Abstract
Existing algorithms for learning Boolean networks (BNs) have time complexities of at least O(N · n0:7(k+1)), where n is the number of variables, N is the number of samples and k is the number of inputs in Boolean functions. Some recent studies propose more efficient methods with O(N · n2) time complexities. However, these methods can only be used to learn monotonic BNs, and their performances are not satisfactory when the sample size is small. In this paper, we mathematically prove that OR/AND BNs, where the variables are related with logical OR/AND operations, can be found with the time complexity of O(k·(N+ logn)·n2), if there are enough noiseless training samples randomly generated from a uniform distribution. We also demonstrate that our method can successfully learn most BNs, whose variables are not related with exclusive OR and Boolean equality operations, with the same order of time complexity for learning OR/AND BNs, indicating our method has good efficiency for learning general BNs other than monotonic BNs. When the datasets are noisy, our method can still successfully identify most BNs with the same efficiency. When compared with two existing methods with the same settings, our method achieves a better comprehensive performance than both of them, especially for small training sample sizes. More importantly, our method can be used to learn all BNs. However, of the two methods that are compared, one can only be used to learn monotonic BNs, and the other one has a much worse time complexity than our method. In conclusion, our results demonstrate that Boolean networks can be learned with improved time complexities.
Highlights
Gene Regulatory Networks (GRNs) are believed to be the underlying mechanisms that control different gene expression patterns
We prove that the Discrete Function Learning (DFL) algorithm can learn the OR/AND Boolean networks (BNs) with the O(k · (N + log n) · n2 )
Our experiments demonstrate that the DFL algorithm can successfully learn > 95% BNs of k = 3 with the same complexity of O(k ·
Summary
Gene Regulatory Networks (GRNs) are believed to be the underlying mechanisms that control different gene expression patterns. Different regulatory modules are weaved together into complex GRNs, which give specific outputs, i.e., different gene expression patterns (like in developmental processes), depending on their inputs, i.e., the current status of the cell. Boolean networks (BNs) [4] as models have received much attention in reconstructing GRNs from gene expression data sets [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. In Boolean functions, we use “∨”, “∧”, “¬”, “ ” and “ ≡” to represent logical OR, AND, INVERT ( named NOT or SWITCH), exclusive OR (XOR) and Boolean equality (the inversion of exclusive OR, XNOR), respectively.
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