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UMAP
Uniform Manifold Approximation and Projection, a nonlinear dimensionality reduction algorithm that preserves both local and global data structures in a low-dimensional space.
Differential manifold
Topological space that locally resembles Euclidean space, based on manifold theory to model the intrinsic structure of high-dimensional data.
Fuzzy simplicial set
Mathematical structure generalizing simplicial sets by assigning weights to relationships between points, allowing a fuzzy representation of neighborhood relationships in UMAP.
Spectral embedding
Embedding technique based on eigenvalue decomposition of similarity matrices, used in UMAP to initialize projection optimization.
Force-directed layout
Visualization algorithm simulating physical forces between points to optimize their positioning, applied in UMAP to minimize divergence between spaces.
Stochastic optimization
Optimization method using random samples to minimize a cost function, employed by UMAP to adjust low-dimensional coordinates.
Local structure
Immediate proximity relationships between data points in the original space, preserved by UMAP to maintain natural data groupings.
Global structure
Large-scale relationships between data clusters and regions, maintained by UMAP to preserve the overall topology of the dataset.
Geodesic distance
Distance measure following the curvature of the data manifold, used by UMAP to calculate the true distances between points in the intrinsic space.
k-nearest neighborhood
Set of the k closest points to a given point according to a defined metric, fundamental for building the neighborhood graph in UMAP.
Cross-entropy
Loss function measuring the divergence between probability distributions, optimized by UMAP to align high and low-dimensional spaces.
Manifold learning
Machine learning paradigm discovering the underlying manifold structure of data, of which UMAP is a modern implementation.
Neighborhood graph
Data structure representing proximity relationships between points, built by UMAP to model the local topology of data.
Barycentric coordinates
Representation of a point as a weighted combination of reference points, used by UMAP for initialization and interpolation of projections.
Custom metric
User-defined distance function to measure similarity between points, supported by UMAP to adapt the algorithm to specific domains.
Adaptive density
UMAP's ability to dynamically adjust local resolution based on data density, preserving structures in both dense and sparse regions.
Local minima
Suboptimal equilibrium points in the optimization landscape, avoided by UMAP thanks to advanced initialization and optimization techniques.
Laplace transform
Mathematical operator applied to the neighborhood graph in UMAP to capture geometric and topological properties of the data.
n_neighbors hyperparameter
Parameter controlling the size of the local neighborhood in UMAP, influencing the balance between preservation of local and global structures.
min_dist hyperparameter
Parameter regularizing the compactness of clusters in UMAP, controlling the minimum distance between points in the projected space.