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Concept in mathematics
Sparse approximation (also known as sparse representation) theory deals with sparse solutions for systems of linear equations. Techniques for finding
Sparse_approximation
Concept in numerical linear algebra
(abbreviated as ILU) of a matrix is a sparse approximation of the LU factorization often used as a preconditioner. Consider a sparse linear system A x = b {\displaystyle
Incomplete_LU_factorization
Property of artificial neural networks
In the field of machine learning, the universal approximation theorems (UATs) state that neural networks with a certain structure can, in principle, approximate
Universal approximation theorem
Universal_approximation_theorem
Approximation of a matrix's Cholesky factorization
Cholesky factorization of a symmetric positive definite matrix is a sparse approximation of the Cholesky factorization. An incomplete Cholesky factorization
Incomplete Cholesky factorization
Incomplete_Cholesky_factorization
special cases of the sparse general Vecchia approximation. These methods approximate the true model in a way the covariance matrix is sparse. Typically, each
Gaussian process approximations
Gaussian_process_approximations
Representation learning method
problem of finding an optimal sparse coding R with a given dictionary š is known as sparse approximation (or sometimes just sparse coding problem). A number
Sparse_dictionary_learning
Multidimensional data algorithm
Matching pursuit (MP) is a sparse approximation algorithm which finds the "best matching" projections of multidimensional data onto the span of an over-complete
Matching_pursuit
Signal processing technique
sensitive materials. Low-density parity-check code Noiselet Sparse approximation Sparse coding Verification-based message-passing algorithms in compressed
Compressed_sensing
Technique in numerical linear algebra
In mathematics, low-rank approximation refers to the process of approximating a given matrix by a matrix of lower rank. More precisely, it is a minimization
Low-rank_approximation
Method by which information is represented in the brain
roughly 100,000 neurons. Other models are based on matching pursuit, a sparse approximation algorithm which finds the "best matching" projections of multidimensional
Neural_coding
Type of artificial neural network
feedforward neural networks for classification, regression, clustering, sparse approximation, compression and feature learning with a single layer or multiple
Extreme_learning_machine
Approximation method
hierarchical matrices (H-matrices) are used as data-sparse approximations of non-sparse matrices. While a sparse matrix of dimension n {\displaystyle n} can be
Hierarchical_matrix
Cholesky factorization ā sparse approximation to the Cholesky factorization Incomplete LU factorization ā sparse approximation to the LU factorization
List of numerical analysis topics
List_of_numerical_analysis_topics
Function defined by multiple sub-functions
shearlets have been used as a representation system to provide sparse approximations of this model class in 2D and 3D. Piecewise defined functions are
Piecewise_function
weighted completion time Block Sorting (Sorting by Block Moves) Sparse approximation Variations of the Steiner tree problem. Specifically, with the discretized
List_of_NP-complete_problems
Unrelated vertices in graphs
different when restricted to special classes of graphs. For instance, for sparse graphs (graphs in which the number of edges is at most a constant times
Independent set (graph theory)
Independent_set_(graph_theory)
American mathematician (born 1977)
the California Institute of Technology. He is known for work on sparse approximation, numerical linear algebra, and random matrix theory. Tropp studied
Joel_Tropp
Analytical expression in statistics
Laplace's approximation or the quadratic approximation (QUAP) provides an analytical expression for a posterior probability distribution by fitting a Gaussian
Laplace's_approximation
related fields, relaxation is a modeling strategy. A relaxation is an approximation of a difficult problem by a nearby problem that is easier to solve.
Relaxation_(approximation)
Originally, shearlets were introduced in 2006 for the analysis and sparse approximation of functions f ā L 2 ( R 2 ) {\displaystyle f\in L^{2}(\mathbb {R}
Shearlet
Numerical approximation algorithm
improving approximate solutions for a class of problems, in which the i-th approximation (called an "iterate") is derived from the previous ones. A specific
Iterative_method
Process in machine learning and statistics
"Submodular meets Spectral: Greedy Algorithms for Subset Selection, Sparse Approximation and Dictionary Selection". arXiv:1102.3975 [stat.ML]. Liu, Yuzong;
Feature_selection
meaningful gene sets Statistical learning theory Regularization Sparse approximation Proximal gradient methods Convex analysis Feature selection Rosasco
Structured sparsity regularization
Structured_sparsity_regularization
Russian-American mathematician
and 2015 in the summer seminar on "Applied Harmonic Analysis and Sparse Approximation" at Oberwolfach. He is the author or co-author of over 100 articles
Gregory_Beylkin
Discrete Fourier transform algorithm
computes such transformations by factorizing the DFT matrix into a product of sparse (mostly zero) factors. As a result, it manages to reduce the complexity
Fast_Fourier_transform
Statistical model allowing for frequent zero values
Zero-truncated Poisson distribution Compound Poisson distribution Sparse approximation Hurdle model pscl, glmmTMB and brms R packages Bilder, Christopher;
Zero-inflated_model
Optimization algorithm
differentiable or subdifferentiable). It can be regarded as a stochastic approximation of gradient descent optimization, since it replaces the actual gradient
Stochastic_gradient_descent
Statistical method
coefficient multiplying the approximation-error term is exactly one. Their bound therefore allows comparison with sparse approximations even when the underlying
Lasso_(statistics)
decisions about how to construct the approximation. More technically, general versions of the approximation lead to a sparse Cholesky factor of the precision
Vecchia_approximation
Technique to solve partial differential equations
admissible solutions, increasing the generalizability of the function approximation. This way, embedding this prior information into a neural network results
Physics-informed neural networks
Physics-informed_neural_networks
Dictionary learning algorithm
[betterĀ sourceĀ needed] Sparse approximation Singular value decomposition Matrix norm k-means clustering Low-rank approximation Michal Aharon; Michael
K-SVD
Problem in combinatorial optimization
algorithm using dynamic programming. There is a fully polynomial-time approximation scheme, which uses the pseudo-polynomial time algorithm as a subroutine
Knapsack_problem
Numerical method for solving boundary value problems
Poisson's equation or the Laplace's equation. The PGD algorithm computes an approximation of the solution of the BVP by successive enrichment. This means that
Proper generalized decomposition
Proper_generalized_decomposition
Topics referred to by the same term
a function of many variables Hierarchical matrix, a data-sparse approximation of a non-sparse matrix Hilbert matrix, a square matrix with entries being
H-matrix
Image processing method
Redondo, Rafael; Perrinet, Laurent; Cristobal, Gabriel (2007). "Sparse approximation of images inspired from the functional architecture of the primary
Edge_detection
American electrical engineer and academic
of Manitoba (1987) KnownĀ for Wavelet theory, Compressive sensing, Sparse approximation, Machine learning, Deep learning, Open educational resources Awards
Richard_Baraniuk
Optimization problem
Least-squares spectral analysis Matching pursuit Sparse approximation Natarajan, B. K. (April 1995). "Sparse Approximate Solutions to Linear Systems". SIAM
Basis_pursuit
British mathematician (1925ā2015)
won the Fields Medal for proving Roth's theorem on the Diophantine approximation of algebraic numbers. He was also a winner of the De Morgan Medal and
Klaus_Roth
Task of computing complete subgraphs
independent sets in sparse graphs, a case that does not make sense for the complementary clique problem, there has also been work on approximation algorithms that
Clique_problem
Method for solving continuous operator problems (such as differential equations)
method, one also gives the name along with typical assumptions and approximation methods used: RitzāGalerkin method (after Walther Ritz) typically assumes
Galerkin_method
Type of activation function
paper also introduces a few faster approximations for GELU. The first approximation follows from an approximation for Ī¦(x): f(x)ā12x(1+tanhā”[2/Ļ(x+0.044715x3)])
Rectified_linear_unit
Statistical analysis technique
large-scale dataset, including sparse principal component analysis and sparse matrix approximation. nsprcomp - R package for sparse and/or non-negative PCA based
Sparse_PCA
deconvolution, are ill-posed. Variants of this method have been used also in sparse approximation problems and compressed sensing settings. Landweber, L. (1951). "An
Landweber_iteration
Mathematical model of memory
utilizes SDM for storing sparse distributed representations of the data. SDMs provide a linear, local function approximation scheme, designed to work
Sparse_distributed_memory
Discrete Fourier transform algorithm
The sparse Fourier transform (SFT) is a kind of discrete Fourier transform (DFT) for handling big data signals. Specifically, it is used in GPS synchronization
Sparse_Fourier_transform
Methods for numerical approximations
(in contrast to discrete mathematics), and typically use numerical approximation in addition to symbolic manipulation. Numerical analysis finds application
Numerical_analysis
Discrete analog of a derivative
differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation. The difference
Finite_difference
Technique to make a model more generalizable and transferable
discourage complex models: L1 regularization (also called LASSO) leads to sparse models by adding a penalty based on the absolute value of coefficients.
Regularization_(mathematics)
Methods of calculating definite integrals
from the approximation. An important part of the analysis of any numerical integration method is to study the behavior of the approximation error as a
Numerical_integration
Value in matrix theory
S2CIDĀ 18432970. Joel A. Tropp (2004). "Greed is good: Algorithmic results for sparse approximation" (PDF). Welch, L. R. (1974). "Lower bounds on the maximum cross-correlation
Mutual coherence (linear algebra)
Mutual_coherence_(linear_algebra)
Concept in linear algebra
sampling theory, operator theory, harmonic analysis, nonlinear sparse approximation, pseudodifferential operators, wireless communications, geophysics
Overcompleteness
Smoothed ramp function
ln ā” ( 1 + e x ) . {\displaystyle f(x)=\ln(1+e^{x}).} It is a smooth approximation (in fact, an analytic function) to the ramp function, which is known
Softplus
Numerical method for solving physical or engineering problems
equations are often partial differential equations (PDEs). To explain the approximation of this process, FEM is commonly introduced as a special case of the
Finite_element_method
Turkish-American mathematician (born 1958)
principal component analysis of first-order autoregressive process, sparse approximation, digital watermarking, financial signal processing and quantitative
Ali_Akansu
Partition of a graph's nodes into 2 disjoint subsets
both sparse (few edges crossing the cut) and balanced (close to a bisection). The problem is known to be NP-hard, and the best known approximation algorithm
Cut_(graph_theory)
Sylvain Fischer, Rafael Redondo, Laurent Perrinet, Gabriel Cristobal. Sparse approximation of images inspired from the functional architecture of the primary
Log_Gabor_filter
Type of mathematical function
and thus have sparse differentiation matrices Bump function: Radial basis functions are typically used to build up function approximations of the form where
Radial_basis_function
Partition of a graph's nodes into cliques
there can be no polynomial time approximation algorithm for any Īµ > 0 that, on n-vertex graphs, achieves an approximation ratio better than n1 ā Īµ. In graphs
Clique_cover
Approximation method in quantum physics
the HartreeāFock wave function and energy of the system. HartreeāFock approximation is an instance of mean-field theory, where neglecting higher-order fluctuations
HartreeāFock_method
Connectivity measure in graph theory
and to the star height of a regular language. It has also found use in sparse matrix computations (see Bodlaender et al. 1995) and logic (Rossman 2008)
Cycle_rank
Method of data analysis
low rank approximation (Appendix B). arXiv:1410.6801. Bibcode:2014arXiv1410.6801C. Hui Zou; Trevor Hastie; Robert Tibshirani (2006). "Sparse principal
Principal_component_analysis
Optimization problem in computer science
"Mining of Massive Datasets, Ch. 3". Weber, Roger; Blott, Stephen. "An Approximation-Based Data Structure for Similarity Search" (PDF). S2CIDĀ 14613657. Archived
Nearest_neighbor_search
Method in approximation theory
Radial basis function (RBF) interpolation is an advanced method in approximation theory for constructing high-order accurate interpolants of unstructured
Radial basis function interpolation
Radial_basis_function_interpolation
Quantum chemistry rule regarding vibronic transitions
the momentum is zero. Classically, the FranckāCondon principle is the approximation that an electronic transition is most likely to occur without changes
FranckāCondon_principle
al. based on the observation by Bast et al. that any road network has a sparse set of "transit nodes", such that driving from a point A to a sufficiently
Highway_dimension
Longest distance between two vertices
Vassilevska Williams, Virginia (2013), "Fast approximation algorithms for the diameter and radius of sparse graphs", in Boneh, Dan; Roughgarden, Tim; Feigenbaum
Diameter_(graph_theory)
Mathematical optimization algorithm
sparse systems that are too large to be handled by a direct implementation or other direct methods such as the Cholesky decomposition. Large sparse systems
Conjugate_gradient_method
tree is an efficient method for constructing a second-order product approximation of a joint probability distribution, first described in a paper by Chow
ChowāLiu_tree
Optimization algorithm
1016/0022-247X(78)90137-3. Clarkson, K. L. (2010). "Coresets, sparse greedy approximation, and the Frank-Wolfe algorithm". ACM Transactions on Algorithms
FrankāWolfe_algorithm
Model-free reinforcement learning algorithm
with (linear) function approximation. The advantage of Greedy GQ is that convergence is guaranteed even when function approximation is used to estimate the
Q-learning
American mathematician
with Needell recognized in particular for her contributions to sparse approximation, signal processing, and stochastic optimization. She is also the
Deanna_Needell
Annual award by the National Academy of Sciences
wavelets and sampling techniques and their impact on data analysis and sparse approximation. Shanhui Fan (2007, optical science) For innovative research on the
William O. Baker Award for Initiatives in Research
William_O._Baker_Award_for_Initiatives_in_Research
Element of a basis for a function space
preferable for approximation and numerical computation. In finite element methods, local support is particularly useful because it produces sparse systems."Approximation
Basis_function
randomized rounding is a widely used approach for designing and analyzing approximation algorithms. Many combinatorial optimization problems are computationally
Randomized_rounding
Technique to reduce dimensionality of points in Euclidean space
random projection preserves distances well, but empirical results are sparse. They have been applied to many natural language tasks under the name random
Random_projection
Mathematical optimization problem
"Forward Backward Algorithm". Archived from the original on February 16, 2014. A list of BPDN solvers at the sparse- and low-rank approximation wiki.
Basis_pursuit_denoising
Measure of similarity and diversity between sets
where binary or binarized data are used. Both the exact solution and approximation methods are available for hypothesis testing with the Jaccard index
Jaccard_index
Technique for shaping training datasets
may become prohibitively slow to compute. Online algorithms and sparse approximations of the manifold may help in this case. Manifold learning Manifold
Manifold_regularization
Algorithms for matrix decomposition
Non-negative matrix factorization (NMF or NNMF), also non-negative matrix approximation is a group of algorithms in multivariate analysis and linear algebra
Non-negative matrix factorization
Non-negative_matrix_factorization
Probabilistic graphical representation of causal relationships
NP-hard. This result prompted research on approximation algorithms with the aim of developing a tractable approximation to probabilistic inference. In 1993
Bayesian_network
Lower bound on the log-likelihood of some observed data
p ā {\displaystyle p^{*}} exactly, forcing us to search for a good approximation. That is, we define a sufficiently large parametric family { p Īø } Īø
Evidence_lower_bound
Decision problem in computer science
where r is a number in (0,1) called the approximation ratio. The following very simple algorithm has an approximation ratio of 1/2: Order the inputs by descending
Subset_sum_problem
Nonlinear Software Package
available. It employs a sparse sequential quadratic programming (SQP) algorithm with limited-memory quasi-Newton approximations to the Hessian of the Lagrangian
SNOPT
Eigenvalue algorithm
matrix A {\displaystyle A} by a vector, so it is effective for a very large sparse matrix with appropriate implementation. The speed of convergence is like
Power_iteration
Statistical significance test
(e.g., p-value) can be calculated exactly, rather than relying on an approximation that becomes exact in the limit as the sample size grows to infinity
Fisher's_exact_test
Purely statistical model of language
/s\rangle \mid {\text{house}})} In a trigram (nĀ =Ā 3) language model, the approximation is P ( I, saw, the, red, house ) ā P ( I ā£ āØ s ā© , āØ s ā© ) P ( saw ā£
Word_n-gram_language_model
way to obtain the perceptual-based sound localization is from the sparse approximations of the anthropometric features. Perceptual-based sound localization
Perceptual-based 3D sound localization
Perceptual-based_3D_sound_localization
applications. Joel A. Tropp (2004). "Greed is good: Algorithmic results for sparse approximation" (PDF). IEEE Trans. Inform. Theory. 50 (10): 2231ā2242. doi:10.1109/TIT
Babel_function
Statistical principle about ratio of effects to causes
as the 80:20 rule, the law of the vital few and the principle of factor sparsity) states that, for many outcomes, roughly 80% of consequences come from
Pareto_principle
Subfield of computational complexity theory
Vassilevska Williams, Virginia (2013). "Fast approximation algorithms for the diameter and radius of sparse graphs". Proceedings of STOC 2013. Henzinger
Fine_grained_complexity
Criterion for model selection
Gideon E. Schwarz and published in a 1978 paper, as a large-sample approximation to the Bayes factor. The BIC is formally defined as B I C = k ln ā” (
Bayesian information criterion
Bayesian_information_criterion
Mathematical algorithm
algorithm. It allows one to find an approximate eigenvector when an approximation to a corresponding eigenvalue is already known. The method is conceptually
Inverse_iteration
Romanian scientist and emeritus professor
and contributed to nine US patents Bruckstein's research topics include sparse representations, various aspects of computer vision, pattern recognition
Alfred_Marcel_Bruckstein
Hypothesis in machine learning
result, unstructured sparsity, where zeros are distributed irregularly throughout weight matrices, requires specialized sparse accelerators or structured
Lottery_ticket_hypothesis
Field of machine learning
and algorithms for their exact computation, and less with learning or approximation (particularly in the absence of a mathematical model of the environment)
Reinforcement_learning
Projection of data onto lower-dimensional manifolds
includes a quality of data approximation and some penalty terms for the bending of the manifold. The popular initial approximations are generated by linear
Nonlinear dimensionality reduction
Nonlinear_dimensionality_reduction
Number used to approximate the square root of 2
the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins ā 1/1ā , ā 3/2ā , ā 7/5ā , ā 17/12ā , and
Pell_number
random variable cannot be simulated exactly, but there is a sequence of approximations G0,G1,ā¦,GL with increasing accuracy, but also increasing cost, that
Multilevel_Monte_Carlo_method
Subset of a graph's nodes such that all other nodes link to at least one
that can compute Ī³(G) for all graphs G. However, there are efficient approximation algorithms, as well as efficient exact algorithms for certain graph
Dominating_set
Artificial neural network node function
be a universal function approximator. This is known as the Universal Approximation Theorem. The identity activation function does not satisfy this property
Activation_function
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