Lijing Lin

Haiping Liu, Lijing Lin, Jingyuan Sun et al.

Rotary Position Embedding (RoPE) is widely adopted in large language models (LLMs) due to its efficient encoding of relative positions with strong extrapolation capabilities. However, while its application in higher-dimensional input domains, such as 2D images, have been explored in several attempts, a unified theoretical framework is still lacking. To address this, we propose a systematic mathematical framework for RoPE grounded in Lie group and Lie algebra theory. We derive the necessary and sufficient conditions for any valid $N$-dimensional RoPE based on two core properties of RoPE - relativity and reversibility. We demonstrate that RoPE can be characterized as a basis of a maximal abelian subalgebra (MASA) in the special orthogonal Lie algebra, and that the commonly used axis-aligned block-diagonal RoPE, where each input axis is encoded by an independent 2x2 rotation block, corresponds to the maximal toral subalgebra. Furthermore, we reduce spatial inter-dimensional interactions to a change of basis, resolved by learning an orthogonal transformation. Our experiment results suggest that inter-dimensional interactions should be balanced with local structure preservation. Overall, our framework unifies and explains existing RoPE designs while enabling principled extensions to higher-dimensional modalities and tasks.

Lijing Lin, Matthew Sperrin, David A. Jenkins et al.

Background and Aims: The methods with which prediction models are usually developed mean that neither the parameters nor the predictions should be interpreted causally. However, when prediction models are used to support decision making, there is often a need for predicting outcomes under hypothetical interventions. We aimed to identify published methods for developing and validating prediction models that enable risk estimation of outcomes under hypothetical interventions, utilizing causal inference: their main methodological approaches, underlying assumptions, targeted estimands, and potential pitfalls and challenges with using the method, and unresolved methodological challenges. Methods: We systematically reviewed literature published by December 2019, considering papers in the health domain that used causal considerations to enable prediction models to be used for predictions under hypothetical interventions. Results: We identified 4919 papers through database searches and a further 115 papers through manual searches, of which 13 were selected for inclusion, from both the statistical and the machine learning literature. Most of the identified methods for causal inference from observational data were based on marginal structural models and g-estimation. Conclusions: There exist two broad methodological approaches for allowing prediction under hypothetical intervention into clinical prediction models: 1) enriching prediction models derived from observational studies with estimated causal effects from clinical trials and meta-analyses; and 2) estimating prediction models and causal effects directly from observational data. These methods require extending to dynamic treatment regimes, and consideration of multiple interventions to operationalise a clinical decision support system. Techniques for validating 'causal prediction models' are still in their infancy.