An External Definition of the One-Hot Constraint and Fast QUBO Generation for High-Performance Combinatorial Clustering

Abstract

Recently, a clustering method using a combinatorial optimization problem, called combinatorial clustering, has been drawing attention due to the rapid spreads of quantum annealing and simulated annealing. Combinatorial clustering is performed by minimizing an objective function under a condition to satisfy a one-hot constraint. The objective function and the constraint function are generally formulated to a unified objective function of a QUBO (Quadratic Unconstrained Binary Optimization) problem using the method of the Lagrange multiplier. The coefficients of the QUBO function can be represented by a square matrix, which is called the QUBO matrix.Although the Lagrange multiplier needs to be large enough to avoid violating the constraint, it is usually hard to be set appropriately due to the limitation of the bit precision. For example, the latest quantum annealer can handle values represented by only six or fewer bits. Even conventional computing systems cannot control the larger value of the Lagrange multiplier as the number of data points increases. Besides, the execution time for combinatorial clustering increases exponentially as the problem size increases. This is because the time for the QUBO matrix generation is long and a dominant factor of the total execution time when the problem size is large.To solve these problems, this paper proposes combinatorial clustering that overcomes the limitation of the method of the Lagrange multiplier. The proposed method uses a QUBO solver

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“…Its value is only 0 or 1, and different types are stored in different spaces. [59][60][61] That is to say, One-Hot coding is a vector of length N, in which, in any case, only one component of each vector is 1, and the other components are 0, thus ensuring uniqueness.…”