## 3 Methods
It may reconstruct or produce outputs across several modalities, facilitating cross-modal tasks such as producing an image from a text prompt. It generates diverse and richer outputs but requires higher training with a larger dataset. In summary, unimodal VAEs are ideal for specific tasks, while multimodal VAEs are suited for comprehensive applications integrating diverse data types. In the VAE model, encoders and decoders are essential for data creation, compression, and transformation activities. The encoder processes input data and compresses it into a  
Figure 1: Architecture of unimodal and multimodal approaches in VAE. reduced-dimensional representation of smaller, simpler data known as the latent vector. This compressed approach preserves vital information while eliminating unnecessary details in the latent space. Mathematically, it associates the input data (x) with a latent vector (y), as presented in Equation (1). Each xi represents data modality for a multimodal approach. Then, the decoder's function is to utilize the information from the latent space to reconstruct the original input data. Mathematically, the decoder transforms y generated via the encoder into a reconstructed output (\(\overline{\mathrm{x}}\)), as presented in Equation (2). \[\mathrm{Latent~{}vector~{}(y)=f_{encoder}[x_{1},x_{2},x_{3},x_{4}.\ldots x_{m}]} \tag{1}\]  
\[\mathrm{Reconstructed~{}output~{}data~{}(\overline{\mathrm{x}})=f_{decoder}[y]} \tag{2}\]  
An autoencoder is an unsupervised learning model that encompasses both the encoder and decoder to train itself to reduce the difference between x and \(\overline{\mathrm{x}}\). The autoencoder employs the mean square error (MSE) loss function to determine the variation of \(\overline{\mathrm{x}}\) from the original input data and several input data points or specifications (n), as shown in Equation (3). The autoencoder loss function measures the reconstruction accuracy but fails to generate new or diverse data. \[\mathrm{Loss~{}function}=\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\overline{x}_{i})^{2} \tag{3}\]  
VAE is an extension of the autoencoder with an introduction of randomness (\(\epsilon\)) and probabilistic distribution to the system, allowing the generation of new and diverse outputs, as shown in Figure 2. In VAE, x is encoded as two probabilistic distribution parameters, mean (\(\mu\)) and standard deviation (\(\sigma\)), to the latent vector (y), as shown in Equation (4). VAE uses a structured latent space with a reparameterization (\(\theta\)) method to maintain differentiability. The decoder reconstructs the input as \(\overline{\mathrm{x}}\) or generates new data (z) from the latent vector, represented in Equation (5). This iterates for several epochs until convergence is achieved. \[\mathrm{Latent~{}vector~{}(y)=\mu+(\sigma\times\epsilon),\,\epsilon\sim(0,1)} \tag{4}\]  
\[\mathrm{Generated~{}output~{}data~{}(z)=g_{0}[y]} \tag{5}\]  
A feedback-based VAE is a modified version of traditional VAE that incorporates an iterative feedback mechanism to enhance the quality and relevance of generated outputs. The reconstructed outputs may not always align with the desired results due to the stochastic nature of latent space sampling in the standard VAE technique. By integrating a feedback loop, the VAE model can iteratively refine its outputs, adjusting model parameters or latent variables accordingly. The feedback mechanism helps users minimize discrepancies between generated data and expected output, allowing them to propagate feedback error (\(\alpha\)) back through the network to update encoder and decoder weights. This process helps the model learn more accurate mappings between output data and feedback latent vector (\(\hat{y}\)), improving reconstruction and generation capabilities, as presented in  
Figure 2: The framework of a generic VAE model. Equation (6). The feedback-based approach benefits applications requiring high reliability in data reconstruction, optimization, adaptive augmentation, and personalized design generation.