Embedding Self-Correction as an Inherent Ability in Large Language Models for Enhanced Mathematical Reasoning

Accurate mathematical reasoning with Large Language Models (LLMs) is crucial in revolutionizing domains that heavily rely on such reasoning. However, LLMs often encounter difficulties in certain aspects of mathematical reasoning, leading to flawed reasoning and erroneous results. To mitigate these issues, we introduce a novel mechanism, the Chain of Self-Correction (CoSC), specifically designed to embed self-correction as an inherent ability in LLMs, enabling them to validate and rectify their own results. The CoSC mechanism operates through a sequence of self-correction stages. In each stage, the LLMs generate a program to address a given problem, execute this program using program-based tools to obtain an output, subsequently verify this output. Based on the verification, the LLMs either proceed to the next correction stage or finalize the answer. This iterative self-correction process allows the LLMs to refine their reasoning steps and improve the accuracy of their mathematical reasoning. To enable the CoSC mechanism at a low cost, we employ a two-phase finetuning approach. In the first phase, the LLMs are trained with a relatively small volume of seeding data generated from GPT-4, establishing an initial CoSC capability. In the second phase, the CoSC capability is further enhanced by training with a larger volume of self-generated data using the trained model in the first phase, without relying on the paid GPT-4. Our comprehensive experiments demonstrate that CoSC significantly improves performance on traditional mathematical datasets among existing open-source LLMs. Notably, our CoSC-Code-34B model achieved a 53.5% score on MATH, the most challenging mathematical reasoning dataset in the public domain, surpassing the performance of well-established models such as ChatGPT, GPT-4, and even multi-modal LLMs like GPT-4V, Gemini-1.0 Pro, and Gemini-1.0 Ultra.

DPF-Net: Combining Explicit Shape Priors in Deformable Primitive Field for Unsupervised Structural Reconstruction of 3D Objects

Unsupervised methods for reconstructing structures face significant challenges in capturing the geometric details with consistent structures among diverse shapes of the same category. To address this issue, we present a novel unsupervised structural reconstruction method, named DPF-Net, based on a new Deformable Primitive Field (DPF) representation, which allows for high-quality shape reconstruction using parameterized geometric primitives. We design a two-stage shape reconstruction pipeline which consists of a primitive generation module and a primitive deformation module to approximate the target shape of each part progressively. The primitive generation module estimates the explicit orientation, position, and size parameters of parameterized geometric primitives, while the primitive deformation module predicts a dense deformation field based on a parameterized primitive field to recover shape details. The strong shape prior encoded in parameterized geometric primitives enables our DPF-Net to extract high-level structures and recover fine-grained shape details consistently. The experimental results on three categories of objects in diverse shapes demonstrate the effectiveness and generalization ability of our DPF-Net on structural reconstruction and shape segmentation.