Multi-Operational Mathematical Derivations in Latent Space

TL;DR

This work investigates whether neural encoders can approximate and compose multiple mathematical operators within a single latent space to enable equational reasoning. It introduces two joint-embedding paradigms—projection and translation—that couple expression and operation encoders to model latent transformations, and it evaluates them on a Symbolic-engineered dataset of 1.7M derivation steps drawn from 61K premises across 6 operators, using GNNs, CNNs, RNNs, and Transformers. A key finding is that the translation paradigm enhances cross-operational inference and supports multi-step latent derivations, while intra-operational discrimination can be achieved with conventional expression encoders; architectural choices strongly influence results beyond mere model size. The study also shows that graph-based encoders generalise better to longer expressions and that sequential models excel at multi-step reasoning under translation, with the authors releasing the dataset to spur further research in latent-space mathematical reasoning.

Abstract

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.

Figures (6)

• Figure 1: Can neural encoders learn to approximate multiple mathematical operators in latent space? Given a premise $x$, we investigate the problem of applying a sequence of latent operations ($t_1,\ldots,t_n$) to derive valid mathematical expressions ($y_1,\ldots,y_n$).
• Figure 2: Overview of the proposed joint-embedding predictive architectures for latent multi-operational derivation (left). Schematic workflow for multi-step inference and latent propagation of mathematical operations (right).
• Figure 3: Typical training dynamics of different multi-operational paradigms (MAP on the dev set).
• Figure 4: 2D projection of the latent space before and after an operation-specific transformation. The visualization supports the crucial role of the multi-operational paradigm for cross-operational inference, showing, at the same time, that intra-operational inference concerns larger regions and can be achieved in the original expression encoder.
• Figure 5: Multi-step derivations in latent space with different multi-operational paradigms and neural encoders.