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Geometric property of a pair of sets of points in Euclidean geometry
In Euclidean geometry, linear separability is a property of two sets of points. This is most easily visualized in two dimensions (the Euclidean plane)
Linear_separability
Algorithm for supervised learning of binary classifiers
converge regardless of (prior) knowledge of linear separability of the data set. In the linearly separable case, it will solve the training problem – if
Perceptron
Topics referred to by the same term
Look up separable in Wiktionary, the free dictionary. Separability may refer to: Separable algebra, a generalization to associative algebras of the notion
Separability
Type of feedforward neural network
in layers, notable for being able to distinguish data that is not linearly separable. Modern neural networks are trained using backpropagation and are
Multilayer_perceptron
Statement in computational learning theory
it as counting function theorem. Let the number of homogeneously linearly separable sets of N {\displaystyle N} points in d {\displaystyle d} dimensions
Cover's_theorem
Type of vector space in math
separable Hilbert spaces are therefore isometrically isomorphic to the square-summable sequence space, ℓ 2 . {\displaystyle \ell ^{2}.} Separability was
Hilbert_space
Set of methods for supervised statistical learning
finite-dimensional space, it often happens that the sets to discriminate are not linearly separable in that space. For this reason, it was proposed that the original
Support_vector_machine
Kirchberger's theorem is a theorem in discrete geometry, on linear separability. The two-dimensional version of the theorem states that, if a finite set
Kirchberger's_theorem
Type of algebraic field extension
field is separable. It follows that most extensions that are considered in mathematics are separable. Nevertheless, the concept of separability is important
Separable_extension
Hypersurface used by a classification algorithm
is a hyperplane, then the classification problem is linear, and the classes are linearly separable. Decision boundaries are not always clear cut. That
Decision_boundary
Difficulties arising when analyzing data with many aspects ("dimensions")
dimension. Moreover, this linear functional can be selected in the form of the simplest linear Fisher discriminant. This separability theorem was proven for
Curse_of_dimensionality
Dimension of the column space of a matrix
In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. This corresponds to the maximal number
Rank_(linear_algebra)
Topological space with a dense countable subset
on a separable space whose image is a subset of a Hausdorff space is determined by its values on the countable dense subset. Contrast separability with
Separable_space
Convention of symbols representing language
diacritics can be characterized as less linear than those without. In the initial historical distinction, linear writing systems (e.g. the Phoenician alphabet)
Writing_system
Method used in statistics, pattern recognition, and other fields
concentration inequality for product probability spaces). Data separability by classical linear discriminants simplifies the problem of error correction for
Linear_discriminant_analysis
Overview of and topical guide to machine learning
support vector machine Leslie P. Kaelbling Linear genetic programming Linear predictor function Linear separability Linkurious Lior Ron (business executive)
Outline_of_machine_learning
Type of artificial neural network
three layers, notable for being able to distinguish data that is not linearly separable. Examples of other feedforward networks include convolutional neural
Feedforward_neural_network
Grouping a set of objects by similarity
clusters in a data set Parallel coordinates Structured data analysis Linear separability Driver and Kroeber (1932). "Quantitative Expression of Cultural Relationships"
Cluster_analysis
Density-based data clustering algorithm
implemented using a database index for better performance, or using a slow linear scan: RangeQuery(DB, distFunc, Q, eps) { Neighbors N := empty list for each
DBSCAN
Technique for solving differential equations
differential equation for the unknown f ( x ) {\displaystyle f(x)} is separable if it can be written in the form d d x f ( x ) = g ( x ) h ( f ( x ) )
Separation_of_variables
modifiable layers learned internal representations to classify non-linearly separable pattern classes. Subsequent developments in hardware and hyperparameter
History of artificial neural networks
History_of_artificial_neural_networks
Neuromorphic data-processing model
nanowire networks, diffractive media) to perform computations. Non-Linear Separability: Unlike traditional perceptrons, which fail on problems like the
Receptron
About polynomials in several variables
MR 1336641. Wang, Stuart Sui-Sheng (August 1980), "A Jacobian criterion for separability", Journal of Algebra, 65 (2): 453–494, doi:10.1016/0021-8693(80)90233-1
Jacobian_conjecture
Optimization algorithm for artificial neural networks
with 2 learnable layers, and it learned to classify patterns not linearly separable. Modern backpropagation was first published by Seppo Linnainmaa as
Backpropagation
Vector quantization algorithm minimizing the sum of squared deviations
and kernel k-means, which uses kernel functions to identify non-linearly separable clusters. The most common algorithm uses an iterative refinement technique
K-means_clustering
Topics referred to by the same term
and as such are supplementary Linear separability, the property of two sets of points, where they are linearly separable if there exists at least one line
Linear_(disambiguation)
Mathematical function conceived as a crude model
0 {\displaystyle y(t+1)=0} otherwise. It can be used to represent linearly separable boolean functions (for example, AND, OR, NOR) but not, for example
Artificial_neuron
Fourteenth letter in the Greek alphabet
mapping. Used in Support Vector machines in cases where the data is not linearly separable. Used in Microelectronics to represent the distance from a p-n junction
Xi_(letter)
Iterative method for finding a linear decision boundary
non-separability. The convergence rate depends on the choice of the learning rate parameter ρ {\displaystyle \rho } and the degree of linear separability
Ho–Kashyap_algorithm
Multivariate statistical technique
(PCA) using techniques of kernel methods. Using a kernel, the originally linear operations of PCA are performed in a reproducing kernel Hilbert space. Recall
Kernel principal component analysis
Kernel_principal_component_analysis
True when either but not both inputs are true
modeling the XOR function requires a second layer because XOR is not a linearly separable function. Similarly, XOR can be used in generating entropy pools for
Exclusive_or
Convolutional neural network structure
problem in another paper, and showed that even though the problem is linearly separable, single-layer networks exhibited poor generalization capabilities
LeNet
Computational model used in machine learning
models used single-layer perceptrons, which were restricted to solving linearly separable problems. These limitations were highlighted in the book Perceptrons
Neural network (machine learning)
Neural_network_(machine_learning)
state must satisfy in order for it to be separable. In other words, the reduction criterion is a separability criterion. It was first proved and independently
Reduction_criterion
Normed vector space that is complete
normed spaces are separable Banach spaces and any two Banach spaces of the same finite dimension are linearly homeomorphic. Every separable infinite–dimensional
Banach_space
Mathematical expression for linear operators
specifically linear algebra, the Jordan–Chevalley decomposition, named after Camille Jordan and Claude Chevalley, expresses a linear operator in a unique
Jordan–Chevalley decomposition
Jordan–Chevalley_decomposition
Book by Marvin Minsky and Seymour Papert
names included linearly separable logic, linear-input logic, threshold logic, majority logic, and voting logic. Hardware for realizing linear threshold logic
Perceptrons_(book)
In mathematics, vector space of linear forms
corresponding dual vector space (or just dual space for short) consisting of all linear forms on V , {\displaystyle V,} together with the vector space structure
Dual_space
Numerical method for differential equations
In numerical analysis, the local linearization (LL) method is a general strategy for designing numerical integrators for differential equations based
Local_linearization_method
Principle in statistical learning theory
such as linear classifiers. Nevertheless, it can be solved efficiently when the minimal empirical risk is zero, i.e., data is linearly separable.[citation
Empirical_risk_minimization
Type of differential equation
PDE is called linear if it is linear in the unknown and its derivatives. For example, for a function u of x and y, a second order linear PDE is of the
Partial_differential_equation
Area of mathematics
isomorphic to ℓ 2 ( ℵ 0 ) {\displaystyle \ell ^{\,2}(\aleph _{0})\,} . Separability being important for applications, functional analysis of Hilbert spaces
Functional_analysis
Artificial neural network algorithm
converges to the correct classification if: the training data is linearly separable* η {\displaystyle \eta } is sufficiently small (though smaller η {\displaystyle
Learning_rule
Method of data analysis
mass of two or more classes. The linear discriminant analysis is an alternative which is optimized for class separability. Some properties of PCA include:[page needed]
Principal_component_analysis
Linear optimal control technique
In control theory, the linear–quadratic–Gaussian (LQG) control problem is one of the most fundamental optimal control problems, and it can also be operated
Linear–quadratic–Gaussian control
Linear–quadratic–Gaussian_control
In mathematics, linear maps form an important class of "simple" functions which preserve the algebraic structure of linear spaces and are often used as
Discontinuous_linear_map
Construct in quantum information theory
entangled state from separable ones. Entanglement witnesses can be linear or nonlinear functionals of the density matrix. If linear, then they can also
Entanglement_witness
Form of decision making in machine learning
using the concept of piecewise-linear separability of datasets to categorize data. Two datasets are linearly separable if their convex hulls do not intersect
Multi-surface_method
Relation among continuous functions
boundedness principle states that a pointwise bounded family of continuous linear operators between Banach spaces is equicontinuous. Let X and Y be two metric
Equicontinuity
Soviet and Ukrainian mathematician
recognition he developed a theory of linear discriminant rules where he investigated the problems of linear separability of any number of sets in n-dimensional
Yuri_Petunin
Cluster analysis problem
space. It is believed that the data become more linearly separable in the feature space, and hence, linear algorithms can be applied on the data with a higher
Determining the number of clusters in a data set
Determining_the_number_of_clusters_in_a_data_set
Vector space with a notion of nearness
it is Hausdorff; importantly, "separated" does not mean separable. The topological and linear algebraic structures can be tied together even more closely
Topological_vector_space
Class of economic models
the assumption of weak separability, under which groups of goods or inputs can be treated as composite aggregates. Separability allows a high-dimensional
Computable general equilibrium
Computable_general_equilibrium
convolution of Lucas numbers 1881 = tricapped prism number 1882 = number of linearly separable Boolean functions in 4 variables 1883 = number of conjugacy classes
1000_(number)
Type of ordinary differential equation
derivatives. In the case of linear differential equations, this means that there are no constant terms. The solutions of any linear ordinary differential equation
Homogeneous differential equation
Homogeneous_differential_equation
Class of error-correcting code
In coding theory, a linear code is an error-correcting code for which any linear combination of codewords is also a codeword. Linear codes are traditionally
Linear_code
Mathematical concept
certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used
Weak_topology
Mathematical concept
In constrained least squares one solves a linear least squares problem with an additional constraint on the solution. This means, the unconstrained equation
Constrained_least_squares
Notation for quantum states
Bra–ket notation or Dirac notation is a mathematical notation for linear algebra and linear operators on complex vector spaces together with their dual spaces
Bra–ket_notation
Represents a function with diffusion properties useful in cryptography
An MDS matrix (maximum distance separable) is a matrix representing a function with certain diffusion properties that have useful applications in cryptography
MDS_matrix
The real numbers or their cardinality
y ∈ C such that x < y, then there exists z ∈ S such that x < z < y. (separability axiom) C has no first element and no last element. (Unboundedness axiom)
Continuum_(set_theory)
Ring that is also a vector space or a module
of A, sometimes called the bidimension of A, measures the failure of separability. Let A be a finite-dimensional algebra over a field k. Then A is an Artinian
Associative_algebra
calculations changes with the system size. Size consistency (or strict separability) is a property that guarantees the consistency of the energy behaviour
Size consistency and size extensivity
Size_consistency_and_size_extensivity
condition that a state must satisfy in order to be separable. In other words, it is a separability criterion. Consider a quantum mechanical system composed
Range_criterion
Result on the topology of operators on an infinite-dimensional, complex Hilbert space
mathematician Nicolaas Kuiper, for the case of a separable Hilbert space; the restriction of separability was later lifted. The same result, but for the
Kuiper's_theorem
Upper bound in coding theory
regarding Komamiya (1953). Linear block codes that achieve equality in the Singleton bound are called MDS (maximum distance separable) codes. Examples of such
Singleton_bound
Dual pair of vector spaces
b(x,\,\cdot \,)} is a linear functional on Y {\displaystyle Y} and every b ( ⋅ , y ) {\displaystyle b(\,\cdot \,,y)} is a linear functional on X {\displaystyle
Dual_system
Vector space with generalized dot product
sesquilinear forms with linearity in the second argument rather than the first. Then the first argument becomes conjugate linear, rather than the second
Inner_product_space
analysis, spaces of linear maps between two vector spaces can be endowed with a variety of topologies. Studying space of linear maps and these topologies
Topologies on spaces of linear maps
Topologies_on_spaces_of_linear_maps
Type of continuous linear operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator that behaves, in several important respects, like a finite-dimensional
Compact_operator
List of concepts in artificial intelligence
in layers, notable for being able to distinguish data that is not linearly separable. multi-swarm optimization A variant of particle swarm optimization
Glossary of artificial intelligence
Glossary_of_artificial_intelligence
Differential equation containing derivatives with respect to only one variable
the modeled process is random. A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function
Ordinary differential equation
Ordinary_differential_equation
Statistics dataset
The use of multiple measurements in taxonomic problems as an example of linear discriminant analysis. It is sometimes called Anderson's Iris data set because
Iris_flower_data_set
trivially related to the linear entropy S L {\displaystyle S_{L}\,} of a state by γ = 1 − S L . {\displaystyle \gamma =1-S_{L}\,.} The linear entropy is a lower
Purity_(quantum_mechanics)
Robotics problem on coordinating two parts of a robot
Rotation is represented using quaternions, allowing for a linear solution to be found. While separable methods are useful, any error in the estimation for the
Hand–eye_calibration_problem
Mathematical operation in signal processing
the separability of the filter, this will require approximately X Y J K {\displaystyle XYJK} multiplications and additions. If the separability of the
Multidimensional discrete convolution
Multidimensional_discrete_convolution
Solid-state welding process
(RFW) being the oldest of the methods. W. Richter patented the method of linear friction welding (LFW) process in 1924 in England and 1929 in the Weimar
Friction_welding
Theorems connecting continuity to closure of graphs
connecting the continuity of a linear operator to a topological property of their graph. Precisely, the theorem states that a linear operator between two Banach
Closed graph theorem (functional analysis)
Closed_graph_theorem_(functional_analysis)
{\displaystyle f:X\to B} is said to be weakly measurable if, for every continuous linear functional g : B → K , {\displaystyle g:B\to \mathbb {K} ,} the function
Weakly_measurable_function
Theorem on extension of bounded linear functionals
Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole
Hahn–Banach_theorem
a real, separable reflexive Banach space X into its continuous dual space X∗ is automatically surjective. That is, for each continuous linear functional
Browder–Minty_theorem
Field extension that is not algebraic
the other hand. This analogy can be made more formal, by observing that linear independence in vector spaces and algebraic independence in field extensions
Transcendental_extension
Matrix decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix into a rotation, followed by a scaling, followed
Singular_value_decomposition
Square matrix constructed from a monic polynomial
In linear algebra, the Frobenius companion matrix of the monic polynomial p ( x ) = c 0 + c 1 x + ⋯ + c n − 1 x n − 1 + x n {\displaystyle p(x)=c_{0}+c_{1}x+\cdots
Companion_matrix
Statistical models used for static panel data
inherently nonlinear, the use of the linear index and the exponential link function lead to multiplicative separability, more specifically E[yit ∨ xi1...
Fixed-effect_Poisson_model
Graph whose induced subgraphs preserve distance
discrete mathematics, a distance-hereditary graph (also called a completely separable graph) is a graph in which the distances in any connected induced subgraph
Distance-hereditary_graph
Mathematical function
K-linear transformation of this vector space into itself. The trace, TrL/K(α), is defined as the trace (in the linear algebra sense) of this linear transformation
Field_trace
Parallel computing paradigm
universal CNN processors. The original CNN processors can only perform linearly separable Boolean functions. By translating functions from digital logic or
Cellular_neural_network
Compact operator for which a finite trace can be defined
mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite
Trace_class
Topological complex vector space
adjoint. A particular case is that of a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties: A is
C*-algebra
Consumer demand model
using product category spending and brand prices alone. Assuming weak separability of consumer preferences, the optimal allocation of expenditure among
Almost_ideal_demand_system
Neural network technology
bandpass receptive fields, which could be recreated by fitting sparse linear codes for natural scenes. This was later found to also occur in the lowest-level
Convolutional_layer
Any entity that can be measured
input. In quantum mechanics, observables correspond to linear self-adjoint operators on a separable complex Hilbert space representing the quantum state
Observable
non-separable Banach spaces. A slightly modified version also commonly referred to as the Sobczyk theorem, deals with the extension of a bounded linear operator
Sobczyk's_theorem
Collection of statistical models
most common of which uses a linear model that relates the response to the treatments and blocks. Note that the model is linear in parameters but may be nonlinear
Analysis_of_variance
A separable partial differential equation can be broken into a set of equations of lower dimensionality (fewer independent variables) by a method of separation
Separable partial differential equation
Separable_partial_differential_equation
Function with a multiplicative scaling behaviour
is any real number. Let V and W be two vector spaces over a field F. A linear cone in V is a subset C of V such that s x ∈ C {\displaystyle sx\in C} for
Homogeneous_function
Property of materials with both viscous and elastic characteristics under deformation
suffix -elasticity. Linear viscoelasticity is behavior in which the function is separable in both creep response and load. All linear viscoelastic models
Viscoelasticity
Type of reservoir computer
control over the process. If a reservoir has fading memory and input separability, with help of a readout, it can be proven the liquid state machine is
Liquid_state_machine
Family of computer vision models designed for efficient inference on mobile devices
MobileNetV2 was published in March 2019. It uses inverted residual layers and linear bottlenecks. Inverted residuals modify the traditional residual block structure
MobileNet
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