Flask is a micro web framework written in Python. It is classified as a microframework because it does not require particular tools or libraries. It has no database abstraction layer, form validation, or any other components where pre-existing third-party libraries provide common functions. However, Flask supports extensions that can add application features as if they were implemented in Flask itself. Extensions exist for object-relational mappers, form validation, upload handling, various open authentication technologies and several common framework related tools. Applications that use the Flask framework include Pinterest and LinkedIn. == History == Flask was created by Armin Ronacher of Pocoo, an international group of Python enthusiasts formed in 2004. According to Ronacher, the idea was originally an April Fool's joke that was popular enough to make into a serious application. The name is a play on the earlier Bottle framework. When Ronacher and Georg Brandl created a bulletin board system written in Python in 2004, the Pocoo projects Werkzeug and Jinja were developed. In April 2016, the Pocoo team was disbanded and development of Flask and related libraries passed to the newly formed Pallets project. Flask has become popular among Python enthusiasts. As of October 2020, it has the second-most number of stars on GitHub among Python web-development frameworks, only slightly behind Django, and was voted the most popular web framework in the Python Developers Survey for years between and including 2018 and 2022. == Components == The microframework Flask is part of the Pallets Projects (formerly Pocoo), and based on several others of them, all under a BSD license. === Werkzeug === Werkzeug (German for "tool") is a utility library for the Python programming language for Web Server Gateway Interface (WSGI) applications. Werkzeug can instantiate objects for request, response, and utility functions. It can be used as the basis for a custom software framework and supports Python 2.7 and 3.5 and later. === Jinja === Jinja, also by Ronacher, is a template engine for the Python programming language. Similar to the Django web framework, it handles templates in a sandbox. === MarkupSafe === MarkupSafe is a string handling library for the Python programming language. The eponymous MarkupSafe type extends the Python string type and marks its contents as "safe"; combining MarkupSafe with regular strings automatically escapes the unmarked strings, while avoiding double escaping of already marked strings. === ItsDangerous === ItsDangerous is a safe data serialization library for the Python programming language. It is used to store the session of a Flask application in a cookie without allowing users to tamper with the session contents. === Click === Click is a Python package used by Flask to create command-line interfaces (CLI) by providing a simple and composable way to define commands, arguments, and options. == Features == Development server and debugger Integrated support for unit testing RESTful request dispatching Uses Jinja templating Support for secure cookies (client side sessions) 100% WSGI 1.0 compliant Unicode-based Complete documentation Google App Engine compatibility Extensions available to extend functionality == Example == The following code shows a simple web application that displays "Hello World!" when visited: === Render Template with Flask === ==== Jinja in HTML for the Render Template ====
Cloud-native computing
Cloud native computing is an approach in software development that utilizes cloud computing to "build and run scalable applications in modern, dynamic environments such as public, private, and hybrid clouds". These technologies, such as containers, microservices, serverless functions, cloud native processors and immutable infrastructure, deployed via declarative code are common elements of this architectural style. Cloud native technologies focus on minimizing users' operational burden. Cloud native techniques "enable loosely coupled systems that are resilient, manageable, and observable. Combined with robust automation, they allow engineers to make high-impact changes frequently and predictably with minimal toil." This independence contributes to the overall resilience of the system, as issues in one area do not necessarily cripple the entire application. Additionally, such systems are easier to manage, and monitor, given their modular nature, which simplifies tracking performance and identifying issues. Frequently, cloud-native applications are built as a set of microservices that run in Open Container Initiative compliant containers, such as Containerd, and may be orchestrated in Kubernetes and managed and deployed using DevOps and Git CI workflows (although there is a large amount of competing open source that supports cloud-native development). The advantage of using containers is the ability to package all software needed to execute into one executable package. The container runs in a virtualized environment, which isolates the contained application from its environment.
Dynamic Bayesian network
A dynamic Bayesian network (DBN) is a Bayesian network (BN) which relates variables to each other over adjacent time steps. == History == A dynamic Bayesian network (DBN) is often called a "two-timeslice" BN (2TBN) because it says that at any point in time T, the value of a variable can be calculated from the internal regressors and the immediate prior value (time T-1). DBNs were developed by Paul Dagum in the early 1990s at Stanford University's Section on Medical Informatics. Dagum developed DBNs to unify and extend traditional linear state-space models such as Kalman filters, linear and normal forecasting models such as ARMA and simple dependency models such as hidden Markov models into a general probabilistic representation and inference mechanism for arbitrary nonlinear and non-normal time-dependent domains. Today, DBNs are common in robotics, and have shown potential for a wide range of data mining applications. For example, they have been used in speech recognition, digital forensics, protein sequencing, and bioinformatics. DBN is a generalization of hidden Markov models and Kalman filters. DBNs are conceptually related to probabilistic Boolean networks and can, similarly, be used to model dynamical systems at steady-state.
Homogeneity blockmodeling
In mathematics applied to analysis of social structures, homogeneity blockmodeling is an approach in blockmodeling, which is best suited for a preliminary or main approach to valued networks, when a prior knowledge about these networks is not available. This is because homogeneity blockmodeling emphasizes the similarity of link (tie) strengths within the blocks over the pattern of links. In this approach, tie (link) values (or statistical data computed on them) are assumed to be equal (homogenous) within blocks. This approach to the generalized blockmodeling of valued networks was first proposed by Aleš Žiberna in 2007 with the basic idea, "that the inconsistency of an empirical block with its ideal block can be measured by within block variability of appropriate values". The newly–formed ideal blocks, which are appropriate for blockmodeling of valued networks, are then presented together with the definitions of their block inconsistencies. Similar approach to the homogeneity blockmodeling, dealing with direct approach for structural equivalence, was previously suggested by Stephen P. Borgatti and Martin G. Everett (1992).
Growth function
The growth function, also called the shatter coefficient or the shattering number, measures the richness of a set family or class of functions. It is especially used in the context of statistical learning theory, where it is used to study properties of statistical learning methods. The term 'growth function' was coined by Vapnik and Chervonenkis in their 1968 paper, where they also proved many of its properties. It is a basic concept in machine learning. == Definitions == === Set-family definition === Let H {\displaystyle H} be a set family (a set of sets) and C {\displaystyle C} a set. Their intersection is defined as the following set-family: H ∩ C := { h ∩ C ∣ h ∈ H } {\displaystyle H\cap C:=\{h\cap C\mid h\in H\}} The intersection-size (also called the index) of H {\displaystyle H} with respect to C {\displaystyle C} is | H ∩ C | {\displaystyle |H\cap C|} . If a set C m {\displaystyle C_{m}} has m {\displaystyle m} elements then the index is at most 2 m {\displaystyle 2^{m}} . If the index is exactly 2m then the set C {\displaystyle C} is said to be shattered by H {\displaystyle H} , because H ∩ C {\displaystyle H\cap C} contains all the subsets of C {\displaystyle C} , i.e.: | H ∩ C | = 2 | C | , {\displaystyle |H\cap C|=2^{|C|},} The growth function measures the size of H ∩ C {\displaystyle H\cap C} as a function of | C | {\displaystyle |C|} . Formally: Growth ( H , m ) := max C : | C | = m | H ∩ C | {\displaystyle \operatorname {Growth} (H,m):=\max _{C:|C|=m}|H\cap C|} === Hypothesis-class definition === Equivalently, let H {\displaystyle H} be a hypothesis-class (a set of binary functions) and C {\displaystyle C} a set with m {\displaystyle m} elements. The restriction of H {\displaystyle H} to C {\displaystyle C} is the set of binary functions on C {\displaystyle C} that can be derived from H {\displaystyle H} : H C := { ( h ( x 1 ) , … , h ( x m ) ) ∣ h ∈ H , x i ∈ C } {\displaystyle H_{C}:=\{(h(x_{1}),\ldots ,h(x_{m}))\mid h\in H,x_{i}\in C\}} The growth function measures the size of H C {\displaystyle H_{C}} as a function of | C | {\displaystyle |C|} : Growth ( H , m ) := max C : | C | = m | H C | {\displaystyle \operatorname {Growth} (H,m):=\max _{C:|C|=m}|H_{C}|} == Examples == 1. The domain is the real line R {\displaystyle \mathbb {R} } . The set-family H {\displaystyle H} contains all the half-lines (rays) from a given number to positive infinity, i.e., all sets of the form { x > x 0 ∣ x ∈ R } {\displaystyle \{x>x_{0}\mid x\in \mathbb {R} \}} for some x 0 ∈ R {\displaystyle x_{0}\in \mathbb {R} } . For any set C {\displaystyle C} of m {\displaystyle m} real numbers, the intersection H ∩ C {\displaystyle H\cap C} contains m + 1 {\displaystyle m+1} sets: the empty set, the set containing the largest element of C {\displaystyle C} , the set containing the two largest elements of C {\displaystyle C} , and so on. Therefore: Growth ( H , m ) = m + 1 {\displaystyle \operatorname {Growth} (H,m)=m+1} . The same is true whether H {\displaystyle H} contains open half-lines, closed half-lines, or both. 2. The domain is the segment [ 0 , 1 ] {\displaystyle [0,1]} . The set-family H {\displaystyle H} contains all the open sets. For any finite set C {\displaystyle C} of m {\displaystyle m} real numbers, the intersection H ∩ C {\displaystyle H\cap C} contains all possible subsets of C {\displaystyle C} . There are 2 m {\displaystyle 2^{m}} such subsets, so Growth ( H , m ) = 2 m {\displaystyle \operatorname {Growth} (H,m)=2^{m}} . 3. The domain is the Euclidean space R n {\displaystyle \mathbb {R} ^{n}} . The set-family H {\displaystyle H} contains all the half-spaces of the form: x ⋅ ϕ ≥ 1 {\displaystyle x\cdot \phi \geq 1} , where ϕ {\displaystyle \phi } is a fixed vector. Then Growth ( H , m ) = Comp ( n , m ) {\displaystyle \operatorname {Growth} (H,m)=\operatorname {Comp} (n,m)} , where Comp is the number of components in a partitioning of an n-dimensional space by m hyperplanes. 4. The domain is the real line R {\displaystyle \mathbb {R} } . The set-family H {\displaystyle H} contains all the real intervals, i.e., all sets of the form { x ∈ [ x 0 , x 1 ] | x ∈ R } {\displaystyle \{x\in [x_{0},x_{1}]|x\in \mathbb {R} \}} for some x 0 , x 1 ∈ R {\displaystyle x_{0},x_{1}\in \mathbb {R} } . For any set C {\displaystyle C} of m {\displaystyle m} real numbers, the intersection H ∩ C {\displaystyle H\cap C} contains all runs of between 0 and m {\displaystyle m} consecutive elements of C {\displaystyle C} . The number of such runs is ( m + 1 2 ) + 1 {\displaystyle {m+1 \choose 2}+1} , so Growth ( H , m ) = ( m + 1 2 ) + 1 {\displaystyle \operatorname {Growth} (H,m)={m+1 \choose 2}+1} . == Polynomial or exponential == The main property that makes the growth function interesting is that it can be either polynomial or exponential - nothing in-between. The following is a property of the intersection-size: If, for some set C m {\displaystyle C_{m}} of size m {\displaystyle m} , and for some number n ≤ m {\displaystyle n\leq m} , | H ∩ C m | ≥ Comp ( n , m ) {\displaystyle |H\cap C_{m}|\geq \operatorname {Comp} (n,m)} - then, there exists a subset C n ⊆ C m {\displaystyle C_{n}\subseteq C_{m}} of size n {\displaystyle n} such that | H ∩ C n | = 2 n {\displaystyle |H\cap C_{n}|=2^{n}} . This implies the following property of the Growth function. For every family H {\displaystyle H} there are two cases: The exponential case: Growth ( H , m ) = 2 m {\displaystyle \operatorname {Growth} (H,m)=2^{m}} identically. The polynomial case: Growth ( H , m ) {\displaystyle \operatorname {Growth} (H,m)} is majorized by Comp ( n , m ) ≤ m n + 1 {\displaystyle \operatorname {Comp} (n,m)\leq m^{n}+1} , where n {\displaystyle n} is the smallest integer for which Growth ( H , n ) < 2 n {\displaystyle \operatorname {Growth} (H,n)<2^{n}} . == Other properties == === Trivial upper bound === For any finite H {\displaystyle H} : Growth ( H , m ) ≤ | H | {\displaystyle \operatorname {Growth} (H,m)\leq |H|} since for every C {\displaystyle C} , the number of elements in H ∩ C {\displaystyle H\cap C} is at most | H | {\displaystyle |H|} . Therefore, the growth function is mainly interesting when H {\displaystyle H} is infinite. === Exponential upper bound === For any nonempty H {\displaystyle H} : Growth ( H , m ) ≤ 2 m {\displaystyle \operatorname {Growth} (H,m)\leq 2^{m}} I.e, the growth function has an exponential upper-bound. We say that a set-family H {\displaystyle H} shatters a set C {\displaystyle C} if their intersection contains all possible subsets of C {\displaystyle C} , i.e. H ∩ C = 2 C {\displaystyle H\cap C=2^{C}} . If H {\displaystyle H} shatters C {\displaystyle C} of size m {\displaystyle m} , then Growth ( H , C ) = 2 m {\displaystyle \operatorname {Growth} (H,C)=2^{m}} , which is the upper bound. === Cartesian intersection === Define the Cartesian intersection of two set-families as: H 1 ⨂ H 2 := { h 1 ∩ h 2 ∣ h 1 ∈ H 1 , h 2 ∈ H 2 } {\displaystyle H_{1}\bigotimes H_{2}:=\{h_{1}\cap h_{2}\mid h_{1}\in H_{1},h_{2}\in H_{2}\}} . Then: Growth ( H 1 ⨂ H 2 , m ) ≤ Growth ( H 1 , m ) ⋅ Growth ( H 2 , m ) {\displaystyle \operatorname {Growth} (H_{1}\bigotimes H_{2},m)\leq \operatorname {Growth} (H_{1},m)\cdot \operatorname {Growth} (H_{2},m)} === Union === For every two set-families: Growth ( H 1 ∪ H 2 , m ) ≤ Growth ( H 1 , m ) + Growth ( H 2 , m ) {\displaystyle \operatorname {Growth} (H_{1}\cup H_{2},m)\leq \operatorname {Growth} (H_{1},m)+\operatorname {Growth} (H_{2},m)} === VC dimension === The VC dimension of H {\displaystyle H} is defined according to these two cases: In the polynomial case, VCDim ( H ) = n − 1 {\displaystyle \operatorname {VCDim} (H)=n-1} = the largest integer d {\displaystyle d} for which Growth ( H , d ) = 2 d {\displaystyle \operatorname {Growth} (H,d)=2^{d}} . In the exponential case VCDim ( H ) = ∞ {\displaystyle \operatorname {VCDim} (H)=\infty } . So VCDim ( H ) ≥ d {\displaystyle \operatorname {VCDim} (H)\geq d} if-and-only-if Growth ( H , d ) = 2 d {\displaystyle \operatorname {Growth} (H,d)=2^{d}} . The growth function can be regarded as a refinement of the concept of VC dimension. The VC dimension only tells us whether Growth ( H , d ) {\displaystyle \operatorname {Growth} (H,d)} is equal to or smaller than 2 d {\displaystyle 2^{d}} , while the growth function tells us exactly how Growth ( H , m ) {\displaystyle \operatorname {Growth} (H,m)} changes as a function of m {\displaystyle m} . Another connection between the growth function and the VC dimension is given by the Sauer–Shelah lemma: If VCDim ( H ) = d {\displaystyle \operatorname {VCDim} (H)=d} , then: for all m {\displaystyle m} : Growth ( H , m ) ≤ ∑ i = 0 d ( m i ) {\displaystyle \operatorname {Growth} (H,m)\leq \sum _{i=0}^{d}{m \choose i}} In particular, for all m > d + 1 {\displaystyle m>d+1} : Growth ( H , m ) ≤ ( e m / d ) d = O ( m d ) {\displaystyle \operatorname {Growth} (H,m)\leq (
Color management
Color management is the process of ensuring consistent and accurate colors across various devices, such as monitors, printers, and cameras. It involves the use of color profiles, which are standardized descriptions of how colors should be displayed or reproduced. Color management is necessary because different devices have different color capabilities and characteristics. For example, a monitor may display colors differently than a printer can reproduce them. Without color management, the same image may appear differently on different devices, leading to inconsistencies and inaccuracies. To achieve color management, a color profile is created for each device involved in the color workflow. This profile describes the device's color capabilities and characteristics, such as its color gamut (range of colors it can display or reproduce) and color temperature. These profiles are then used to translate colors between devices, ensuring consistent and accurate color reproduction. Color management is particularly important in industries such as graphic design, photography, and printing, where accurate color representation is crucial. It helps to maintain color consistency throughout the entire workflow, from capturing an image to displaying or printing it. Parts of color management are implemented in the operating system (OS), helper libraries, the application, and devices. The type of color profile that is typically used is called an ICC profile. A cross-platform view of color management is the use of an ICC-compatible color management system. The International Color Consortium (ICC) is an industry consortium that has defined: an open standard for a Color Matching Module (CMM) at the OS level color profiles for: devices, including DeviceLink profiles that transform one device profile (color space) to another device profile without passing through an intermediate color space, such as LAB, more accurately preserving color working spaces, the color spaces in which color data is meant to be manipulated There are other approaches to color management besides using ICC profiles. This is partly due to history and partly because of other needs than the ICC standard covers. The film and broadcasting industries make use of some of the same concepts, but they frequently rely on more limited boutique solutions. The film industry, for instance, often uses 3D LUTs (lookup table) to represent a complete color transformation for a specific RGB encoding. At the consumer level, system wide color management is available in most of Apple's products (macOS, iOS, iPadOS, watchOS). Microsoft Windows lacks system wide color management and virtually all applications do not employ color management. Windows' media player API is not color space aware, and if applications want to color manage videos manually, they have to incur significant performance and power consumption penalties. Android supports system wide color management, but most devices ship with color management disabled. == Overview == Characterize. Every color-managed device requires a personalized table, or "color profile," which characterizes the color response of that particular device. Standardize. Each color profile describes these colors relative to a standardized set of reference colors (the "Profile Connection Space"). Translate. Color-managed software then uses these standardized profiles to translate color from one device to another. This is usually performed by a color management module (CMM). == Hardware == === Characterization === To describe the behavior of various output devices, they must be compared (measured) in relation to a standard color space. Often a step called linearization is performed first, to undo the effect of gamma correction that was done to get the most out of limited 8-bit color paths. Instruments used for measuring device colors include colorimeters and spectrophotometers. As an intermediate result, the device gamut is described in the form of scattered measurement data. The transformation of the scattered measurement data into a more regular form, usable by the application, is called profiling. Profiling is a complex process involving mathematics, intense computation, judgment, testing, and iteration. After the profiling is finished, an idealized color description of the device is created. This description is called a profile. === Calibration === Calibration is like characterization, except that it can include the adjustment of the device, as opposed to just the measurement of the device. Color management is sometimes sidestepped by calibrating devices to a common standard color space such as sRGB; when such calibration is done well enough, no color translations are needed to get all devices to handle colors consistently. This avoidance of the complexity of color management was one of the goals in the development of sRGB. == Color profiles == === Embedding === Image formats themselves (such as TIFF, JPEG, PNG, EPS, PDF, and SVG) may contain embedded color profiles but are not required to do so by the image format. The International Color Consortium standard was created to bring various developers and manufacturers together. The ICC standard permits the exchange of output device characteristics and color spaces in the form of metadata. This allows the embedding of color profiles into images as well as storing them in a database or a profile directory. === Working spaces === Working spaces, such as sRGB, Adobe RGB or ProPhoto are color spaces that facilitate good results while editing. For instance, pixels with equal values of R,G,B should appear neutral. Using a large (gamut) working space will lead to posterization, while using a small working space will lead to clipping. This trade-off is a consideration for the critical image editor. == Color transformation == Color transformation, or color space conversion, is the transformation of the representation of a color from one color space to another. This calculation is required whenever data is exchanged inside a color-managed chain and carried out by a Color Matching Module. Transforming profiled color information to different output devices is achieved by referencing the profile data into a standard color space. It makes it easier to convert colors from one device to a selected standard color space and from that to the colors of another device. By ensuring that the reference color space covers the many possible colors that humans can see, this concept allows one to exchange colors between many different color output devices. Color transformations can be represented by two profiles (source profile and target profile) or by a devicelink profile. In this process there are approximations involved which make sure that the image keeps its important color qualities and also gives an opportunity to control on how the colors are being changed. === Profile connection space === In the terminology of the International Color Consortium, a translation between two color spaces can go through a profile connection space (PCS): Color Space 1 → PCS (CIELAB or CIEXYZ) → Color space 2; conversions into and out of the PCS are each specified by a profile. === Gamut mapping === In nearly every translation process, we have to deal with the fact that the color gamut of different devices vary in range which makes an accurate reproduction impossible. They therefore need some rearrangement near the borders of the gamut. Some colors must be shifted to the inside of the gamut, as they otherwise cannot be represented on the output device and would simply be clipped. This so-called gamut mismatch occurs for example, when we translate from the RGB color space with a wider gamut into the CMYK color space with a narrower gamut range. In this example, the dark highly saturated purplish-blue color of a typical computer monitor's "blue" primary is impossible to print on paper with a typical CMYK printer. The nearest approximation within the printer's gamut will be much less saturated. Conversely, an inkjet printer's "cyan" primary, a saturated mid-brightness blue, is outside the gamut of a typical computer monitor. The color management system can utilize various methods to achieve desired results and give experienced users control of the gamut mapping behavior. ==== Rendering intent ==== When the gamut of source color space exceeds that of the destination, saturated colors are liable to become clipped (inaccurately represented), or more formally burned. The color management module can deal with this problem in several ways. The ICC specification includes four different rendering intents, listed below. Before the actual rendering intent is carried out, one can temporarily simulate the rendering by soft proofing. It is a useful tool as it predicts the outcome of the colors and is available as an application in many color management systems: Absolute colorimetric Absolute colorimetry and relative colorimetry actually use the same table but differ in the adjust
Vapnik–Chervonenkis dimension
In Vapnik–Chervonenkis theory, the Vapnik–Chervonenkis (VC) dimension is a measure of the size (capacity, complexity, expressive power, richness, or flexibility) of a class of sets. The notion can be extended to classes of binary functions. It is defined as the cardinality of the largest set of points that the function class can shatter—that is, for which all possible binary labelings can be realized by some function in the class. It was originally defined by Vladimir Vapnik and Alexey Chervonenkis. Informally, the capacity of a classification model is related to how complicated it can be. For example, consider the thresholding of a high-degree polynomial: if the polynomial evaluates above zero, that point is classified as positive, otherwise as negative. A high-degree polynomial can be wiggly, so that it can fit a given set of training points well. Such a polynomial has a high capacity. A much simpler alternative is to threshold a linear function. This function may not fit the training set well, because it has a low capacity. This notion of capacity is made rigorous below. == Definitions == === VC dimension of a set-family === Let C = { C } C ∈ C {\displaystyle {\mathcal {C}}=\{C\}_{C\in {\mathcal {C}}}} be a family of sets (also called set family, collection of sets or set of sets) and X {\displaystyle X} a set. Their intersection is defined as the following set family: C ∩ X := { C ∩ X ∣ C ∈ C } . {\displaystyle {\mathcal {C}}\cap X:=\{C\cap X\mid C\in {\mathcal {C}}\}.} Here typically X {\displaystyle X} and each C ∈ C {\displaystyle C\in {\mathcal {C}}} are subsets of a big "universe" of possibilities U {\displaystyle U} where intersection takes place. We say that a set X {\displaystyle X} is shattered by C {\displaystyle {\mathcal {C}}} if P ( X ) = C ∩ X {\displaystyle {\mathcal {P}}(X)={\mathcal {C}}\cap X} i.e. the set of intersections contains (hence is equal to) all the subsets of X {\displaystyle X} . For finite sets X {\displaystyle X} this is equivalent to | C ∩ X | = 2 | X | . {\displaystyle |{\mathcal {C}}\cap X|=2^{|X|}.} The VC dimension D {\displaystyle D} of C {\displaystyle {\mathcal {C}}} is the cardinality of the largest set that is shattered by C {\displaystyle {\mathcal {C}}} . If arbitrarily large sets can be shattered, the VC dimension of C {\displaystyle {\mathcal {C}}} is ∞ {\displaystyle \infty } . === VC dimension of a classification model === A binary classification model f {\displaystyle f} with some parameter vector θ {\displaystyle \theta } is said to shatter a set of generally positioned data points ( x 1 , x 2 , … , x n ) {\displaystyle (x_{1},x_{2},\ldots ,x_{n})} if, for every assignment of labels to those points, there exists a θ {\displaystyle \theta } such that the model f {\displaystyle f} makes no errors when evaluating that set of data points. The VC dimension of a model f {\displaystyle f} is the maximum number of points that can be arranged so that f {\displaystyle f} shatters them. More formally, it is the maximum cardinal D {\displaystyle D} such that there exists a generally positioned data point set of cardinality D {\displaystyle D} that can be shattered by f {\displaystyle f} . == Examples == f {\displaystyle f} is a constant classifier (with no parameters); Its VC dimension is 0 since it cannot shatter even a single point. In general, the VC dimension of a finite classification model, which can return at most 2 d {\displaystyle 2^{d}} different classifiers, is at most d {\displaystyle d} (this is an upper bound on the VC dimension; the Sauer–Shelah lemma gives a lower bound on the dimension). f {\displaystyle f} is a single-parametric threshold classifier on real numbers; i.e., for a certain threshold θ {\displaystyle \theta } , the classifier f θ {\displaystyle f_{\theta }} returns 1 if the input number is larger than θ {\displaystyle \theta } and 0 otherwise. The VC dimension of f {\displaystyle f} is 1 because: (a) It can shatter a single point. For every point x {\displaystyle x} , a classifier f θ {\displaystyle f_{\theta }} labels it as 0 if θ > x {\displaystyle \theta >x} and labels it as 1 if θ < x {\displaystyle \theta