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Persistent Homology Group
In persistent homology, a persistent homology group is a multiscale analog of a homology group that captures information about the evolution of topological features across a filtration of spaces. While the ordinary homology group represents nontrivial homology classes of an individual topological space, the persistent homology group tracks only those classes that remain nontrivial across multiple parameters in the underlying filtration. Analogous to the ordinary Betti number, the ranks of the persistent homology groups are known as the persistent Betti numbers. Persistent homology groups were first introduced by Herbert Edelsbrunner, David Letscher, and Afra Zomorodian in a 2002 paper ''Topological Persistence and Simplification'', one of the foundational papers in the fields of persistent homology and topological data analysis, based largely on the persistence barcodes and the persistence algorithm, that were first described by Serguei Barannikov in the 1994 paper. Since t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Persistent Homology
:''See homology for an introduction to the notation.'' Persistent homology is a method for computing topological features of a space at different spatial resolutions. More persistent features are detected over a wide range of spatial scales and are deemed more likely to represent true features of the underlying space rather than artifacts of sampling, noise, or particular choice of parameters. To find the persistent homology of a space, the space must first be represented as a simplicial complex. A distance function on the underlying space corresponds to a filtration of the simplicial complex, that is a nested sequence of increasing subsets. Definition Formally, consider a real-valued function on a simplicial complex f:K \rightarrow \mathbb that is non-decreasing on increasing sequences of faces, so f(\sigma) \leq f(\tau) whenever \sigma is a face of \tau in K. Then for every a \in \mathbb the sublevel set K_a=f^((-\infty, a]) is a subcomplex of ''K'', and the ordering of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotonic Function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus and analysis In calculus, a function f defined on a subset of the real numbers with real values is called ''monotonic'' if and only if it is either entirely non-increasing, or entirely non-decreasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease. A function is called ''monotonically increasing'' (also ''increasing'' or ''non-decreasing'') if for all x and y such that x \leq y one has f\!\left(x\right) \leq f\!\left(y\right), so f preserves the order (see Figure 1). Likewise, a function is called ''monotonically decreasing'' (also ''decreasing'' or ''non-increasing'') if, whenever x \leq y, then f\!\left(x\right) \geq f\!\left(y\ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plane (mathematics)
In mathematics, a plane is a Euclidean space, Euclidean (flatness (mathematics), flat), two-dimensional surface (mathematics), surface that extends indefinitely. A plane is the two-dimensional analogue of a point (geometry), point (zero dimensions), a line (geometry), line (one dimension) and three-dimensional space. Planes can arise as Euclidean subspace, subspaces of some higher-dimensional space, as with one of a room's walls, infinitely extended, or they may enjoy an independent existence in their own right, as in the setting of two-dimensional Euclidean geometry. Sometimes the word ''plane'' is used more generally to describe a two-dimensional surface (mathematics), surface, for example the hyperbolic plane and elliptic plane. When working exclusively in two-dimensional Euclidean space, the definite article is used, so ''the'' plane refers to the whole space. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory, and graph of a function, graphing are p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiset
In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that element in the multiset. As a consequence, an infinite number of multisets exist which contain only elements and , but vary in the multiplicities of their elements: * The set contains only elements and , each having multiplicity 1 when is seen as a multiset. * In the multiset , the element has multiplicity 2, and has multiplicity 1. * In the multiset , and both have multiplicity 3. These objects are all different when viewed as multisets, although they are the same set, since they all consist of the same elements. As with sets, and in contrast to tuples, order does not matter in discriminating multisets, so and denote the same multiset. To distinguish between sets and multisets, a notation that incorporates square brackets is s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extended Real Number Line
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on infinities and the various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration. The affinely extended real number system is denoted \overline or \infty, +\infty/math> or It is the Dedekind–MacNeille completion of the real numbers. When the meaning is clear from context, the symbol +\infty is often written simply as Motivation Limits It is often useful to describe the behavior of a function f, as either the argument x or the function value f gets "infinitely large" in some sense. For example, consider the function f defined by :f(x) = \frac. The graph of this function has a horizontal asymptote at y = 0. Geometrically, when moving increasingly farther to the right along t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval (mathematics)
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other examples of intervals are the set of numbers such that , the set of all real numbers \R, the set of nonnegative real numbers, the set of positive real numbers, the empty set, and any singleton (set of one element). Real intervals play an important role in the theory of integration, because they are the simplest sets whose "length" (or "measure" or "size") is easy to define. The concept of measure can then be extended to more complicated sets of real numbers, leading to the Borel measure and eventually to the Lebesgue measure. Intervals are central to interval arithmetic, a general numerical computing technique that automatically provides guaranteed enclosures for arbitrary formulas, even in the presence of uncertainties, mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Image (mathematics)
In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) f". Similarly, the inverse image (or preimage) of a given subset B of the codomain of f, is the set of all elements of the domain that map to the members of B. Image and inverse image may also be defined for general binary relations, not just functions. Definition The word "image" is used in three related ways. In these definitions, f : X \to Y is a function from the set X to the set Y. Image of an element If x is a member of X, then the image of x under f, denoted f(x), is the value of f when applied to x. f(x) is alternatively known as the output of f for argument x. Given y, the function f is said to "" or "" if there exists some x in the function's domain such that f(x) = y. Similarly, given a set S, f is said to "" if there exi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Persistence Module
A persistence module is a mathematical structure in persistent homology and topological data analysis that formally captures the persistence of Topology, topological features of an object across a range of scale parameters. A persistence module often consists of a collection of Homology (mathematics), homology groups (or vector spaces if using Field (mathematics), field coefficients) corresponding to a Filtration (mathematics), filtration of topological spaces, and a collection of linear maps induced by the Inclusion map, inclusions of the filtration. The concept of a persistence module was first introduced in 2005 as an application of graded modules over polynomial rings, thus importing well-developed algebraic ideas from classical commutative algebra theory to the setting of persistent homology. Since then, persistence modules have been one of the primary algebraic structures studied in the field of applied topology. Definition Single Parameter Persistence Modules Let T be a t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
