## 2 Methods  
### CqSL Model Overview  
We propose a model that combines the concept of variational generative and adversarial learning, and disentangled interpretation learning in a semi-supervised learning scheme, which is suited for domain-adapted segmentation as well as reconstruction. We define the learning task as follows: given an (unknown) data distribution \(p(x,y)\) over images and segmentation masks, we define a source domain having a training set, \(\mathcal{D}_{\mathcal{L}}=\{(x_{l}^{\prime},y_{l}^{\prime})\}_{j=1}^{n_{l}}\) with \(n_{l}\) labeled examples, and another domain having a training set, \(\mathcal{D}_{\mathcal{U}\mathcal{L}}=\{(x_{l}^{\prime\prime})\}_{j=1}^{n_{l}}\) with \(n_{ul}\) unlabeled examples, which are sampled as independent, identically distributed variables from \(p(x,y)\) and \(p(x)\) distribution. Empirically, we want to minimize the target risk \(\in_{t}(\phi,\theta)=min_{\phi,\theta}\)\(\mathcal{L}_{\mathcal{L}}(\mathcal{D}_{\mathcal{L}},(\phi,\theta))+\gamma \mathcal{L}_{\mathcal{U}\mathcal{L}}(\mathcal{D}_{\mathcal{U}\mathcal{L}},( \phi,\theta))\), where \(\mathcal{L}_{\mathcal{L}}\) is the supervised loss for segmentation, \(\mathcal{L}_{\mathcal{U}\mathcal{L}}\) is the unsupervised loss defined on unlabeled images and \(\phi,\theta\) denotes the learnable parameters of the overall network. We propose to solve the task by learning domain-specific and domain-invariant features that are discriminative of the semgentor and reconstructor. Figure 3 shows the proposed model comprised of five components--(1) disentanglement component, (2) a disentangled variational autoencoder (DVAE), (3) a mask segmentor identifier (SI), (4) a mask discriminator identifier (DI), and (5) a reconstructor \(R\). The disentangler \(D\) (Figure 3a) is designed to factorize the representation of an image pair into two parts: a shared spatial representation (skeleton, \(SK_{e}\)) that captures the common information between images and an exclusive non-spatial representation (sentiency, \(S_{e}\)) that contains the specific information of each image. The skeleton block \(SK_{e}\) is a modified U-Net++ [42] type architecture (EPU-Net++) (Figure 4 and Section 2.1.1) and is responsible for capturing the domain-invariant features \((f_{SK})\). The sentiency block \(S_{e}\) is a DVAE (Figure 3b) type architecture, which takes both the input image and the domain-invariant features \((f_{SK})\) as the input to map domain-specific features \((f_{SE})\) using the reparameterized trick [43]. The reconstruction block consists of two decoders: the SPADE-based decoder takes the \((f_{SE})\) feature from the sentiency block and proceeds directly to the reconstructor \(R\) (Figure 3d), while the FiLM-based decoder works as another disentangler, which untanglesa segmentor identifier \((SI)\) (Figure 3c), used for segmentation and extracted features, which then proceed directly to the reconstructor \(R\). The reconstructor \(R\) aims to recover the original image from both \((f_{SK},f_{SE})\). A mutual information minimizer (Figure 3a block) is applied between \((SK_{e}\) and \(S_{e})\) to enhance the disentanglement. A supervised trainer is trained on the labeled data to predict the segmentation mask distribution optimizing a supervised loss. An unsupervised trainer is trained on the unlabeled data, optimizing unsupervised losses (Algorithm 1 specifies the overall training procedure). Both the unsupervised and supervised trainers share the same block, as mentioned above. Figure 3: Illustration of _CqSL_**framework:** Our model makes use of both labeled as well as unlabeled images. The first block (**a**) crops the input images to a specific dimension. Then, we disentangle the latent features of the images via a disentangled block. An input image is first encoded to a multi-channel spatial representation, \(S\&d_{n=1,2..8}\). Then, \(S\&d_{n}\) can be fed into a segmentation network \(SI\) to generate a multi-class segmentation mask.