# Fuzzy Windows with Gaussian Processed Labels for Ordinal Image Scoring Tasks
## 2 Related Work
### Soft Labels and Gaussian Processes  
Hard labels. This type of label is traditionally used as a one-hot vector. For example, the encoding \(h_{i}=[0;1;0]\) means that \(x_{i}\) is annotated to be the second class and \(y_{i}=2\). However, the undesirable application in classifying ordinal images is its disadvantage, as many of them are ambiguous or have an unclear \"borderline\". Thus, it brings difficulties to researchers as to which class they should belong to [27]. These hard labels tend to create an artificial gap that ponderously defines the borderline, and this instinctual drawback might then reduce the ability of the network to adapt [28].  
Soft labels. By contrast, soft labels annotate categories by representing corresponding classes with a probability vector. For example, the encoding \(h_{i}=[0.1;0.7;0.2]\) indicates that \(P(Y=2|X=x_{i})=0.7\). Thus, the value of the true label has been switched from 1 to 0.7 [27]. Instead of using a single bit, soft labels with probabilities can provide extra information to the training models [29]. Meanwhile, they have information inheritance, which can resist disturbance during inference [30; 31; 32].  
Gaussian Processes. Gaussian process approaches for ordinal regression have been studied based on support vector machines [33], deep neural networks [7], and deep learning models with Gaussian distribution labels [34; 35; 36]. One partial label machine learning study used the Gaussian Processed (GP) approach to address and disambiguate the vague labeling information conveyed by the training data [8]. It assumed that there was already an unobservable latent function depending on the Gaussian process in the feature space of each class label. The essential problem, however, is that manually affected and annotated ambiguous labels have been ignored. The reason is that the Gaussian distribution, at times, could not really represent realistic labels without logic deblurring.  
For facial-age detection, many technical articles, for example, RCNN [37; 38; 22; 39], Distribution Learning (DLDL) [9; 30], and DLDL [2], reported that all of them trend towards the implicit use of the learning label distribution method, and data distribution is a Gaussian-like distribution.  
The distribution of facial ages is represented by a Gaussian distribution for which a lookup table is generated beforehand to store multi-part integrals [40]. These integrals can explain the probability of whether an input image should belong to the true chronological age of a given person whose multiple age samples have been provided. In [10], a label distribution learning with a normal distribution variance \(\sigma\) was used and proposed \(p_{\mu}(y,\sigma)\) to represent the \(k\)-th \((k\in[0,99])\) element of \(p(y,\sigma)\):  
\[p_{\mu}(y,\sigma)=1/\sqrt{2\pi\sigma^{2}}e^{-\frac{(y-y)^{2}}{2\sigma^{2}}}, \tag{5}\]  
where \(p_{\mu}\) is the probability that the true age is \(\mu\) years old. It represents the connection between the classes \(\mu\) and \(y\) in a normal distribution view. The optimal \(\sigma\) in each iteration depends on the optimal model parameter \(\theta^{*}\):  
\[\theta^{*}(\sigma)=argm_{k}L_{KL}(H,y_{true},\theta,\sigma), \tag{6}\]  
where \(L_{KL}(H,y_{true},\theta^{*},\sigma)\) denotes the train loss. \(H\) is the training input image, while \(y_{true}\) is its label. \(KL\) is the Kullback-Leibler divergence.