K-nearest neighbors (KNN) is a supervised learning algorithm used for classification and regression. The algorithm works by finding the k closest data points (neighbors) to a given test data point and making a prediction based on their labels/values. The prediction is typically the average (for regression) or the majority class label (for classification) among the k nearest neighbors.

In mathematical terms, KNN is a non-parametric method. Given a training dataset of N labeled points in a d-dimensional feature space, where each point is represented by its d feature values and a class label, the KNN algorithm works as follows:

For a new test data point with feature values x, the Euclidean or other distance metric is used to calculate the distance between x and each of the N training data points.

The K nearest neighbors are selected based on the distances, where K is a user-defined parameter.

For a classification problem, the K nearest neighbors are assigned to their respective class labels and the majority class label is used as the prediction for x. This can be represented as:

- For a regression problem, the K nearest neighbors are used to predict the value of x by taking the average of their labels. This can be represented as:

In conclusion, KNN is a simple yet powerful algorithm that is easy to understand and implement. It does not make any assumptions about the underlying data distribution and is suitable for a wide range of applications. However, the performance of KNN can be affected by the choice of K and the distance metric used. It is also important to preprocess the data and normalize the features to prevent the influence of one feature on the results. In addition, KNN is not recommended for large datasets as the computation time increases linearly with the size of the dataset. Overall, KNN is a good starting point for many classification and regression problems, but it is important to evaluate its performance and consider other algorithms if needed.

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