## 3 Materials and Methods
### Pareto Optimization  
Pareto optimization is a multi-objective optimization technique used to find a set of optimal solutions that best balance multiple conflicting objectives. In the context of EMG feature selection, we use Pareto optimization to identify the optimal subset of EMG features that best represent the underlying signal while minimizing the number of features required for classification. To do this, a Pareto front is first constructed by generating multiple solutions that represent different trade-offs between performance and feature number. Each solution is evaluated on the basis of its classification accuracy and the number of features it employs. The set of all solutions that cannot be improved in one objective without sacrificing performance in another objective is called the Pareto front.  
The Pareto front combines non-dominated multi-objective optimization solutions. Given a set of solutions \(S=s_{1},s_{2},\ldots,s_{n}\) with \(m\) objectives \(f_{1},f_{2},\ldots,f_{m}\), the Pareto front \(P^{*}\) is defined as follows. Let \(F(x)=(f_{1}(x),f_{2}(x),...,f_{m}(x))\) be the m-dimensional objective function vector and \(X\) be the decision variable vector. The Pareto front \(P^{*}\) is defined as: \(P^{*}=x\in X\) [there is no other \(x^{\prime}\in X\) such that \(f_{j}(x^{\prime})<=f_{j}(x)\forall j\in 1,2,...,m\) with at least one strict inequality. In other words, a solution \(s_{i}\) is on the Pareto front if there does not exist any other solution \(s_{j}\) that is better than \(s_{i}\) in all objectives. The Pareto front contains all non-dominated solutions, which means that there is no solution in the front that is worse than another solution in any objective. Constructing the Pareto front involves finding all non-dominated solutions in the solution set \(S\). Once the Pareto front is constructed, we select a solution that best fits their needs based on their preference for accuracy versus the number of features used. This approach ensures that the selected solution is not only accurate but also parsimonious and less prone to overfitting. We present the EMG feature selection procedure using Pareto optimization as Algorithm 1.  
```
Input: EMG dataset \(D\), maximum number of features \(m\) Output: Set of Pareto-optimal feature subsets \(P\) \(F\leftarrow\) set of all possible feature subsets of size \(\leq m\) \(P\leftarrow\emptyset\) \(S\leftarrow\) random subset of features of size \(m\) while not stopping criterion met do \(P\leftarrow\)\(P\cup\) non-dominated solutions in \(S\) \(S\leftarrow\) next subset in \(F\) endwhile  Return \(P\)
```  
**Algorithm 1** EMG feature selection using Pareto optimization  
The above algorithm takes as input an EMG dataset \(D\) and the maximum number of features to consider \(m\). It first generates a set \(F\) of all possible feature subsets of size \(\leq m\). Itthen initializes an empty set \(P\) to store the Pareto-optimal feature subsets. The algorithm then randomly selects a subset \(S\) of \(m\) features and begins an iterative process to generate the Pareto-optimal feature subsets. At each iteration, it adds the non-dominated solutions in \(S\) to the set \(P\). It then updates \(S\) to the next subset in \(F\), and the process continues until a stopping criterion is met. Finally, the algorithm returns the set of Pareto-optimal subsets of features \(P\).