featurevectors

A feature vector is a fixed-length numeric representation of an object used in machine learning and data analysis. It is typically written as x = [x1, x2, ..., xd], where d is the number of features and x belongs to the real coordinate space R^d. Vectors can be dense (most components nonzero) or sparse (many zeros) depending on the domain and representation.

Purpose and use: Feature vectors provide a standardized input for learning algorithms. They enable mathematical operations such as distances, similarities, and inner products that underpin methods like k-nearest neighbors, linear models, support vector machines, and kernel methods. They also serve as the input to many optimization problems in prediction and clustering tasks.

Construction: Features may be handcrafted based on domain knowledge or learned automatically from data. Examples include computer vision features such as SIFT or HOG, text representations like bag-of-words or TF-IDF, and audio features such as Mel-frequency cepstral coefficients (MFCCs). In time-series or sensor data, features can be simple statistics or more complex transforms.

Processing and reduction: Feature vectors are often normalized or standardized to ensure comparable scales across components. Dimensionality reduction techniques (e.g., PCA, ICA) or learned representations (e.g., autoencoders, neural network embeddings) reduce d while preserving important information.

Embeddings and representations: Learned feature vectors are sometimes called embeddings, including word embeddings, image embeddings, or latent representations from neural networks. They capture salient properties of data in a compact, continuous space.

Applications and challenges: Feature vectors are central to classification, regression, clustering, ranking, and anomaly detection. Challenges include selecting meaningful features, managing high dimensionality, and maintaining interpretability.