Vector distance is crucial in various fields such as mathematics, physics, engineering, and computer science. They are used to measure physical quantities, analyze data, identify similarities, and determine the relationships between vectors. This article provides an overview of vector distance and its applications in data science.
What Is Vector Distance?
Vector distance, also known as distance metric or similarity metric, is a mathematical function used to quantify the similarity or dissimilarity between two vectors. These vectors can represent various datasets, and vector distance helps to understand how close or far apart vectors are in feature space. Therefore, vector distance is essential in various machine learning algorithms, enabling these algorithms to make decisions based on the relationships between vectors.
For measuring distance, we can choose between geometric distance measurement and statistical distance measurement, depending on the type of data. Features may have different data types (e.g., real values, boolean values, categorical values), and the data may be multidimensional or consist of geographical data.
What Are the Applications of Vector Distance in Machine Learning?
Never underestimate the power of vector distance. It has a wide range of applications in the field of machine learning.
Firstly, in clustering tasks, vector distance helps to group similar vectors into clusters. Algorithms such as k-means, hierarchical clustering, and DBSCAN rely on vector distance to determine which vectors belong to the same cluster.
In classification tasks, algorithms like kNN classification determine the class of a vector by calculating the distance to its k nearest neighbor vectors, thus assigning the vector to the cluster with the most neighbors. In Natural Language Processing (NLP), vector distance is used to calculate document similarity, perform sentiment analysis, and cluster text documents.
In the data preprocessing step, vector distance is crucial for feature scaling, normalization, and outlier removal, ensuring that data can better fit into machine learning algorithms.
In neural network training, vector distance serves as a loss function or regularization term, encouraging a certain relationship to be maintained between the output vector and the target vector, thereby improving model performance. In anomaly detection tasks, measuring the distance between vectors and the central cluster or other vectors can identify anomalies or outliers, which are considered anomalies due to their distance from the majority of vectors.
Dimensionality reduction techniques such as UMAP and t-SNE utilize vector distance to create low-dimensional representations from high-dimensional data while maintaining pairwise distances as much as possible, aiding in data visualization and understanding.
In summary, vector distance is the cornerstone of many machine learning tasks and applications. Choosing the right vector distance is crucial for the capability of algorithms and their ability to capture the relationships between vector data.
What Are the Types of Vector Similarity?
1. Euclidean Distance
Euclidean distance measures the shortest distance between two real-valued vectors. Due to its intuitiveness, simplicity, and good results for many use cases, it is the most commonly used distance metric and the default distance metric for many applications.
Euclidean distance is also known as the l2 norm, and its calculation is:
Python code is as follows:
from scipy.spatial import distance
distance.euclidean(vector_1, vector_2)Euclidean distance has two main drawbacks. Firstly, the distance measurement is not suitable for data in dimensions higher than 2D or 3D. Secondly, if we do not normalize and/or standardize the features, the distance may be biased due to different units.
2. Manhattan Distance
Manhattan distance, also known as taxicab or city block distance, is calculated based on the distance between two real-valued vectors when a person can only move at right angles. This distance metric is commonly used for discrete and binary attributes, allowing for a true path.
Manhattan distance is based on the l1 norm, and the calculation formula is:
Python code is as follows:
from scipy.spatial import distance
distance.cityblock(vector_1, vector_2)Manhattan distance has two main drawbacks. It is not as intuitive as Euclidean distance in high-dimensional spaces, and it does not show the possible shortest path. While this may not be a problem, we should be aware that this is not the shortest distance.
3. Chebyshev Distance
Chebyshev distance is also known as chessboard distance because it measures the maximum distance between two real-valued vectors in any dimension. It is often used in warehouse logistics, where the longest path determines the time required to travel from one point to another.
Chebyshev distance is calculated using the l-infinity norm:
Python code is as follows:
from scipy.spatial import distance
distance.chebyshev(vector_1, vector_2)Chebyshev distance has very specific use cases, so it is rarely used.
4. Minkowski Distance
Minkowski distance is a generalization of the distance metrics mentioned above. It can be used for the same use cases while providing high flexibility. We can choose the p value to find the most suitable distance metric.
The calculation of Minkowski distance is:
Python code is as follows:
from scipy.spatial import distance
distance.minkowski(vector_1, vector_2, p)Since Minkowski distance represents different distance metrics, it shares the same main drawbacks, such as issues in high-dimensional spaces and dependence on feature units. Additionally, the flexibility of the p value can also be a disadvantage, as it may reduce computational efficiency due to the need for multiple calculations to find the correct p value.
5. Cosine Similarity and Distance
Cosine similarity is a measure of direction, determined by the cosine of the angle between two vectors, and ignores the size of the vectors. Cosine similarity is commonly used in high dimensions where the size of the data is not important, such as in recommendation systems or text analysis.
Cosine similarity can range from -1 (opposite direction) to 1 (same direction), and is calculated as:
Cosine similarity is often used in positive space ranging from 0 to 1. Cosine distance is calculated as 1 minus cosine similarity, and ranges between 0 (similar values) and 1 (different values).
Python code is as follows:
from scipy.spatial import distance
distance.cosine(vector_1, vector_2)The main drawback of cosine distance is that it does not consider size but only the direction of the vectors. Therefore, it does not fully account for the differences in values.
6. Haversine Distance
Haversine distance measures the shortest distance between two points on a sphere. It is commonly used in navigation, where longitude, latitude, and curvature all affect the calculations.
The formula for Haversine distance is as follows:
where r is the radius of the sphere, and φ and λ are the latitude and longitude.
Python code is as follows:
from sklearn.metrics.pairwise import haversine_distances
haversine_distances([vector_1, vector_2])The main drawback of Haversine distance is the assumption of a spherical shape, which rarely occurs.
7. Hamming Distance
Hamming distance measures the difference between two binary vectors or strings.
It compares the vectors element-wise and averages the number of differences. If two vectors are the same, the resulting distance is 0; if the two vectors are completely different, the resulting distance is 1.
Python code is as follows:
from scipy.spatial import distance
distance.hamming(vector_1, vector_2)Hamming distance has two main drawbacks. The distance measurement can only compare vectors of the same length, and it cannot provide the magnitude of the differences. Therefore, it is not recommended to use Hamming distance when the magnitude of the differences is important.
Statistical distance measurements can be used for hypothesis testing, goodness of fit tests, classification tasks, or outlier detection.
8. Jaccard Index and Distance
The Jaccard index is used to determine the similarity between two sample sets. It reflects how many one-to-one matches exist compared to the entire dataset. The Jaccard index is commonly used for binary data, such as comparing predictions of deep learning models for image recognition with labeled data, or comparing text patterns in documents based on word overlap.
The calculation of Jaccard distance is:
Python code is as follows:
from scipy.spatial import distance
distance.jaccard(vector_1, vector_2)The main drawback of the Jaccard index and distance is its strong influence from the scale of the data, meaning the weight of each item is inversely proportional to the size of the dataset.
9. Sørensen-Dice Index
The Sørensen-Dice index is similar to the Jaccard index, measuring the similarity and diversity of sample sets. This index is more intuitive as it calculates the percentage of overlap. The Sørensen-Dice index is commonly used in image segmentation and text similarity analysis.
The calculation formula is as follows:
Python code is as follows:
from scipy.spatial import distance
distance.dice(vector_1, vector_2)The main drawback is also significantly influenced by the size of the dataset.
Popular Software Libraries for Using Vector Distance
Applications of Faiss Vector Retrieval Library
Faiss is an efficient vector retrieval library developed by the Facebook AI research team, widely used for similarity search and clustering of high-dimensional vectors. Here are some common operations in Faiss:
These operations cover the basic use cases of Faiss, including creating indexes, adding and searching for vectors, and compatibility with GPUs. Depending on the actual application needs, you can choose the appropriate index type and optimization method.
Overall, the value of vector database distance lies in its core role in efficiently retrieving and processing large-scale data.
By quantifying the similarity or dissimilarity between vectors, vector distance enables machine learning algorithms to accurately classify, cluster, and predict data in feature space. It not only improves accuracy in areas such as natural language processing, image recognition, and anomaly detection but also plays a key role in data preprocessing and dimensionality reduction.
Vector distance is a fundamental tool for achieving intelligent data analysis and decision-making, greatly advancing AI and machine learning technologies.
References:
-
https://zilliz.com.cn/glossary/%E5%90%91%E9%87%8F%E8%B7%9D%E7%A6%BB
-
https://segmentfault.com/a/1190000042705356