assumption $F$ leaves the bounding sphere   \n219 $S ( r )$ invariant, and $L _ { D F }$ is close to zero.   \n220 Sketch of the proof of Theorem 3.9 The idea of the proof is depicted in Figure 5. Thanks to Federer’s   \n221 result (Theorem 2.6), we know that the reach of the maximal empty weakly tangent ball $B ( c , \\rho )$   \n222 does not change too much under the ambient diffeomorphism $F$ . This gives a lower bound on the   \n223 radius of every maximal empty weakly tangent ball of the image of this ball — the set $F ( B ( c , \\rho ) )$ .   \n224 We show that in the interior of $F ( B ( { \\dot { c } } , \\rho ) )$ , the radii of the maximal empty weakly tangent balls   \n225 of $F ( B ( c , \\rho ) )$ are close to $\\rho$ . One of these balls is also empty weakly tangent to $F ( S )$ at $F ( p )$ ,   \n226 though not necessarily maximal. We denote its centre by $c ^ { \\prime }$ . Since we can apply the same argument   \n227 for the map $F ^ { - 1 }$ , we find an upper and lower bound on the radius of the maximal weakly tangent   \n228 ball $B ( c ^ { \\prime } , \\rho ^ { \\prime } )$ of $F ( S )$ at $F ( p )$ that is also weakly tangent to $F ( B ( c , \\rho ) )$ , or equivalently tangent to   \n229 $D _ { p } F ( T )$ .   \n230 While this bound on the difference of the radii is essentially a bound on the distance $\\left| \\left| c - p \\right| - \\left| c ^ { \\prime } - \\right| \\right|$   \n231 $F ( p ) | |$ between the points $c - p$ and $c ^ { \\prime } - F ( p )$ , the bound $\\varepsilon _ { 2 }$ on $\\| D F - 1 \\|$ allows one to bound the   \n232 angle between the vectors $c - p$ and $c ^ { \\prime } - F ( p )$ . With the assumption (2) we can then derive a bound   \n233 the distance between the points $c$ and $c ^ { \\prime }$ . Finally, thanks to [19, Theorem 4.8 (6)] (Lemma 2.3) this   \n234 induces a bound on the Hausdorff distance between the (closure of the) two medial axes $\\operatorname { a x } ( S )$ and   \n235 $\\operatorname { a x } ( F ( S ) )$ . □ ",
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+        "type": "text",
+        "text": "",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "It was a surprise to the authors that no assumption on the set (apart from closedness) needed to be made, and that the techniques used were that simple and well established; they go back to Federer [19]. In fact, the authors at first envisioned a far more elaborate argument assuming the set had positive $\\mu$ -reach [11]. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4 Quantifying 240 $C ^ { 1 , 1 }$ diffeomorphisms as deviations from identity ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "241 In this section we reformulate the main result in terms of norms on Banach spaces. This reformulation   \n242 offers a more theoretical insight, and we believe the reformulated bounds are easier to work with in   \n243 certain applications. Indeed, in the context of practical numerical computations, a bound on the   \n244 Lipschitz constant of an operator — or, at least, a modulus of continuity — allows to control the   \n245 condition number. This control is particularly useful when we calculate with objects such as the   \n246 medial axis,whose (numerical) stability is often problematic in practice. ",
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+    {
+        "type": "text",
+        "text": "As we will see below, for this reformulation we somewhat strengthen our assumptions. ",
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+        "type": "text",
+        "text": "248 We decompose a diffeomorphism $F$ into the identity map $\\mathbb { 1 } _ { \\mathbb { R } ^ { d } }$ on $\\mathbb { R } ^ { d }$ , and a displacement field $\\varphi$   \n249 $F = \\mathbb { 1 } _ { \\mathbb { R } ^ { d } } + \\varphi$ . For the choice of the displacement field, we restrict ourselves to the vector space $\\mathcal { U }$ of   \n250 all $C ^ { 1 , 1 }$ maps $\\varphi$ from $\\mathbb { R } ^ { d }$ to $\\mathbb { R } ^ { d }$ whose restriction to the exterior $\\mathbb { R } ^ { d } \\backslash B ( r )$ of a certain bounding ball   \n251 $B ( r )$ equals 0.3 ",
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+        "type": "text",
+        "text": "52 A natural norm associated to $\\mathcal { U }$ is one that makes it a Banach space. A typical choice, inherited from general Banach spaces of 53 $C ^ { 1 , 1 }$ functions, would be for example, for $\\varphi \\in { \\mathcal { U } }$ , ",
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+        "img_path": "images/dc897c8f17a3383873735a73365d60f4d1af3887650a9c0ff55b1755a9989562.jpg",
+        "text": "$$\n\\| \\varphi \\| _ { C ^ { 1 , 1 } } = \\operatorname* { m a x } \\left( \\| \\varphi \\| _ { \\infty } , \\| D \\varphi \\| _ { \\infty } , \\operatorname { L i p } ( D \\varphi ) \\right) .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "54 Here we used the following notation: ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "• $\\begin{array} { r } { \\| \\varphi \\| _ { \\infty } = \\operatorname* { s u p } _ { x \\in \\mathbb { R } ^ { d } } | \\varphi ( x ) | } \\end{array}$ denotes the sup norm on $x \\mapsto | \\varphi ( x ) |$ , where $| \\cdot |$ is the Euclidean norm in $\\mathbb { R } ^ { d }$ ,   \n• $\\| D \\varphi \\| _ { \\infty } = \\operatorname* { s u p } _ { x \\in \\mathbb { R } ^ { d } } \\| D \\varphi ( x ) \\|$ denotes the sup norm on $x \\mapsto \\| D \\varphi ( x ) \\|$ , where $\\| D \\varphi ( x ) \\|$ is the operator norm induced by the Euclidean norm on $\\mathbb { R } ^ { d }$ . ",
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+    {
+        "type": "text",
+        "text": "• We write $\\mathrm { L i p } ( D \\varphi )$ for the Lipschitz semi-norm of $D \\varphi$ . The Lipschitz semi-norms of $\\varphi$ and $D \\varphi$ are defined as ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "and ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c83e56150c242c1ba23fdd3931709b7b9207431d192d732d0a4c2c2cc89d8335.jpg",
+        "text": "$$\n\\begin{array} { c } { \\displaystyle \\mathrm { L i p } ( \\varphi ) = \\displaystyle \\operatorname* { s u p } _ { \\substack { x , y \\in \\mathbb { R } ^ { d } , x \\neq y } } \\frac { \\vert \\varphi ( y ) - \\varphi ( x ) \\vert } { \\vert y - x \\vert } , } \\\\ { \\displaystyle \\mathrm { L i p } ( D \\varphi ) = \\displaystyle \\operatorname* { s u p } _ { \\substack { x , y \\in \\mathbb { R } ^ { d } , x \\neq y } } \\frac { \\Vert D \\varphi ( y ) - D \\varphi ( x ) \\Vert } { \\vert y - x \\vert } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "262 The norm defined in (4) makes $\\mathcal { U }$ into a Banach space, since every Cauchy sequence in $\\mathcal { U }$ has a limit   \n263 in $\\mathcal { U }$ . In addition, any function $\\varphi \\in { \\mathcal { U } }$ satisfies: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/f6154a857e3019d9bc5daf68bd6bdd312301d98133a27275a1e5469aed2dfafa.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathrm { L i p } ( \\varphi ) = \\| D \\varphi \\| _ { \\infty } , } \\\\ & { \\| D \\varphi \\| _ { \\infty } \\leq r \\mathrm { L i p } ( D \\varphi ) , } \\\\ & { \\| \\varphi \\| _ { \\infty } \\leq r \\mathrm { L i p } ( \\varphi ) \\leq r ^ { 2 } \\mathrm { L i p } ( D \\varphi ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "since the restriction of $\\varphi$ to $\\mathbb { R } ^ { d } \\setminus B ( r )$ is 0. This in turn yields that $\\mathrm { L i p } ( D \\varphi ) \\ \\leq \\ \\| \\varphi \\| _ { C ^ { 1 , 1 } } \\ \\leq$ $\\operatorname* { m a x } ( 1 , r , r ^ { 2 } ) \\operatorname { L i p } ( D \\varphi )$ . Thus, in $\\mathcal { U }$ , the norm $\\varphi \\mapsto \\operatorname { L i p } ( D \\varphi )$ is equivalent to the norm $\\varphi \\mapsto \\| \\varphi \\| _ { C ^ { 1 , 1 } }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We can now state slightly less general version of Theorem 3.9 in terms of the Banach space $( \\mathcal { U } , \\varphi \\mapsto$ $\\mathrm { L i p } ( D \\varphi )$ ) . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Theorem 4.1 Let $S \\subseteq \\mathbb { R } ^ { d }$ be bounded by the ball $B ( r )$ of radius $r > 0$ , such that $S ( \\boldsymbol { r } ) = \\partial B ( \\boldsymbol { r } ) \\subseteq$ $\\mathcal { S } _ { \\mathbf { \\Omega } }$ . Let further $F$ be a $C ^ { 1 , 1 }$ diffeomorphism from $\\mathbb { R } ^ { d }$ to itself that leaves the set $\\mathbb { R } ^ { d } \\backslash B ( r )$ invariant, and define two displacement fields $\\varphi , \\tilde { \\varphi } \\in \\mathcal { U }$ such that $F = \\mathbb { 1 } _ { \\mathbb { R } ^ { d } } + \\varphi$ and ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/e4d957f8ebebf8552ea12dd86e16eb5c94a404420265535bc2bb6e2d2a400056.jpg",
+        "text": "$$\n\\left( \\mathbb { 1 } _ { \\mathbb { R } ^ { d } } + \\tilde { \\varphi } \\right) \\circ \\left( \\mathbb { 1 } _ { \\mathbb { R } ^ { d } } + \\varphi \\right) = \\mathbb { 1 } _ { \\mathbb { R } ^ { d } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "271 Define $\\varepsilon = \\operatorname* { m a x } \\left( \\mathrm { L i p } ( D \\varphi ) , \\mathrm { L i p } ( D \\tilde { \\varphi } ) \\right)$ . ",
+        "bbox": [
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+        "type": "text",
+        "text": "272 If $r \\varepsilon ~ \\leq ~ 1 / 4$ , the Hausdorff distance between the medial axes of the set √ $s$ and its image 273 $F ( S )$ is bounded by $d _ { H } ( \\operatorname { a x } ( S ) , \\operatorname { a x } ( F ( S ) ) ) ~ \\leq ~ \\left( 1 + { \\sqrt { 5 0 } } \\right) r ^ { 2 } \\varepsilon + { \\mathcal { O } } \\left( r ^ { 3 } \\varepsilon ^ { 2 } \\right)$ . In particular, 274 $d _ { H } ( \\operatorname { a x } ( S ) , \\operatorname { a x } ( F ( S ) ) ) = { \\mathcal { O } } \\left( r ^ { 2 } \\varepsilon \\right)$ . ",
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+        "type": "text",
+        "text": "275 Sketch of the proof Essentially, the proof consists of rewriting Theorem 3.9 in terms of the language   \n76 developed in this section. □   \n277 Remark 4.2 Observe that the bound $\\mathcal { O } \\left( r ^ { 2 } \\varepsilon \\right)$ is consistent with a scaling by factor $\\lambda \\colon S \\mapsto \\lambda S$   \n278 $F ( \\cdot ) \\mapsto \\lambda F ( \\cdot / \\lambda )$ . Under such a scaling, the radius $r$ is multiplied by $\\lambda$ , while the Lipschitz   \n279 constant $\\mathrm { L i p } ( D \\varphi )$ — and therefore $\\varepsilon$ — is divided by $\\lambda$ . Furthermore, the Hausdorff distance   \n280 $d _ { H } ( \\operatorname { a x } ( S ) , \\operatorname { a x } ( F ( S ) ) )$ increases by a factor $\\lambda .$ . By considering a diffeomorphism that translates the   \n281 set $ { \\boldsymbol { S } } \\setminus  { \\boldsymbol { S } } (  { \\boldsymbol { r } } )$ while keeping the bounding sphere $S ( r )$ fixed, we see that this bound is asymptotically   \n282 optimal. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "5 Conclusion and future work ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We proved the Hausdorff stability of the medial axis of a closed set without any further assumption on it (as explained in Remark 2.1, the existence of the bounding sphere serves to formulate the main result in a clean way). ",
+        "bbox": [
+            173,
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "With regard to applications, our result is the first step towards providing a provably correct image recognition in particular in the context of astrophysics. The next step is to produce physics-informed models for the medial axis as occurring in astronomical data. ",
+        "bbox": [
+            171,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "On the mathematical side, we conclude with a conjecture generalizing our result. We believe that our result generalizes to compact Riemannian manifolds with bounded curvature. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Conjecture 5.1 Let $\\mathcal { M }$ be a compact Riemannian manifold with bounded sectional curvature4 and S a closed subset of M. Then the medial axis (also called cut locus [26]) of $s$ in $\\mathcal { M }$ is Lipschitz stable under diffeomorphisms of $\\mathcal { M }$ . ",
+        "bbox": [
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+    {
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+        "text": "References [1] Eddie Aamari and Alexander Knop. Statistical query complexity of manifold estimation. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021, pages 116–122, New York, NY, USA, 2021. Association for Computing Machinery. [2] Eddie Aamari and Clément Levrard. Stability and minimax optimality of tangential Delaunay complexes for manifold reconstruction. Discrete & Computational Geometry, 59:923–971, 2018. [3] N. Amenta and M. Bern. Surface reconstruction by Voronoi filtering. Discrete & Computational Geometry, 22(4):481–504, Dec 1999. [4] Nina Amenta, Sunghee Choi, and Ravi Krishna Kolluri. The power crust. In Proceedings of the sixth ACM symposium on Solid modeling and applications, pages 249–266, 2001. [5] Dominique Attali, Jean-Daniel Boissonnat, and Herbert Edelsbrunner. Stability and computation of medial axes - a state-of-the-art report. In Torsten Möller, Bernd Hamann, and Robert D. Russell, editors, Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration, pages 109–125, Berlin, Heidelberg, 2009. Springer Berlin Heidelberg. [6] Dominique Attali and Annick Montanvert. Computing and simplifying 2d and 3d continuous skeletons. Computer vision and image understanding, 67(3):261–273, 1997. [7] Matthias Bartelmann. Gravitational lensing. Classical and Quantum Gravity, 27(23):233001, nov 2010. [8] M. Berger. A Panoramic View of Riemannian Geometry. Springer-Verlag, 2003. [9] W. Blaschke. Kreis und Kugel. Verlag von Veit und Comp., 1916. [10] Erin Chambers, Ellen Gasparovic, and Kathryn Leonard. Medial fragments for segmentation of articulating objects in images. Research in Shape Analysis: WiSH2, Sirince, Turkey, June 2016, pages 1–15, 2018.   \n320 [11] F. Chazal, D. Cohen-Steiner, and A. Lieutier. A sampling theory for compact sets in Euclidean space. 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[18] Ilke Demir, Camilla Hahn, Kathryn Leonard, Geraldine Morin, Dana Rahbani, Athina Panotopoulou, Amelie Fondevilla, Elena Balashova, Bastien Durix, and Adam Kortylewski. SkelNetOn 2019: Dataset and challenge on deep learning for geometric shape understanding. In 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), pages 1143–1151, 2019. [19] H. Federer. Curvature measures. Transactions of the America mathematical Society, 93:418–491, 1959.   \n40 [20] Charles Fefferman, Sergei Ivanov, Yaroslav Kurylev, Matti Lassas, and Hariharan Narayanan. Fitting a putative manifold to noisy data. In Conference On Learning Theory, pages 688–720. PMLR, 2018.   \n43 [21] Charles Fefferman, Sergei Ivanov, Matti Lassas, and Hariharan Narayanan. Fitting a manifold of large reach to noisy data. arXiv preprint arXiv:1910.05084, 2019.   \n45 [22] Charles Fefferman, Sergei Ivanov, Matti Lassas, and Hariharan Narayanan. Reconstruction of a Riemannian manifold from noisy intrinsic distances. SIAM Journal on Mathematics of Data Science, 2(3):770–808, 2020.   \n48 [23] J. D. Fernie. The period-luminosity relation: A historical review. Publications of the Astronomical Society of the Pacific, 81(483):707, dec 1969. [24] Ellen Gasparovic. The Blum medial linking structure for multi-region analysis. PhD thesis, The University of North Carolina at Chapel Hill, 2012. [25] Seng-Beng Ho and Charles R Dyer. Shape smoothing using medial axis properties. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-8(4):512–520, 1986. [26] Vitali Kapovitch and Alexander Lytchak. Remarks on manifolds with two-sided curvature bounds. Analysis and Geometry in Metric Spaces, 9(1):53–64, 2021. [27] Jean-Claude Latombe. Robot motion planning, volume 124. Springer Science & Business Media, 2012.   \n[28] André Lieutier. Any open bounded subset of 58 $\\mathbb { R } ^ { n }$ has the same homotopy type as its medial axis. Computer-Aided Design, 36(11):1029 – 1046, 2004. Solid Modeling Theory and Applications. [29] André Lieutier and Mathijs Wintraecken. Hausdorff and gromov-hausdorff stable subsets of the medial axis. Proceedings of the 55th ACM Symposium on Theory of Computing (STOC 2023), 2023. [30] John N Mather. Distance from a submanifold in euclidean-space. In Proceedings of symposia in pure mathematics, volume 40, pages 199–216. AMER MATHEMATICAL SOC 201 CHARLES ST, PROVIDENCE, RI 02940-2213, 1983.   \n66 [31] Phillip James Edwin Peebles. Principles of physical cosmology, volume 27. Princeton university press, 1993. [32] Punam K Saha, Gunilla Borgefors, and Gabriella Sanniti di Baja. A survey on skeletonization algorithms and their applications. Pattern recognition letters, 76:3–12, 2016.   \n70 [33] Doron Shaked and Alfred M. Bruckstein. Pruning medial axes. Computer Vision and Image Understanding, 69(2):156 – 169, 1998. [34] Barak Sober and David Levin. Manifold approximation by moving least-squares projection (MMLS). Constructive Approximation, 52(3):433–478, 2020. [35] Andrea Tagliasacchi, Thomas Delame, Michela Spagnuolo, Nina Amenta, and Alexandru Telea. 3d skeletons: A state-of-the-art report. In Computer Graphics Forum, volume 35, pages 573–597. Wiley Online Library, 2016. [36] Zhongwei Tang, Rafael Grompone Von Gioi, Pascal Monasse, and Jean-Michel Morel. A precision analysis of camera distortion models. IEEE Transactions on Image Processing, 26(6):2694–2704, 2017.   \n80 [37] R. Thom. Sur le cut-locus d’une variété plongée. Journal of Differential Geometry, 6(4):577– 586, 1972.   \n82 [38] Martijn van Manen. Maxwell strata and caustics. In Singularities In Geometry And Topology, pages 787–824. World Scientific, 2007. ",
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+        "text": "384 [39] C. T. C. Wall. Geometric properties of generic differentiable manifolds. In Jacob Palis and   \n385 Manfredo do Carmo, editors, Geometry and Topology, pages 707–774, Berlin, Heidelberg, 1977.   \n386 Springer Berlin Heidelberg.   \n387 [40] Franz-Erich Wolter. Cut locus and medial axis in global shape interrogation and representation.   \n388 1993.   \n389 [41] Yajie Yan, Kyle Sykes, Erin Chambers, David Letscher, and Tao Ju. Erosion thickness on   \n390 medial axes of 3d shapes. ACM Transactions on Graphics, 35(4):38:1–38:12, July 2016. ",
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+# On Embeddings for Numerical Features in Tabular Deep Learning
+Yury Gorishniy∗ Yandex
+Ivan Rubachev HSE, Yandex
+Artem Babenko Yandex
+# Abstract
+Recently, Transformer-like deep architectures have shown strong performance on tabular data problems. Unlike traditional models, e.g., MLP, these architectures map scalar values of numerical features to high-dimensional embeddings before mixing them in the main backbone. In this work, we argue that embeddings for numerical features are an underexplored degree of freedom in tabular DL, which allows constructing more powerful DL models and competing with gradient boosted decision trees (GBDT) on some GBDT-friendly benchmarks (that is, where GBDT outperforms conventional DL models). We start by describing two conceptually different approaches to building embedding modules: the first one is based on a piecewise linear encoding of scalar values, and the second one utilizes periodic activations. Then, we empirically demonstrate that these two approaches can lead to significant performance boosts compared to the embeddings based on conventional blocks such as linear layers and ReLU activations. Importantly, we also show that embedding numerical features is beneficial for many backbones, not only for Transformers. Specifically, after proper embeddings, simple MLP-like models can perform on par with the attention-based architectures. Overall, we highlight embeddings for numerical features as an important design aspect with good potential for further improvements in tabular DL. The source code is available at https://github.com/Yura52/tabular-dl-num-embeddings.
+# 1 Introduction
+Tabular data problems are currently a final frontier for deep learning (DL) research. While the most recent breakthroughs in NLP, vision, and speech are achieved by deep models [12], their success in the tabular domain is not convincing yet. Despite a large number of proposed architectures for tabular DL [2, 3, 13, 17, 21, 24, 31, 39, 40], the performance gap between them and the “shallow” ensembles of decision trees, like GBDT, often remains significant [13, 36].
+The recent line of works [13, 24, 39] reduce this performance gap by successfully adapting the Transformer architecture [45] for the tabular domain. Compared to traditional models, like MLP or ResNet, the proposed Transformer-like architectures have a specific way to handle numerical features of the data. Namely, they map scalar values of numerical features to high-dimensional embedding vectors, which are then mixed by the self-attention modules. Beyond transformers, mapping numerical features to vectors was also employed in different forms in the click-through rate (CTR) prediction problems [8, 14, 40]. Nevertheless, the literature is mostly focused on developing more powerful backbones while keeping the design of embedding modules relatively simple. In particular, the existing architectures [13, 14, 24, 39, 40] construct embeddings for numerical features using quite restrictive parametric mappings, e.g., linear functions, which can lead to suboptimal performance. In this work, we demonstrate that the embedding step has a substantial impact on the model effectiveness, and its proper design can significantly improve tabular DL models.
+Specifically, we describe two different building blocks suitable for constructing embeddings for numerical features. The first one is a piecewise linear encoding that produces alternative initial representations for the original scalar values and is based on feature binning, a long-existing preprocessing technique [11]. The second one relies on periodic activation functions, which is inspired by their usage in implicit neural representations [28, 38, 42], NLP [41, 45] and CV tasks [25]. The first approach is simple, interpretable and non-differentiable, while the second demonstrates better results on average. We observe that DL models equipped with our embedding schemes successfully compete with GBDT on GBDT-friendly benchmarks and achieve the new state-of-the-art on tabular DL.
+As another important finding, we demonstrate that the step of embedding the numerical features is universally beneficial for different deep architectures, not only for Transformer-like ones. In particular, we show, that after proper embeddings, simple MLP-like architectures often provide the performance comparable to the state-of-the-art attention-based models. Overall, our work demonstrates the large impact of the embeddings of numerical features on the tabular DL performance and shows the potential of investigating more advanced embedding schemes in future research.
+To sum up, our contributions are as follows:
+1. We demonstrate that embedding schemes for numerical features are an underexplored research question in tabular DL. Namely, we show that more expressive embedding schemes can provide substantial performance improvements over prior models.
+2. We show that the profit from embedding numerical features is not specific for Transformerlike architectures, and proper embedding schemes benefit traditional models as well.
+3. On a number of public benchmarks, we achieve the new state-of-the-art on tabular DL.
+# 2 Related work
+Tabular deep learning. During several recent years, the community has proposed a large number of deep models for tabular data [2, 3, 13, 15, 17, 21, 24, 31, 39, 40, 46]. However, when systematically evaluated, these models do not consistently outperform the ensembles of decision trees, such as GBDT (Gradient Boosting Decision Tree) [7, 19, 32], which are typically the top-choice in various ML competitions [13, 36]. Moreover, several recent works have shown that the proposed sophisticated architectures are not superior to properly tuned simple models, like MLP and ResNet [13, 18]. In this work, unlike the prior literature, we do not aim to propose a new backbone architecture. Instead, we focus on more accurate ways to handle numerical features, and our developments can be potentially combined with any model, including traditional MLPs and more recent Transformer-like ones.
+Transformers in tabular DL. Due to the tremendous success of Transformers for different domains [10, 45], several recent works adapt their self-attention design for tabular DL as well [13, 17, 24, 39]. Compared to existing alternatives, applying self-attention modules to the numerical features of tabular data requires mapping the scalar values of these features to high-dimensional embedding vectors. So far, the existing architectures perform this “scalar” “vector” mapping by relatively simple computational blocks, which, in practice, can limit the model expressiveness. For instance, the recent FT-Transformer architecture [13] employs only a single linear layer. In our experiments, we demonstrate that such embedding schemes can provide suboptimal performance, and more advanced schemes often lead to substantial profit.
+CTR Prediction. In CTR prediction problems, objects are represented by numerical and categorical features, which makes this field highly relevant to tabular data problems. In several works, numerical features are handled in some non-trivial way while not being the central part of the research [8, 40]. Recently, however, a more advanced scheme has been proposed in Guo et al. [14]. Nevertheless, it is still based on linear layers and conventional activation functions, which we found to be suboptimal in our evaluation.
+Feature binning. Binning is a discretization technique that converts numerical features to categorical features. Namely, for a given feature, its value range is split into bins (intervals), after which the original feature values are replaced with discrete descriptors (e.g. bin indices or one-hot vectors) of the corresponding bins. We point to the work by Dougherty et al. [11], which performs an overview of some classic approaches to binning and can serve as an entry point to the relevant literature on the topic. In our work, however, we utilize bins in a different way. Specifically, we use their edges to construct lossless piecewise linear representations of the original scalar values. It turns out that this simple and interpretable representations can provide substantial benefit to deep models on several tabular problems.
+Periodic activations. Recently, periodic activation functions have become a key component in processing coordinates-like inputs, which is required in many applications. Examples include NLP [45], CV [25], implicit neural representations [28, 38, 42]. In our work, we show that periodic activations can be used to construct powerful embedding modules for numerical features in tabular data problems. Contrary to some of the aforementioned papers, where components of the multidimensional coordinates are mixed (e.g. with linear layers) before passing them to periodic functions [38, 42], we find it crucial to embed each feature separately before mixing them in the main backbone.
+# 3 Embeddings for numerical features
+In this section, we describe the general framework for what we call "embeddings for numerical features" and the main building blocks used in the experimental comparison in section 4.
+Notation. For a given supervised learning problem on tabular data, we denote the dataset as $\left\{ \left( x ^ { j } , \ y ^ { j } \right) \right\} _ { j = 1 } ^ { n }$ where $y ^ { j } \in \mathbb { Y }$ represents the object’s label and $x ^ { j } = \left( x ^ { j \left( n u m \right) } , x ^ { j \left( c a t \right) } \right) \in \mathbb { X }$ represents the object’s features (numerical and categorical). xj(num)i , in turn, denotes the i-th numerical feature of the -th object. Depending on the context, the index can be omitted. The dataset is split into three disjoint parts: $\overline { { 1 , n } } \overline { { = J _ { t r a i n } } } \cup J _ { v a l } \cup J _ { t e s t }$ , where the “train” part is used for training, the “validation” part is used for early stopping and hyperparameter tuning, and the “test” part is used for the final evaluation.
+# 3.1 General framework
+We formalize the notion of "embeddings for numerical features" as $z _ { i } = f _ { i } ( ( x _ { i } ^ { ( n u m ) } ) \in \mathbb { R } ^ { d _ { i } }$ where $f _ { i } ( x )$ is the embedding function for the $i$ -th numerical feature, $z _ { i }$ is the embedding of the $i$ -th numerical feature and $d _ { i }$ is the dimensionality of the embedding. Importantly, the proposed framework implies that embeddings for all features are computed independently of each other. Note that the function $f _ { i }$ can depend on parameters that are trained as a part of the whole model or in some other fashion (e.g. before the main optimization). In this work, we consider only embedding schemes where the embedding functions for all features are of the same functional form. We never share parameters of embedding functions of different features.
+The subsequent use of the embeddings depends on the model backbone. For MLP-like architectures, they are concatenated into one flat vector (see Appendix A for illustrations). For Transformer-based architectures, no extra step is performed and the embeddings are passed as is, so the usage is defined by the original architectures.
+# 3.2 Piecewise linear encoding
+While vanilla MLP is known to be a universal approximator [9, 16], in practice, due to optimization peculiarities, it has limitations in its learning capabilities [34]. However, the recent work by Tancik et al. [42] uncovers the case where changing the input space alleviates the above issue. This observation motivates us to check if changing the representations of the original scalar values of numerical features can improve the learning capabilities of tabular DL models.
+At this point, we try to start simple and turn to "classical" machine learning techniques. Namely, we take inspiration from the one-hot encoding algorithm that is widely and successfully used for representing discrete entities such as categorical features in tabular data problems or tokens in NLP. We note that the one-hot representation can be seen as an opposite solution to the scalar representation in terms of the trade-off between parameter efficiency and expressivity. To check whether the onehot-like approach can be beneficial for tabular DL models, we design a continuous alternative to the one-hot encoding (since the vanilla one-hot encoding is barely applicable to numerical features).
+Formally, for the $i \cdot$ -th numerical feature, we split its value range into the disjoint set of $T ^ { i }$ intervals $B _ { 1 } ^ { i }$ , . . . , $B _ { T } ^ { i }$ , which we call bins: $B _ { t } ^ { i } = [ b _ { t - 1 } ^ { i } , b _ { t } ^ { i } )$ . The splitting algorithm is an important implementation detail that we discuss later. From now on, we omit the feature index $i$ for simplicity. Once the bins are determined, we define the encoding scheme as in Equation 1:
+$$
+\mathsf { P L E } ( x ) = [ e _ { 1 } , ~ . ~ . ~ . , ~ e _ { T } ] \in \mathbb { R } ^ { T }
+$$
+$$
+e _ { t } = \left\{ \begin{array} { l l } { 0 , } & { x < b _ { t - 1 } \mathrm { ~ A N D ~ } t > 1 } \\ { 1 , } & { x \geq b _ { t } \mathrm { ~ A N D ~ } t < T } \\ { \frac { x - b _ { t - 1 } } { b _ { t } - b _ { t - 1 } } , } & { \mathrm { o t h e r w i s e } } \end{array} \right.
+$$
+where PLE stands for “peicewise linear encoding”. We provide the visualization in Figure 1.
+![](images/ac479be40b57aa351199b81061f9a1933898465f23d6b971cf0fb15c89fb5d38.jpg)
+Figure 1: The piecewise linear encoding (PLE) in action for $T = 4$ (see Equation 1).
+Note that:
+• PLE produces alternative initial representations for the numerical features and can be viewed as a preprocessing strategy. These representations are computed once and then used instead of the original scalar values during the main optimization.
+• For $T = 1$ , the PLE-representation is effectively equivalent to the scalar representation.
+• Contrary to categorical features, numerical features are ordered; we express that by setting to 1 the components corresponding to bins with the right boundaries lower than the given feature value (this approach resembles how labels are encoded in ordinal regression problems).
+• The cases $( x < b _ { 0 } )$ ) and $( x \geq b _ { T } )$ ) are also covered by Equation 1 (which leads to $( e _ { 1 } \leq 0 )$ ) and $( e _ { T } \ge 1 )$ respectively).
+• The choice to make the representation piecewise linear is itself a subject for discussion. We analyze some alternatives in subsection 5.2.
+• PLE can be viewed as feature preprocessing, which is additionally discussed in subsection 5.3.
+A note on attention-based models. While the described PLE-representations can be passed to MLPlike models as is, attention-based models are inherently invariant to the order of input embeddings, so one additional step is required to add the information about feature indices to the obtained encodings. Technically, we observe that it is enough to place one linear layer after PLE(without sharing weights between features). Conceptually, however, this solution has a clear semantic interpretation. Namely, it is equivalent to allocating one trainable embedding $v _ { t } \in \mathbb { R } ^ { d }$ for each bin $B _ { t }$ and obtaining the final feature embedding by aggregating the embeddings of its bins with $e _ { t }$ as weights, plus bias $v _ { 0 }$ . Formally: $\begin{array} { r } { f _ { i } \left( x \right) = v _ { 0 } + \sum _ { t = 1 } ^ { T } e _ { t } \cdot v _ { t } = \mathrm { L i n e a r } \left( \mathrm { P L E } \left( x \right) \right) } \end{array}$ .
+In the following two sections, we describe two simple algorithms for building bins suitable for PLE. Namely, we rely on the classic binning algorithms [11] and one of the two algorithms is unsupervised, while another one utilizes labels for constructing bins.
+# 3.2.1 Obtaining bins from quantiles
+A natural baseline way to construct the bins for PLE is by splitting value ranges according to the uniformly chosen empirical quantiles of the corresponding individual feature distributions. Formally, for the $i$ -th feature: $b _ { t } = \mathbb { Q } _ { \frac { t } { T } } \left( \{ x _ { i } ^ { j \left( n u m \right) } \} _ { j \in J _ { t r a i n } } \right)$ , where $\mathsf { Q }$ is the empirical quantile function. Trivial bins of zero size are removed. In subsection D.1, we demonstrate the usefulness of the proposed scheme on the synthetic GBDT-friendly dataset described in section 5.1 in Gorishniy et al. [13].
+# 3.2.2 Building target-aware bins
+In fact, there are also supervised approaches that employ training labels for constructing bins [11]. Intuitively, such target-aware algorithms aim to produce bins that correspond to relatively narrow ranges of possible target values. The supervised approach used in our work is identical in its spirit to the $" \mathrm { C } 4 . 5$ Discretization" algorithm from Kohavi and Sahami [23]. In a nutshell, for each feature, we recursively split its value range in a greedy manner using target as guidance, which is equivalent to building a decision tree (which uses for growing only this one feature and the target) and treating the regions corresponding to its leaves as the bins for PLE (see the illustration in Figure 4). Additionally, we define $\begin{array} { r } { b _ { 0 } ^ { i } = \operatorname* { m i n } _ { j \in J _ { t r a i n } } x _ { i } ^ { j } } \end{array}$ and $b _ { T } ^ { i } = \operatorname* { m a x } _ { j \in J _ { t r a i n } } x _ { i } ^ { j }$ .
+# 3.3 Periodic activation functions
+Recall that in subsection 3.2 the work by Tancik et al. [42] was used as a starting point of our motivation for developing PLE. Thus, we also try to adapt the original work itself for tabular data problems. Our variation differs in two aspects. First, we take into account the fact the embedding framework described in subsection 3.1 forbids mixing features during the embedding process (see subsection D.2 for additional discussion). Second, we train the pre-activation coefficients instead of keeping them fixed. As a result, our approach is rather close to Li et al. [25] with the number of “groups” equal to the number of numerical features. We formalize the described scheme in Equation 2,
+$$
+f _ { i } ( x ) = { \mathsf { P e r i o d i c } } ( x ) = { \mathsf { c o n c a t } } [ { \mathsf { s i n } } ( v ) , { \mathsf { c o s } } ( v ) ] , \qquad v = [ 2 \pi c _ { 1 } x , \ldots , 2 \pi c _ { k } x ]
+$$
+where $c _ { i }$ are trainable parameters initialized from $\mathcal { N } ( 0 , \sigma )$ . We observe that $\sigma$ is an important hyperparameter. Both $\sigma$ and $k$ are tuned using validation sets.
+# 3.4 Simple differentiable layers
+In the context of Deep Learning, embedding numerical features with conventional differentiable layers (e.g. linear layers, ReLU activation, etc.) is a natural approach. In fact, this technique is already used on its own in the recently proposed attention-based architectures [13, 24, 39] and in some models for CTR prediction problems [14, 40]. However, we also note that such conventional modules can be used on top of the components described in subsection 3.2 and subsection 3.3. In section 4, we find that such combinations often lead to better results.
+# 4 Experiments
+In this section, we empirically evaluate the techniques discussed in section 3 and compare them with Gradient Boosted Decision Trees to check the status quo of the “DL vs GBDT” competition.
+# 4.1 Datasets
+Table 1: Dataset properties. “RMSE” denotes root-mean-square error, “Acc.” denotes accuracy.
+<table><tr><td></td><td>GE</td><td>CH</td><td>CA</td><td>HO</td><td>AD</td><td>OT</td><td>HI</td><td>FB</td><td>SA</td><td>CO</td><td>MI</td></tr><tr><td>#objects</td><td>9873</td><td>10000</td><td>20640</td><td>22784</td><td>48842</td><td>61878</td><td>98049</td><td>197080</td><td>200000</td><td>581012</td><td>1200192</td></tr><tr><td>#num.features</td><td>32</td><td>10</td><td>8</td><td>16</td><td>6</td><td>93</td><td>28</td><td>50</td><td>200</td><td>54</td><td>136</td></tr><tr><td>#cat.features metric</td><td>0 Acc.</td><td>1 Acc.</td><td>0</td><td>0</td><td>8</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td></tr><tr><td>#classes</td><td>5</td><td></td><td>RMSE</td><td>RMSE</td><td>Acc.</td><td>Acc.</td><td>Acc.</td><td>RMSE</td><td>Acc.</td><td>Acc.</td><td>RMSE</td></tr><tr><td></td><td></td><td>2</td><td>1</td><td></td><td>2</td><td>9</td><td>2</td><td></td><td>2</td><td>7</td><td>1</td></tr><tr><td>majority class</td><td>29%</td><td>79%</td><td>1</td><td>1</td><td>76%</td><td>26%</td><td>52%</td><td>1</td><td>89%</td><td>48%</td><td>1</td></tr></table>
+We use eleven public datasets mostly from the previous works on tabular DL and Kaggle competitions. Importantly, we focus on the middle and large scale tasks, and our benchmark is biased towards GBDT-friendly problems, since, as of now, closing the gap with GBDT models on such tasks is one of the main challenges for tabular DL. The main dataset properties are summarized in Table 1 and the used sources and additional details are provided in Appendix C.
+# 4.2 Implementation details
+We mostly follow Gorishniy et al. [13] in terms of the hyperparameter tuning, training and evaluation protocols. Nevertheless, for completeness, we list all the details in Appendix E. In the next paragraph, we describe the implementation details specific to embeddings for numerical features.
+Embeddings for numerical features. If linear layers are used, we tune their output dimensions. The PLE hyperparameters are the same for all features. For quantile-based PLE, we tune the number of quantiles. For target-aware PLE, we tune the following parameters for decision trees: the maximum number of leaves, the minimum number of items per leaf, and the minimum information gain required for making a split when growing the tree. For the Periodic module (see Equation 2), we tune $\sigma$ and $k$ (these hyperparameters are the same for all features).
+# 4.3 Model names
+In the experiments, we consider different combinations of backbones and embeddings. For convenience, we use the “Backbone-Embedding” pattern to name the models, where “Backbone” denotes the backbone (e.g. MLP, ResNet, Transformer) and “Embedding” denotes the embedding type. See Table 2 for all considered embedding modules. Note that:
+• Periodic is defined in Equation 2. • $\mathrm { P L E _ { q } }$ denotes the quantile-based PLE. $\mathrm { P L E } _ { \mathrm { t } }$ denotes the target-aware PLE. • Linear− denotes bias-free linear layer. LReLU denotes leaky ReLU. AutoDis was proposed in Guo et al. [14] • “Transformer-L” is equivalent to FTTransformer [13].
+Table 2: Embedding names. See subsection 4.3
+<table><tr><td>Name</td><td>Embedding function(fi)</td></tr><tr><td>L</td><td>Linear</td></tr><tr><td>LR</td><td>ReLUoLinear</td></tr><tr><td>LRLR</td><td>ReLUoLinearoReLUoLinear</td></tr><tr><td>Q</td><td>PLEq</td></tr><tr><td>Q-L</td><td>Linear o PLEq</td></tr><tr><td>Q-LR</td><td>ReLUo Linear o PLEq</td></tr><tr><td>Q-LRLR</td><td>ReLUoLinear o ReLUo Linear o PLEq</td></tr><tr><td>T</td><td>PLEt</td></tr><tr><td>T-L</td><td>Linear o PLEt</td></tr><tr><td>T-LR</td><td>ReLUo Linear o PLEt</td></tr><tr><td>T-LRLR</td><td>ReLUoLinearoReLUoLinearoPLEt</td></tr><tr><td>P</td><td>Periodic</td></tr><tr><td>PL</td><td>LinearoPeriodic</td></tr><tr><td>PLR</td><td>ReLUoLinearoPeriodic</td></tr><tr><td>PLRLR</td><td>ReLUoLinearoReLUoLinearoPeriodic</td></tr><tr><td>AutoDis</td><td>Linear oSoftMaxo Linear_oLReLUoLinear_</td></tr></table>
+# 4.4 Simple differentiable embedding modules
+Table 3: Results for MLP equipped with simple embedding modules (see subsection 4.3). The metric values averaged over 15 random seeds are reported. The standard deviations are provided in Appendix F. We consider one result to be better than another if its mean score is better and its standard deviation is less than the difference. For each dataset, top results are in bold. Notation: ↓ corresponds to RMSE, $\uparrow$ corresponds to accuracy
+<table><tr><td></td><td>GE↑（</td><td></td><td></td><td>CH↑ CA↓ HO↓ AD↑ OT↑ HI↑ FB↓ SA↑ CO↑MI↓</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>MLP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.6320.856 0.495 3.204 0.854 0.818 0.720 5.686 0.912 0.964 0.747</td><td></td></tr><tr><td>MLP-L(</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.639 0.861 0.4753.123 0.856 0.820 0.723 5.684 0.916 0.963 0.748</td><td></td></tr><tr><td>MLP-LR 0.642 0.860 0.471 3.084 0.857 0.819 0.726 5.625 0.923 0.963 0.746</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>
+We start by evaluating embedding modules consisting of “conventional” differentiable layers (linear layers, ReLU activations, etc.). The results are summarized in Table 3.
+# The main takeaways:
+• first and foremost, the results indicate that MLP can benefit from embedding modules. Thus, we conclude that this backbone is worth attention when it comes to evaluating embedding modules. • the simple LR module leads to modest, but consistent improvements when applied to MLP.
+Interestingly, the “redundant” MLP-L configuration also tends to outperform the vanilla MLP. Although the improvements are not dramatic, the special property of this architecture is that the linear embedding module can be fused together with the first linear layer of MLP after training, which completely removes the overhead. As for LRLR and AutoDis, we observe that these heavy modules do not justify the extra costs (see the results in Appendix F).
+# 4.5 Piecewise linear encoding
+In this section, we evaluate the encoding scheme described in subsection 3.2. The results are summarized in Table 4.
+# The main takeaways:
+• The piecewise linear encoding is often beneficial for both types of architectures (MLP and Transformer) and the profit can be significant (for example, see the CA and AD datasets). • Adding differentiable components on top of the PLE can improve the performance. Though, the most expensive modifications such as $\mathsf { Q }$ -LRLR and T-LRLR are not worth it (see Appendix F).
+Note that the benchmark is biased towards GBDT-friendly problems, so the typical superiority of tree-based bins over quantile-based bins, which can be observed in Table 4, may not generalize to more DL-friendly datasets. Thus, we do not make any general claims about the relative advantages of the two schemes here.
+Table 4: Results for MLP and Transformer with embedding modules based on the piecewise linear encoding (subsection 3.2). Notation follows Table 3 and Table 2. The best results are defined separately for the MLP and Transformer backbones.
+<table><tr><td></td><td>GE↑</td><td>CH↑</td><td>CA↓</td><td>HO↓</td><td>AD个</td><td>OT↑</td><td>HI↑</td><td>FB↓</td><td>SA↑</td><td>CO →</td><td>MI↓</td></tr><tr><td>MLP</td><td>0.632</td><td>0.856</td><td>0.495</td><td>3.204</td><td>0.854</td><td>0.818</td><td>0.720</td><td>5.686</td><td>0.912</td><td>0.964</td><td>0.747</td></tr><tr><td>MLP-Q</td><td>0.653</td><td>0.854</td><td>0.464</td><td>3.163</td><td>0.859</td><td>0.816</td><td>0.721</td><td>5.766</td><td>0.922</td><td>0.968</td><td>0.750</td></tr><tr><td>MLP-T</td><td>0.647</td><td>0.861</td><td>0.447</td><td>3.149</td><td>0.864</td><td>0.821</td><td>0.720</td><td>5.577</td><td>0.923</td><td>0.967</td><td>0.749</td></tr><tr><td>MLP-Q-LR</td><td>0.646</td><td>0.857</td><td>0.455</td><td>3.184</td><td>0.863</td><td>0.811</td><td>0.720</td><td>5.394</td><td>0.923</td><td>0.969</td><td>0.747</td></tr><tr><td>MLP-T-LR</td><td>0.640</td><td>0.861</td><td>0.439</td><td>3.207</td><td>0.868</td><td>0.818</td><td>0.724</td><td>5.508</td><td>0.924</td><td>0.968</td><td>0.747</td></tr><tr><td>Transformer-L</td><td>0.632</td><td>0.860</td><td>0.465</td><td>3.239</td><td>0.858</td><td>0.817</td><td>0.725</td><td>5.602</td><td>0.924</td><td>0.971</td><td>0.746</td></tr><tr><td>Transformer-Q-L</td><td>0.659</td><td>0.856</td><td>0.451</td><td>3.319</td><td>0.867</td><td>0.812</td><td>0.729</td><td>5.741</td><td>0.924</td><td>0.973</td><td>0.747</td></tr><tr><td>Transformer-T-L</td><td>0.663</td><td>0.861</td><td>0.454</td><td>3.197</td><td>0.871</td><td>0.817</td><td>0.726</td><td>5.803</td><td>0.924</td><td>0.974</td><td>0.747</td></tr><tr><td>Transformer-Q-LR</td><td>0.659</td><td>0.857</td><td>0.448</td><td>3.270</td><td>0.867</td><td>0.812</td><td>0.723</td><td>5.683</td><td>0.923</td><td>0.972</td><td>0.748</td></tr><tr><td>Transformer-T-LR</td><td>0.665（</td><td>0.860</td><td>0.442</td><td>3.219</td><td>0.870</td><td>0.818</td><td>0.729</td><td>5.699</td><td>0.924</td><td>0.973</td><td>0.747</td></tr></table>
+# 4.6 Periodic activation functions
+Table 5: Results for MLP and Transformer with embedding modules based on periodic activations (subsection 3.3). Notation follows Table 3 and Table 2. The best results are defined separately for the MLP and Transformer backbones.
+<table><tr><td></td><td>GE个</td><td>CH↑</td><td>CA↓</td><td>HO↓</td><td>AD↑</td><td>OT↑</td><td>HI↑</td><td>FB←</td><td>SA↑</td><td>co↑</td><td>MI↓</td></tr><tr><td>MLP</td><td>0.632</td><td>0.856</td><td>0.495</td><td>3.204</td><td>0.854</td><td>0.818</td><td>0.720</td><td>5.686</td><td>0.912</td><td>0.964</td><td>0.747</td></tr><tr><td>MLP-P</td><td>0.631</td><td>0.860</td><td>0.489</td><td>3.129</td><td>0.869</td><td>0.807</td><td>0.723</td><td>5.845</td><td>0.923</td><td>0.968</td><td>0.747</td></tr><tr><td>MLP-PL</td><td>0.641</td><td>0.859</td><td>0.467</td><td>3.113</td><td>0.868</td><td>0.819</td><td>0.727</td><td>5.530</td><td>0.924</td><td>0.969</td><td>0.746</td></tr><tr><td>MLP-PLR</td><td>0.674</td><td>0.857</td><td>0.467</td><td>3.050</td><td>0.870</td><td>0.819</td><td>0.728</td><td>5.525</td><td>0.924</td><td>0.970</td><td>0.746</td></tr><tr><td>Transformer-L</td><td>0.632</td><td>0.860</td><td>0.465</td><td>3.239</td><td>0.858</td><td>0.817</td><td>0.725</td><td>5.602</td><td>0.924</td><td>0.971</td><td>0.746</td></tr><tr><td>Transformer-PLR</td><td>0.646</td><td>0.863</td><td>0.464</td><td>3.162</td><td>0.870</td><td>0.814</td><td>0.730</td><td>5.760</td><td>0.924</td><td>0.972</td><td>0.746</td></tr></table>
+In this section, we evaluate embedding modules based on periodic activation functions as described in subsection 3.3. The results are reported in Table 5.
+The main takeaway: on average, MLP-P is superior to the vanilla MLP. However, adding a differentiable component on top of the Periodic module should be the default strategy (which is in line with Li et al. [25]). Indeed, MLP-PLR and MLP-PL provide meaningful improvements over MLP-P (e.g. see GE, CA, HO) and even “fix” MLP-P where it is inferior to MLP (OT, FB).
+Although MLP-PLR is usually superior to MLP-PL, we note that in the latter case the last linear layer of the embedding module is “redundant” in terms of expressivity and can be fused with the first linear layer of the backbone after training, which, in theory, can lead to a more lightweight model. Finally, we observe that MLP-PLRLR and MLP-PLR do not differ significantly enough to justify the extra cost of the PLRLR module (see Appendix F).
+# 4.7 Comparing DL models and GBDT
+In this section, we perform a big comparison of different approaches to identify the best embedding modules and backbones, as well as to check if embeddings for numerical features allow DL models to compete with GBDT on more tasks than before. Importantly, we compare ensembles of DL models against ensembles of GBDT, since Gradient Boosting is essentially an ensembling technique, so such comparison will be fairer. Note that we focus only on the best metric values without taking efficiency into account, so we only check if DL models are conceptually ready to compete with GBDT.
+We consider three backbones: MLP, ResNet, and Transformer, since they are reported to be representative of what baseline DL backbones are currently capable of [13, 18, 24, 39]. Note that we do not include the attention-based models that also apply attention on the level of objects [24, 35, 39], since this non-parametric component is orthogonal to the central topic of our work. The results are summarized in Table 6.
+Table 6: Results for ensembles of GBDT, the baseline DL models and their modifications using different types of embeddings for numerical features. Notation follows Table 3 and Table 2. Due to the limited precision, some different values are represented with the same figures.
+<table><tr><td></td><td>GE↑</td><td>CH↑</td><td>CA↓</td><td>HO↓</td><td>AD↑</td><td>OT↑</td><td>HI↑</td><td>FB↓</td><td>SA↑</td><td>CO→</td><td>MI↓</td><td></td><td>Avg.Rank</td></tr><tr><td>CatBoost</td><td>0.692</td><td>0.861</td><td>0.430</td><td>3.093</td><td>0.873</td><td>0.825</td><td>0.727</td><td></td><td>5.226</td><td>0.924</td><td>0.967</td><td>0.741</td><td>3.6±2.9</td></tr><tr><td>XGBoost</td><td>0.683</td><td>0.859</td><td>0.434</td><td>3.152</td><td>0.875</td><td>0.827</td><td>0.726</td><td>5.338</td><td>0.919</td><td>0.969</td><td></td><td>0.742</td><td>4.6±2.7</td></tr><tr><td>MLP</td><td>0.665</td><td>0.856</td><td>0.486</td><td>3.109</td><td>0.856</td><td>0.822</td><td>0.727</td><td>5.616</td><td>0.913</td><td>0.968</td><td></td><td>0.746</td><td>8.5±2.6</td></tr><tr><td>MLP-LR</td><td>0.679</td><td>0.861</td><td>0.463</td><td>3.012</td><td>0.859</td><td>0.826</td><td>0.731</td><td>5.477</td><td>0.924</td><td>0.972</td><td></td><td>0.744</td><td>5.5 ± 2.7</td></tr><tr><td>MLP-Q-LR</td><td>0.682</td><td>0.859</td><td>0.433</td><td>3.080</td><td>0.867</td><td>0.818</td><td>0.724</td><td>5.144</td><td>0.924</td><td></td><td>0.974</td><td>0.745</td><td>5.1 ± 1.9</td></tr><tr><td>MLP-T-LR</td><td>0.673</td><td>0.861</td><td>0.435</td><td>3.099</td><td>0.870</td><td>0.821</td><td>0.727</td><td>5.409</td><td>0.924</td><td></td><td>0.973</td><td>0.746</td><td>5.1 ± 1.7</td></tr><tr><td>MLP-PLR</td><td>0.700</td><td>0.858</td><td>0.453</td><td>2.975</td><td>0.874</td><td>0.830</td><td>0.734</td><td>5.388</td><td>0.924</td><td></td><td>0.975</td><td>0.743</td><td>3.0±2.4</td></tr><tr><td>ResNet</td><td>0.690</td><td>0.861</td><td>0.483</td><td>3.081</td><td>0.856</td><td>0.821</td><td>0.734</td><td>5.482</td><td>0.918</td><td></td><td>0.968</td><td>0.745</td><td>6.7±3.3</td></tr><tr><td>ResNet-LR</td><td>0.672</td><td>0.862</td><td>0.450</td><td>2.992</td><td>0.859</td><td>0.822</td><td>0.733</td><td>5.415</td><td>0.923</td><td></td><td>0.971</td><td>0.743</td><td>5.6±2.7</td></tr><tr><td>ResNet-Q-LR</td><td>0.674</td><td>0.859</td><td>0.427</td><td>3.066</td><td>0.868</td><td>0.815</td><td>0.729</td><td>5.309</td><td></td><td>0.923</td><td>0.976</td><td>0.746</td><td>4.7 ± 2.0</td></tr><tr><td>ResNet-T-LR</td><td>0.683</td><td>0.862</td><td>0.425</td><td>3.030</td><td>0.872</td><td>0.822</td><td>0.731</td><td>5.471</td><td></td><td>0.923</td><td>0.975</td><td>0.744</td><td>4.1 ± 1.9</td></tr><tr><td>ResNet-PLR</td><td>0.691</td><td>0.861</td><td>0.443</td><td>3.040</td><td>0.874</td><td>0.825</td><td>0.734</td><td>5.400</td><td></td><td>0.924</td><td>0.975</td><td>0.743</td><td>3.2 ±1.3</td></tr><tr><td>Transformer-L</td><td>0.668</td><td>0.861</td><td>0.455</td><td>3.188</td><td>0.860</td><td>0.824</td><td>0.727</td><td></td><td>5.434</td><td>0.924</td><td>0.973</td><td>0.743</td><td>5.9±2.2</td></tr><tr><td>Transformer-LR</td><td>0.666</td><td>0.861</td><td>0.446</td><td>3.193</td><td>0.861</td><td>0.824</td><td>0.733</td><td></td><td>5.430</td><td>0.924</td><td>0.973</td><td>0.743</td><td>5.2 ±2.2</td></tr><tr><td>Transformer-Q-LR</td><td>0.690</td><td>0.857</td><td>0.425</td><td>3.143</td><td>0.868</td><td>0.818</td><td>0.726</td><td></td><td>5.471</td><td>0.924</td><td>0.975</td><td>0.744</td><td>4.4 ± 2.2</td></tr><tr><td>Transformer-T-LR</td><td>0.686</td><td>0.862</td><td>0.423</td><td>3.149</td><td>0.871</td><td>0.823</td><td>0.733</td><td></td><td>5.515</td><td>0.924</td><td>0.976</td><td>0.744</td><td>3.7±2.2</td></tr><tr><td>Transformer-PLR</td><td>0.686</td><td>0.864</td><td>0.449</td><td>3.091</td><td>0.873</td><td>0.823</td><td>0.734</td><td></td><td>5.581</td><td>0.924</td><td>0.975</td><td>0.743</td><td>3.9 ±2.5</td></tr></table>
+# The main takeaways for DL models:
+• For most datasets, embeddings for numerical features can provide noticeable improvements for three different backbones. Although the average rank is not a good metric for making subtle conclusions, we highlight the impressive difference in average ranks between the MLP and MLP-PLR models.
+• The simplest LR embedding is a good baseline solution: although the performance gains are not dramatic, its main advantage is consistency (e.g. see MLP vs MLP-LR).
+• The PLR module provides the best average performance. Empirically, we observe $\sigma$ (see Equation 2) to be an important hyperparameter that should be tuned.
+• Piecewise linear encoding (PLE) allows building well performing embeddings (e.g. T-LR, Q-LR). In addition to that, PLE itself is worth attention because of its simplicity, interpretability and efficiency (no computationally expensive periodic functions).
+• Importantly, after the MLP-like architectures are coupled with embeddings for numerical features, they perform on par with the Transformer-based models.
+The main takeaway for the “DL vs GBDT” competition: embeddings for numerical features is a significant design aspect that has a great potential for improving DL models and closing the gap with GBDT on GBDT-friendly tasks. Let us illustrate this claim with several observations:
+• The benchmark is initially biased to GBDT-friendly problems, which can be observed by comparing GBDT solutions with the vanilla DL models (MLP, ResNet, Transformer-L). • However, for the vast majority of the “backbone & dataset” pairs, proper embeddings are the only thing needed to close the gap with GBDT. Exceptions (rather formal) include the MI dataset and the following pairs: “ResNet & GE”, “Transformer & FB”, “Transformer & GE”, “Transformer & OT”. • Additionally, to the best of our knowledge, it is the first time when DL models perform on par with GBDT on the well-known California Housing and Adult datasets.
+That said, compared to GBDT models, efficiency can still be an issue for the considered DL architectures. In any case, the trade-off completely depends on the specific use case and requirements.
+# 5 Analysis
+# 5.1 Comparing model sizes
+To quantify the effect of embeddings for numerical features on model sizes, we report the parameter counts in Table 7. Overall, introducing embeddings for numerical features can cause non-negligible overhead in terms of model size. Importantly, the overhead in terms of size does not translate to the same overhead in terms of training times and throughput. For example, the almost 2000-fold increase in the parameter count for MLP-LR on the CH dataset results in only 1.5-fold increase in training times. Finally, in practice, we observe that coupling MLP and ResNet with embedding modules leads to architectures that are still faster than Transformer-based models.
+Table 7: Parameter counts for MLP with different embedding modules. All the models are tuned and the corresponding backbones are not identical in their sizes, so we take into account the fact that different approaches require a different number of parameters to realize their full potential.
+<table><tr><td></td><td>GE</td><td>CH</td><td>CA</td><td>HO</td><td>AD</td><td>OT</td><td>HI</td><td>FB</td><td>SA</td><td>CO</td><td>MI</td></tr><tr><td>MLP</td><td>2.0M</td><td>1.5K</td><td>43.5K</td><td>3.6M</td><td>5.3M</td><td>479.9K</td><td>25.8K</td><td>937.3K</td><td>5.8M</td><td>3.2M</td><td>276.5K</td></tr><tr><td>MLP-LR</td><td>×2.52</td><td>×1931.03</td><td>×25.05</td><td>×1.28</td><td>×0.35</td><td>×12.53</td><td>×68.16</td><td>×4.76</td><td>×1.58</td><td>×0.72</td><td>×15.79</td></tr><tr><td>MLP-T</td><td>×1.58</td><td>×14.13</td><td>×7.97</td><td>×0.43</td><td>×0.04</td><td>×2.27</td><td>×5.85</td><td>×0.47</td><td>×0.59</td><td>×0.74</td><td>×3.85</td></tr><tr><td>MLP-T-LR</td><td>×1.61</td><td>×463.55</td><td>×6.80</td><td>×0.23</td><td>×0.16</td><td>×2.52</td><td>×113.22</td><td>×3.43</td><td>×0.41</td><td>×0.35</td><td>×8.47</td></tr><tr><td>MLP-PLR</td><td>×1.73</td><td>×250.24</td><td>×12.94</td><td>×1.07</td><td>×0.66</td><td>×8.05</td><td>×110.57</td><td>×4.93</td><td>×0.64</td><td>×0.44</td><td>×9.57</td></tr></table>
+# 5.2 Ablation study
+Table 8: Comparing piecewise linear encoding (PLE) with the two variations described in subsection 5.2. Notation follows Table 3 and Table 2.
+<table><tr><td></td><td>GE↑</td><td>CH↑</td><td>CA↓1</td><td>HO←</td><td>AD↑</td><td>OT↑</td><td>HI↑</td><td>FB←</td></tr><tr><td>MLP-Q(piecewise linear) 0.653 0.854</td><td></td><td></td><td>0.464</td><td></td><td></td><td>3.163 0.859 0.816 0.721</td><td></td><td>5.766</td></tr><tr><td>MLP-Q (binary)</td><td>0.652</td><td>0.815</td><td>0.462</td><td>3.200</td><td>0.860</td><td>）0.810</td><td>0.720</td><td>5.748</td></tr><tr><td>MLP-Q (one-blob)</td><td>0.613 0.851（</td><td></td><td></td><td></td><td></td><td>0.461 3.187 0.857 0.808</td><td>30.719</td><td>5.645</td></tr><tr><td>MLP-T (piecewise linear)</td><td>0.647 0.861</td><td></td><td>0.447 3.149</td><td></td><td></td><td>90.8640.821（</td><td>0.720</td><td>5.577</td></tr><tr><td>MLP-T (binary)</td><td>0.639</td><td>90.855</td><td></td><td></td><td></td><td>0.4643.163 0.869 0.813</td><td>0.718</td><td>5.572</td></tr><tr><td>MLP-T (one-blob)</td><td>0.622</td><td>0.858</td><td>0.464</td><td>3.158</td><td>0.870</td><td>0.809</td><td>0.724</td><td>5.475</td></tr></table>
+In this section, we compare two alternative binning-based encoding schemes with PLE (see subsection 3.2). The first one ("thermometer" [6]) sets the value 1 instead of the piecewise linear term (see Equation 1). The second one is a generalized version of the one-blob encoding [29] (see subsection E.1 for details). The tuning and evaluation protocols are the same as in subsection 4.2. The results in table Table 8 indicate that making the binning-based encoding piecewise linear is a good default strategy.
+# 5.3 Piecewise linear encoding as a feature preprocessing technique
+It is known that data preprocessing, such as standardization or quantile transformation, is often crucial for DL models for achieving competitive performance. Moreover, the performance can significantly vary between different types of preprocessing. At the same time, PLE-representations contain only values from [0, 1] and they are invariant to shifting and scaling, which makes PLE itself a general feature preprocessing technique potentially suitable for DL models without the need to use traditional preprocessing first.
+To illustrate that, for datasets where the quantile transformation was used in section 4, we reevaluate the tuned configurations of MLP, MLP-Q, and MLP-T with different preprocessing policies and report the results in Table 9 (note that standardization is equivalent to no preprocessing for models with PLE).
+Table 9: Results for MLP and MLP with PLE for different types of data preprocessing. Solutions using PLE are significantly less sensitive to data preprocessing. Notation follows Table 3 and Table 2.
+<table><tr><td></td><td>GE↑</td><td>CH↑</td><td>CA↓HO↓AD↑]</td><td></td><td>HI↑</td><td>FB↓</td><td>SA↑</td><td>CO↑1</td><td>MI↓</td></tr><tr><td>MLP (none)</td><td>0.565</td><td></td><td>0.796 1.118 5.328 0.808 0.707</td><td></td><td></td><td>13.125</td><td>0.911</td><td></td><td>0.948 0.844</td></tr><tr><td>MLP (standard)</td><td>0.629</td><td></td><td>0.855 0.509 3.3030.855（</td><td></td><td>0.721</td><td>5.919</td><td>0.912</td><td></td><td>0.9630.754</td></tr><tr><td>MLP (quantile)</td><td>0.632</td><td></td><td>0.8560.495 3.204 0.8540.720</td><td></td><td></td><td>5.686</td><td>0.912</td><td></td><td>0.9640.747</td></tr><tr><td>MLP-Q (none)</td><td></td><td></td><td>0.654 0.851 0.463 3.162 0.860 0.721</td><td></td><td></td><td>5.889</td><td></td><td></td><td>0.922 0.968 0.754</td></tr><tr><td>MLP-Q(quantile) 0.653 0.854 0.464 3.163 0.859 0.721</td><td></td><td></td><td></td><td></td><td></td><td>5.766</td><td></td><td></td><td>0.922 0.968 0.750</td></tr><tr><td>MLP-T (none)</td><td></td><td></td><td>0.644 0.860 0.447 3.175 0.865 0.721</td><td></td><td></td><td>5.598</td><td></td><td></td><td>0.9230.968 0.749</td></tr><tr><td>MLP-T (quantile)</td><td></td><td></td><td>0.647 0.861 0.447 3.149 0.864 0.720</td><td></td><td></td><td>5.577</td><td>0.923</td><td></td><td>0.9670.749</td></tr></table>
+First, the vanilla MLP often becomes unusable without preprocessing. Second, for the vanilla MLP, it can be important to choose one specific type of preprocessing (CA, HO, FB, MI), which is less pronounced for MLP-Q and not the case for MLP-T (though, this specific observation can be the property of the benchmarks, not of MLP-T). Overall, the results indicate that models using PLE are less sensitive to the initial preprocessing compared to the vanilla MLP. This is an additional benefit of PLE-representations for practitioners since the aspect of preprocessing becomes less critical with PLE.
+# 5.4 The “feature engineering” perspective
+Table 10: The comparison of the effects of Periodic-based modules for XGBoost and MLP
+<table><tr><td>CA↓HO↓HI↑</td></tr><tr><td>XGBoost 0.436 3.160 0.724 XGBoost with Periodic 0.441 3.184 0.724</td></tr><tr><td>MLP 0.4953.204 0.720</td></tr><tr><td>MLP-PL</td></tr><tr><td>0.467 3.113 0.727</td></tr></table>
+At first sight, feature embeddings may resemble feature engineering and should be suitable for all kinds of models. However, the proposed embedding schemes are motivated by DL-specific aspects of training (see the motivational parts of subsection 3.2 and subsection 3.3). While our methods are likely to transfer well to models with similar training properties (e.g. to linear models since those are a special case of deep models), it is not the case in general. To illustrate that, we try adopting the Periodic module for XGBoost by fixing the random coefficients from Equation 2. We also keep the original features instead of dropping them. The tuning and evaluation protocols are the same as in subsection 4.2. The results in Table 10 show that this technique, while being useful for DL models, does not provide any benefits for XGBoost.
+# 6 Conclusion & Future work
+In this work, we have demonstrated that embeddings for numerical features are an important design aspect of tabular DL architectures. Namely, it allows existing DL backbones to achieve noticeably better results and significantly reduce the gap with Gradient Boosted Decision Trees. We have described two approaches illustrating this phenomenon, one using the piecewise linear encoding of original scalar values, and another using periodic functions. We have also shown that traditional MLP-like models coupled with embeddings can perform on par with attention-based models.
+Nevertheless, we have only scratched the surface of the new direction. For example, it is still to be explained how exactly the discussed embedding modules help optimization on the fundamental level. Additionally, we have considered only schemes where the same functional transformation was applied to all features, which may be a suboptimal choice.
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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] See the analysis in subsection 5.1.
+(c) Did you discuss any potential negative societal impacts of your work? [N/A] The work focuses on a generic aspect of deep learning models.
+(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] We do not include theoretical results.
+(b) Did you include complete proofs of all theoretical results? [N/A] We do not include theoretical results.
+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] See the supplementary material.
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] The supplementary material includes the script used to create data splits. The hyperparameters are either explicitly described in subsection 4.2 and supplementary material, or tuned as described in subsection 4.2.
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We provide standard deviations in the supplementary material, see Table 18 and see Table 19
+(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] The experiment reports included in the supplementary material provide the information about the used hardware and execution times.
+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] See Appendix C.
+(b) Did you mention the license of the assets? [Yes] In the README.md file in the supplementary material, we refer to the original licenses of the used datasets.
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A] We do not provide new datasets.
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] We use publicly available datasets.
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] We use publicly available datasets.
+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] We did not use crowdsourcing. We did not conduct research with human subjects.
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] We did not use crowdsourcing. We did not conduct research with human subjects.
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] We did not use crowdsourcing. We did not conduct research with human subjects.

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+    {
+        "type": "text",
+        "text": "On Embeddings for Numerical Features in Tabular Deep Learning ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Yury Gorishniy∗ Yandex ",
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ivan Rubachev HSE, Yandex ",
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Artem Babenko Yandex ",
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+        "text": "Recently, Transformer-like deep architectures have shown strong performance on tabular data problems. Unlike traditional models, e.g., MLP, these architectures map scalar values of numerical features to high-dimensional embeddings before mixing them in the main backbone. In this work, we argue that embeddings for numerical features are an underexplored degree of freedom in tabular DL, which allows constructing more powerful DL models and competing with gradient boosted decision trees (GBDT) on some GBDT-friendly benchmarks (that is, where GBDT outperforms conventional DL models). We start by describing two conceptually different approaches to building embedding modules: the first one is based on a piecewise linear encoding of scalar values, and the second one utilizes periodic activations. Then, we empirically demonstrate that these two approaches can lead to significant performance boosts compared to the embeddings based on conventional blocks such as linear layers and ReLU activations. Importantly, we also show that embedding numerical features is beneficial for many backbones, not only for Transformers. Specifically, after proper embeddings, simple MLP-like models can perform on par with the attention-based architectures. Overall, we highlight embeddings for numerical features as an important design aspect with good potential for further improvements in tabular DL. The source code is available at https://github.com/Yura52/tabular-dl-num-embeddings. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+            606,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Tabular data problems are currently a final frontier for deep learning (DL) research. While the most recent breakthroughs in NLP, vision, and speech are achieved by deep models [12], their success in the tabular domain is not convincing yet. Despite a large number of proposed architectures for tabular DL [2, 3, 13, 17, 21, 24, 31, 39, 40], the performance gap between them and the “shallow” ensembles of decision trees, like GBDT, often remains significant [13, 36]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The recent line of works [13, 24, 39] reduce this performance gap by successfully adapting the Transformer architecture [45] for the tabular domain. Compared to traditional models, like MLP or ResNet, the proposed Transformer-like architectures have a specific way to handle numerical features of the data. Namely, they map scalar values of numerical features to high-dimensional embedding vectors, which are then mixed by the self-attention modules. Beyond transformers, mapping numerical features to vectors was also employed in different forms in the click-through rate (CTR) prediction problems [8, 14, 40]. Nevertheless, the literature is mostly focused on developing more powerful backbones while keeping the design of embedding modules relatively simple. In particular, the existing architectures [13, 14, 24, 39, 40] construct embeddings for numerical features using quite restrictive parametric mappings, e.g., linear functions, which can lead to suboptimal performance. In this work, we demonstrate that the embedding step has a substantial impact on the model effectiveness, and its proper design can significantly improve tabular DL models. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Specifically, we describe two different building blocks suitable for constructing embeddings for numerical features. The first one is a piecewise linear encoding that produces alternative initial representations for the original scalar values and is based on feature binning, a long-existing preprocessing technique [11]. The second one relies on periodic activation functions, which is inspired by their usage in implicit neural representations [28, 38, 42], NLP [41, 45] and CV tasks [25]. The first approach is simple, interpretable and non-differentiable, while the second demonstrates better results on average. We observe that DL models equipped with our embedding schemes successfully compete with GBDT on GBDT-friendly benchmarks and achieve the new state-of-the-art on tabular DL. ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "As another important finding, we demonstrate that the step of embedding the numerical features is universally beneficial for different deep architectures, not only for Transformer-like ones. In particular, we show, that after proper embeddings, simple MLP-like architectures often provide the performance comparable to the state-of-the-art attention-based models. Overall, our work demonstrates the large impact of the embeddings of numerical features on the tabular DL performance and shows the potential of investigating more advanced embedding schemes in future research. ",
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+    {
+        "type": "text",
+        "text": "To sum up, our contributions are as follows: ",
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+    {
+        "type": "text",
+        "text": "1. We demonstrate that embedding schemes for numerical features are an underexplored research question in tabular DL. Namely, we show that more expressive embedding schemes can provide substantial performance improvements over prior models.   \n2. We show that the profit from embedding numerical features is not specific for Transformerlike architectures, and proper embedding schemes benefit traditional models as well.   \n3. On a number of public benchmarks, we achieve the new state-of-the-art on tabular DL. ",
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+    {
+        "type": "text",
+        "text": "2 Related work ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 1
+    },
+    {
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+        "text": "Tabular deep learning. During several recent years, the community has proposed a large number of deep models for tabular data [2, 3, 13, 15, 17, 21, 24, 31, 39, 40, 46]. However, when systematically evaluated, these models do not consistently outperform the ensembles of decision trees, such as GBDT (Gradient Boosting Decision Tree) [7, 19, 32], which are typically the top-choice in various ML competitions [13, 36]. Moreover, several recent works have shown that the proposed sophisticated architectures are not superior to properly tuned simple models, like MLP and ResNet [13, 18]. In this work, unlike the prior literature, we do not aim to propose a new backbone architecture. Instead, we focus on more accurate ways to handle numerical features, and our developments can be potentially combined with any model, including traditional MLPs and more recent Transformer-like ones. ",
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+    {
+        "type": "text",
+        "text": "Transformers in tabular DL. Due to the tremendous success of Transformers for different domains [10, 45], several recent works adapt their self-attention design for tabular DL as well [13, 17, 24, 39]. Compared to existing alternatives, applying self-attention modules to the numerical features of tabular data requires mapping the scalar values of these features to high-dimensional embedding vectors. So far, the existing architectures perform this “scalar” “vector” mapping by relatively simple computational blocks, which, in practice, can limit the model expressiveness. For instance, the recent FT-Transformer architecture [13] employs only a single linear layer. In our experiments, we demonstrate that such embedding schemes can provide suboptimal performance, and more advanced schemes often lead to substantial profit. ",
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "CTR Prediction. In CTR prediction problems, objects are represented by numerical and categorical features, which makes this field highly relevant to tabular data problems. In several works, numerical features are handled in some non-trivial way while not being the central part of the research [8, 40]. Recently, however, a more advanced scheme has been proposed in Guo et al. [14]. Nevertheless, it is still based on linear layers and conventional activation functions, which we found to be suboptimal in our evaluation. ",
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+    {
+        "type": "text",
+        "text": "Feature binning. Binning is a discretization technique that converts numerical features to categorical features. Namely, for a given feature, its value range is split into bins (intervals), after which the original feature values are replaced with discrete descriptors (e.g. bin indices or one-hot vectors) of the corresponding bins. We point to the work by Dougherty et al. [11], which performs an overview of some classic approaches to binning and can serve as an entry point to the relevant literature on the topic. In our work, however, we utilize bins in a different way. Specifically, we use their edges to construct lossless piecewise linear representations of the original scalar values. It turns out that this simple and interpretable representations can provide substantial benefit to deep models on several tabular problems. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Periodic activations. Recently, periodic activation functions have become a key component in processing coordinates-like inputs, which is required in many applications. Examples include NLP [45], CV [25], implicit neural representations [28, 38, 42]. In our work, we show that periodic activations can be used to construct powerful embedding modules for numerical features in tabular data problems. Contrary to some of the aforementioned papers, where components of the multidimensional coordinates are mixed (e.g. with linear layers) before passing them to periodic functions [38, 42], we find it crucial to embed each feature separately before mixing them in the main backbone. ",
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+    {
+        "type": "text",
+        "text": "3 Embeddings for numerical features ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we describe the general framework for what we call \"embeddings for numerical features\" and the main building blocks used in the experimental comparison in section 4. ",
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+        "type": "text",
+        "text": "Notation. For a given supervised learning problem on tabular data, we denote the dataset as $\\left\\{ \\left( x ^ { j } , \\ y ^ { j } \\right) \\right\\} _ { j = 1 } ^ { n }$ where $y ^ { j } \\in \\mathbb { Y }$ represents the object’s label and $x ^ { j } = \\left( x ^ { j \\left( n u m \\right) } , x ^ { j \\left( c a t \\right) } \\right) \\in \\mathbb { X }$ represents the object’s features (numerical and categorical). xj(num)i , in turn, denotes the i-th numerical feature of the -th object. Depending on the context, the index can be omitted. The dataset is split into three disjoint parts: $\\overline { { 1 , n } } \\overline { { = J _ { t r a i n } } } \\cup J _ { v a l } \\cup J _ { t e s t }$ , where the “train” part is used for training, the “validation” part is used for early stopping and hyperparameter tuning, and the “test” part is used for the final evaluation. ",
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+        "type": "text",
+        "text": "3.1 General framework ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We formalize the notion of \"embeddings for numerical features\" as $z _ { i } = f _ { i } ( ( x _ { i } ^ { ( n u m ) } ) \\in \\mathbb { R } ^ { d _ { i } }$ where $f _ { i } ( x )$ is the embedding function for the $i$ -th numerical feature, $z _ { i }$ is the embedding of the $i$ -th numerical feature and $d _ { i }$ is the dimensionality of the embedding. Importantly, the proposed framework implies that embeddings for all features are computed independently of each other. Note that the function $f _ { i }$ can depend on parameters that are trained as a part of the whole model or in some other fashion (e.g. before the main optimization). In this work, we consider only embedding schemes where the embedding functions for all features are of the same functional form. We never share parameters of embedding functions of different features. ",
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+    {
+        "type": "text",
+        "text": "The subsequent use of the embeddings depends on the model backbone. For MLP-like architectures, they are concatenated into one flat vector (see Appendix A for illustrations). For Transformer-based architectures, no extra step is performed and the embeddings are passed as is, so the usage is defined by the original architectures. ",
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+    {
+        "type": "text",
+        "text": "3.2 Piecewise linear encoding ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "While vanilla MLP is known to be a universal approximator [9, 16], in practice, due to optimization peculiarities, it has limitations in its learning capabilities [34]. However, the recent work by Tancik et al. [42] uncovers the case where changing the input space alleviates the above issue. This observation motivates us to check if changing the representations of the original scalar values of numerical features can improve the learning capabilities of tabular DL models. ",
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+    {
+        "type": "text",
+        "text": "At this point, we try to start simple and turn to \"classical\" machine learning techniques. Namely, we take inspiration from the one-hot encoding algorithm that is widely and successfully used for representing discrete entities such as categorical features in tabular data problems or tokens in NLP. We note that the one-hot representation can be seen as an opposite solution to the scalar representation in terms of the trade-off between parameter efficiency and expressivity. To check whether the onehot-like approach can be beneficial for tabular DL models, we design a continuous alternative to the one-hot encoding (since the vanilla one-hot encoding is barely applicable to numerical features). ",
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+    {
+        "type": "text",
+        "text": "Formally, for the $i \\cdot$ -th numerical feature, we split its value range into the disjoint set of $T ^ { i }$ intervals $B _ { 1 } ^ { i }$ , . . . , $B _ { T } ^ { i }$ , which we call bins: $B _ { t } ^ { i } = [ b _ { t - 1 } ^ { i } , b _ { t } ^ { i } )$ . The splitting algorithm is an important implementation detail that we discuss later. From now on, we omit the feature index $i$ for simplicity. Once the bins are determined, we define the encoding scheme as in Equation 1: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b687c961a6ae96047028d71ddc1736199d29395151ce786d890a7d35e07479d0.jpg",
+        "text": "$$\n\\mathsf { P L E } ( x ) = [ e _ { 1 } , ~ . ~ . ~ . , ~ e _ { T } ] \\in \\mathbb { R } ^ { T }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/f5202934799064ad4e2b586a8831329f8e87577c4126eb491750a214dfb1b550.jpg",
+        "text": "$$\ne _ { t } = \\left\\{ \\begin{array} { l l } { 0 , } & { x < b _ { t - 1 } \\mathrm { ~ A N D ~ } t > 1 } \\\\ { 1 , } & { x \\geq b _ { t } \\mathrm { ~ A N D ~ } t < T } \\\\ { \\frac { x - b _ { t - 1 } } { b _ { t } - b _ { t - 1 } } , } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where PLE stands for “peicewise linear encoding”. We provide the visualization in Figure 1. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/ac479be40b57aa351199b81061f9a1933898465f23d6b971cf0fb15c89fb5d38.jpg",
+        "image_caption": [
+            "Figure 1: The piecewise linear encoding (PLE) in action for $T = 4$ (see Equation 1). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Note that: ",
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+    {
+        "type": "text",
+        "text": "• PLE produces alternative initial representations for the numerical features and can be viewed as a preprocessing strategy. These representations are computed once and then used instead of the original scalar values during the main optimization.   \n• For $T = 1$ , the PLE-representation is effectively equivalent to the scalar representation.   \n• Contrary to categorical features, numerical features are ordered; we express that by setting to 1 the components corresponding to bins with the right boundaries lower than the given feature value (this approach resembles how labels are encoded in ordinal regression problems).   \n• The cases $( x < b _ { 0 } )$ ) and $( x \\geq b _ { T } )$ ) are also covered by Equation 1 (which leads to $( e _ { 1 } \\leq 0 )$ ) and $( e _ { T } \\ge 1 )$ respectively).   \n• The choice to make the representation piecewise linear is itself a subject for discussion. We analyze some alternatives in subsection 5.2.   \n• PLE can be viewed as feature preprocessing, which is additionally discussed in subsection 5.3. ",
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+    {
+        "type": "text",
+        "text": "A note on attention-based models. While the described PLE-representations can be passed to MLPlike models as is, attention-based models are inherently invariant to the order of input embeddings, so one additional step is required to add the information about feature indices to the obtained encodings. Technically, we observe that it is enough to place one linear layer after PLE(without sharing weights between features). Conceptually, however, this solution has a clear semantic interpretation. Namely, it is equivalent to allocating one trainable embedding $v _ { t } \\in \\mathbb { R } ^ { d }$ for each bin $B _ { t }$ and obtaining the final feature embedding by aggregating the embeddings of its bins with $e _ { t }$ as weights, plus bias $v _ { 0 }$ . Formally: $\\begin{array} { r } { f _ { i } \\left( x \\right) = v _ { 0 } + \\sum _ { t = 1 } ^ { T } e _ { t } \\cdot v _ { t } = \\mathrm { L i n e a r } \\left( \\mathrm { P L E } \\left( x \\right) \\right) } \\end{array}$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the following two sections, we describe two simple algorithms for building bins suitable for PLE. Namely, we rely on the classic binning algorithms [11] and one of the two algorithms is unsupervised, while another one utilizes labels for constructing bins. ",
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+    {
+        "type": "text",
+        "text": "3.2.1 Obtaining bins from quantiles ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "A natural baseline way to construct the bins for PLE is by splitting value ranges according to the uniformly chosen empirical quantiles of the corresponding individual feature distributions. Formally, for the $i$ -th feature: $b _ { t } = \\mathbb { Q } _ { \\frac { t } { T } } \\left( \\{ x _ { i } ^ { j \\left( n u m \\right) } \\} _ { j \\in J _ { t r a i n } } \\right)$ , where $\\mathsf { Q }$ is the empirical quantile function. Trivial bins of zero size are removed. In subsection D.1, we demonstrate the usefulness of the proposed scheme on the synthetic GBDT-friendly dataset described in section 5.1 in Gorishniy et al. [13]. ",
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+    {
+        "type": "text",
+        "text": "3.2.2 Building target-aware bins ",
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+    {
+        "type": "text",
+        "text": "In fact, there are also supervised approaches that employ training labels for constructing bins [11]. Intuitively, such target-aware algorithms aim to produce bins that correspond to relatively narrow ranges of possible target values. The supervised approach used in our work is identical in its spirit to the $\" \\mathrm { C } 4 . 5$ Discretization\" algorithm from Kohavi and Sahami [23]. In a nutshell, for each feature, we recursively split its value range in a greedy manner using target as guidance, which is equivalent to building a decision tree (which uses for growing only this one feature and the target) and treating the regions corresponding to its leaves as the bins for PLE (see the illustration in Figure 4). Additionally, we define $\\begin{array} { r } { b _ { 0 } ^ { i } = \\operatorname* { m i n } _ { j \\in J _ { t r a i n } } x _ { i } ^ { j } } \\end{array}$ and $b _ { T } ^ { i } = \\operatorname* { m a x } _ { j \\in J _ { t r a i n } } x _ { i } ^ { j }$ . ",
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+    {
+        "type": "text",
+        "text": "3.3 Periodic activation functions ",
+        "text_level": 1,
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Recall that in subsection 3.2 the work by Tancik et al. [42] was used as a starting point of our motivation for developing PLE. Thus, we also try to adapt the original work itself for tabular data problems. Our variation differs in two aspects. First, we take into account the fact the embedding framework described in subsection 3.1 forbids mixing features during the embedding process (see subsection D.2 for additional discussion). Second, we train the pre-activation coefficients instead of keeping them fixed. As a result, our approach is rather close to Li et al. [25] with the number of “groups” equal to the number of numerical features. We formalize the described scheme in Equation 2, ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ec7e2c8973a78840a2c075d5460ae5ac5b71157c85269c4bd2773fd3a848c942.jpg",
+        "text": "$$\nf _ { i } ( x ) = { \\mathsf { P e r i o d i c } } ( x ) = { \\mathsf { c o n c a t } } [ { \\mathsf { s i n } } ( v ) , { \\mathsf { c o s } } ( v ) ] , \\qquad v = [ 2 \\pi c _ { 1 } x , \\ldots , 2 \\pi c _ { k } x ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $c _ { i }$ are trainable parameters initialized from $\\mathcal { N } ( 0 , \\sigma )$ . We observe that $\\sigma$ is an important hyperparameter. Both $\\sigma$ and $k$ are tuned using validation sets. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.4 Simple differentiable layers ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            289,
+            403,
+            304
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In the context of Deep Learning, embedding numerical features with conventional differentiable layers (e.g. linear layers, ReLU activation, etc.) is a natural approach. In fact, this technique is already used on its own in the recently proposed attention-based architectures [13, 24, 39] and in some models for CTR prediction problems [14, 40]. However, we also note that such conventional modules can be used on top of the components described in subsection 3.2 and subsection 3.3. In section 4, we find that such combinations often lead to better results. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            416,
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+            434
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we empirically evaluate the techniques discussed in section 3 and compare them with Gradient Boosted Decision Trees to check the status quo of the “DL vs GBDT” competition. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1 Datasets ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            491,
+            272,
+            506
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/019a24bfbf3e04acc8507aaa9e5bae63b68777c4e29870a6da4e2d33fd579d5e.jpg",
+        "table_caption": [
+            "Table 1: Dataset properties. “RMSE” denotes root-mean-square error, “Acc.” denotes accuracy. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>GE</td><td>CH</td><td>CA</td><td>HO</td><td>AD</td><td>OT</td><td>HI</td><td>FB</td><td>SA</td><td>CO</td><td>MI</td></tr><tr><td>#objects</td><td>9873</td><td>10000</td><td>20640</td><td>22784</td><td>48842</td><td>61878</td><td>98049</td><td>197080</td><td>200000</td><td>581012</td><td>1200192</td></tr><tr><td>#num.features</td><td>32</td><td>10</td><td>8</td><td>16</td><td>6</td><td>93</td><td>28</td><td>50</td><td>200</td><td>54</td><td>136</td></tr><tr><td>#cat.features metric</td><td>0 Acc.</td><td>1 Acc.</td><td>0</td><td>0</td><td>8</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td></tr><tr><td>#classes</td><td>5</td><td></td><td>RMSE</td><td>RMSE</td><td>Acc.</td><td>Acc.</td><td>Acc.</td><td>RMSE</td><td>Acc.</td><td>Acc.</td><td>RMSE</td></tr><tr><td></td><td></td><td>2</td><td>1</td><td></td><td>2</td><td>9</td><td>2</td><td></td><td>2</td><td>7</td><td>1</td></tr><tr><td>majority class</td><td>29%</td><td>79%</td><td>1</td><td>1</td><td>76%</td><td>26%</td><td>52%</td><td>1</td><td>89%</td><td>48%</td><td>1</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We use eleven public datasets mostly from the previous works on tabular DL and Kaggle competitions. Importantly, we focus on the middle and large scale tasks, and our benchmark is biased towards GBDT-friendly problems, since, as of now, closing the gap with GBDT models on such tasks is one of the main challenges for tabular DL. The main dataset properties are summarized in Table 1 and the used sources and additional details are provided in Appendix C. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2 Implementation details ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We mostly follow Gorishniy et al. [13] in terms of the hyperparameter tuning, training and evaluation protocols. Nevertheless, for completeness, we list all the details in Appendix E. In the next paragraph, we describe the implementation details specific to embeddings for numerical features. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Embeddings for numerical features. If linear layers are used, we tune their output dimensions. The PLE hyperparameters are the same for all features. For quantile-based PLE, we tune the number of quantiles. For target-aware PLE, we tune the following parameters for decision trees: the maximum number of leaves, the minimum number of items per leaf, and the minimum information gain required for making a split when growing the tree. For the Periodic module (see Equation 2), we tune $\\sigma$ and $k$ (these hyperparameters are the same for all features). ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.3 Model names ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In the experiments, we consider different combinations of backbones and embeddings. For convenience, we use the “Backbone-Embedding” pattern to name the models, where “Backbone” denotes the backbone (e.g. MLP, ResNet, Transformer) and “Embedding” denotes the embedding type. See Table 2 for all considered embedding modules. Note that: ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• Periodic is defined in Equation 2. • $\\mathrm { P L E _ { q } }$ denotes the quantile-based PLE. $\\mathrm { P L E } _ { \\mathrm { t } }$ denotes the target-aware PLE. • Linear− denotes bias-free linear layer. LReLU denotes leaky ReLU. AutoDis was proposed in Guo et al. [14] • “Transformer-L” is equivalent to FTTransformer [13]. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/7a482184896d8bb37b8f2ff7e8356cc2ba827309efd45bed297c5089fbef124b.jpg",
+        "table_caption": [
+            "Table 2: Embedding names. See subsection 4.3 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Name</td><td>Embedding function(fi)</td></tr><tr><td>L</td><td>Linear</td></tr><tr><td>LR</td><td>ReLUoLinear</td></tr><tr><td>LRLR</td><td>ReLUoLinearoReLUoLinear</td></tr><tr><td>Q</td><td>PLEq</td></tr><tr><td>Q-L</td><td>Linear o PLEq</td></tr><tr><td>Q-LR</td><td>ReLUo Linear o PLEq</td></tr><tr><td>Q-LRLR</td><td>ReLUoLinear o ReLUo Linear o PLEq</td></tr><tr><td>T</td><td>PLEt</td></tr><tr><td>T-L</td><td>Linear o PLEt</td></tr><tr><td>T-LR</td><td>ReLUo Linear o PLEt</td></tr><tr><td>T-LRLR</td><td>ReLUoLinearoReLUoLinearoPLEt</td></tr><tr><td>P</td><td>Periodic</td></tr><tr><td>PL</td><td>LinearoPeriodic</td></tr><tr><td>PLR</td><td>ReLUoLinearoPeriodic</td></tr><tr><td>PLRLR</td><td>ReLUoLinearoReLUoLinearoPeriodic</td></tr><tr><td>AutoDis</td><td>Linear oSoftMaxo Linear_oLReLUoLinear_</td></tr></table>",
+        "bbox": [
+            472,
+            147,
+            821,
+            344
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.4 Simple differentiable embedding modules ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            366,
+            501,
+            381
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/06df660a4beed008676e71c8829dccb6b35b901072c27c85bd79d1f2191cb795.jpg",
+        "table_caption": [
+            "Table 3: Results for MLP equipped with simple embedding modules (see subsection 4.3). The metric values averaged over 15 random seeds are reported. The standard deviations are provided in Appendix F. We consider one result to be better than another if its mean score is better and its standard deviation is less than the difference. For each dataset, top results are in bold. Notation: ↓ corresponds to RMSE, $\\uparrow$ corresponds to accuracy "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>GE↑（</td><td></td><td></td><td>CH↑ CA↓ HO↓ AD↑ OT↑ HI↑ FB↓ SA↑ CO↑MI↓</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>MLP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.6320.856 0.495 3.204 0.854 0.818 0.720 5.686 0.912 0.964 0.747</td><td></td></tr><tr><td>MLP-L(</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.639 0.861 0.4753.123 0.856 0.820 0.723 5.684 0.916 0.963 0.748</td><td></td></tr><tr><td>MLP-LR 0.642 0.860 0.471 3.084 0.857 0.819 0.726 5.625 0.923 0.963 0.746</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": [
+            212,
+            488,
+            782,
+            555
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We start by evaluating embedding modules consisting of “conventional” differentiable layers (linear layers, ReLU activations, etc.). The results are summarized in Table 3. ",
+        "bbox": [
+            176,
+            569,
+            823,
+            597
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The main takeaways: ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            598,
+            321,
+            611
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• first and foremost, the results indicate that MLP can benefit from embedding modules. Thus, we conclude that this backbone is worth attention when it comes to evaluating embedding modules. • the simple LR module leads to modest, but consistent improvements when applied to MLP. ",
+        "bbox": [
+            189,
+            617,
+            825,
+            660
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Interestingly, the “redundant” MLP-L configuration also tends to outperform the vanilla MLP. Although the improvements are not dramatic, the special property of this architecture is that the linear embedding module can be fused together with the first linear layer of MLP after training, which completely removes the overhead. As for LRLR and AutoDis, we observe that these heavy modules do not justify the extra costs (see the results in Appendix F). ",
+        "bbox": [
+            174,
+            666,
+            825,
+            736
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.5 Piecewise linear encoding ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            752,
+            392,
+            766
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section, we evaluate the encoding scheme described in subsection 3.2. The results are summarized in Table 4. ",
+        "bbox": [
+            176,
+            777,
+            823,
+            805
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The main takeaways: ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            805,
+            323,
+            819
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• The piecewise linear encoding is often beneficial for both types of architectures (MLP and Transformer) and the profit can be significant (for example, see the CA and AD datasets). • Adding differentiable components on top of the PLE can improve the performance. Though, the most expensive modifications such as $\\mathsf { Q }$ -LRLR and T-LRLR are not worth it (see Appendix F). ",
+        "bbox": [
+            192,
+            825,
+            825,
+            882
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Note that the benchmark is biased towards GBDT-friendly problems, so the typical superiority of tree-based bins over quantile-based bins, which can be observed in Table 4, may not generalize to more DL-friendly datasets. Thus, we do not make any general claims about the relative advantages of the two schemes here. ",
+        "bbox": [
+            173,
+            90,
+            825,
+            147
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/69e59696d5403818735f5af95ca783bd8012241b572399cc9f516cf56bc6d4f9.jpg",
+        "table_caption": [
+            "Table 4: Results for MLP and Transformer with embedding modules based on the piecewise linear encoding (subsection 3.2). Notation follows Table 3 and Table 2. The best results are defined separately for the MLP and Transformer backbones. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>GE↑</td><td>CH↑</td><td>CA↓</td><td>HO↓</td><td>AD个</td><td>OT↑</td><td>HI↑</td><td>FB↓</td><td>SA↑</td><td>CO →</td><td>MI↓</td></tr><tr><td>MLP</td><td>0.632</td><td>0.856</td><td>0.495</td><td>3.204</td><td>0.854</td><td>0.818</td><td>0.720</td><td>5.686</td><td>0.912</td><td>0.964</td><td>0.747</td></tr><tr><td>MLP-Q</td><td>0.653</td><td>0.854</td><td>0.464</td><td>3.163</td><td>0.859</td><td>0.816</td><td>0.721</td><td>5.766</td><td>0.922</td><td>0.968</td><td>0.750</td></tr><tr><td>MLP-T</td><td>0.647</td><td>0.861</td><td>0.447</td><td>3.149</td><td>0.864</td><td>0.821</td><td>0.720</td><td>5.577</td><td>0.923</td><td>0.967</td><td>0.749</td></tr><tr><td>MLP-Q-LR</td><td>0.646</td><td>0.857</td><td>0.455</td><td>3.184</td><td>0.863</td><td>0.811</td><td>0.720</td><td>5.394</td><td>0.923</td><td>0.969</td><td>0.747</td></tr><tr><td>MLP-T-LR</td><td>0.640</td><td>0.861</td><td>0.439</td><td>3.207</td><td>0.868</td><td>0.818</td><td>0.724</td><td>5.508</td><td>0.924</td><td>0.968</td><td>0.747</td></tr><tr><td>Transformer-L</td><td>0.632</td><td>0.860</td><td>0.465</td><td>3.239</td><td>0.858</td><td>0.817</td><td>0.725</td><td>5.602</td><td>0.924</td><td>0.971</td><td>0.746</td></tr><tr><td>Transformer-Q-L</td><td>0.659</td><td>0.856</td><td>0.451</td><td>3.319</td><td>0.867</td><td>0.812</td><td>0.729</td><td>5.741</td><td>0.924</td><td>0.973</td><td>0.747</td></tr><tr><td>Transformer-T-L</td><td>0.663</td><td>0.861</td><td>0.454</td><td>3.197</td><td>0.871</td><td>0.817</td><td>0.726</td><td>5.803</td><td>0.924</td><td>0.974</td><td>0.747</td></tr><tr><td>Transformer-Q-LR</td><td>0.659</td><td>0.857</td><td>0.448</td><td>3.270</td><td>0.867</td><td>0.812</td><td>0.723</td><td>5.683</td><td>0.923</td><td>0.972</td><td>0.748</td></tr><tr><td>Transformer-T-LR</td><td>0.665（</td><td>0.860</td><td>0.442</td><td>3.219</td><td>0.870</td><td>0.818</td><td>0.729</td><td>5.699</td><td>0.924</td><td>0.973</td><td>0.747</td></tr></table>",
+        "bbox": [
+            197,
+            222,
+            795,
+            376
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.6 Periodic activation functions ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            405,
+            411,
+            420
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/388ba8b5b6a3d53f3edc9d97292876afcc5ad823f9a83fb15b7c343a208a46c8.jpg",
+        "table_caption": [
+            "Table 5: Results for MLP and Transformer with embedding modules based on periodic activations (subsection 3.3). Notation follows Table 3 and Table 2. The best results are defined separately for the MLP and Transformer backbones. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>GE个</td><td>CH↑</td><td>CA↓</td><td>HO↓</td><td>AD↑</td><td>OT↑</td><td>HI↑</td><td>FB←</td><td>SA↑</td><td>co↑</td><td>MI↓</td></tr><tr><td>MLP</td><td>0.632</td><td>0.856</td><td>0.495</td><td>3.204</td><td>0.854</td><td>0.818</td><td>0.720</td><td>5.686</td><td>0.912</td><td>0.964</td><td>0.747</td></tr><tr><td>MLP-P</td><td>0.631</td><td>0.860</td><td>0.489</td><td>3.129</td><td>0.869</td><td>0.807</td><td>0.723</td><td>5.845</td><td>0.923</td><td>0.968</td><td>0.747</td></tr><tr><td>MLP-PL</td><td>0.641</td><td>0.859</td><td>0.467</td><td>3.113</td><td>0.868</td><td>0.819</td><td>0.727</td><td>5.530</td><td>0.924</td><td>0.969</td><td>0.746</td></tr><tr><td>MLP-PLR</td><td>0.674</td><td>0.857</td><td>0.467</td><td>3.050</td><td>0.870</td><td>0.819</td><td>0.728</td><td>5.525</td><td>0.924</td><td>0.970</td><td>0.746</td></tr><tr><td>Transformer-L</td><td>0.632</td><td>0.860</td><td>0.465</td><td>3.239</td><td>0.858</td><td>0.817</td><td>0.725</td><td>5.602</td><td>0.924</td><td>0.971</td><td>0.746</td></tr><tr><td>Transformer-PLR</td><td>0.646</td><td>0.863</td><td>0.464</td><td>3.162</td><td>0.870</td><td>0.814</td><td>0.730</td><td>5.760</td><td>0.924</td><td>0.972</td><td>0.746</td></tr></table>",
+        "bbox": [
+            200,
+            500,
+            794,
+            607
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we evaluate embedding modules based on periodic activation functions as described in subsection 3.3. The results are reported in Table 5. ",
+        "bbox": [
+            176,
+            625,
+            823,
+            652
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The main takeaway: on average, MLP-P is superior to the vanilla MLP. However, adding a differentiable component on top of the Periodic module should be the default strategy (which is in line with Li et al. [25]). Indeed, MLP-PLR and MLP-PL provide meaningful improvements over MLP-P (e.g. see GE, CA, HO) and even “fix” MLP-P where it is inferior to MLP (OT, FB). ",
+        "bbox": [
+            173,
+            652,
+            825,
+            708
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Although MLP-PLR is usually superior to MLP-PL, we note that in the latter case the last linear layer of the embedding module is “redundant” in terms of expressivity and can be fused with the first linear layer of the backbone after training, which, in theory, can lead to a more lightweight model. Finally, we observe that MLP-PLRLR and MLP-PLR do not differ significantly enough to justify the extra cost of the PLRLR module (see Appendix F). ",
+        "bbox": [
+            174,
+            713,
+            825,
+            785
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.7 Comparing DL models and GBDT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            801,
+            450,
+            816
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we perform a big comparison of different approaches to identify the best embedding modules and backbones, as well as to check if embeddings for numerical features allow DL models to compete with GBDT on more tasks than before. Importantly, we compare ensembles of DL models against ensembles of GBDT, since Gradient Boosting is essentially an ensembling technique, so such comparison will be fairer. Note that we focus only on the best metric values without taking efficiency into account, so we only check if DL models are conceptually ready to compete with GBDT. ",
+        "bbox": [
+            174,
+            827,
+            825,
+            911
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We consider three backbones: MLP, ResNet, and Transformer, since they are reported to be representative of what baseline DL backbones are currently capable of [13, 18, 24, 39]. Note that we do not include the attention-based models that also apply attention on the level of objects [24, 35, 39], since this non-parametric component is orthogonal to the central topic of our work. The results are summarized in Table 6. ",
+        "bbox": [
+            174,
+            90,
+            825,
+            160
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/81afadf7bdcb6f23b8a98bc86294f0f10e6cd43905a4289fbc8c850ca8c05054.jpg",
+        "table_caption": [
+            "Table 6: Results for ensembles of GBDT, the baseline DL models and their modifications using different types of embeddings for numerical features. Notation follows Table 3 and Table 2. Due to the limited precision, some different values are represented with the same figures. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>GE↑</td><td>CH↑</td><td>CA↓</td><td>HO↓</td><td>AD↑</td><td>OT↑</td><td>HI↑</td><td>FB↓</td><td>SA↑</td><td>CO→</td><td>MI↓</td><td></td><td>Avg.Rank</td></tr><tr><td>CatBoost</td><td>0.692</td><td>0.861</td><td>0.430</td><td>3.093</td><td>0.873</td><td>0.825</td><td>0.727</td><td></td><td>5.226</td><td>0.924</td><td>0.967</td><td>0.741</td><td>3.6±2.9</td></tr><tr><td>XGBoost</td><td>0.683</td><td>0.859</td><td>0.434</td><td>3.152</td><td>0.875</td><td>0.827</td><td>0.726</td><td>5.338</td><td>0.919</td><td>0.969</td><td></td><td>0.742</td><td>4.6±2.7</td></tr><tr><td>MLP</td><td>0.665</td><td>0.856</td><td>0.486</td><td>3.109</td><td>0.856</td><td>0.822</td><td>0.727</td><td>5.616</td><td>0.913</td><td>0.968</td><td></td><td>0.746</td><td>8.5±2.6</td></tr><tr><td>MLP-LR</td><td>0.679</td><td>0.861</td><td>0.463</td><td>3.012</td><td>0.859</td><td>0.826</td><td>0.731</td><td>5.477</td><td>0.924</td><td>0.972</td><td></td><td>0.744</td><td>5.5 ± 2.7</td></tr><tr><td>MLP-Q-LR</td><td>0.682</td><td>0.859</td><td>0.433</td><td>3.080</td><td>0.867</td><td>0.818</td><td>0.724</td><td>5.144</td><td>0.924</td><td></td><td>0.974</td><td>0.745</td><td>5.1 ± 1.9</td></tr><tr><td>MLP-T-LR</td><td>0.673</td><td>0.861</td><td>0.435</td><td>3.099</td><td>0.870</td><td>0.821</td><td>0.727</td><td>5.409</td><td>0.924</td><td></td><td>0.973</td><td>0.746</td><td>5.1 ± 1.7</td></tr><tr><td>MLP-PLR</td><td>0.700</td><td>0.858</td><td>0.453</td><td>2.975</td><td>0.874</td><td>0.830</td><td>0.734</td><td>5.388</td><td>0.924</td><td></td><td>0.975</td><td>0.743</td><td>3.0±2.4</td></tr><tr><td>ResNet</td><td>0.690</td><td>0.861</td><td>0.483</td><td>3.081</td><td>0.856</td><td>0.821</td><td>0.734</td><td>5.482</td><td>0.918</td><td></td><td>0.968</td><td>0.745</td><td>6.7±3.3</td></tr><tr><td>ResNet-LR</td><td>0.672</td><td>0.862</td><td>0.450</td><td>2.992</td><td>0.859</td><td>0.822</td><td>0.733</td><td>5.415</td><td>0.923</td><td></td><td>0.971</td><td>0.743</td><td>5.6±2.7</td></tr><tr><td>ResNet-Q-LR</td><td>0.674</td><td>0.859</td><td>0.427</td><td>3.066</td><td>0.868</td><td>0.815</td><td>0.729</td><td>5.309</td><td></td><td>0.923</td><td>0.976</td><td>0.746</td><td>4.7 ± 2.0</td></tr><tr><td>ResNet-T-LR</td><td>0.683</td><td>0.862</td><td>0.425</td><td>3.030</td><td>0.872</td><td>0.822</td><td>0.731</td><td>5.471</td><td></td><td>0.923</td><td>0.975</td><td>0.744</td><td>4.1 ± 1.9</td></tr><tr><td>ResNet-PLR</td><td>0.691</td><td>0.861</td><td>0.443</td><td>3.040</td><td>0.874</td><td>0.825</td><td>0.734</td><td>5.400</td><td></td><td>0.924</td><td>0.975</td><td>0.743</td><td>3.2 ±1.3</td></tr><tr><td>Transformer-L</td><td>0.668</td><td>0.861</td><td>0.455</td><td>3.188</td><td>0.860</td><td>0.824</td><td>0.727</td><td></td><td>5.434</td><td>0.924</td><td>0.973</td><td>0.743</td><td>5.9±2.2</td></tr><tr><td>Transformer-LR</td><td>0.666</td><td>0.861</td><td>0.446</td><td>3.193</td><td>0.861</td><td>0.824</td><td>0.733</td><td></td><td>5.430</td><td>0.924</td><td>0.973</td><td>0.743</td><td>5.2 ±2.2</td></tr><tr><td>Transformer-Q-LR</td><td>0.690</td><td>0.857</td><td>0.425</td><td>3.143</td><td>0.868</td><td>0.818</td><td>0.726</td><td></td><td>5.471</td><td>0.924</td><td>0.975</td><td>0.744</td><td>4.4 ± 2.2</td></tr><tr><td>Transformer-T-LR</td><td>0.686</td><td>0.862</td><td>0.423</td><td>3.149</td><td>0.871</td><td>0.823</td><td>0.733</td><td></td><td>5.515</td><td>0.924</td><td>0.976</td><td>0.744</td><td>3.7±2.2</td></tr><tr><td>Transformer-PLR</td><td>0.686</td><td>0.864</td><td>0.449</td><td>3.091</td><td>0.873</td><td>0.823</td><td>0.734</td><td></td><td>5.581</td><td>0.924</td><td>0.975</td><td>0.743</td><td>3.9 ±2.5</td></tr></table>",
+        "bbox": [
+            196,
+            233,
+            797,
+            450
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The main takeaways for DL models: ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            467,
+            426,
+            481
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "• For most datasets, embeddings for numerical features can provide noticeable improvements for three different backbones. Although the average rank is not a good metric for making subtle conclusions, we highlight the impressive difference in average ranks between the MLP and MLP-PLR models. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "• The simplest LR embedding is a good baseline solution: although the performance gains are not dramatic, its main advantage is consistency (e.g. see MLP vs MLP-LR). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "• The PLR module provides the best average performance. Empirically, we observe $\\sigma$ (see Equation 2) to be an important hyperparameter that should be tuned. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "• Piecewise linear encoding (PLE) allows building well performing embeddings (e.g. T-LR, Q-LR). In addition to that, PLE itself is worth attention because of its simplicity, interpretability and efficiency (no computationally expensive periodic functions). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "• Importantly, after the MLP-like architectures are coupled with embeddings for numerical features, they perform on par with the Transformer-based models. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The main takeaway for the “DL vs GBDT” competition: embeddings for numerical features is a significant design aspect that has a great potential for improving DL models and closing the gap with GBDT on GBDT-friendly tasks. Let us illustrate this claim with several observations: ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "• The benchmark is initially biased to GBDT-friendly problems, which can be observed by comparing GBDT solutions with the vanilla DL models (MLP, ResNet, Transformer-L). • However, for the vast majority of the “backbone & dataset” pairs, proper embeddings are the only thing needed to close the gap with GBDT. Exceptions (rather formal) include the MI dataset and the following pairs: “ResNet & GE”, “Transformer & FB”, “Transformer & GE”, “Transformer & OT”. • Additionally, to the best of our knowledge, it is the first time when DL models perform on par with GBDT on the well-known California Housing and Adult datasets. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "That said, compared to GBDT models, efficiency can still be an issue for the considered DL architectures. In any case, the trade-off completely depends on the specific use case and requirements. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 Analysis ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.1 Comparing model sizes ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            121,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To quantify the effect of embeddings for numerical features on model sizes, we report the parameter counts in Table 7. Overall, introducing embeddings for numerical features can cause non-negligible overhead in terms of model size. Importantly, the overhead in terms of size does not translate to the same overhead in terms of training times and throughput. For example, the almost 2000-fold increase in the parameter count for MLP-LR on the CH dataset results in only 1.5-fold increase in training times. Finally, in practice, we observe that coupling MLP and ResNet with embedding modules leads to architectures that are still faster than Transformer-based models. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/f8b2d961ee829c431a61b8a08d2619eee635cc4943c1dbc643ce2ef964c3f340.jpg",
+        "table_caption": [
+            "Table 7: Parameter counts for MLP with different embedding modules. All the models are tuned and the corresponding backbones are not identical in their sizes, so we take into account the fact that different approaches require a different number of parameters to realize their full potential. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>GE</td><td>CH</td><td>CA</td><td>HO</td><td>AD</td><td>OT</td><td>HI</td><td>FB</td><td>SA</td><td>CO</td><td>MI</td></tr><tr><td>MLP</td><td>2.0M</td><td>1.5K</td><td>43.5K</td><td>3.6M</td><td>5.3M</td><td>479.9K</td><td>25.8K</td><td>937.3K</td><td>5.8M</td><td>3.2M</td><td>276.5K</td></tr><tr><td>MLP-LR</td><td>×2.52</td><td>×1931.03</td><td>×25.05</td><td>×1.28</td><td>×0.35</td><td>×12.53</td><td>×68.16</td><td>×4.76</td><td>×1.58</td><td>×0.72</td><td>×15.79</td></tr><tr><td>MLP-T</td><td>×1.58</td><td>×14.13</td><td>×7.97</td><td>×0.43</td><td>×0.04</td><td>×2.27</td><td>×5.85</td><td>×0.47</td><td>×0.59</td><td>×0.74</td><td>×3.85</td></tr><tr><td>MLP-T-LR</td><td>×1.61</td><td>×463.55</td><td>×6.80</td><td>×0.23</td><td>×0.16</td><td>×2.52</td><td>×113.22</td><td>×3.43</td><td>×0.41</td><td>×0.35</td><td>×8.47</td></tr><tr><td>MLP-PLR</td><td>×1.73</td><td>×250.24</td><td>×12.94</td><td>×1.07</td><td>×0.66</td><td>×8.05</td><td>×110.57</td><td>×4.93</td><td>×0.64</td><td>×0.44</td><td>×9.57</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.2 Ablation study ",
+        "text_level": 1,
+        "bbox": [
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+            431,
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+            446
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/ad479bd5b7a1f5bdb432dc1d92aebe1683d3ba433b5ef38577b5abf8f121c6f9.jpg",
+        "table_caption": [
+            "Table 8: Comparing piecewise linear encoding (PLE) with the two variations described in subsection 5.2. Notation follows Table 3 and Table 2. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>GE↑</td><td>CH↑</td><td>CA↓1</td><td>HO←</td><td>AD↑</td><td>OT↑</td><td>HI↑</td><td>FB←</td></tr><tr><td>MLP-Q(piecewise linear) 0.653 0.854</td><td></td><td></td><td>0.464</td><td></td><td></td><td>3.163 0.859 0.816 0.721</td><td></td><td>5.766</td></tr><tr><td>MLP-Q (binary)</td><td>0.652</td><td>0.815</td><td>0.462</td><td>3.200</td><td>0.860</td><td>）0.810</td><td>0.720</td><td>5.748</td></tr><tr><td>MLP-Q (one-blob)</td><td>0.613 0.851（</td><td></td><td></td><td></td><td></td><td>0.461 3.187 0.857 0.808</td><td>30.719</td><td>5.645</td></tr><tr><td>MLP-T (piecewise linear)</td><td>0.647 0.861</td><td></td><td>0.447 3.149</td><td></td><td></td><td>90.8640.821（</td><td>0.720</td><td>5.577</td></tr><tr><td>MLP-T (binary)</td><td>0.639</td><td>90.855</td><td></td><td></td><td></td><td>0.4643.163 0.869 0.813</td><td>0.718</td><td>5.572</td></tr><tr><td>MLP-T (one-blob)</td><td>0.622</td><td>0.858</td><td>0.464</td><td>3.158</td><td>0.870</td><td>0.809</td><td>0.724</td><td>5.475</td></tr></table>",
+        "bbox": [
+            230,
+            512,
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+            626
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this section, we compare two alternative binning-based encoding schemes with PLE (see subsection 3.2). The first one (\"thermometer\" [6]) sets the value 1 instead of the piecewise linear term (see Equation 1). The second one is a generalized version of the one-blob encoding [29] (see subsection E.1 for details). The tuning and evaluation protocols are the same as in subsection 4.2. The results in table Table 8 indicate that making the binning-based encoding piecewise linear is a good default strategy. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.3 Piecewise linear encoding as a feature preprocessing technique ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            739,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "It is known that data preprocessing, such as standardization or quantile transformation, is often crucial for DL models for achieving competitive performance. Moreover, the performance can significantly vary between different types of preprocessing. At the same time, PLE-representations contain only values from [0, 1] and they are invariant to shifting and scaling, which makes PLE itself a general feature preprocessing technique potentially suitable for DL models without the need to use traditional preprocessing first. ",
+        "bbox": [
+            174,
+            766,
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+            849
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To illustrate that, for datasets where the quantile transformation was used in section 4, we reevaluate the tuned configurations of MLP, MLP-Q, and MLP-T with different preprocessing policies and report the results in Table 9 (note that standardization is equivalent to no preprocessing for models with PLE). ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/178a278036c941afc3953c2f6c3e9014cc7199850232c111c56345d4090d7a44.jpg",
+        "table_caption": [
+            "Table 9: Results for MLP and MLP with PLE for different types of data preprocessing. Solutions using PLE are significantly less sensitive to data preprocessing. Notation follows Table 3 and Table 2. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>GE↑</td><td>CH↑</td><td>CA↓HO↓AD↑]</td><td></td><td>HI↑</td><td>FB↓</td><td>SA↑</td><td>CO↑1</td><td>MI↓</td></tr><tr><td>MLP (none)</td><td>0.565</td><td></td><td>0.796 1.118 5.328 0.808 0.707</td><td></td><td></td><td>13.125</td><td>0.911</td><td></td><td>0.948 0.844</td></tr><tr><td>MLP (standard)</td><td>0.629</td><td></td><td>0.855 0.509 3.3030.855（</td><td></td><td>0.721</td><td>5.919</td><td>0.912</td><td></td><td>0.9630.754</td></tr><tr><td>MLP (quantile)</td><td>0.632</td><td></td><td>0.8560.495 3.204 0.8540.720</td><td></td><td></td><td>5.686</td><td>0.912</td><td></td><td>0.9640.747</td></tr><tr><td>MLP-Q (none)</td><td></td><td></td><td>0.654 0.851 0.463 3.162 0.860 0.721</td><td></td><td></td><td>5.889</td><td></td><td></td><td>0.922 0.968 0.754</td></tr><tr><td>MLP-Q(quantile) 0.653 0.854 0.464 3.163 0.859 0.721</td><td></td><td></td><td></td><td></td><td></td><td>5.766</td><td></td><td></td><td>0.922 0.968 0.750</td></tr><tr><td>MLP-T (none)</td><td></td><td></td><td>0.644 0.860 0.447 3.175 0.865 0.721</td><td></td><td></td><td>5.598</td><td></td><td></td><td>0.9230.968 0.749</td></tr><tr><td>MLP-T (quantile)</td><td></td><td></td><td>0.647 0.861 0.447 3.149 0.864 0.720</td><td></td><td></td><td>5.577</td><td>0.923</td><td></td><td>0.9670.749</td></tr></table>",
+        "bbox": [
+            227,
+            137,
+            767,
+            268
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "First, the vanilla MLP often becomes unusable without preprocessing. Second, for the vanilla MLP, it can be important to choose one specific type of preprocessing (CA, HO, FB, MI), which is less pronounced for MLP-Q and not the case for MLP-T (though, this specific observation can be the property of the benchmarks, not of MLP-T). Overall, the results indicate that models using PLE are less sensitive to the initial preprocessing compared to the vanilla MLP. This is an additional benefit of PLE-representations for practitioners since the aspect of preprocessing becomes less critical with PLE. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5.4 The “feature engineering” perspective ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            398,
+            478,
+            414
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/6064cdcf2a4d59ec5956745d10aa5964e8f9cc7d0cdd971fad22215271d6a4a0.jpg",
+        "table_caption": [
+            "Table 10: The comparison of the effects of Periodic-based modules for XGBoost and MLP "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>CA↓HO↓HI↑</td></tr><tr><td>XGBoost 0.436 3.160 0.724 XGBoost with Periodic 0.441 3.184 0.724</td></tr><tr><td>MLP 0.4953.204 0.720</td></tr><tr><td>MLP-PL</td></tr><tr><td>0.467 3.113 0.727</td></tr></table>",
+        "bbox": [
+            356,
+            469,
+            638,
+            556
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "At first sight, feature embeddings may resemble feature engineering and should be suitable for all kinds of models. However, the proposed embedding schemes are motivated by DL-specific aspects of training (see the motivational parts of subsection 3.2 and subsection 3.3). While our methods are likely to transfer well to models with similar training properties (e.g. to linear models since those are a special case of deep models), it is not the case in general. To illustrate that, we try adopting the Periodic module for XGBoost by fixing the random coefficients from Equation 2. We also keep the original features instead of dropping them. The tuning and evaluation protocols are the same as in subsection 4.2. The results in Table 10 show that this technique, while being useful for DL models, does not provide any benefits for XGBoost. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 Conclusion & Future work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            719,
+            429,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this work, we have demonstrated that embeddings for numerical features are an important design aspect of tabular DL architectures. Namely, it allows existing DL backbones to achieve noticeably better results and significantly reduce the gap with Gradient Boosted Decision Trees. We have described two approaches illustrating this phenomenon, one using the piecewise linear encoding of original scalar values, and another using periodic functions. We have also shown that traditional MLP-like models coupled with embeddings can perform on par with attention-based models. ",
+        "bbox": [
+            174,
+            752,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Nevertheless, we have only scratched the surface of the new direction. For example, it is still to be explained how exactly the discussed embedding modules help optimization on the fundamental level. Additionally, we have considered only schemes where the same functional transformation was applied to all features, which may be a suboptimal choice. ",
+        "bbox": [
+            174,
+            843,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            90,
+            267,
+            106
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
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Bayesian optimization is superior to random search for machine learning hyperparameter tuning: Analysis of the black-box optimization challenge 2020. arXiv, https://arxiv.org/abs/2104.10201v1, 2021.   \n[44] J. Vanschoren, J. N. van Rijn, B. Bischl, and L. Torgo. Openml: networked science in machine learning. arXiv, 1407.7722v1, 2014.   \n[45] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin. Attention is all you need. In NIPS, 2017.   \n[46] R. Wang, B. Fu, G. Fu, and M. Wang. Deep & cross network for ad click predictions. In ADKDD, 2017. ",
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+# CROSSFORMER: TRANSFORMER UTILIZING CROSSDIMENSION DEPENDENCY FOR MULTIVARIATE TIME SERIES FORECASTING
+Yunhao Zhang & Junchi Yan∗
+MoE Key Lab of Artificial Intelligence, Shanghai Jiao Tong University and Shanghai AI Lab
+{zhangyunhao, yanjunchi}@sjtu.edu.cn
+Code: https://github.com/Thinklab-SJTU/Crossformer
+# ABSTRACT
+Recently many deep models have been proposed for multivariate time series (MTS) forecasting. In particular, Transformer-based models have shown great potential because they can capture long-term dependency. However, existing Transformerbased models mainly focus on modeling the temporal dependency (cross-time dependency) yet often omit the dependency among different variables (crossdimension dependency), which is critical for MTS forecasting. To fill the gap, we propose Crossformer, a Transformer-based model utilizing cross-dimension dependency for MTS forecasting. In Crossformer, the input MTS is embedded into a 2D vector array through the Dimension-Segment-Wise (DSW) embedding to preserve time and dimension information. Then the Two-Stage Attention (TSA) layer is proposed to efficiently capture the cross-time and cross-dimension dependency. Utilizing DSW embedding and TSA layer, Crossformer establishes a Hierarchical Encoder-Decoder (HED) to use the information at different scales for the final forecasting. Extensive experimental results on six real-world datasets show the effectiveness of Crossformer against previous state-of-the-arts.
+# 1 INTRODUCTION
+Multivariate time series (MTS) are time series with multiple dimensions, where each dimension represents a specific univariate time series (e.g. a climate feature of weather). MTS forecasting aims to forecast the future value of MTS using their historical values. MTS forecasting benefits the decision-making of downstream tasks and is widely used in many fields including weather (Angryk et al., 2020), energy (Demirel et al., 2012), finance (Patton, 2013), etc. With the development of deep learning, many models have been proposed and achieved superior performances in MTS forecasting (Lea et al., 2017; Qin et al., 2017; Flunkert et al., 2017; Rangapuram et al., 2018; Li et al., 2019a; Wu et al., 2020; Li et al., 2021). Among them, the recent Transformer-based models (Li et al., 2019b; Zhou et al., 2021; Wu et al., 2021a; Liu et al., 2021a; Zhou et al., 2022; Chen et al., 2022) show great potential thanks to their ability to capture long-term temporal dependency (cross-time dependency).
+Besides cross-time dependency, the cross-dimension dependency is also critical for MTS forecasting, i.e. for a specific dimension, information from associated series in other dimensions may improve prediction. For example, when predicting future temperature, not only the historical temperature, but also historical wind speed helps to forecast. Some previous neural models explicitly capture the cross-dimension dependency, i.e. preserving the information of dimensions in the latent feature space and using convolution neural network (CNN) (Lai et al., 2018) or graph neural network (GNN) (Wu et al., 2020; Cao et al., 2020) to capture their dependency. However, recent Transformer-based models only implicitly utilize this dependency by embedding. In general, Transformer-based models embed data points in all dimensions at the same time step into a feature vector and try to capture dependency among different time steps (like Fig. 1 (b)). In this way, cross-time dependency is well captured, but cross-dimension dependency is not, which may limit their forecasting capability.
+To fill the gap, we propose Crossformer, a Transformer-based model that explicitly utilizes crossdimension dependency for MTS forecasting. Specifically, we devise Dimension-Segment-Wise (DSW) embedding to process the historical time series. In DSW embedding, the series in each dimension is first partitioned into segments and then embedded into feature vectors. The output of DSW embedding is a 2D vector array where the two axes correspond to time and dimension. Then we propose the Two-Stage-Attention (TSA) layer to efficiently capture the cross-time and cross-dimension dependency among the 2D vector array. Using DSW embedding and TSA layer, Crossformer establishes a Hierarchical Encoder-Decoder (HED) for forecasting. In HED, each layer corresponds to a scale. The encoder’s upper layer merges adjacent segments output by the lower layer to capture the dependency at a coarser scale. Decoder layers generate predictions at different scales and add them up as the final prediction. The contributions of this paper are:
+1) We dive into the existing Transformer-based models for MTS forecasting and figure out that the cross-dimension dependency is not well utilized: these models simply embed data points of all dimensions at a specific time step into a single vector and focus on capturing the cross-time dependency among different time steps. Without adequate and explicit mining and utilization of cross-dimension dependency, their forecasting capability is empirically shown limited.
+2) We develop Crossformer, a Transformer model utilizing cross-dimension dependency for MTS forecasting. This is one of the few transformer models (perhaps the first to our best knowledge) that explicitly explores and utilizes cross-dimension dependency for MTS forecasting.
+3) Extensive experimental results on six real-world benchmarks show the effectiveness of our Crossformer against previous state-of-the-arts. Specifically, Crossformer ranks top-1 among the 9 models for comparison on 36 out of the 58 settings of varying prediction lengths and metrics and ranks top-2 on 51 settings.
+# 2 RELATED WORKS
+Multivariate Time Series Forecasting. MTS forecasting models can be roughly divided into statistical and neural models. Vector auto-regressive (VAR) model (Kilian & LAtkepohl ˜ , 2017) and Vector auto-regressive moving average (VARMA) are typical statistical models, which assume linear cross-dimension and cross-time dependency. With the development of deep learning, many neural models have been proposed and often empirically show better performance than statistical ones. TCN (Lea et al., 2017) and DeepAR (Flunkert et al., 2017) treat the MTS data as a sequence of vectors and use CNN/RNN to capture the temporal dependency. LSTnet (Lai et al., 2018) employs CNN to capture cross-dimension dependency and RNN for cross-time dependency. Another category of works use graph neural networks (GNNs) to capture the cross-dimension dependency explicitly for forecasting (Li et al., 2018; Yu et al., 2018; Cao et al., 2020; Wu et al., 2020). For example, MTGNN (Wu et al., 2020) uses temporal convolution and graph convolution layers to capture crosstime and cross-dimension dependency. These neural models capture the cross-time dependency through CNN or RNN, which have difficulty in modeling long-term dependency.
+Transformers for MTS Forecasting. Transformers (Vaswani et al., 2017) have achieved success in natural language processing (NLP) (Devlin et al., 2019), vision (CV) (Dosovitskiy et al., 2021) and speech processing (Dong et al., 2018). Recently, many Transformer-based models have been proposed for MTS forecasting and show great potential (Li et al., 2019b; Zhou et al., 2021; Wu et al., 2021a; Liu et al., 2021a; Zhou et al., 2022; Du et al., 2022). LogTrans (Li et al., 2019b) proposes the LogSparse attention that reduces the computation complexity of Transformer from ${ \dot { O } } ( { \dot { L } } ^ { 2 } )$ to $O \left( L ( \log L ) ^ { 2 } \right)$ . Informer (Zhou et al., 2021) utilizes the sparsity of attention score through KL divergence estimation and proposes ProbSparse self-attention which achieves $O ( L \log L )$ complexity. Autoformer (Wu et al., 2021a) introduces a decomposition architecture with an Auto-Correlation mechanism to Transformer, which also achieves the $O ( L \log L )$ complexity. Pyraformer (Liu et al., 2021a) introduces a pyramidal attention module that summarizes features at different resolutions and models the temporal dependencies of different ranges with the complexity of $O ( L )$ . FEDformer (Zhou et al., 2022) proposes that time series have a sparse representation in frequency domain and develop a frequency enhanced Transformer with the $O ( L )$ complexity. Preformer (Du et al., 2022) divides the embedded feature vector sequence into segments and utilizes segment-wise correlation-based attention for forecasting. These models mainly focus on reducing the complexity of cross-time dependency modeling, but omits the cross-dimension dependency which is critical for MTS forecasting.
+![](images/58f91577761d114749db0b9c2411629ff28fc541aac4585ce6c559b60c4f406b.jpg)
+Figure 1: Illustration for our DSW embedding. (a) Self-attention scores from a 2-layer Transformer trained on ETTh1, showing that MTS data tends to be segmented. (b) Embedding method of previous Transformer-based models (Li et al., 2019b; Zhou et al., 2021; Wu et al., 2021a; Liu et al., 2021a): data points in different dimensions at the same step are embedded into a vector. (c) DSW embedding of Crossformer: in each dimension, nearby points over time form a segment for embedding.
+Vision Transformers. Transformer is initially applied to NLP for sequence modeling, recent works apply transformer to CV tasks to process images (Dosovitskiy et al., 2021; Touvron et al., 2021; Liu et al., 2021b; Chen et al., 2021; Han et al., 2021). These works achieve state-of-the-art performance on various tasks in CV and inspire our work. ViT (Dosovitskiy et al., 2021) is one of the pioneers of vision transformers. The basic idea of ViT is to split an image into non-overlapping medium-sized patches, then it rearranges these patches into a sequence to be input to the Transformer. The idea of partitioning images into patches inspires our DSW embedding where MTS is split into dimensionwise segments. Swin Transformer (Liu et al., 2021b) performs local attention within a window to reduce the complexity and builds hierarchical feature maps by merging image patches. Readers can refer to the recent survey (Han et al., 2022) for comprehensive study on vision transformers.
+# 3 METHODOLOGY
+In multivariate time series forecasting, one aims to predict the future value of time series $\mathbf { x } _ { T + 1 : T + \tau } \in$ Rτ×D given the history $\mathbf { x } _ { 1 : T } \in \mathbb { R } ^ { \tilde { T } \times D }$ , where $\tau$ , $T$ is the number of time steps in the future and past, respectively2. $D > 1$ is the number of dimensions. A natural assumption is that these $D$ series are associated (e.g. climate features of weather), which helps to improve the forecasting accuracy. To utilize the cross-dimension dependency, in Section 3.1, we embed the MTS using Dimension-Segment-Wise (DSW) embedding. In Section 3.2, we propose a Two-Stage Attention (TSA) layer to efficiently capture the dependency among the embedded segments. In Section 3.3, using DSW embedding and TSA layer, we construct a hierarchical encoder-decoder (HED) to utilize information at different scales for final forecasting.
+# 3.1 DIMENSION-SEGMENT-WISE EMBEDDING
+To motivate our approach, we first analyze the embedding methods of the previous Transformer-based models for MTS forecasting (Zhou et al., 2021; Wu et al., 2021a; Liu et al., 2021a; Zhou et al., 2022). As shown in Fig. 1 (b), existing methods embed data points at the same time step into a vector: $\mathbf { x } _ { t }  \mathbf { h } _ { t } , \mathbf { x } _ { t } \in \bar { \mathbb { R } } ^ { D } , \mathbf { h } _ { t } \in \mathbb { R } ^ { d _ { m o d e l } }$ , where $\mathbf { x } _ { t }$ represents all the data points in $D$ dimensions at step $t$ . In this way, the input $\mathbf { x } _ { 1 : T }$ is embedded into $T$ vectors $\{ \mathbf { h } _ { 1 } , \mathbf { h } _ { 2 } , \dots , \mathbf { h } _ { T } \}$ . Then the dependency among the $T$ vectors is captured for forecasting. Therefore, previous Transformer-based models mainly capture cross-time dependency, while the cross-dimension dependency is not explicitly captured during embedding, which limits their forecasting capability.
+Transformer was originally developed for NLP (Vaswani et al., 2017), where each embedded vector represents an informative word. For MTS, a single value at a step alone provides little information.
+While it forms informative pattern with nearby values in time domain. Fig. 1 (a) shows a typical attention score map of original Transformer for MTS forecasting. We can see that attention values have a tendency to segment, i.e. close data points have similar attention weights.
+Based on the above two points, we argue that an embedded vector should represent a series segment of single dimension (Fig. 1 (c)), rather than the values of all dimensions at single step (Fig. 1 (b)). To this end, we propose Dimension-Segment-Wise (DSW) embedding where the points in each dimension are divided into segments of length $L _ { s e g }$ and then embedded:
+$$
+\begin{array} { r l } & { \mathbf { x } _ { 1 : T } = \left\{ \mathbf { x } _ { i , d } ^ { ( s ) } \vert 1 \leq i \leq \frac { T } { L _ { s e g } } , 1 \leq d \leq D \right\} } \\ & { \mathbf { x } _ { i , d } ^ { ( s ) } = \left\{ x _ { t , d } \vert ( i - 1 ) \times L _ { s e g } < t \leq i \times L _ { s e g } \right\} } \end{array}
+$$
+where x(s)i,d $\mathbf { x } _ { i , d } ^ { ( s ) } \in \mathbb { R } ^ { L _ { s e g } }$ is the $i$ -th segment in dimension $d$ with length $L _ { s e g }$ . For convenience, we assume that $T , \tau$ are divisible by $L _ { s e g }$ . Then each segment is embedded into a vector using linear projection added with a position embedding:
+$$
+\mathbf { h } _ { i , d } = \mathbf { E x } _ { i , d } ^ { ( s ) } + \mathbf { E } _ { i , d } ^ { ( p o s ) }
+$$
+where $\mathbf { E } \in \mathbb { R } ^ { d _ { m o d e l } \times L _ { s e g } }$ denotes the learnable projection matrix, and ${ \bf E } _ { i , d } ^ { ( p o s ) } \in \mathbb { R } ^ { d _ { m o d e l } }$ denotes the learnable position embedding for position $( i , d )$ . After embedding, we obtain a 2D vector array $\begin{array} { r } { \mathbf { H } = \left\{ \mathbf { h } _ { i , d } | 1 \leq i \leq \frac { T } { L _ { s e g } } , 1 \leq d \leq D \right\} } \end{array}$ , where each $\mathbf { h } _ { i , d }$ represents a univariate time series segment. The idea of segmentation is also used in Du et al. (2022), which splits the embedded 1D vector sequence into segments to compute the Segment-Correlation in order to enhance locality and reduce computation complexity. However, like other Transformers for MTS forecasting, it does not explicitly capture cross-dimension dependency.
+# 3.2 TWO-STAGE ATTENTION LAYER
+For the obtained 2D array $\mathbf { H }$ , one can flatten it into a 1D sequence so that it can be input to a canonical Transformer like ViT (Dosovitskiy et al., 2021) does in vision. While we have specific considerations: 1) Different from images where the axes of height and width are interchangeable, the axes of time and dimension for MTS have different meanings and thus should be treated differently. 2) Directly applying self-attention on 2D array will cause the complexity of O(D2 T 2L2 ) , which is unaffordable for large $D$ . Therefore, we propose the Two-Stage Attention (TSA) Layer to capture cross-time and cross-dimension dependency among the 2D vector array, as sketched in Fig. 2 (a).
+Cross-Time Stage Given a 2D array $\mathbf { Z } \in \mathbb { R } ^ { L \times D \times d _ { m o d e l } }$ as the input of the TSA Layer, where $L$ and $D$ are the number of segments and dimensions, respectively. $\mathbf { Z }$ here can be the output of DSW embedding or lower TSA layers. For convenience, in the following, we use $\mathbf { Z } _ { i , }$ : to denote the vectors of all dimensions at time step $i$ , $\mathbf { Z } _ { : , d }$ for those of all time steps in dimension $d$ . In the cross-time stage, we directly apply multi-head self-attention (MSA) to each dimension:
+$$
+\begin{array} { r l } & { \hat { \mathbf { Z } } _ { : , d } ^ { t i m e } = \mathrm { L a y e r N o r m } \Big ( \mathbf { Z } _ { : , d } + \mathrm { M S } \mathbb { A } ^ { t i m e } ( \mathbf { Z } _ { : , d } , \mathbf { Z } _ { : , d } , \mathbf { Z } _ { : , d } ) \Big ) } \\ & { \mathbf { Z } ^ { t i m e } = \mathrm { L a y e r N o r m } \left( \hat { \mathbf { Z } } ^ { t i m e } + \mathrm { M L P } ( \hat { \mathbf { Z } } ^ { t i m e } ) \right) } \end{array}
+$$
+where $1 \leq d \leq D$ and LayerNorm denotes layer normalization as widely adopted in Vaswani et al. (2017); Dosovitskiy et al. (2021); Zhou et al. (2021), MLP denotes a multi-layer (two in this paper) feedforward network, $\mathtt { M S A } ( \mathbf { Q } , \mathbf { K } , \mathbf { V } )$ denotes the multi-head self-attention (Vaswani et al., 2017) layer where $\mathbf { Q } , \mathbf { K } , \mathbf { V }$ serve as queries, keys and values. All dimensions $1 \leq d \leq D _ { \cdot }$ share the same MSA layer. $\hat { \mathbf { Z } } ^ { t i m e } , \mathbf { Z } ^ { t i m e }$ denotes the output of the MSA and MLP.
+The computation complexity of cross-time stage is $O ( D L ^ { 2 } )$ . After this stage, the dependency among time segments in the same dimension is captured in ${ \bf Z } ^ { t i m e }$ . Then ${ \bf Z } ^ { t i \bar { m } e }$ becomes the input of Cross-Dimension Stage to capture cross-dimension dependency.
+![](images/189c8be9df5923b18b60424b48ac199e9ced3d1f325af4ddb7f49da8ea8efbfe.jpg)
+Figure 2: The TSA layer. (a) Two-Stage Attention Layer to process a 2D vector array representing multivariate time series: each vector refers to a segment of the original series. The whole vector array goes through the Cross-Time Stage and Cross-Dimension Stage to get corresponding dependency. (b) Directly using MSA in Cross-Dimension Stage to build the $D$ -to- $D$ connection results in $O ( D ^ { 2 } )$ complexity. (c) Router mechanism for Cross-Dimension Stage: a small fixed number (c) of “routers” gather information from all dimensions and then distribute the gathered information. The complexity is reduced to $O ( 2 c D ) = O ( D )$ .
+Cross-Dimension Stage We can use a large $L _ { s e g }$ for long sequence in DSW Embedding to reduce the number of segments $L$ in cross-time stage. While in Cross-Dimension Stage, we can not partition dimensions and directly apply MSA will cause the complexity of $O ( D ^ { 2 } )$ (as shown in Fig. 2 (b)), which is unaffordable for datasets with large $D$ . Instead, we propose the router mechanism for potentially large $D$ . As shown in Fig. 2 (c), we set a small fixed number $c < < D$ ) of learnable vectors for each time step $i$ as routers. These routers first aggregate messages from all dimensions by using routers as query in MSA and vectors of all dimensions as key and value. Then routers distribute the received messages among dimensions by using vectors of dimensions as query and aggregated messages as key and value. In this way, the all-to-all connection among $D$ dimensions are built:
+$$
+\begin{array} { r l } & { \quad \mathbf { B } _ { i , : } = \mathbb { M } \mathbb { S } \mathbb { A } _ { 1 } ^ { d i m } ( \mathbf { R } _ { i , : } , \mathbf { Z } _ { i , : } ^ { t i m e } , \mathbf { Z } _ { i , : } ^ { t i m e } ) , 1 \leq i \leq L } \\ & { \quad \overline { { \mathbf { Z } } } _ { i , : } ^ { d i m } = \mathbb { M } \mathbb { S } \mathbb { A } _ { 2 } ^ { d i m } ( \mathbf { Z } _ { i , : } ^ { t i m e } , \mathbf { B } _ { i , : } , \mathbf { B } _ { i , : } ) , 1 \leq i \leq L } \\ & { \quad \hat { \mathbf { Z } } ^ { d i m } = \mathbb { L } \mathbb { a } \mathbb { Y } \mathrm { e r N o r m } \left( \mathbf { Z } ^ { t i m e } + \overline { { \mathbf { Z } } } ^ { d i m } \right) } \\ & { \quad \mathbf { Z } ^ { d i m } = \mathbb { L } \mathbb { a } \mathbb { Y } \mathrm { e r N o r m } \left( \hat { \mathbf { Z } } ^ { d i m } + \mathbb { M } \mathbf { L } \mathbf { P } ( \hat { \mathbf { Z } } ^ { d i m } ) \right) } \end{array}
+$$
+where $\mathbf { R } \in \mathbb { R } ^ { L \times c \times d _ { m o d e l } }$ $\dot { } c$ is a constant) is the learnable vector array serving as routers. B ∈ RL×c×dmodel is the aggregated messages from all dimensions. Zdim denotes output of the router mechanism. All time steps $( 1 ~ \leq ~ i ~ \leq ~ L )$ share the same $\mathbf { M S A } _ { 1 } ^ { d i m }$ , $\mathbf { M S A } _ { 2 } ^ { d i m }$ . $\hat { \mathbf { Z } } ^ { d i m } , \mathbf { Z } ^ { d i m }$ denote output of skip connection and MLP respectively. The router mechanism reduce the complexity from $O ( D ^ { 2 } L )$ to $O ( D L )$ .
+Adding up Eq. 3 and Eq. 4, we model the two stages as:
+$$
+\mathbf { Y } = \mathbf { Z } ^ { d i m } = \mathrm { T S A } ( \mathbf { Z } )
+$$
+![](images/c5d80c2619c68f099818b7e2189b44e63ea3d8b90bac0ec5cafd752f6cb747fb.jpg)
+Figure 3: Architecture of the Hierarchical Encoder-Decoder in Crossformer with 3 encoder layers. The length of each vector denotes the covered time range. The encoder (left) uses TSA layer and segment merging to capture dependency at different scales: a vector in upper layer covers a longer range, resulting in dependency at a coarser scale. Exploring different scales, the decoder (right) makes the final prediction by forecasting at each scale and adding them up.
+where Z, $\mathbf { Y } \in \mathbb { R } ^ { L \times D \times d _ { m o d e l } }$ denotes the input and output vector array of TSA layer, respectively. Note that the overall computation complexity of the
+TSA layer is $O ( D L ^ { 2 } + D L ) = O ( D L ^ { 2 } )$ . After the Cross-Time and Cross-Dimension Stages, every two segments (i.e. ${ \bf Z } _ { i _ { 1 } , d _ { 1 } } , { \bf Z } _ { i _ { 2 } , d _ { 2 } } )$ in $\mathbf { Z }$ are connected, as such both cross-time and cross-dimension dependencies are captured in $\mathbf { Y }$ .
+# 3.3 HIERARCHICAL ENCODER-DECODER
+Hierarchical structures are widely used in Transformers for MTS forecasting to capture information at different scales (Zhou et al., 2021; Liu et al., 2021a). In this section, we use the proposed DSW embedding, TSA layer and segment merging to construct a Hierarchical Encoder-Decoder (HED). As shown in Fig. 3, the upper layer utilizes information at a coarser scale for forecasting. Forecasting values at different scales are added to output the final result.
+Encoder In each layer of the encoder (except the first layer), every two adjacent vectors in time domain are merged to obtain the representation at a coarser level. Then a TSA layer is applied to capture dependency at this scale. This process is modeled as ${ \bf Z } ^ { e n c , l } = \mathrm { E n c o d e r } ( { \bf Z } ^ { e \bar { n } c , l - 1 } )$ :
+$$
+\left\{ \begin{array} { l l } { l = 1 : } & { \hat { \mathbf { Z } } ^ { e n c , l } = \mathbf { H } } \\ { l > 1 : } & { \hat { \mathbf { Z } } _ { i , d } ^ { e n c , l } = \mathbf { M } [ \mathbf { Z } _ { 2 i - 1 , d } ^ { e n c , l - 1 } \cdot \mathbf { Z } _ { 2 i , d } ^ { e n c , l - 1 } ] , 1 \le i \le \frac { L _ { l - 1 } } { 2 } , 1 \le d \le D } \\ { } & { \mathbf { Z } ^ { e n c , l } = \mathrm { T S A } ( \hat { \mathbf { Z } } ^ { e n c , l } ) } \end{array} \right.
+$$
+where $\mathbf { H }$ denotes the 2D array obtained by DSW embedding; ${ \bf Z } ^ { e n c , l }$ denotes the output of the $l$ -th encoder layer; $\textbf { M } \in \ \mathbb { R } ^ { d _ { m o d e l } \times 2 d _ { m o d e l } }$ denotes a learnable matrix for segment merging; $[ \cdot ]$ denotes the concatenation operation; $L _ { l - 1 }$ denotes the number of segments in each dimension in layer $l - 1$ , if it is not divisible by 2, we pad ${ \bf Z } ^ { e n c , l - 1 }$ to the proper length; $\hat { \mathbf { Z } } ^ { e n c , l }$ denotes the array after segment merging in the $i$ -th layer. Suppose there are $N$ layers in the encoder, we use ${ \bf { Z } } ^ { e n c , 0 } , { \bf { Z } } ^ { e n c , \tilde { 1 } } , \ldots , { \bf { Z } } ^ { e n c , N } , \left( { \bf { Z } } ^ { e n c , 0 } = \bf { H } \right)$ to represent the $N + 1$ outputs of the encoder. The complexity of each encoder layer is $\begin{array} { r } { O ( D \frac { T ^ { 2 } } { L _ { s e g } ^ { 2 } } ) } \end{array}$
+Decoder Obtaining the $N + 1$ feature arrays output by the encoder, we use $N + 1$ layers (indexed by $0 , 1 , \ldots , N )$ in decoder for forecasting. Layer $l$ takes the $l$ -th encoded array as input, then outputs a decoded 2D array of layer $l$ . This process is summarized as ${ \bf Z } ^ { d e c , l } = \mathrm { D e c o } \dot { { \bf d e r } } ( { \bf Z } ^ { d e c , l - 1 } , { \bf Z } ^ { e n c , \dot { l } } )$ :
+$$
+\begin{array} { r l } & { \left\{ \begin{array} { l l } { l = 0 : } & { \tilde { \mathbf { Z } } ^ { d e c , l } } \\ { l > 0 : } & { \tilde { \mathbf { Z } } ^ { d e c , l } } \end{array} \right. = \mathrm { T S } \mathbb { A } ( \mathbf { E } ^ { ( d e c ) } ) } \\ & { \left. \begin{array} { l l } { \overline { { \mathbf { Z } } } _ { : , d } ^ { d e c , l } = \mathbb { M } \mathbb { S } \mathbb { A } \left( \mathbf { \tilde { Z } } _ { : , d } ^ { d e c , l } \right) } \\ { \overline { { \mathbf { Z } } } _ { : , d } ^ { d e c , l } = \mathbb { M } \mathbb { S } \mathbb { A } \left( \mathbf { \tilde { Z } } _ { : , d } ^ { d e c , l } , \mathbf { Z } _ { : , d } ^ { e n c , l } , \mathbf { Z } _ { : , d } ^ { e n c , l } \right) , 1 \leq d \leq D } \end{array} \right. } \\ & { \left. \begin{array} { r l } { \tilde { \mathbf { Z } } ^ { d e c , l } = \mathrm { L a y e r N o r n } \left( \tilde { \mathbf { Z } } ^ { d e c , l } + \mathbf { \overline { { Z } } } ^ { d e c , l } \right) } \\ { \mathbf { Z } ^ { d e c , l } = \mathrm { L a y e r N o r n } \left( \hat { \mathbf { Z } } ^ { d e c , l } + \mathbb { M L P } ( \hat { \mathbf { Z } } ^ { d e c , l } ) \right) } \end{array} \right. } \end{array}
+$$
+where $\mathbf { E } ^ { ( d e c ) } \in \mathbb { R } ^ { \frac { \tau } { L _ { s e g } } \times D \times d _ { m o d e l } }$ denotes the learnable position embedding for decoder. $\tilde { \mathbf { Z } } ^ { d e c , l }$ is the output of TSA. The MSA layer takes $\tilde { \mathbf { Z } } _ { : , d } ^ { d e c , l }$ as query and ${ \bf Z } _ { : , d } ^ { e n c , l }$ as the key and value to build the connection between encoder and decoder. The output of MSA is denoted as $\overline { { \mathbf { Z } } } _ { : , d } ^ { d e c , l }$ . $\hat { \mathbf { Z } } ^ { d e c , l } , \mathbf { Z } ^ { d e c , l }$ denote the output of skip connection and MLP respectively. We use ${ \mathbf { Z } } ^ { d e c , 0 }$ , ${ \bf Z } ^ { e n c , 1 } , \ldots , { \bf Z } ^ { d e c , N }$ to represent the decoder output. The complexity of each decoder layer is O D τ(T +τ)L2
+Linear projection is applied to each layer’s output to yield the prediction of this layer. Layer predictions are summed to make the final prediction (for $l = 0 , \ldots , N )$ :
+$$
+\begin{array} { r l } { \mathrm { ~ o r ~ } l = 0 , \dots , N : \mathbf { x } _ { i , d } ^ { ( s ) , l } = \mathbf { W } ^ { l } \mathbf { Z } _ { i , d } ^ { d e c , l } } & { \quad \mathbf { x } _ { T + 1 : T + \tau } ^ { p r e d , l } = \left\{ \mathbf { x } _ { i , d } ^ { ( s ) , l } \vert 1 \leq i \leq \frac { \tau } { L _ { s e g } } , 1 \leq d \leq D \right\} } \\ & { \quad \qquad \quad \mathbf { x } _ { T + 1 : T + \tau } ^ { p r e d } = \displaystyle \sum _ { l = 0 } ^ { N } \mathbf { x } _ { T + 1 : T + \tau } ^ { p r e d , l } } \end{array}
+$$
+where $\mathbf { W } ^ { l } ~ \in ~ \mathbb { R } ^ { L _ { s e g } \times d _ { m o d e l } }$ is a learnable matrix to project a vector to a time series segment. $\mathbf { x } _ { i , d } ^ { ( s ) , l } \in \mathbb { R } ^ { L _ { s e g } }$ denotes the $i$ -th segment in dimension $d$ of the prediction. All the segments in layer
+Table 1: MSE/MAE with different prediction lengths. Bold/underline indicates the best/second. Results of LSTMa, LSTnet, Transformer, Informer on the first 4 datasets are from Zhou et al. (2021).
+<table><tr><td colspan="2">Models</td><td colspan="2">LSTMa</td><td colspan="2">LSTnet</td><td colspan="2">MTGNN</td><td colspan="2">Transformer</td><td colspan="2">Informer</td><td colspan="2">Autoformer</td><td colspan="2">Pyraformer</td><td colspan="2">FEDformer</td><td colspan="2">Crossformer</td></tr><tr><td colspan="2">Metric</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td></tr><tr><td rowspan="2">FTLLI</td><td>24 48 168</td><td>0.650 0.720 1.212</td><td>0.624 0.675 0.867</td><td>1.293 1.456 1.997</td><td>0.901 0.960 1.214</td><td>0.336 0.386 0.466</td><td>0.393 0.429 0.474</td><td>0.620 0.692 0.947</td><td>0.577 0.671 0.797</td><td>0.577 0.685 0.931</td><td>0.549 0.625 0.752</td><td>0.439 0.429 0.493</td><td>0.440 0.442 0.479</td><td>0.493 0.554 0.781</td><td>0.507 0.544 0.675</td><td>0.318 0.342 0.412</td><td>0.384 0.396 0.449</td><td>0.305 0.352</td><td>0.367 0.394 0.441</td></tr><tr><td>336 720 24</td><td>1.424 1.960 0.621</td><td>0.994 1.322 0.629</td><td>2.655 2.143 1.968</td><td>1.369 1.380 1.170</td><td>0.736 0.916 0.260</td><td>0.643 0.750 0.324</td><td>1.094 1.241 0.306</td><td>0.813 0.917 0.371</td><td>1.128 1.215 0.323</td><td>0.873 0.896 0.369</td><td>0.509 0.539 0.410</td><td>0.492 0.537 0.428</td><td>0.912 0.993 0.310</td><td>0.747 0.792 0.371</td><td>0.456 0.521 0.290</td><td>0.474 0.515 0.364</td><td>0.410 0.440 0.519 0.211</td><td>0.461 0.524 0.293</td></tr><tr><td>[LL</td><td>48 96 288 672</td><td>1.392 1.339 1.740 2.736</td><td>0.939 0.913 1.124 1.555</td><td>1.999 2.762 1.257 1.917</td><td>1.215 1.542 2.076 2.941</td><td>0.386 0.428 0.469 0.620</td><td>0.408 0.446 0.488 0.571</td><td>0.465 0.681 1.162 1.231</td><td>0.470 0.612 0.879 1.103</td><td>0.494 0.678 1.056 1.192</td><td>0.503 0.614 0.786 0.926</td><td>0.485 0.502 0.604 0.607</td><td>0.464 0.476 0.522 0.530</td><td>0.465 0.520 0.729 0.980</td><td>0.464 0.504 0.657 0.678</td><td>0.342 0.366 0.398 0.455</td><td>0.396 0.412 0.433 0.464</td><td>0.300 0.320 0.404 0.569</td><td>0.352 0.373 0.427 0.528</td></tr><tr><td>HLM</td><td>24 48 168 336 720</td><td>0.546 0.829 1.038 1.657 1.536</td><td>0.570 0.677 0.835 1.059 1.109</td><td>0.615 0.660 0.748 0.782 0.851</td><td>0.545 0.589 0.647 0.683 0.757</td><td>0.307 0.388 0.498 0.506 0.510</td><td>0.356 0.422 0.512 0.523 0.527</td><td>0.349 0.386 0.613 0.707 0.834</td><td>0.397 0.433 0.582 0.634 0.741</td><td>0.335 0.395 0.608 0.702 0.831</td><td>0.381 0.459 0.567 0.620 0.731 0.587</td><td>0.363 0.456 0.574 0.600</td><td>0.396 0.462 0.548 0.571 0.570</td><td>0.301 0.376 0.519 0.539 0.547</td><td>0.359 0.421 0.521 0.543 0.553</td><td>0.357 0.428 0.564 0.533 0.562</td><td>0.412 0.458 0.541 0.536 0.557</td><td>0.294 0.370 0.473 0.495 0.526</td><td>0.343 0.411 0.494 0.515 0.542</td></tr><tr><td rowspan="6">R</td><td>48 168 336</td><td>0.486 0.572 0.574</td><td>0.602</td><td>0.369 0.394</td><td>0.445 0.476</td><td>0.173 0.236</td><td>0.280 0.320</td><td>0.334 0.353</td><td>0.399 0.420</td><td>0.344 0.393 0.368 0.424</td><td>0.241 0.299</td><td></td><td>0.351 0.387</td><td>0.478 0.452</td><td>0.471 0.455</td><td>0.229 0.263</td><td>0.338 0.361</td><td>0.156 0.231 0.323</td><td>0.255 0.309 0.369</td></tr><tr><td>720</td><td>0.886 0.795 1.095</td><td>0.419 0.556</td><td>0.477 0.565</td><td>0.328 0.422</td><td>0.373 0.410</td><td>0.381 0.391</td><td>0.439 0.438</td><td>0.381 0.406</td><td>0.431 0.443</td><td>0.375 0.377</td><td>0.428 0.434</td><td>0.463 0.480</td><td>0.456 0.461</td><td>0.305 0.372</td><td>0.386 0.434</td><td>0.404</td><td>0.423</td></tr><tr><td></td><td>1.128</td><td>0.605</td><td>0.599</td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.366</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>4.771</td><td>1.335</td><td>4.975</td><td>1.660</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><td></td><td></td></tr><tr><td></td><td>4.220</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>3.101</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>24</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><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>960</td><td>1.676 1.591</td><td></td><td></td><td></td><td>0.471</td><td>0.451</td><td>0.492</td><td>0.550</td><td>0.460</td><td>0.548</td><td></td><td>0.426</td><td>0.550</td><td>0.489</td><td>0.393</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><td></td><td></td><td></td><td></td><td></td><td></td><td>0.449</td><td>0.433</td><td>0.438</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><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><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>4.265</td><td>1.387</td><td>3.954</td><td>1.323</td><td>4.588</td><td>1.462</td><td></td><td>1.238</td><td>3.970</td><td>1.338</td><td>2.687</td><td>1.147</td><td>3.041</td><td>1.186</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>1.659</td><td>4.777</td><td>1.496</td><td>4.167</td><td>1.360</td><td>4.845</td><td>1.496</td><td></td><td>1.270</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>I</td><td>36</td><td>1.427</td><td></td><td>5.322</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>3.397</td><td></td><td>4.377</td><td>1.410</td><td>2.887</td><td>1.160</td><td>3.406</td><td>1.232</td></tr><tr><td></td><td></td><td></td><td></td><td>5.425</td><td>1.632</td><td>5.333</td><td>1.592</td><td>4.746</td><td>1.463</td><td>4.865</td><td>1.516</td><td>2.947</td><td>1.203</td><td>4.811</td><td>1.503</td><td>2.797</td><td>1.155</td><td>3.459</td><td>1.221</td></tr><tr><td></td><td>48</td><td>4.945 1.462</td><td></td><td></td><td></td><td>5.070</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>2.809</td><td>1.163</td><td>3.640</td><td>1.305</td></tr><tr><td>Yteee</td><td>60</td><td>5.176</td><td>1.504 0.378</td><td>5.477 0.648</td><td>1.675 0.403</td><td>0.506</td><td>1.552 0.278</td><td>5.219 0.597</td><td>1.553 0.332</td><td>5.212 0.608</td><td>1.576 0.334</td><td>3.019 0.550</td><td>1.202 0.363</td><td>5.204 0.606</td><td>1.588 0.338</td></table>
+$l$ are rearranged to get the layer prediction $\mathbf { x } _ { T + 1 : T + \tau } ^ { p r e d , l }$ . Predictions of all the layers are summed to obtain the final forecasting xpredT +1:T +τ .
+# 4 EXPERIMENTS
+# 4.1 PROTOCOLS
+Datasets We conduct experiments on six real-world datasets following Zhou et al. (2021); Wu et al. (2021a). 1) ETTh1 (Electricity Transformer Temperature-hourly), 2) ETTm1 (Electricity Transformer Temperature-minutely), 3) WTH (Weather), 4) ECL (Electricity Consuming Load), 5) ILI (Influenza-Like Illness), 6) Traffic. The train/val/test splits for the first four datasets are same as Zhou et al. (2021), the last two are split by the ratio of 0.7:0.1:0.2 following Wu et al. (2021a).
+Baselines We use the following popular models for MTS forecasting as baselines:1) LSTMa (Bahdanau et al., 2015), 2) LSTnet (Lai et al., 2018), 3) MTGNN (Wu et al., 2020), and recent Transformer-based models for MTS forecasting: 4) Transformer (Vaswani et al., 2017), 5) Informer (Zhou et al., 2021), 6) Autoformer (Wu et al., 2021a), 7) Pyraformer (Liu et al., 2021a) and 8) FEDformer (Zhou et al., 2022).
+Setup We use the same setting as in Zhou et al. (2021): train/val/test sets are zero-mean normalized with the mean and std of training set. On each dataset, we evaluate the performance over the changing future window size $\tau$ . For each $\tau$ , the past window size $T$ is regarded as a hyper-parameter to search which is a common protocol in recent MTS transformer literature (Zhou et al., 2021; Liu et al., 2021a). We roll the whole set with stride $= 1$ to generate different input-output pairs. The Mean Square Error (MSE) and Mean Absolute Error (MAE) are used as evaluation metrics. All experiments are repeated for 5 times and the mean of the metrics reported. Our Crossformer only utilize the past series to forecast the future, while baseline models use additional covariates such as hour-of-the-day. Details about datasets, baselines, implementation, hyper-parameters are shown in Appendix A.
+# 4.2 MAIN RESULTS
+As shown in Table 1, Crossformer shows leading performance on most datasets, as well as on different prediction length settings, with the 36 top-1 and 51 top-2 cases out of 58 in total. It is worth noting that, perhaps due to the explicit use of cross-dimension dependency via GNN, MTGNN outperforms many Transformer-based baselines. While MTGNN has been rarely compared in existing transformers for MTS forecasting literatures. FEDformer and Autoformer outperform our model on ILI. We conjecture this is because the size of dataset ILI is small and these two models introduce the prior knowledge of sequence decomposition into the network structure which makes them perform well when the data is limited. Crossformer still outperforms other baselines on this dataset.
+Table 2: Component ablation of Crossformer: DSW embedding, TSA layer and HED on ETTh1.
+<table><tr><td>Models</td><td colspan="2">Transformer</td><td colspan="2">DSW</td><td colspan="2">DSW+TSA</td><td colspan="2">DSW+HED</td><td colspan="2">DSW+TSA+HED</td></tr><tr><td>Metric</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td></tr><tr><td>24</td><td>0.620</td><td>0.577</td><td>0.373</td><td>0.418</td><td>0.322</td><td>0.373</td><td>0.406</td><td>0.454</td><td>0.305</td><td>0.367</td></tr><tr><td>48</td><td>0.692</td><td>0.671</td><td>0.456</td><td>0.479</td><td>0.365</td><td>0.403</td><td>0.493</td><td>0.512</td><td>0.352</td><td>0.394</td></tr><tr><td>168</td><td>0.947</td><td>0.797</td><td>0.947</td><td>0.731</td><td>0.473</td><td>0.479</td><td>0.614</td><td>0.583</td><td>0.410</td><td>0.441</td></tr><tr><td>336</td><td>1.094</td><td>0.813</td><td>0.969</td><td>0.752</td><td>0.553</td><td>0.534</td><td>0.788</td><td>0.676</td><td>0.440</td><td>0.461</td></tr><tr><td>720</td><td>1.241</td><td>0.971</td><td>1.086</td><td>0.814</td><td>0.636</td><td>0.599</td><td>0.841</td><td>0.717</td><td>0.519</td><td>0.524</td></tr></table>
+![](images/a6cb92446b54acd5345629019f18eecb486532ff4a89ef6b420ad7722a99d282.jpg)
+Figure 4: Evaluation on hyper-parameter impact and computational efficiency. (a) MSE against hyperparameter segment length $L _ { s e g }$ in DSW embedding on ETTh1. (b) MSE against hyper-parameter number of routers $c$ in the Cross-Dimension Stage of TSA layer on ETTh1. (c) Memory occupation against the input length $T$ on ETTh1. (d) Memory occupation against number of dimensions $D$ on synthetic datasets with different number of dimensions.
+# 4.3 ABLATION STUDY
+In our approach, there are three components: DSW embedding, TSA layer and HED. We perform ablation study on the ETTh1 dataset in line with Zhou et al. (2021); Liu et al. (2021a). We use Transformer as the baseline and $\mathbf { D S W + T S A + H E D }$ to denote Crossformer without ablation. Three ablation versions are compared: 1) DSW 2) DSW $+ ^ { \prime }$ TSA 3) $\mathbf { D S W + H E D }$ .
+We analyze the results shown in Table 2. 1) DSW performs better than Transformer on most settings. The only difference between DSW and Transformer is the embedding method, which indicates the usefulness of DSW embedding and the importance of cross-dimension dependency. 2) TSA constantly improves the forecasting accuracy. This suggests that it is reasonable to treat time and dimension differently. Moreover, TSA makes it possible to use Crossformer on datasets where the number of dimensions is large (e.g. $D = 8 6 2$ for dataset Traffic). 3) Comparing $\mathrm { D S W + H E D }$ with DSW, HED decreases the forecasting accuracy when prediction length is short but increases it for long term prediction. The possible reason is that information at different scales is helpful to long term prediction. 4) Combining DSW, TSA and HED, our Crossformer yields best results on all settings.
+# 4.4 EFFECT OF HYPER-PARAMETERS
+We evaluate the effect of two hyper-parameters: segment length $L _ { s e g }$ in Eq. 1) and number of routers in TSA $\dot { c }$ in Cross-Dimension Stage of TSA) on the ETTh1 dataset. Segment Length: In Fig. 4(a), we prolong the segment length from 4 to 24 and evaluate MSE with different prediction windows. For short-term forecasting $( \tau = 2 4 , 4 8 )$ ), smaller segment yields relevantly better results, but the prediction accuracy is stable. For long-term forecasting $\tau \geq 1 6 8 )$ , prolonging the segment length from 4 to 24 causes the MSE to decrease. This indicates that long segments should be used for long-term forecasting. We further prolong the segment length to 48 for $\tau = 3 3 6$ , 720, the MSE is slightly larger than that of 24. The possible reason is that 24 hours exactly matches the daily period of this dataset, while 48 is too coarse to capture fine-grained information. Number of Routers in TSA Layer: Number of Routers $c$ controls the information bandwidth among all dimensions. As Fig. 4(b) shows, the performance of Crossformer is stable w.r.t to $c$ for $\tau \leq 3 3 6$ . For $\tau = 7 2 0$ , the MSE is large when $c = 3$ but decreases and stabilizes when $c \geq 5$ . In pratice, we set $c = 1 0$ to balance the prediction accuracy and computation efficiency.
+# 4.5 COMPUTATIONAL EFFICIENCY ANALYSIS
+The theoretical complexity per layer of Transformer-based models is compared in Table 3. The complexity of Crossformer encoder is quadratic w.r.t $T$ . However, for long-term prediction where large $L _ { s e q }$ is used, the coefficient $\frac { 1 } { L _ { s e q } ^ { 2 } }$ term can significantly reduce its practical complexity. We evaluate the memory occupation of these models on ETTh1.4 We set the prediction window $\tau = 3 3 6$ and prolong input length $T$ . For Crossformer, $L _ { s e g }$ is set to 24, which is the best value for $\tau \geq 1 6 8$
+Table 3: Computation complexity per layer of Transformer-based models. $T$ denotes the length of past series, $\tau$ denotes the length of prediction window, $D$ denotes the number of dimensions, $L _ { s e g }$ denotes the segment length of DSW embedding in Crossformer.
+<table><tr><td>Method</td><td>Encoder layer</td><td>Decoder layer</td></tr><tr><td>Transformer (Vaswani et al., 2017)</td><td>O（T2）</td><td>O(T(T+T))</td></tr><tr><td>Informer (Zhou et al., 2021)</td><td>O(TlogT)</td><td>O(t(T+logT))</td></tr><tr><td>Autoformer (Wu et al.,2021a)</td><td>O(TlogT)</td><td>0（（+T）log（+T））</td></tr><tr><td>Pyraformer (Liu et al.,2021a)</td><td>O(T)</td><td>O(T(T+T))</td></tr><tr><td>FEDformer (Zhou et al.,2022)</td><td>O(T)</td><td>0（+）</td></tr><tr><td>Crossformer (Ours)</td><td>（T2） 0</td><td>（(+T）） 0</td></tr></table>
+(see Fig. 4 (a)). The result in Fig. 4 (c) shows that Crossformer achieves the best efficiency among the five methods within the tested length range. Theoretically, Informer, Autoformer and FEDformer are more efficient when $T$ approaches infinity. In practice, Crossformer performs better when $T$ is not extremely large (e.g. $T \leq 1 0 ^ { 4 }$ ).
+We also evaluate the memory occupation w.r.t the number of dimensions $D$ . For baseline models where cross-dimension dependency is not modeled explicitly, $D$ has little effect. Therefore, we compare Crossformer with its ablation versions in Section 4.3. We also evaluate the TSA layers that directly use MSA in Cross-Dimension Stage without the Router mechanism, denoted as TSA(w/o Router). Fig. 4 (d) shows that Crossformer without TSA layer (DSW and $\mathrm { D S W + H E D } )$ has quadratic complexity w.r.t $D$ . TSA(w/o Router) helps to reduce complexity and the Router mechanism further makes the complexity linear, so that Crossformer can process data with $D = 3 0 0$ . Moreover, HED can slightly reduce the memory cost and we analyze this is because there are less vectors in upper layers after segment merging (see Fig. 3). Besides memory occupation, the actual running time evaluation is shown in Appendix B.6.
+# 5 CONCLUSIONS AND FUTURE WORK
+We have proposed Crossformer, a Transformer-based model utilizing cross-dimension dependency for multivariate time-series (MTS) forecasting. Specifically, the Dimension-Segment-Wise (DSW) embedding embeds the input data into a 2D vector array to preserve the information of both time and dimension. The Two-Stage-Attention (TSA) layer is devised to capture the cross-time and crossdimension dependency of the embedded array. Using DSW embedding and TSA layer, a Hierarchical Encoder-Decoder (HED) is devised to utilize the information at different scales. Experimental results on six real-world datasets show its effectiveness over previous state-of-the-arts.
+We analyzed the limitations of our work and briefly discuss some directions for future research: 1) In Cross-Dimension Stage, we build a simple full connection among dimensions, which may introduce noise on high-dimensional datasets. Recent sparse and efficient Graph Transformers (Wu et al., 2022) can benefit our TSA layer on this problem. 2) A concurrent work (Zeng et al., 2023) which was accepted after the submission of this work received our attention. It questions the effectiveness of Transformers for MTS forecasting and proposes DLinear that outperforms all Transformers including our Crossformer on three of the six datasets (details are in Appendix B.2). It argues the main reason is that MSA in Transformer is permutation-invariant. Therefore, enhancing the ordering preserving capability of Transformers is a promising direction to overcome this shortcoming . 3) Considering datasets used in MTS analysis are much smaller and simpler than those used in vision and texts, besides new models, large datasets with various patterns are also needed for future research.
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+# A DETAILS OF EXPERIMENTS
+# A.1 BENCHMARKING DATASETS
+We conduct experiments on the following six real-world datasets following Zhou et al. (2021); Wu et al. (2021a):
+1) ETTh1 (Electricity Transformer Temperature-hourly) contains 7 indicators of an electricity transformer in two years, including oil temperature, useful load, etc. Data points are recorded every hour and train/val/test is 12/4/4 months.
+2) ETTm1 (Electricity Transformer Temperature-minutely) contains the same indicators as ETTh1 but data points are recorded every 15 miniutes. Train/val/test split is same as ETTh1.
+3) WTH (Weather) contains 12 meteorological indicators in U.S. in 4 years, including visibility, wind speed, etc. Train/val/test is 28/10/10 months.
+4) ECL (Electricity Consuming Load) contains hourly electricity consumption (in Kwh) of 321 clients in two years. Train/val/test is 15/3/4 months.
+5) ILI (Influenza-Like Illness) contains 7 weekly recorded indicators of patients data from Centers for Disease Control and Prevention of the United States between between 2002 and 2021. The ratio of train/validation/test splits is 0.7:0.1:0.2.
+6) Traffic contains hourly road occupancy rates measured by 862 sensors on San Francisco Bay area freeways in 2 years. The ratio of train/validation/test splits is 0.7:0.1:0.2.
+The train/val/test splits for ETTh1, ETTm1, WTH, ECL are same as Zhou et al. (2021), for ILI and Traffic are same as Wu et al. (2021a).
+The first four datasets are publicly available at https://github.com/zhouhaoyi/ Informer2020 and the last two are publicly available at https://github.com/thuml/ Autoformer.
+# A.2 BASELINE METHODS
+We briefly describe the selected baselines:
+1) LSTMa (Bahdanau et al., 2015) treats the input MTS as a sequence of multi-dimensional vectors. It builds an encoder-decoder using RNN and automatically aligns target future steps with their relevant past.
+2) LSTnet (Lai et al., 2018) uses CNN to extract cross-dimension dependency and short term crosstime dependency. The long-term cross-time dependency is captured through RNN. The source code is available at https://github.com/laiguokun/LSTNet.
+3) MTGNN (Wu et al., 2020) explicitly utilizes cross-dimension dependency using GNN. A graph learning layer learns a graph structure where each node represents one dimension in MTS. Then graph convolution modules are interleaved with temporal convolution modules to explicitly capture cross-dimension and cross-time dependency respectively. The source code is available at https://github.com/nnzhan/MTGNN.
+4) Transformer is closed to the original Transformer (Vaswani et al., 2017) that uses self-attention mechanism to capture cross-time dependency. The Informer-style one-step generative decoder is used for forecasting, therefore this is denoted as Informer† in Informer (Zhou et al., 2021).
+5) Informer (Zhou et al., 2021) is a Transformer-based model using the ProbSparse self-attention to capture cross-time dependency for forecasting. The source code of Transformer and Informer is available at https://github.com/zhouhaoyi/Informer2020.
+6) Autoformer (Wu et al., 2021a) is a Transformer-based model using decomposition architecture with Auto-Correlation mechanism to capture cross-time dependency for forecasting. The source code is available at https://github.com/thuml/Autoformer.
+7) Pyraformer (Liu et al., 2021a) is a Transformer-based model learning multi-resolution representation of the time series by the pyramidal attention module to capture cross-time dependency for forecasting. The source code is available at https://github.com/alipay/Pyraformer.
+8) FEDformer (Zhou et al., 2022) is a Transformer-based model that uses the seasonal-trend decomposition with frequency enhanced blocks to capture cross-time dependency for forecasting. The source code is available at https://github.com/MAZiqing/FEDformer.
+A.3 HYPER-PARAMETER SELECTION AND IMPLEMENTATION DETAILS
+# A.3.1 MAIN EXPERIMENTS
+For the main experiments, we use the Crossformer with 3 encoder layers. The number of routers in TSA layer $c$ is set to 10. For dataset ETTh1, ETTm1, WTH and ILI, dimension of hidden state $d _ { m o d e l }$ is set to 256, the head number of multi-head attention is set to 4; For dataset ECL and Traffic, dimension of hidden state $d _ { m o d e l }$ is set to 64, the head number of multi-head attention is set to 2. The segment length $L _ { s e g }$ is chosen from $\{ 6 , 1 2 , 2 4 \}$ via grid search. We use MSE as loss function and batch size is set to 32. Adam optimizer is used for training and the initial learning rate is chosen from $\{ 5 \mathrm { e } \mathrm { - } 3$ , 1e-3, 5e-4, 1e-4, 5e-5, 1e- $\{ 5 \}$ via grid search. The total number of epochs is 20. If the validation loss does not decreases within three epochs, the training process will stop early.
+For baseline models, if the original papers conduct experiments on the dataset we use, the hyperparameters (except input length $T$ ) recommended in the original papers are used, including the number of layers, dimension of hidden states, etc. Otherwise, the hyper-parameters are chosen through grid search using the validation set.
+Following Zhou et al. (2021), on datasets ETTh1, WTH, ECL and Traffic, for different prediction length $\tau$ , the input length $T$ is chosen from $\{ 2 4 , 4 8 , 9 6 , 1 6 8 , 3 3 6 , 7 2 0 \}$ ; on ETTm1, the input length is chosen from $\{ 2 4 , 4 8 , 9 6 , 1 9 2 , 2 8 8 , 6 7 2 \}$ ; on ILI, the input length is chosen from $\{ 2 4 , 3 6 , 4 8 , 6 0 \}$ .
+All models including Crossformer and baselines are implemented in PyTorch and trained on a single NVIDIA Quadro RTX 8000 GPU with 48GB memory.
+# A.3.2 EFFICIENCY ANALYSIS
+To evaluate the computational efficiency w.r.t the input length $T$ in Figure 4(c) of the main paper, we align the hyper-parameters of all Transformer-based models as follows: prediction length $\tau$ is set to 336, number of encoder layers is set to 2, dimension of hidden state $d _ { m o d e l }$ is set to 256, the head number of multi-head attention is set to 4.
+To evaluate the computational efficiency w.r.t the number of dimensions $D$ in Figure 4(d) of the main paper, we align the hyper-parameters of ablation versions of Crossformer as follows as: both input length $T$ and prediction length $\tau$ are set to 336, number of encoder layers is set to 3, $d _ { m o d e l }$ is set to 64, the head number of multi-head attention is set to 2.
+Experiments in the computational efficiency analysis section are conducted on a single NVIDIA GeForce RTX 2080Ti GPU with 11GB memory.
+# A.4 DETAILS OF ABLATION VERSIONS OF CROSSFORMER
+We describe the models we used in ablation study below:
+1) DSW represents Crossformer without TSA and HED. The input is embedded by DSW embedding and flatten into a 1D sequence to be input to the original Transformer. The only difference between this model and the Transformer is the embedding method.
+2) $\mathbf { D S W + T S A }$ represents Crossformer without HED. Compared with Crossformer, the encoder does not use segment merging to capture dependency at different scales. The decoder takes the final output of encoder (i.e. ${ \bf Z } ^ { e n c , N }$ ) as input instead of using encoder’s output at each scale.
+3) $\mathbf { D S W + H E D }$ represents Crossformer without TSA. In each encoder layer and decoder layer, the 2D vector array is flatten into a 1D sequence to be input to the original self-attention layer for dependency capture.
+# B EXTRA EXPERIMENTAL RESULTS
+# B.1 SHOWCASES OF MAIN RESULTS
+Figure 5 shows the forecasting cases of three dimensions of the ETTm1 dataset with prediction length $\tau = 2 8 8$ . For dimension “HUFL”, all the five models capture the periodic pattern, but Crossformer is the closest to the ground truth. For “HULL”, Pyraformer fails to capture the periodic pattern from the noisy data. For “LUFL” where the data has no clear periodic pattern, MTGNN, FEDformer and Crossformer capture its trend and show significantly better results than the other two models.
+![](images/20e4632f8d52a9fc09110bbdc1e1167db8ab0cdfd800d60199633eb1a3f524c0.jpg)
+Figure 5: Forecasting cases of three dimensions: High UseFul Load (HUFL), High UseLess Load (HULL) and Low UseFul Load (LUFL) of the ETTm1 dataset with prediction length $\tau = 2 8 8$ . The red / blue curves stand for the ground truth / prediction. Each row represents one model and each column represents one dimension.
+Figure 6 shows the forecasting cases of three dimensions of the WTH dataset with prediction length $\tau = 3 3 6$ . For dimension “DBT”, all the five models capture the periodic pattern. For “DPT”, Autoformer and FEDformer fails to capture increasing trend of the data. For “WD”, all models capture the periodic pattern from the noisy data, and the cruves output by MTGNN and Crossformer are sharper than the other three models.
+# B.2 COMPARISON WITH EXTRA METHODS
+We further compare with two additional concurrent methods which were either not peerreviewed (Grigsby et al., 2022) or were accepted after the submission of this work (Zeng et al., 2023): 1) STformer (Grigsby et al., 2022), a Transformer-based model that directly flattens the multivariate time-series $\mathbf { x } _ { 1 : T } \in \mathbb { R } ^ { T \times D }$ into a 1D sequence to be input to Transformers; 2) DLinear (Zeng et al., 2023), a simple linear model with seasonal-trend decomposition that challenges Transformer-based models for MTS forecasting. Results are shown in Table 4 and LSTMa and LSTnet are omitted as they are not competitive with other models.
+The basic idea of STformer is similar to our Crossformer: both of them extend the 1-D attention to 2- D. The explicit utilization of cross-dimension dependency makes STformer competitive with previous Transformer-based models on ETTh1, ETTm1 and WTH, especially for short-term prediction. However, STformer directly flattens the raw 2-D time series into a 1-D sequence to be input to the
+![](images/af22d7bb2883fa9d0a2cecb1beb4c67313bd7fecd7ba5b7c10ffa191350dfb39.jpg)
+Figure 6: Forecasting cases of three dimensions: Dry Bulb Temperature (DBT), Dew Point Temperature (DPT) and Wind Direction (WD) of the WTH dataset with prediction length $\tau = 3 3 6$ . The red / blue curves stand for the ground truth / prediction. Each row represents one model and each column represents one dimension.
+Transformer. This straightforward method does not distinguish the time and dimension axes and is computationally inefficient. Therefore, besides the good performance for short-term prediction, STformer has difficulty in long-term prediction and encounters the out-of-memory (OOM) problem on high-dimensional datasets (ECL and Traffic). While Crossformer uses the DSW embedding to capture local dependency and reduce the complexity. The TSA layer with the router mechanism is devised to deal with the heterogeneity of time and dimension axis and further improve efficiency.
+DLinear is on par with our Crossformer on ETTh1 and ETTm1 $\tau \leq 9 6 $ ); has similar performance with FEDformer on ILI; performs worse than Crossformer on WTH; outperforms all Transformerbased models including our Crossformer on ETTm1 $\tau \geq 2 8 8 $ ), ECL and Traffic. Considering its simplicity, the performance is impressive. Based on the results, we analyze the limitations of Crossformer and propose some directions to improve it in the future:
+1) In Cross-Dimension Stage of TSA layer, we simply build an all-to-all connection among $D$ dimensions with the router mechanism. Besides capturing the cross-dimension dependency, this full connection also introduces noise, especially for high-dimensional dataset. We think high-dimensional data has the sparse property: each dimension is only relevant to a small fraction of all dimensions. Therefore, utilizing the sparsity to reduce noise and improve the computation efficiency of the TSA layer could be a promising direction.
+2) Authors of DLinear (Zeng et al., 2023) argue that the Transformer-based models have difficulty in preserving ordering information because the attention mechanism is permutation-invariant and the absolute position embedding injected into the model is not enough for time series forecasting, which is an order-sensitive task. Although Yun et al. (2020) theoretically proves that Transformers with trainable positional embedding are universal approximators of sequence-to-sequence functions, the ordering information still needs to be enhanced in practice. We think that relative position encoding in texts (Ke et al., 2021; Dufter et al., 2022) and vision (Wu et al., 2021b) could be useful for ordering information enhancement.
+Table 4: MSE/MAE comparison with extra methods: STformer (Grigsby et al., 2022) and DLinear (Zeng et al., 2023). Bold/underline indicates the best/second. OOM indicates out-of-memory problem. Gray background marks the CNN-GNN-based model; yellow marks Transformer-based models where cross-dimension dependency is omitted; blue marks Transformer-based models explicitly utilizing cross-dimension dependency; red marks the linear model with series decomposition.
+<table><tr><td colspan="2">Models</td><td colspan="2">MTGNN</td><td colspan="2">Transformer</td><td colspan="2">Informer</td><td colspan="2">Autoformer</td><td colspan="2">Pyraformer</td><td colspan="2">FEDformer</td><td colspan="2">STformer</td><td colspan="2">Crossformer</td><td colspan="2">DLinear</td></tr><tr><td colspan="2">Metric</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td></tr><tr><td rowspan="4">FLL</td><td>24 48</td><td>0.336 0.386</td><td>0.393 0.429</td><td>0.620 0.692</td><td>0.577 0.671</td><td>0.577 0.685</td><td>0.549 0.625</td><td>0.439 0.429</td><td>0.440 0.442</td><td>0.493 0.554</td><td>0.507 0.544</td><td>0.318 0.342</td><td>0.384 0.396</td><td>0.368</td><td>0.441</td><td>0.305 0.352</td><td>0.367 0.312</td><td></td><td>0.355 0.383</td></tr><tr><td>168</td><td>0.466 0.474</td><td>0.947</td><td></td><td></td><td>0.931</td><td>0.752</td><td>0.493</td><td>0.479</td><td>0.781</td><td>0.675</td><td>0.412</td><td></td><td>0.445</td><td>0.465</td><td></td><td>0.394</td><td>0.352</td><td>0.430</td></tr><tr><td>336</td><td>0.736</td><td></td><td>0.797</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.449</td><td>0.652</td><td>0.608</td><td>0.410 0.440</td><td>0.441 0.461</td><td>0.416 0.450</td><td></td></tr><tr><td>720</td><td>0.643 0.750</td><td>1.094 1.241</td><td>0.813 0.917</td><td>1.128</td><td>0.873 0.896</td><td>0.509 0.539</td><td>0.492</td><td>0.912</td><td>0.747</td><td>0.456</td><td>0.474</td><td>1.069</td><td>0.806</td><td></td><td></td><td></td><td>0.452</td><td></td></tr><tr><td rowspan="8">[LL</td><td>24</td><td>0.916</td><td></td><td></td><td></td><td>1.215</td><td></td><td></td><td>0.537</td><td>0.993</td><td>0.792</td><td>0.521</td><td>0.515</td><td>1.071</td><td>0.817</td><td>0.519</td><td>0.524</td><td>0.486</td><td>0.501</td></tr><tr><td>48</td><td>0.260 0.324</td><td>0.306 0.465</td><td></td><td>0.371</td><td>0.323</td><td>0.369 0.503</td><td>0.410</td><td>0.428</td><td>0.310</td><td>0.371</td><td>0.290</td><td>0.364</td><td>0.278</td><td>0.348</td><td>0.211</td><td>0.293</td><td>0.217</td><td>0.289</td></tr><tr><td>96</td><td>0.386 0.428</td><td>0.408 0.446</td><td>0.681</td><td>0.470 0.612</td><td>0.494 0.678</td><td>0.614</td><td>0.485</td><td>0.464</td><td>0.465</td><td>0.464</td><td>0.342</td><td>0.396</td><td>0.445</td><td>0.458</td><td>0.300</td><td>0.352</td><td>0.278</td><td>0.330 0.354</td></tr><tr><td>288</td><td>0.469</td><td>0.488</td><td>1.162</td><td>0.879</td><td>1.056</td><td>0.786</td><td>0.502 0.604</td><td>0.476 0.522</td><td>0.520 0.729</td><td>0.504 0.657</td><td>0.366 0.398</td><td>0.412</td><td>0.420</td><td>0.455</td><td>0.320</td><td>0.373</td><td>0.310 0.369</td><td>0.386</td></tr><tr><td>672</td><td>0.620</td><td>0.571</td><td>1.231</td><td>1.103</td><td>1.192</td><td>0.926</td><td>0.607</td><td>0.530</td><td>0.980</td><td>0.678</td><td>0.455</td><td>0.433 0.464</td><td>0.733</td><td>0.597</td><td>0.404 0.569</td><td>0.427 0.528</td><td>0.416</td><td>0.417</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><td></td><td></td><td>0.777</td><td>0.625</td><td></td><td></td><td></td><td></td></tr><tr><td>24 48</td><td>0.307 0.388</td><td>0.356</td><td>0.349 0.386</td><td>0.397 0.433</td><td>0.335 0.395</td><td>0.381 0.459</td><td>0.363 0.456</td><td>0.396 0.462</td><td>0.301 0.376</td><td>0.359 0.421</td><td>0.357 0.428</td><td>0.412 0.458</td><td>0.307</td><td>0.359</td><td>0.294</td><td>0.343</td><td>0.357</td><td>0.391 0.444</td></tr><tr><td>168 336</td><td>0.498 0.506</td><td>0.422 0.512 0.523</td><td>0.613</td><td>0.582</td><td>0.608</td><td>0.567</td><td>0.574</td><td>0.548</td><td>0.519</td><td>0.521</td><td>0.564</td><td>0.541</td><td>0.381 0.497</td><td>0.416 0.502</td><td>0.370 0.473</td><td>0.411 0.494</td><td>0.425 0.515</td><td>0.516</td></tr><tr><td rowspan="7">HLM</td><td>720</td><td>0.510</td><td>0.527</td><td>0.707 0.834</td><td>0.634 0.741</td><td>0.702 0.831</td><td>0.620 0.731</td><td>0.600 0.587</td><td>0.571 0.570</td><td>0.539 0.547</td><td>0.543 0.553</td><td>0.533 0.562</td><td>0.536 0.557</td><td>0.566 0.589</td><td>0.564 0.582</td><td>0.495 0.526</td><td>0.515 0.542</td><td>0.536 0.582</td><td>0.537 0.571</td></tr><tr><td>48</td><td>0.173 0.280</td><td>0.334</td><td></td><td>0.399</td><td>0.344</td><td>0.393</td><td>0.241</td><td>0.351</td><td>0.478</td><td></td><td>0.229</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>168</td><td>0.236 0.320</td><td>0.353</td><td></td><td>0.420</td><td>0.368</td><td>0.424</td><td>0.299</td><td>0.387</td><td>0.452</td><td>0.471 0.455</td><td>0.263</td><td>0.338 0.361</td><td>0.356</td><td>0.432</td><td>0.156 0.231</td><td>0.255</td><td>0.155</td><td>0.258 0.287</td></tr><tr><td>336</td><td>0.328 0.373</td><td></td><td>0.381</td><td>0.439</td><td>0.381</td><td>0.431</td><td>0.375</td><td>0.428</td><td>0.463</td><td>0.456</td><td>0.305</td><td>0.386</td><td></td><td>0.5160.527 00M</td><td>0.323</td><td>0.309 0.369</td><td>0.195 0.238</td><td>0.316</td></tr><tr><td>720</td><td>0.422</td><td>0.410</td><td>0.391</td><td>0.438</td><td>0.406</td><td>0.443</td><td>0.377</td><td>0.434</td><td>0.480</td><td>0.461</td><td>0.372</td><td>0.434</td><td></td><td>0OM</td><td>0.404</td><td>0.423</td><td>0.272</td><td>0.346</td></tr><tr><td>960</td><td>0.471</td><td>0.451</td><td>0.492</td><td>0.550</td><td>0.460</td><td>0.548</td><td>0.366</td><td>0.426</td><td>0.550</td><td>0.489</td><td>0.393</td><td>0.449</td><td></td><td>0OM</td><td>0.433</td><td>0.438</td><td>0.299</td><td>0.367</td></tr><tr><td rowspan="4">目</td><td>24</td><td></td><td>3.954</td><td></td><td>1.323</td><td></td><td>1.462</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1.186</td><td>2.940</td><td>1.205</td></tr><tr><td>36</td><td>4.265 1.387 4.777 1.496</td><td>4.167</td><td></td><td>1.360</td><td>4.588 4.845</td><td>1.496</td><td>3.101 3.397</td><td>1.238 1.270</td><td>3.970 4.377</td><td>1.338 1.410</td><td>2.687 2.887</td><td>1.147 1.160</td><td>3.150 3.512</td><td>1.232 1.243</td><td>3.041 3.406</td></table>
+3) The sizes of datasets used for time series forecasting are much smaller than those for texts and vision, and the patterns in time series datasets are also simpler. Considering vision transformers surpass inductive bias and achieves excellent results compared to CNNs after pre-trained on large amounts of data (Dosovitskiy et al., 2021), Transformers for time series may also require large-size datasets with various patterns to exploit their full potential.
+As quoted from the paper, authors mentioned that DLinear “does not model correlations among variates”. Therefore, incorporating cross-dimension dependency into DLinear to further improve prediction accuracy is also a promising direction. Moreover, our DSW embedding to enhance locality and HED to capture dependency at different scales can also be potentially useful to further inspire and enhance DLinear.
+# B.3 ABLATION STUDY OF THE ROUTER MECHANISM
+The ablation study of the three main components of Crossformer is shown in Sec. 4.3. In this section, we conduct an ablation study of the router mechanism, a sub-module in TSA layer, and evaluate its impact on prediction accuracy. It should be noticed that the router mechanism is mainly proposed to reduce the computation complexity when $D$ is large. Results are shown in Table 5. Adding TSA(w/o Router) constantly improves the prediction accuracy of DSW and ${ \mathrm { D S W / H E D } }$ , showing the necessity of capturing cross-time and cross-dimension dependency in two different stages. For short term prediction $\tau \leq 1 6 8 )$ ), the performances of TSA(w/o Router) and TSA are similar, no matter whether HED is used or not. For long term prediction $\tau \geq 3 3 6$ ), the router mechanism slightly improves the prediction accuracy. The possible reason is that we set separate routers for each time step, which helps capture long-term dependency that varies over time.
+Table 5: Complementary results to ablation study in Table 2. TSA(w/o Router) denotes TSA layer without the router mechanism that directly uses MSA in the Cross-Dimension Stage.
+<table><tr><td rowspan="2">Models</td><td colspan="2">DSW</td><td colspan="2">DSW+ TSA(w/o Router)</td><td colspan="2">DSW+TSA</td><td colspan="2">DSW+HED</td><td colspan="2">DSW+HED+ TSA(w/o Router)</td><td colspan="2">DSW+TSA+HED</td></tr><tr><td>Metric MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td></tr><tr><td>24</td><td>0.373</td><td>0.418</td><td>0.320</td><td>0.376</td><td>0.322</td><td>0.373</td><td>0.406</td><td>0.454</td><td>0.311</td><td>0.375</td><td>0.305</td><td>0.367</td></tr><tr><td>48</td><td>0.456</td><td>0.479</td><td>0.356</td><td>0.396</td><td>0.365</td><td>0.403</td><td>0.493</td><td>0.512</td><td>0.363</td><td>0.406</td><td>0.352</td><td>0.394</td></tr><tr><td>168</td><td>0.947</td><td>0.731</td><td>0.487</td><td>0.493</td><td>0.473</td><td>0.479</td><td>0.614</td><td>0.583</td><td>0.416</td><td>0.444</td><td>0.410</td><td>0.441</td></tr><tr><td>336</td><td>0.969</td><td>0.752</td><td>0.585</td><td>0.564</td><td>0.553</td><td>0.534</td><td>0.788</td><td>0.676</td><td>0.487</td><td>0.499</td><td>0.440</td><td>0.461</td></tr><tr><td>720</td><td>1.086</td><td>0.814</td><td>0.665</td><td>0.615</td><td>0.636</td><td>0.599</td><td>0.841</td><td>0.717</td><td>0.540</td><td>0.542</td><td>0.519</td><td>0.524</td></tr></table>
+![](images/36262eb1daf2a88c1f76e947cc3a85115cd100aa510cccb0a99dfa8add182a70.jpg)
+Figure 7: Attention scores calculated by the decoder of the ablation version of Crossformer (i.e. DSW) on dataset ETTh1. The input length, prediction length and segment length are set as $T =$ 168, $\tau = 2 4 , L _ { s e g } = 6$ . The $\mathbf { X }$ axis in each sub-figure represents the time steps serve as keys in attention mechanism, while the y axis denotes dimensions. Brighter color denotes higher attention weights.
+# B.4 DEPENDENCY VISUALIZATION
+As the attention scores computed by Crossformer are abstract and hard to visualize, we visualize scores computed by the ablation version, DSW, in Figure 7. In addition to cross-time dependency that other Transformer models can compute, Crossformer also provides information about crossdimension dependency. As shown in Figure 7, when predicting Dim #1, the model focus on both Dim #1 and #3. When predicting Dim #5, instead of focus on Dim #5 itself, more attention is paid to Dim #4.
+# B.5 HIERARCHICAL PREDICTION PATTERN VISUALIZATION
+Figure 8 shows the hierarchical prediction patterns output by our HED. The top prediction layer, Layer 3, captures the low frequency general trend and periodic pattern of the future value. By adding predictions at finer scales, finer high frequency patterns are added and the prediction get closer to the ground truth curve.
+# B.6 RUNNING TIME EFFICIENCY ANALYSIS
+In the main paper, we show the memory occupation w.r.t input length $T$ and number of dimensions $D$ . Here we evaluate the running time. Figure 9 (a) shows the running time per batch of Crossformer and other Transformer-based models w.r.t input length $T$ . FEDformer is much slower than other Transformer-based models. Crossformer achieves the best computation speed among the five methods within the tested length range.
+![](images/579e34db0fe8505dd44552ed91bf2f6e9cc93f2a150eb6b8d5b9967aab55f548.jpg)
+Figure 8: Hierarchical prediction visualization of ETTm1 with dimension HUFL and prediction length $\tau = 2 8 8$ . From top left to bottom right, we gradually add layer predictions at finer scales.
+![](images/7b7af163249cfc6bf6382d4b95a798eb7f6406602acc97eb7b0e86268a17e56a.jpg)
+Figure 9: Evaluation on computational speed. (a) Running time per batch w.r.t the input length $T$ on ETTh1. (b) Running time per batch w.r.t number of dimensions $D$ on synthetic datasets by different numbers of dimensions.
+Figure 9 (b) shows the running time per batch of Crossformer and its ablation versions w.r.t the number of dimensions $D$ . Crossformers without TSA layer (DSW and $\mathrm { D S W + H E D } )$ ) are faster when $D$ is small $\left( D \leq 3 0 \right)$ ). However, they have difficulty processing high-dimensional MTS due to the quadratic complexity w.r.t $D$ . Indeed, for a single NVIDIA GeForce RTX 2080Ti GPU with 11GB memory, DSW and $\mathrm { D S W + H E D }$ encounters the out-of-memory (OOM) problem when $D > 5 0$ Moreover, TSA(w/o Router) encounter the OOM problem when $D > 2 0 0$ .
+# C DISCUSSION ON THE SELECTION OF HYPER-PARAMETERS
+We recommend to first determine the segment length $L _ { s e g }$ , as it is related to both the model performance and computation efficiency. The general idea is to use small $L _ { s e g }$ for short-term prediction and large $L _ { s e g }$ for long-term prediction. Some priors about the data also help to select $L _ { s e g }$ . For example, if the hourly sampled data has a daily period, it is better to set $L _ { s e g } = 2 4$ . Next, we select the number of layers for encoder and decoder $N$ . Crossformer with larger $N$ can utilize information of more scales, but also requires more computing resources. The number of routers in TSA layer $c$ can be set to 5 or 10 to balance the prediction accuracy and computation efficiency. Finally, dimension of hidden states $d _ { m o d e l }$ and head number of multi-head attention can be determined based on the available computing resources.
+Table 6: MSE and MAE evaluation with different segment lengths on ETTm1 dataset. \* denotes segment length used in the main text, which is a divisor of $T , \tau$ .
+<table><tr><td>Metric</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE MAE</td></tr><tr><td> Segment Length L seg</td><td>5</td><td></td><td>6*</td><td></td><td>7</td></tr><tr><td>T= 288,τ = 48</td><td>0.291</td><td>0.349</td><td>0.300</td><td>0.352</td><td>0.284 0.346</td></tr><tr><td>Segment Length Lseg</td><td>22</td><td></td><td>24*</td><td></td><td>26</td></tr><tr><td>T= 672,T = 288</td><td>0.401</td><td>0.424</td><td>0.404</td><td>0.427</td><td>0.409 0.429</td></tr></table>
+# D SUPPLEMENTARY DESIGN TO CROSSFORMER
+# D.1 HANDLING INDIVISIBLE LENGTH
+In the main paper, we assume that the input length $T$ and prediction length $\tau$ are divisible by segment length $L _ { s e g }$ . In this section, we use padding mechanism to handle cases where the assumption is not satisfied.
+If $T$ is not divisible by $L _ { s e g }$ , we have $( k _ { 1 } - 1 ) L _ { s e g } < T < k _ { 1 } L _ { s e g }$ for some $k _ { 1 }$ . We pad $k _ { 1 } L _ { s e g } - T$ duplicated $\mathbf { x } _ { 1 }$ in front of $\mathbf { x } _ { \mathrm { 1 : } T }$ to get $\mathbf { x } _ { 1 : T } ^ { \prime }$ :
+$$
+\mathbf { x } _ { 1 : T } ^ { \prime } = [ \underbrace { \mathbf { x } _ { 1 } , \dots , \mathbf { x } _ { 1 } } _ { k _ { 1 } L _ { s e g } - T } , \mathbf { x } _ { 1 : T } ]
+$$
+where $[ , ]$ denotes the concatenation operation. $\mathbf { x } _ { 1 : T } ^ { \prime } \in \mathbb { R } ^ { k _ { 1 } L _ { s e g } \times D }$ can be input to the encoder of Crossformer.
+If $\tau$ is not divisible by $L _ { s e g }$ , we have $( k _ { 2 } - 1 ) L _ { s e g } < \tau < k _ { 2 } L _ { s e g }$ for some $k _ { 2 }$ . We set the learnable position embeddioutput in shape of der as . Then $\mathbf { E } ^ { ( d e c ) } \in \mathbb { R } ^ { k _ { 2 } \times D \times d _ { m o d e l } }$ and input it ttput is used as der to get an. $\mathbb { R } ^ { k _ { 2 } L _ { s e g } \times D }$ $\tau$ $\mathbf { x } _ { T + 1 : T + \tau } ^ { p r e d }$
+We conduct experiment on ETTm1 dataset to evaluate the effect of indivisible length. Results in Table 6 show that with padding mechanism, indivisible length does not degrade model performance, for both short-term prediction and long-term prediction.
+# D.2 INCORPORATING COVARIATES
+In the main text, we only use historical series $\mathbf { x } _ { \mathrm { 1 : } T }$ to forecast the future $\mathbf { x } _ { T + 1 : T + \tau }$ . In this section, we try to incorporate covariates $\mathbf { c } _ { 1 : T + \tau }$ into Crossformer. We use a straightforward method: first embed the covariates into point-wise vectors $\left\{ { \bf d } _ { 1 } , { \bf d } _ { 2 } , \ldots , { \bf d } _ { T + \tau } \right\}$ like previous Transformer-based models do (Zhou et al., 2021; Wu et al., 2021a; Liu et al., 2021a). Then, merge the point-wise vectors into segment-wise vectors using learnable linear combination. Finally, add the segment-wise vectors to each dimension of the 2D vector array obtained by DSW embedding:
+$$
+\begin{array} { c } { \displaystyle \mathbf { c } _ { t } \to \mathbf { d } _ { t } , 1 \leq t \leq T } \\ { \displaystyle \mathbf { d } _ { i } ^ { ( s ) } = \sum _ { \substack { 0 < j \leq L _ { s e g } } } \alpha _ { j } \mathbf { d } _ { ( i - 1 ) \times L _ { s e g } + j } , \quad 1 \leq i \leq \frac { T } { L _ { s e g } } } \\ { \displaystyle \mathbf { h } _ { i , d } ^ { c o v } = \mathbf { h } _ { i , d } + \mathbf { d } _ { i } ^ { ( s ) } , \quad 1 \leq i \leq \frac { T } { L _ { s e g } } , \quad 1 \leq d \leq D } \end{array}
+$$
+where denotes embedding method for point-wise covariates. $\alpha _ { j } , 1 \le j \le L _ { s e g }$ denotes learnable factors for linear combination. $\mathbf { d } _ { i } ^ { ( s ) }$ denotes the segment-wise covariate embedding. ${ \bf h } _ { i , d } ^ { c o v }$ denotes the embedded vector with covariate information for the $i$ -th segment in dimension $d$ , where $\mathbf { h } _ { i , d }$ is the embedded vector obtained from DSW embedding in the main text. The processing for the input of the decoder is similar, the segment-wise covariate embedding is added to the position embedding for decoder, i.e. ${ \bf E } ^ { ( d e c ) }$ .
+Table 7: MSE and MAE evaluation of Crossformer without/with covariates on ETTh1 dataset.
+<table><tr><td>Models</td><td>Crossformer</td><td>Crossformer+Cov</td></tr><tr><td>Metric</td><td>MSE MAE</td><td>MSE MAE</td></tr><tr><td>24</td><td>0.305 0.367</td><td>0.308 0.368</td></tr><tr><td>48</td><td>0.352 0.394</td><td>0.358 0.399</td></tr><tr><td>168</td><td>0.410 0.441</td><td>0.412 0.440</td></tr><tr><td>336</td><td>0.440 0.461</td><td>0.438 0.465</td></tr><tr><td>720</td><td>0.519 0.524</td><td>0.522 0.531</td></tr></table>
+We conduct experiments on ETTh1 dataset to evaluate the effect of covariates. Hour-of-the-day, dayof-the-week, day-of-the-month and day-of-the-year are used as covariates. Results in Table 7 show that incorporating covariates does not improve the performance of Crossformer. The possible reason is this straightforward embedding method does not cooperate well with Crossformer. Incorporating covariates into Crossformer to further improve prediction accuracy is still an open problem.

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