High-Order Epistasis Detection Using Factorization Machine with Quadratic Optimization Annealing and MDR-Based Evaluation

TL;DR

We address the computational burden of detecting high-order epistasis in large genotype spaces by casting epistasis detection as a black-box optimization problem and solving it with FMQA, using MDR-derived CER as the objective and a factorization-machine surrogate optimized via an Ising machine. The FMQA framework is mapped to a QUBO for efficient binary optimization, with a constraint term ensuring a fixed interaction order and neighborhood exploration to improve search. Empirical results on simulated datasets show ground-truth epistasis is identified across interaction orders $d=3,4,5$ and varying numbers of loci, with relatively few evaluations, though performance degrades for the no marginal effects (eNME) setting at larger $N$ and for the toughest cases. The work offers a scalable, efficient approach to high-order epistasis detection that can extend to real datasets and other biomarker discovery tasks by reducing the need for exhaustive MDR searches and enabling broader exploratory searches in genomic spaces.

Abstract

Detecting high-order epistasis is a fundamental challenge in genetic association studies due to the combinatorial explosion of candidate locus combinations. Although multifactor dimensionality reduction (MDR) is a widely used method for evaluating epistasis, exhaustive MDR-based searches become computationally infeasible as the number of loci or the interaction order increases. In this paper, we define the epistasis detection problem as a black-box optimization problem and solve it with a factorization machine with quadratic optimization annealing (FMQA). We propose an efficient epistasis detection method based on FMQA, in which the classification error rate (CER) computed by MDR is used as a black-box objective function. Experimental evaluations were conducted using simulated case-control datasets with predefined high-order epistasis. The results demonstrate that the proposed method successfully identified ground-truth epistasis across various interaction orders and the numbers of genetic loci within a limited number of iterations. These results indicate that the proposed method is effective and computationally efficient for high-order epistasis detection.

Figures (2)

• Figure 1: An example of the case--control dataset. Each column corresponds to a genetic locus, and each entry represents the genotype of an individual at that locus. Genotypes are denoted using standard diploid notation, each locus has three possible genotypes, where uppercase and lowercase letters indicate different alleles (e.g., AA and aa denote homozygous genotypes, and Aa denotes a heterozygous genotype with one copy of each allele). Each row corresponds to one individual sample, which is labeled as either a case or a control according to the phenotype.
• Figure 2: Flow of the proposed method. The black arrows and gray triangles denote the flow for the proposed method and for CER computation using MDR, respectively. This figure illustrates an example with $d=2$, where the solution obtained by the Ising machine proposes an epistasis candidate between locus $1$ and locus $2$ that affects the case status.