GRIL: A $2$-parameter Persistence Based Vectorization for Machine Learning

$1$-parameter persistent homology, a cornerstone in Topological Data Analysis (TDA), studies the evolution of topological features such as connected components and cycles hidden in data. It has been applied to enhance the representation power of deep learning models, such as Graph Neural Networks (GNNs). To enrich the representations of topological features, here we propose to study $2$-parameter persistence modules induced by bi-filtration functions. In order to incorporate these representations into machine learning models, we introduce a novel vector representation called Generalized Rank Invariant Landscape \textsc{Gril} for $2$-parameter persistence modules. We show that this vector representation is $1$-Lipschitz stable and differentiable with respect to underlying filtration functions and can be easily integrated into machine learning models to augment encoding topological features. We present an algorithm to compute the vector representation efficiently. We also test our methods on synthetic and benchmark graph datasets, and compare the results with previous vector representations of $1$-parameter and $2$-parameter persistence modules.

Humans are still the best lossy image compressors

Lossy image compression has been studied extensively in the context of typical loss functions such as RMSE, MS-SSIM, etc. However, it is not well understood what loss function might be most appropriate for human perception. Furthermore, the availability of massive public image datasets appears to have hardly been exploited in image compression. In this work, we perform compression experiments in which one human describes images to another, using publicly available images and text instructions. These image reconstructions are rated by human scorers on the Amazon Mechanical Turk platform and compared to reconstructions obtained by existing image compressors. In our experiments, the humans outperform the state of the art compressor WebP in the MTurk survey on most images, which shows that there is significant room for improvement in image compression for human perception. The images, results and additional data is available at https://compression.stanford.edu/human-compression.