Autoformalization is the task of automatically translating mathematical content written in natural language to a formal language expression. The growing language interpretation capabilities of Large Language Models (LLMs), including in formal languages, are lowering the barriers for autoformalization. However, LLMs alone are not capable of consistently and reliably delivering autoformalization, in particular as the complexity and specialization of the target domain grows. As the field evolves into the direction of systematically applying autoformalization towards large mathematical libraries, the need to improve syntactic, terminological and semantic control increases. This paper proposes the coordinated use of three mechanisms, most-similar retrieval augmented generation (MS-RAG), denoising steps, and auto-correction with syntax error feedback (Auto-SEF) to improve autoformalization quality. The empirical analysis, across different models, demonstrates that these mechanisms can deliver autoformalizaton results which are syntactically, terminologically and semantically more consistent. These mechanisms can be applied across different LLMs and have shown to deliver improve results across different model types.

This paper investigates the possibility of approximating multiple mathematical operations in latent space for expression derivation. To this end, we introduce different multi-operational representation paradigms, modelling mathematical operations as explicit geometric transformations. By leveraging a symbolic engine, we construct a large-scale dataset comprising 1.7M derivation steps stemming from 61K premises and 6 operators, analysing the properties of each paradigm when instantiated with state-of-the-art neural encoders.Specifically, we investigate how different encoding mechanisms can approximate expression manipulation in latent space, exploring the trade-off between learning different operators and specialising within single operations, as well as the ability to support multi-step derivations and out-of-distribution generalisation. Our empirical analysis reveals that the multi-operational paradigm is crucial for disentangling different operators, while discriminating the conclusions for a single operation is achievable in the original expression encoder. Moreover, we show that architectural choices can heavily affect the training dynamics, structural organisation, and generalisation of the latent space, resulting in significant variations across paradigms and classes of encoders.

Injecting desired geometric properties into text representations has attracted a lot of attention. A property that has been argued for, due to its better utilisation of representation space, is isotropy. In parallel, VAEs have been successful in areas of NLP, but are known for their sub-optimal utilisation of the representation space. To address an aspect of this, we investigate the impact of injecting isotropy during training of VAEs. We achieve this by using an isotropic Gaussian posterior (IGP) instead of the ellipsoidal Gaussian posterior. We illustrate that IGP effectively encourages isotropy in the representations, inducing a more discriminative latent space. Compared to vanilla VAE, this translates into a much better classification performance, robustness to input perturbation, and generative behavior. Additionally, we offer insights about the representational properties encouraged by IGP.

To highlight the challenges of achieving representation disentanglement for text domain in an unsupervised setting, in this paper we select a representative set of successfully applied models from the image domain. We evaluate these models on 6 disentanglement metrics, as well as on downstream classification tasks and homotopy. To facilitate the evaluation, we propose two synthetic datasets with known generative factors. Our experiments highlight the existing gap in the text domain and illustrate that certain elements such as representation sparsity (as an inductive bias), or representation coupling with the decoder could impact disentanglement. To the best of our knowledge, our work is the first attempt on the intersection of unsupervised representation disentanglement and text, and provides the experimental framework and datasets for examining future developments in this direction.