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Process of reducing the number of random variables under consideration
Dimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the
Dimensionality_reduction
Projection of data onto lower-dimensional manifolds
Nonlinear dimensionality reduction (NLDR), also known as manifold learning, is any of various related techniques that aim to project high-dimensional data
Nonlinear dimensionality reduction
Nonlinear_dimensionality_reduction
sufficiency. Dimension reduction has long been a primary goal of regression analysis. Given a response variable y and a p-dimensional predictor vector x {\displaystyle
Sufficient dimension reduction
Sufficient_dimension_reduction
Theoretical effect in physics
Dynamical dimensional reduction or spontaneous dimensional reduction is the apparent reduction in the number of spacetime dimensions as a function of the
Dynamical dimensional reduction
Dynamical_dimensional_reduction
or quantum field theory, dimensional reduction is the limit of a compactified theory where the size of the compact dimension(s) goes to zero. For a system
Dimensional_reduction
Posits ability to interpolate within latent manifolds
many high-dimensional data sets that occur in the real world actually lie along low-dimensional latent manifolds inside that high-dimensional space. As
Manifold_hypothesis
Subset of artificial intelligence
hypothesis proposes that high-dimensional data sets lie along low-dimensional manifolds, and many dimensionality reduction techniques make this assumption
Machine_learning
Non-parametric classification method
feature vectors in reduced-dimension space. This process is also called low-dimensional embedding. For very-high-dimensional datasets (e.g. when performing
K-nearest_neighbors_algorithm
Multifactor dimensionality reduction (MDR) is a statistical approach, also used in machine learning automatic approaches, for detecting and characterizing
Multifactor dimensionality reduction
Multifactor_dimensionality_reduction
Partial differential equations whose solutions are instantons
surpassed by Seiberg–Witten invariants. Through the process of dimensional reduction, the Yang–Mills equations may be used to derive other important
Yang–Mills_equations
Simplifying data to facilitate analysis
meaningful. Dimensionality reduction helps reduce noise in the data and allows for easier visualization, such as the example below where 3-dimensional data is
Data_reduction
Circle on Earth's surface enclosing majority of the human population
circles can be used for studying population changes over time, dimensional reduction and measuring population centralization. The world’s tightest cluster
Valeriepieris_circle
Indication of topological symmetry groups to topological condensed matter
fact equivalent to a d + 1 {\displaystyle d+1} sphere wedged with lower-dimensional spheres), which are not included in this table. Furthermore, the table
Periodic table of topological insulators and topological superconductors
Periodic_table_of_topological_insulators_and_topological_superconductors
Topics referred to by the same term
treatment adequacy Dimensional reduction, the limit of a compactified theory where the size of the compact dimension goes to zero Reduction criterion, in quantum
Reduction
Dimensionality reduction method
In computational mechanics, Guyan reduction, is a dimensionality reduction method which reduces the number of degrees of freedom by ignoring the inertial
Guyan_reduction
Technique for dimensionality reduction
It is a nonlinear dimensionality reduction technique for embedding high-dimensional data for visualization in a low-dimensional space of two or three
T-distributed stochastic neighbor embedding
T-distributed_stochastic_neighbor_embedding
German mathematician
Maryland, College Park, where he received his PhD in 1989. His thesis Dimensional Reduction for Elliptic Boundary Value Problems was written under the supervision
Christoph_Schwab
Nonlinear dimensionality reduction method
Isomap is a nonlinear dimensionality reduction method. It is one of several widely used low-dimensional embedding methods. Isomap is used for computing
Isomap
Method of data analysis
Principal component analysis (PCA) is a linear dimensionality reduction technique with applications in exploratory data analysis, visualization and data
Principal_component_analysis
Set of related ordination techniques used in information visualization
dimensions, N, an MDS algorithm places each object into N-dimensional space (a lower-dimensional representation) such that the between-object distances are
Multidimensional_scaling
Difficulties arising when analyzing data with many aspects ("dimensions")
high-dimensional spaces that do not occur in low-dimensional settings such as the three-dimensional physical space of everyday experience. The expression
Curse_of_dimensionality
Technique to uniquely identify a video recording
Video fingerprinting or video hashing are a class of dimension reduction techniques in which a system identifies, extracts and then summarizes characteristic
Digital_video_fingerprinting
Neural network that learns efficient data encoding in an unsupervised manner
representation (encoding) for a set of data, typically for dimensionality reduction, to generate lower-dimensional embeddings for subsequent use by other machine
Autoencoder
Study of vector bundles, principal bundles, and fibre bundles
Werner Nahm, are obtained as the dimensional reduction of the anti-self-duality in four dimensions to one dimension, by imposing translational invariance
Gauge_theory_(mathematics)
Principle in theoretical physics
distant two-dimensional surface." As pointed out by Raphael Bousso, Thorn observed in 1978 that string theory admits a lower-dimensional description from
Holographic_principle
Algorithmic technique using hashing
as a way to reduce the dimensionality of high-dimensional data; high-dimensional input items can be reduced to low-dimensional versions while preserving
Locality-sensitive_hashing
Measurable property or characteristic
feature space. In order to reduce the dimensionality of the feature space, a number of dimensionality reduction techniques can be employed. Higher-level
Feature_(machine_learning)
Machine learning technique useful for dimensionality reduction
learning technique used to produce a low-dimensional (typically two-dimensional) representation of a higher-dimensional data set while preserving the topological
Self-organizing_map
Computational technique
the methods. Typically, the steps in the algorithm consist of dimensionality reduction to reduce the complexity of the data, trajectory building to determine
Trajectory_inference
Type of large language model
Problems Classification Generative modeling Regression Clustering Dimensionality reduction Density estimation Anomaly detection Data cleaning AutoML Association
Generative pre-trained transformer
Generative_pre-trained_transformer
Method for dimension reduction in statistics
exponentially with high-dimensional data (as p grows), reducing the number of dimensions can make the operation computable. Dimensionality reduction aims to achieve
Sliced_inverse_regression
Embedding of data within a manifold based on a similarity function
drawn, making the construction of a latent space an example of dimensionality reduction, which can also be viewed as a form of data compression. Latent
Latent_space
Supergravity in eleven dimensions
four-dimensional supergravity with one gravitino. One important direction in the supergravity program was to try to construct four-dimensional N = 8
Eleven-dimensional supergravity
Eleven-dimensional_supergravity
uses semidefinite programming to perform non-linear dimensionality reduction of high-dimensional vectorial input data. It is motivated by the observation
Semidefinite_embedding
Method in natural language processing
numbers. Methods to generate this mapping include neural networks, dimensionality reduction on the word co-occurrence matrix, probabilistic models, explainable
Word_embedding
Unit of information
cleaning Outlier Winsorizing Truncation Missing data Data reduction Dimensionality reduction Principal component analysis Factor analysis Time-series preprocessing
Data
Technique to reduce dimensionality of points in Euclidean space
projection, the original d {\displaystyle d} -dimensional data is projected to a k {\displaystyle k} -dimensional subspace, by multiplying on the left by a
Random_projection
Representation learning technique
representation learning technique that maps complex, high-dimensional data into a lower-dimensional vector space of numerical vectors. It also denotes the
Embedding_(machine_learning)
Framework of superstring theory
eleven-dimensional supergravity. Although a complete formulation of M-theory is not known, such a formulation should describe two- and five-dimensional objects
M-theory
Statistical technique
the corresponding k {\displaystyle k} dimensional derived covariates. Thus the k {\displaystyle k} dimensional principal components provide the best linear
Principal component regression
Principal_component_regression
Geometric algorithm
Diffusion maps is a dimensionality reduction or feature extraction algorithm introduced by Coifman and Lafon which computes a family of embeddings of a
Diffusion_map
Set of eigenvectors used in the computer vision problem of human face recognition
all images used to construct the covariance matrix. This produces dimension reduction by allowing the smaller set of basis images to represent the original
Eigenface
Overview of and topical guide to machine learning
classifier Binary classifier Linear classifier Hierarchical classifier Dimensionality reduction Canonical correlation analysis (CCA) Factor analysis Feature extraction
Outline_of_machine_learning
Least variables needed to represent data
intrinsic dimension can be used as a lower bound of what dimension it is possible to compress a data set into through dimension reduction, but it can
Intrinsic_dimension
Approach to dimensionality reduction
hyperspectral cubes (3D/4D). The mapping from a high-dimensional vector space to a set of lower dimensional vector spaces is a multilinear projection. When
Multilinear_subspace_learning
General relativity in M-theory
could consider the 10-dimensional theory on a nontrivial circle bundle over M9. Dimensional reduction still leads to a 9-dimensional theory on M9, but with
Higher-dimensional supergravity
Higher-dimensional_supergravity
Multivariate statistical technique
captured by each principal component. This is useful for data dimensionality reduction and it could also be applied to KPCA. However, in practice there
Kernel principal component analysis
Kernel_principal_component_analysis
TurboQuant – online vector quantization algorithm for compressing high-dimensional vectors in large language model inference AlphaDev, AlphaEvolve, AlphaTensor
List of artificial intelligence algorithms
List_of_artificial_intelligence_algorithms
Set of learning techniques in machine learning
learning is often to discover low-dimensional features that capture some structure underlying the high-dimensional input data. When the feature learning
Representation_learning
Data analysis process
several football teams over several years is a two-dimensional data set. In many disciplines, two-dimensional data sets are also called panel data. While, strictly
Multidimensional_analysis
Method of a dimension reduction
Count sketch is a type of dimensionality reduction that is particularly efficient in statistics, machine learning and algorithms. It was invented by Moses
Count_sketch
Grouping a set of objects by similarity
high-dimensional data, many methods fail due to the curse of dimensionality, which renders particular distance functions problematic in high-dimensional spaces
Cluster_analysis
Points with more than one closest boundary point
surface reconstruction for physical models, and for dimensional reduction of complex models. In any dimension, the medial axis of a bounded open set is homotopy
Medial_axis
Reverse-engineering neural networks
subfield combined various techniques such as feature visualization, dimensionality reduction, and attribution with human-computer interaction methods to analyze
Mechanistic_interpretability
Deep learning generative model to encode data representation
variational distribution. As it maps from a known input space to the low-dimensional latent space, it is called the encoder. The decoder is the second neural
Variational_autoencoder
Similarity measure for number sequences
techniques available to any Euclidean space, notably standard dimensionality reduction techniques. This normalised form distance is often used within
Cosine_similarity
Data analysis technique
in a data set. It does this by representing data as points in a low-dimensional Euclidean space. The procedure thus appears to be the counterpart of
Multiple correspondence analysis
Multiple_correspondence_analysis
Mathematical result
of points from high-dimensional into low-dimensional Euclidean space. The lemma states that a set of points in a high-dimensional space can be embedded
Johnson–Lindenstrauss_lemma
Equations describing magnetic monopoles
extensively by Michael Atiyah and Nigel Hitchin. The equations are a dimensional reduction of the self-dual Yang–Mills equations from four dimensions to three
Bogomolny_equations
Study of health and disease within a population
physical or mental illnesses could, therefore, be purchased at steep reduction. Following the Philippine-American war, the United States government instituted
Epidemiology
Method in statistics
statistical method for dimensionality reduction (the transformation of data from a high-dimensional space into a low-dimensional space). SM can be used
Semantic_mapping_(statistics)
Superconformal Yang–Mills theory
N=1 dimensional super Yang–Mills theory, and the lower dimensional cases D=6, N=2 and D=4, N=4 may be derived from this via dimensional reduction. In
N = 4 supersymmetric Yang–Mills theory
N_=_4_supersymmetric_Yang–Mills_theory
Form of dimensional reduction
Freund–Rubin compactification is a form of dimensional reduction in which a field theory in d-dimensional spacetime, containing gravity and some field
Freund–Rubin_compactification
Method used in statistics, pattern recognition, and other fields
combination may be used as a linear classifier, or, more commonly, for dimensionality reduction before later classification. LDA is closely related to analysis
Linear_discriminant_analysis
Machine learning technique
Problems Classification Generative modeling Regression Clustering Dimensionality reduction Density estimation Anomaly detection Data cleaning AutoML Association
Attention_(machine_learning)
Machine learning algorithm
discretization before being applied. The variance reduction of a node N is defined as the total reduction of the variance of the target variable Y due to
Decision_tree_learning
Statistical method
such as T-distributed stochastic neighbor embedding and nonlinear dimensionality reduction. The third group includes model-based ordination methods, which
Ordination_(statistics)
rigorous nonlinear dimensionality reduction in dynamical systems. The reduction of a high-dimensional phase space to a lower-dimensional manifold can lead
Spectral_submanifold
Smooth approximation of one-hot arg max
standard (K−1)-simplex, cutting the dimension by one (the range is a (K−1)-dimensional simplex in K-dimensional space), due to the linear constraint
Softmax_function
Ten-dimensional supergravity
discovery of eleven-dimensional supergravity in 1978 led to the derivation of many lower dimensional supergravities through dimensional reduction of this theory
Type_IIA_supergravity
Political science analytical tool
that alternative choices can be projected on a basic, low-dimensional (often two-dimensional) Euclidean space. Second, within that space, individuals have
NOMINATE_(scaling_method)
Machine learning strategy
User-centered labeling strategies: Learning is accomplished by applying dimensionality reduction to graphs and figures like scatter plots. Then the user is asked
Active learning (machine learning)
Active_learning_(machine_learning)
Method for solving continuous operator problems (such as differential equations)
unchanged and only the spaces have changed. Reducing the problem to a finite-dimensional vector subspace allows us to numerically compute u n {\displaystyle u_{n}}
Galerkin_method
Machine-learning and computational-neuroscience conference
k-means Fuzzy Expectation–maximization (EM) DBSCAN OPTICS Mean shift Dimensionality reduction Factor analysis Exploratory CCA ICA NMF PCA PGD SDL Structured
Conference on Neural Information Processing Systems
Conference_on_Neural_Information_Processing_Systems
Technique in numerical linear algebra
This problem was originally solved by Erhard Schmidt in the infinite dimensional context of integral operators (although his methods easily generalize
Low-rank_approximation
Modern theory of gravitation that combines supersymmetry and general relativity
higher-dimensional supergravity theories via dimensional reduction (e.g. N=1, 11-dimensional supergravity is dimensionally reduced on T7 to 4-dimensional, ungauged
Supergravity
In multivariate analysis, canonical correspondence analysis (CCA) is an ordination technique that determines axes from the response data as a unimodal
Canonical correspondence analysis
Canonical_correspondence_analysis
American ice hockey executive
Inorganic chemistry, Solid-state chemistry, Nanotechnology Thesis Dimensional reduction: directed synthesis of metal-anion frameworks and heterometallic
Eric_Tulsky
Conversational software
Problems Classification Generative modeling Regression Clustering Dimensionality reduction Density estimation Anomaly detection Data cleaning AutoML Association
Chatbot
Signal processing computational method
signals. Given M signal mixtures in an M-dimensional space, GSO project these data points onto an (M-1)-dimensional space by using the weight vector. We can
Independent component analysis
Independent_component_analysis
Academic conference in machine learning
k-means Fuzzy Expectation–maximization (EM) DBSCAN OPTICS Mean shift Dimensionality reduction Factor analysis Exploratory CCA ICA NMF PCA PGD SDL Structured
International Conference on Learning Representations
International_Conference_on_Learning_Representations
Computational methods in biology
graph. Joint dimension reduction aims to reduce the complexity of multi-omics data by projecting observations onto a lower dimensional latent space such
Single-cell multi-omics integration
Single-cell_multi-omics_integration
Vector quantization algorithm minimizing the sum of squared deviations
through manual inspection. K-means has also been combined with dimensionality reduction techniques to classify large numbers of unlabeled survey objects
K-means_clustering
Technique in theoretical physics
vacua or type IIB string theory vacua with or without D-branes. Dimensional reduction Dean Rickles (2014). A Brief History of String Theory: From Dual
Compactification_(physics)
Process in machine learning and statistics
which finds low-dimensional projections of the data that score highly: the features that have the largest projections in the lower-dimensional space are then
Feature_selection
Science fiction book trilogy by Liu Cixin
eleven-dimensional protons dimensionally unfolded down to two-dimensional protons with Trisolaran particle accelerators. While in the two-dimensional form
Remembrance_of_Earth's_Past
Stratification of a genetic population based on allele frequencies
interpretation as source populations. Genetic data are high dimensional and dimensionality reduction techniques can capture population structure. Principal
Population structure (genetics)
Population_structure_(genetics)
Measure of variation in statistics
random vector in multiple dimensions in the same way that σ does in one dimension. A scalar random variable x with variance σ2 can be written as x=σz, where
Standard_deviation
Concept in machine learning
Advani, Madhu S.; Saxe, Andrew M.; Sompolinsky, Haim (2020-12-01). "High-dimensional dynamics of generalization error in neural networks". Neural Networks
Double_descent
Clustering methods
(eigenvalues) of the similarity matrix of the data to perform dimensionality reduction before clustering in fewer dimensions. The similarity matrix is
Spectral_clustering
Study of the deformation of solids that touch each other
problems can be solved with the method of dimensionality reduction (MDR). In this method, the initial three-dimensional system is replaced with a contact of
Contact_mechanics
Dimensionality reduction algorithm
In data science, dynamic mode decomposition (DMD) is a dimensionality reduction algorithm developed by Peter J. Schmid and Joern Sesterhenn in 2008. Given
Dynamic_mode_decomposition
Machine learning technique
Problems Classification Generative modeling Regression Clustering Dimensionality reduction Density estimation Anomaly detection Data cleaning AutoML Association
Mixture_of_experts
Machine learning paradigm
of the more general strategy of dimensionality reduction, which seeks to map the input data into a lower-dimensional space prior to running the supervised
Supervised_learning
Analysis of datasets using techniques from topology
contains relevant information. Real high-dimensional data is typically sparse, and tends to have relevant low dimensional features. One task of TDA is to provide
Topological_data_analysis
Subfield of artificial intelligence
Problems Classification Generative modeling Regression Clustering Dimensionality reduction Density estimation Anomaly detection Data cleaning AutoML Association
Neuro-symbolic_AI
Field of machine learning
function approximation starts with a mapping ϕ that assigns a finite-dimensional vector to each state-action pair. Then, the action values of a state-action
Reinforcement_learning
Technique in mathematical modeling
numerical simulations, due to complexity and large size (dimension). Model order reduction aims to lower the computational complexity of such problems
Model_order_reduction
Measure of statistical dispersion
cleaning Outlier Winsorizing Truncation Missing data Data reduction Dimensionality reduction Principal component analysis Factor analysis Time-series preprocessing
Interquartile_range
Probability distribution
application of a quantile function to uniform samples; e.g., in the multi-dimensional applications basis of copula-dependency.[citation needed] In the case
Student's_t-distribution
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