Self-Similarity Priors:
Neural Collages as Differentiable Fractal Representations

CollageSteps.mp4

Many patterns in nature exhibit self-similarity: they can be compactly described via self-referential transformations.

In this work, we investigate the role of learning in the automated discovery of self-similarity and in its utilization for downstream tasks. We design a novel class of implicit operators, Neural Collages, which (1) represent data as the parameters of a self-referential, structured transformation, and (2) employ hypernetworks to amortize the cost of finding these parameters to a single forward pass.

We investigate how to leverage the representations produced by Neural Collages in various tasks:

• Lossy image compression
• Deep generative modeling
• Image fractalization

The Upshot

We introduce a contractive operator as a layer. One application of a Collage Operator involves tokenizing the input domain into two partitions: sources and targets. The source tokens are combined into target tokens, which are then "stitched together" (as a collage). The parameters of a Collage operator relate parts of the input to itself, and can thus be used to achieve high rates of compression in self-similar data.

The machinery behind collage operators is inspired by fractal compression schemes. We provide a self-contained differentiable implementation of a simple fractal compression scheme in jax_src/compress/fractal.py. The script scripts/fractal_compress_img.py can be used to fractal compress a batch of images using this method.

Codebase

The codebase is organized as follows. We provide a simple implementation of a Collage Operator and related utilities under torch_src/. The bulk of the experiments has been carried out in jax. Under scripts/ we provide training and evalaution scripts for the three main experiments. The lossy image compression experiment is performed on a slice of the aerial dataset described in the paper, which can be found at this link.

Citing this work

If you found the paper or this codebase useful, please consider citing:

@article{poli2022self,
    title={Self-Similarity Priors: Neural Collages as Differentiable Fractal Representations},
    author={Poli, Michael and Xu, Winnie and Massaroli, Stefano and Meng, Chenlin and Kim, Kuno and Ermon, Stefano}, 
    journal={arXiv preprint arXiv:2204.07673}, 
    year={2022}
      }