Mixture of Neighboring Fragments
With the learned expert feature extractors, the next step is to perform sample selection.
Given a sample (x,y) , let f be a fragment close to y and f^{+} be its contrasting pair.
Intuitively, we consider a sample clean if the expert trained on (\mathcal{D}_{f},\mathcal{D}_{f^{+}}) strongly predicts that x belongs to a fragment f .
However, since the expert feature extractor is a binary classifier only trained using a contrasting pair of fragments, we utilize all experts’ opinions to obtain a more robust prediction. Specifically, we deem a sample as clean if the experts exhibit a consensus response ( Neighborhood Agreement ) for fragments close to y ( Fragment Prior ).

Based on this intuition, we formulate Mixture of Experts (MoE) (Jacobs et al., 1991 ) model, where the sampling probability of a datapoint (x,y) is defined as

 p(s|x,y,\mathcal{D}_{1\ldots F};\Theta) =\sum_{f}^{F}\rho_{f}(y)\alpha_{f}(x;\mathcal{D}_{1\ldots F},\Theta), (1)

where \Theta denotes parameters of all feature extractors, \rho_{f} is the fragment prior (mixture weight), and \alpha_{f} is the neighborhood agreement (a binary vote of whether x belongs to the fragment f ).
Based on the intuition above, \rho_{f}(y) is large when the fragment f is close to y , and \alpha_{f}(x)\in\{0,1\} is 1 if x is likely to belong to the fragment f .

Fragment Prior. For a sample (x,y) , we compute the prior \rho_{f}(y) of a fragment f ,
using a softmax weighting of each fragment f with respect to its relative distance to y :

\rho_{f}(y)=\frac{\exp(g_{f}(y))}{\sum_{f^{\prime}}^{F}\exp(g_{f^{\prime}}(y))}, (2)

where g_{f}(y)=\text{range}(Y)/(|y-\bar{Y}_{f}|) , \text{range}(Y)=\max(Y)-\min(Y) is the label range, and \bar{Y}_{f} is the mean label value of fragment f .
Since \text{range}(Y) is a constant for a given dataset, g_{f}(y) rapidly decreases when the mean value of fragment f is far from y in the continuous label space.
From the MoE perspective, the fragment prior can be regarded as soft gating that depends on y .

Neighborhood Agreement. Given a sample (x,y) and a fragment f , we need to determine whether x belongs to f .
The simplest approach is to use the expert trained using (\mathcal{D}_{f},\mathcal{D}_{f^{+}}) to classify whether x belongs to f or f^{+} , where f^{+} is the contrasting fragment of f .
Based on the classification output h(x;\theta_{f,f^{+}})\in\{f,f^{+}\} , we define self-agreement as:

\alpha^{self}_{f}=[h(x;\theta_{f,f^{+}})=f] (3)

where [A] is the Iverson bracket outputting 1 if A is true, and 0 otherwise.
Since training with noisy labels often results in suboptimal calibration (Bae et al., 2022 ; Zong et al., 2024 ) , we use discrete classification output for \alpha^{\text{self}}_{f} rather than continuous probabilistic one.

Since the expert \theta_{f,f^{+}} is only trained to discriminate between f and its contrasting fragment f^{+} , it is better to utilize other experts to obtain a more robust prediction.
For example, consider contrastive fragment pairs \{(1,4),(2,5),(3,6)\} as in Fig. 2 .
If x is more likely to belong to fragment 2 than 5 , then it should be more likely to belong to 1 than 4 and 3 than 6 .
Thus, we consider agreement of neighboring fragments f_{L} (left) and f_{R} (right) to obtain neighborhood agreement \alpha_{f}(x;\mathcal{D}_{1\ldots F},\Theta) :

\alpha_{f}(x;\mathcal{D}_{1\ldots F},\Theta)=\alpha^{self}_{f}\cdot\alpha^{ngb}_{f},\hskip 3.0pt where\ \ \alpha^{ngb}_{f}=\left[\alpha^{self}_{f_{L}}\lor\alpha^{self}_{f_{R}}\right]

Intuitively, \alpha_{f} is 1 if the fragment f is more likely for x than f^{+} ( \alpha^{\text{self}}_{f}=1 ) and either f ’s left or right fragment is more likely for x than its respective contrasting fragment ( \alpha^{\text{ngb}}_{f}=1 ).

In practice, we implement two variants of the agreements in Eq.( 3 – 4 ) using the feature extractor’s binary classifier and a K -nearest neighbor classifier on the learned feature space.
These two classifiers respectively consider predictive and representational aspects of the expert feature extractor and effectively work as an ensemble, as shown in Appendix G.7 .
As a result, we compute two versions of sample probability in Eq.( 1 ), and use the union of the sampled clean dataset \mathcal{S} for training of the regression model.
Algorithm G.12 in Appendix summarizes the overall procedure.