Image as First-Order Norm+Linear Autoregression: Unveiling Mathematical Invariance

Abstract

This paper introduces a novel mathematical property applicable to diverse images, referred to as FINOLA (First-Order Norm+Linear Autoregressive). FINOLA represents each image in the latent space as a first-order autoregressive process, in which each regression step simply applies a shared linear model on the normalized value of its immediate neighbor. This intriguing property reveals a mathematical invariance that transcends individual images. Expanding from image grids to continuous coordinates, we unveil the presence of two underlying partial differential equations. We validate the FINOLA property from two distinct angles: image reconstruction and self-supervised learning. Firstly, we demonstrate the ability of FINOLA to auto-regress up to a 256x256 feature map (the same resolution to the image) from a single vector placed at the center, successfully reconstructing the original image by only using three 3x3 convolution layers as decoder. Secondly, we leverage FINOLA for self-supervised learning by employing a simple masked prediction approach. Encoding a single unmasked quadrant block, we autoregressively predict the surrounding masked region. Remarkably, this pre-trained representation proves highly effective in image classification and object detection tasks, even when integrated into lightweight networks, all without the need for extensive fine-tuning. The code will be made publicly available.