In this document, we provide additional details and results to the main paper. The document is structured as follows: A.1 Loss Analysis - Analysis of the unimodal and multimodal latent regularization loss across different distributions and an ablation study on the proposed loss function. A.2 Image Generation - In this section, we compare VQVAE model with our method, provide detailed descriptions of the dataset, network architecture, and implementation details of the image generation experiments in the main paper. A.3 Modelling Discrete Structures - In this section, we describe the experimental and implementation details of the discrete data structure experiments in the main paper. A.5 Additional Qualitative Analysis - More examples of the randomly generated samples of MNIST, FASHIONMNIST, SVHN and CELEBA images.

We explore the minimax optimal error associated with a demographic parityconstrained regression problem within the context of a linear model. Our proposed model encompasses a broader range of discriminatory bias sources compared to the model presented by Chzhen and Schreuder [6]. Our analysis reveals that the minimax optimal error for the demographic parity-constrained regression problem under our model is characterized by Θ(dM/n), where ndenotes the sample size, d represents the dimensionality, and M signifies the number of demographic groups arising from sensitive attributes. Moreover, we demonstrate that the minimax error increases in conjunction with a larger bias present in the model.

We explore the minimax optimal error associated with a demographic parityconstrained regression problem within the context of a linear model. Our proposed model encompasses a broader range of discriminatory bias sources compared to the model presented by Chzhen and Schreuder [6]. Our analysis reveals that the minimax optimal error for the demographic parity-constrained regression problem under our model is characterized by Θ(dM/n), where ndenotes the sample size, d represents the dimensionality, and M signifies the number of demographic groups arising from sensitive attributes. Moreover, we demonstrate that the minimax error increases in conjunction with a larger bias present in the model.