# An Explainable Scheme for Memorization of Noisy Instances by Downstream Evaluation
## 3 Methods
### Memorization Matrix  
Inspired by [PERSON] [21] and [PERSON] [23], we adopt the influence function and extend the memorization attribution as derived and shown in Equations (5)-(24) [21; 23] to analyze the impact of introducing noisy training data on the downstream model. In general, letting \(D_{i}=(x_{i},y_{i})\) be a training data point mixed with noise, where \(x_{i}\) is a instance with its label \(y_{i}\), and \(n\) is the number of total instances, and letting \(L(D_{i},\theta)\) be the model's loss of \(D_{i}\) under the downstream model parameter \(\theta\), the empirical risk (or mean loss) \(R(\theta)\) can be defined as (5). \[\text{Empirical risk}:\ R(\theta)=\frac{1}{n}\!\sum_{i=1}^{n}L(D_{i},\theta) \tag{5}\]  
Then the second-order derivative of \(R(\theta)\) is a Hessian matrix, denoted by \(H_{\hat{\theta}}\) as (6), where \(\hat{\theta}=argmin_{\theta\in\odot}R(\theta)\). \[H_{\hat{\theta}}\stackrel{{\text{\tiny{\raisebox{-0.86 pt}{$ \bigcirc$}}}}}{{=}}\
abla^{2}R\big{(}\hat{\theta}\big{)}=\frac{1}{n}\!\sum_{i= 1}^{n}\
abla_{\theta}^{2}L\big{(}D_{i},\hat{\theta}\big{)} \tag{6}\]  
We then assume a baseline instance \(x^{\prime}\) and a normal instance \(x\) with their label \(y\), where \(x^{\prime}\) acts as a reference data point that does not affect any other instance, i.e., in practice, it is a blank image in this study, and \(x\) acts as a specific image mixed with noise. To evaluate the influence of the data point \(D^{\prime}=(x^{\prime},y)\) in place of the data point \(D=(x,y)\), let \(\hat{\theta}_{e,D^{\prime},-D}\) denote the optimized model parameters obtained by training the model with an objective function defined in (7) by up-weighting the loss impacted by \(D^{\prime}\) and down-weighting the loss impacted by \(D\) with a weighting factor \(\epsilon\). \[\hat{\theta}_{e,D^{\prime},-D}=argmin_{\theta\in\odot}\big{(}R(\theta)- \epsilon L(D,\theta)+\epsilon L\big{(}D^{\prime},\theta\big{)}\big{)} \tag{7}\]  
The primary purpose of the analysis is to find the optimal model parameters \(\hat{\theta}_{e,D^{\prime},-D}\) (i.e., to train an optimal downstream model) that make the value of the objective function depicted in (7) minimal when replacing the data point \(D\) with the data point \(D^{\prime}\) under a very small change of weighting factor \(\epsilon\). To evaluate this, we would first deal with a simplified form considering only the down-weighting part, as shown in (8). \[\hat{\theta}_{\epsilon,-D}=argmin_{\theta\in\in\mathbb{\tiny{(}}}(R(\theta)- \epsilon L(D,\theta)) \tag{8}\]  
Let \(\Delta_{\epsilon}=\hat{\theta}-\hat{\theta}_{\epsilon,-D}\) be a variable representing the small amount of parameter changes. Since \(\hat{\theta}\) does not depend on \(\epsilon\), we have  
\[\frac{d\hat{\theta}_{\epsilon,-D}}{d\epsilon}=\frac{d\Delta_{\epsilon}}{d \epsilon}. \tag{9}\]  
To evaluate the minimizer \(\hat{\theta}_{\epsilon,-D}\) of the objective function in (8), let its first-order derivative be 0, as shown in (10). \[0=\
abla R\big{(}\hat{\theta}_{\epsilon,-D}\big{)}-\epsilon\
abla L\big{(}D, \hat{\theta}_{\epsilon,-D}\big{)} \tag{10}\]  
Let \(f(\theta)=\
abla R(\theta)-\epsilon\
abla L(D,\theta)\). By Tayler expansion on \(f(\theta)\), the Equation (10) can be reformed to Equations (11) and (12).