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matrix regularization generalizes notions of vector regularization to cases where the object to be learned is a matrix. The purpose of regularization
Matrix_regularization
Regularization technique for ill-posed problems
estimator. LASSO estimator is another regularization method in statistics. Elastic net regularization Matrix regularization L-curve In statistics, the method
Ridge_regression
Technique to make a model more generalizable and transferable
strong connection between regularization methods and Bayesian approaches for solving such ill-posed problems. Although regularization procedures can be divided
Regularization_(mathematics)
Topics referred to by the same term
Look up regularization, regularisation, or regularizations in Wiktionary, the free dictionary. Regularization may refer to: Regularization (linguistics)
Regularization
Filling in missing entries of a matrix
point of view, the matrix completion problem is an application of matrix regularization which is a generalization of vector regularization. For example, in
Matrix_completion
Spectral regularization is any of a class of regularization techniques used in machine learning to control the impact of noise and prevent overfitting
Regularization by spectral filtering
Regularization_by_spectral_filtering
Method used in mathematical physics
not always possible to define a regularization such that the limit of ε going to zero is independent of the regularization. In this case, one says that the
Regularization_(physics)
Matrix decomposition
and generalization of the extension method of covariance matrix inversion by regularization". Imaging Spectrometry IX. Proceedings of SPIE. 5159: 299
Eigendecomposition of a matrix
Eigendecomposition_of_a_matrix
Matrix representation of a graph
theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix, or discrete Laplacian, is a matrix representation of a
Laplacian_matrix
Mathematical procedure
assigning different regularization weights to the latent factors based on items' popularity and users' activeness. The idea behind matrix factorization is
Matrix factorization (recommender systems)
Matrix_factorization_(recommender_systems)
Matrix decomposition
complex matrix into a rotation, followed by a scaling, followed by another rotation. It generalizes the eigendecomposition of a square normal matrix with
Singular_value_decomposition
Technique for shaping training datasets
Manifold regularization adds a second regularization term, the intrinsic regularizer, to the ambient regularizer used in standard Tikhonov regularization. Under
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
Concept in regression analysis mathematics
Regularized least squares (RLS) is a family of methods for solving the least-squares problem while using regularization to further constrain the resulting
Regularized_least_squares
Overview of and topical guide to machine learning
approximation Low-rank matrix approximations MATLAB MIMIC (immunology) MXNet Mallet (software project) Manifold regularization Margin-infused relaxed
Outline_of_machine_learning
Most widely known generalized inverse of a matrix
A^{+}} of a matrix A {\displaystyle A} , often called the pseudoinverse, is the most widely known generalization of the inverse matrix. It was independently
Moore–Penrose_inverse
estimator can be derived both from a regularization and a Bayesian perspective. The main assumption in the regularization perspective is that the set of functions
Bayesian interpretation of kernel regularization
Bayesian_interpretation_of_kernel_regularization
Statistical method
also Lasso, LASSO or L1 regularization) is a regression analysis method that performs both variable selection and regularization in order to enhance the
Lasso_(statistics)
No-go theorem concerning chirality of regularized fermions
generalized to all possible regularization schemes, not just lattice regularization. This general no-go theorem states that no regularized chiral fermion theory
Nielsen–Ninomiya_theorem
Visualization method for regularization
corresponds to heavy regularization (small solution norm but large residual), while the steep part corresponds to light regularization (small residual but
L-curve
Approximation method in statistics
functions. In some contexts, a regularized version of the least squares solution may be preferable. Tikhonov regularization (or ridge regression) adds a
Least_squares
Process of calculating the causal factors that produced a set of observations
knowledge Seismic inversion – Geophysical process Tikhonov regularization – Regularization technique for ill-posed problemsPages displaying short descriptions
Inverse_problem
Signal processing technique
projection matrix P of the fan-beam geometry, which is constrained by the data fidelity term. This may contain noise and artifacts as no regularization is performed
Compressed_sensing
Linear dependency situation in a regression model
perfect collinearity, the design matrix X {\displaystyle X} has less than full rank, and therefore the moment matrix X T X {\displaystyle X^{\mathsf {T}}X}
Multicollinearity
Approximations used in machine learning
Low-rank matrix approximations are essential tools in the application of kernel methods to large-scale learning problems. Kernel methods (for instance
Low-rank_matrix_approximation
Method for model fitting in statistics
generalized least squares, when all the off-diagonal entries of the covariance matrix of the errors are null. The fit of a model to a data point is measured by
Weighted_least_squares
closely related to the first-order term in Taylor's theorem Hadamard regularization Hadamard's gamma function Hadamard multiplication theorem – for the
List of things named after Jacques Hadamard
List_of_things_named_after_Jacques_Hadamard
Method for estimating the unknown parameters in a linear regression model
power", in that they tend to overfit the data. As a result, some kind of regularization must typically be used to prevent unreasonable solutions coming out
Ordinary_least_squares
Statistical estimator
{\displaystyle L_{1}} penalty, it performs regularization to give a sparse estimate for the precision matrix. In the case of multivariate Gaussian distributions
Graphical_lasso
Algorithm for modelling sequential data
layer:UnEmbed(x)=softmax(xW+b)The matrix has shape (demb,|V|). Some architectures use the transpose of the embedding matrix M as the un-embedding matrix W in order to avoid
Transformer_(deep_learning)
Regularization method for artificial neural networks
Dropout is a regularization technique for reducing overfitting in artificial neural networks by preventing complex co-adaptations on training data. The
Dropout_(neural_networks)
Statistical learning theory
likewise independent of v {\displaystyle v} . For the second term (the regularization term), since v {\displaystyle v} is orthogonal to ∑ i = 1 n α i φ (
Representer_theorem
Gilbert. It is a regularization method for obtaining meaningful solutions to ill-posed inverse problems. Where other regularization methods, such as the
Backus–Gilbert_method
Image noise reducing technique
can be achieved by this regularization but it also introduces blurring effect, which is the main drawback of regularization. A prior knowledge of noise
Anisotropic_diffusion
Type of feedforward neural network
noisy inputs. L1 with L2 regularization can be combined; this is called elastic net regularization. Another form of regularization is to enforce an absolute
Convolutional_neural_network
Quantum field theory
ISBN 978-0-19-920846-3. Primary source: 't Hooft, G.; Veltman, M. (1972). "Regularization and renormalization of gauge fields". Nuclear Physics B. 44 (1): 189–213
Yang–Mills_theory
Method of modelling contact between solids
it practically applicable. This novel regularization, known as HuHu regularization, is a general regularization technique for finite elements which has
Third_medium_contact_method
Class of algorithms for pattern analysis
machines is infinite dimensional but only requires a finite dimensional matrix from user-input according to the representer theorem. Kernel machines are
Kernel_method
and other metrics. Regularization perspectives on support-vector machines interpret SVM as a special case of Tikhonov regularization, specifically Tikhonov
Regularization perspectives on support vector machines
Regularization_perspectives_on_support_vector_machines
Force resulting from the quantisation of a field
computed using Euler–Maclaurin summation with a regularizing function (e.g., exponential regularization) not so anomalous as |ωn|−s in the above. Casimir's
Casimir_effect
Dutch theoretical physicist
include: a proof that gauge theories are renormalizable; dimensional regularization; and the holographic principle. 't Hooft was born in Den Helder on 5
Gerard_'t_Hooft
Set of methods for supervised statistical learning
regularization penalty ℛ(f)=λk‖f‖ℋ, and solving the new optimization problem f̂=argminf∈ℋε̂(f)+ℛ(f). This approach is called Tikhonov regularization.
Support_vector_machine
Optimization algorithm for artificial neural networks
rather than a diagonal matrix. Since matrix multiplication is linear, the derivative of multiplying by a matrix is just the matrix: (Wx)′=W. One may notice
Backpropagation
Class of algorithms for solving constrained optimization problems
together with extensions involving non-quadratic regularization functions (e.g., entropic regularization). This combined study gives rise to the "exponential
Augmented_Lagrangian_method
Optimization algorithm
Like the original BFGS, L-BFGS uses an estimate of the inverse Hessian matrix to steer its search through variable space, but where BFGS stores a dense
Limited-memory_BFGS
Statistical technique
corresponding to the higher eigenvalues of the sample variance-covariance matrix of the explanatory variables) are selected as regressors. However, for the
Principal component regression
Principal_component_regression
Flaw in mathematical modelling
model to better capture the underlying patterns in the data. Regularization: Regularization is a technique used to prevent overfitting by adding a penalty
Overfitting
Neural network that learns efficient data encoding in an unsupervised manner
enforce. The contractive regularization loss itself is defined as the expected square of Frobenius norm of the Jacobian matrix of the encoder activations
Autoencoder
2017 paper conceptualizing the transformer-based deep learning architecture
achieving the comparatively lowest training cost. Hyperparameters and regularization - For their 100M-parameter Transformer model, the authors increased
Attention_Is_All_You_Need
Solving multiple machine learning tasks at the same time
learning works because regularization induced by requiring an algorithm to perform well on a related task can be superior to regularization that prevents overfitting
Multi-task_learning
Method of machine learning
through empirical risk minimization or regularized empirical risk minimization (usually Tikhonov regularization). The choice of loss function here gives
Online_machine_learning
Volume rendering technique
through future improvements like better culling approaches, antialiasing, regularization, and compression techniques. Extending 3D Gaussian splatting to dynamic
Gaussian_splatting
Types of special mathematical functions
T(m,s,x)=G_{m-1,\,m}^{\,m,\,0}\!\left(\left.{\begin{matrix}0,0,\dots ,0\\s-1,-1,\dots ,-1\end{matrix}}\;\right|\,x\right).} This particular special case
Incomplete_gamma_function
Statistical model used in machine learning
neural networks. To regularize the flow f, one can impose regularization losses. The paper proposed the following regularization loss based on optimal
Flow-based_generative_model
Connection between correlation functions and the S-matrix
Lehmann–Symanzik–Zimmermann (LSZ) reduction formula is a method to calculate S-matrix elements (the scattering amplitudes) from the time-ordered correlation functions
LSZ_reduction_formula
Set of machine learning methods
is an error function and R is a regularization term. E is typically the square loss function (Tikhonov regularization) or the hinge loss function (for
Multiple_kernel_learning
least squares. The use of a priori parameter covariance matrix is akin to Tikhonov regularization If rank deficiency is encountered, it can often be rectified
Least-squares_adjustment
Set of learning techniques in machine learning
error, an L1 regularization on the representing weights for each data point (to enable sparse representation of data), and an L2 regularization on the parameters
Representation_learning
Machine learning system
gradient descent (SGD) BFGS Conjugate gradient Regularization (L1 norm, L2 norm, & elastic net regularization) Flexible input - input features may be: Binary
Vowpal_Wabbit
Tool for characterizing the response properties of a neuron
I} denotes the identity matrix and λ {\displaystyle \lambda } is the ridge parameter controlling the amount of regularization. This procedure has a simple
Spike-triggered_average
Deep learning software
subprogram (BLAS) operations like dot product, matrix–vector multiplication, matrix–matrix multiplication and matrix product. The following exemplifies using
Torch_(machine_learning)
Algorithm used to solve non-linear least squares problems
{\beta }}\right)}\right].} A similar damping factor appears in Tikhonov regularization, which is used to solve linear ill-posed problems, as well as in ridge
Levenberg–Marquardt_algorithm
Generalization of graph theory
extensively used in machine learning tasks as the data model and classifier regularization. The applications include recommender system (communities as hyperedges)
Hypergraph
Theory of the strong nuclear interactions
because space and time break down. This was the more radical approach of S-matrix theory. James Bjorken proposed that pointlike partons would imply certain
Quantum_chromodynamics
Dutch theoretical physicist (1931–2021)
Asteroid 9492 Veltman is named in his honor. Chiral anomaly Pauli–Villars regularization Veltman, M. "Perturbation Theory of Massive Yang-Mills Fields", Utrecht
Martinus_J._G._Veltman
Brand of data center GPUs by AMD
increase in TFLOPS. Since CDNA3 it is also able to use structured sparsity regularization for a 2× increase in TFLOPS for all data types. In CDNA4 the speedup
AMD_Instinct
Study of high-dimensional data
1214/009053606000001523. MR 2382644. S2CID 88524200. Zou, Hui; Hastie, Trevor (2005). "Regularization and Variable Selection via the Elastic Net". Journal of the Royal Statistical
High-dimensional_statistics
Extracting features from raw data for machine learning
linear system Feature explosion can be limited via techniques such as regularization, kernel methods, and feature selection. Feature templates (abstract
Feature_engineering
Method for solving certain optimization problems
|}y_{i}-X_{i}{\boldsymbol {\beta }}^{(t)}{\big |}}}.} To avoid dividing by zero, regularization must be done, so in practice the formula is w i ( t ) = 1 max { δ ,
Iteratively reweighted least squares
Iteratively_reweighted_least_squares
Recursive filter
version of the Kalman filter for large problems (essentially, the covariance matrix is replaced by the sample covariance), and it is now an important data assimilation
Ensemble_Kalman_filter
Statistics concept
In statistics, sometimes the covariance matrix of a multivariate random variable is not known but has to be estimated. Estimation of covariance matrices
Estimation of covariance matrices
Estimation_of_covariance_matrices
Pictorial representation of the behavior of subatomic particles
obtained from a Lagrangian by Feynman rules. Dimensional regularization is a method for regularizing integrals in the evaluation of Feynman diagrams; it assigns
Feynman_diagram
Statistical estimation technique
requires knowledge of the covariance matrix for the residuals. If this is unknown, estimating the covariance matrix gives the method of feasible generalized
Generalized_least_squares
Type of artificial neural network
𝐱N)] and 𝐓 is the training data target matrix: 𝐓=[𝐭1⋮𝐭N] Generally speaking, ELM is a kind of regularization neural networks but with non-tuned hidden
Extreme_learning_machine
Optimization algorithm
of applying weight decay directly, it is implemented by adding an L2 regularization term to the loss function, of form λ12‖w‖2. This term then produces
Stochastic_gradient_descent
Statistical method
direction in the Y space. PLS regression is particularly suited when the matrix of predictors has more variables than observations, and when there is multicollinearity
Partial least squares regression
Partial_least_squares_regression
Signal processing computational method
how to obtain non-linear ICA or source separation as a by-product of regularization (1999). Their method does not require a priori knowledge about the number
Independent component analysis
Independent_component_analysis
Expectation value of time-ordered quantum operators
theory where they can be used to calculate various observables such as S-matrix elements, although they are not themselves observables. This is because
Correlation function (quantum field theory)
Correlation_function_(quantum_field_theory)
Machine learning technique
fine-tuning models. The basic idea is to design a low-rank matrix that is then added to the original matrix. An adapter, in this context, is a collection of low-rank
Fine-tuning_(deep_learning)
Type of feedforward neural network
signal variance. To prevent overfitting, regularization techniques are standard. Weight decay (L2 regularization) adds a penalty for large weights, while
Multilayer_perceptron
codes. The regularization and kernel theory literature for vector-valued functions followed in the 2000s. While the Bayesian and regularization perspectives
Kernel methods for vector output
Kernel_methods_for_vector_output
Mathematical optimization problem
formulation is NP-hard. Either types of basis pursuit denoising solve a regularization problem with a trade-off between having a small residual (making y {\displaystyle
Basis_pursuit_denoising
Probability distribution
information matrix for the four parameter case is positive-definite only for α, β > 2 (for further discussion, see section on Fisher information matrix, four
Beta_distribution
Process in machine learning and statistics
l_{1}} -regularization techniques, such as sparse regression, LASSO, and l 1 {\displaystyle l_{1}} -SVM Regularized trees, e.g. regularized random forest
Feature_selection
Type of artificial neural network
each subsequent layer: xℓ+1=F(x1,x2,…,xℓ−1,xℓ) Stochastic depth is a regularization method that randomly drops a subset of layers and lets the signal propagate
Residual_neural_network
Gradient boosting machine learning library
Gradient Boosting) is an open-source software library which provides a regularizing gradient boosting framework for C++, Java, Python, R, Julia, Perl, and
XGBoost
Potential problem in computer-based information systems
various recommendation models benefit from this strategy. Differentiating regularization weights can be integrated with the other cold start mitigating strategies
Cold start (recommender systems)
Cold_start_(recommender_systems)
Mathematical concept
errors that may be corrected simultaneously. Overdetermined system Regularization (mathematics) Biswa Nath Datta (4 February 2010). Numerical Linear Algebra
Underdetermined_system
Generalized method of moments estimator in econometrics
Z)^{-1}Z'\Delta y} where Z {\displaystyle Z} is the instrument matrix for Δ R {\displaystyle \Delta R} . The matrix Ω {\displaystyle \Omega } can be calculated from
Arellano–Bond_estimator
Gamma matrices for arbitrary Clifford algebras
general identities are ubiquitous in loop calculations due to dimensional regularization. The Γ matrices can be constructed recursively, first in all even dimensions
Higher-dimensional gamma matrices
Higher-dimensional_gamma_matrices
Statistics concept
_{m}x_{i}^{m}+\varepsilon _{i}\ (i=1,2,\dots ,n)} can be expressed in matrix form in terms of a design matrix X {\displaystyle \mathbf {X} } , a response vector y →
Polynomial_regression
Least squares approximation of linear functions to data
residuals. Numerical methods for linear least squares include inverting the matrix of the normal equations and orthogonal decomposition methods. Consider the
Linear_least_squares
Asymmetry of classical and quantum action
invariance, a Pauli–Villars regularization of such diagrams is possible while preserving the symmetry. Whenever the regularization of a diagram is consistent
Anomaly_(physics)
Distribution of singular values of large rectangular random matrices
If X {\displaystyle X} denotes an m × n {\displaystyle m\times n} random matrix whose entries are independent identically distributed random variables with
Marchenko–Pastur_distribution
Statistical classification in machine learning
of a word in a document (see document-term matrix). In such cases, the classifier should be well-regularized. There are two broad classes of methods for
Linear_classifier
Mathematical concept
standard deviation corresponds to the Discrepancy Principle for automatic regularization. Non-negative least squares (NNLS): The vector β {\displaystyle {\boldsymbol
Constrained_least_squares
Linear function of explanatory variables used to predict a dependent variable
cases by eliminating one of the dummy variables, and/or introduce a regularization constraint (which necessitates a more powerful, typically iterative
Linear_predictor_function
Technique in machine learning
This has been shown to work in many domains, most likely as a form of regularization. There are several major variations in how the technique is applied:
Curriculum_learning
In applied statistics, optimal estimation is a regularized matrix inverse method based on Bayes' theorem. It is used very commonly in the geosciences,
Optimal_estimation
Lowest possible energy of a quantum system or field
observed spectra. Then just a year later in 1925, with the development of matrix mechanics in Werner Heisenberg's article "Quantum theoretical re-interpretation
Zero-point_energy
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MATRIX REGULARIZATION
MATRIX REGULARIZATION
MATRIX REGULARIZATION
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