# Enhanced Label Noise Filtering with Multiple Voting
## 2 Related Work
Therefore, the decision function of MVFilter can be described as y(x) = vote\({}_{2}\)(vote\({}_{1}\)(E\({}_{i}\)E\({}_{i}\)), vote\({}_{1}\)(E\({}_{i}\)E\({}_{i}\)),..., vote\({}_{1}\)(E\({}_{i}\)E\({}_{i}\))). In this function, vote\({}_{1}\) is the voting policy used by each SVFilter; vote\({}_{2}\) is the voting policy used by MVFilter; E\({}^{i}\) is the subset containing x obtained from the ith SVFilter. Usually, the vote\({}_{1}\) policy can either be based on majority voting or consensus voting. For the vote\({}_{2}\) policy, we have developed three policies: majority voting, consensus voting, and one-time veto. One-time vote means that if at least one SVFilter tagsdata as mislabeled, then the MVFilter will tag these data as mislabeled. In the MVFilter, different vote\({}_{1}\) and vote\({}_{2}\) policies can be combined to make various algorithms. As the example of MVFilter, the MF\({}_{MF}\)[40] algorithm is presented in Algorithm 2, which utilizes majority voting for both vote\({}_{1}\) and vote\({}_{2}\). ```
Algorithm: majority filtering (MF) Input: E (training set) Parameter: n (number of subsets), y (number of learning algorithms), \(\text{A}_{1}\), \(\text{A}_{2}\),..., \(\text{A}_{y}\)(y kinds of learning algorithms) Output: A (detected noisy subset of E) (1) form n disjoint almost equally sized subset of E\({}_{i}\), where \(\bigcup\limits_{i}E_{i}=E\) (2) \(A\leftarrow\emptyset\) for i\(\ldots,\)n do (4) form \(E_{i}\leftarrow E\setminus E_{i}\) for j\(\leftarrow\)1\(\ldots,\)y do (6) induce \(\text{H}_{j}\) based on examples in E\({}_{i}\) and \(\text{A}_{j}\) (7) end for (8) for every \(e\in E_{i}\)do (9) \(ErrorCounter\gets 0\) for i\(=\)1\(\ldots,\)y do (10) if i\(\text{H}_{i}\) incorrectly classifies e then \(ErrorCounter\gets ErrorCounter+1\) end for
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**Algorithm 1** Majority filtering algorithm. ```
MajorityFiltering_MajorityFiltering (MF\({}_{MF}\)) Input: E (training set) Parameter: n (number of subsets), y (number of learning algorithms), t (number of times of subsets partitioning), \(\text{A}_{1}\), \(\text{A}_{2}\),..., \(\text{A}_{y}\)(y kinds of learning algorithms) Output: A (detected noisy subset of E) (1) for j\(=\)1\(\ldots,\)t do (2) form n disjoint almost equally sized subset of E\({}_{pi}\), where \(\bigcup\limits_{i}E_{pi}=E\) (3) \(A^{p}\leftarrow\emptyset\) for i\(=\)1\(\ldots,\)n do (5) form \(E_{i}\leftarrow\)\(\sum\limits_{i}E_{pi}\) (6) for j\(=\)1\(\ldots,\)y do (7) induce \(\text{H}_{pi}\) based on examples in E\({}_{i}\) and \(\text{A}_{j}\) end for (9) for every \(e\in E_{pi}\)do (10) \(ErrorCounter\gets 0\) for j\(=\)1\(\ldots,\)y do (11) if j\({}_{pi}\) incorrectly classifies e then \(ErrorCounter\gets ErrorCounter+1\) end for (12) if \(ErrorCounter>\frac{y}{2}\), then \(A^{p}\gets A^{p}\cup\{e\}\) end for (13) end for (14) end for (15) end for (16)\(A\leftarrow\emptyset\) (20) for every \(e\in E\)do (21) \(ErrorCounter\gets 0\) (22) for j\(=\)1\(\ldots,\)y do (23) if e\(\in A^{p}\)then \(ErrorCounter\gets ErrorCounter+1\) end for (24) if \(ErrorCounter>\frac{y}{2}\), then \(A\gets A\cup\{e\}\) end for
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**Algorithm 2**MF\({}_{MF}\) algorithm.