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In the field of mathematics known as convex analysis, the indicator function of a set is a convex function that indicates the membership (or non-membership)
Indicator function (convex analysis)
Indicator_function_(convex_analysis)
Mathematics of convex functions and sets
Convex analysis is the branch of mathematics that studies convex sets, convex functions, and their applications to optimization, functional analysis, variational
Convex_analysis
Notion from the theory of entire functions
mathematics known as complex analysis, the indicator function of an entire function indicates the rate of growth of the function in different directions.
Indicator function (complex analysis)
Indicator_function_(complex_analysis)
Mathematical function characterizing set membership
characteristic function in convex analysis, which is defined as if using the reciprocal of the standard definition of the indicator function. A related concept
Indicator_function
Type of mathematical function
In convex analysis, a non-negative function f: Rn → R+ is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it
Logarithmically concave function
Logarithmically_concave_function
Smallest convex set containing a given set
In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined
Convex_hull
Mathematical function having a characteristic S-shaped curve or sigmoid curve
asymptotes as x → ± ∞ {\displaystyle x\rightarrow \pm \infty } . A sigmoid function is convex for values less than a particular point, and it is concave for values
Sigmoid_function
Objects that generalize functions
Distributions (or generalized functions) are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible
Distribution (mathematical analysis)
Distribution_(mathematical_analysis)
Generalized function whose value is zero everywhere except at zero
In mathematical analysis, the Dirac delta function (or δ {\displaystyle {\boldsymbol {\delta }}} distribution), also known as the unit impulse, is a generalized
Dirac_delta_function
Study of uncertainty in the output of a mathematical model or system
development of better models. The object of study for sensitivity analysis is a function f {\displaystyle f} , (called "mathematical model" or "programming
Sensitivity_analysis
Theorem in convex analysis
&{\text{otherwise}}\end{cases}}} be the indicator function for a cone C . {\displaystyle C.} Then the convex conjugate, f ∗ ( x ∗ ) = δ ( x ∗ | C ∘ )
Bipolar_theorem
Form of projection
simultaneously several convex constraints. Let f i {\displaystyle f_{i}} be the indicator function of non-empty closed convex set C i {\displaystyle C_{i}}
Proximal_gradient_method
Sums vector sets A and B by adding each vector in A to each vector in B
Minkowski inequality, the function hK+pL is again positive homogeneous and convex and hence the support function of a compact convex set. This definition is
Minkowski_addition
Property of functions which is weaker than continuity
in convex analysis. Given a convex (extended real) function, the epigraph might not be closed. But the lower semicontinuous hull of a convex function is
Semi-continuity
locally convex spaces ( X , X ∗ ) {\displaystyle \left(X,X^{*}\right)} and ( Y , Y ∗ ) {\displaystyle \left(Y,Y^{*}\right)} . Then given the function f :
Duality_gap
Function spaces generalizing finite-dimensional p norm spaces
that the function F : [ 0 , ∞ ) → R {\displaystyle F:[0,\infty )\to \mathbb {R} } defined by F ( t ) = t p {\displaystyle F(t)=t^{p}} is convex, which by
Lp_space
Concept in machine learning
relatively tight, convex upper bound on the 0–1 indicator function. Specifically, the hinge loss equals the 0–1 indicator function when sgn ( f ( x
Loss functions for classification
Loss_functions_for_classification
Property of a mathematical function
\partial _{+}f(0)=1.} If a function is semi-differentiable at a point a, it implies that it is continuous at a. The indicator function 1[0,∞) is right differentiable
Semi-differentiability
Statistical method
interpretations including in terms of geometry, Bayesian statistics and convex analysis. The LASSO is closely related to basis pursuit denoising. Lasso was
Lasso_(statistics)
Method of data analysis
factor analysis will give erroneous results. It has been asserted that the relaxed solution of k-means clustering, specified by the cluster indicators, is
Principal_component_analysis
Diagnostic plot of binary classifier ability
t 1 ) ] {\textstyle {\textbf {1}}[f(t_{0})<f(t_{1})]} denotes an indicator function which returns 1 if f ( t 0 ) < f ( t 1 ) {\displaystyle f(t_{0})<f(t_{1})}
Receiver operating characteristic
Receiver_operating_characteristic
Primal-Dual algorithm optimization for convex problems
designed to efficiently solve convex optimization problems that involve the minimization of a non-smooth cost function composed of a data fidelity term
Chambolle–Pock_algorithm
then defined by approximating the function by linear combinations of indicator functions. inverse The inverse function theorem gives a necessary and sufficient
Glossary of real and complex analysis
Glossary_of_real_and_complex_analysis
Mathematical concept
-Lipschitz gradient. When every f i {\displaystyle f_{i}} is convex the function is convex, and an ε {\displaystyle \varepsilon } -optimal point is reachable
Multi-objective_optimization
Method for finding stationary points of a function
the second derivative is positive, the quadratic approximation is a convex function of t {\displaystyle t} , and its minimum can be found by setting the
Newton's method in optimization
Newton's_method_in_optimization
Constructions in nonsmooth analysis
{\displaystyle C\subseteq H} be a non-empty, closed, convex set and consider its convex indicator function δ C ( x ) = { 0 x ∈ C + ∞ x ∈ H ∖ C . {\displaystyle
Normal cone (variational analysis)
Normal_cone_(variational_analysis)
Game where groups of players may enforce cooperative behaviour
are reversed, so that we say the cost game is convex if the characteristic function is submodular. Convex cooperative games have many nice properties:
Cooperative_game_theory
Integral expressing the amount of overlap of one function as it is shifted over another
functional analysis), convolution is a mathematical operation on two functions f {\displaystyle f} and g {\displaystyle g} that produces a third function f ∗
Convolution
Optical device which transmits and refracts light
the Latin name of the lentil (a seed of a lentil plant), because a double-convex lens is lentil-shaped. The lentil also gives its name to a geometric figure
Lens
Approximate power law relating animal metabolic rate to mass
surface area and scale with power 2⁄3. Basal metabolic rate is then the convex combination of these two effects: if the proportion of useful work is f
Kleiber's_law
locally convex spaces ( X , X ∗ ) {\displaystyle \left(X,X^{*}\right)} and ( Y , Y ∗ ) {\displaystyle \left(Y,Y^{*}\right)} . Then given the function f :
Perturbation_function
Measure that is 1 if and only if a specified element is in the set
the extreme points of the convex set of probability measures on X. The name is a back-formation from the Dirac delta function; considered as a Schwartz
Dirac_measure
Fundamental theorem in probability theory and statistics
density is a function constant inside a given convex body and vanishing outside; it corresponds to the uniform distribution on the convex body, which explains
Central_limit_theorem
Statistical distribution for dependence between random variables
\rightarrow [0,\infty )\ } is a continuous, strictly decreasing and convex function such that ψ ( 1 ; θ ) = 0 , {\displaystyle \ \psi (1;\theta )=0\
Copula_(statistics)
Fundamental construction of differential calculus
and subgradient are generalizations of the derivative to convex functions used in convex analysis. In commutative algebra, Kähler differentials are universal
Generalizations of the derivative
Generalizations_of_the_derivative
Variable representing a random phenomenon
probability density function of a CURV X ∼ U [ a , b ] {\displaystyle X\sim \operatorname {U} [a,b]} is given by the indicator function of its interval
Random_variable
Framework for machine learning
statistics and functional analysis. Statistical learning theory deals with the statistical inference problem of finding a predictive function based on data. Statistical
Statistical_learning_theory
Generalization of finite measure to Banach spaces
{\displaystyle \mu (A)=\chi _{A}} where χ A {\displaystyle \chi _{A}} is the indicator function of A . {\displaystyle A.} Depending on where μ {\displaystyle \mu
Vector_measure
Mathematical function for the probability a given outcome occurs in an experiment
cumulative distribution function admits a decomposition as the convex sum of the three according cumulative distribution functions. A discrete probability
Probability_distribution
Theorem in geometry
be much larger than r(x). Sometimes in the context of a convex geometry, the radius function has a different meaning, here we follow the terminology of
Brunn–Minkowski_theorem
In Euclidean space, a measure of that set's "size"
Potential theory – Harmonic functions as solutions to Laplace's equation Choquet theory – Area of functional analysis and convex analysis Brélot, Marcel (1967)
Capacity_of_a_set
calculus Ampheck Analysis of Boolean functions Balanced Boolean function Bent function Boolean algebras canonically defined Boolean function Boolean matrix
List of Boolean algebra topics
List_of_Boolean_algebra_topics
Evolutionary algorithm
derivative-free methods for numerical optimization of non-linear or non-convex continuous optimization problems. They belong to the class of evolutionary
CMA-ES
Topological vector spaces
In mathematical analysis, the spaces of test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application
Spaces of test functions and distributions
Spaces_of_test_functions_and_distributions
Mathematical function often applied to matrices
or monotone vector fields in nonlinear analysis, and strong ellipticity in differential operators on function spaces, subject to specific boundary conditions
Logarithmic_norm
Space which has no holes through it
transformed into each other. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected
Simply_connected_space
{\displaystyle w} is the indicator function of a given shape, this is the same as the axiality. Lassak, Marek (2002), "Approximation of convex bodies by axially
Axiality_(geometry)
Nonparametric estimate of cumulative hazard
idea of the hazard rate shape. A concave shape is an indicator for infant mortality while a convex shape indicates wear out mortality. It can be used for
Nelson–Aalen_estimator
Part of the abdomen between the rib cage and hips
circumference may be a better indicator of overall health. Some research suggests waist circumference can be predicted from brain function, therefore capturing
Waist
the function is convex. Well-known examples of convex functions include the quadratic function x 2 {\displaystyle x^{2}} and the exponential function e
Glossary_of_calculus
Sequential model-based optimization of expensive black-box functions
sequential design is also used in reliability analysis to estimate a probability of failure. For a limit-state function f {\displaystyle f} , random input X {\displaystyle
Bayesian_optimization
Type of mathematical measure
continuous real-valued functions with compact support on X form a vector space K(X) = Cc(X), which can be given a natural locally convex topology. Indeed,
Radon_measure
is a matrix regularization penalty. The function R ( W ) {\displaystyle R(W)} is typically chosen to be convex and is often selected to enforce sparsity
Matrix_regularization
Motion of a curve based on its curvature
evolution. If the curve is non-convex, its total absolute curvature decreases monotonically, until it becomes convex. Once convex, the isoperimetric ratio of
Curve-shortening_flow
correlation Multiple correspondence analysis Multiple discriminant analysis Multiple-indicator kriging Multiple Indicator Cluster Survey Multiple of the median
List_of_statistics_articles
Concept in statistics
the indicator functions. In cases where the Cesàro limit does not exist this function can actually be defined as the Banach limit of the indicator functions
Exchangeable_random_variables
Vector quantization algorithm minimizing the sum of squared deviations
computes the distance from the datum to each centroid, or simply an indicator function for the nearest centroid, or some smooth transformation of the distance
K-means_clustering
System of logic in computer science
processing Video processing and computer vision Failure Mode And Effect Analysis Function approximation and estimation Control systems Freeware MATLAB implementations
Type-2_fuzzy_sets_and_systems
Class of ecological models
surviving species, and the right-hand side resembles the expression for a convex cone with apex R ⋆ {\displaystyle \mathbf {R} ^{\star }} and whose generating
Consumer-resource_model
Autonomic nervous system control of sweat gland activity
TST% acts as an indicator of the severity of neurologic impairment. When used in conjunction with postganglionic sudomotor function testing, such as
Sudomotor
Solving multiple machine learning tasks at the same time
cardinality of group r, and I {\displaystyle \mathbb {I} } is the indicator function). For example, people in different political parties (groups) might
Multi-task_learning
Left-invariant (or right-invariant) measure on locally compact topological group
functions, though it only works for compact groups. The idea is that given a function f {\displaystyle f} on a compact group, one can find a convex combination
Haar_measure
Summary statistic of variability
\left[\varphi (Y)\right]} , where φ {\displaystyle \varphi } is a convex function, this implies for Y = | X − μ | {\displaystyle Y=\vert X-\mu \vert
Average_absolute_deviation
Differential equation exhibiting high rate of dissipation
numerical analysis. For example, in optimization, gradient methods have similar limitations. Using the steepest descent method to minimize a convex functional
Stiff_equation
Ecological relationship between species' range sizes and abundance
definition, see Fig. 2 and ALA). The EOO can best be thought of as the minimum convex polygon encompassing all known normal occurrences of a particular species
Occupancy–abundance relationship
Occupancy–abundance_relationship
Mathematical concept
family of functions being dominated in L 1 {\displaystyle L^{1}} which is central in dominated convergence. Several textbooks on real analysis and measure
Uniform_integrability
Generalization of mass, length, area and volume
The Dirac measure δa (cf. Dirac delta function) is given by δa(S) = χS(a), where χS is the indicator function of S . {\displaystyle S.} The measure of
Measure_(mathematics)
Ecological concept
receive less or more sun than the opposite side (depending on hemisphere and convex or concave relief). Other factors such as light intensity, soil composition
Edge_effects
Euler measure of a polyhedral set equals the Euler integral of its indicator function. By induction, it is easy to show that independent of dimension, the
Euler_measure
Process of calculating the causal factors that produced a set of observations
points. The forward map being nonlinear, the data misfit function is likely to be non-convex, making local minimization techniques inefficient. Several
Inverse_problem
Inequality between integrals in Lp spaces
gave another proof as part of a work developing the concept of convex and concave functions and introducing Jensen's inequality, which was in turn named
Hölder's_inequality
Average value of a random variable
field of mathematical analysis and its applications to probability theory. Jensen's inequality: Let f: R → R be a convex function and X a random variable
Expected_value
Archaeological site in South Africa
1007/s12520-020-01176-1 Deacon, H. J., and Janette Deacon. "The Hafting, Function and Distribution of Small Convex Scrapers with an Example from Boomplaas Cave." The South
Boomplaas_Cave
Model in probability theory
_{F}\right]=0,} where χ F {\displaystyle \chi _{F}} denotes the indicator function of the event F {\displaystyle F} . In Grimmett and Stirzaker's Probability
Martingale (probability theory)
Martingale_(probability_theory)
Algorithms for matrix decomposition
Data Analysis. 52 (8): 3913–3927. doi:10.1016/j.csda.2008.01.011. Archived from the original (PDF) on 2016-03-04. C Ding, T Li, MI Jordan, Convex and semi-nonnegative
Non-negative matrix factorization
Non-negative_matrix_factorization
Sets whose elements have degrees of membership
membership function valued in the real unit interval [ 0 , 1 ] {\displaystyle [0,1]} . Fuzzy sets generalize classical sets, since the indicator functions (aka
Fuzzy_set
Projection of data onto lower-dimensional manifolds
Scaling performs multidimensional scaling in local regions, and then uses convex optimization to fit all the pieces together. Nonlinear PCA (NLPCA) uses
Nonlinear dimensionality reduction
Nonlinear_dimensionality_reduction
Discipline within engineering design
method of moments Statistical interference quality function deployment Failure mode and effects analysis Interval finite element Stochastic modeling First-order
Probabilistic_design
Theory that attempts to blend economics and ergodic theory
agent specified by a convex utility function is more risk-seeking than an expected wealth maximizer, and a concave utility function implies greater risk
Ergodicity_economics
Method of calculating an investment's rate of return
}}n\geq 1.\,} In this case the NPV of the payment stream is a convex, strictly decreasing function of interest rate. There is always a single unique solution
Internal_rate_of_return
Genus of theropod dinosaurs
surfaces. Like many tyrannosauroids, the lacrimal of Nanotyrannus has a convex cornual process, a structure which would have supported a keratinous structure
Nanotyrannus
Non-closing over-pressure relief device
forward-acting disc. The dome is inverted and the pressure is now loaded on the convex side of the disc. Once the reversal threshold is met, the dome will collapse
Rupture_disc
Hematoma usually associated with traumatic brain injury
crescent-shaped, with a concave surface away from the skull. However, they can have a convex appearance, especially in the early stages of bleeding. This may cause difficulty
Subdural_hematoma
Laboratory equipment to measure liquid volume
through molecular forces. This forces the liquid surface to develop either a convex or concave shape, depending on the type of the liquid in the cylinder. Reading
Graduated_cylinder
Research topic in computational geometry
reconstruction strategy can be employed. This method states that the indicator function, a function that determines which points in space belong to the surface
Geometry_processing
Statistical distance measure
Mahalanobis distance has also been used in ecological niche modelling, as the convex elliptical shape of the distances relates well to the concept of the fundamental
Mahalanobis_distance
Expected value of a random variable given that certain conditions are known to occur
everywhere. Any simple function is a finite linear combination of indicator functions. By linearity the above property holds for simple functions: if X n {\displaystyle
Conditional_expectation
Geometric space with four dimensions
there are 6 convex regular 4-polytopes, the analogs of the Platonic solids. Relaxing the conditions for regularity generates a further 58 convex uniform 4-polytopes
Four-dimensional_space
Probability distribution
applied in acoustic analysis to assess damage to gears, as the kurtosis of the beta distribution has been reported to be a good indicator of the condition
Beta_distribution
Statistical amount
{y}}_{i}} . As above, we can add a tick mark if multiplying by the indicator function. I.e.: y ˇ i ′ = I i y ˇ i = I i y i π i {\displaystyle {\check
Weighted_arithmetic_mean
Computation machine that uses continuously varying data technology
numbers; a board, a set of nails, and a rubber band as a model of finding the convex hull of a set of points; and strings tied together as a model of finding
Analog_computer
Field of economics and game theory
utility-sharing games with convex utility functions. Mirrlees (1971) introduces a setting in which the transfer function t() is easy to solve for. Due
Mechanism_design
Approach to optimizing robustness to failure
info-gap robustness function is local, and has restricted quantitative value in some cases. However, a major purpose of decision analysis is to provide focus
Info-gap_decision_theory
Stone tool
sexual selection as fitness indicators. Kohn in his book As We Know It wrote that the hand axe is "a highly visible indicator of fitness, and so becomes
Hand_axe
probability / Bay Sunrise problem The Doctrine of Chances B-convex space Conditional event algebra Error function Goodman–Nguyen–van Fraassen algebra List of mathematical
Catalog of articles in probability theory
Catalog_of_articles_in_probability_theory
Genus of large, heavily armored mammals
shield that had a smoothly convex exterior surface without ornamentation. Each osteoderm has a rugose and slightly convex dorsal surface, with ornamentation
Glyptodon
Field of knowledge
concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used
Mathematics
Class of shelled marine molluscs
the concave side of the shell and viscera are anatomically dorsal. The convex side has to be divided into anteriorly ventral and dorsally posterior portions
Tusk_shell
Set of random variables
variables. Usually, the feature functions f k , i {\displaystyle f_{k,i}} are defined such that they are indicators of the clique's configuration, i
Markov_random_field
Genus of theropod dinosaur from the Early Jurassic
from the side, bulbous at the front, and its outer surface became less convex from snout to naris (bony nostril). The nostrils were placed further back
Dilophosaurus
INDICATOR FUNCTION-CONVEX-ANALYSIS
INDICATOR FUNCTION-CONVEX-ANALYSIS
INDICATOR FUNCTION-CONVEX-ANALYSIS
INDICATOR FUNCTION-CONVEX-ANALYSIS
INDICATOR FUNCTION-CONVEX-ANALYSIS
INDICATOR FUNCTION-CONVEX-ANALYSIS
INDICATOR FUNCTION-CONVEX-ANALYSIS
INDICATOR FUNCTION-CONVEX-ANALYSIS
INDICATOR FUNCTION-CONVEX-ANALYSIS