How is KNN different from k-means?

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KNN (K-Nearest Neighbors) and k-means are both machine learning algorithms, but they serve different purposes and have distinct characteristics. Here are the main differences between KNN and k-means:

Purpose:
- KNN is a supervised learning algorithm used for classification and regression tasks. It predicts the class or value of a new data point based on the majority class or average value of its k nearest neighbors.
- k-means is an unsupervised learning algorithm used for clustering. It aims to partition n data points into k clusters, where each data point belongs to the cluster with the nearest mean (centroid).
Input Data:
- KNN requires labeled data for training, where each data point has a corresponding class label or target value.
- k-means works with unlabeled data and does not require any prior knowledge of class labels.
Output:
- KNN predicts the class label or target value for a new data point based on its neighbors.
- k-means assigns each data point to one of the k clusters based on its proximity to the cluster centroids.
Algorithm:
- KNN is a lazy learning algorithm. It does not build a model during training but instead stores the training data. Classification or regression is performed at the time of prediction by finding the k nearest neighbors.
- k-means is an iterative algorithm that minimizes the sum of squared distances between data points and their assigned cluster centroids. It iteratively updates the cluster centroids until convergence.
Distance Metric:
- KNN typically uses distance metrics such as Euclidean distance, Manhattan distance, or cosine similarity to measure the similarity between data points.
- k-means commonly uses Euclidean distance to measure the distance between data points and cluster centroids.
Hyperparameters:
- The main hyperparameter in KNN is the value of k, which determines the number of neighbors considered for classification or regression.
- In k-means, the main hyperparameter is the number of clusters (k) to be formed.
Scalability:
- KNN can be computationally expensive for large datasets, as it requires calculating distances between the new data point and all training data points during prediction.
- k-means is generally more scalable and can handle larger datasets efficiently, as it only needs to update cluster centroids during each iteration.

In summary, KNN is a supervised learning algorithm used for classification and regression, while k-means is an unsupervised learning algorithm used for clustering. KNN predicts based on the majority class or average value of neighboring data points, while k-means assigns data points to clusters based on their proximity to cluster centroids.

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Let's dive into the details of Euclidean distance, Manhattan distance, and cosine similarity, and then discuss why k-means commonly uses Euclidean distance.

Euclidean Distance:
- Euclidean distance is the straight-line distance between two points in Euclidean space.
- It is calculated as the square root of the sum of the squared differences between the corresponding coordinates of two points.
- Formula: d(x, y) = sqrt(sum((x_i - y_i)^2))
- Euclidean distance is sensitive to the scale of the features and is commonly used when the data has continuous or numerical attributes.
Manhattan Distance:
- Manhattan distance, also known as L1 distance or city block distance, is the sum of the absolute differences between the coordinates of two points.
- It is called Manhattan distance because it resembles the distance a car would drive in a city laid out in a grid-like pattern, where it can only move vertically or horizontally.
- Formula: d(x, y) = sum(|x_i - y_i|)
- Manhattan distance is less sensitive to outliers compared to Euclidean distance and is often used when the data has categorical or ordinal attributes.
Cosine Similarity:
- Cosine similarity measures the cosine of the angle between two vectors in a high-dimensional space.
- It is commonly used to measure the similarity between text documents or other high-dimensional data points.
- Formula: similarity(x, y) = (x · y) / (||x|| * ||y||)
- Cosine similarity ranges from -1 to 1, where 1 indicates high similarity (vectors pointing in the same direction), 0 indicates orthogonality (vectors are perpendicular), and -1 indicates dissimilarity (vectors pointing in opposite directions).
- Cosine similarity is insensitive to the magnitude of the vectors and focuses on their orientation.

Now, let's discuss why k-means commonly uses Euclidean distance:

Compatibility with Centroid Calculation:
- In k-means, the objective is to minimize the sum of squared distances between data points and their assigned cluster centroids.
- Euclidean distance aligns well with this objective because it allows for the calculation of the mean (centroid) of a cluster by averaging the coordinates of the data points within that cluster.
- The centroid update step in k-means involves calculating the mean of the data points assigned to each cluster, and Euclidean distance facilitates this calculation.
Sensitivity to Scale:
- Euclidean distance is sensitive to the scale of the features, which means that features with larger values will have a greater impact on the distance calculation.
- In k-means, it is common to normalize or standardize the features before applying the algorithm to ensure that all features contribute equally to the distance calculation.
- By using Euclidean distance and normalizing the features, k-means can effectively cluster data points based on their relative distances in the feature space.
Geometric Interpretation:
- Euclidean distance has a clear geometric interpretation, as it represents the straight-line distance between two points in Euclidean space.
- This geometric interpretation aligns well with the concept of clustering, where data points are grouped based on their proximity to each other.
- Euclidean distance provides a natural and intuitive measure of similarity between data points in a geometric sense.

While Euclidean distance is commonly used in k-means, it's important to note that other distance metrics, such as Manhattan distance or cosine similarity, can also be used depending on the nature of the data and the specific problem at hand.

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When performing k-means clustering on text data using embedding vectors, it is generally recommended to use cosine similarity instead of Euclidean distance. Here's why:

Capturing Semantic Similarity:
- Embedding models, such as Word2Vec, GloVe, or BERT, are designed to capture semantic relationships between words or documents in a high-dimensional vector space.
- Cosine similarity measures the cosine of the angle between two vectors, which effectively captures the semantic similarity between the corresponding texts.
- Texts with similar semantic meaning will have embedding vectors pointing in similar directions, resulting in high cosine similarity scores.
Insensitivity to Vector Magnitude:
- Cosine similarity is insensitive to the magnitude of the embedding vectors and focuses solely on their orientation.
- This is advantageous because the magnitude of the embedding vectors can vary depending on factors such as the length of the text or the specific embedding model used.
- By using cosine similarity, you can compare the similarity between texts based on their semantic content, regardless of the vector magnitudes.
Handling High-Dimensional Spaces:
- Text embedding vectors often have high dimensionality, typically ranging from hundreds to thousands of dimensions.
- In high-dimensional spaces, Euclidean distance can suffer from the "curse of dimensionality," where the distance between points becomes less meaningful as the number of dimensions increases.
- Cosine similarity, on the other hand, is less affected by high dimensionality and can still effectively measure the similarity between vectors in high-dimensional spaces.
Alignment with Text Similarity Measures:
- Cosine similarity is commonly used as a similarity measure in various text-related tasks, such as information retrieval, document clustering, and text classification.
- Using cosine similarity for k-means clustering of text embeddings aligns with the standard practices and allows for consistent comparison and interpretation of results.

To use cosine similarity with k-means clustering, you can modify the distance calculation step in the algorithm. Instead of using Euclidean distance, you can calculate the cosine similarity between each data point and the cluster centroids. The goal would be to assign each text to the cluster with the highest cosine similarity.

It's important to note that when using cosine similarity, the centroid update step in k-means needs to be adjusted accordingly. Instead of calculating the mean of the assigned data points, you would calculate the mean of the normalized embedding vectors to obtain the new centroid.

In summary, when performing k-means clustering on text data using embedding vectors, it is recommended to use cosine similarity instead of Euclidean distance. Cosine similarity captures the semantic similarity between texts, is insensitive to vector magnitude, handles high-dimensional spaces well, and aligns with standard text similarity measures.