In machine learning and statistics, dimensionality reduction or dimension reduction is the process of reducing the number of random variables under consideration, and can be divided into feature selection and feature extraction.
Feature extraction transforms the data in the high-dimensional space to a space of fewer dimensions. The data transformation may be linear, as in principal component analysis (PCA), but many nonlinear dimensionality reduction techniques also exist. For multidimensional data, tensor representation can be used in dimensionality reduction through multilinear subspace learning.
The main linear technique for dimensionality reduction, principal component analysis, performs a linear mapping of the data to a lower dimensional space in such a way that the variance of the data in the low-dimensional representation is maximized. In practice, the correlation matrix of the data is constructed and the eigenvectors on this matrix are computed. The eigenvectors that correspond to the largest eigenvalues (the principal components) can now be used to reconstruct a large fraction of the variance of the original data. Moreover, the first few eigenvectors can often be interpreted in terms of the large-scale physical behavior of the system. The original space (with dimension of the number of points) has been reduced (with data loss, but hopefully retaining the most important variance) to the space spanned by a few eigenvectors.
Principal component analysis can be employed in a nonlinear way by means of the kernel trick. The resulting technique is capable of constructing nonlinear mappings that maximize the variance in the data. The resulting technique is entitled kernel PCA. Other prominent nonlinear techniques include manifold learning techniques such as Isomap, locally linear embedding (LLE), Hessian LLE, Laplacian eigenmaps, and LTSA. These techniques construct a low-dimensional data representation using a cost function that retains local properties of the data, and can be viewed as defining a graph-based kernel for Kernel PCA. More recently, techniques have been proposed that, instead of defining a fixed kernel, try to learn the kernel using semidefinite programming. The most prominent example of such a technique is maximum variance unfolding (MVU). The central idea of MVU is to exactly preserve all pairwise distances between nearest neighbors (in the inner product space), while maximizing the distances between points that are not nearest neighbors. A dimensionality reduction technique that is sometimes used in neuroscience is maximally informative dimensions, which finds a lower dimensional representation of a dataset such that as much information as possible about the original data is preserved.
An alternative approach to neighborhood preservation is through the minimization of a cost function that measures differences between distances in the input and output spaces. Important examples of such techniques include classical multidimensional scaling (which is identical to PCA), Isomap (which uses geodesic distances in the data space), diffusion maps (which uses diffusion distances in the data space), t-SNE (which minimizes the divergence between distributions over pairs of points), and curvilinear component analysis.
A different approach to nonlinear dimensionality reduction is through the use of autoencoders, a special kind of feed-forward neural networks with a bottle-neck hidden layer. The training of deep encoders is typically performed using a greedy layer-wise pre-training (e.g., using a stack of restricted Boltzmann machines) that is followed by a finetuning stage based on backpropagation.
For high-dimensional datasets (i.e. with number of dimensions more than 10), dimension reduction is usually performed prior to applying a K-nearest neighbors algorithm(k-NN) in order to avoid the effects of the curse of dimensionality. 
Feature extraction and dimension reduction can be combined in one step using principal component analysis (PCA), linear discriminant analysis (LDA), or canonical correlation analysis (CCA) techniques as a pre-processing step followed by clustering by K-NN on feature vectors in reduced-dimension space. In machine learning this process is also called low-dimensional embedding.
For very high-dimensional datasets (e.g. when performing similarity search on live video streams, DNA data or high dimensional Time series) running a fast approximate K-NN search using locality sensitive hashing, "random projections", "sketches"  or other high-dimensional similarity search techniques from the VLDB toolbox might be the only feasible option.
Adaptive dimension reduction
Many dimension reduction techniques do the reduction once for all.
Adaptive dimension reduction combines dimension reduction and unsupervised learning (clustering) together to improve the reduced data (subspace) adaptively.
- Nearest neighbor search
- Information gain in decision trees
- Semidefinite embedding
- Multifactor dimensionality reduction
- Multilinear subspace learning
- Multilinear PCA
- Singular value decomposition
- Latent semantic analysis
- Semantic mapping
- Topological data analysis
- Locality sensitive hashing
- Sufficient dimension reduction
- Data transformation (statistics)
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