Dense image prediction is a collection of computer vision tasks, which produce a pixel-wise label map for a given image. Such tasks range from low-level vision to high-level vision, including edge detection, saliency detection  and semantic segmentation . To address these tasks, Long propose fully convolutional networks (FCNs), which follow an encoder-decoder style 
. The encoder is transformed from a pre-trained image classifier, while the decoder combines low-level and high-level features of the encoder to generate the final output. As follows, various methods based on FCNs are proposed, which focus on adjusting the architecture of the decoder to achieve a better fusion of multi-level features[39, 31, 20, 16, 12, 40, 43, 29]. However, designing the architecture remains a laborious task, which requires lots of expert knowledge and takes ample time.
Inspired by the success of automatically designing network architectures for image classification [48, 49, 23, 25], we propose to extend it into dense image prediction. However, directly porting the methods from image classification is not sufficient because (1) dense image prediction requires to produce a pixel-wise label map, while image classification only needs to predict one label, (2) the key to dense image prediction is encoding multi-level features, while image classification merely needs global features, and (3) dense image prediction usually requires a pre-trained image classifier, while the network for image classification can be designed and trained from scratch. Thus, we face two major challenges: (1) We require to design a novel search space for segmentation to combine multi-level features and pre-trained image classifiers. (2) The feature maps in segmentation are usually in high resolution, bringing in heavy computation complexity and memory footprint. Thus, the proposed algorithm needs to automatically design a network that is time and memory efficient.
To solve the first challenge, we transform a pre-trained image classifier into a densely connected network named Fully Dense Network (FDN) as the search space, which defines a large set of possible final architectures. FDN follows the encode-decode style, where the encoder is converted from the classifier and the decoder consists of learnable dense connections between different features (Figure 0(a)). To solve the second challenge, we employ gradient descent to search the optimal connectivity from FDN with a novel loss function. The proposed loss forces the weight of each connection to be binary and the connectivity to be sparse (Figure 0(b)). After training, we prune FDN to obtain the final architecture with sparse connectivity (Figure 0(c)), which is time and memory efficient. As a result, we propose a novel method named SparseMask, which aims at automatically searching an efficient architecture for dense image prediction. Particularly, the proposed method focuses on designing the connectivity for the decoder to achieve a better fusion of low-level and high-level features.
To the best of our knowledge, the most related work to our method is MaskConnect 
, which focuses on image classification. Compared to it, our method is unique in three aspects: (1) We propose a novel sparse loss, which forces the weight of each connection to be binary and the connectivity to be sparse. Such a loss allows a decoder stage to have an arbitrary number of input features. The number can be adjusted adaptively for different decoder stages, resulting in a larger and more flexible search space. As for MaskConnect, the number of inputs is fixed. (2) We propose an efficient way to concatenate multiple feature maps with different spatial resolutions instead of simply padding and summation in MaskConnect. (3) Following the encoder-decoder style, we design a search space for dense image prediction tasks, which differs notably from classification in MaskConnect.
To validate the effectiveness of our method, we take a comprehensive ablation study on the Pascal VOC 2012 benchmark . Results show that our method discovers an architecture that outperforms the baseline methods by a large margin within 18 GPU-hours. We then transfer the architecture to other pre-trained image classifiers, datasets and tasks. Experiments show that the discovered connectivity has a good generalization ability, which achieves competitive performance compared to the state-of-the-art methods, while runs more than three times faster with less than half parameters.
In summary, we propose a differentiable connectivity learning method, which automatically designs an efficient architecture for dense image prediction. Our contributions are three-folds, (1) we propose Fully Dense Network to define the search space, which is dedicated for dense image prediction, (2) we introduce a novel loss function to force the dense connections to be sparse, and (3) we conduct comprehensive experiments to validate the effectiveness of our method as well as the generalization ability of the discovered connectivity.
2 Related Work
2.1 Architecture Search
Our work is motivated by differentiable architecture search [33, 36, 25, 1], which is based on the continuous relaxation of the architecture representation, allowing efficient search with gradient descent. [33, 36] propose a grid-like network as the search space, while 
relax the search space to be continuous and train the network by solving a bilevel optimization problem. Other works in architecture search focus on employing reinforcement learning[4, 48]30, 38, 24], and sequential model-based optimization [28, 23] to search the discrete space.
Recently, [33, 36, 11] propose to embed a large number of architectures in a grid-like network for semantic segmentation, which have to be trained from scratch. Instead, our method utilizes the pre-trained image classifier and defines a much larger search space of the connectivity. Our work is also complementary to , which constructs a recursive search space for dense image prediction and employs random search  to discover the best architecture. Such a method focuses on extracting multi-scale information from the high-level features, while ours aims at the fusion of low-level and high-level features. Besides, the search space and search approach are significantly different.
2.2 Dense Image Prediction
Currently, there are two prominent directions for dense image prediction. [39, 31, 20, 16, 12, 40, 43, 29] propose an encoder-decoder network for combining low-level and high-level features, while [6, 45, 41, 8, 44, 42] utilize dilated convolutions to keep the receptive field size and design a multi-scale context module to process high-level features.
In this paper, we follow the encoder-decoder style, of which the key is to design a connectivity structure to fuse low-level and high-level features.  introduce skip connections to construct U-Net, which combines the encoder features and the corresponding decoder activations. Alternatively,  aggregate multiple side outputs to generate the final result.  introduce short connections to the skip-layer structures, while  enhance the U-Net architecture with an additional bottom-up path. Differently, we aim at automatically designing an efficient connectivity structure with sparse connections.
In this paper, we aim at automatically transforming a pre-trained image classifier into an efficient fully convolutional network (FCN) for dense image prediction. Concretely, given a pre-trained image classifier, we follow the three steps as shown in Figure 1. (1) Transform the classifier into a densely connected network with learnable connections, Fully Dense Network (FDN). See Section 3.1. (2) Employ gradient descent to train the FDN with a novel loss function, which forces the dense connections to be sparse. See Section 3.2. (3) Prune the well-trained FDN to obtain the final network architecture. See Section 3.3.
3.1 Fully Dense Network
We follow the encoder-decoder style [3, 31] in dense image prediction and design a network named Fully Dense Network (FDN) as the search space. As shown in Figure 0(a), the encoder is a pre-trained image classifier, while the decoder combines multi-level features of the encoder with learnable connections.
3.1.1 The Encoder
We take a pre-trained image classifier as the input, which is composed of multiple convolution layers, a global average pooling layer 
, and a multi-layer perceptron (MLP). To form the encoder, we simply drop the MLP and keep the rest of the network unchanged. As shown in Figure0(a), the encoder consists of three convolution stages, as well as a global average pooling layer. Each convolution stage contains multiple convolution blocks, such as residual block  and inception block . Because the features inside a stage usually have the same spatial and channel dimensions, it’s reasonable to assume that the feature generated by the last block contains the most information. We thus restrict the decoder from accessing the other features. Concretely, the input features of the decoder is limited to the last feature of each encoder stage and the feature after global average pooling, which are noted as and respectively. is an index ranging from to , where is the number of convolution stages.
3.1.2 The Decoder
We focus on designing the architecture of the decoder, which includes the connectivity between the encoder and the decoder, as well as the connections inside the decoder. The first problem is to decide the number of decoder stages automatically. To achieve that, we initialize the decoder with a large number of stages and employ the search algorithm to select the most important ones. Concretely, the decoder is initialized with stages, which has the same number of stages as the encoder. The feature generated by each stage is noted as , where is an index ranging from to , as shown in Figure 0(a). Additionally, and have the same spatial dimensions by our design.
The second problem is to automatically choose the input features for each decoder stage. Inspired by DenseNet , we propose to initialize the decoder to a densely connected network with learnable connections, which contains a large set of possible final architectures. Many classic architectures in dense image prediction is a subset of the proposed network, such as U-net . As shown in Figure 0(a), the input features of stage contains three parts, which are (), () and . Our method then selects the most important features for each stage automatically. Notably, the proposed FDN is significantly different from DenseNet in four aspects. (1) FDN is densely connected in network-level while DenseNet only has block-level dense connections. (2) FDN contains input features from both inside and outside the decoder, following the encoder-decoder style. (3) The spatial dimensions of each input feature are different in FDN. (4) FDN contains learnable dense connections, which becomes sparser as the training progresses.
The last problem is to efficiently combine the input features inside each decoder stage. In MaskConnect, the input features are padded to the largest resolution and summed into one feature, which is then processed by a convolution block. To make the fusion more flexible, we propose to concatenate all the input features channel-wisely and then apply a convolution block to produce the output. However, all the features require to be up-sampled to the same spatial dimensions before concatenating, which occupies large memory space and costs many computation resources. To reduce memory usage and speed up computation, we show that (1) concatenating the features and then applying convolution is equal to applying convolution to each feature and then take a summation, and (2) the order of bilinear upsampling and point-wise convolution is changeable.111The proofs are shown in the supplementary material. Thus, the architecture can be reformulated as Equation 1,
where is bilinear upsampling, represents a convolution block, and is the learnable weight for each connection.
3.2 Learning Sparse Connections
The proposed FDN contains possible final architectures. Concretely, there’re architectures when the encoder has stages. Our goal is to automatically pick the optimal connectivity out of all the possible ones. To achieve that, we require to (1) select the most important stages from the ones, and (2) choose the input features for each selected stage, as shown in Section 3.1. In practice, the first problem can be reduced to the second one. Concretely, we first select the input features for all the stages, and then remove the stages without in-connections or out-connections, as shown in Figure 0(b) and 0(c). Thus, we focus on automatically choosing the input features for each stage.
As shown in Equation 1, each connection contains a learnable weight , which is a binary indicator. If , the feature is chosen as the input of the -th stage. However, directly optimizing over binary numbers is data inefficient, which requires lots of computation resources . Alternatively, we propose to relax to be a continuous number between and , which is then optimized by gradient descent in a differentiable manner.
The weight is desired to satisfy two requirements, (1) needs to approximate binary numbers, of which the value is around or , and (2) requires to be sparse, in which most values are close to . To achieve that, we propose a novel loss function, which forces to be binary and the connections to be sparse, as shown in Equation 2.
where represents the mean, is element-wise multiplication, and is a hyper parameter that controls the sparsity ratio. As shown in Figure 2, pushes close to either or , resulting in a binary-like value. forces the mean of to be close to . When is close to , most values in also tend to be , which makes the connections sparse. The final loss is shown in Equation 3,
where controls the balance between the task-oriented loss and the sparse loss.
Notably, itself cannot represent the importance of the input feature when it is relaxed to be a continuous number. Because the amplitude of also has an influence on the final value of . To remove the influence of
, we introduce batch normalization (BN) into Equation 1, which normalizes the amplitude of into as shown in Equation 4:
3.3 Pruning Fully Dense Network
To obtain the final architecture, we prune FDN according to the following rules: (1) Drop all connections whose , where is a predefined threshold. (2) Drop all stages that do not have any input features. (3) Drop all stages of which the generated feature is not used by other stages. After pruning, the network is then trained on the target dataset under the supervision of only. Notably, and in Equation 4 are removed when training the discovered architecture.
In this section, we first describe the experimental details of automatically designing a network architecture with our method. We then conduct an extensive experiment to evaluate our method and the discovered architecture. To show the generalization ability, we conduct a thorough experiment by transferring the discovered connectivity structure to other pre-trained image classifiers, datasets, and dense image prediction tasks.
4.1 Learning Sparse Connectivity
The experiment is designed to automatically search a network architecture for semantic segmentation with the proposed method. Concretely, we employ our method on the PASCAL VOC 2012 benchmark , which contains 20 foreground object classes and one background class. This dataset contains , and pixelwise labeled images for training (train), validation (val), and testing (test) respectively. We augment the train set with the extra annotations provided by , resulting in training images (trainaug). In our experiment, the trainaug set is used for training, while the val set is used for testing.
where represents the number of elements in . With this setting, each decoder stage tends to select two input features. in Equation 3 is set to . The task-related loss is set to the pixel-wise cross entropy loss.
As for training, we follow the protocol presented in . Concretely, We set the learning rates to (encoder) and (decoder) initially, which decrease to as the ”poly” strategy. For data augmentation, we randomly scale (from to ) and left-right flip the input images. The images are then cropped to and grouped with batch size . We train the network for epochs with SGD, of which the momentum is set to and weight decay is set to 4e-5. Notably, the training is finished within 18 hours on a single Nvidia P100 GPU with G memory.
After training, we prune the network with and train the final architecture following the same protocol. The automatically designed architecture is shown in Figure 3. The feature after global average pooling plays an essential role in semantic segmentation, as each stage takes it as the input. The high-level features with small spatial sizes and the low-level features with large spatial dimensions are also very important, which provides the semantic information and pixel-wise location information respectively. The middle-level information is less useful compared to other features.
The discovered architecture is evaluated on the val set with mean intersection-over-union (mIoU) as the metric. As shown in Table 1, our architecture outperforms the strong baseline FCN  and the state-of-the-art method Deeplab-V3  by a large margin. Besides, our method runs much faster than Deeplab-V3 with fewer parameters. Notably, all the methods employ MobileNet-V2 as the backbone and follow the same training and testing protocol. Besides, no multi-scale testing and left-right flipping are applied to the test images.
|Method||mIoU||#Params222Params comes from two parts: the encoder and the decoder.||FPS333In this paper, FPS (test phase) is measured on a Nvidia Titan-Xp GPU with a image as input.|
|Deeplab-V3 ||72.51%444w/o COCO pre-training, multi-scale evaluation and deep supervision.||2.22+0.67M||50|
|Ours w/ (FDN)555After learning with , none of the connections are removed.||72.72%||2.22+1.93M||37|
4.2 Ablation Study
Models in the Search Space
Our search space is huge, which contains various classical encoder-decoder architectures. To show the architecture discovered by our method is better than the others in the search space, we select two well-known architectures to compare, namely U-Net  and SparseACN , as shown in Table 1. U-net is a classical architecture that follows the encoder-decoder style. Compare to it, the connectivity pattern of our model is more expressive, which combines more than two input features for a decoder stage and have a better fusion of multi-scale information. As a result, our method outperforms U-net by a large margin. SparseACN proposes a predefined sparse connection pattern for densely connected networks, which shows a significant parameter and speed advantages without performance loss. Compared to the predefined connection pattern, the connectivity of our model is more flexible and sparser. Experiments show that the discovered model outperforms SparseACN in mIOU, the number of parameters, and FPS, which demonstrates the advantage of learnable connections. We also compare with the whole network FDN without pruning any connections. Results show that our method achieves a superior performance.
The Sparse Loss
To show the effect of the proposed sparse loss, we compare it with MaskConnect  and the widely used loss. As shown in Table 1, the architecture discovered by our method outperforms that of MaskConnect a large margin, while the number of parameters and speed are comparable. We attribute it to the flexibility of the proposed sparse loss, which enables the search in a much larger space. Concretely, our method allows an arbitrary number of input features for a decoder stage, while in MaskConnect, each decoder stage contains exactly input features. Thus, our method can discover more expressive connection patterns in a search space with fewer constraints.
The architecture designed by loss also performs worse than ours. Moreover, it runs nearly times slower and contains times more parameters. We also notice that the weights learned with loss are all larger than the threshold , resulting in connections after pruning. As for our method, there’re only connections in the final architecture, which shows that our loss function is more effective at learning sparse connectivity. As a result, our method can discover architectures with higher FPS and fewer parameters. Besides, the proposed sparse loss is also good at obtaining binary values. In our method, the largest weight of the dropped connections (1.342e-4) is about smaller than the smallest weight of the reserved connections (2.666e-2). Since the weights of all the dropped connections are close to , the influence of these connections can be ignored.
To further show the effectiveness of our loss function, we visualize the weight of each connection. As shown in Figure 4, learned with our method is shown in the top-left corner, of which most squares are close to red or green. This indicates that our method is good at approximating binary values. Besides, the proposed loss is also helpful to sparsity, as most squares are in red. The bottom-right corner presents the weights trained with loss. The color of most squares is between red and green, which means that loss has little effect to approximate binary numbers. Besides, there’re only a few squares in red, which shows that loss has a marginal effect to sparsity.
In Equation 4, we employ a BN layer to normalize the feature before multiplying with . To verify its effectiveness, we conduct an experiment by removing the introduced BN layers. As a result, the mIoU of the discovered architecture () becomes much lower than that with BN layers (), which shows the importance of Equation 4.
|Ours (Res101) 666http://host.robots.ox.ac.uk:8080/anonymous/WDAEVT.html||85.4%||45+ 7M||39.2||0.6+1.3G|
4.3 Transfer to Other Backbones
Our method automatically designed a network architecture based on MobileNet-V2. In this section, we transfer the discovered architecture to other image classifiers (also known as backbones), such as VGG nets  and ResNets . The key is to transfer the sparse connectivity structure. However, there’re no direct correspondences between the features of different backbones. Instead, we propose to match the features according to the spatial dimensions. If a match fails, we simply drop the corresponding stage. By following such rules, we transfer the automatically designed connectivity to other image classifiers. As shown in Figure 5, the encoder contains convolution stages, in which a down-sampling operation is employed followed by multiple convolution blocks. Then a global average pooling layer is used to extract global information. Such an encoder represents many widely used CNNs including VGG nets and ResNets. Compared to Figure 3, the connectivity structure is similar except that one stage is removed from the decoder because there is no corresponding feature.
To evaluate the performance of the transferred architecture, we conduct an experiment on the PASCAL VOC 2012 benchmark. The training and testing protocol is a little different from Section 4.1. Concretely, we have three steps for training, (1) train the network on MS COCO dataset  with learning rate for epochs, (2) train the network on the trainval set of  with learning rate for epochs, and (3) train the network on the trainval set of the original VOC dataset with learning rate for epochs. The learning rate for the decoder in each step is times larger than the above learning rate. After training, we evaluate the model on the test set of PASCAL VOC 2012 benchmark with multi-scale inputs and left-right flipping.
The results are shown in Table 2. Our method achieves competitive performance in mIoU compared to the state-of-the-art methods, although the architecture is transferred from MobileNet-V2 and not designed specifically for ResNet101. Moreover, our decoder requires half the parameters and runs more than times faster. When training the network, our method occupies much fewer GPU memory. Concretely, our method can be trained on a Nvidia Titan-Xp GPU with batch size , while other methods like EncNet are limited to images. The qualitative results on the val set of the original VOC benchmark are shown in Figure 6 (a).
4.4 Transfer to Other Datasets
The architecture is optimized on the Pascal VOC 2012 benchmark. We now employ it on another semantic segmentation dataset ADE20K  to verify its generalization ability. ADE20K dataset is a scene parsing benchmark, which contains 150 stuff/object categories. The dataset includes 20K/2K/3K images for training (train), validation (val) and testing (test).
We train our network on the train set for 120 epochs with learning rate . We then evaluate the model on the val set and report the pixel-wise accuracy (PixAcc) and mIoU in Table 3. Our method achieves comparable results to the state-of-the-art PSPNet and EncNet, while requires much fewer parameters and runs much faster. We then fine-tune our network on the trainval set for another 20 epochs with learning rate . The outputs on the test set are submitted to the evaluation server. As shown in Table 4, our method outperforms the baseline by a large margin and achieves competitive results compared to EncNet. The qualitative results on the val set are shown in Figure 6 (b).
4.5 Transfer to Other Tasks
4.5.1 Saliency Detection
MSRA-B  is a widely used dataset for saliency detection, which contains images with a large variation. There are , and images used for training (train), validation (val) and testing (test) respectively. After training on the train set, we evaluate our method on the test set. The performance is reported in Table 5. Our method is significantly better than the FCN baseline. Even compared to the state-of-the-art DSS, our method achieves comparable result with only parameters. Besides, our method runs nearly times faster than DSS. The qualitative results on the test set are shown in Figure 6 (c).
4.5.2 Edge Detection
BSDS500  contains training (train), validation (val) and test (test) images, which is a widely used dataset in edge detection. The trainval set is used for training, which is augmented in the same way as . When evaluating on the test set, standard non-maximum suppression (NMS)  is applied to thin the detected edges. The results are reported in Table 6, where our method outperforms the baseline method HED in two metrics (OIS and AP) and achieves the same ODS. The qualitative results on the test set are shown in Figure 6 (d).
We presented SparseMask, a novel method that automatically designs an efficient network architecture for dense image prediction in a differentiable manner, which follows the encoder-decoder style and focuses on the connectivity structure. Concretely, we transformed a pre-trained image classifier into Fully Dense Network, which contains a large set of possible final architectures and learnable dense connections. With the supervision of the proposed sparse loss, the weight of each connection is pushed to either or , resulting in a network architecture with sparse connectivity. Experiments show that the resulted architecture achieved competitive results on two semantic segmentation datasets, which requires much fewer parameters and runs more than times faster than the state-of-the-art methods. Besides, the discovered sparse connectivity structure is compatible with various backbones and generalizes well to many other datasets and dense image prediction tasks.
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We present two theorems in Section 3.1.2, of which the proofs are given in this section.
Concatenating the features and then applying convolution is equal to applying convolution to each feature and then take a summation.
Given input features with shape , the concatenated feature is noted as with shape , where . The corresponding convolution kernel is noted as with shape , which can be split into weights with shape . The output feature is represented as following:
The order of bilinear upsampling and point-wise convolution is changeable.
The input feature is with shape , while the corresponding convolution kernel is with shape . The output features is then represented as following:
where and . and is calculated as follows:
Fully Dense Network with MobileNet V2
Figure 7 presents the Fully Dense Network based on MobileNet V2. The inputs to the red circle (U) are multiple sets of features, while the output is the union of all the sets. F is the decoder stage, which takes a set of features as input.