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these functions produce a scalar output. Recent development of kernel methods for functions with vector-valued output is due, at least in part, to interest
Kernel methods for vector output
Kernel_methods_for_vector_output
Class of algorithms for pattern analysis
learning, kernel machines are a class of algorithms for pattern analysis, whose best known member is the support-vector machine (SVM). These methods involve
Kernel_method
Set of methods for supervised statistical learning
Shawe-Taylor, John (2000). An Introduction to Support Vector Machines and other kernel-based learning methods. Cambridge University Press. ISBN 0-521-78019-5
Support_vector_machine
Overview of and topical guide to machine learning
k-nearest neighbors algorithm Kernel methods for vector output Kernel principal component analysis Learning vector quantization Leabra Linde–Buzo–Gray
Outline_of_machine_learning
Type of kernel induced by artificial neural networks
It allows ANNs to be studied using theoretical tools from kernel methods. In general, a kernel is a positive-semidefinite symmetric function of two inputs
Neural_tangent_kernel
Solving multiple machine learning tasks at the same time
Foundation model General game playing Human-based genetic algorithm Kernel methods for vector output Multiple-criteria decision analysis Multi-objective optimization
Multi-task_learning
Computation routine
kernel. Matrix–vector multiplication General-purpose computing on graphics processing units#Kernels "Hypergraph Partitioning Based Models and Methods
Sparse matrix–vector multiplication
Sparse_matrix–vector_multiplication
Algorithm for modelling sequential data
fixed-size output vector, which is then processed by another recurrent network into an output. If the input is long, then the output vector would not be
Transformer_(deep_learning)
Machine learning technique
assigned to each word in a sentence. More generally, attention encodes vectors called token embeddings across a fixed-width sequence that can range from
Attention_(machine_learning)
Classification problem where multiple labels may be assigned to each instance
classification methods. kernel methods for vector output neural networks: BP-MLL is an adaptation of the popular back-propagation algorithm for multi-label
Multi-label_classification
have extended kernel methods to handle multiple outputs, as seen in multi-task learning. The mathematical framework for kernel methods typically involves
Bayesian interpretation of kernel regularization
Bayesian_interpretation_of_kernel_regularization
Subset of artificial intelligence
function, or kernel, that models how pairs of points relate to each other depending on their locations. Given a set of observed points, or input–output examples
Machine_learning
Type of machine learning model
the documents into vectors, then finding the documents with vectors (usually stored in a vector database) most similar to the vector of the query. The
Large_language_model
Type of feedforward neural network
type of feedforward neural network that learns features via filter (or kernel) optimization. This type of deep learning network has been applied to process
Convolutional_neural_network
Class of nonparametric methods
machine learning, the kernel embedding of distributions (also called the kernel mean or mean map) comprises a class of nonparametric methods in which a probability
Kernel embedding of distributions
Kernel_embedding_of_distributions
Algorithm for supervised learning of binary classifiers
algorithm for supervised learning of binary classifiers. A binary classifier is a function that can decide whether or not an input, represented by a vector of
Perceptron
notions of vector regularization to cases where the object to be learned is a matrix. The purpose of regularization is to enforce conditions, for example
Matrix_regularization
Software that manages computer hardware resources
system. Memory protection enables the kernel to limit a process' access to the computer's memory. Various methods of memory protection exist, including
Operating_system
Method of machine learning
requirements independent of training data size). For many formulations, for example nonlinear kernel methods, true online learning is not possible, though
Online_machine_learning
Automated recognition of patterns and regularities in data
These feature vectors can be seen as defining points in an appropriate multidimensional space, and methods for manipulating vectors in vector spaces can
Pattern_recognition
System call for device-specific input/output operations
number 1, and write() number 4. The system call vector is then used to find the desired kernel function for the request. In this way, conventional operating
Ioctl
Machine learning paradigm
1] interval). Methods that employ a distance function, such as nearest neighbor methods and support-vector machines with Gaussian kernels, are particularly
Supervised_learning
Machine learning calibration technique
Platt, John (1999). "Probabilistic outputs for support vector machines and comparisons to regularized likelihood methods". Advances in Large Margin Classifiers
Platt_scaling
Multivariate statistical technique
statistics, kernel principal component analysis (kernel PCA) is an extension of principal component analysis (PCA) using techniques of kernel methods. Using
Kernel principal component analysis
Kernel_principal_component_analysis
Concept in machine learning
is the width of the kernel. This definition can be rephrased as a matrix-vector product in terms of tensors that express the kernel, data and inverse transform
Tensor_(machine_learning)
Vectorizing features using a hash function
analogy to the kernel trick), is a fast and space-efficient way of vectorizing features, i.e. turning arbitrary features into indices in a vector or matrix
Feature_hashing
Statistical model
some desired kernel, and sample from that Gaussian. For solution of the multi-output prediction problem, Gaussian process regression for vector-valued function
Gaussian_process
Machine learning technique
x ( 0 ) {\displaystyle x^{(0)}} is the input vector, x ( 1 ) {\displaystyle x^{(1)}} is the output vector from the first module, etc. BatchNorm is a module
Normalization (machine learning)
Normalization_(machine_learning)
Mathematical function, in linear algebra
then we can conveniently use it to compute the vector output of f {\displaystyle f} for any vector in V {\displaystyle V} . To get M {\displaystyle
Linear_map
Paradigm in machine learning
or kernel for the data in an unsupervised first step. Then supervised learning proceeds from only the labeled examples. In this vein, some methods learn
Weak_supervision
Machine learning technique
layer Output layer PNN is often used in classification problems. When an input is present, the first layer computes the distance from the input vector to
Probabilistic_neural_network
Mapping involving integration between function spaces
two variables, that is called the kernel or nucleus of the transform. Some kernels have an associated inverse kernel K − 1 ( u , t ) {\displaystyle K^{-1}(u
Integral_transform
Branch of mathematics
called vectors, and elements of F are called scalars. The first operation, vector addition, takes any two vectors v and w and outputs a third vector v +
Linear_algebra
Categorization of data using statistics
perceptron algorithm Support vector machine – Set of methods for supervised statistical learning Linear discriminant analysis – Method used in statistics, pattern
Statistical_classification
Tree-based ensemble machine learning methods
forest and kernel methods. He pointed out that random forests trained using i.i.d. random vectors in the tree construction are equivalent to a kernel acting
Random_forest
Process of reducing the number of random variables under consideration
analysis using kernel function operator. The underlying theory is close to the support-vector machines (SVM) insofar as the GDA method provides a mapping
Dimensionality_reduction
Framework for machine learning
{\displaystyle X} to be the vector space of all possible inputs, and Y {\displaystyle Y} to be the vector space of all possible outputs. Statistical learning
Statistical_learning_theory
Memory unit used in neural networks
\odot } denotes the Hadamard product. Initially, for t = 0 {\displaystyle t=0} , the output vector is h 0 = 0 {\displaystyle h_{0}=0} . z t = σ ( W z
Gated_recurrent_unit
Computer vision framework
with a vector field kernel k {\displaystyle \mathbf {k} } where The vector field kernel k {\displaystyle \textstyle \mathbf {k} } has vectors that always
Gradient_vector_flow
Method of data analysis
space are a sequence of p {\displaystyle p} unit vectors, where the i {\displaystyle i} -th vector is the direction of a line that best fits the data
Principal_component_analysis
Concept in regression analysis mathematics
input vectors, and Y {\displaystyle Y} a n × 1 {\displaystyle n\times 1} vector where the entries are corresponding outputs. In terms of vectors, the kernel
Regularized_least_squares
sparsity and structured sparsity regularization methods seek to exploit the assumption that the output variable Y {\displaystyle Y} (i.e., response, or
Structured sparsity regularization
Structured_sparsity_regularization
Neural network technology
The size of the kernel is a hyperparameter that affects the network's behavior. For a 2D input x {\displaystyle x} and a 2D kernel w {\displaystyle w}
Convolutional_layer
Projection of data onto lower-dimensional manifolds
standard kernels. For example, it is known to perform poorly with these kernels on the Swiss roll manifold. However, one can view certain other methods that
Nonlinear dimensionality reduction
Nonlinear_dimensionality_reduction
Non-parametric classification method
neighbor methods, although user-perceived usefulness may be similar or higher in some cases. k-NN is a special case of a variable-bandwidth, kernel density
K-nearest_neighbors_algorithm
Covariance and correlation
{\displaystyle K_{g}=[k(g,T_{0}(g)),k(g,T_{1}(g)),\dots ,k(g,T_{N-1}(g))]} is a vector of kernel functions k ( ⋅ , ⋅ ) : C M × C M → R {\displaystyle k(\cdot ,\cdot
Cross-correlation
(required for HDD; SCPH-300xx to 500xx only) Emotion Engine (EE) includes an on-chip Serial I/O port (SIO) used internally by the EE's kernel to output debugging
PlayStation 2 technical specifications
PlayStation_2_technical_specifications
Classification of Artificial Neural Networks (ANNs)
an RBF leads naturally to kernel methods such as support vector machines (SVM) and Gaussian processes (the RBF is the kernel function). All three approaches
Types of artificial neural networks
Types_of_artificial_neural_networks
Optimization algorithm for artificial neural networks
{\displaystyle x} : input (vector of features) y {\displaystyle y} : target output For classification, output will be a vector of class probabilities (e
Backpropagation
Concepts from linear algebra
algebra, an eigenvector (/ˈaɪɡən-/ EYE-gən-) or characteristic vector is a (nonzero) vector that has its direction unchanged (or reversed) by a given linear
Eigenvalues_and_eigenvectors
Computer programming paradigm
the sources and kernel. For simplicity, there's a 1:1 mapping between input and output data but this does not need to be. Applied kernels can also be much
Stream_processing
Statistical classification in machine learning
and use. If the input feature vector to the classifier is a real vector x → {\displaystyle {\vec {x}}} , then the output score is y = f ( w → ⋅ x → ) =
Linear_classifier
Integral expressing the amount of overlap of one function as it is shifted over another
incompatibility (help). Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1
Convolution
Computer operating system
community-created "stop-gap" update for BeOS 5.0.3 in 2002, featuring open source replacement for some BeOS components. The kernel of NewOS, for x86, SuperH, and PowerPC
Haiku_(operating_system)
Recurrent neural network architecture
input/update gate's activation vector o t ∈ ( 0 , 1 ) h {\displaystyle o_{t}\in {(0,1)}^{h}} : output gate's activation vector h t ∈ ( − 1 , 1 ) h {\displaystyle
Long_short-term_memory
Supervised machine learning techniques
Structured prediction or structured output learning is an umbrella term for supervised machine learning techniques that involves predicting structured
Structured_prediction
Machine learning model for vision processing
<CLS> in the input side, and the corresponding output vector is used as the only input of the final output MLP head. The special token is an architectural
Vision_transformer
Proving or disproving the correctness of certain intended algorithms
formal methods of mathematics. Formal verification is a key incentive for formal specification of systems, and is at the core of formal methods. It represents
Formal_verification
Method of a dimension reduction
count sketches, rather than the mean. These properties allow use for explicit kernel methods, bilinear pooling in neural networks and is a cornerstone in
Count_sketch
Algorithm
random projection P . {\displaystyle P.} The vector x k − 1 − x k {\displaystyle x_{k-1}-x_{k}} is in the kernel of P k . {\displaystyle P_{k}.} It is orthogonal
Kaczmarz_method
Method for visualizing vector fields
the vector length) is used to determine the hue, while the grayscale LIC output determines the brightness. Different choices of convolution kernels and
Line_integral_convolution
Statistical method
widely used algorithm appropriate for the vector Y case. It estimates T as an orthonormal matrix. (Caution: the t vectors in the code below may not be normalized
Partial least squares regression
Partial_least_squares_regression
Study of uncertainty in the output of a mathematical model or system
functional outputs: Generally introduced for single-output codes, sensitivity analysis extends to cases where the output Y {\displaystyle Y} is a vector or function
Sensitivity_analysis
Smooth approximation of one-hot arg max
exponentials. The normalization ensures that the sum of the components of the output vector σ ( z ) {\displaystyle \sigma (\mathbf {z} )} is 1. The term "softmax"
Softmax_function
Type of artificial neural network
probability of an observation. Capsnets replace scalar-output feature detectors with vector-output capsules and max-pooling with routing-by-agreement. Because
Capsule_neural_network
Representation of a signal as a rectangular wave with varying duty cycle
the primary methods of controlling the output of solar panels to that which can be utilized by a battery. PWM is particularly suited for running inertial
Pulse-width_modulation
Machine learning method
commonly used machine learning methods. In particular, black box methods, such as multilayer perceptron and support vector machine, had good accuracy but
Logic_learning_machine
Computational model used in machine learning
minimize cost. Evolutionary methods, gene expression programming, simulated annealing, expectation–maximization, non-parametric methods and particle swarm optimization
Neural network (machine learning)
Neural_network_(machine_learning)
Technique for setting initial values of trainable parameters in a neural network
neural networks (CNNs) are called kernels and biases, and this article also describes these. We discuss the main methods of initialization in the context
Weight_initialization
Distribution over functions corresponding to an infinitely wide Bayesian neural network
Bibcode:2020arXiv200610540H. Cho, Youngmin; Saul, Lawrence K. (2009). "Kernel Methods for Deep Learning". Neural Information Processing Systems. 22: 342–350
Neural network Gaussian process
Neural_network_Gaussian_process
Matrix with a multiplicative inverse
gives the identity matrix. A matrix can be viewed as a rule for transforming vectors. For example, a real n × n {\displaystyle n\times n} matrix A {\displaystyle
Invertible_matrix
Models used to produce word embeddings
is a technique in natural language processing for obtaining vector representations of words. These vectors capture information about the meaning of the
Word2vec
Deep learning method
{\displaystyle *} is the Markov kernel convolution. A data-augmentation method is defined to be invertible if its Markov kernel K trans {\displaystyle K_{\text{trans}}}
Generative adversarial network
Generative_adversarial_network
Machine learning technique
{\displaystyle w} , which takes input x {\displaystyle x} and produces a vector of outputs ( w ( x ) 1 , . . . , w ( x ) n ) {\displaystyle (w(x)_{1},...,w(x)_{n})}
Mixture_of_experts
Machine learning problem
Platt, John (1999). "Probabilistic outputs for support vector machines and comparisons to regularized likelihood methods". Advances in Large Margin Classifiers
Probabilistic_classification
Medical imaging technique of the heart
of alternating the size of the kernel and search region to adapt to different resolution requirement. However, vector Doppler is less computationally
Doppler_echocardiography
Calculation of complex statistical distributions
) {\displaystyle (X_{n})} with transition kernel K ( x , y ) {\displaystyle K(x,y)} is φ-irreducible if, for every A ∈ B ( X ) {\displaystyle A\in {\mathcal
Markov_chain_Monte_Carlo
Technique for the generative modeling of a continuous probability distribution
to estimating z {\displaystyle z} . Therefore, let the network output a noise vector ϵ θ ( x t , t ) {\displaystyle \epsilon _{\theta }(x_{t},t)} , and
Diffusion_model
Machine learning technique
(inconsistently rewarding similar outputs) reward functions. RLHF was not the first successful method of using human feedback for reinforcement learning, but
Reinforcement learning from human feedback
Reinforcement_learning_from_human_feedback
Use of a GPU for computations typically assigned to CPUs
the kernels to be run on them.[dubious – discuss] For each element we can only read from the input, perform operations on it, and write to the output. It
General-purpose computing on graphics processing units
General-purpose_computing_on_graphics_processing_units
Tasks in machine learning
training data set often consists of pairs of an input vector (or scalar) and the corresponding output vector (or scalar), where the answer key is commonly denoted
Training, validation, and test data sets
Training,_validation,_and_test_data_sets
Paradigm in machine learning that uses no classification labels
network. In contrast to supervised methods' dominant use of backpropagation, unsupervised learning also employs other methods including: Hopfield learning rule
Unsupervised_learning
Deep learning library
machine-learning library written in C++ and CUDA, supporting methods including neural networks, support vector machines (SVM), hidden Markov models, etc. Around
PyTorch
Type of large language model
allocate more computation time analyzing the problem before generating an output, and are called reasoning models. In 2025, GPT-5 was released with a router
Generative pre-trained transformer
Generative_pre-trained_transformer
Piece of information about the content of an image
as the elements of one single vector, commonly referred to as a feature vector. The set of all possible feature vectors constitutes a feature space. A
Feature_(computer_vision)
Euclidean space without distance and angles
associated vector space. For every affine homomorphism E → F {\displaystyle E\to F} , the image is isomorphic to the quotient of E by the kernel of the associated
Affine_space
Machine learning methods using multiple input modalities
into vectors, and treating them like embedding vector of tokens in a standard transformer. Conformer and later Whisper follow the same pattern for speech
Multimodal_learning
Class of artificial neural network
input vector h t {\displaystyle h_{t}} : hidden layer vector s t {\displaystyle s_{t}} : "state" vector, y t {\displaystyle y_{t}} : output vector W {\displaystyle
Recurrent_neural_network
Reverse-engineering neural networks
Mechanistic interpretability employs causal methods to understand how internal model components influence outputs, often using formal tools from causality
Mechanistic_interpretability
Extracting features from raw data for machine learning
Feature explosion can be limited via techniques such as regularization, kernel methods, and feature selection. Automation of feature engineering is a research
Feature_engineering
Architectural motif in neural networks for aggregating information
called "filter size" (aka "kernel size") and "stride". Sometimes, it is necessary to use a different filter size and stride for horizontal and vertical directions
Pooling_layer
Process in machine learning and statistics
commonly used with Support Vector Machines to repeatedly construct a model and remove features with low weights. Embedded methods are a catch-all group of
Feature_selection
Research field that lies at the intersection of machine learning and computer security
gradient calculation that requires only the model's output predictions alone. By generating many random vectors in all directions, denoted as u b {\textstyle
Adversarial_machine_learning
is the output vector. Where applicable, the kernel function is denoted by k {\displaystyle k} , and the n × n {\displaystyle n\times n} kernel matrix
Regularization by spectral filtering
Regularization_by_spectral_filtering
Machine learning technique
its gradient. Many supervised learning problems involve an output variable y and a vector of input variables x, related to each other with some probabilistic
Gradient_boosting
System with self-optimizing transfer function
general idea behind Volterra LMS and Kernel LMS is to replace data samples by different nonlinear algebraic expressions. For Volterra LMS this expression is
Adaptive_filter
Type of reservoir computer
are for the synapses that connect the hidden neurons to output neurons. Thus, the error function is quadratic with respect to the parameter vector and
Echo_state_network
Loss function in machine learning
Yoram (2001). "On the algorithmic implementation of multiclass kernel-based vector machines" (PDF). Journal of Machine Learning Research. 2: 265–292
Hinge_loss
2019 text-generating language model
e. an interface that allowed input and provided output, not the source code itself) was allowed for selected press outlets on announcement. One commonly-cited
GPT-2
Matrix decomposition
left- and right-singular vectors of singular value 0 {\displaystyle 0} comprise all unit vectors in the cokernel and kernel, respectively, of M {\displaystyle
Singular_value_decomposition
KERNEL METHODS-FOR-VECTOR-OUTPUT
KERNEL METHODS-FOR-VECTOR-OUTPUT
Male
Scandinavian
Scandinavian form of German Werner, VERNER means "Warin warrior," i.e. "covered warrior."
Female
English
Variant spelling of English Muriel, MERIEL means "sea-bright."
Female
English
Variant form of English Keren, KERENA means "horn (of an animal)."Â
Male
Polish
Polish form of Greek Methodios, METODY means "method."
Male
English
Roman Latin name VICTOR means "conqueror."Â
Male
Polish
Polish form of Roman Latin Cornelius, KORNELI means "of a horn."
Male
Scandinavian
 Scandinavian form of Roman Latin Victor, VIKTOR means "conqueror." Compare with another form of Viktor.
Surname or Lastname
Americanized form of German Herrle.English and Irish
Americanized form of German Herrle.English and Irish : variant of Harrell.
Male
English
 Anglicized form of Scottish Gaelic Eachann, HECTOR means "brown horse." Compare with another form of Hector.
Male
Slovene
Slovene form of Greek Methodios, METOD means "method."
Male
Greek
(Μεθόδιος) Greek name derived from methodos, METHODIOS means "method."
Male
Slovene
Slovene form of Greek Bartholomaios, JERNEJ means "son of Talmai."
Male
Romanian
Romanian form of Greek Kornelios, CORNEL means "of a horn."
Male
English
Middle English form of Anglo-Saxon Cenhelm, KENELM means "keen protection."Â
Male
Portuguese
Galician-Portuguese form of Roman Latin Victor, VITOR means "conqueror."
Female
English
Medieval English contracted form of Roman Latin Petronel, PERONEL means "little rock."
Boy/Male
Spanish
Victor.
Surname or Lastname
Swedish
Swedish : ornamental name formed with the common surname suffix -ell. The first element is unexplained, possibly from a place-name.English, Scottish, and northern Irish : unexplained; possibly a respelling of Scottish Kerneil, a habitational name from Carneil in Carnock, Fife.
Male
Arthurian
, sir Hector de Maris; (defender).
Male
Portuguese
Portuguese form of Latin Hector, HEITOR means "defend; hold fast."
KERNEL METHODS-FOR-VECTOR-OUTPUT
KERNEL METHODS-FOR-VECTOR-OUTPUT
Girl/Female
Hindu
Peacock- modified
Boy/Male
Hindu, Indian, Telugu
Revolution
Boy/Male
Tamil
Wise
Girl/Female
Tamil
Lightening
Girl/Female
Greek
Holy one.
Surname or Lastname
English
English : variant spelling of the habitational name Aughton, from any of three places, in Lancashire, East and South Yorkshire, named Aughton, from Old English as Äc ‘oak’ + tÅ«n ‘settlement’.Possibly French : there are several places in France named Authon and it could be a habitational name from any of these.
Boy/Male
Teutonic American
Famous wolf.
Girl/Female
Hindu
Star
Girl/Female
Indian
Girl/Female
Indian
Gods gift
KERNEL METHODS-FOR-VECTOR-OUTPUT
KERNEL METHODS-FOR-VECTOR-OUTPUT
KERNEL METHODS-FOR-VECTOR-OUTPUT
KERNEL METHODS-FOR-VECTOR-OUTPUT
KERNEL METHODS-FOR-VECTOR-OUTPUT
n.
The ratio of one vector to another in length, no regard being had to the direction of the two vectors; -- so called because considered as a stretching factor in changing one vector into another. See Versor.
v. t.
To put or keep in a kennel.
n.
Same as Radius vector.
n.
An orderly procedure or process; regular manner of doing anything; hence, manner; way; mode; as, a method of teaching languages; a method of improving the mind.
n.
See Kimnel.
a.
Pertaining to a rector or a rectory; rectoral.
n.
One who observes method.
n.
Classification; a mode or system of classifying natural objects according to certain common characteristics; as, the method of Theophrastus; the method of Ray; the Linnaean method.
n.
The art and principles of method.
n.
The central, substantial or essential part of anything; the gist; the core; as, the kernel of an argument.
a.
Of or pertaining to the sect of Methodists; as, Methodist hymns; a Methodist elder.
a.
Full of kernels; resembling kernels; of the nature of kernels.
n.
See Weanel.
n.
Any species of the genus Cornus, as C. florida, the flowering cornel; C. stolonifera, the osier cornel; C. Canadensis, the dwarf cornel, or bunchberry.
n.
A single seed or grain; as, a kernel of corn.
imp. & p. p.
of Kernel
imp. & p. p.
of Kern
n.
A directed quantity, as a straight line, a force, or a velocity. Vectors are said to be equal when their directions are the same their magnitudes equal. Cf. Scalar.
n.
The essential part of a seed; all that is within the seed walls; the edible substance contained in the shell of a nut; hence, anything included in a shell, husk, or integument; as, the kernel of a nut. See Illust. of Endocarp.
v. i.
To harden or ripen into kernels; to produce kernels.