Exploring Invariance in Images through One-way Wave Equations

In this paper, we empirically demonstrate that natural images can be reconstructed with high fidelity from compressed representations using a simple first-order norm-plus-linear autoregressive (FINOLA) process—without relying on explicit positional information. Through systematic analysis, we observe that the learned coefficient matrices ($\mathbf{A}$ and $\mathbf{B}$) in FINOLA are typically invertible, and their product, $\mathbf{AB}^{-1}$, is diagonalizable across training runs. This structure enables a striking interpretation: FINOLA’s latent dynamics resemble a system of one-way wave equations evolving in a compressed latent space. Under this framework, each image corresponds to a unique solution of these equations. This offers a new perspective on image invariance, suggesting that the underlying structure of images may be governed by simple, invariant dynamic laws. Our findings shed light on a novel avenue for understanding and modeling visual data through the lens of latent-space dynamics and wave propagation.

We explored a surprising and elegant way to reconstruct images using only a small amount of compressed information — without needing to know the exact positions of pixels. Our method, called FINOLA, learns to rebuild images using simple mathematical rules that work like "ripples" spreading out from a central point.What’s fascinating is that, when we look closely at how this method works under the hood, we see patterns that resemble wave equations — the same kind used to describe sound or light. It turns out that all images might follow a shared set of these wave-like rules, with each specific image corresponding to its own unique “ripple pattern.”This discovery gives us a new way to think about how images are structured. Instead of seeing images as just grids of color, we can start to understand them as dynamic patterns governed by underlying laws. Our findings may open up new possibilities for how machines understand, compress, and generate visual data.