Abstract Visual Reasoning: An Algebraic Approach for Solving Raven's Progressive Matrices
This addresses the problem of abstract reasoning for AI systems, representing a strong specific gain in performance.
The paper tackles abstract visual reasoning by introducing an algebraic framework that reduces problem-solving to algebraic computation, achieving 93.2% accuracy on the I-RAVEN dataset, outperforming the previous state-of-the-art of 77.0% and human performance of 84.4%.
We introduce algebraic machine reasoning, a new reasoning framework that is well-suited for abstract reasoning. Effectively, algebraic machine reasoning reduces the difficult process of novel problem-solving to routine algebraic computation. The fundamental algebraic objects of interest are the ideals of some suitably initialized polynomial ring. We shall explain how solving Raven's Progressive Matrices (RPMs) can be realized as computational problems in algebra, which combine various well-known algebraic subroutines that include: Computing the Gröbner basis of an ideal, checking for ideal containment, etc. Crucially, the additional algebraic structure satisfied by ideals allows for more operations on ideals beyond set-theoretic operations. Our algebraic machine reasoning framework is not only able to select the correct answer from a given answer set, but also able to generate the correct answer with only the question matrix given. Experiments on the I-RAVEN dataset yield an overall $93.2\%$ accuracy, which significantly outperforms the current state-of-the-art accuracy of $77.0\%$ and exceeds human performance at $84.4\%$ accuracy.