This allows us to learn high-precision embeddings without using BigFloats, and enables us to store the resulting embeddings with fewer bits.

As recommended by the NeurIPS dataset and benchmark track, we documented QM1B and intended uses through the Datasheets for Datasets framework [1]. The goal of dataset datasheets as outlined by [1] is to provide a standardized process for documentating datasets. The authors of [1] present a list of carefully selected questions which dataset authors should answer. We hope our answers to these questions will facilitate better communication between us (the dataset creators) and future users of QM1B. For what purpose was the dataset created? Prior gaussian-based Density Functional Theory (DFT) datasets contained fewer than 20 million training examples.

In this work, we treat variational flows as dynamical systems, and leverage shadowing theory to elucidate this behavior via theoretical guarantees on the error of sampling, density evaluation, and ELBO estimation.

Generalization for streaming and distributed (big) data is trivial.

Renormalize algorithm to reduce the number of components.Algorithm 4: Scale-Expansion, modified from [4] Input: m-components expansion (a More importantly, we show in Alg. At the start of the training, we train models with an initial "burn-in" phase We mention an interesting tuning result here, take the training of the halfspace model over the WordNet Mammal for example, we varies the learning rates for different batchsize as shown in Table. 1. We found that, if trained with a larger batchsize, when the learning rate is adjusted (increased) properly, the embedding performance of the converged model with a large batchsize can nearly match the best performance of the converged model with a smaller batchsize.

Hyperbolic embeddings achieve excellent performance when embedding hierarchical data structures like synonym or type hierarchies, but they can be limited by numerical error when ordinary floating-point numbers are used to represent points in hyperbolic space. Standard models such as the Poincar{\'e} disk and the Lorentz model have unbounded numerical error as points get far from the origin. To address this, we propose a new model which uses an integer-based tiling to represent \emph{any} point in hyperbolic space with provably bounded numerical error. This allows us to learn high-precision embeddings without using BigFloats, and enables us to store the resulting embeddings with fewer bits. We evaluate our tiling-based model empirically, and show that it can both compress hyperbolic embeddings (down to $2\%$ of a Poincar{\'e} embedding on WordNet Nouns) and learn more accurate embeddings on real-world datasets.

The computational model can be assessed through comparison of these predictions to test data generated from an experiment.

Existing post-training quantization methods for large language models (LLMs) offer remarkable success. However, the increasingly marginal performance gains suggest that existing quantization strategies are insufficient to support the development of more compressed models. To inspire new directions for future research, this paper introduces the concept of null space into LLMs quantization. We argue that the quantization error can be effectively alleviated by constraining the post-quantization weight perturbation to lie within the null space of input activations. To prove this idea, we propose a plug-and-play null space projection module for existing milestone PTQ baselines named Q2N. Specifically, we first design an efficient and accurate null space projection approximation method tailored to the characteristics of LLMs. Subsequently, we theoretically derive a closed-form solution for an equivalent vector of the obtained projection matrix, which satisfies practical inference condition while avoiding additional memory overhead. Extensive experiments are conducted on various state-of-the-art LLMs (LLaMA3, DeepSeek, Qwen3) and baselines, demonstrating the effectiveness of both our Q2N and the perspective of null space optimization for LLMs quantization. We view this paper the first step to further alleviate the quantization error based on the insights of null space, hoping it inspiring future researchers to design more advanced quantization methods. Codes are available at https://github.com/zjq0455/q2n.