Learning representations that generalize to novel compositions of known concepts is crucial for bridging the gap between human and machine perception. One prominent effort is learning object-centric representations, which are widely conjectured to enable compositional generalization. Yet, it remains unclear when this conjecture will be true, as a principled theoretical or empirical understanding of compositional generalization is lacking. In this work, we investigate when compositional generalization is guaranteed for object-centric representations through the lens of identifiability theory. We show that autoencoders that satisfy structural assumptions on the decoder and enforce encoder-decoder consistency will learn object-centric representations that provably generalize compositionally. We validate our theoretical result and highlight the practical relevance of our assumptions through experiments on synthetic image data.

While dataset condensation effectively enhances training efficiency, its application in on-device scenarios brings unique challenges. 1) Due to the fluctuating computational resources of these devices, there's a demand for a flexible dataset size that diverges from a predefined size. 2) The limited computational power on devices often prevents additional condensation operations. These two challenges connect to the "subset degradation problem" in traditional dataset condensation: a subset from a larger condensed dataset is often unrepresentative compared to directly condensing the whole dataset to that smaller size. In this paper, we propose Multisize Dataset Condensation (MDC) by **compressing $N$ condensation processes into a single condensation process to obtain datasets with multiple sizes.** Specifically, we introduce an "adaptive subset loss" on top of the basic condensation loss to mitigate the "subset degradation problem". Our MDC method offers several benefits: 1) No additional condensation process is required; 2) reduced storage requirement by reusing condensed images. Experiments validate our findings on networks including ConvNet, ResNet and DenseNet, and datasets including SVHN, CIFAR-10, CIFAR-100 and ImageNet. For example, we achieved 5.22%-6.40% average accuracy gains on condensing CIFAR-10 to ten images per class. Code is available at: [https://github.com/he-y/Multisize-Dataset-Condensation](https://github.com/he-y/Multisize-Dataset-Condensation).

Summarizing book-length documents ($>$100K tokens) that exceed the context window size of large language models (LLMs) requires first breaking the input document into smaller chunks and then prompting an LLM to merge, update, and compress chunk-level summaries. Despite the complexity and importance of this task, it has yet to be meaningfully studied due to the challenges of evaluation: existing book-length summarization datasets (e.g., BookSum) are in the pretraining data of most public LLMs, and existing evaluation methods struggle to capture errors made by modern LLM summarizers. In this paper, we present the first study of the coherence of LLM-based book-length summarizers implemented via two prompting workflows: (1) hierarchically merging chunk-level summaries, and (2) incrementally updating a running summary. We obtain 1193 fine-grained human annotations on GPT-4 generated summaries of 100 recently-published books and identify eight common types of coherence errors made by LLMs. Because human evaluation is expensive and time-consuming, we develop an automatic metric, BooookScore, that measures the proportion of sentences in a summary that do not contain any of the identified error types. BooookScore has high agreement with human annotations and allows us to systematically evaluate the impact of many other critical parameters (e.g., chunk size, base LLM) while saving \$15K USD and 500 hours in human evaluation costs. We find that closed-source LLMs such as GPT-4 and Claude 2 produce summaries with higher BooookScore than those generated by open-source models. While LLaMA 2 falls behind other models, Mixtral achieves performance on par with GPT-3.5-Turbo. Incremental updating yields lower BooookScore but higher level of detail than hierarchical merging, a trade-off sometimes preferred by annotators. We release code and annotations to spur more principled research on book-length summarization.