AI Assistant Grok

AI Assistant Grok — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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  Foodsi

  Foodsi is a Polish mobile application that connects customers with restaurants, convenience stores, bakeries and cafes that have a surplus of food, allowing its users to buy the surplus at a reduced price. The service launched in 2019 in Warsaw and has expanded to other major cities in Poland. In 2023, a new feature was introduced in the app, allowing users to buy packages not only with self-pickup but also with delivery. The products range has also been expanded to include unsold magazines, cosmetics or plants. == History == The company was created in 2019 in Poland by Mateusz Kowalczyk and Jakub Fryszczyn. During studies in their home country and abroad, when they made a living working in restaurants and bakeries, they recognized the problem and the scale of food waste. They launched the application by themselves, having previously raised PLN 100,000 on their own for the purpose. Initially, Foodsi was an Android-only app, but over time, an IOS version was developed. In 2022, the startup raised PLN 6 million in a seed round from VC companies including CofounderZone and Status Starter, as well as private investors such as founders of Pyszne.pl. As of December 2023, it claimed more than 5000 businesses, serving over 1,5 million users, have saved nearly 3 million bags of food. == Purpose == Foodsi aims to significantly reduce food waste, which contributes to the Sustainable Development Goals. The application bridges the gap between the customers who are looking for shopping deals and the companies that want to reduce surplus products but are unable to sell them at a normal price. This allows the customers to buy unsold products for as little as 30% of the normal price. The company claims that every 4 out of 5 packages are sold on average. As of 2019 Foodsi employed more than 30 people. By 2024 it was more than 50. For now, Foodsi operates in major Polish cities such as Warsaw, Kraków, Trójmiasto, Wrocław, Poznań etc. However, in the upcoming years, Foodsi plans to expand to other countries. == Use == To start selling surplus, a company must leave Foodsi its contact information to register in the system. Registration in the app is completely free of charge. Then, companies offer available packages anticipating what won’t be sold and post them in the app along with the price so that users can buy them and pick them up. Companies can put their packages in the app at any time during the day. Users can pick up packages from bakeries, grocery stores, restaurants, but also florists and beauty stores. Foodsi charges a small commission on each package from the cooperating companies. If a user wants to start ordering packages from Foodsi, he or she needs to install the app on their mobile phone (Android or IOS) and register an account. The app displays a list of restaurants and other venues available in a specific region set by the user's location. Customers can see the price, address, distance and time range for package pickup. Packages are usually in the form of so-called 'surprise-packages', meaning that customers do not know specifically what kind of food/product will be inside. Some restaurants offer a choice of different package sizes. Prices are up to 70% lower than those of the original products. Customers have to show up at the restaurant to pick up the package using their phone at a time specified in the app. == Awards == Auler All-Stars 2025 - 3rd place Deloitte Technology Fast 50 - 2025 Central Europe Executive Club - Innowacja Roku: Żywność i Rolnictwo - Wyróżnienie (2025) Stena Circular Economy Award - Lider Gospodarki Obiegu Zamkniętego (2025) - wyróżnienie w kategorii start-up wdrażający GOZ na rynku polskim 255th place in the international poll FoodTech 500 2025 Finalist for the EY Entrepreneur Of The Year™ 2025 Wpływowi 2024 - Laureat w kategorii “Zrównoważony rozwój” Supplier of the Year 2024 - XXII Food & Business Forum Supplier of the Year 2024 - VII Sweets & Coffee Forum Innovative Leader 2024 - Leader in Food / Food-Tech Category - Executive Summit “Orzeł Innowacji - Start-up z potencjałem Polska-Świat” (Rzeczpospolita, 2024) 102nd place in the international poll FoodTech 500 2024 Auler 2023 Startup of the Year 2023 according to money.pl Start(up) w zrównoważoną przyszłość Kongresu Kompas ESG 2023 Marka Godna Zaufania according to My Company Polska 2023 184th place in the international poll FoodTech 500 2023 In 2023, Foodsi co-founder Mateusz Kowalczyk was recognized by Forbes magazine and included in its "30 before 30" list.

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  Optical character recognition

  Optical character recognition (OCR) or optical character reader is the electronic or mechanical conversion of images of typed, handwritten or printed text into machine-encoded text, whether from a scanned document, a photo of a document, a scene photo (for example the text on signs and billboards in a landscape photo) or from subtitle text superimposed on an image (for example: from a television broadcast). Widely used as a form of data entry from printed paper data records – whether passport documents, invoices, bank statements, computerized receipts, business cards, mail, printed data, or any suitable documentation – it is a common method of digitizing printed texts so that they can be electronically edited, searched, stored more compactly, displayed online, and used in machine processes such as cognitive computing, machine translation, (extracted) text-to-speech, key data and text mining. OCR is a field of research in pattern recognition, artificial intelligence and computer vision. Early versions needed to be trained with images of each character, and worked on one font at a time. Advanced systems capable of producing a high degree of accuracy for most fonts are now common, and with support for a variety of image file format inputs. Some systems are capable of reproducing formatted output that closely approximates the original page including images, columns, and other non-textual components. == History == Early optical character recognition may be traced to technologies involving telegraphy and creating reading devices for the blind. In 1914, Emanuel Goldberg developed a machine that read characters and converted them into standard telegraph code. Concurrently, Edmund Fournier d'Albe developed the Optophone, a handheld scanner that when moved across a printed page, produced tones that corresponded to specific letters or characters. In the late 1920s and into the 1930s, Emanuel Goldberg developed what he called a "Statistical Machine" for searching microfilm archives using an optical code recognition system. In 1931, he was granted US Patent number 1,838,389 for the invention. The patent was acquired by IBM. === Visually impaired users === In 1974, Ray Kurzweil started the company Kurzweil Computer Products, Inc. and continued development of omni-font OCR, which could recognize text printed in virtually any font. (Kurzweil is often credited with inventing omni-font OCR, but it was in use by companies, including CompuScan, in the late 1960s and 1970s.) Kurzweil used the technology to create a reading machine for blind people to have a computer read text to them out loud. The device included a CCD-type flatbed scanner and a text-to-speech synthesizer. On January 13, 1976, the finished product was unveiled during a widely reported news conference headed by Kurzweil and the leaders of the National Federation of the Blind. In 1978, Kurzweil Computer Products began selling a commercial version of the optical character recognition computer program. LexisNexis was one of the first customers, and bought the program to upload legal paper and news documents onto its nascent online databases. Two years later, Kurzweil sold his company to Xerox, which eventually spun it off as Scansoft, which merged with Nuance Communications. In the 2000s, OCR was made available online as a service (WebOCR), in a cloud computing environment, and in mobile applications like real-time translation of foreign-language signs on a smartphone. With the advent of smartphones and smartglasses, OCR can be used in internet connected mobile device applications that extract text captured using the device's camera. These devices that do not have built-in OCR functionality will typically use an OCR API to extract the text from the image file captured by the device. The OCR API returns the extracted text, along with information about the location of the detected text in the original image back to the device app for further processing (such as text-to-speech) or display. Various commercial and open source OCR systems are available for most common writing systems, including Latin, Cyrillic, Arabic, Hebrew, Indic, Bengali (Bangla), Devanagari, Tamil, Chinese, Japanese, and Korean characters. == Applications == OCR engines have been developed into software applications specializing in various subjects such as receipts, invoices, checks, and legal billing documents. The software can be used for: Entering data for business documents, e.g. checks, passports, invoices, bank statements and receipts Automatic number-plate recognition Passport recognition and information extraction in airports Automatically extracting key information from insurance documents Traffic-sign recognition Extracting business card information into a contact list Creating textual versions of printed documents, e.g. book scanning for Project Gutenberg Making electronic images of printed documents searchable, e.g. Google Books Converting handwriting in real-time to control a computer (pen computing) Defeating or testing the robustness of CAPTCHA anti-bot systems, though these are specifically designed to prevent OCR. Assistive technology for blind and visually impaired users Writing instructions for vehicles by identifying CAD images in a database that are appropriate to the vehicle design as it changes in real time Making scanned documents searchable by converting them to PDFs == Types == Optical character recognition (OCR) – targets typewritten text, one glyph or character at a time. Optical word recognition – targets typewritten text, one word at a time (for languages that use a space as a word divider). Usually just called "OCR". Intelligent character recognition (ICR) – also targets handwritten printscript or cursive text one glyph or character at a time, usually involving machine learning. Intelligent word recognition (IWR) – also targets handwritten printscript or cursive text, one word at a time. This is especially useful for languages where glyphs are not separated in cursive script. OCR is generally an offline process, which analyses a static document. There are cloud based services which provide an online OCR API service. Handwriting movement analysis can be used as input to handwriting recognition. Instead of merely using the shapes of glyphs and words, this technique is able to capture motion, such as the order in which segments are drawn, the direction, and the pattern of putting the pen down and lifting it. This additional information can make the process more accurate. This technology is also known as "online character recognition", "dynamic character recognition", "real-time character recognition", and "intelligent character recognition". == Techniques == === Pre-processing === OCR software often pre-processes images to improve the chances of successful recognition. Techniques include: De-skewing – if the document was not aligned properly when scanned, it may need to be tilted a few degrees clockwise or counterclockwise in order to make lines of text perfectly horizontal or vertical. Despeckling – removal of positive and negative spots, smoothing edges Binarization – conversion of an image from color or greyscale to black-and-white (called a binary image because there are two colors). The task is performed as a simple way of separating the text (or any other desired image component) from the background. The task of binarization is necessary since most commercial recognition algorithms work only on binary images, as it is simpler to do so. In addition, the effectiveness of binarization influences to a significant extent the quality of character recognition, and careful decisions are made in the choice of the binarization employed for a given input image type; since the quality of the method used to obtain the binary result depends on the type of image (scanned document, scene text image, degraded historical document, etc.). Line removal – Cleaning up non-glyph boxes and lines Layout analysis or zoning – Identification of columns, paragraphs, captions, etc. as distinct blocks. Especially important in multi-column layouts and tables. Line and word detection – Establishment of a baseline for word and character shapes, separating words as necessary. Script recognition – In multilingual documents, the script may change at the level of the words and hence, identification of the script is necessary, before the right OCR can be invoked to handle the specific script. Character isolation or segmentation – For per-character OCR, multiple characters that are connected due to image artifacts must be separated; single characters that are broken into multiple pieces due to artifacts must be connected. Normalization of aspect ratio and scale Segmentation of fixed-pitch fonts is accomplished relatively simply by aligning the image to a uniform grid based on where vertical grid lines will least often intersect black areas. For proportional fonts, more sophisticated techniques are needed because whitespace bet

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  T-distributed stochastic neighbor embedding

  t-distributed stochastic neighbor embedding (t-SNE) is a statistical method for visualizing high-dimensional data by giving each datapoint a location in a two or three-dimensional map. It is based on Stochastic Neighbor Embedding originally developed by Geoffrey Hinton and Sam Roweis, where Laurens van der Maaten and Hinton proposed the t-distributed variant. It is a nonlinear dimensionality reduction technique for embedding high-dimensional data for visualization in a low-dimensional space of two or three dimensions. Specifically, it models each high-dimensional object by a two- or three-dimensional point in such a way that similar objects are modeled by nearby points and dissimilar objects are modeled by distant points with high probability. The t-SNE algorithm comprises two main stages. First, t-SNE constructs a probability distribution over pairs of high-dimensional objects in such a way that similar objects are assigned a higher probability while dissimilar points are assigned a lower probability. Second, t-SNE defines a similar probability distribution over the points in the low-dimensional map, and it minimizes the Kullback–Leibler divergence (KL divergence) between the two distributions with respect to the locations of the points in the map. While the original algorithm uses the Euclidean distance between objects as the base of its similarity metric, this can be changed as appropriate. A Riemannian variant is UMAP. t-SNE has been used for visualization in a wide range of applications, including genomics, computer security research, natural language processing, music analysis, cancer research, bioinformatics, geological domain interpretation, and biomedical signal processing. For a data set with n {\displaystyle n} elements, t-SNE runs in O ( n 2 ) {\displaystyle O(n^{2})} time and requires O ( n 2 ) {\displaystyle O(n^{2})} space. == Details == Given a set of N {\displaystyle N} high-dimensional objects x 1 , … , x N {\displaystyle \mathbf {x} _{1},\dots ,\mathbf {x} _{N}} , t-SNE first computes probabilities p i j {\displaystyle p_{ij}} that are proportional to the similarity of objects x i {\displaystyle \mathbf {x} _{i}} and x j {\displaystyle \mathbf {x} _{j}} , as follows. For i ≠ j {\displaystyle i\neq j} , define p j ∣ i = exp ⁡ ( − ‖ x i − x j ‖ 2 / 2 σ i 2 ) ∑ k ≠ i exp ⁡ ( − ‖ x i − x k ‖ 2 / 2 σ i 2 ) {\displaystyle p_{j\mid i}={\frac {\exp(-\lVert \mathbf {x} _{i}-\mathbf {x} _{j}\rVert ^{2}/2\sigma _{i}^{2})}{\sum _{k\neq i}\exp(-\lVert \mathbf {x} _{i}-\mathbf {x} _{k}\rVert ^{2}/2\sigma _{i}^{2})}}} and set p i ∣ i = 0 {\displaystyle p_{i\mid i}=0} . Note the above denominator ensures ∑ j p j ∣ i = 1 {\displaystyle \sum _{j}p_{j\mid i}=1} for all i {\displaystyle i} . As van der Maaten and Hinton explained: "The similarity of datapoint x j {\displaystyle x_{j}} to datapoint x i {\displaystyle x_{i}} is the conditional probability, p j | i {\displaystyle p_{j|i}} , that x i {\displaystyle x_{i}} would pick x j {\displaystyle x_{j}} as its neighbor if neighbors were picked in proportion to their probability density under a Gaussian centered at x i {\displaystyle x_{i}} ." Now define p i j = p j ∣ i + p i ∣ j 2 N {\displaystyle p_{ij}={\frac {p_{j\mid i}+p_{i\mid j}}{2N}}} This is motivated because p i {\displaystyle p_{i}} and p j {\displaystyle p_{j}} from the N samples are estimated as 1/N, so the conditional probability can be written as p i ∣ j = N p i j {\displaystyle p_{i\mid j}=Np_{ij}} and p j ∣ i = N p j i {\displaystyle p_{j\mid i}=Np_{ji}} . Since p i j = p j i {\displaystyle p_{ij}=p_{ji}} , you can obtain previous formula. Also note that p i i = 0 {\displaystyle p_{ii}=0} and ∑ i , j p i j = 1 {\displaystyle \sum _{i,j}p_{ij}=1} . The bandwidth of the Gaussian kernels σ i {\displaystyle \sigma _{i}} is set in such a way that the entropy of the conditional distribution equals a predefined entropy using the bisection method. As a result, the bandwidth is adapted to the density of the data: smaller values of σ i {\displaystyle \sigma _{i}} are used in denser parts of the data space. The entropy increases with the perplexity of this distribution P i {\displaystyle P_{i}} ; this relation is seen as P e r p ( P i ) = 2 H ( P i ) {\displaystyle Perp(P_{i})=2^{H(P_{i})}} where H ( P i ) {\displaystyle H(P_{i})} is the Shannon entropy H ( P i ) = − ∑ j p j | i log 2 ⁡ p j | i . {\displaystyle H(P_{i})=-\sum _{j}p_{j|i}\log _{2}p_{j|i}.} The perplexity is a hand-chosen parameter of t-SNE, and as the authors state, "perplexity can be interpreted as a smooth measure of the effective number of neighbors. The performance of SNE is fairly robust to changes in the perplexity, and typical values are between 5 and 50.". Since the Gaussian kernel uses the Euclidean distance ‖ x i − x j ‖ {\displaystyle \lVert x_{i}-x_{j}\rVert } , it is affected by the curse of dimensionality, and in high dimensional data when distances lose the ability to discriminate, the p i j {\displaystyle p_{ij}} become too similar (asymptotically, they would converge to a constant). It has been proposed to adjust the distances with a power transform, based on the intrinsic dimension of each point, to alleviate this. t-SNE aims to learn a d {\displaystyle d} -dimensional map y 1 , … , y N {\displaystyle \mathbf {y} _{1},\dots ,\mathbf {y} _{N}} (with y i ∈ R d {\displaystyle \mathbf {y} _{i}\in \mathbb {R} ^{d}} and d {\displaystyle d} typically chosen as 2 or 3) that reflects the similarities p i j {\displaystyle p_{ij}} as well as possible. To this end, it measures similarities q i j {\displaystyle q_{ij}} between two points in the map y i {\displaystyle \mathbf {y} _{i}} and y j {\displaystyle \mathbf {y} _{j}} , using a very similar approach. Specifically, for i ≠ j {\displaystyle i\neq j} , define q i j {\displaystyle q_{ij}} as q i j = ( 1 + ‖ y i − y j ‖ 2 ) − 1 ∑ k ∑ l ≠ k ( 1 + ‖ y k − y l ‖ 2 ) − 1 {\displaystyle q_{ij}={\frac {(1+\lVert \mathbf {y} _{i}-\mathbf {y} _{j}\rVert ^{2})^{-1}}{\sum _{k}\sum _{l\neq k}(1+\lVert \mathbf {y} _{k}-\mathbf {y} _{l}\rVert ^{2})^{-1}}}} and set q i i = 0 {\displaystyle q_{ii}=0} . Herein a heavy-tailed Student t-distribution (with one-degree of freedom, which is the same as a Cauchy distribution) is used to measure similarities between low-dimensional points in order to allow dissimilar objects to be modeled far apart in the map. The locations of the points y i {\displaystyle \mathbf {y} _{i}} in the map are determined by minimizing the (non-symmetric) Kullback–Leibler divergence of the distribution P {\displaystyle P} from the distribution Q {\displaystyle Q} , that is: K L ( P ∥ Q ) = ∑ i ≠ j p i j log ⁡ p i j q i j {\displaystyle \mathrm {KL} \left(P\parallel Q\right)=\sum _{i\neq j}p_{ij}\log {\frac {p_{ij}}{q_{ij}}}} The minimization of the Kullback–Leibler divergence with respect to the points y i {\displaystyle \mathbf {y} _{i}} is performed using gradient descent. The result of this optimization is a map that reflects the similarities between the high-dimensional inputs. == Output == While t-SNE plots often seem to display clusters, the visual clusters can be strongly influenced by the chosen parameterization (especially the perplexity) and so a good understanding of the parameters for t-SNE is needed. Such "clusters" can be shown to even appear in structured data with no clear clustering, and so may be false findings. Similarly, the size of clusters produced by t-SNE is not informative, and neither is the distance between clusters. Thus, interactive exploration may be needed to choose parameters and validate results. It has been shown that t-SNE can often recover well-separated clusters, and with special parameter choices, approximates a simple form of spectral clustering. == Software == A C++ implementation of Barnes-Hut is available on the github account of one of the original authors. The R package Rtsne implements t-SNE in R. ELKI contains tSNE, also with Barnes-Hut approximation scikit-learn, a popular machine learning library in Python implements t-SNE with both exact solutions and the Barnes-Hut approximation. Tensorboard, the visualization kit associated with TensorFlow, also implements t-SNE (online version) The Julia package TSne implements t-SNE

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  Ilastik

  ilastik is free open source software for image classification and segmentation. No previous experience in image processing is required to run the software. Since 2018 ilastik is further developed and maintained by Anna Kreshuk's group at European Molecular Biology Laboratory. == Features == ilastik allows user to annotate an arbitrary number of classes in images with a mouse interface. Using these user annotations and the generic (nonlinear) image features, the user can train a random forest classifier. Trained ilastik classifiers can be applied new data not included in the training set in ilastik via its batch processing functionality, or without using the graphical user interface, in headless mode. ilastik can be integrated into various related tools: Pre-trained workflows can be executed directly from ImageJ/Fiji using the ilastik-ImageJ plugin. Pre-trained ilastik Pixel Classification workflows can be run directly in Python with the ilastik Python package, which is available via conda. ilastik has a CellProfiler module to use ilastik classifiers to process images within a CellProfiler framework. == History == ilastik was first released in 2011 by scientists at the Heidelberg Collaboratory for Image Processing (HCI), University of Heidelberg. == Application == The Interactive Learning and Segmentation Toolkit Carving Cell classification and neuron classification Synapse detection Cell tracking Neural Network Classification == Resources == ilastik project is hosted on GitHub. It is a collaborative project, any contributions such as comments, bug reports, bug fixes or code contributions are welcome. The ilastik team can be contacted for user support on the image.sc forum.

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  Static program analysis

  In computer science, static program analysis (also known as static analysis or static simulation) is the analysis of computer programs performed without executing them, in contrast with dynamic program analysis, which is performed on programs during their execution in the integrated environment. The term is usually applied to analysis performed by an automated tool, with human analysis typically being called "program understanding", program comprehension, or code review. In the last of these, software inspection and software walkthroughs are also used. In most cases the analysis is performed on some version of a program's source code, and, in other cases, on some form of its object code. Two leading approaches to resource certification have been Static Analysis (SA) and Implicit Computational Complexity (ICC). SA is algorithmic in nature: it focuses on a broad programming language of choice, and seeks to determine by syntactic means whether given programs in that language are feasible. In contrast, ICC attempts to create from the outset specialized programming languages or methods that delineate a complexity class. Thus, SA's focus is on compile time, making no demand on the programmer; whereas ICC is a language-design discipline." The discipline of static analysis should not be confused with linting, which is the process of checking for coding style mistakes. == Rationale == The sophistication of the analysis performed by tools varies from those that only consider the behaviour of individual statements and declarations, to those that include the complete source code of a program in their analysis. The uses of the information obtained from the analysis vary from highlighting possible coding errors (e.g., the lint tool) to formal methods that mathematically prove properties about a given program (e.g., its behaviour matches that of its specification). Software metrics and reverse engineering can be described as forms of static analysis. Deriving software metrics and static analysis are increasingly deployed together, especially in creation of embedded systems, by defining so-called software quality objectives. A growing commercial use of static analysis is in the verification of properties of software used in safety-critical computer systems and locating potentially vulnerable code. For example, the following industries have identified the use of static code analysis as a means of improving the quality of increasingly sophisticated and complex software: Medical software: The US Food and Drug Administration (FDA) has identified the use of static analysis for medical devices. Nuclear software: In the UK the Office for Nuclear Regulation (ONR) recommends the use of static analysis on reactor protection systems. Aviation software (in combination with dynamic analysis). Automotive & Machines (functional safety features form an integral part of each automotive product development phase, ISO 26262, section 8). A study in 2012 by VDC Research reported that 28.7% of the embedded software engineers surveyed use static analysis tools and 39.7% expect to use them within 2 years. A study from 2010 found that 60% of the interviewed developers in European research projects made at least use of their basic IDE built-in static analyzers. However, only about 10% employed an additional other (and perhaps more advanced) analysis tool. In the application security industry the name static application security testing (SAST) is also used. SAST is an important part of Security Development Lifecycles (SDLs) such as the SDL defined by Microsoft and a common practice in software companies. == Tool types == The OMG (Object Management Group) published a study regarding the types of software analysis required for software quality measurement and assessment. This document on "How to Deliver Resilient, Secure, Efficient, and Easily Changed IT Systems in Line with CISQ Recommendations" describes three levels of software analysis. Unit Level Analysis that takes place within a specific program or subroutine, without connecting to the context of that program. Technology Level Analysis that takes into account interactions between unit programs to get a more holistic and semantic view of the overall program in order to find issues and avoid obvious false positives. System Level Analysis that takes into account the interactions between unit programs, but without being limited to one specific technology or programming language. A further level of software analysis can be defined. Mission/Business Level Analysis that takes into account the business/mission layer terms, rules and processes that are implemented within the software system for its operation as part of enterprise or program/mission layer activities. These elements are implemented without being limited to one specific technology or programming language and in many cases are distributed across multiple languages, but are statically extracted and analyzed for system understanding for mission assurance. == Formal methods == Formal methods is the term applied to the analysis of software (and computer hardware) whose results are obtained purely through the use of rigorous mathematical methods. The mathematical techniques used include denotational semantics, axiomatic semantics, operational semantics, and abstract interpretation. By a straightforward reduction to the halting problem, it is possible to prove that (for any Turing complete language), finding all possible run-time errors in an arbitrary program (or more generally any kind of violation of a specification on the final result of a program) is undecidable: there is no mechanical method that can always answer truthfully whether an arbitrary program may or may not exhibit runtime errors. This result dates from the works of Church, Gödel and Turing in the 1930s (see: Halting problem and Rice's theorem). As with many undecidable questions, one can still attempt to give useful approximate solutions. Some of the implementation techniques of formal static analysis include: Abstract interpretation, to model the effect that every statement has on the state of an abstract machine (i.e., it 'executes' the software based on the mathematical properties of each statement and declaration). This abstract machine over-approximates the behaviours of the system: the abstract system is thus made simpler to analyze, at the expense of incompleteness (not every property true of the original system is true of the abstract system). If properly done, though, abstract interpretation is sound (every property true of the abstract system can be mapped to a true property of the original system). Data-flow analysis, a lattice-based technique for gathering information about the possible set of values; Hoare logic, a formal system with a set of logical rules for reasoning rigorously about the correctness of computer programs. There is tool support for some programming languages (e.g., the SPARK programming language (a subset of Ada) and the Java Modeling Language—JML—using ESC/Java and ESC/Java2, Frama-C WP (weakest precondition) plugin for the C language extended with ACSL (ANSI/ISO C Specification Language) ). Model checking, considers systems that have finite state or may be reduced to finite state by abstraction; Symbolic execution, as used to derive mathematical expressions representing the value of mutated variables at particular points in the code. Nullable reference analysis == Data-driven static analysis == Data-driven static analysis leverages extensive codebases to infer coding rules and improve the accuracy of the analysis. For instance, one can use all Java open-source packages available on GitHub to learn good analysis strategies. The rule inference can use machine learning techniques. It is also possible to learn from a large amount of past fixes and warnings. == Remediation == Static analyzers produce warnings. For certain types of warnings, it is possible to design and implement automated remediation techniques. For example, Logozzo and Ball have proposed automated remediations for C# cccheck.

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  PVLV

  The primary value learned value (PVLV) model is a possible explanation for the reward-predictive firing properties of dopamine (DA) neurons. It simulates behavioral and neural data on Pavlovian conditioning and the midbrain dopaminergic neurons that fire in proportion to unexpected rewards. It is an alternative to the temporal-differences (TD) algorithm. It is used as part of Leabra.

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  ID3 algorithm

  In decision tree learning, ID3 (Iterative Dichotomiser 3) is a greedy algorithm invented by Ross Quinlan used to generate a decision tree from a dataset. ID3 is the precursor to the C4.5 algorithm. The 3 in the name is meant to signify that this was Quinlan's third attempt at a model based on entropy-based splitting, and the term dichotimser is a misnomer as it implies a binary split, but the ID3 algorithm can split on multi-valued attributes. == Algorithm == The ID3 algorithm begins with the original set S {\displaystyle S} as the root node. On each iteration of the algorithm, it iterates through every unused attribute of the set S {\displaystyle S} and calculates the entropy H ( S ) {\displaystyle \mathrm {H} {(S)}} or the information gain I G ( S ) {\displaystyle IG(S)} of that attribute. It then selects the attribute which has the smallest entropy (or largest information gain) value. The set S {\displaystyle S} is then split or partitioned by the selected attribute to produce subsets of the data. (For example, a node can be split into child nodes based upon the subsets of the population whose ages are less than 50, between 50 and 100, and greater than 100.) The algorithm continues to recurse on each subset, considering only attributes never selected before. Recursion on a subset may stop in one of these cases: every element in the subset belongs to the same class; in which case the node is turned into a leaf node and labelled with the class of the examples. there are no more attributes to be selected, but the examples still do not belong to the same class. In this case, the node is made a leaf node and labelled with the most common class of the examples in the subset. there are no examples in the subset, which happens when no example in the parent set was found to match a specific value of the selected attribute. An example could be the absence of a person among the population with age over 100 years. Then a leaf node is created and labelled with the most common class of the examples in the parent node's set. Throughout the algorithm, the decision tree is constructed with each non-terminal node (internal node) representing the selected attribute on which the data was split, and terminal nodes (leaf nodes) representing the class label of the final subset of this branch. === Summary === Calculate the entropy of every attribute a {\displaystyle a} of the data set S {\displaystyle S} . Partition ("split") the set S {\displaystyle S} into subsets using the attribute for which the resulting entropy after splitting is minimized; or, equivalently, information gain is maximum. Make a decision tree node containing that attribute. Recurse on subsets using the remaining attributes. === Properties === ID3 does not guarantee an optimal solution. It can converge upon local optima. It uses a greedy strategy by selecting the locally best attribute to split the dataset on each iteration. The algorithm's optimality can be improved by using backtracking during the search for the optimal decision tree at the cost of possibly taking longer. ID3 can overfit the training data. To avoid overfitting, smaller decision trees should be preferred over larger ones. This algorithm usually produces small trees, but it does not always produce the smallest possible decision tree. ID3 is harder to use on continuous data than on factored data (factored data has a discrete number of possible values, thus reducing the possible branch points). If the values of any given attribute are continuous, then there are many more places to split the data on this attribute, and searching for the best value to split by can be time-consuming. === Usage === The ID3 algorithm is used by training on a data set S {\displaystyle S} to produce a decision tree which is stored in memory. At runtime, this decision tree is used to classify new test cases (feature vectors) by traversing the decision tree using the features of the datum to arrive at a leaf node. == The ID3 metrics == === Entropy === Entropy H ( S ) {\displaystyle \mathrm {H} {(S)}} is a measure of the amount of uncertainty in the (data) set S {\displaystyle S} (i.e. entropy characterizes the (data) set S {\displaystyle S} ). H ( S ) = ∑ x ∈ X − p ( x ) log 2 ⁡ p ( x ) {\displaystyle \mathrm {H} {(S)}=\sum _{x\in X}{-p(x)\log _{2}p(x)}} Where, S {\displaystyle S} – The current dataset for which entropy is being calculated This changes at each step of the ID3 algorithm, either to a subset of the previous set in the case of splitting on an attribute or to a "sibling" partition of the parent in case the recursion terminated previously. X {\displaystyle X} – The set of classes in S {\displaystyle S} p ( x ) {\displaystyle p(x)} – The proportion of the number of elements in class x {\displaystyle x} to the number of elements in set S {\displaystyle S} When H ( S ) = 0 {\displaystyle \mathrm {H} {(S)}=0} , the set S {\displaystyle S} is perfectly classified (i.e. all elements in S {\displaystyle S} are of the same class). In ID3, entropy is calculated for each remaining attribute. The attribute with the smallest entropy is used to split the set S {\displaystyle S} on this iteration. Entropy in information theory measures how much information is expected to be gained upon measuring a random variable; as such, it can also be used to quantify the amount to which the distribution of the quantity's values is unknown. A constant quantity has zero entropy, as its distribution is perfectly known. In contrast, a uniformly distributed random variable (discretely or continuously uniform) maximizes entropy. Therefore, the greater the entropy at a node, the less information is known about the classification of data at this stage of the tree; and therefore, the greater the potential to improve the classification here. As such, ID3 is a greedy heuristic performing a best-first search for locally optimal entropy values. Its accuracy can be improved by preprocessing the data. === Information gain === Information gain I G ( A ) {\displaystyle IG(A)} is the measure of the difference in entropy from before to after the set S {\displaystyle S} is split on an attribute A {\displaystyle A} . In other words, how much uncertainty in S {\displaystyle S} was reduced after splitting set S {\displaystyle S} on attribute A {\displaystyle A} . I G ( S , A ) = H ( S ) − ∑ t ∈ T p ( t ) H ( t ) = H ( S ) − H ( S | A ) . {\displaystyle IG(S,A)=\mathrm {H} {(S)}-\sum _{t\in T}p(t)\mathrm {H} {(t)}=\mathrm {H} {(S)}-\mathrm {H} {(S|A)}.} Where, H ( S ) {\displaystyle \mathrm {H} (S)} – Entropy of set S {\displaystyle S} T {\displaystyle T} – The subsets created from splitting set S {\displaystyle S} by attribute A {\displaystyle A} such that S = ⋃ t ∈ T t {\displaystyle S=\bigcup _{t\in T}t} p ( t ) {\displaystyle p(t)} – The proportion of the number of elements in t {\displaystyle t} to the number of elements in set S {\displaystyle S} H ( t ) {\displaystyle \mathrm {H} (t)} – Entropy of subset t {\displaystyle t} In ID3, information gain can be calculated (instead of entropy) for each remaining attribute. The attribute with the largest information gain is used to split the set S {\displaystyle S} on this iteration.

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  Correspondence analysis

  Correspondence analysis (CA) is a multivariate statistical technique proposed by Herman Otto Hartley (Hirschfeld) and later developed by Jean-Paul Benzécri. It is conceptually similar to principal component analysis, but applies to categorical rather than continuous data. In a manner similar to principal component analysis, it provides a means of displaying or summarising a set of data in two-dimensional graphical form. Its aim is to display in a biplot any structure hidden in the multivariate setting of the data table. As such it is a technique from the field of multivariate ordination. Since the variant of CA described here can be applied either with a focus on the rows or on the columns it should in fact be called simple (symmetric) correspondence analysis. It is traditionally applied to the contingency table of a pair of nominal variables where each cell contains either a count or a zero value. If more than two categorical variables are to be summarized, a variant called multiple correspondence analysis should be chosen instead. CA may also be applied to binary data given the presence/absence coding represents simplified count data i.e. a 1 describes a positive count and 0 stands for a count of zero. Depending on the scores used CA preserves the chi-square distance between either the rows or the columns of the table. Because CA is a descriptive technique, it can be applied to tables regardless of a significant chi-squared test. Although the χ 2 {\displaystyle \chi ^{2}} statistic used in inferential statistics and the chi-square distance are computationally related they should not be confused since the latter works as a multivariate statistical distance measure in CA while the χ 2 {\displaystyle \chi ^{2}} statistic is in fact a scalar not a metric. == Details == Like principal components analysis, correspondence analysis creates orthogonal components (or axes) and, for each item in a table i.e. for each row, a set of scores (sometimes called factor scores, see Factor analysis). Correspondence analysis is performed on the data table, conceived as matrix C of size m × n where m is the number of rows and n is the number of columns. In the following mathematical description of the method capital letters in italics refer to a matrix while letters in italics refer to vectors. Understanding the following computations requires knowledge of matrix algebra. === Preprocessing === Before proceeding to the central computational step of the algorithm, the values in matrix C have to be transformed. First compute a set of weights for the columns and the rows (sometimes called masses), where row and column weights are given by the row and column vectors, respectively: w m = 1 n C C 1 , w n = 1 n C 1 T C . {\displaystyle w_{m}={\frac {1}{n_{C}}}C\mathbf {1} ,\quad w_{n}={\frac {1}{n_{C}}}\mathbf {1} ^{T}C.} Here n C = ∑ i = 1 n ∑ j = 1 m C i j {\displaystyle n_{C}=\sum _{i=1}^{n}\sum _{j=1}^{m}C_{ij}} is the sum of all cell values in matrix C, or short the sum of C, and 1 {\displaystyle \mathbf {1} } is a column vector of ones with the appropriate dimension. Put in simple words, w m {\displaystyle w_{m}} is just a vector whose elements are the row sums of C divided by the sum of C, and w n {\displaystyle w_{n}} is a vector whose elements are the column sums of C divided by the sum of C. The weights are transformed into diagonal matrices W m = diag ⁡ ( 1 / w m ) {\displaystyle W_{m}=\operatorname {diag} (1/{\sqrt {w_{m}}})} and W n = diag ⁡ ( 1 / w n ) {\displaystyle W_{n}=\operatorname {diag} (1/{\sqrt {w_{n}}})} where the diagonal elements of W n {\displaystyle W_{n}} are 1 / w n {\displaystyle 1/{\sqrt {w_{n}}}} and those of W m {\displaystyle W_{m}} are 1 / w m {\displaystyle 1/{\sqrt {w_{m}}}} respectively i.e. the vector elements are the inverses of the square roots of the masses. The off-diagonal elements are all 0. Next, compute matrix P {\displaystyle P} by dividing C {\displaystyle C} by its sum P = 1 n C C . {\displaystyle P={\frac {1}{n_{C}}}C.} In simple words, Matrix P {\displaystyle P} is just the data matrix (contingency table or binary table) transformed into portions i.e. each cell value is just the cell portion of the sum of the whole table. Finally, compute matrix S {\displaystyle S} , sometimes called the matrix of standardized residuals, by matrix multiplication as S = W m ( P − w m w n ) W n {\displaystyle S=W_{m}(P-w_{m}w_{n})W_{n}} Note, the vectors w m {\displaystyle w_{m}} and w n {\displaystyle w_{n}} are combined in an outer product resulting in a matrix of the same dimensions as P {\displaystyle P} . In words the formula reads: matrix outer ⁡ ( w m , w n ) {\displaystyle \operatorname {outer} (w_{m},w_{n})} is subtracted from matrix P {\displaystyle P} and the resulting matrix is scaled (weighted) by the diagonal matrices W m {\displaystyle W_{m}} and W n {\displaystyle W_{n}} . Multiplying the resulting matrix by the diagonal matrices is equivalent to multiply the i-th row (or column) of it by the i-th element of the diagonal of W m {\displaystyle W_{m}} or W n {\displaystyle W_{n}} , respectively. === Interpretation of preprocessing === The vectors w m {\displaystyle w_{m}} and w n {\displaystyle w_{n}} are the row and column masses or the marginal probabilities for the rows and columns, respectively. Subtracting matrix outer ⁡ ( w m , w n ) {\displaystyle \operatorname {outer} (w_{m},w_{n})} from matrix P {\displaystyle P} is the matrix algebra version of double centering the data. Multiplying this difference by the diagonal weighting matrices results in a matrix containing weighted deviations from the origin of a vector space. This origin is defined by matrix outer ⁡ ( w m , w n ) {\displaystyle \operatorname {outer} (w_{m},w_{n})} . In fact matrix outer ⁡ ( w m , w n ) {\displaystyle \operatorname {outer} (w_{m},w_{n})} is identical with the matrix of expected frequencies in the chi-squared test. Therefore S {\displaystyle S} is computationally related to the independence model used in that test. But since CA is not an inferential method the term independence model is inappropriate here. === Orthogonal components === The table S {\displaystyle S} is then decomposed by a singular value decomposition as S = U Σ V ∗ {\displaystyle S=U\Sigma V^{}\,} where U {\displaystyle U} and V {\displaystyle V} are the left and right singular vectors of S {\displaystyle S} and Σ {\displaystyle \Sigma } is a square diagonal matrix with the singular values σ i {\displaystyle \sigma _{i}} of S {\displaystyle S} on the diagonal. Σ {\displaystyle \Sigma } is of dimension p ≤ ( min ( m , n ) − 1 ) {\displaystyle p\leq (\min(m,n)-1)} hence U {\displaystyle U} is of dimension m×p and V {\displaystyle V} is of n×p. As orthonormal vectors U {\displaystyle U} and V {\displaystyle V} fulfill U ∗ U = V ∗ V = I {\displaystyle U^{}U=V^{}V=I} . In other words, the multivariate information that is contained in C {\displaystyle C} as well as in S {\displaystyle S} is now distributed across two (coordinate) matrices U {\displaystyle U} and V {\displaystyle V} and a diagonal (scaling) matrix Σ {\displaystyle \Sigma } . The vector space defined by them has as number of dimensions p, that is the smaller of the two values, number of rows and number of columns, minus 1. === Inertia === While a principal component analysis may be said to decompose the (co)variance, and hence its measure of success is the amount of (co-)variance covered by the first few PCA axes - measured in eigenvalue -, a CA works with a weighted (co-)variance which is called inertia. The sum of the squared singular values is the total inertia I {\displaystyle \mathrm {I} } of the data table, computed as I = ∑ i = 1 p σ i 2 . {\displaystyle \mathrm {I} =\sum _{i=1}^{p}\sigma _{i}^{2}.} The total inertia I {\displaystyle \mathrm {I} } of the data table can also computed directly from S {\displaystyle S} as I = ∑ i = 1 n ∑ j = 1 m s i j 2 . {\displaystyle \mathrm {I} =\sum _{i=1}^{n}\sum _{j=1}^{m}s_{ij}^{2}.} The amount of inertia covered by the i-th set of singular vectors is ι i {\displaystyle \iota _{i}} , the principal inertia. The higher the portion of inertia covered by the first few singular vectors i.e. the larger the sum of the principal inertiae in comparison to the total inertia, the more successful a CA is. Therefore, all principal inertia values are expressed as portion ϵ i {\displaystyle \epsilon _{i}} of the total inertia ϵ i = σ i 2 / ∑ i = 1 p σ i 2 {\displaystyle \epsilon _{i}=\sigma _{i}^{2}/\sum _{i=1}^{p}\sigma _{i}^{2}} and are presented in the form of a scree plot. In fact a scree plot is just a bar plot of all principal inertia portions ϵ i {\displaystyle \epsilon _{i}} . === Coordinates === To transform the singular vectors to coordinates which preserve the chi-square distances between rows or columns an additional weighting step is necessary. The resulting coordinates are called principal coordinates in CA text books. If principal coordinates are used for

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  Enterprise mobile application

  The term enterprise mobile application is used in the context of mobile apps created/brought by individual organizations for their workers to carry out the functions required to run the organization. It is the process of building a mobile application for the requirements of an enterprise. An enterprise mobile application belonging to an organization is expected to be used by only the workers of that organization. The definition of enterprise mobile application does not include the mobile apps that an organization create for its customers or consumers of the products or services generated by the organization. == Example == An organization, whether for-profit or non-profit, may create a mobile app for its members to track inventory levels of supplies they distribute to their target communities or materials used in product manufacturing. Such a mobile app comes under the definition of enterprise mobile application. However, the same organization may also create another mobile app to sell their products to end users or spread awareness of their services to various communities, and that mobile app would not come under definition of enterprise mobile application. == Enterprise mobile solution providers == Enterprise Mobile solution providers create and develop apps for individual organizations that can buy instead of creating the apps themselves. Reasons for Organizations buying the apps include time and cost savings, technical expertise. Today Enterprise Mobility is playing track role for enterprise transformation. Today, enterprises needs productivity is a fast way. Enterprise mobility helps business owners to build their work in a progressive way by assisting enterprise mobility solutions.

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  Multinomial logistic regression

  In statistics, multinomial logistic regression is a classification method that generalizes logistic regression to multiclass problems, i.e. with more than two possible discrete outcomes. That is, it is a model that is used to predict the probabilities of the different possible outcomes of a categorically distributed dependent variable, given a set of independent variables (which may be real-valued, binary-valued, categorical-valued, etc.). Multinomial logistic regression is known by a variety of other names, including polytomous LR, multiclass LR, softmax regression, multinomial logit (mlogit), the maximum entropy (MaxEnt) classifier, and the conditional maximum entropy model. == Background == Multinomial logistic regression is used when the dependent variable in question is nominal (equivalently categorical, meaning that it falls into any one of a set of categories that cannot be ordered in any meaningful way) and for which there are more than two categories. Some examples would be: Which major will a college student choose, given their grades, stated likes and dislikes, etc.? Which blood type does a person have, given the results of various diagnostic tests? In a hands-free mobile phone dialing application, which person's name was spoken, given various properties of the speech signal? Which candidate will a person vote for, given particular demographic characteristics? Which country will a firm locate an office in, given the characteristics of the firm and of the various candidate countries? These are all statistical classification problems. They all have in common a dependent variable to be predicted that comes from one of a limited set of items that cannot be meaningfully ordered, as well as a set of independent variables (also known as features, explanators, etc.), which are used to predict the dependent variable. Multinomial logistic regression is a particular solution to classification problems that use a linear combination of the observed features and some problem-specific parameters to estimate the probability of each particular value of the dependent variable. The best values of the parameters for a given problem are usually determined from some training data (e.g. some people for whom both the diagnostic test results and blood types are known, or some examples of known words being spoken). == Assumptions == The multinomial logistic model assumes that data are case-specific; that is, each independent variable has a single value for each case. As with other types of regression, there is no need for the independent variables to be statistically independent from each other (unlike, for example, in a naive Bayes classifier); however, collinearity is assumed to be relatively low, as it becomes difficult to differentiate between the impact of several variables if this is not the case. If the multinomial logit is used to model choices, it relies on the assumption of independence of irrelevant alternatives (IIA), which is not always desirable. This assumption states that the odds of preferring one class over another do not depend on the presence or absence of other "irrelevant" alternatives. For example, the relative probabilities of taking a car or bus to work do not change if a bicycle is added as an additional possibility. This allows the choice of K alternatives to be modeled as a set of K − 1 independent binary choices, in which one alternative is chosen as a "pivot" and the other K − 1 compared against it, one at a time. The IIA hypothesis is a core hypothesis in rational choice theory; however numerous studies in psychology show that individuals often violate this assumption when making choices. An example of a problem case arises if choices include a car and a blue bus. Suppose the odds ratio between the two is 1 : 1. Now if the option of a red bus is introduced, a person may be indifferent between a red and a blue bus, and hence may exhibit a car : blue bus : red bus odds ratio of 1 : 0.5 : 0.5, thus maintaining a 1 : 1 ratio of car : any bus while adopting a changed car : blue bus ratio of 1 : 0.5. Here the red bus option was not in fact irrelevant, because a red bus was a perfect substitute for a blue bus. If the multinomial logit is used to model choices, it may in some situations impose too much constraint on the relative preferences between the different alternatives. It is especially important to take into account if the analysis aims to predict how choices would change if one alternative were to disappear (for instance if one political candidate withdraws from a three candidate race). Other models like the nested logit or the multinomial probit may be used in such cases as they allow for violation of the IIA. == Model == === Introduction === There are multiple equivalent ways to describe the mathematical model underlying multinomial logistic regression. This can make it difficult to compare different treatments of the subject in different texts. The article on logistic regression presents a number of equivalent formulations of simple logistic regression, and many of these have analogues in the multinomial logit model. The idea behind all of them, as in many other statistical classification techniques, is to construct a linear predictor function that constructs a score from a set of weights that are linearly combined with the explanatory variables (features) of a given observation using a dot product: score ⁡ ( X i , k ) = β k ⋅ X i , {\displaystyle \operatorname {score} (\mathbf {X} _{i},k)={\boldsymbol {\beta }}_{k}\cdot \mathbf {X} _{i},} where Xi is the vector of explanatory variables describing observation i, βk is a vector of weights (or regression coefficients) corresponding to outcome k, and score(Xi, k) is the score associated with assigning observation i to category k. In discrete choice theory, where observations represent people and outcomes represent choices, the score is considered the utility associated with person i choosing outcome k. The predicted outcome is the one with the highest score. The difference between the multinomial logit model and numerous other methods, models, algorithms, etc. with the same basic setup (the perceptron algorithm, support vector machines, linear discriminant analysis, etc.) is the procedure for determining (training) the optimal weights/coefficients and the way that the score is interpreted. In particular, in the multinomial logit model, the score can directly be converted to a probability value, indicating the probability of observation i choosing outcome k given the measured characteristics of the observation. This provides a principled way of incorporating the prediction of a particular multinomial logit model into a larger procedure that may involve multiple such predictions, each with a possibility of error. Without such means of combining predictions, errors tend to multiply. For example, imagine a large predictive model that is broken down into a series of submodels where the prediction of a given submodel is used as the input of another submodel, and that prediction is in turn used as the input into a third submodel, etc. If each submodel has 90% accuracy in its predictions, and there are five submodels in series, then the overall model has only 0.95 = 59% accuracy. If each submodel has 80% accuracy, then overall accuracy drops to 0.85 = 33% accuracy. This issue is known as error propagation and is a serious problem in real-world predictive models, which are usually composed of numerous parts. Predicting probabilities of each possible outcome, rather than simply making a single optimal prediction, is one means of alleviating this issue. === Setup === The basic setup is the same as in logistic regression, the only difference being that the dependent variables are categorical rather than binary, i.e. there are K possible outcomes rather than just two. The following description is somewhat shortened; for more details, consult the logistic regression article. ==== Data points ==== Specifically, it is assumed that we have a series of N observed data points. Each data point i (ranging from 1 to N) consists of a set of M explanatory variables x1,i ... xM,i (also known as independent variables, predictor variables, features, etc.), and an associated categorical outcome Yi (also known as dependent variable, response variable), which can take on one of K possible values. These possible values represent logically separate categories (e.g. different political parties, blood types, etc.), and are often described mathematically by arbitrarily assigning each a number from 1 to K. The explanatory variables and outcome represent observed properties of the data points, and are often thought of as originating in the observations of N "experiments" — although an "experiment" may consist of nothing more than gathering data. The goal of multinomial logistic regression is to construct a model that explains the relationship between the explanatory variables and the outcome, so tha

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  Least-squares support vector machine

  Least-squares support-vector machines (LS-SVM) for statistics and in statistical modeling, are least-squares versions of support-vector machines (SVM), which are a set of related supervised learning methods that analyze data and recognize patterns, and which are used for classification and regression analysis. In this version one finds the solution by solving a set of linear equations instead of a convex quadratic programming (QP) problem for classical SVMs. Least-squares SVM classifiers were proposed by Johan Suykens and Joos Vandewalle. LS-SVMs are a class of kernel-based learning methods. == From support-vector machine to least-squares support-vector machine == Given a training set { x i , y i } i = 1 N {\displaystyle \{x_{i},y_{i}\}_{i=1}^{N}} with input data x i ∈ R n {\displaystyle x_{i}\in \mathbb {R} ^{n}} and corresponding binary class labels y i ∈ { − 1 , + 1 } {\displaystyle y_{i}\in \{-1,+1\}} , the SVM classifier, according to Vapnik's original formulation, satisfies the following conditions: { w T ϕ ( x i ) + b ≥ 1 , if y i = + 1 , w T ϕ ( x i ) + b ≤ − 1 , if y i = − 1 , {\displaystyle {\begin{cases}w^{T}\phi (x_{i})+b\geq 1,&{\text{if }}\quad y_{i}=+1,\\w^{T}\phi (x_{i})+b\leq -1,&{\text{if }}\quad y_{i}=-1,\end{cases}}} which is equivalent to y i [ w T ϕ ( x i ) + b ] ≥ 1 , i = 1 , … , N , {\displaystyle y_{i}\left[{w^{T}\phi (x_{i})+b}\right]\geq 1,\quad i=1,\ldots ,N,} where ϕ ( x ) {\displaystyle \phi (x)} is the nonlinear map from original space to the high- or infinite-dimensional space. === Inseparable data === In case such a separating hyperplane does not exist, we introduce so-called slack variables ξ i {\displaystyle \xi _{i}} such that { y i [ w T ϕ ( x i ) + b ] ≥ 1 − ξ i , i = 1 , … , N , ξ i ≥ 0 , i = 1 , … , N . {\displaystyle {\begin{cases}y_{i}\left[{w^{T}\phi (x_{i})+b}\right]\geq 1-\xi _{i},&i=1,\ldots ,N,\\\xi _{i}\geq 0,&i=1,\ldots ,N.\end{cases}}} According to the structural risk minimization principle, the risk bound is minimized by the following minimization problem: min J 1 ( w , ξ ) = 1 2 w T w + c ∑ i = 1 N ξ i , {\displaystyle \min J_{1}(w,\xi )={\frac {1}{2}}w^{T}w+c\sum \limits _{i=1}^{N}\xi _{i},} Subject to { y i [ w T ϕ ( x i ) + b ] ≥ 1 − ξ i , i = 1 , … , N , ξ i ≥ 0 , i = 1 , … , N , {\displaystyle {\text{Subject to }}{\begin{cases}y_{i}\left[{w^{T}\phi (x_{i})+b}\right]\geq 1-\xi _{i},&i=1,\ldots ,N,\\\xi _{i}\geq 0,&i=1,\ldots ,N,\end{cases}}} To solve this problem, we could construct the Lagrangian function: L 1 ( w , b , ξ , α , β ) = 1 2 w T w + c ∑ i = 1 N ξ i − ∑ i = 1 N α i { y i [ w T ϕ ( x i ) + b ] − 1 + ξ i } − ∑ i = 1 N β i ξ i , {\displaystyle L_{1}(w,b,\xi ,\alpha ,\beta )={\frac {1}{2}}w^{T}w+c\sum \limits _{i=1}^{N}{\xi _{i}}-\sum \limits _{i=1}^{N}\alpha _{i}\left\{y_{i}\left[{w^{T}\phi (x_{i})+b}\right]-1+\xi _{i}\right\}-\sum \limits _{i=1}^{N}\beta _{i}\xi _{i},} where α i ≥ 0 , β i ≥ 0 ( i = 1 , … , N ) {\displaystyle \alpha _{i}\geq 0,\ \beta _{i}\geq 0\ (i=1,\ldots ,N)} are the Lagrangian multipliers. The optimal point will be in the saddle point of the Lagrangian function, and then we obtain By substituting w {\displaystyle w} by its expression in the Lagrangian formed from the appropriate objective and constraints, we will get the following quadratic programming problem: max Q 1 ( α ) = − 1 2 ∑ i , j = 1 N α i α j y i y j K ( x i , x j ) + ∑ i = 1 N α i , {\displaystyle \max Q_{1}(\alpha )=-{\frac {1}{2}}\sum \limits _{i,j=1}^{N}{\alpha _{i}\alpha _{j}y_{i}y_{j}K(x_{i},x_{j})}+\sum \limits _{i=1}^{N}\alpha _{i},} where K ( x i , x j ) = ⟨ ϕ ( x i ) , ϕ ( x j ) ⟩ {\displaystyle K(x_{i},x_{j})=\left\langle \phi (x_{i}),\phi (x_{j})\right\rangle } is called the kernel function. Solving this QP problem subject to constraints in (1), we will get the hyperplane in the high-dimensional space and hence the classifier in the original space. === Least-squares SVM formulation === The least-squares version of the SVM classifier is obtained by reformulating the minimization problem as min J 2 ( w , b , e ) = μ 2 w T w + ζ 2 ∑ i = 1 N e i 2 , {\displaystyle \min J_{2}(w,b,e)={\frac {\mu }{2}}w^{T}w+{\frac {\zeta }{2}}\sum \limits _{i=1}^{N}e_{i}^{2},} subject to the equality constraints y i [ w T ϕ ( x i ) + b ] = 1 − e i , i = 1 , … , N . {\displaystyle y_{i}\left[{w^{T}\phi (x_{i})+b}\right]=1-e_{i},\quad i=1,\ldots ,N.} The least-squares SVM (LS-SVM) classifier formulation above implicitly corresponds to a regression interpretation with binary targets y i = ± 1 {\displaystyle y_{i}=\pm 1} . Using y i 2 = 1 {\displaystyle y_{i}^{2}=1} , we have ∑ i = 1 N e i 2 = ∑ i = 1 N ( y i e i ) 2 = ∑ i = 1 N e i 2 = ∑ i = 1 N ( y i − ( w T ϕ ( x i ) + b ) ) 2 , {\displaystyle \sum \limits _{i=1}^{N}e_{i}^{2}=\sum \limits _{i=1}^{N}(y_{i}e_{i})^{2}=\sum \limits _{i=1}^{N}e_{i}^{2}=\sum \limits _{i=1}^{N}\left(y_{i}-(w^{T}\phi (x_{i})+b)\right)^{2},} with e i = y i − ( w T ϕ ( x i ) + b ) . {\displaystyle e_{i}=y_{i}-(w^{T}\phi (x_{i})+b).} Notice, that this error would also make sense for least-squares data fitting, so that the same end results holds for the regression case. Hence the LS-SVM classifier formulation is equivalent to J 2 ( w , b , e ) = μ E W + ζ E D {\displaystyle J_{2}(w,b,e)=\mu E_{W}+\zeta E_{D}} with E W = 1 2 w T w {\displaystyle E_{W}={\frac {1}{2}}w^{T}w} and E D = 1 2 ∑ i = 1 N e i 2 = 1 2 ∑ i = 1 N ( y i − ( w T ϕ ( x i ) + b ) ) 2 . {\displaystyle E_{D}={\frac {1}{2}}\sum \limits _{i=1}^{N}e_{i}^{2}={\frac {1}{2}}\sum \limits _{i=1}^{N}\left(y_{i}-(w^{T}\phi (x_{i})+b)\right)^{2}.} Both μ {\displaystyle \mu } and ζ {\displaystyle \zeta } should be considered as hyperparameters to tune the amount of regularization versus the sum squared error. The solution does only depend on the ratio γ = ζ / μ {\displaystyle \gamma =\zeta /\mu } , therefore the original formulation uses only γ {\displaystyle \gamma } as tuning parameter. We use both μ {\displaystyle \mu } and ζ {\displaystyle \zeta } as parameters in order to provide a Bayesian interpretation to LS-SVM. The solution of LS-SVM regressor will be obtained after we construct the Lagrangian function: { L 2 ( w , b , e , α ) = J 2 ( w , e ) − ∑ i = 1 N α i { [ w T ϕ ( x i ) + b ] + e i − y i } , = 1 2 w T w + γ 2 ∑ i = 1 N e i 2 − ∑ i = 1 N α i { [ w T ϕ ( x i ) + b ] + e i − y i } , {\displaystyle {\begin{cases}L_{2}(w,b,e,\alpha )\;=J_{2}(w,e)-\sum \limits _{i=1}^{N}\alpha _{i}\left\{{\left[{w^{T}\phi (x_{i})+b}\right]+e_{i}-y_{i}}\right\},\\\quad \quad \quad \quad \quad \;={\frac {1}{2}}w^{T}w+{\frac {\gamma }{2}}\sum \limits _{i=1}^{N}e_{i}^{2}-\sum \limits _{i=1}^{N}\alpha _{i}\left\{\left[w^{T}\phi (x_{i})+b\right]+e_{i}-y_{i}\right\},\end{cases}}} where α i ∈ R {\displaystyle \alpha _{i}\in \mathbb {R} } are the Lagrange multipliers. The conditions for optimality are { ∂ L 2 ∂ w = 0 → w = ∑ i = 1 N α i ϕ ( x i ) , ∂ L 2 ∂ b = 0 → ∑ i = 1 N α i = 0 , ∂ L 2 ∂ e i = 0 → α i = γ e i , i = 1 , … , N , ∂ L 2 ∂ α i = 0 → y i = w T ϕ ( x i ) + b + e i , i = 1 , … , N . {\displaystyle {\begin{cases}{\frac {\partial L_{2}}{\partial w}}=0\quad \to \quad w=\sum \limits _{i=1}^{N}\alpha _{i}\phi (x_{i}),\\{\frac {\partial L_{2}}{\partial b}}=0\quad \to \quad \sum \limits _{i=1}^{N}\alpha _{i}=0,\\{\frac {\partial L_{2}}{\partial e_{i}}}=0\quad \to \quad \alpha _{i}=\gamma e_{i},\;i=1,\ldots ,N,\\{\frac {\partial L_{2}}{\partial \alpha _{i}}}=0\quad \to \quad y_{i}=w^{T}\phi (x_{i})+b+e_{i},\,i=1,\ldots ,N.\end{cases}}} Elimination of w {\displaystyle w} and e {\displaystyle e} will yield a linear system instead of a quadratic programming problem: [ 0 1 N T 1 N Ω + γ − 1 I N ] [ b α ] = [ 0 Y ] , {\displaystyle \left[{\begin{matrix}0&1_{N}^{T}\\1_{N}&\Omega +\gamma ^{-1}I_{N}\end{matrix}}\right]\left[{\begin{matrix}b\\\alpha \end{matrix}}\right]=\left[{\begin{matrix}0\\Y\end{matrix}}\right],} with Y = [ y 1 , … , y N ] T {\displaystyle Y=[y_{1},\ldots ,y_{N}]^{T}} , 1 N = [ 1 , … , 1 ] T {\displaystyle 1_{N}=[1,\ldots ,1]^{T}} and α = [ α 1 , … , α N ] T {\displaystyle \alpha =[\alpha _{1},\ldots ,\alpha _{N}]^{T}} . Here, I N {\displaystyle I_{N}} is an N × N {\displaystyle N\times N} identity matrix, and Ω ∈ R N × N {\displaystyle \Omega \in \mathbb {R} ^{N\times N}} is the kernel matrix defined by Ω i j = ϕ ( x i ) T ϕ ( x j ) = K ( x i , x j ) {\displaystyle \Omega _{ij}=\phi (x_{i})^{T}\phi (x_{j})=K(x_{i},x_{j})} . === Kernel function K === For the kernel function K(•, •) one typically has the following choices: Linear kernel : K ( x , x i ) = x i T x , {\displaystyle K(x,x_{i})=x_{i}^{T}x,} Polynomial kernel of degree d {\displaystyle d} : K ( x , x i ) = ( 1 + x i T x / c ) d , {\displaystyle K(x,x_{i})=\left({1+x_{i}^{T}x/c}\right)^{d},} Radial basis function RBF kernel : K ( x , x i ) = exp ⁡ ( − ‖ x − x i ‖ 2 / σ 2 ) , {\displaystyle K(x,x_{i})=\exp \left({-\left\|{x-x_{i}}\right\|^{2}/\sigma ^{2}}\right),} MLP kernel : K ( x , x i ) = tanh ⁡ ( k x i T x + θ ) , {\displaystyle K(x,x_{i})=\tanh \left({k

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  Error tolerance (PAC learning)

  In PAC learning, error tolerance refers to the ability of an algorithm to learn when the examples received have been corrupted in some way. In fact, this is a very common and important issue since in many applications it is not possible to access noise-free data. Noise can interfere with the learning process at different levels: the algorithm may receive data that have been occasionally mislabeled, or the inputs may have some false information, or the classification of the examples may have been maliciously adulterated. == Notation and the Valiant learning model == In the following, let X {\displaystyle X} be our n {\displaystyle n} -dimensional input space. Let H {\displaystyle {\mathcal {H}}} be a class of functions that we wish to use in order to learn a { 0 , 1 } {\displaystyle \{0,1\}} -valued target function f {\displaystyle f} defined over X {\displaystyle X} . Let D {\displaystyle {\mathcal {D}}} be the distribution of the inputs over X {\displaystyle X} . The goal of a learning algorithm A {\displaystyle {\mathcal {A}}} is to choose the best function h ∈ H {\displaystyle h\in {\mathcal {H}}} such that it minimizes e r r o r ( h ) = P x ∼ D ( h ( x ) ≠ f ( x ) ) {\displaystyle error(h)=P_{x\sim {\mathcal {D}}}(h(x)\neq f(x))} . Let us suppose we have a function s i z e ( f ) {\displaystyle size(f)} that can measure the complexity of f {\displaystyle f} . Let Oracle ( x ) {\displaystyle {\text{Oracle}}(x)} be an oracle that, whenever called, returns an example x {\displaystyle x} and its correct label f ( x ) {\displaystyle f(x)} . When no noise corrupts the data, we can define learning in the Valiant setting: Definition: We say that f {\displaystyle f} is efficiently learnable using H {\displaystyle {\mathcal {H}}} in the Valiant setting if there exists a learning algorithm A {\displaystyle {\mathcal {A}}} that has access to Oracle ( x ) {\displaystyle {\text{Oracle}}(x)} and a polynomial p ( ⋅ , ⋅ , ⋅ , ⋅ ) {\displaystyle p(\cdot ,\cdot ,\cdot ,\cdot )} such that for any 0 < ε ≤ 1 {\displaystyle 0<\varepsilon \leq 1} and 0 < δ ≤ 1 {\displaystyle 0<\delta \leq 1} it outputs, in a number of calls to the oracle bounded by p ( 1 ε , 1 δ , n , size ( f ) ) {\displaystyle p\left({\frac {1}{\varepsilon }},{\frac {1}{\delta }},n,{\text{size}}(f)\right)} , a function h ∈ H {\displaystyle h\in {\mathcal {H}}} that satisfies with probability at least 1 − δ {\displaystyle 1-\delta } the condition error ( h ) ≤ ε {\displaystyle {\text{error}}(h)\leq \varepsilon } . In the following we will define learnability of f {\displaystyle f} when data have suffered some modification. == Classification noise == In the classification noise model a noise rate 0 ≤ η < 1 2 {\displaystyle 0\leq \eta <{\frac {1}{2}}} is introduced. Then, instead of Oracle ( x ) {\displaystyle {\text{Oracle}}(x)} that returns always the correct label of example x {\displaystyle x} , algorithm A {\displaystyle {\mathcal {A}}} can only call a faulty oracle Oracle ( x , η ) {\displaystyle {\text{Oracle}}(x,\eta )} that will flip the label of x {\displaystyle x} with probability η {\displaystyle \eta } . As in the Valiant case, the goal of a learning algorithm A {\displaystyle {\mathcal {A}}} is to choose the best function h ∈ H {\displaystyle h\in {\mathcal {H}}} such that it minimizes e r r o r ( h ) = P x ∼ D ( h ( x ) ≠ f ( x ) ) {\displaystyle error(h)=P_{x\sim {\mathcal {D}}}(h(x)\neq f(x))} . In applications it is difficult to have access to the real value of η {\displaystyle \eta } , but we assume we have access to its upperbound η B {\displaystyle \eta _{B}} . Note that if we allow the noise rate to be 1 / 2 {\displaystyle 1/2} , then learning becomes impossible in any amount of computation time, because every label conveys no information about the target function. Definition: We say that f {\displaystyle f} is efficiently learnable using H {\displaystyle {\mathcal {H}}} in the classification noise model if there exists a learning algorithm A {\displaystyle {\mathcal {A}}} that has access to Oracle ( x , η ) {\displaystyle {\text{Oracle}}(x,\eta )} and a polynomial p ( ⋅ , ⋅ , ⋅ , ⋅ ) {\displaystyle p(\cdot ,\cdot ,\cdot ,\cdot )} such that for any 0 ≤ η ≤ 1 2 {\displaystyle 0\leq \eta \leq {\frac {1}{2}}} , 0 ≤ ε ≤ 1 {\displaystyle 0\leq \varepsilon \leq 1} and 0 ≤ δ ≤ 1 {\displaystyle 0\leq \delta \leq 1} it outputs, in a number of calls to the oracle bounded by p ( 1 1 − 2 η B , 1 ε , 1 δ , n , s i z e ( f ) ) {\displaystyle p\left({\frac {1}{1-2\eta _{B}}},{\frac {1}{\varepsilon }},{\frac {1}{\delta }},n,size(f)\right)} , a function h ∈ H {\displaystyle h\in {\mathcal {H}}} that satisfies with probability at least 1 − δ {\displaystyle 1-\delta } the condition e r r o r ( h ) ≤ ε {\displaystyle error(h)\leq \varepsilon } . == Statistical query learning == Statistical Query Learning is a kind of active learning problem in which the learning algorithm A {\displaystyle {\mathcal {A}}} can decide if to request information about the likelihood P f ( x ) {\displaystyle P_{f(x)}} that a function f {\displaystyle f} correctly labels example x {\displaystyle x} , and receives an answer accurate within a tolerance α {\displaystyle \alpha } . Formally, whenever the learning algorithm A {\displaystyle {\mathcal {A}}} calls the oracle Oracle ( x , α ) {\displaystyle {\text{Oracle}}(x,\alpha )} , it receives as feedback probability Q f ( x ) {\displaystyle Q_{f(x)}} , such that Q f ( x ) − α ≤ P f ( x ) ≤ Q f ( x ) + α {\displaystyle Q_{f(x)}-\alpha \leq P_{f(x)}\leq Q_{f(x)}+\alpha } . Definition: We say that f {\displaystyle f} is efficiently learnable using H {\displaystyle {\mathcal {H}}} in the statistical query learning model if there exists a learning algorithm A {\displaystyle {\mathcal {A}}} that has access to Oracle ( x , α ) {\displaystyle {\text{Oracle}}(x,\alpha )} and polynomials p ( ⋅ , ⋅ , ⋅ ) {\displaystyle p(\cdot ,\cdot ,\cdot )} , q ( ⋅ , ⋅ , ⋅ ) {\displaystyle q(\cdot ,\cdot ,\cdot )} , and r ( ⋅ , ⋅ , ⋅ ) {\displaystyle r(\cdot ,\cdot ,\cdot )} such that for any 0 < ε ≤ 1 {\displaystyle 0<\varepsilon \leq 1} the following hold: Oracle ( x , α ) {\displaystyle {\text{Oracle}}(x,\alpha )} can evaluate P f ( x ) {\displaystyle P_{f(x)}} in time q ( 1 ε , n , s i z e ( f ) ) {\displaystyle q\left({\frac {1}{\varepsilon }},n,size(f)\right)} ; 1 α {\displaystyle {\frac {1}{\alpha }}} is bounded by r ( 1 ε , n , s i z e ( f ) ) {\displaystyle r\left({\frac {1}{\varepsilon }},n,size(f)\right)} A {\displaystyle {\mathcal {A}}} outputs a model h {\displaystyle h} such that e r r ( h ) < ε {\displaystyle err(h)<\varepsilon } , in a number of calls to the oracle bounded by p ( 1 ε , n , s i z e ( f ) ) {\displaystyle p\left({\frac {1}{\varepsilon }},n,size(f)\right)} . Note that the confidence parameter δ {\displaystyle \delta } does not appear in the definition of learning. This is because the main purpose of δ {\displaystyle \delta } is to allow the learning algorithm a small probability of failure due to an unrepresentative sample. Since now Oracle ( x , α ) {\displaystyle {\text{Oracle}}(x,\alpha )} always guarantees to meet the approximation criterion Q f ( x ) − α ≤ P f ( x ) ≤ Q f ( x ) + α {\displaystyle Q_{f(x)}-\alpha \leq P_{f(x)}\leq Q_{f(x)}+\alpha } , the failure probability is no longer needed. The statistical query model is strictly weaker than the PAC model: any efficiently SQ-learnable class is efficiently PAC learnable in the presence of classification noise, but there exist efficient PAC-learnable problems such as parity that are not efficiently SQ-learnable. == Malicious classification == In the malicious classification model an adversary generates errors to foil the learning algorithm. This setting describes situations of error burst, which may occur when for a limited time transmission equipment malfunctions repeatedly. Formally, algorithm A {\displaystyle {\mathcal {A}}} calls an oracle Oracle ( x , β ) {\displaystyle {\text{Oracle}}(x,\beta )} that returns a correctly labeled example x {\displaystyle x} drawn, as usual, from distribution D {\displaystyle {\mathcal {D}}} over the input space with probability 1 − β {\displaystyle 1-\beta } , but it returns with probability β {\displaystyle \beta } an example drawn from a distribution that is not related to D {\displaystyle {\mathcal {D}}} . Moreover, this maliciously chosen example may strategically selected by an adversary who has knowledge of f {\displaystyle f} , β {\displaystyle \beta } , D {\displaystyle {\mathcal {D}}} , or the current progress of the learning algorithm. Definition: Given a bound β B < 1 2 {\displaystyle \beta _{B}<{\frac {1}{2}}} for 0 ≤ β < 1 2 {\displaystyle 0\leq \beta <{\frac {1}{2}}} , we say that f {\displaystyle f} is efficiently learnable using H {\displaystyle {\mathcal {H}}} in the malicious classification model, if there exist a learning algorithm A {\displaystyle {\mathcal {A}}} that has access to Oracle ( x , β ) {\displaystyle {\text{Oracle}}(x,\beta )}

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  Smart data capture

  Smart data capture (SDC), also known as 'intelligent data capture' or 'automated data capture', describes the branch of technology concerned with using computer vision techniques like optical character recognition (OCR), barcode scanning, object recognition and other similar technologies to extract and process information from semi-structured and unstructured data sources. IDC characterize smart data capture as an integrated hardware, software, and connectivity strategy to help organizations enable the capture of data in an efficient, repeatable, scalable, and future-proof way. Data is captured visually from barcodes, text, IDs and other objects - often from many sources simultaneously - before being converted and prepared for digital use, typically by artificial intelligence-powered software. An important feature of SDC is that it focuses not just on capturing data more efficiently but serving up easy-to-access, actionable insights at the instant of data collection to both frontline and desk-based workers, aiding decision-making and making it a two-way process. Smart data capture automates and accelerates capture, applying insights in real time and automating processes based on extracted input. Smart data capture is designed to be repeatable and scalable to reduce low-level manual tasks and eliminate human error. To achieve this goal, smart data capture solutions are often made available using specialist software installed on commodity hardware such as smartphones. However, some solutions may rely on specialized hardware such as dedicated scanning devices, wearables or shop floor robots. == Differences from OCR == Optical character recognition applications are typically concerned with the actual data capture process; they are intended to faithfully reproduce text, words, letters and symbols from a printed document. Smart data capture is multimodal, capable of extracting data from a wider range of semi-structured and unstructured sources, going beyond basic text recognition to offer a wider scope of applications. By extending functionality to provide actionable insights at the point of capture, SDC is also a two-way process (capture-display), while OCR is more commonly one-way (capture only), primarily used for data input. Smart data capture solutions typically have two parts: Data capture (which includes OCR, barcode scanning, object recognition) Functionality that then uses this data to provide actionable insights at the point of capture. == Applications == Smart data capture can be applied to almost any industry and application that requires visual information capture and interpretation. This may include: Retail Warehouse inventory control Logistics, handling and shipping Manufacturing Field service Healthcare Transport and travel Fraud detection

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  Linear genetic programming

  "Linear genetic programming" is unrelated to "linear programming". Linear genetic programming (LGP) is a particular method of genetic programming wherein computer programs in a population are represented as a sequence of register-based instructions from an imperative programming language or machine language. The adjective "linear" stems from the fact that each LGP program is a sequence of instructions and the sequence of instructions is normally executed sequentially. Like in other programs, the data flow in LGP can be modeled as a graph that will visualize the potential multiple usage of register contents and the existence of structurally noneffective code (introns) which are two main differences of this genetic representation from the more common tree-based genetic programming (TGP) variant. Like other Genetic Programming methods, Linear genetic programming requires the input of data to run the program population on. Then, the output of the program (its behaviour) is judged against some target behaviour, using a fitness function. However, LGP is generally more efficient than tree genetic programming due to its two main differences mentioned above: Intermediate results (stored in registers) can be reused and a simple intron removal algorithm exists that can be executed to remove all non-effective code prior to programs being run on the intended data. These two differences often result in compact solutions and substantial computational savings compared to the highly constrained data flow in trees and the common method of executing all tree nodes in TGP. Furthermore, LGP naturally has multiple outputs by defining multiple output registers and easily cooperates with control flow operations. Linear genetic programming has been applied in many domains, including system modeling and system control with considerable success. Linear genetic programming should not be confused with linear tree programs in tree genetic programming, program composed of a variable number of unary functions and a single terminal. Note that linear tree GP differs from bit string genetic algorithms since a population may contain programs of different lengths and there may be more than two types of functions or more than two types of terminals. == Examples of LGP programs == Because LGP programs are basically represented by a linear sequence of instructions, they are simpler to read and to operate on than their tree-based counterparts. For example, a simple program written to solve a Boolean function problem with 3 inputs (in R1, R2, R3) and one output (in R0), could read like this: R1, R2, R3 have to be declared as input (read-only) registers, while R0 and R4 are declared as calculation (read-write) registers. This program is very simple, having just 5 instructions. But mutation and crossover operators could work to increase the length of the program, as well as the content of each of its instructions. Note that one instruction is non-effective or an intron (marked), since it does not impact the output register R0. Recognition of those instructions is the basis for the intron removal algorithm which is used analyze code prior to execution. Technically, this happens by copying an individual and then run the intron removal once. The copy with removed introns is then executed as many times as dictated by the number of training cases. Notably, the original individual is left intact, so as to continue participating in the evolutionary process. It is only the copy that is executed that is compressed by removing these "structural" introns. Another simple program, this one written in the LGP language Slash/A looks like a series of instructions separated by a slash: By representing such code in bytecode format, i.e. as an array of bytes each representing a different instruction, one can make mutation operations simply by changing an element of such an array.

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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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