Principal Component Analysis (PCA), a classical dimensionality reduction technique, and 2D Gaussian representation, an adaptation of 3D Gaussian Splatting for image representation, offer distinct approaches to modeling visual data. We present EigenGS, a novel method that bridges these paradigms through an efficient transformation pipeline connecting eigenspace and image-space Gaussian representations. Our approach enables instant initialization of Gaussian parameters for new images without requiring per-image optimization from scratch, dramatically accelerating convergence. EigenGS introduces a frequency-aware learning mechanism that encourages Gaussians to adapt to different scales, effectively modeling varied spatial frequencies and preventing artifacts in high-resolution reconstruction. Extensive experiments demonstrate that EigenGS not only achieves superior reconstruction quality compared to direct 2D Gaussian fitting but also reduces the necessary parameter count and training time. The results highlight EigenGS’s effectiveness and generalization ability across images with varying resolutions and diverse categories, making Gaussian-based image representation both high-quality and viable for real-time applications.

Normalizing flow comprises of a series of invertible transformations. With careful design in transformations, it can generate images or speeches with fast sampling speed. Inference can also be efficient in maximum likelihood manner. In addition to generating scenes or human faces, it can be used to transform probability distributions. On the other hand, learning latent structures of sentences in a global manner is always challenging. Variational autoencoder (VAE) is haunted by the issue of posterior collapse, where the latent space is poorly learned. To improve inference and generation of VAE in learning sequence data, we propose the amortized flow posterior variational recurrent autoencoder (AFP-VRAE). Variational recurrent autoencoder (VRAE) has RNN based encoder and decoder and learns global representations of sentences. To learn latent space that well preserves the semantic information of data, we use the normalizing flow to generate flexible variational distributions. Furthermore, we adopt the amortized regularization to encode similar embeddings to neighboring latent representations, and we use the skip connections to reinforce the representations to predict every output directly. The benefits can be shown in the experiments as we evaluate the models for language modeling, sentiment analysis and document summarization. AFP-VRAE reports good results on variational modeling for sequence data.

Variational autoencoder (VAE) is a popular latent variable model for data generation. However, in natural language applications, VAE suffers from the posterior collapse in optimization procedure where the model posterior likely collapses to a standard Gaussian prior which disregards latent semantics from sequence data. The recurrent decoder accordingly generates du-plicate or noninformative sequence data. To tackle this issue, this paper adopts the Gaussian mixture prior for latent variable, and simultaneously fulfills the amortized regularization in encoder and skip connection in decoder. The noise robust prior, learned from the amortized encoder, becomes semantically meaningful. The prediction of sequence samples, due to skip connection, becomes contextually precise at each time. The amortized mixture prior (AMP) is then formulated in construction of variational recurrent autoencoder (VRAE) for sequence generation. Experiments on different tasks show that AMP-VRAE can avoid the posterior collapse, learn the meaningful latent features and improve the inference and generation for semantic representation.