Building a Better Bijection between Classes of Compositions

A bijective proof is given for the following theorem: The number of compositions of n into parts congruent to a (mod b) equals the number of compositions of n + b - a into parts congruent to b (mod a) that are greater than or equal to b. The bijection is then shown to preserve palindromicity

Neural Image Compression: Generalization, Robustness, and Spectral Biases

Recent advances in neural image compression (NIC) have produced models that are starting to outperform classic codecs. While this has led to growing excitement about using NIC in real-world applications, the successful adoption of any machine learning system in the wild requires it to generalize (and be robust) to unseen distribution shifts at deployment. Unfortunately, current research lacks comprehensive datasets and informative tools to evaluate and understand NIC performance in real-world settings. To bridge this crucial gap, first, this paper presents a comprehensive benchmark suite to evaluate the out-of-distribution (OOD) performance of image compression methods. Specifically, we provide CLIC-C and Kodak-C by introducing 15 corruptions to the popular CLIC and Kodak benchmarks. Next, we propose spectrally-inspired inspection tools to gain deeper insight into errors introduced by image compression methods as well as their OOD performance. We then carry out a detailed performance comparison of several classic codecs and NIC variants, revealing intriguing findings that challenge our current understanding of the strengths and limitations of NIC. Finally, we corroborate our empirical findings with theoretical analysis, providing an in-depth view of the OOD performance of NIC and its dependence on the spectral properties of the data. Our benchmarks, spectral inspection tools, and findings provide a crucial bridge to the real-world adoption of NIC. We hope that our work will propel future efforts in designing robust and generalizable NIC methods. Code and data will be made available at https://github.com/klieberman/ood_nic.Comment: NeurIPS 202

"Understanding Robustness Lottery": A Geometric Visual Comparative Analysis of Neural Network Pruning Approaches

Deep learning approaches have provided state-of-the-art performance in many applications by relying on large and overparameterized neural networks. However, such networks have been shown to be very brittle and are difficult to deploy on resource-limited platforms. Model pruning, i.e., reducing the size of the network, is a widely adopted strategy that can lead to a more robust and compact model. Many heuristics exist for model pruning, but empirical studies show that some heuristics improve performance whereas others can make models more brittle or have other side effects. This work aims to shed light on how different pruning methods alter the network's internal feature representation and the corresponding impact on model performance. To facilitate a comprehensive comparison and characterization of the high-dimensional model feature space, we introduce a visual geometric analysis of feature representations. We decomposed and evaluated a set of critical geometric concepts from the common adopted classification loss, and used them to design a visualization system to compare and highlight the impact of pruning on model performance and feature representation. The proposed tool provides an environment for in-depth comparison of pruning methods and a comprehensive understanding of how model response to common data corruption. By leveraging the proposed visualization, machine learning researchers can reveal the similarities between pruning methods and redundant in robustness evaluation benchmarks, obtain geometric insights about the differences between pruned models that achieve superior robustness performance, and identify samples that are robust or fragile to model pruning and common data corruption to model pruning and data corruption but also obtain insights and explanations on how some pruned models achieve superior robustness performance

Restricted Compositions

An in depth study on many families of restricted compositions. General results are presented for proving infinite families of congruences and enumerating sequences defined by rational generating functions

A Bijection between Two Classes of Restricted Compositions

Two proofs, one using generating functions, the other bijective, are given for the following theorem: The number of compositions of n into parts congruent to 1 (mod k) equals the number of compositions of n + k − 1 into parts greater than k − 1. This bijection is then proven to hold for palindromic compositions. A more general theorem is presented in conclusion

Construction of a Full Row-Rank Matrix System for Multiple Scanning Directions in Computed Tomography

A full row-rank system matrix generated by scans along two directions in discrete tomography was recently studied. In this paper, we generalize the result to multiple directions. Let Ax = h be a reduced binary linear system generated by scans along three directions. Using geometry, it is shown in this paper that the linearly dependent rows of the system matrix A can be explicitly identified and a full row-rank matrix can be obtained after the removal of those rows. The results could be extended to any number of multiple directions. Therefore, certain software packages requiring a full row-rank system matrix can be adopted to reconstruct an image. Meanwhile, the cost of computation is reduced by using a full row-rank matrix

Zeroth-Order SciML: Non-intrusive Integration of Scientific Software with Deep Learning

Using deep learning (DL) to accelerate and/or improve scientific workflows can yield discoveries that are otherwise impossible. Unfortunately, DL models have yielded limited success in complex scientific domains due to large data requirements. In this work, we propose to overcome this issue by integrating the abundance of scientific knowledge sources (SKS) with the DL training process. Existing knowledge integration approaches are limited to using differentiable knowledge source to be compatible with first-order DL training paradigm. In contrast, our proposed approach treats knowledge source as a black-box in turn allowing to integrate virtually any knowledge source. To enable an end-to-end training of SKS-coupled-DL, we propose to use zeroth-order optimization (ZOO) based gradient-free training schemes, which is non-intrusive, i.e., does not require making any changes to the SKS. We evaluate the performance of our ZOO training scheme on two real-world material science applications. We show that proposed scheme is able to effectively integrate scientific knowledge with DL training and is able to outperform purely data-driven model for data-limited scientific applications. We also discuss some limitations of the proposed method and mention potentially worthwhile future directions