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  Color quantization

  In computer graphics, color quantization or color image quantization is quantization applied to color spaces; it is a process that reduces the number of distinct colors used in an image, usually with the intention that the new image should be as visually similar as possible to the original image. Computer algorithms to perform color quantization on bitmaps have been studied since the 1970s. Color quantization is critical for displaying images with many colors on devices that can only display a limited number of colors, usually due to memory limitations, and enables efficient compression of certain types of images. The name "color quantization" is primarily used in computer graphics research literature; in applications, terms such as optimized palette generation, optimal palette generation, or decreasing color depth are used. Some of these are misleading, as the palettes generated by standard algorithms are not necessarily the best possible. == Algorithms == Most standard techniques treat color quantization as a problem of clustering points in three-dimensional space, where the points represent colors found in the original image and the three axes represent the three color channels. Almost any three-dimensional clustering algorithm can be applied to color quantization, and vice versa. After the clusters are located, typically the points in each cluster are averaged to obtain the representative color that all colors in that cluster are mapped to. The three color channels are usually red, green, and blue, but another popular choice is the Lab color space, in which Euclidean distance is more consistent with perceptual difference. The most popular algorithm by far for color quantization, invented by Paul Heckbert in 1979, is the median cut algorithm. Many variations on this scheme are in use. Before this time, most color quantization was done using the population algorithm or population method, which essentially constructs a histogram of equal-sized ranges and assigns colors to the ranges containing the most points. A more modern popular method is clustering using octrees, first conceived by Gervautz and Purgathofer and improved by Xerox PARC researcher Dan Bloomberg. If the palette is fixed, as is often the case in real-time color quantization systems such as those used in operating systems, color quantization is usually done using the "straight-line distance" or "nearest color" algorithm, which simply takes each color in the original image and finds the closest palette entry, where distance is determined by the distance between the two corresponding points in three-dimensional space. In other words, if the colors are ( r 1 , g 1 , b 1 ) {\displaystyle (r_{1},g_{1},b_{1})} and ( r 2 , g 2 , b 2 ) {\displaystyle (r_{2},g_{2},b_{2})} , we want to minimize the Euclidean distance: ( r 1 − r 2 ) 2 + ( g 1 − g 2 ) 2 + ( b 1 − b 2 ) 2 . {\displaystyle {\sqrt {(r_{1}-r_{2})^{2}+(g_{1}-g_{2})^{2}+(b_{1}-b_{2})^{2}}}.} This effectively decomposes the color cube into a Voronoi diagram, where the palette entries are the points and a cell contains all colors mapping to a single palette entry. There are efficient algorithms from computational geometry for computing Voronoi diagrams and determining which region a given point falls in; in practice, indexed palettes are so small that these are usually overkill. Color quantization is frequently combined with dithering, which can eliminate unpleasant artifacts such as banding that appear when quantizing smooth gradients and give the appearance of a larger number of colors. Some modern schemes for color quantization attempt to combine palette selection with dithering in one stage, rather than perform them independently. A number of other much less frequently used methods have been invented that use entirely different approaches. The Local K-means algorithm, conceived by Oleg Verevka in 1995, is designed for use in windowing systems where a core set of "reserved colors" is fixed for use by the system and many images with different color schemes might be displayed simultaneously. It is a post-clustering scheme that makes an initial guess at the palette and then iteratively refines it. In the early days of color quantization, the k-means clustering algorithm was deemed unsuitable because of its high computational requirements and sensitivity to initialization. In 2011, M. Emre Celebi reinvestigated the performance of k-means as a color quantizer. He demonstrated that an efficient implementation of k-means outperforms a large number of color quantization methods. The high-quality but slow NeuQuant algorithm reduces images to 256 colors by training a Kohonen neural network "which self-organises through learning to match the distribution of colours in an input image. Taking the position in RGB-space of each neuron gives a high-quality colour map in which adjacent colours are similar." It is particularly advantageous for images with gradients. Finally, one of the newer methods is spatial color quantization, conceived by Puzicha, Held, Ketterer, Buhmann, and Fellner of the University of Bonn, which combines dithering with palette generation and a simplified model of human perception to produce visually impressive results even for very small numbers of colors. It does not treat palette selection strictly as a clustering problem, in that the colors of nearby pixels in the original image also affect the color of a pixel. See sample images. == History and applications == In the early days of PCs, it was common for video adapters to support only 2, 4, 16, or (eventually) 256 colors due to video memory limitations; they preferred to dedicate the video memory to having more pixels (higher resolution) rather than more colors. Color quantization helped to justify this tradeoff by making it possible to display many high color images in 16- and 256-color modes with limited visual degradation. Many operating systems automatically perform quantization and dithering when viewing high color images in a 256 color video mode, which was important when video devices limited to 256 color modes were dominant. Modern computers can now display millions of colors at once, far more than can be distinguished by the human eye, limiting this application primarily to mobile devices and legacy hardware. Nowadays, color quantization is mainly used in GIF and PNG images. GIF, for a long time the most popular lossless and animated bitmap format on the World Wide Web, only supports up to 256 colors, necessitating quantization for many images. Some early web browsers constrained images to use a specific palette known as the web colors, leading to severe degradation in quality compared to optimized palettes. PNG images support 24-bit color, but can often be made much smaller in filesize without much visual degradation by application of color quantization, since PNG files use fewer bits per pixel for palettized images. The infinite number of colors available through the lens of a camera is impossible to display on a computer screen; thus converting any photograph to a digital representation necessarily involves some quantization. Practically speaking, 24-bit color is sufficiently rich to represent almost all colors perceivable by humans with sufficiently small error as to be visually identical (if presented faithfully), within the available color space. However, the digitization of color, either in a camera detector or on a screen, necessarily limits the available color space. Consequently there are many colors that may be impossible to reproduce, regardless of how many bits are used to represent the color. For example, it is impossible in typical RGB color spaces (common on computer monitors) to reproduce the full range of green colors that the human eye is capable of perceiving. With the few colors available on early computers, different quantization algorithms produced very different-looking output images. As a result, a lot of time was spent on writing sophisticated algorithms to be more lifelike. === Quantization for image compression === Many image file formats support indexed color. A whole-image palette typically selects 256 "representative" colors for the entire image, where each pixel references any one of the colors in the palette, as in the GIF and PNG file formats. A block palette typically selects 2 or 4 colors for each block of 4x4 pixels, used in BTC, CCC, S2TC, and S3TC. === Editor support === Many bitmap graphics editors contain built-in support for color quantization, and will automatically perform it when converting an image with many colors to an image format with fewer colors. Most of these implementations allow the user to set exactly the number of desired colors. Examples of such support include: Photoshop's Mode→Indexed Color function supplies a number of quantization algorithms ranging from the fixed Windows system and Web palettes to the proprietary Local and Global algorithms for generating palettes suited to a particu

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  International Conference on Computer Vision

  The International Conference on Computer Vision (ICCV) is a research conference sponsored by the Institute of Electrical and Electronics Engineers (IEEE) held every other year. It is considered to be one of the top conferences in computer vision, alongside CVPR and ECCV, and it is held on years in which ECCV is not. The conference is usually spread over four to five days. Typically, experts in the focus areas give tutorial talks on the first day, then the technical sessions (and poster sessions in parallel) follow. Recent conferences have also had an increasing number of focused workshops and a commercial exhibition. == Awards == === Azriel Rosenfeld Lifetime Achievement Award === The Azriel Rosenfeld Award, or Azriel Rosenfeld Lifetime Achievement Award, recognizes researchers who have made significant contributions to the field of computer vision over their careers. It is named in memory of computer scientist and mathematician Azriel Rosenfeld. The following people have received this award: === Helmholtz Prize === The ICCV Helmholtz Prize, known as the Test of Time Award before 2013, is awarded every other year at the ICCV, recognizing ICCV papers from ten or more years earlier that had a significant impact on computer vision research. Winners are selected by the IEEE Computer Society's Technical Committee on Pattern Analysis and Machine Intelligence. The award is named after the 19th century physician and physicist Hermann von Helmholtz, and the ICCV's award is not related to the various Helmholtz Prizes in physics, or the Hermann von Helmholtz Prize in neuroscience. === Marr Prize === The ICCV best-paper award is the Marr Prize, named after British neuroscientist David Marr. === Mark Everingham Prize === The Mark Everingham Prize is an award given yearly by the Technical Committee on Pattern Analysis and Machine Intelligence of the IEEE Computer Society at the IEEE International Conference on Computer Vision or the European Conference on Computer Vision to commemorate the late Mark Everingham, "one of the rising stars of computer vision", and to encourage others to follow in his footsteps by acting to further progress in the computer vision community as a whole. The prize is given to a researcher, or a team of researchers, who have made a selfless contribution of significant benefit to other members of the computer vision community. The Mark Everingham Prize for Rigorous Evaluation was an award given in 2012 at the British Machine Vision Conference. === PAMI Distinguished Researcher Award === The PAMI Distinguished Researcher Award (until 2013 called Significant Researcher Award) is awarded to candidates whose research projects have significantly contributed to the progress of computer vision. Awards are made based on major research contributions, as well as the role of those contributions in influencing and inspiring other research. Candidates are nominated by the community. The following people have received this award: == Conference list == The conference is usually held in the Spring in various international locations.

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  Apache Giraph

  Apache Giraph is an Apache project to perform graph processing on big data. Giraph utilizes Apache Hadoop's MapReduce implementation to process graphs. Facebook used Giraph with some performance improvements to analyze one trillion edges using 200 machines in 4 minutes. Giraph is based on a paper published by Google about its own graph processing system called Pregel. It can be compared to other Big Graph processing libraries such as Cassovary. As of September 2023, it is no longer actively developed.

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  Blockmodeling

  Blockmodeling is a set or a coherent framework, that is used for analyzing social structure and also for setting procedure(s) for partitioning (clustering) social network's units (nodes, vertices, actors), based on specific patterns, which form a distinctive structure through interconnectivity. It is primarily used in statistics, machine learning and network science. As an empirical procedure, blockmodeling assumes that all the units in a specific network can be grouped together to such extent to which they are equivalent. Regarding equivalency, it can be structural, regular or generalized. Using blockmodeling, a network can be analyzed using newly created blockmodels, which transforms large and complex network into a smaller and more comprehensible one. At the same time, the blockmodeling is used to operationalize social roles. While some contend that the blockmodeling is just clustering methods, Bonacich and McConaghy state that "it is a theoretically grounded and algebraic approach to the analysis of the structure of relations". Blockmodeling's unique ability lies in the fact that it considers the structure not just as a set of direct relations, but also takes into account all other possible compound relations that are based on the direct ones. The principles of blockmodeling were first introduced by Francois Lorrain and Harrison C. White in 1971. Blockmodeling is considered as "an important set of network analytic tools" as it deals with delineation of role structures (the well-defined places in social structures, also known as positions) and the discerning the fundamental structure of social networks. According to Batagelj, the primary "goal of blockmodeling is to reduce a large, potentially incoherent network to a smaller comprehensible structure that can be interpreted more readily". Blockmodeling was at first used for analysis in sociometry and psychometrics, but has now spread also to other sciences. == Definition == A network as a system is composed of (or defined by) two different sets: one set of units (nodes, vertices, actors) and one set of links between the units. Using both sets, it is possible to create a graph, describing the structure of the network. During blockmodeling, the researcher is faced with two problems: how to partition the units (e.g., how to determine the clusters (or classes), that then form vertices in a blockmodel) and then how to determine the links in the blockmodel (and at the same time the values of these links). In the social sciences, the networks are usually social networks, composed of several individuals (units) and selected social relationships among them (links). Real-world networks can be large and complex; blockmodeling is used to simplify them into smaller structures that can be easier to interpret. Specifically, blockmodeling partitions the units into clusters and then determines the ties among the clusters. At the same time, blockmodeling can be used to explain the social roles existing in the network, as it is assumed that the created cluster of units mimics (or is closely associated with) the units' social roles. Blockmodeling can thus be defined as a set of approaches for partitioning units into clusters (also known as positions) and links into blocks, which are further defined by the newly obtained clusters. A block (also blockmodel) is defined as a submatrix, that shows interconnectivity (links) between nodes, present in the same or different clusters. Each of these positions in the cluster is defined by a set of (in)direct ties to and from other social positions. These links (connections) can be directed or undirected; there can be multiple links between the same pair of objects or they can have weights on them. If there are not any multiple links in a network, it is called a simple network. A matrix representation of a graph is composed of ordered units, in rows and columns, based on their names. The ordered units with similar patterns of links are partitioned together in the same clusters. Clusters are then arranged together so that units from the same clusters are placed next to each other, thus preserving interconnectivity. In the next step, the units (from the same clusters) are transformed into a blockmodel. With this, several blockmodels are usually formed, one being core cluster and others being cohesive; a core cluster is always connected to cohesive ones, while cohesive ones cannot be linked together. Clustering of nodes is based on the equivalence, such as structural and regular. The primary objective of the matrix form is to visually present relations between the persons included in the cluster. These ties are coded dichotomously (as present or absent), and the rows in the matrix form indicate the source of the ties, while the columns represent the destination of the ties. Equivalence can have two basic approaches: the equivalent units have the same connection pattern to the same neighbors or these units have same or similar connection pattern to different neighbors. If the units are connected to the rest of network in identical ways, then they are structurally equivalent. Units can also be regularly equivalent, when they are equivalently connected to equivalent others. With blockmodeling, it is necessary to consider the issue of results being affected by measurement errors in the initial stage of acquiring the data. == Different approaches == Regarding what kind of network is undergoing blockmodeling, a different approach is necessary. Networks can be one–mode or two–mode. In the former all units can be connected to any other unit and where units are of the same type, while in the latter the units are connected only to the unit(s) of a different type. Regarding relationships between units, they can be single–relational or multi–relational networks. Further more, the networks can be temporal or multilevel and also binary (only 0 and 1) or signed (allowing negative ties)/values (other values are possible) networks. Different approaches to blockmodeling can be grouped into two main classes: deterministic blockmodeling and stochastic blockmodeling approaches. Deterministic blockmodeling is then further divided into direct and indirect blockmodeling approaches. Among direct blockmodeling approaches are: structural equivalence and regular equivalence. Structural equivalence is a state, when units are connected to the rest of the network in an identical way(s), while regular equivalence occurs when units are equally related to equivalent others (units are not necessarily sharing neighbors, but have neighbour that are themselves similar). Indirect blockmodeling approaches, where partitioning is dealt with as a traditional cluster analysis problem (measuring (dis)similarity results in a (dis)similarity matrix), are: conventional blockmodeling, generalized blockmodeling: generalized blockmodeling of binary networks, generalized blockmodeling of valued networks and generalized homogeneity blockmodeling, prespecified blockmodeling. According to Brusco and Steinley (2011), the blockmodeling can be categorized (using a number of dimensions): deterministic or stochastic blockmodeling, one–mode or two–mode networks, signed or unsigned networks, exploratory or confirmatory blockmodeling. == Blockmodels == Blockmodels (sometimes also block models) are structures in which: vertices (e.g., units, nodes) are assembled within a cluster, with each cluster identified as a vertex; from such vertices a graph can be constructed; combinations of all the links (ties), represented in a block as a single link between positions, while at the same time constructing one tie for each block. In a case, when there are no ties in a block, there will be no ties between the two positions that define the block. Computer programs can partition the social network according to pre-set conditions. When empirical blocks can be reasonably approximated in terms of ideal blocks, such blockmodels can be reduced to a blockimage, which is a representation of the original network, capturing its underlying 'functional anatomy'. Thus, blockmodels can "permit the data to characterize their own structure", and at the same time not seek to manifest a preconceived structure imposed by the researcher. Blockmodels can be created indirectly or directly, based on the construction of the criterion function. Indirect construction refers to a function, based on "compatible (dis)similarity measure between paris of units", while the direct construction is "a function measuring the fit of real blocks induced by a given clustering to the corresponding ideal blocks with perfect relations within each cluster and between clusters according to the considered types of connections (equivalence)". === Types === Blockmodels can be specified regarding the intuition, substance or the insight into the nature of the studied network; this can result in such models as follows: parent-child role systems, organizational hierarchies, systems of

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  Sketchpad

  Sketchpad (a.k.a. Robot Draftsman) is a computer program written by Ivan Sutherland in 1963 in the course of his PhD thesis, for which he received the Turing Award in 1988, and the Kyoto Prize in 2012. It pioneered human–computer interaction (HCI), and is considered the ancestor of modern computer-aided design (CAD) programs and as a major breakthrough in the development of computer graphics in general. For example, Sketchpad inspired the graphical user interface (GUI) and object-oriented programming. Using the program, Sutherland showed that computer graphics could be used for both artistic and technical purposes and for demonstrating a novel method of human–computer interaction. == History == See History of the graphical user interface for a more detailed discussion of GUI development. == Software == Sketchpad was the earliest program ever to use a complete graphical user interface. The clever way the program organizes its geometric data pioneered the use of master (objects) and occurrences (instances) in computing and pointed forward to object-oriented programming. The main idea was to have master drawings which can be instantiated into many duplicates. When a master drawing is changed, then all instances change also. This was the first known form of an entity component system: for example instead of encapsulating points inside of a line object, the points are stored in a ring buffer as described in pages 48 to 52 of the paper, and the line only points to them. This allowed moving one point to alter all the shapes that use it in a single operation. The structures in Sketchpad were also able to store pointers to functions, to achieve a different behavior depending on the kind of object. In figure 3.8 of the paper, the "instances generic block" stores several "subroutine entries" which are pointers to functions: "display", "howbig" etc. This was an early form of virtual functions. Geometric constraints was another major invention in Sketchpad, letting a user easily constrain geometric properties in the drawing: for instance, the length of a line or the angle between two lines could be fixed. As a trade magazine said, clearly Sutherland "broke new ground in 3D computer modeling and visual simulation, the basis for computer graphics and CAD/CAM". Very few programs can be called precedents for his achievements. Patrick J. Hanratty is sometimes called the "father of CAD/CAM" and wrote PRONTO, a numerical control language at General Electric in 1957, and wrote CAD software while working for General Motors beginning in 1961. Sutherland wrote in his thesis that Bolt, Beranek and Newman had a "similar program" and T-Square was developed by Peter Samson and one or more fellow MIT students in 1962, both for the PDP-1. The Computer History Museum holds program listings for Sketchpad. == Hardware == Sketchpad ran on the MIT Lincoln Laboratory TX-2 (1958) computer at the Massachusetts Institute of Technology (MIT), which had 64k of 36-bit words. The user drew on the computer monitor screen with the recently invented light pen, which relayed information on its position by computing at what time the light from the scanning cathode-ray tube screen is detected. To configure the initial position of the light pen, the word INK was displayed on the screen, which, upon tapping, initialised the program with a white cross to continue keeping track of the pen's movement relative to its prior position. Of the 36 bits available to store each display spot in the display file, 20 gave the coordinates of that spot for the display system and the remaining 16 gave the address of the n-component element responsible for adding that spot to display. The TX-2 was an experimental machine and the hardware changed often (on Wednesdays, according to Sutherland). By 1975, the light pen and the cathode-ray tube with which it had been used had been removed. == Publications == The Sketchpad program was part and parcel of Sutherland's Ph.D. thesis at MIT and peripherally related to the Computer-Aided Design project at that time. Sketchpad: A Man-Machine Graphical Communication System.

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  One-shot learning (computer vision)

  One-shot learning is an object categorization problem, found mostly in computer vision. Whereas most machine learning-based object categorization algorithms require training on hundreds or thousands of examples, one-shot learning aims to classify objects from one, or only a few, examples. The term few-shot learning is also used for these problems, especially when more than one example is needed. == Motivation == The ability to learn object categories from few examples, and at a rapid pace, has been demonstrated in humans. It is estimated that a child learns almost all of the 10 ~ 30 thousand object categories in the world by age six. This is due not only to the human mind's computational power, but also to its ability to synthesize and learn new object categories from existing information about different, previously learned categories. Given two examples from two object categories: one, an unknown object composed of familiar shapes, the second, an unknown, amorphous shape; it is much easier for humans to recognize the former than the latter, suggesting that humans make use of previously learned categories when learning new ones. The key motivation for solving one-shot learning is that systems, like humans, can use knowledge about object categories to classify new objects. == Background == As with most classification schemes, one-shot learning involves three main challenges: Representation: How should objects and categories be described? Learning: How can such descriptions be created? Recognition: How can a known object be filtered from enveloping clutter, irrespective of occlusion, viewpoint, and lighting? One-shot learning differs from single object recognition and standard category recognition algorithms in its emphasis on knowledge transfer, which makes use of previously learned categories. Model parameters: Reuses model parameters, based on the similarity between old and new categories. Categories are first learned on numerous training examples, then new categories are learned using transformations of model parameters from those initial categories or selecting relevant parameters for a classifier. Feature sharing: Shares parts or features of objects across categories. One algorithm extracts "diagnostic information" in patches from already learned categories by maximizing the patches' mutual information, and then applies these features to the learning of a new category. A dog category, for example, may be learned in one shot from previous knowledge of horse and cow categories, because dog objects may contain similar distinguishing patches. Contextual information: Appeals to global knowledge of the scene in which the object appears. Such global information can be used as frequency distributions in a conditional random field framework to recognize objects. Alternatively context can consider camera height and scene geometry. Algorithms of this type have two advantages. First, they learn object categories that are relatively dissimilar; and second, they perform well in ad hoc situations where an image has not been hand-cropped and aligned. == Theory == The Bayesian one-shot learning algorithm represents the foreground and background of images as parametrized by a mixture of constellation models. During the learning phase, the parameters of these models are learned using a conjugate density parameter posterior and variational Bayesian expectation–maximization (VBEM). In this stage the previously learned object categories inform the choice of model parameters via transfer by contextual information. For object recognition on new images, the posterior obtained during the learning phase is used in a Bayesian decision framework to estimate the ratio of p(object | test, train) to p(background clutter | test, train) where p is the probability of the outcome. === Bayesian framework === Given the task of finding a particular object in a query image, the overall objective of the Bayesian one-shot learning algorithm is to compare the probability that object is present vs the probability that only background clutter is present. If the former probability is higher, the algorithm reports the object's presence, otherwise the algorithm reports its absence. To compute these probabilities, the object class must be modeled from a set of (1 ~ 5) training images containing examples. To formalize these ideas, let I {\displaystyle I} be the query image, which contains either an example of the foreground category O f g {\displaystyle O_{fg}} or only background clutter of a generic background category O b g {\displaystyle O_{bg}} . Also let I t {\displaystyle I_{t}} be the set of training images used as the foreground category. The decision of whether I {\displaystyle I} contains an object from the foreground category, or only clutter from the background category is: R = p ( O f g | I , I t ) p ( O b g | I , I t ) = p ( I | I t , O f g ) p ( O f g ) p ( I | I t , O b g ) p ( O b g ) , {\displaystyle R={\frac {p(O_{fg}|I,I_{t})}{p(O_{bg}|I,I_{t})}}={\frac {p(I|I_{t},O_{fg})p(O_{fg})}{p(I|I_{t},O_{bg})p(O_{bg})}},} where the class posteriors p ( O f g | I , I t ) {\displaystyle p(O_{fg}|I,I_{t})} and p ( O b g | I , I t ) {\displaystyle p(O_{bg}|I,I_{t})} have been expanded by Bayes' theorem, yielding a ratio of likelihoods and a ratio of object category priors. We decide that the image I {\displaystyle I} contains an object from the foreground class if R {\displaystyle R} exceeds a certain threshold T {\displaystyle T} . We next introduce parametric models for the foreground and background categories with parameters θ {\displaystyle \theta } and θ b g {\displaystyle \theta _{bg}} respectively. This foreground parametric model is learned during the learning stage from I t {\displaystyle I_{t}} , as well as prior information of learned categories. The background model we assume to be uniform across images. Omitting the constant ratio of category priors, p ( O f g ) p ( O b g ) {\displaystyle {\frac {p(O_{fg})}{p(O_{bg})}}} , and parametrizing over θ {\displaystyle \theta } and θ b g {\displaystyle \theta _{bg}} yields R ∝ ∫ p ( I | θ , O f g ) p ( θ | I t , O f g ) d θ ∫ p ( I | θ b g , O b g ) p ( θ b g | I t , O b g ) d θ b g = ∫ p ( I | θ ) p ( θ | I t , O f g ) d θ ∫ p ( I | θ b g ) p ( θ b g | I t , O b g ) d θ b g {\displaystyle R\propto {\frac {\int {p(I|\theta ,O_{fg})p(\theta |I_{t},O_{fg})}d\theta }{\int {p(I|\theta _{bg},O_{bg})p(\theta _{bg}|I_{t},O_{bg})}d\theta _{bg}}}={\frac {\int {p(I|\theta )p(\theta |I_{t},O_{fg})}d\theta }{\int {p(I|\theta _{bg})p(\theta _{bg}|I_{t},O_{bg})}d\theta _{bg}}}} , having simplified p ( I | θ , O f g ) {\displaystyle p(I|\theta ,O_{fg})} and p ( I | θ , O b g ) {\displaystyle p(I|\theta ,O_{bg})} to p ( I | θ f g ) {\displaystyle p(I|\theta _{fg})} and p ( I | θ b g ) . {\displaystyle p(I|\theta _{bg}).} The posterior distribution of model parameters given the training images, p ( θ | I t , O f g ) {\displaystyle p(\theta |I_{t},O_{fg})} is estimated in the learning phase. In this estimation, one-shot learning differs sharply from more traditional Bayesian estimation models that approximate the integral as δ ( θ M L ) {\displaystyle \delta (\theta ^{ML})} . Instead, it uses a variational approach using prior information from previously learned categories. However, the traditional maximum likelihood estimation of the model parameters is used for the background model and the categories learned in advance through training. === Object category model === For each query image I {\displaystyle I} and training images I t {\displaystyle I_{t}} , a constellation model is used for representation. To obtain this model for a given image I {\displaystyle I} , first a set of N interesting regions is detected in the image using the Kadir–Brady saliency detector. Each region selected is represented by a location in the image, X i {\displaystyle X_{i}} and a description of its appearance, A i {\displaystyle A_{i}} . Letting X = ∑ i = 1 N X i , A = ∑ i = 1 N A i {\displaystyle X=\sum _{i=1}^{N}X_{i},A=\sum _{i=1}^{N}A_{i}} and X t {\displaystyle X_{t}} and A t {\displaystyle A_{t}} the analogous representations for training images, the expression for R becomes: R ∝ ∫ p ( X , A | θ , O f g ) p ( θ | X t , A t , O f g ) d θ ∫ p ( X , A | θ b g , O b g ) p ( θ b g | X t , A t , O b g ) d θ b g = ∫ p ( X , A | θ ) p ( θ | X t , A t , O f g ) d θ ∫ p ( X , A | θ b g ) p ( θ b g | X t , A t , O b g ) d θ b g {\displaystyle R\propto {\frac {\int {p(X,A|\theta ,O_{fg})p(\theta |X_{t},A_{t},O_{fg})}d\theta }{\int {p(X,A|\theta _{bg},O_{bg})p(\theta _{bg}|X_{t},A_{t},O_{bg})}d\theta _{bg}}}={\frac {\int {p(X,A|\theta )p(\theta |X_{t},A_{t},O_{fg})}d\theta }{\int {p(X,A|\theta _{bg})p(\theta _{bg}|X_{t},A_{t},O_{bg})}\,d\theta _{bg}}}} The likelihoods p ( X , A | θ ) {\displaystyle p(X,A|\theta )} and p ( X , A | θ b g ) {\displaystyle p(X,A|\theta _{bg})} are represented as mixtures of constellation models. A typical constellation model has

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  Perceptron

  In machine learning, the perceptron is an algorithm for supervised learning of binary classifiers. A binary classifier is a function that can decide whether or not an input, represented by a vector of numbers, belongs to some specific class. It is a type of linear classifier, i.e. a classification algorithm that makes its predictions based on a linear predictor function combining a set of weights with the feature vector. == History == The artificial neuron and artificial neural network were invented in 1943 by Warren McCulloch and Walter Pitts in their seminal paper "A Logical Calculus of the Ideas Immanent in Nervous Activity". In 1957, Frank Rosenblatt was at the Cornell Aeronautical Laboratory. He simulated the perceptron on an IBM 704. Later, he obtained funding by the Information Systems Branch of the United States Office of Naval Research and the Rome Air Development Center, to build a custom-made computer, the Mark I Perceptron. It was first publicly demonstrated on 23 June 1960. The machine was "part of a previously secret four-year NPIC [the US' National Photographic Interpretation Center] effort from 1963 through 1966 to develop this algorithm into a useful tool for photo-interpreters". Rosenblatt described the details of the perceptron in a 1958 paper. His organization of a perceptron is constructed of three kinds of cells ("units"): S, A, R, which stand for "sensory", "association" and "response". He presented at the first international symposium on AI, Mechanisation of Thought Processes, which took place in 1958 November. Rosenblatt's project was funded under Contract Nonr-401(40) "Cognitive Systems Research Program", which lasted from 1959 to 1970, and Contract Nonr-2381(00) "Project PARA" ("PARA" means "Perceiving and Recognition Automata"), which lasted from 1957 to 1963. In 1959, the Institute for Defense Analysis awarded his group a $10,000 contract. By September 1961, the ONR awarded further $153,000 worth of contracts, with $108,000 committed for 1962. The ONR research manager, Marvin Denicoff, stated that ONR, instead of ARPA, funded the Perceptron project, because the project was unlikely to produce technological results in the near or medium term. Funding from ARPA go up to the order of millions dollars, while from ONR are on the order of 10,000 dollars. Meanwhile, the head of IPTO at ARPA, J.C.R. Licklider, was interested in 'self-organizing', 'adaptive' and other biologically-inspired methods in the 1950s; but by the mid-1960s he was openly critical of these, including the perceptron. Instead he strongly favored the logical AI approach of Simon and Newell. === Mark I Perceptron machine === The perceptron was intended to be a machine, rather than a program, and while its first implementation was in software for the IBM 704, it was subsequently implemented in custom-built hardware as the Mark I Perceptron with the project name "Project PARA", designed for image recognition. The machine is currently in Smithsonian National Museum of American History. The Mark I Perceptron had three layers. One version was implemented as follows: An array of 400 photocells arranged in a 20x20 grid, named "sensory units" (S-units), or "input retina". Each S-unit can connect to up to 40 A-units. A hidden layer of 512 perceptrons, named "association units" (A-units). An output layer of eight perceptrons, named "response units" (R-units). Rosenblatt called this three-layered perceptron network the alpha-perceptron, to distinguish it from other perceptron models he experimented with. The S-units are connected to the A-units randomly (according to a table of random numbers) via a plugboard (see photo), to "eliminate any particular intentional bias in the perceptron". The connection weights are fixed, not learned. Rosenblatt was adamant about the random connections, as he believed the retina was randomly connected to the visual cortex, and he wanted his perceptron machine to resemble human visual perception. The A-units are connected to the R-units, with adjustable weights encoded in potentiometers, and weight updates during learning were performed by electric motors.The hardware details are in an operators' manual. In a 1958 press conference organized by the US Navy, Rosenblatt made statements about the perceptron that caused a heated controversy among the fledgling AI community; based on Rosenblatt's statements, The New York Times reported the perceptron to be "the embryo of an electronic computer that [the Navy] expects will be able to walk, talk, see, write, reproduce itself and be conscious of its existence." The Photo Division of Central Intelligence Agency, from 1960 to 1964, studied the use of Mark I Perceptron machine for recognizing militarily interesting silhouetted targets (such as planes and ships) in aerial photos. === Principles of Neurodynamics (1962) === Rosenblatt described his experiments with many variants of the Perceptron machine in a book Principles of Neurodynamics (1962). The book is a published version of the 1961 report. Among the variants are: "cross-coupling" (connections between units within the same layer) with possibly closed loops, "back-coupling" (connections from units in a later layer to units in a previous layer), four-layer perceptrons where the last two layers have adjustable weights (and thus a proper multilayer perceptron), incorporating time-delays to perceptron units, to allow for processing sequential data, analyzing audio (instead of images). The machine was shipped from Cornell to Smithsonian in 1967, under a government transfer administered by the Office of Naval Research. === Perceptrons (1969) === Although the perceptron initially seemed promising, it was quickly proved that perceptrons could not be trained to recognise many classes of patterns. This caused the field of neural network research to stagnate for many years, before it was recognised that a feedforward neural network with two or more layers (also called a multilayer perceptron) had greater processing power than perceptrons with one layer (also called a single-layer perceptron). Single-layer perceptrons are only capable of learning linearly separable patterns. For a classification task with some step activation function, a single node will have a single line dividing the data points forming the patterns. More nodes can create more dividing lines, but those lines must somehow be combined to form more complex classifications. A second layer of perceptrons, or even linear nodes, are sufficient to solve many otherwise non-separable problems. In 1969, a famous book entitled Perceptrons by Marvin Minsky and Seymour Papert showed that it was impossible for these classes of network to learn an XOR function. It is often incorrectly believed that they also conjectured that a similar result would hold for a multi-layer perceptron network. However, this is not true, as both Minsky and Papert already knew that multi-layer perceptrons were capable of producing an XOR function. (See the page on Perceptrons (book) for more information.) Nevertheless, the often-miscited Minsky and Papert text caused a significant decline in interest and funding of neural network research. It took ten more years until neural network research experienced a resurgence in the 1980s. This text was reprinted in 1987 as "Perceptrons - Expanded Edition" where some errors in the original text are shown and corrected. === Subsequent work === Rosenblatt continued working on perceptrons despite diminishing funding. The last attempt was Tobermory, built between 1961 and 1967, built for speech recognition. It occupied an entire room. It had 4 layers with 12,000 weights implemented by toroidal magnetic cores. By the time of its completion, simulation on digital computers had become faster than purpose-built perceptron machines. He died in a boating accident in 1971. A simulation program for neural networks was written for IBM 7090/7094, and was used to study various pattern recognition applications, such as character recognition, particle tracks in bubble-chamber photographs; phoneme, isolated word, and continuous speech recognition; speaker verification; and center-of-attention mechanisms for image processing. The kernel perceptron algorithm was already introduced in 1964 by Aizerman et al. Margin bounds guarantees were given for the Perceptron algorithm in the general non-separable case first by Freund and Schapire (1998), and more recently by Mohri and Rostamizadeh (2013) who extend previous results and give new and more favorable L1 bounds. The perceptron is a simplified model of a biological neuron. While the complexity of biological neuron models is often required to fully understand neural behavior, research suggests a perceptron-like linear model can produce some behavior seen in real neurons. The solution spaces of decision boundaries for all binary functions and learning behaviors are studied in. == Definition == In the modern sense, the perceptron is an algori

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  Radial basis function network

  In the field of mathematical modeling, a radial basis function network is an artificial neural network that uses radial basis functions as activation functions. The output of the network is a linear combination of radial basis functions of the inputs and neuron parameters. Radial basis function networks have many uses, including function approximation, time series prediction, classification, and system control. They were first formulated in a 1988 paper by Broomhead and Lowe, both researchers at the Royal Signals and Radar Establishment. == Network architecture == Radial basis function (RBF) networks typically have three layers: an input layer, a hidden layer with a non-linear RBF activation function and a linear output layer. The input can be modeled as a vector of real numbers x ∈ R n {\displaystyle \mathbf {x} \in \mathbb {R} ^{n}} . The output of the network is then a scalar function of the input vector, φ : R n → R {\displaystyle \varphi :\mathbb {R} ^{n}\to \mathbb {R} } , and is given by φ ( x ) = ∑ i = 1 N a i ρ ( | | x − c i | | ) {\displaystyle \varphi (\mathbf {x} )=\sum _{i=1}^{N}a_{i}\rho (||\mathbf {x} -\mathbf {c} _{i}||)} where N {\displaystyle N} is the number of neurons in the hidden layer, c i {\displaystyle \mathbf {c} _{i}} is the center vector for neuron i {\displaystyle i} , and a i {\displaystyle a_{i}} is the weight of neuron i {\displaystyle i} in the linear output neuron. Functions that depend only on the distance from a center vector are radially symmetric about that vector, hence the name radial basis function. In the basic form, all inputs are connected to each hidden neuron. The norm is typically taken to be the Euclidean distance (although the Mahalanobis distance appears to perform better with pattern recognition) and the radial basis function is commonly taken to be Gaussian ρ ( ‖ x − c i ‖ ) = exp ⁡ [ − β i ‖ x − c i ‖ 2 ] {\displaystyle \rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}=\exp \left[-\beta _{i}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert ^{2}\right]} . The Gaussian basis functions are local to the center vector in the sense that lim | | x | | → ∞ ρ ( ‖ x − c i ‖ ) = 0 {\displaystyle \lim _{||x||\to \infty }\rho (\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert )=0} i.e. changing parameters of one neuron has only a small effect for input values that are far away from the center of that neuron. Given certain mild conditions on the shape of the activation function, RBF networks are universal approximators on a compact subset of R n {\displaystyle \mathbb {R} ^{n}} . This means that an RBF network with enough hidden neurons can approximate any continuous function on a closed, bounded set with arbitrary precision. The parameters a i {\displaystyle a_{i}} , c i {\displaystyle \mathbf {c} _{i}} , and β i {\displaystyle \beta _{i}} are determined in a manner that optimizes the fit between φ {\displaystyle \varphi } and the data. === Normalization === ==== Normalized architecture ==== In addition to the above unnormalized architecture, RBF networks can be normalized. In this case the mapping is φ ( x ) = d e f ∑ i = 1 N a i ρ ( ‖ x − c i ‖ ) ∑ i = 1 N ρ ( ‖ x − c i ‖ ) = ∑ i = 1 N a i u ( ‖ x − c i ‖ ) {\displaystyle \varphi (\mathbf {x} )\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\sum _{i=1}^{N}a_{i}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}{\sum _{i=1}^{N}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}}=\sum _{i=1}^{N}a_{i}u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}} where u ( ‖ x − c i ‖ ) = d e f ρ ( ‖ x − c i ‖ ) ∑ j = 1 N ρ ( ‖ x − c j ‖ ) {\displaystyle u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}{\sum _{j=1}^{N}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{j}\right\Vert {\big )}}}} is known as a normalized radial basis function. ==== Theoretical motivation for normalization ==== There is theoretical justification for this architecture in the case of stochastic data flow. Assume a stochastic kernel approximation for the joint probability density P ( x ∧ y ) = 1 N ∑ i = 1 N ρ ( ‖ x − c i ‖ ) σ ( | y − e i | ) {\displaystyle P\left(\mathbf {x} \land y\right)={1 \over N}\sum _{i=1}^{N}\,\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}\,\sigma {\big (}\left\vert y-e_{i}\right\vert {\big )}} where the weights c i {\displaystyle \mathbf {c} _{i}} and e i {\displaystyle e_{i}} are exemplars from the data and we require the kernels to be normalized ∫ ρ ( ‖ x − c i ‖ ) d n x = 1 {\displaystyle \int \rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}\,d^{n}\mathbf {x} =1} and ∫ σ ( | y − e i | ) d y = 1 {\displaystyle \int \sigma {\big (}\left\vert y-e_{i}\right\vert {\big )}\,dy=1} . The probability densities in the input and output spaces are P ( x ) = ∫ P ( x ∧ y ) d y = 1 N ∑ i = 1 N ρ ( ‖ x − c i ‖ ) {\displaystyle P\left(\mathbf {x} \right)=\int P\left(\mathbf {x} \land y\right)\,dy={1 \over N}\sum _{i=1}^{N}\,\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}} and The expectation of y given an input x {\displaystyle \mathbf {x} } is φ ( x ) = d e f E ( y ∣ x ) = ∫ y P ( y ∣ x ) d y {\displaystyle \varphi \left(\mathbf {x} \right)\ {\stackrel {\mathrm {def} }{=}}\ E\left(y\mid \mathbf {x} \right)=\int y\,P\left(y\mid \mathbf {x} \right)dy} where P ( y ∣ x ) {\displaystyle P\left(y\mid \mathbf {x} \right)} is the conditional probability of y given x {\displaystyle \mathbf {x} } . The conditional probability is related to the joint probability through Bayes' theorem P ( y ∣ x ) = P ( x ∧ y ) P ( x ) {\displaystyle P\left(y\mid \mathbf {x} \right)={\frac {P\left(\mathbf {x} \land y\right)}{P\left(\mathbf {x} \right)}}} which yields φ ( x ) = ∫ y P ( x ∧ y ) P ( x ) d y {\displaystyle \varphi \left(\mathbf {x} \right)=\int y\,{\frac {P\left(\mathbf {x} \land y\right)}{P\left(\mathbf {x} \right)}}\,dy} . This becomes φ ( x ) = ∑ i = 1 N e i ρ ( ‖ x − c i ‖ ) ∑ i = 1 N ρ ( ‖ x − c i ‖ ) = ∑ i = 1 N e i u ( ‖ x − c i ‖ ) {\displaystyle \varphi \left(\mathbf {x} \right)={\frac {\sum _{i=1}^{N}e_{i}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}{\sum _{i=1}^{N}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}}=\sum _{i=1}^{N}e_{i}u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}} when the integrations are performed. === Local linear models === It is sometimes convenient to expand the architecture to include local linear models. In that case the architectures become, to first order, φ ( x ) = ∑ i = 1 N ( a i + b i ⋅ ( x − c i ) ) ρ ( ‖ x − c i ‖ ) {\displaystyle \varphi \left(\mathbf {x} \right)=\sum _{i=1}^{N}\left(a_{i}+\mathbf {b} _{i}\cdot \left(\mathbf {x} -\mathbf {c} _{i}\right)\right)\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}} and φ ( x ) = ∑ i = 1 N ( a i + b i ⋅ ( x − c i ) ) u ( ‖ x − c i ‖ ) {\displaystyle \varphi \left(\mathbf {x} \right)=\sum _{i=1}^{N}\left(a_{i}+\mathbf {b} _{i}\cdot \left(\mathbf {x} -\mathbf {c} _{i}\right)\right)u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}} in the unnormalized and normalized cases, respectively. Here b i {\displaystyle \mathbf {b} _{i}} are weights to be determined. Higher order linear terms are also possible. This result can be written φ ( x ) = ∑ i = 1 2 N ∑ j = 1 n e i j v i j ( x − c i ) {\displaystyle \varphi \left(\mathbf {x} \right)=\sum _{i=1}^{2N}\sum _{j=1}^{n}e_{ij}v_{ij}{\big (}\mathbf {x} -\mathbf {c} _{i}{\big )}} where e i j = { a i , if i ∈ [ 1 , N ] b i j , if i ∈ [ N + 1 , 2 N ] {\displaystyle e_{ij}={\begin{cases}a_{i},&{\mbox{if }}i\in [1,N]\\b_{ij},&{\mbox{if }}i\in [N+1,2N]\end{cases}}} and v i j ( x − c i ) = d e f { δ i j ρ ( ‖ x − c i ‖ ) , if i ∈ [ 1 , N ] ( x i j − c i j ) ρ ( ‖ x − c i ‖ ) , if i ∈ [ N + 1 , 2 N ] {\displaystyle v_{ij}{\big (}\mathbf {x} -\mathbf {c} _{i}{\big )}\ {\stackrel {\mathrm {def} }{=}}\ {\begin{cases}\delta _{ij}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )},&{\mbox{if }}i\in [1,N]\\\left(x_{ij}-c_{ij}\right)\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )},&{\mbox{if }}i\in [N+1,2N]\end{cases}}} in the unnormalized case and in the normalized case. Here δ i j {\displaystyle \delta _{ij}} is a Kronecker delta function defined as δ i j = { 1 , if i = j 0 , if i ≠ j {\displaystyle \delta _{ij}={\begin{cases}1,&{\mbox{if }}i=j\\0,&{\mbox{if }}i\neq j\end{cases}}} . == Training == RBF networks are typically trained from pairs of input and target values x ( t ) , y ( t ) {\displaystyle \mathbf {x} (t),y(t)} , t = 1 , … , T {\displaystyle t=1,\dots ,T} by a two-step algorithm. In the first step, the center vectors c i {\displaystyle \mathbf {c} _{i}} of the RBF functions in the hidden layer

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  Quack.com

  Quack.com was an early voice portal company. The domain name later was used for Quack, an iPad search application from AOL. == History == It was founded in 1998 by Steven Woods, Jeromy Carriere and Alex Quilici as a Pittsburgh, Pennsylvania, USA, based voice portal infrastructure company named Quackware. Quack was the first company to try to create a voice portal: a consumer-based destination "site" in which consumers could not only access information by voice alone, but also complete transactions. Quackware launched a beta phone service in 1999 that allowed consumers to purchase books from sites such as Amazon and CDs from sites such as CDNow by answering a short set of questions. Quack followed with a set of information services from movie listings (inspired by, but expanding upon, Moviefone) to news, weather and stock quotes. This concept introduced a series of lookalike startups including Tellme Networks which raised more money than any Internet startup in history on a similar concept. Quack received its first venture funding from HDL Capital in 1999 and moved operations to Mountain View in Silicon Valley, California in 1999. A deal with Lycos was announced in May 2000. In September 2000 Quack was acquired for $200 million by America Online (AOL) and moved onto the Netscape campus with what was left of the Netscape team. Quack was attacked in the Canadian press for being representative of the Canadian "brain drain" to the US during the Internet bubble, focusing its recruiting efforts on the University of Waterloo, hiring more than 50 engineers from Waterloo in less than 10 months. Quack competitor Tellme Networks raised enormous funds in what became a highly competitive market in 2000, with the emergence of more than a dozen additional competitors in a 12-month period. Following its acquisition by America Online in an effort led by Ted Leonsis to bring Quack into AOL Interactive, the Quack voice service became AOLbyPhone as one of AOL's "web properties" along with MapQuest, Moviefone and others. Quack secured several patents that underlie the technical challenges of delivering interactive voice services. Constructing a voice portal required integrations and innovations not only in speech recognition and speech generation, but also in databases, application specification, constraint-based reasoning and artificial intelligence and computational linguistics. "Quack"'s name derived from the company goal of providing not only voice-based services, but more broadly "Quick Ubiquitous Access to Consumer Knowledge". The patents assigned to Quack.com include: System and method for voice access to Internet-based information, System and method for advertising with an Internet Voice Portal and recognizing the axiom that in interactive voice systems one must "know the set of possible answers to a question before asking it". System and method for determining if one web site has the same information as another web site. Quack.com was spoofed in The Simpsons in March 2002 in the episode "Blame It on Lisa" in which a "ComQuaak" sign is replaced by another equally crazy telecom company name. == 2010 onwards == In July 2010, quack.com became the focus of a new AOL iPad application, that was a web search experience. The product delivers web results and blends in picture, video and Twitter results. It enables you to preview the web results before you go to the site, search within each result, and flip through the results pages, making full use of the iPad's touch screen features. The iPad app was free via iTunes, but support discontinued in 2012.

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  Modern Hopfield network

  Modern Hopfield networks (also known as Dense Associative Memories) are generalizations of the classical Hopfield networks that break the linear scaling relationship between the number of input features and the number of stored memories. This is achieved by introducing stronger non-linearities (either in the energy function or neurons’ activation functions) leading to super-linear (even an exponential) memory storage capacity as a function of the number of feature neurons. The network still requires a sufficient number of hidden neurons. The key theoretical idea behind the modern Hopfield networks is to use an energy function and an update rule that is more sharply peaked around the stored memories in the space of neuron’s configurations compared to the classical Hopfield network. == Classical Hopfield networks == Hopfield networks are recurrent neural networks with dynamical trajectories converging to fixed point attractor states and described by an energy function. The state of each model neuron i {\textstyle i} is defined by a time-dependent variable V i {\displaystyle V_{i}} , which can be chosen to be either discrete or continuous. A complete model describes the mathematics of how the future state of activity of each neuron depends on the known present or previous activity of all the neurons. In the original Hopfield model of associative memory, the variables were binary, and the dynamics were described by a one-at-a-time update of the state of the neurons. An energy function quadratic in the V i {\displaystyle V_{i}} was defined, and the dynamics consisted of changing the activity of each single neuron i {\displaystyle i} only if doing so would lower the total energy of the system. This same idea was extended to the case of V i {\displaystyle V_{i}} being a continuous variable representing the output of neuron i {\displaystyle i} , and V i {\displaystyle V_{i}} being a monotonic function of an input current. The dynamics became expressed as a set of first-order differential equations for which the "energy" of the system always decreased. The energy in the continuous case has one term which is quadratic in the V i {\displaystyle V_{i}} (as in the binary model), and a second term which depends on the gain function (neuron's activation function). While having many desirable properties of associative memory, both of these classical systems suffer from a small memory storage capacity, which scales linearly with the number of input features. == Discrete variables == A simple example of the Modern Hopfield network can be written in terms of binary variables V i {\displaystyle V_{i}} that represent the active V i = + 1 {\displaystyle V_{i}=+1} and inactive V i = − 1 {\displaystyle V_{i}=-1} state of the model neuron i {\displaystyle i} . E = − ∑ μ = 1 N mem F ( ∑ i = 1 N f ξ μ i V i ) {\displaystyle E=-\sum \limits _{\mu =1}^{N_{\text{mem}}}F{\Big (}\sum \limits _{i=1}^{N_{f}}\xi _{\mu i}V_{i}{\Big )}} In this formula the weights ξ μ i {\textstyle \xi _{\mu i}} represent the matrix of memory vectors (index μ = 1... N mem {\displaystyle \mu =1...N_{\text{mem}}} enumerates different memories, and index i = 1... N f {\displaystyle i=1...N_{f}} enumerates the content of each memory corresponding to the i {\displaystyle i} -th feature neuron), and the function F ( x ) {\displaystyle F(x)} is a rapidly growing non-linear function. The update rule for individual neurons (in the asynchronous case) can be written in the following form V i ( t + 1 ) = sign ⁡ [ ∑ μ = 1 N mem ( F ( ξ μ i + ∑ j ≠ i ξ μ j V j ( t ) ) − F ( − ξ μ i + ∑ j ≠ i ξ μ j V j ( t ) ) ) ] {\displaystyle V_{i}^{(t+1)}=\operatorname {sign} {\bigg [}\sum \limits _{\mu =1}^{N_{\text{mem}}}{\bigg (}F{\Big (}\xi _{\mu i}+\sum \limits _{j\neq i}\xi _{\mu j}V_{j}^{(t)}{\Big )}-F{\Big (}-\xi _{\mu i}+\sum \limits _{j\neq i}\xi _{\mu j}V_{j}^{(t)}{\Big )}{\bigg )}{\bigg ]}} which states that in order to calculate the updated state of the i {\textstyle i} -th neuron the network compares two energies: the energy of the network with the i {\displaystyle i} -th neuron in the ON state and the energy of the network with the i {\displaystyle i} -th neuron in the OFF state, given the states of the remaining neuron. The updated state of the i {\displaystyle i} -th neuron selects the state that has the lowest of the two energies. In the limiting case when the non-linear energy function is quadratic F ( x ) = x 2 {\displaystyle F(x)=x^{2}} these equations reduce to the familiar energy function and the update rule for the classical binary Hopfield network. The memory storage capacity of these networks can be calculated for random binary patterns. For the power energy function F ( x ) = x n {\displaystyle F(x)=x^{n}} the maximal number of memories that can be stored and retrieved from this network without errors is given by N mem max ≈ 1 2 ( 2 n − 3 ) ! ! N f n − 1 ln ⁡ ( N f ) {\displaystyle N_{\text{mem}}^{\max }\approx {\frac {1}{2(2n-3)!!}}{\frac {N_{f}^{n-1}}{\ln(N_{f})}}} For an exponential energy function F ( x ) = e x {\textstyle F(x)=e^{x}} the memory storage capacity is exponential in the number of feature neurons N mem max ≈ 2 N f / 2 {\displaystyle N_{\text{mem}}^{\max }\approx 2^{N_{f}/2}} == Continuous variables == Modern Hopfield networks or Dense Associative Memories can be best understood in continuous variables and continuous time. Consider the network architecture, shown in Fig.1, and the equations for the neurons' state evolutionwhere the currents of the feature neurons are denoted by x i {\textstyle x_{i}} , and the currents of the memory neurons are denoted by h μ {\displaystyle h_{\mu }} ( h {\displaystyle h} stands for hidden neurons). There are no synaptic connections among the feature neurons or the memory neurons. A matrix ξ μ i {\displaystyle \xi _{\mu i}} denotes the strength of synapses from a feature neuron i {\displaystyle i} to the memory neuron μ {\displaystyle \mu } . The synapses are assumed to be symmetric, so that the same value characterizes a different physical synapse from the memory neuron μ {\displaystyle \mu } to the feature neuron i {\displaystyle i} . The outputs of the memory neurons and the feature neurons are denoted by f μ {\displaystyle f_{\mu }} and g i {\displaystyle g_{i}} , which are non-linear functions of the corresponding currents. In general these outputs can depend on the currents of all the neurons in that layer so that f μ = f ( { h μ } ) {\displaystyle f_{\mu }=f(\{h_{\mu }\})} and g i = g ( { x i } ) {\textstyle g_{i}=g(\{x_{i}\})} . It is convenient to define these activation function as derivatives of the Lagrangian functions for the two groups of neuronsThis way the specific form of the equations for neuron's states is completely defined once the Lagrangian functions are specified. Finally, the time constants for the two groups of neurons are denoted by τ f {\displaystyle \tau _{f}} and τ h {\displaystyle \tau _{h}} , I i {\displaystyle I_{i}} is the input current to the network that can be driven by the presented data. General systems of non-linear differential equations can have many complicated behaviors that can depend on the choice of the non-linearities and the initial conditions. For Hopfield networks, however, this is not the case - the dynamical trajectories always converge to a fixed point attractor state. This property is achieved because these equations are specifically engineered so that they have an underlying energy function The terms grouped into square brackets represent a Legendre transform of the Lagrangian function with respect to the states of the neurons. If the Hessian matrices of the Lagrangian functions are positive semi-definite, the energy function is guaranteed to decrease on the dynamical trajectory This property makes it possible to prove that the system of dynamical equations describing temporal evolution of neurons' activities will eventually reach a fixed point attractor state. In certain situations one can assume that the dynamics of hidden neurons equilibrates at a much faster time scale compared to the feature neurons, τ h ≪ τ f {\textstyle \tau _{h}\ll \tau _{f}} . In this case the steady state solution of the second equation in the system (1) can be used to express the currents of the hidden units through the outputs of the feature neurons. This makes it possible to reduce the general theory (1) to an effective theory for feature neurons only. The resulting effective update rules and the energies for various common choices of the Lagrangian functions are shown in Fig.2. In the case of log-sum-exponential Lagrangian function the update rule (if applied once) for the states of the feature neurons is the attention mechanism commonly used in many modern AI systems (see Ref. for the derivation of this result from the continuous time formulation). == Relationship to classical Hopfield network with continuous variables == Classical formulation of continuous Hopfield networks can be understood as a

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  Santa Fe Trail problem

  The Santa Fe Trail problem is a genetic programming exercise in which artificial ants search for food pellets according to a programmed set of instructions. The layout of food pellets in the Santa Fe Trail problem has become a standard for comparing different genetic programming algorithms and solutions. One method for programming and testing algorithms on the Santa Fe Trail problem is by using the NetLogo application. There is at least one case of a student creating a Lego robotic ant to solve the problem.

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  LPBoost

  Linear Programming Boosting (LPBoost) is a supervised classifier from the boosting family of classifiers. LPBoost maximizes a margin between training samples of different classes, and thus also belongs to the class of margin classifier algorithms. Consider a classification function f : X → { − 1 , 1 } , {\displaystyle f:{\mathcal {X}}\to \{-1,1\},} which classifies samples from a space X {\displaystyle {\mathcal {X}}} into one of two classes, labelled 1 and -1, respectively. LPBoost is an algorithm for learning such a classification function, given a set of training examples with known class labels. LPBoost is a machine learning technique especially suited for joint classification and feature selection in structured domains. == LPBoost overview == As in all boosting classifiers, the final classification function is of the form f ( x ) = ∑ j = 1 J α j h j ( x ) , {\displaystyle f({\boldsymbol {x}})=\sum _{j=1}^{J}\alpha _{j}h_{j}({\boldsymbol {x}}),} where α j {\displaystyle \alpha _{j}} are non-negative weightings for weak classifiers h j : X → { − 1 , 1 } {\displaystyle h_{j}:{\mathcal {X}}\to \{-1,1\}} . Each individual weak classifier h j {\displaystyle h_{j}} may be just a little bit better than random, but the resulting linear combination of many weak classifiers can perform very well. LPBoost constructs f {\displaystyle f} by starting with an empty set of weak classifiers. Iteratively, a single weak classifier to add to the set of considered weak classifiers is selected, added and all the weights α {\displaystyle {\boldsymbol {\alpha }}} for the current set of weak classifiers are adjusted. This is repeated until no weak classifiers to add remain. The property that all classifier weights are adjusted in each iteration is known as totally-corrective property. Early boosting methods, such as AdaBoost do not have this property and converge slower. == Linear program == More generally, let H = { h ( ⋅ ; ω ) | ω ∈ Ω } {\displaystyle {\mathcal {H}}=\{h(\cdot ;\omega )|\omega \in \Omega \}} be the possibly infinite set of weak classifiers, also termed hypotheses. One way to write down the problem LPBoost solves is as a linear program with infinitely many variables. The primal linear program of LPBoost, optimizing over the non-negative weight vector α {\displaystyle {\boldsymbol {\alpha }}} , the non-negative vector ξ {\displaystyle {\boldsymbol {\xi }}} of slack variables and the margin ρ {\displaystyle \rho } is the following. min α , ξ , ρ − ρ + D ∑ n = 1 ℓ ξ n sb.t. ∑ ω ∈ Ω y n α ω h ( x n ; ω ) + ξ n ≥ ρ , n = 1 , … , ℓ , ∑ ω ∈ Ω α ω = 1 , ξ n ≥ 0 , n = 1 , … , ℓ , α ω ≥ 0 , ω ∈ Ω , ρ ∈ R . {\displaystyle {\begin{array}{cl}{\underset {{\boldsymbol {\alpha }},{\boldsymbol {\xi }},\rho }{\min }}&-\rho +D\sum _{n=1}^{\ell }\xi _{n}\\{\textrm {sb.t.}}&\sum _{\omega \in \Omega }y_{n}\alpha _{\omega }h({\boldsymbol {x}}_{n};\omega )+\xi _{n}\geq \rho ,\qquad n=1,\dots ,\ell ,\\&\sum _{\omega \in \Omega }\alpha _{\omega }=1,\\&\xi _{n}\geq 0,\qquad n=1,\dots ,\ell ,\\&\alpha _{\omega }\geq 0,\qquad \omega \in \Omega ,\\&\rho \in {\mathbb {R} }.\end{array}}} Note the effects of slack variables ξ ≥ 0 {\displaystyle {\boldsymbol {\xi }}\geq 0} : their one-norm is penalized in the objective function by a constant factor D {\displaystyle D} , which—if small enough—always leads to a primal feasible linear program. Here we adopted the notation of a parameter space Ω {\displaystyle \Omega } , such that for a choice ω ∈ Ω {\displaystyle \omega \in \Omega } the weak classifier h ( ⋅ ; ω ) : X → { − 1 , 1 } {\displaystyle h(\cdot ;\omega ):{\mathcal {X}}\to \{-1,1\}} is uniquely defined. When the above linear program was first written down in early publications about boosting methods it was disregarded as intractable due to the large number of variables α {\displaystyle {\boldsymbol {\alpha }}} . Only later it was discovered that such linear programs can indeed be solved efficiently using the classic technique of column generation. === Column generation for LPBoost === In a linear program a column corresponds to a primal variable. Column generation is a technique to solve large linear programs. It typically works in a restricted problem, dealing only with a subset of variables. By generating primal variables iteratively and on-demand, eventually the original unrestricted problem with all variables is recovered. By cleverly choosing the columns to generate the problem can be solved such that while still guaranteeing the obtained solution to be optimal for the original full problem, only a small fraction of columns has to be created. ==== LPBoost dual problem ==== Columns in the primal linear program corresponds to rows in the dual linear program. The equivalent dual linear program of LPBoost is the following linear program. max λ , γ γ sb.t. ∑ n = 1 ℓ y n h ( x n ; ω ) λ n + γ ≤ 0 , ω ∈ Ω , 0 ≤ λ n ≤ D , n = 1 , … , ℓ , ∑ n = 1 ℓ λ n = 1 , γ ∈ R . {\displaystyle {\begin{array}{cl}{\underset {{\boldsymbol {\lambda }},\gamma }{\max }}&\gamma \\{\textrm {sb.t.}}&\sum _{n=1}^{\ell }y_{n}h({\boldsymbol {x}}_{n};\omega )\lambda _{n}+\gamma \leq 0,\qquad \omega \in \Omega ,\\&0\leq \lambda _{n}\leq D,\qquad n=1,\dots ,\ell ,\\&\sum _{n=1}^{\ell }\lambda _{n}=1,\\&\gamma \in \mathbb {R} .\end{array}}} For linear programs the optimal value of the primal and dual problem are equal. For the above primal and dual problems, the optimal value is equal to the negative 'soft margin'. The soft margin is the size of the margin separating positive from negative training instances minus positive slack variables that carry penalties for margin-violating samples. Thus, the soft margin may be positive although not all samples are linearly separated by the classification function. The latter is called the 'hard margin' or 'realized margin'. ==== Convergence criterion ==== Consider a subset of the satisfied constraints in the dual problem. For any finite subset we can solve the linear program and thus satisfy all constraints. If we could prove that of all the constraints which we did not add to the dual problem no single constraint is violated, we would have proven that solving our restricted problem is equivalent to solving the original problem. More formally, let γ ∗ {\displaystyle \gamma ^{}} be the optimal objective function value for any restricted instance. Then, we can formulate a search problem for the 'most violated constraint' in the original problem space, namely finding ω ∗ ∈ Ω {\displaystyle \omega ^{}\in \Omega } as ω ∗ = argmax ω ∈ Ω ∑ n = 1 ℓ y n h ( x n ; ω ) λ n . {\displaystyle \omega ^{}={\underset {\omega \in \Omega }{\textrm {argmax}}}\sum _{n=1}^{\ell }y_{n}h({\boldsymbol {x}}_{n};\omega )\lambda _{n}.} That is, we search the space H {\displaystyle {\mathcal {H}}} for a single decision stump h ( ⋅ ; ω ∗ ) {\displaystyle h(\cdot ;\omega ^{})} maximizing the left hand side of the dual constraint. If the constraint cannot be violated by any choice of decision stump, none of the corresponding constraint can be active in the original problem and the restricted problem is equivalent. ==== Penalization constant ==== D {\displaystyle D} The positive value of penalization constant D {\displaystyle D} has to be found using model selection techniques. However, if we choose D = 1 ℓ ν {\displaystyle D={\frac {1}{\ell \nu }}} , where ℓ {\displaystyle \ell } is the number of training samples and 0 < ν < 1 {\displaystyle 0<\nu <1} , then the new parameter ν {\displaystyle \nu } has the following properties. ν {\displaystyle \nu } is an upper bound on the fraction of training errors; that is, if k {\displaystyle k} denotes the number of misclassified training samples, then k ℓ ≤ ν {\displaystyle {\frac {k}{\ell }}\leq \nu } . ν {\displaystyle \nu } is a lower bound on the fraction of training samples outside or on the margin. == Algorithm == Input: Training set X = { x 1 , … , x ℓ } {\displaystyle X=\{{\boldsymbol {x}}_{1},\dots ,{\boldsymbol {x}}_{\ell }\}} , x i ∈ X {\displaystyle {\boldsymbol {x}}_{i}\in {\mathcal {X}}} Training labels Y = { y 1 , … , y ℓ } {\displaystyle Y=\{y_{1},\dots ,y_{\ell }\}} , y i ∈ { − 1 , 1 } {\displaystyle y_{i}\in \{-1,1\}} Convergence threshold θ ≥ 0 {\displaystyle \theta \geq 0} Output: Classification function f : X → { − 1 , 1 } {\displaystyle f:{\mathcal {X}}\to \{-1,1\}} Initialization Weights, uniform λ n ← 1 ℓ , n = 1 , … , ℓ {\displaystyle \lambda _{n}\leftarrow {\frac {1}{\ell }},\quad n=1,\dots ,\ell } Edge γ ← 0 {\displaystyle \gamma \leftarrow 0} Hypothesis count J ← 1 {\displaystyle J\leftarrow 1} Iterate h ^ ← argmax ω ∈ Ω ∑ n = 1 ℓ y n h ( x n ; ω ) λ n {\displaystyle {\hat {h}}\leftarrow {\underset {\omega \in \Omega }{\textrm {argmax}}}\sum _{n=1}^{\ell }y_{n}h({\boldsymbol {x}}_{n};\omega )\lambda _{n}} if ∑ n = 1 ℓ y n h ^ ( x n ) λ n + γ ≤ θ {\displaystyle \sum _{n=1}^{\ell }y_{n}{\hat {h}}({\boldsymbol {x}}_{n})\lambda _{n}+\gamma \leq \theta } then break h J ← h ^ {\displaystyle h_{J}\leftarrow {\hat {h}}} J

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  Content as a service

  Content as a service (CaaS) or managed content as a service (MCaaS) is a service-oriented model, where the service provider delivers the content on demand to the service consumer via web services that are licensed under subscription. The content is hosted by the service provider centrally in the cloud and offered to a number of consumers that need the content delivered into any applications or system, hence content can be demanded by the consumers as and when required. Content as a Service is a way to provide raw content (in other words, without the need for a specific human compatible representation, such as HTML) in a way that other systems can make use of it. Content as a Service is not meant for direct human consumption, but rather for other platforms to consume and make use of the content according to their particular needs. This happens usually on the cloud, with a centralized platform which can be globally accessible and provides a standard format for your content. With Content as a Service, you centralize your content into a single repository, where you can manage it, categorize it, make it available to others, search for it, or do whatever you wish with it. == Overview == The content delivered typically could be one or more of the following The technical terminology related to equipment or spares that is required to procure or design the materials The industrial terminology of the equipment or spares Technical values pertaining to various types, specifications, applications, characteristics of equipment or spares Sourcing information which will help in procurement or supply-chain management of equipment or spares Descriptive specifications of equipment or spares based on the product reference number or identifier UNSPSC codes or industry practiced classifications ISO, IEC compliant terminology Ontology or Technical Dictionary of products & services Predefined content for specific business needs The term "Content as a service" (CaaS) is considered to be part of the nomenclature of cloud computing service models & Service-oriented architecture along with Software as a service (SaaS), Infrastructure as a service (IaaS), and Platform as a service (PaaS).

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  Nonlinear dimensionality reduction

  Nonlinear dimensionality reduction (NLDR), also known as manifold learning, is any of various related techniques that aim to project high-dimensional data, potentially existing across non-linear manifolds which cannot be adequately captured by linear decomposition methods, onto lower-dimensional latent manifolds, with the goal of either visualizing the data in the low-dimensional space, or learning the mapping (either from the high-dimensional space to the low-dimensional embedding or vice versa) itself. The techniques described below can be understood as generalizations of linear decomposition methods used for dimensionality reduction, such as singular value decomposition and principal component analysis. == Applications of NLDR == High dimensional data can be hard for machines to work with, requiring significant time and space for analysis. It also presents a challenge for humans, since it's hard to visualize or understand data in more than three dimensions. Reducing the dimensionality of a data set, while keeping its essential features relatively intact, can make algorithms more efficient and allow analysts to visualize trends and patterns. The reduced-dimensional representations of data are often referred to as "intrinsic variables". This description implies that these are the values from which the data was produced. For example, consider a dataset that contains images of a letter 'A', which has been scaled and rotated by varying amounts. Each image has 32×32 pixels. Each image can be represented as a vector of 1024 pixel values. Each row is a sample on a two-dimensional manifold in 1024-dimensional space (a Hamming space). The intrinsic dimensionality is two, because two variables (rotation and scale) were varied in order to produce the data. Information about the shape or look of a letter 'A' is not part of the intrinsic variables because it is the same in every instance. Nonlinear dimensionality reduction will discard the correlated information (the letter 'A') and recover only the varying information (rotation and scale). By comparison, if principal component analysis, which is a linear dimensionality reduction algorithm, is used to reduce this same dataset into two dimensions, the resulting values are not so well organized. This demonstrates that the high-dimensional vectors (each representing a letter 'A') that sample this manifold vary in a non-linear manner. It should be apparent, therefore, that NLDR has several applications in the field of computer-vision. For example, consider a robot that uses a camera to navigate in a closed static environment. The images obtained by that camera can be considered to be samples on a manifold in high-dimensional space, and the intrinsic variables of that manifold will represent the robot's position and orientation. Invariant manifolds are of general interest for model order reduction in dynamical systems. In particular, if there is an attracting invariant manifold in the phase space, nearby trajectories will converge onto it and stay on it indefinitely, rendering it a candidate for dimensionality reduction of the dynamical system. While such manifolds are not guaranteed to exist in general, the theory of spectral submanifolds (SSM) gives conditions for the existence of unique attracting invariant objects in a broad class of dynamical systems. Active research in NLDR seeks to unfold the observation manifolds associated with dynamical systems to develop modeling techniques. Some of the more prominent nonlinear dimensionality reduction techniques are listed below. == Important concepts == === Sammon's mapping === Sammon's mapping is one of the first and most popular NLDR techniques. === Self-organizing map === The self-organizing map (SOM, also called Kohonen map) and its probabilistic variant generative topographic mapping (GTM) use a point representation in the embedded space to form a latent variable model based on a non-linear mapping from the embedded space to the high-dimensional space. These techniques are related to work on density networks, which also are based around the same probabilistic model. === Kernel principal component analysis === Perhaps the most widely used algorithm for dimensional reduction is kernel PCA. PCA begins by computing the covariance matrix of the m × n {\displaystyle m\times n} matrix X {\displaystyle \mathbf {X} } C = 1 m ∑ i = 1 m x i x i T . {\displaystyle C={\frac {1}{m}}\sum _{i=1}^{m}{\mathbf {x} _{i}\mathbf {x} _{i}^{\mathsf {T}}}.} It then projects the data onto the first k eigenvectors of that matrix. By comparison, KPCA begins by computing the covariance matrix of the data after being transformed into a higher-dimensional space, C = 1 m ∑ i = 1 m Φ ( x i ) Φ ( x i ) T . {\displaystyle C={\frac {1}{m}}\sum _{i=1}^{m}{\Phi (\mathbf {x} _{i})\Phi (\mathbf {x} _{i})^{\mathsf {T}}}.} It then projects the transformed data onto the first k eigenvectors of that matrix, just like PCA. It uses the kernel trick to factor away much of the computation, such that the entire process can be performed without actually computing Φ ( x ) {\displaystyle \Phi (\mathbf {x} )} . Of course Φ {\displaystyle \Phi } must be chosen such that it has a known corresponding kernel. Unfortunately, it is not trivial to find a good kernel for a given problem, so KPCA does not yield good results with some problems when using standard kernels. For example, it is known to perform poorly with these kernels on the Swiss roll manifold. However, one can view certain other methods that perform well in such settings (e.g., Laplacian Eigenmaps, LLE) as special cases of kernel PCA by constructing a data-dependent kernel matrix. KPCA has an internal model, so it can be used to map points onto its embedding that were not available at training time. === Principal curves and manifolds === Principal curves and manifolds give the natural geometric framework for nonlinear dimensionality reduction and extend the geometric interpretation of PCA by explicitly constructing an embedded manifold, and by encoding using standard geometric projection onto the manifold. This approach was originally proposed by Trevor Hastie in his 1984 thesis, which he formally introduced in 1989. This idea has been explored further by many authors. How to define the "simplicity" of the manifold is problem-dependent, however, it is commonly measured by the intrinsic dimensionality and/or the smoothness of the manifold. Usually, the principal manifold is defined as a solution to an optimization problem. The objective function includes a quality of data approximation and some penalty terms for the bending of the manifold. The popular initial approximations are generated by linear PCA and Kohonen's SOM. === Laplacian eigenmaps === Laplacian eigenmaps uses spectral techniques to perform dimensionality reduction. This technique relies on the basic assumption that the data lies in a low-dimensional manifold in a high-dimensional space. This algorithm cannot embed out-of-sample points, but techniques based on Reproducing kernel Hilbert space regularization exist for adding this capability. Such techniques can be applied to other nonlinear dimensionality reduction algorithms as well. Traditional techniques like principal component analysis do not consider the intrinsic geometry of the data. Laplacian eigenmaps builds a graph from neighborhood information of the data set. Each data point serves as a node on the graph and connectivity between nodes is governed by the proximity of neighboring points (using e.g. the k-nearest neighbor algorithm). The graph thus generated can be considered as a discrete approximation of the low-dimensional manifold in the high-dimensional space. Minimization of a cost function based on the graph ensures that points close to each other on the manifold are mapped close to each other in the low-dimensional space, preserving local distances. The eigenfunctions of the Laplace–Beltrami operator on the manifold serve as the embedding dimensions, since under mild conditions this operator has a countable spectrum that is a basis for square integrable functions on the manifold (compare to Fourier series on the unit circle manifold). Attempts to place Laplacian eigenmaps on solid theoretical ground have met with some success, as under certain nonrestrictive assumptions, the graph Laplacian matrix has been shown to converge to the Laplace–Beltrami operator as the number of points goes to infinity. === Isomap === Isomap is a combination of the Floyd–Warshall algorithm with classic Multidimensional Scaling (MDS). Classic MDS takes a matrix of pair-wise distances between all points and computes a position for each point. Isomap assumes that the pair-wise distances are only known between neighboring points, and uses the Floyd–Warshall algorithm to compute the pair-wise distances between all other points. This effectively estimates the full matrix of pair-wise geodesic distances between all of the points. Isomap th

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  Constructing skill trees

  Constructing skill trees (CST) is a hierarchical reinforcement learning algorithm which can build skill trees from a set of sample solution trajectories obtained from demonstration. CST uses an incremental MAP (maximum a posteriori) change point detection algorithm to segment each demonstration trajectory into skills and integrate the results into a skill tree. CST was introduced by George Konidaris, Scott Kuindersma, Andrew Barto and Roderic Grupen in 2010. == Algorithm == CST consists of mainly three parts;change point detection, alignment and merging. The main focus of CST is online change-point detection. The change-point detection algorithm is used to segment data into skills and uses the sum of discounted reward R t {\displaystyle R_{t}} as the target regression variable. Each skill is assigned an appropriate abstraction. A particle filter is used to control the computational complexity of CST. The change point detection algorithm is implemented as follows. The data for times t ∈ T {\displaystyle t\in T} and models Q with prior p ( q ∈ Q ) {\displaystyle p(q\in Q)} are given. The algorithm is assumed to be able to fit a segment from time j + 1 {\displaystyle j+1} to t using model q with the fit probability P ( j , t , q ) {\displaystyle P(j,t,q)_{}^{}} . A linear regression model with Gaussian noise is used to compute P ( j , t , q ) {\displaystyle P(j,t,q)} . The Gaussian noise prior has mean zero, and variance which follows I n v e r s e G a m m a ( v 2 , u 2 ) {\displaystyle \mathrm {InverseGamma} \left({\frac {v}{2}},{\frac {u}{2}}\right)} . The prior for each weight follows N o r m a l ( 0 , σ 2 δ ) {\displaystyle \mathrm {Normal} (0,\sigma ^{2}\delta )} . The fit probability P ( j , t , q ) {\displaystyle P(j,t,q)} is computed by the following equation. P ( j , t , q ) = π − n 2 δ m | ( A + D ) − 1 | 1 2 u v 2 ( y + u ) u + v 2 Γ ( n + v 2 ) Γ ( v 2 ) {\displaystyle P(j,t,q)={\frac {\pi ^{-{\frac {n}{2}}}}{\delta ^{m}}}\left|(A+D)^{-1}\right|^{\frac {1}{2}}{\frac {u^{\frac {v}{2}}}{(y+u)^{\frac {u+v}{2}}}}{\frac {\Gamma ({\frac {n+v}{2}})}{\Gamma ({\frac {v}{2}})}}} Then, CST compute the probability of the changepoint at time j with model q, P t ( j , q ) {\displaystyle P_{t}(j,q)} and P j MAP {\displaystyle P_{j}^{\text{MAP}}} using a Viterbi algorithm. P t ( j , q ) = ( 1 − G ( t − j − 1 ) ) P ( j , t , q ) p ( q ) P j MAP {\displaystyle P_{t}(j,q)=(1-G(t-j-1))P(j,t,q)p(q)P_{j}^{\text{MAP}}} P j MAP = max i , q P j ( i , q ) g ( j − i ) 1 − G ( j − i − 1 ) , ∀ j < t {\displaystyle P_{j}^{\text{MAP}}=\max _{i,q}{\frac {P_{j}(i,q)g(j-i)}{1-G(j-i-1)}},\forall j Read more →