We thank all the reviewers for their insightful and encouraging comments. Per your suggestion, we will update the appendix by adding more explanations about the proof ideas. T, (a) and (b) hold for IFS. Similarly, we can extend convergence results in Theorem 4 in Appendix from FS to IFS. We expect that the approach developed in this paper will fuel this future investigation.

Given the vast number of classifiers that have been (and continue to be) proposed, reliable methods for comparing them are becoming increasingly important. The desire for reliability is broken down into three main aspects: (1) Comparisons should allow for different quality metrics simultaneously.

We study how information propagates in decoder-only Transformers, which are the architectural backbone of most existing frontier large language models (LLMs). We rely on a theoretical signal propagation analysis--specifically, we analyse the representations of the last token in the final layer of the Transformer, as this is the representation used for next-token prediction. Our analysis reveals a representational collapse phenomenon: we prove that certain distinct sequences of inputs to the Transformer can yield arbitrarily close representations in the final token. This effect is exacerbated by the low-precision floating-point formats frequently used in modern LLMs. As a result, the model is provably unable to respond to these sequences in different ways--leading to errors in, e.g., tasks involving counting or copying. Further, we show that decoder-only Transformer language models can lose sensitivity to specific tokens in the input, which relates to the well-known phenomenon of over-squashing in graph neural networks. We provide empirical evidence supporting our claims on contemporary LLMs. Our theory also points to simple solutions towards ameliorating these issues.

We thank the reviewers for their comments, and will incorporate their suggestions to improve the paper. General-purpose embeddings may be valuable for discovering properties of the graph. Attention, learn to solve routing problems! Link prediction based on graph neural networks.

Transformer based large-language models (LLMs) display extreme proficiency with language yet a precise understanding of how they work remains elusive. One way of demystifying transformer predictions would be to describe how they depend on their context in terms of simple template functions. This paper takes a first step in this direction by considering families of functions (i.e.

Given a set of a reference RGBD views of an indoor environment, and a new viewpoint, our goal is to predict the view from that location.

For what purpose was the dataset created? The dataset was created with two primary purposes. First, it serves as a benchmark for einsum libraries, enabling the assessment of both the efficiency in determining contraction paths and the performance in executing einsum expressions. Second, it provides developers with a diverse set of einsum problem instances, thereby facilitating the development of more efficient, general-purpose einsum libraries. The dataset instances were created by the authors.

Modern artificial intelligence and machine learning workflows rely on efficient tensor libraries. However, tuning tensor libraries without considering the actual problems they are meant to execute can lead to a mismatch between expected performance and the actual performance. Einsum libraries are tuned to efficiently execute tensor expressions with only a few, relatively large, dense, floating-point tensors. But, practical applications of einsum cover a much broader range of tensor expressions than those that can currently be executed efficiently. For this reason, we have created a benchmark dataset that encompasses this broad range of tensor expressions, allowing future implementations of einsum to build upon and be evaluated against. In addition, we also provide generators for einsum expressions and converters to einsum expressions in our repository, so that additional data can be generated as needed. The benchmark dataset, the generators and converters are released openly and are publicly available at https://benchmark.einsum.org.

The ability to measure similarity between documents enables intelligent summarization and analysis of large corpora. Past distances between documents suffer from either an inability to incorporate semantic similarities between words or from scalability issues. As an alternative, we introduce hierarchical optimal transport as a meta-distance between documents, where documents are modeled as distributions over topics, which themselves are modeled as distributions over words. We then solve an optimal transport problem on the smaller topic space to compute a similarity score. We give conditions on the topics under which this construction defines a distance, and we relate it to the word mover's distance. We evaluate our technique for k-NN classification and show better interpretability and scalability with comparable performance to current methods at a fraction of the cost.

We show that it improves inception score and FID calculated by unconditional GAN trained by CIFAR-10, STL-10 and a public pre-trained model of conditional GAN trained by ImageNet.