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Johnson-Lindenstrauss Lemma
Fundamental mathematical theorem guaranteeing that points in a high-dimensional space can be projected into a considerably lower-dimensional space while preserving Euclidean distances.
Gaussian Random Projection
Variant of Random Projection using a projection matrix with entries drawn from a normal distribution, offering optimal theoretical guarantees of distance preservation.
Sparse Random Projection
Random Projection method using a sparse matrix with mostly zeros and a few non-zero entries, significantly reducing computation time and memory requirements.
Distance Preservation
Principle according to which distances between pairs of points in the original space are approximately maintained after projection into a lower-dimensional space.
Random Embedding
Linear mapping that probabilistically maps a high-dimensional space to a lower-dimensional space, used to accelerate machine learning algorithms.
Projection Matrix
Matrix used in Random Projection to transform high-dimensional data into a lower-dimensional space, generated randomly according to specific statistical properties.
Pairwise Distances
Distance measures calculated between all pairs of points in a dataset, used as a metric to evaluate the quality of structural preservation in Random Projection.
Random Subspace
Randomly generated lower-dimensional vector subspace into which data are projected when applying the Random Projection technique.
Random Orthogonal Projection
Variant of Random Projection where the projection matrix is orthogonal, better preserving the geometric properties of the original data.
Reduced Feature Space
New feature space of lower dimension obtained after applying Random Projection, where data are represented in a more compact manner.
Norm Preservation
Property of Random Projection ensuring that vector norms are approximately preserved between the original space and the projected space, guaranteeing conservation of data energy.
Target Dimension
Dimension of the destination space after Random Projection, typically much smaller than the original dimension while being sufficient to preserve structural properties.
Random Linear Transformation
Fundamental mathematical operation in Random Projection, where data are linearly transformed using a randomly generated matrix.