In machine learning, the polynomial kernel is a kernel function commonly used with support vector machines (SVMs) and other kernelized models, that represents the similarity of vectors (training samples) in a feature space over polynomials of the original variables, allowing learning of non-linear models. Intuitively, the polynomial kernel looks not only at the given features of input samples to determine their similarity, but also combinations of these. In the context of regression analysis, such combinations are known as interaction features. The (implicit) feature space of a polynomial kernel is equivalent to that of polynomial regression, but without the combinatorial blowup in the number of parameters to be learned. When the input features are binary-valued (booleans), then the features correspond to logical conjunctions of input features. == Definition == For degree-d polynomials, the polynomial kernel is defined as K ( x , y ) = ( x T y + c ) d {\displaystyle K(\mathbf {x} ,\mathbf {y} )=(\mathbf {x} ^{\mathsf {T}}\mathbf {y} +c)^{d}} where x and y are vectors of size n in the input space, i.e. vectors of features computed from training or test samples and c ≥ 0 is a free parameter trading off the influence of higher-order versus lower-order terms in the polynomial. When c = 0, the kernel is called homogeneous. (A further generalized polykernel divides xTy by a user-specified scalar parameter a.) As a kernel, K corresponds to an inner product in a feature space based on some mapping φ: K ( x , y ) = ⟨ φ ( x ) , φ ( y ) ⟩ {\displaystyle K(\mathbf {x} ,\mathbf {y} )=\langle \varphi (\mathbf {x} ),\varphi (\mathbf {y} )\rangle } The nature of φ can be seen from an example. Let d = 2, so we get the special case of the quadratic kernel. After using the multinomial theorem (twice—the outermost application is the binomial theorem) and regrouping, K ( x , y ) = ( ∑ i = 1 n x i y i + c ) 2 = ∑ i = 1 n ( x i 2 ) ( y i 2 ) + ∑ i = 2 n ∑ j = 1 i − 1 ( 2 x i x j ) ( 2 y i y j ) + ∑ i = 1 n ( 2 c x i ) ( 2 c y i ) + c 2 {\displaystyle K(\mathbf {x} ,\mathbf {y} )=\left(\sum _{i=1}^{n}x_{i}y_{i}+c\right)^{2}=\sum _{i=1}^{n}\left(x_{i}^{2}\right)\left(y_{i}^{2}\right)+\sum _{i=2}^{n}\sum _{j=1}^{i-1}\left({\sqrt {2}}x_{i}x_{j}\right)\left({\sqrt {2}}y_{i}y_{j}\right)+\sum _{i=1}^{n}\left({\sqrt {2c}}x_{i}\right)\left({\sqrt {2c}}y_{i}\right)+c^{2}} From this it follows that the feature map is given by: φ ( x ) = ( x n 2 , … , x 1 2 , 2 x n x n − 1 , … , 2 x n x 1 , 2 x n − 1 x n − 2 , … , 2 x n − 1 x 1 , … , 2 x 2 x 1 , 2 c x n , … , 2 c x 1 , c ) {\displaystyle \varphi (x)=\left(x_{n}^{2},\ldots ,x_{1}^{2},{\sqrt {2}}x_{n}x_{n-1},\ldots ,{\sqrt {2}}x_{n}x_{1},{\sqrt {2}}x_{n-1}x_{n-2},\ldots ,{\sqrt {2}}x_{n-1}x_{1},\ldots ,{\sqrt {2}}x_{2}x_{1},{\sqrt {2c}}x_{n},\ldots ,{\sqrt {2c}}x_{1},c\right)} generalizing for ( x T y + c ) d {\displaystyle \left(\mathbf {x} ^{T}\mathbf {y} +c\right)^{d}} , where x ∈ R n {\displaystyle \mathbf {x} \in \mathbb {R} ^{n}} , y ∈ R n {\displaystyle \mathbf {y} \in \mathbb {R} ^{n}} and applying the multinomial theorem: ( x T y + c ) d = ∑ j 1 + j 2 + ⋯ + j n + 1 = d d ! j 1 ! ⋯ j n ! j n + 1 ! x 1 j 1 ⋯ x n j n c j n + 1 d ! j 1 ! ⋯ j n ! j n + 1 ! y 1 j 1 ⋯ y n j n c j n + 1 = φ ( x ) T φ ( y ) {\displaystyle {\begin{alignedat}{2}\left(\mathbf {x} ^{T}\mathbf {y} +c\right)^{d}&=\sum _{j_{1}+j_{2}+\dots +j_{n+1}=d}{\frac {\sqrt {d!}}{\sqrt {j_{1}!\cdots j_{n}!j_{n+1}!}}}x_{1}^{j_{1}}\cdots x_{n}^{j_{n}}{\sqrt {c}}^{j_{n+1}}{\frac {\sqrt {d!}}{\sqrt {j_{1}!\cdots j_{n}!j_{n+1}!}}}y_{1}^{j_{1}}\cdots y_{n}^{j_{n}}{\sqrt {c}}^{j_{n+1}}\\&=\varphi (\mathbf {x} )^{T}\varphi (\mathbf {y} )\end{alignedat}}} The last summation has l d = ( n + d d ) {\displaystyle l_{d}={\tbinom {n+d}{d}}} elements, so that: φ ( x ) = ( a 1 , … , a l , … , a l d ) {\displaystyle \varphi (\mathbf {x} )=\left(a_{1},\dots ,a_{l},\dots ,a_{l_{d}}\right)} where l = ( j 1 , j 2 , . . . , j n , j n + 1 ) {\displaystyle l=(j_{1},j_{2},...,j_{n},j_{n+1})} and a l = d ! j 1 ! ⋯ j n ! j n + 1 ! x 1 j 1 ⋯ x n j n c j n + 1 | j 1 + j 2 + ⋯ + j n + j n + 1 = d {\displaystyle a_{l}={\frac {\sqrt {d!}}{\sqrt {j_{1}!\cdots j_{n}!j_{n+1}!}}}x_{1}^{j_{1}}\cdots x_{n}^{j_{n}}{\sqrt {c}}^{j_{n+1}}\quad |\quad j_{1}+j_{2}+\dots +j_{n}+j_{n+1}=d} == Practical use == Although the RBF kernel is more popular in SVM classification than the polynomial kernel, the latter is quite popular in natural language processing (NLP). The most common degree is d = 2 (quadratic), since larger degrees tend to overfit on NLP problems. Various ways of computing the polynomial kernel (both exact and approximate) have been devised as alternatives to the usual non-linear SVM training algorithms, including: full expansion of the kernel prior to training/testing with a linear SVM, i.e. full computation of the mapping φ as in polynomial regression; basket mining (using a variant of the apriori algorithm) for the most commonly occurring feature conjunctions in a training set to produce an approximate expansion; inverted indexing of support vectors. One problem with the polynomial kernel is that it may suffer from numerical instability: when xTy + c < 1, K(x, y) = (xTy + c)d tends to zero with increasing d, whereas when xTy + c > 1, K(x, y) tends to infinity.
Verbal overshadowing
Verbal overshadowing is a phenomenon where giving a verbal description of sensory input impairs formation of memories of that input. This was first reported by Schooler and Engstler-Schooler (1990) where it was shown that the effects can be observed across multiple domains of cognition which are known to rely on non-verbal knowledge and perceptual expertise. One example of this is memory, which has been known to be influenced by language. Seminal work by Carmichael and collaborators (1932) demonstrated that when verbal labels are connected to non-verbal forms during an individual's encoding process, it could potentially bias the way those forms are reproduced. Because of this, memory performance relying on reportable aspects of memory that encode visual forms should be vulnerable to the effects of verbalization. == Initial findings == Schooler and Engstler-Schooler (1990) were the first to report findings of verbal overshadowing. In their study, participants watched a video of a simulated robbery and were instructed to either verbally describe the robber or engage in a control task. Those who engaged in giving a verbal description were less likely to correctly identify the robber from a test lineup, compared to those who engaged in the control task. A larger effect was detected when the verbal description was provided 20, rather than 5, minutes after the video, and immediately before the test lineup. A meta-analysis by Meissner and Brigham (2001) supported the effects of verbal overshadowing, showing a small but reliably negative effect. == General effects of verbal overshadowing == The effects of verbal overshadowing have been generalized across multiple domains of cognition that are known to rely on non-verbal knowledge and perceptual expertise, such as memory. Memory has been known to be influenced by language. Seminal work by Carmichael and collaborators (1932) demonstrated that labels attached to, or associated with, non-verbal forms during memory encoding can affect the way the forms were subsequently reproduced. Because of this, memory performance that relies on reportable aspects of memory that encode visual forms should be vulnerable to the effects of verbalization. Pelizzon, Brandimonte, and Luccio (2002) found that visual memory representations appear to incorporate visual, spatial, and temporal characteristics. It is explained as follows: With the temporal code (where the only information available is the sequence of the stimuli), performance levels remain high, unless participants are required to retrieve the stimuli in a different order from that used at encoding (visual cue). In this case, performance is significantly impaired, even in the presence of a visual cue. The study showed that order information acts as a link between the two separate representations of figure and background, hence preventing verbal overshadowing at encoding (temporal component) or attenuating its influence at retrieval (spatial component).(p. 960) Hatano, Ueno, Kitagami, and Kawaguchi found that verbal overshadowing is likely to occur when participants verbally described targets in detail. Detailed verbal descriptions resulted in more frequently inaccurate descriptions that in turn created inaccurate representations in the memories of participants. Inaccuracies are also likely to occur when face recognition comes immediately after verbalization. Other forms of non-verbal knowledge affected by verbal overshadowing include the following: [Verbal overshadowing] has also been observed when participants attempt to generate descriptions of other 'difficult-to-describe' stimuli such as colors (Schooler and Engstler-Schooler, 1990) or abstract figures (Brandimonte et al., 1997), or other non-visual tasks such as wine tasting (Melcher and Schooler, 1996), decision making (Wilson and Schooler, 1991), and insight problem-solving. (p. 871) (Schooler et al., 1993) Verbalization of stimuli leads to the disruption of non-reportable processes that are necessary for achieving insight solutions, which are distinct from language processes. Schooler, Ohlsson, and Brooks (1993) found that face recognition requires information that cannot be adequately verbalized, giving rise to difficulty in describing factors in recognition judgments. Subjects were less effective in solving insight problems when compelled to put their thoughts in words, which suggests that language may interfere with thought. The verbal overshadowing effect was not seen when participants engaged in articulatory suppression. Performance was reduced in both the verbal and non-verbal description conditions. This is evidence that verbal encoding plays a role in face recognition. By testing with distracting faces presented between study and test, Lloyd-Jones and Brown (2008) suggested a dual-process approach to recognition memory took place, that verbalization influenced familiarity-based processes at first, but its effects were later seen on recollection, when discrimination between items became more difficult. == Verbal overshadowing in facial recognition == The verbal overshadowing effect can be found for facial recognition because faces are predominately processed in a holistic or configurable manner. (Tanaka & Farah, 1993; Tanaka & Sengco, 1997) Verbalizing one's memory for a face is done using a featural or analytic strategy, leading to a drift from the configurable information about the face and to impaired recognition performance. However, Fallshore & Schooler (1995) found that the verbal overshadowing effect was not found when participants described faces of races different from their own. A study by Brown and Lloyd-Jones (2003) found that there was no verbal overshadowing effect found in car descriptions; it was only seen in facial descriptions. The authors noted that descriptions were no different on any measure including accuracy. It is suggested that less expertise in verbalizing faces rather than cars invokes a stronger shift in verbal and featural processing. This supports the concept of a transfer inappropriate retrieval framework and addresses some limitations of the effect. Wickham and Swift (2006) suggested that the verbal overshadowing effect is not seen in describing all faces, and one aspect that determines this is distinctiveness. Results showed that typical faces produce verbal overshadowing, while distinctive faces did not. In studies of eyewitness reports, variation in response criteria given by participants influenced the quality of the descriptions generated and accuracy on identification task, known as the retrieval-based effect. Face recognition was also impaired when subjects described a familiar face, such as a parent, or when describing a previously seen but novel face. Dodson, Johnson, and Schooler (1997) found that recognition was also impaired when participants were provided with a description of a previously seen face, and they were able to ignore provided versus self-generated descriptions more easily. This finding of verbal overshadowing suggested that eyewitness recognition is not only affected by their own descriptions, but of descriptions heard from others, such other eyewitness testimonies. == Voice recognition == The verbal overshadowing effect has also been found to affect voice identification. Research shows that describing a non-verbal stimuli leads to a decrease in recognition accuracy. In an unpublished study by Schooler, Fiore, Melcher, and Ambadar (1996), participants listened to a tape-recorded voice, after which they were asked either to verbally describe it or to not do so, and then asked to distinguish the voice from 3 similar distractor voices. The results showed that verbal overshadowing impaired accuracy of recognition based on gut feeling, suggesting an overall verbal overshadowing for voice recognition. Due to the forensic relevance of voices heard over the telephone and harassing phone calls that are often a problem for police, Perfect, Hunt, and Harris (2002) examined the influence of three factors on accuracy and confidence in voice recognition from a line-up. They expected to find an effect, because voice represents a class of stimuli that is difficult to describe verbally. This meets Schooler et al.'s (1997) modality mismatch criterion, meaning that describing the speakers age, gender, or accent is difficult, making voice recognition susceptible to the verbal overshadowing phenomenon. It was found that the method of memory encoding had no impact on performance, and that hearing a telephone voice reduced confidence but did not affect accuracy. They also found that providing a verbal description impaired accuracy but had no effect on confidence. The data showed an effect of verbal overshadowing in voice recognition and provided yet another disassociation between confidence and performance. Although there was a difference in confidence level, witnesses were able to identify voices over the telephone as accurately as voices heard direc
Projection-slice theorem
In mathematics, the projection-slice theorem, central slice theorem or Fourier slice theorem in two dimensions states that the results of the following two calculations are equal: Take a two-dimensional function f(r), project (e.g. using the Radon transform) it onto a (one-dimensional) line, and do a Fourier transform of that projection. Take that same function, but do a two-dimensional Fourier transform first, and then slice the function through its origin, parallel to the projection line. In operator terms, if F1 and F2 are the 1- and 2-dimensional Fourier transform operators mentioned above, P1 is the projection operator (which projects a 2-D function onto a 1-D line), S1 is a slice operator (which extracts a 1-D central slice from a function), then F 1 P 1 = S 1 F 2 . {\displaystyle F_{1}P_{1}=S_{1}F_{2}.} This idea can be extended to higher dimensions. This theorem is used, for example, in the analysis of medical CT scans where a "projection" is an x-ray image of an internal organ. The Fourier transforms of these images are seen to be slices through the Fourier transform of the 3-dimensional density of the internal organ, and these slices can be interpolated to build up a complete Fourier transform of that density. The inverse Fourier transform is then used to recover the 3-dimensional density of the object. This technique was first derived by Ronald N. Bracewell in 1956 for a radio-astronomy problem. == The projection-slice theorem in N dimensions == In N dimensions, the projection-slice theorem states that the Fourier transform of the projection of an N-dimensional function f(r) onto an m-dimensional linear submanifold is equal to an m-dimensional slice of the N-dimensional Fourier transform of that function consisting of an m-dimensional linear submanifold through the origin in the Fourier space which is parallel to the projection submanifold. In operator terms: F m P m = S m F N . {\displaystyle F_{m}P_{m}=S_{m}F_{N}.\,} == The generalized Fourier-slice theorem == In addition to generalizing to N dimensions, the projection-slice theorem can be further generalized with an arbitrary change of basis. For convenience of notation, we consider the change of basis to be represented as B, an N-by-N invertible matrix operating on N-dimensional column vectors. Then the generalized Fourier-slice theorem can be stated as F m P m B = S m B − T | B − T | F N {\displaystyle F_{m}P_{m}B=S_{m}{\frac {B^{-T}}{|B^{-T}|}}F_{N}} where B − T = ( B − 1 ) T {\displaystyle B^{-T}=(B^{-1})^{T}} is the transpose of the inverse of the change of basis transform. == Proof in two dimensions == The projection-slice theorem is easily proven for the case of two dimensions. Without loss of generality, we can take the projection line to be the x-axis. There is no loss of generality because if we use a shifted and rotated line, the law still applies. Using a shifted line (in y) gives the same projection and therefore the same 1D Fourier transform results. The rotated function is the Fourier pair of the rotated Fourier transform, for which the theorem again holds. If f(x, y) is a two-dimensional function, then the projection of f(x, y) onto the x axis is p(x) where p ( x ) = ∫ − ∞ ∞ f ( x , y ) d y . {\displaystyle p(x)=\int _{-\infty }^{\infty }f(x,y)\,dy.} The Fourier transform of f ( x , y ) {\displaystyle f(x,y)} is F ( k x , k y ) = ∫ − ∞ ∞ ∫ − ∞ ∞ f ( x , y ) e − 2 π i ( x k x + y k y ) d x d y . {\displaystyle F(k_{x},k_{y})=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(x,y)\,e^{-2\pi i(xk_{x}+yk_{y})}\,dxdy.} The slice is then s ( k x ) {\displaystyle s(k_{x})} s ( k x ) = F ( k x , 0 ) = ∫ − ∞ ∞ ∫ − ∞ ∞ f ( x , y ) e − 2 π i x k x d x d y {\displaystyle s(k_{x})=F(k_{x},0)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(x,y)\,e^{-2\pi ixk_{x}}\,dxdy} = ∫ − ∞ ∞ [ ∫ − ∞ ∞ f ( x , y ) d y ] e − 2 π i x k x d x {\displaystyle =\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }f(x,y)\,dy\right]\,e^{-2\pi ixk_{x}}dx} = ∫ − ∞ ∞ p ( x ) e − 2 π i x k x d x {\displaystyle =\int _{-\infty }^{\infty }p(x)\,e^{-2\pi ixk_{x}}dx} which is just the Fourier transform of p(x). The proof for higher dimensions is easily generalized from the above example. == The FHA cycle == If the two-dimensional function f(r) is circularly symmetric, it may be represented as f(r), where r = |r|. In this case the projection onto any projection line will be the Abel transform of f(r). The two-dimensional Fourier transform of f(r) will be a circularly symmetric function given by the zeroth-order Hankel transform of f(r), which will therefore also represent any slice through the origin. The projection-slice theorem then states that the Fourier transform of the projection equals the slice or F 1 A 1 = H , {\displaystyle F_{1}A_{1}=H,} where A1 represents the Abel-transform operator, projecting a two-dimensional circularly symmetric function onto a one-dimensional line, F1 represents the 1-D Fourier-transform operator, and H represents the zeroth-order Hankel-transform operator. == Extension to fan beam or cone-beam CT == The projection-slice theorem is suitable for CT image reconstruction with parallel beam projections. It does not directly apply to fanbeam or conebeam CT. The theorem was extended to fan-beam and conebeam CT image reconstruction by Shuang-ren Zhao in 1995.
Moving object detection
Moving object detection is a technique used in computer vision and image processing. Multiple consecutive frames from a video are compared by various methods to determine if any moving object is detected. Moving objects detection has been used for wide range of applications like video surveillance, activity recognition, road condition monitoring, airport safety, monitoring of protection along marine border, etc. == Definition == Moving object detection is to recognize the physical movement of an object in a given place or region. By acting segmentation among moving objects and stationary area or region, the moving objects' motion can be tracked and thus analyzed later. To achieve this, consider a video is a structure built upon single frames, moving object detection is to find the foreground moving target(s), either in each video frame or only when the moving target shows the first appearance in the video. == Traditional methods == Among all the traditional moving object detection methods, we could categorize them into four major approaches: Background subtraction, Frame differencing, Temporal Differencing, and Optical Flow. === Frame differencing === Instead of using traditional approach, to use image subtraction operator by subtracting second and images afterwards, the frame differencing method makes comparisons between two successive frames to detect moving targets. === Temporal differencing === The temporal differencing method identifies the moving object by applying pixel-wise difference method with two or three consecutive frames.
Gapo
Gapo is a Vietnamese social networking service based in Hanoi, Vietnam. Users are able to create a personal profile and share text, photos and videos with others on the platform. Users can also use Gapo for live streaming, instant messaging, blogging, and online payments. Gapo was launched in July 2019 by Hà Trung Kiên and Duong Vi Khoa. == History == Gapo was founded in response to calls for Vietnam's Communist-led government to produce a domestic alternative to social media giants like Facebook and Google. Gapo officially launched on July 23, 2019 at an event in Hanoi. The company received 500 billion đồng (US$22 million) in funding from technology corporation G-Group to be utilized in the first phase of development. They also partnered with Sony Music Entertainment to provide music content to its services. == Features == Gapo features a news feed for posting content, livestreaming, instant messaging, and blogging. It also allows users to pay online and access public services. == Reception == Within two days of launch, Gapo received about 200,000 registrations. By September 2019, the user base increased to one million. Upon launch, Gapo experienced significant technical difficulties. Users complained about the inability to sign up for a new account and said that certain functions were not available for use at launch. This issue caused Gapo to temporarily suspend their services in order to perform upgrades and bug fixes. Gapo relaunched the next day, though many users reported that the access speed decreased. The mobile app also received mixed reviews from users in both the App Store and the Google Play Store, with an average rating of 3.1 and 3.5, respectively. Most users found the app to be a knockoff of Facebook, although some users praised the app for being locally developed. === Expert opinions on platform viability === Le Hong Hiep of the ISEAS - Yusof Ishak Institute was doubtful that a Vietnamese-owned social network service could be as powerful as a foreign-based service, stating that Vietnam might not be able to develop a viable social media network to compete with the likes of Facebook or Google. Others, like blogger Ann Chi, said that, due to local players complying with local censorship policy, there is a chance that locals might not trust Gapo and other local services in light of possible surveillance. Regarding the targeted user base figure for the end of 2019 and 2021, experts cautioned that the company might need an additional trillion đồng of funding to reach its planned user base targets. In response, the company stated that Gapo was never meant to compete with Facebook, but instead noted that the main difference between Gapo and Facebook is that Gapo provides a personalized user experience through customization. == Censorship == Gapo has the right to censor posts and news that are deemed offensive and inaccurate by users or not approved by the censorship curators.
List of JavaScript libraries
This is a list of notable JavaScript libraries. == Constraint programming == Cassowary (software) CHR.js == DOM (manipulation) oriented == Google Polymer Dojo Toolkit jQuery MooTools Prototype JavaScript Framework == Graphical/visualization (canvas, SVG, or WebGL related) == AnyChart Apache ECharts Babylon.js Chart.js Cytoscape D3.js Dojo Toolkit FusionCharts Google Charts JointJS p5.js Plotly.js Processing.js Raphaël RGraph SWFObject Teechart Three.js Velocity.js Verge3D Webix == GUI (Graphical user interface) and widget related == Angular (application platform) by Google AngularJS by Google Bootstrap Dojo Widgets Ext JS by Sencha Foundation by ZURB jQuery UI jQWidgets OpenUI5 by SAP Polymer (library) by Google qooxdoo React.js by Meta/Facebook Vue.js Webix WinJS Svelte === No longer actively developed === Glow Lively Kernel Script.aculo.us YUI Library == Pure JavaScript/Ajax == Google Closure Library JsPHP Microsoft's Ajax library MochiKit PDF.js Socket.IO Spry framework Underscore.js == Template systems == jQuery Mobile Mustache Jinja-JS Twig.js == Unit testing == Jasmine Mocha QUnit == Test automation == Playwright Cypress == Web-application related (MVC, MVVM) == Angular (application platform) by Google AngularJS by Google Backbone.js Echo Ember.js Enyo Express.js Ext JS Google Web Toolkit JsRender/JsViews Knockout Meteor Mojito MooTools Next.js Nuxt.js OpenUI5 by SAP Polymer (library) by Google Prototype JavaScript Framework qooxdoo React.js SproutCore svelte Vue.js == Other == Blockly Cannon.js MathJax Modernizr TensorFlow Brain.js
Bandhan Tod
Bandhan Tod is a mobile app to stop child marriage in India's Bihar state through SOS button in the app. When the SOS on Bandhan Tod is activated, the nearest small NGO will attempt to resolve the issue. If the family resists, then the police gets notified. Till now so many child marriages has been cancelled through Bandhan Tod interventions. Bandhan Tod is an initiative of Gender Alliance managed by Prashanti Tiwari to support the state government's efforts to end child marriage and dowry.