AI Coding Unity

AI Coding Unity — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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  Microsoft Clipchamp

  Microsoft Clipchamp is a freemium video editing tool developed by Australian company Clipchamp Pty Ltd., a subsidiary of Microsoft. It is a web-based, non-linear editing software that allows users to import, edit, and export audiovisual material in a web browser window. The application is designed to be easy to use for beginners. Clipchamp has offices in Australia, the Philippines, Germany, and the United States. According to figures published by the company, at the beginning of 2021, it had more than 14 million users worldwide. In September 2021, Clipchamp Pty Ltd. was acquired by Microsoft. It has since been offered in a personal version through a Microsoft account and in a business or education version through a work or school account that is built on OneDrive and SharePoint. == Features == Microsoft Clipchamp has multiple features that allow further creativity and accessibility. Since July 2023, users can drag and drop files from their computer, OneDrive, and SharePoint (images, sound & video files) into a list of all media uploaded or inserted. Users can insert media into the video timeline as many times as they want. Users can replace an image, sound, or video clip with another by dragging and dropping it over the target. There is also a Gap Remover tool that removes gaps in the video. Videos can be trimmed, along with timings that can be edited. The user can crop videos and images, too. Text can be added anywhere on the screen, and can be in many fonts, and the size can be changed, too. Specific text color can be selected using presets or an HSV picker, and specific Text Styles (bold, medium, italics, normal) can be selected. The aspect ratio can also be selected, including 16:9, 9:16, 1:1, 4:5, 2:3, and 21:9. Clipchamp also supports numerous effects and transitions for videos and images. The user can export videos in 480p, 720p, and 1080p for free. Exporting GIFs are possible, while the video has to be 15 seconds or less. Microsoft Clipchamp uses a hybrid model of desktop and online application. In the personal version of Clipchamp (on Windows and in a web browser), video processing is all done locally on the computer and mobile phones, but the app itself runs online as a browser-based web app. This is done by uploading and saving project data and information like file names online but not the associated media files themselves. In the work version of Clipchamp, which is a part of Microsoft 365, media files are still processed locally but are automatically backed up to the user's OneDrive or SharePoint work or school account so that it can be accessed anywhere. This version also has integration with other Microsoft productivity services like Microsoft Teams and Microsoft Stream. == History == Clipchamp Pty Ltd. was founded as a startup company by Alexander Dreiling (current CEO), Dave Hewitt, Tobias Raub and Soeren Balko, in Brisbane, Australia, in 2013. In an interview given to SmartCompany, Dreiling commented that at first, the company was "trying to build an enormous, distributed supercomputer". Among the first software developed by the company's team was a tool for video compression and conversion. 2014 saw the official launch of the first version of the free, audiovisual browser-based software on the Clipchamp platform. When the supercomputer project ground to a halt, the team decided to keep going with the video programming technology, which was, in the words of Dreiling, "a tool that worked on Chromebooks". In June 2016, Clipchamp was valued at 1.1 million dollars, according to the Wall Street Journal. In the same month, the second version of Clipchamp was launched internationally. By 2018, the firm had amassed 6.5 million users, attracting investors such as Steve Baxter, who invested one million dollars. In 2020, Clipchamp set up a base in Seattle, USA, after achieving capital of 13.2 million dollars, from alliances made with investment funds such as Transition Level Investments, Tola Capital, and TEN13, among others. In February 2021, Clipchamp published on its website that it has 14 million users worldwide, registered in 250 countries and territories. At that time, the company announced that it had an audiovisual library of 800,000 files. On September 7, 2021, Microsoft announced the acquisition of Clipchamp. In a press release, they expressed their interest in learning more about the video content creation market. Johnson Winter Slattery advised Microsoft on its acquisition. Clipchamp was integrated as part of Windows 11 beginning on March 9, 2022, as part of Insider Preview Build 22572.

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  AI Paragraph Rewriters Reviews: What Actually Works in 2026

  Looking for the best AI paragraph rewriter? An AI paragraph rewriter is software that uses machine learning to help you get more done — it can save you hours every week by automating repetitive work. Most options offer a generous free tier, with paid plans unlocking higher limits, faster processing, and team features. Whether you are a beginner or a pro, the right AI paragraph rewriter slots into your workflow and pays for itself fast. This guide breaks down the top picks, their pros and cons, and who each one is best for.

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  Baum–Welch algorithm

  In electrical engineering, statistical computing and bioinformatics, the Baum–Welch algorithm is a special case of the expectation–maximization algorithm used to find the unknown parameters of a hidden Markov model (HMM). It makes use of the forward-backward algorithm to compute the statistics for the expectation step. The Baum–Welch algorithm, the primary method for inference in hidden Markov models, is numerically unstable due to its recursive calculation of joint probabilities. As the number of variables grows, these joint probabilities become increasingly small, leading to the forward recursions rapidly approaching values below machine precision. == History == The Baum–Welch algorithm was named after its inventors Leonard E. Baum and Lloyd R. Welch. The algorithm and the Hidden Markov models were first described in a series of articles by Baum and his peers at the IDA Center for Communications Research, Princeton in the late 1960s and early 1970s. One of the first major applications of HMMs was to the field of speech processing. In the 1980s, HMMs were emerging as a useful tool in the analysis of biological systems and information, and in particular genetic information. They have since become an important tool in the probabilistic modeling of genomic sequences. == Description == A hidden Markov model describes the joint probability of a collection of "hidden" and observed discrete random variables. It relies on the assumption that the i-th hidden variable given the (i − 1)-th hidden variable is independent of previous hidden variables, and the current observation variables depend only on the current hidden state. The Baum–Welch algorithm uses the well known EM algorithm to find the maximum likelihood estimate of the parameters of a hidden Markov model given a set of observed feature vectors. Let X t {\displaystyle X_{t}} be a discrete hidden random variable with N {\displaystyle N} possible values (i.e. We assume there are N {\displaystyle N} states in total). We assume the P ( X t ∣ X t − 1 ) {\displaystyle P(X_{t}\mid X_{t-1})} is independent of time t {\displaystyle t} , which leads to the definition of the time-independent stochastic transition matrix A = { a i j } = P ( X t = j ∣ X t − 1 = i ) . {\displaystyle A=\{a_{ij}\}=P(X_{t}=j\mid X_{t-1}=i).} The initial state distribution (i.e. when t = 1 {\displaystyle t=1} ) is given by π i = P ( X 1 = i ) . {\displaystyle \pi _{i}=P(X_{1}=i).} The observation variables Y t {\displaystyle Y_{t}} can take one of K {\displaystyle K} possible values. We also assume the observation given the "hidden" state is time independent. The probability of a certain observation y i {\displaystyle y_{i}} at time t {\displaystyle t} for state X t = j {\displaystyle X_{t}=j} is given by b j ( y i ) = P ( Y t = y i ∣ X t = j ) . {\displaystyle b_{j}(y_{i})=P(Y_{t}=y_{i}\mid X_{t}=j).} Taking into account all the possible values of Y t {\displaystyle Y_{t}} and X t {\displaystyle X_{t}} , we obtain the N × K {\displaystyle N\times K} matrix B = { b j ( y i ) } {\displaystyle B=\{b_{j}(y_{i})\}} where b j {\displaystyle b_{j}} belongs to all the possible states and y i {\displaystyle y_{i}} belongs to all the observations. An observation sequence is given by Y = ( Y 1 = y 1 , Y 2 = y 2 , … , Y T = y T ) {\displaystyle Y=(Y_{1}=y_{1},Y_{2}=y_{2},\ldots ,Y_{T}=y_{T})} . Thus we can describe a hidden Markov chain by θ = ( A , B , π ) {\displaystyle \theta =(A,B,\pi )} . The Baum–Welch algorithm finds a local maximum for θ ∗ = a r g m a x θ ⁡ P ( Y ∣ θ ) {\displaystyle \theta ^{}=\operatorname {arg\,max} _{\theta }P(Y\mid \theta )} (i.e. the HMM parameters θ {\displaystyle \theta } that maximize the probability of the observation). === Algorithm === Set θ = ( A , B , π ) {\displaystyle \theta =(A,B,\pi )} with random initial conditions. They can also be set using prior information about the parameters if it is available; this can speed up the algorithm and also steer it toward the desired local maximum. ==== Forward procedure ==== Let α i ( t ) = P ( Y 1 = y 1 , … , Y t = y t , X t = i ∣ θ ) {\displaystyle \alpha _{i}(t)=P(Y_{1}=y_{1},\ldots ,Y_{t}=y_{t},X_{t}=i\mid \theta )} , the probability of seeing the observations y 1 , y 2 , … , y t {\displaystyle y_{1},y_{2},\ldots ,y_{t}} and being in state i {\displaystyle i} at time t {\displaystyle t} . This is found recursively: α i ( 1 ) = π i b i ( y 1 ) , {\displaystyle \alpha _{i}(1)=\pi _{i}b_{i}(y_{1}),} α i ( t + 1 ) = b i ( y t + 1 ) ∑ j = 1 N α j ( t ) a j i . {\displaystyle \alpha _{i}(t+1)=b_{i}(y_{t+1})\sum _{j=1}^{N}\alpha _{j}(t)a_{ji}.} Since this series converges exponentially to zero, the algorithm will numerically underflow for longer sequences. However, this can be avoided in a slightly modified algorithm by scaling α {\displaystyle \alpha } in the forward and β {\displaystyle \beta } in the backward procedure below. ==== Backward procedure ==== Let β i ( t ) = P ( Y t + 1 = y t + 1 , … , Y T = y T ∣ X t = i , θ ) {\displaystyle \beta _{i}(t)=P(Y_{t+1}=y_{t+1},\ldots ,Y_{T}=y_{T}\mid X_{t}=i,\theta )} that is the probability of the ending partial sequence y t + 1 , … , y T {\displaystyle y_{t+1},\ldots ,y_{T}} given starting state i {\displaystyle i} at time t {\displaystyle t} . We calculate β i ( t ) {\displaystyle \beta _{i}(t)} as, β i ( T ) = 1 , {\displaystyle \beta _{i}(T)=1,} β i ( t ) = ∑ j = 1 N β j ( t + 1 ) a i j b j ( y t + 1 ) . {\displaystyle \beta _{i}(t)=\sum _{j=1}^{N}\beta _{j}(t+1)a_{ij}b_{j}(y_{t+1}).} ==== Update ==== We can now calculate the temporary variables, according to Bayes' theorem: γ i ( t ) = P ( X t = i ∣ Y , θ ) = P ( X t = i , Y ∣ θ ) P ( Y ∣ θ ) = α i ( t ) β i ( t ) ∑ j = 1 N α j ( t ) β j ( t ) , {\displaystyle \gamma _{i}(t)=P(X_{t}=i\mid Y,\theta )={\frac {P(X_{t}=i,Y\mid \theta )}{P(Y\mid \theta )}}={\frac {\alpha _{i}(t)\beta _{i}(t)}{\sum _{j=1}^{N}\alpha _{j}(t)\beta _{j}(t)}},} which is the probability of being in state i {\displaystyle i} at time t {\displaystyle t} given the observed sequence Y {\displaystyle Y} and the parameters θ {\displaystyle \theta } ξ i j ( t ) = P ( X t = i , X t + 1 = j ∣ Y , θ ) = P ( X t = i , X t + 1 = j , Y ∣ θ ) P ( Y ∣ θ ) = α i ( t ) a i j β j ( t + 1 ) b j ( y t + 1 ) ∑ k = 1 N ∑ w = 1 N α k ( t ) a k w β w ( t + 1 ) b w ( y t + 1 ) , {\displaystyle \xi _{ij}(t)=P(X_{t}=i,X_{t+1}=j\mid Y,\theta )={\frac {P(X_{t}=i,X_{t+1}=j,Y\mid \theta )}{P(Y\mid \theta )}}={\frac {\alpha _{i}(t)a_{ij}\beta _{j}(t+1)b_{j}(y_{t+1})}{\sum _{k=1}^{N}\sum _{w=1}^{N}\alpha _{k}(t)a_{kw}\beta _{w}(t+1)b_{w}(y_{t+1})}},} which is the probability of being in state i {\displaystyle i} and j {\displaystyle j} at times t {\displaystyle t} and t + 1 {\displaystyle t+1} respectively given the observed sequence Y {\displaystyle Y} and parameters θ {\displaystyle \theta } . The denominators of γ i ( t ) {\displaystyle \gamma _{i}(t)} and ξ i j ( t ) {\displaystyle \xi _{ij}(t)} are the same ; they represent the probability of making the observation Y {\displaystyle Y} given the parameters θ {\displaystyle \theta } . The parameters of the hidden Markov model θ {\displaystyle \theta } can now be updated: π i ∗ = γ i ( 1 ) , {\displaystyle \pi _{i}^{}=\gamma _{i}(1),} which is the expected frequency spent in state i {\displaystyle i} at time 1 {\displaystyle 1} . a i j ∗ = ∑ t = 1 T − 1 ξ i j ( t ) ∑ t = 1 T − 1 γ i ( t ) , {\displaystyle a_{ij}^{}={\frac {\sum _{t=1}^{T-1}\xi _{ij}(t)}{\sum _{t=1}^{T-1}\gamma _{i}(t)}},} which is the expected number of transitions from state i to state j compared to the expected total number of transitions starting in state i, including from state i to itself. The number of transitions starting in state i is equivalent to the number of times state i is observed in the sequence from t = 1 to t = T − 1. b i ∗ ( v k ) = ∑ t = 1 T 1 y t = v k γ i ( t ) ∑ t = 1 T γ i ( t ) , {\displaystyle b_{i}^{}(v_{k})={\frac {\sum _{t=1}^{T}1_{y_{t}=v_{k}}\gamma _{i}(t)}{\sum _{t=1}^{T}\gamma _{i}(t)}},} where 1 y t = v k = { 1 if y t = v k , 0 otherwise {\displaystyle 1_{y_{t}=v_{k}}={\begin{cases}1&{\text{if }}y_{t}=v_{k},\\0&{\text{otherwise}}\end{cases}}} is an indicator function, and b i ∗ ( v k ) {\displaystyle b_{i}^{}(v_{k})} is the expected number of times the output observations have been equal to v k {\displaystyle v_{k}} while in state i {\displaystyle i} over the expected total number of times in state i {\displaystyle i} . These steps are now repeated iteratively until a desired level of convergence. Note: It is possible to over-fit a particular data set. That is, P ( Y ∣ θ final ) > P ( Y ∣ θ true ) {\displaystyle P(Y\mid \theta _{\text{final}})>P(Y\mid \theta _{\text{true}})} . The algorithm also does not guarantee a global maximum. ==== Multiple sequences ==== The algorithm described thus far assumes a single observed sequence Y = y 1 , … , y T {\displaystyle Y=y_{1},\ldots ,y_{T}} . However, in many situations, there are several sequences observed: Y 1 ,

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  Tomáš Mikolov

  Tomáš Mikolov is a Czech computer scientist working in the field of machine learning. In March 2020, Mikolov became a senior research scientist at the Czech Institute of Informatics, Robotics and Cybernetics. == Career == Mikolov obtained his PhD in Computer Science from Brno University of Technology for his work on recurrent neural network-based language models. He is the lead author of the 2013 paper that introduced the Word2vec technique in natural language processing and is an author on the FastText architecture. Mikolov came up with the idea to generate text from neural language models in 2007 and his RNNLM toolkit was the first to demonstrate the capability to train language models on large corpora, resulting in large improvements over the state of the art. Prior to joining Facebook in 2014, Mikolov worked as a visiting researcher at Johns Hopkins University, Université de Montréal, Microsoft and Google. He left Facebook at some time in 2019/2020 to join the Czech Institute of Informatics, Robotics and Cybernetics. Mikolov has argued that humanity might be at a greater existential risk if an artificial general intelligence is not developed.

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  Plane Finder

  Plane Finder is a United Kingdom-based real-time flight tracking service launched in 2009, that is able to show flight data globally. The data available includes flight numbers, how fast an aircraft is moving, its elevation and destination of travel. Several variants of the service are available as mobile apps including free, premium 3D and augmented reality versions. The flight tracking map and database can be accessed by web browsers. Plane Finder allows registered users to share their ADS-B and MLAT data via the Plane Finder ADS-B Client, available for macOS, Windows and Linux. Plane Finder supports VFR charts from NATS, and was the first major flight tracking app to introduce a replay feature, allowing users to replay flights dating back to 2011. == Flight tracking == Plane Finder collects data from its own global network of receivers, using the following sources. === Automatic dependent surveillance-broadcast (ADS-B) === A network of automatic dependent surveillance-broadcast (ADS-B) receivers gathers aircraft data such as callsign, position and speed. Plane Finder serves to supplement this data with additional information, including aircraft registration/tail number, departure airport, destination, artwork, and photographs. Plane Finder users can apply for an ADS-B receiver in exchange for their flight data. === Multilateration (MLAT) === To deliver aircraft position data where ADS-B is unavailable, Plane Finder uses multilateration (MLAT). Using three or more receivers running Plane Finder client software, monitoring the aircraft simultaneously, the aircraft’s position is calculated using receiver location and accurate timestamps. While European airspace is widely covered, only some parts of North American airspace are covered. === Federal Aviation Administration (FAA) feed === ADS-B is prevalent across Europe and Australia, but not in North America. Where MLAT or ADS-B data is unavailable, a feed from the Federal Aviation Administration provides flight information. The FAA feed covers United States and Canadian airspace, including bordering areas of the Atlantic and Pacific Oceans. === FLARM feed === Plane Finder collects data from a centralised FLARM feed, for monitoring small aircraft and gliders. == Flight data source == The Plane Finder website and database is widely used as an information source to support articles in the media. The Independent used Plane Finder flight tracking to demonstrate to readers the flight path of flight MT2706, which turned back as a result of last minute Egyptian government flight restrictions on 6 November 2015. The Independent also used Plane Finder information to demonstrate a timeline of the speed/altitude of flight 7K 9268, a Russian plane which crashed on 31 October 2015. The BBC cited Plane Finder in regard to the point at which at British Airways flight turned back to Heathrow Airport to make an emergency landing after smoke was seen coming from its engines. Plane Finder data has also been used to create original imagery for the media, such as the Washington Post, which used Plane Finder as a source to show flight patterns immediately after the Brussels bombings in March 2016.

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  Co-occurrence

  In linguistics, co-occurrence or cooccurrence (in older texts often shown with diacritic as coöccurrence) is an above-chance frequency of ordered occurrence of two adjacent terms in a text corpus. Co-occurrence in this linguistic sense can be interpreted as an indicator of semantic proximity or an idiomatic expression. Corpus linguistics and its statistical analyses can reveal (regularity of) patterns of co-occurrences within a language and enable the working out of typical collocations for its lexical items. A co-occurrence restriction is identified when linguistic elements never occur together. Analysis of these restrictions can lead to discoveries about the structure and development of a language. Co-occurrence can be seen an extension of word counting in higher dimensions. Co-occurrence can be quantitatively described using measures like a massive correlation or mutual information. Co-occurrence information and knowledge of co-occurring words may be relevant in analysis of language for the purposes of large language models, part of the emerging field of artificial intelligence, and helpful in word games such as scrabble.

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  Maghi King

  Margaret (Maghi) Daniel King is a retired British computational linguist known for her work on evaluating the quality of machine translation. She is an honorary professor in the Department of Translation Technology of the University of Geneva in Switzerland, and the former director of the Dalle Molle Institute for Semantic and Cognitive Studies at the University of Geneva. == Education and career == King read classics, Ancient History and Philosophy (Greats) at the University of Oxford, worked as a computer programmer, and became a lecturer in the Department of Computation at the University of Manchester Institute of Science and Technology. She moved to the Dalle Molle Institute for Semantic and Cognitive Studies (ISSCO) in 1974. In 1976, ISSCO became part of the University of Geneva, and she continued there, becoming ISSCO's director in 1978. She remained director until her retirement in 2006. == Recognition == King is a Fellow of the European Association for Artificial Intelligence (formerly ECCAI), elected in 1999.

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  Richard Zemel

  Richard Stanley Zemel (born 1963) is a Canadian-American computer scientist and professor at Columbia University, Department of Computer Science, and a leading figure in the field of machine learning and computer vision. Zemel studied the history of science at Harvard University and obtained his B.A. in 1984. He continued his study at the Department of Computer Science of the University of Toronto under the supervision of Geoffrey Hinton. He obtained his M.Sc. and Ph.D. both in computer science in 1989 and 1994, respectively.

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  Color normalization

  Color normalization is a topic in computer vision concerned with artificial color vision and object recognition. In general, the distribution of color values in an image depends on the illumination, which may vary depending on lighting conditions, cameras, and other factors. Color normalization allows for object recognition techniques based on color to compensate for these variations. == Main concepts == === Color constancy === Color constancy is a feature of the human internal model of perception, which provides humans with the ability to assign a relatively constant color to objects even under different illumination conditions. This is helpful for object recognition as well as identification of light sources in an environment. For example, humans see an object approximately as the same color when the sun is bright or when the sun is dim. === Applications === Color normalization has been used for object recognition on color images in the field of robotics, bioinformatics and general artificial intelligence, when it is important to remove all intensity values from the image while preserving color values. One example is in case of a scene shot by a surveillance camera over the day, where it is important to remove shadows or lighting changes on same color pixels and recognize the people that passed. Another example is automated screening tools used for the detection of diabetic retinopathy as well as molecular diagnosis of cancer states, where it is important to include color information during classification. == Known issues == The main issue about certain applications of color normalization is that the result looks unnatural or too distant from the original colors. In cases where there is a subtle variation between important aspects, this can be problematic. More specifically, the side effect can be that pixels become divergent and not reflect the actual color value of the image. A way of combating this issue is to use color normalization in combination with thresholding to correctly and consistently segment a colored image. == Transformations and algorithms == There is a vast array of different transformations and algorithms for achieving color normalization and a limited list is presented here. The performance of an algorithm is dependent on the task and one algorithm which performs better than another in one task might perform worse in another (no free lunch theorem). Additionally, the choice of the algorithm depends on the preferences of the user for the end-result, e.g. they may want a more natural-looking color image. === Grey world === The grey world normalization makes the assumption that changes in the lighting spectrum can be modelled by three constant factors applied to the red, green and blue channels of color. More specifically, a change in illuminated color can be modelled as a scaling α, β and γ in the R, G and B color channels and as such the grey world algorithm is invariant to illumination color variations. Therefore, a constancy solution can be achieved by dividing each color channel by its average value as shown in the following formula: ( α R , β G , γ B ) → ( α R α n ∑ i R , β G β n ∑ i G , γ B γ n ∑ i B ) {\displaystyle \left(\alpha R,\beta G,\gamma B\right)\rightarrow \left({\frac {\alpha R}{{\frac {\alpha }{n}}\sum _{i}R}},{\frac {\beta G}{{\frac {\beta }{n}}\sum _{i}G}},{\frac {\gamma B}{{\frac {\gamma }{n}}\sum _{i}B}}\right)} As mentioned above, grey world color normalization is invariant to illuminated color variations α, β and γ, however it has one important problem: it does not account for all variations of illumination intensity and it is not dynamic; when new objects appear in the scene it fails. To solve this problem there are several variants of the grey world algorithm. Additionally there is an iterative variation of the grey world normalization, however it was not found to perform significantly better. === Histogram equalization === Histogram equalization is a non-linear transform which maintains pixel rank and is capable of normalizing for any monotonically increasing color transform function. It is considered to be a more powerful normalization transformation than the grey world method. The results of histogram equalization tend to have an exaggerated blue channel and look unnatural, due to the fact that in most images the distribution of the pixel values is usually more similar to a Gaussian distribution, rather than uniform. === Histogram specification === Histogram specification transforms the red, green and blue histograms to match the shapes of three specific histograms, rather than simply equalizing them. It refers to a class of image transforms which aims to obtain images of which the histograms have a desired shape. As specified, firstly it is necessary to convert the image so that it has a particular histogram. Assume an image x. The following formula is the equalization transform of this image: y = f ( x ) = ∫ 0 x p x ( u ) d u {\displaystyle y=f(x)=\int \limits _{0}^{x}p_{x}(u)du} Then assume wanted image z. The equalization transform of this image is: y ′ = g ( z ) = ∫ 0 z p z ( u ) d u {\displaystyle y'=g(z)=\int \limits _{0}^{z}p_{z}(u)du} Of course p z ( u ) {\displaystyle p_{z}(u)} is the histogram of the output image. The formula to find the inverse of the above transform is: z = g − 1 ( y ′ ) {\displaystyle z=g^{-1}(y')} Therefore, since images y and y' have the same equalized histogram they are actually the same image, meaning y = y' and the transform from the given image x to the wanted image z is: z = g − 1 ( y ′ ) = g − 1 ( y ) = g − 1 ( f ( x ) ) {\displaystyle z=g^{-1}(y')=g^{-1}(y)=g^{-1}(f(x))} Histogram specification has the advantage of producing more realistic looking images, as it does not exaggerate the blue channel like histogram equalization. === Comprehensive Color Normalization === The comprehensive color normalization is shown to increase localization and object classification results in combination with color indexing. It is an iterative algorithm which works in two stages. The first stage is to use the red, green and blue color space with the intensity normalized, to normalize each pixel. The second stage is to normalize each color channel separately, so that the sum of the color components is equal to one third of the number of pixels. The iterations continue until convergence, meaning no additional changes. Formally: Normalize the color image f ( t ) = [ f i j ( t ) ] i = 1... N , j = 1... M {\displaystyle f^{(t)}=[f_{ij}^{(t)}]_{i=1...N,j=1...M}} which consists of color vectors f i j ( t ) = ( r i j ( t ) , g i j ( t ) , b i j ( t ) ) T . {\displaystyle f_{ij}^{(t)}=(r_{ij}^{(t)},g_{ij}^{(t)},b_{ij}^{(t)})^{T}.} For the first step explained above, compute: S i j := r i j ( t ) + g i j ( t ) + b i j ( t ) {\displaystyle S_{ij}:=r_{ij}^{(t)}+g_{ij}^{(t)}+b_{ij}^{(t)}} which leads to r i j ( t + 1 ) = r i j ( t ) S i j , g i j ( t + 1 ) = g i j ( t ) S i j {\displaystyle r_{ij}^{(t+1)}={\frac {r_{ij}^{(t)}}{S_{ij}}},g_{ij}^{(t+1)}={\frac {g_{ij}^{(t)}}{S_{ij}}}} and b i j ( t + 1 ) = b i j ( t ) S i j . {\displaystyle b_{ij}^{(t+1)}={\frac {b_{ij}^{(t)}}{S_{ij}}}.} For the second step explained above, compute: r ′ = 3 N M ∑ i = 1 N ∑ j = 1 M r i j ( t + 1 ) {\displaystyle r'={\frac {3}{NM}}\sum _{i=1}^{N}\sum _{j=1}^{M}r_{ij}^{(t+1)}} and normalize r i j ( t + 2 ) = r i j ( t + 1 ) r ′ . {\displaystyle r_{ij}^{(t+2)}={\frac {r_{ij}^{(t+1)}}{r'}}.} Of course the same process is done for b' and g'. Then these two steps are repeated until the changes between iteration t and t+2 are less than some set threshold. Comprehensive color normalization, just like the histogram equalization method previously mentioned, produces results that may look less natural due to the reduction in the number of color values.

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  Robert Wilensky

  Robert Wilensky (26 March 1951 – 15 March 2013) was an American computer scientist and professor at the UC Berkeley School of Information, with his main focus of research in artificial intelligence. == Academic career == In 1971, Wilensky received his bachelor's degree in mathematics from Yale University, and in 1978, a Ph.D. in computer science from the same institution. After finishing his thesis, "Understanding Goal-Based Stories", Wilensky joined the faculty from the EECS Department of UC Berkeley. In 1986, he worked as the doctoral advisor of Peter Norvig, who then later published the standard textbook of the field: Artificial Intelligence: A Modern Approach. From 1993 to 1997, Wilensky was the Berkeley Computer Science Division Chair. During this time, he also served as director of the Berkeley Cognitive Science Program, director of the Berkeley Artificial Intelligence Research Project, and board member of the International Computer Science Institute. In 1997, he became a fellow of the Association for Computing Machinery "for research contributions to the areas of natural language processing and digital libraries as well as outstanding leadership in Computer Science." Furthermore, he also was a Fellow of the Association for the Advancement of Artificial Intelligence. He retired from faculty in 2007 and died on Friday, March 15, 2013, of a bacterial infection at the Alta Bates Summit Medical Center. Wilensky was married to Ann Danforth and he is survived by her and their two children, Avi and Eli Wilensky == Research == Throughout his career, Wilensky authored and co-authored over 60 scholarly articles and technical reports on AI, natural language processing, and information dissemination. In addition to his numerous technical publications, Wilensky also published two books on the programming language LISP, LISPcraft and Common LISPcraft, and had almost completed another book manuscript when he suffered a cardiac arrest and stopped writing. Among his publications are: R. Wilensky, (1986-09-17). Common LISPcraft. W. W. Norton & Company. ISBN 9780393955446. T. A. Phelps and R. Wilensky, "Toward active, extensible, networked documents: Multivalent architecture and applications," in Proc. 1st ACM Intl. Conf. on Digital Libraries, E. A. Fox and G. Marchionini, Eds., New York, NY: ACM Press, 1996, pp. 100–108. J. Traupman and R. Wilensky, "Experiments in Improving Unsupervised Word Sense Disambiguation," University of California, Berkeley, Department of EECS, Computer Science Division, Tech. Rep. 03–1227, Feb. 2003. R. Wilensky, Planning and Understanding: A Computational Approach to Human Reasoning, Advanced Book Program, Reading, MA: Addison-Wesley Publishing Co., 1983. R. Wilensky, "Understanding Goal-Based Stories," Yale University, Sep. 1978. B. Kahn and R. Wilensky, "A Framework for Distributed Digital Object Services", May 1995.

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  Brainware

  Brainware was an American software company that marketed Automatic identification and data capture and data extraction products. The company was acquired by Hyland Software in 2017. Brainware originally spun out of Dulles, Virginia-based SER Solutions Inc. in February 2006 when SER was acquired by The Gores Group LLC. From February 2006 to March 2012, Brainware's majority owner was San Francisco-based private equity firm Vista Equity Partners. == History == On March 5, 2012, Lexmark International announced it had acquired the company for a cash price of approximately $148 million. The company was added to Lexmark's Perceptive Software division. On July 10, 2017, Hyland Software finalized its acquisition of the Perceptive Business Unit of Lexmark International, Inc. All enterprise software business assets in the Perceptive business unit, including Perceptive Content (formerly ImageNow), Perceptive Intelligent Capture (formerly Brainware), Acuo VNA, PACSGEAR, Claron, Nolij, Saperion, Pallas Athena, ISYS and Twistage, now operate under Hyland's portfolio of products. Brainware was headquartered in Ashburn, Virginia, USA, with sales, support, professional services and R&D offices in London, UK; Kirchzarten, Germany; and Neuchâtel, Switzerland. The company had partnerships with most major enterprise software providers, including Oracle, SAP and Microsoft, and said its software integrated with most available enterprise content management platforms. Brainware also partnered with a number of hardware providers, including Hewlett-Packard, Fujitsu and OPEX. Brainware's core solution, Distiller, "disrupted the data capture industry by using contextual document data to deliver higher automated processing than earlier technology" said Henry Ijams, Managing Director and Founder, PayStream Advisors. Brainware was awarded a Technology Excellence Award by PayStream Advisors and their Advisory Board to honor those providers who are delivering industry leading solutions. Brainware said its software "could relieve a company of 60 percent to 80 percent of the work of manually keying in information from unstructured documents," and serviced companies such as NEC, Mayo Clinic, Bechtel, Royal Dutch Shell, and Rabobank. In a 2011 comparison report, Real Story Group classifies Brainware as a "Capture Solutions" vendor, competing directly with Kofax and ReadSoft. Brainware and its customers were profiled in publications including Profit Online, Business Finance, imageSource, Managing Automation, Industryweek, Treasury & Risk and others. The company's enterprise search technology has been profiled by InfoWorld.

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  Top 10 AI Pair Programmers Compared (2026)

  Looking for the best AI pair programmer? An AI pair programmer is software that uses machine learning to help you get more done — it can save you hours every week by automating repetitive work. Most options offer a generous free tier, with paid plans unlocking higher limits, faster processing, and team features. Whether you are a beginner or a pro, the right AI pair programmer slots into your workflow and pays for itself fast. Read on for hands-on impressions, pricing tiers, and the standout features that matter.

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  Speech recognition

  Speech recognition (automatic speech recognition (ASR), computer speech recognition, or speech-to-text (STT)) is a sub-field of computational linguistics concerned with methods and technologies that translate spoken language into text or other interpretable forms. Speech recognition applications include voice user interfaces, where the user speaks to a device, which "listens" and processes the audio. Common voice applications include interpreting commands for calling, call routing, home automation, and aircraft control. These applications are called direct voice input. Productivity applications include searching audio recordings, creating transcripts, and dictation. Speech recognition can be used to analyse speaker characteristics, such as identifying native language using pronunciation assessment. Voice recognition (speaker identification) refers to identifying the speaker, rather than speech contents. Recognizing the speaker can simplify the task of translating speech in systems trained on a specific person's voice. It can also be used to authenticate the speaker as part of a security process. == History == Applications for speech recognition developed over many decades, with progress accelerated due to advances in deep learning and the use of big data. These advances are reflected in an increase in academic papers, and greater system adoption. Key areas of growth include vocabulary size, more accurate recognition for unfamiliar speakers (speaker independence), and faster processing speed. === Pre-1970 === 1952 – Bell Labs researchers, Stephen Balashek, R. Biddulph, and K. H. Davis, built Audrey for single-speaker digit recognition. Their system located the formants in the power spectrum of each utterance. 1960 – Gunnar Fant developed and published the source–filter model of speech production. 1962 – IBM's 16-word "Shoebox" machine's speech recognition debuted at the 1962 World's Fair. 1966 – Linear predictive coding, a speech coding method, was proposed by Fumitada Itakura of Nagoya University and Shuzo Saito of Nippon Telegraph and Telephone. 1969 – Funding at Bell Labs came to a halt for several years after the company's head engineer, John R. Pierce, wrote an open letter criticizing speech recognition research. This defunding lasted until Pierce retired and James L. Flanagan took over. Raj Reddy was the first person to work on continuous speech recognition, as a graduate student at Stanford University in the late 1960s. Previous systems required users to pause after each word. Reddy's system issued spoken commands for playing chess. Around this time, Soviet researchers invented the dynamic time warping (DTW) algorithm and used it to create a recognizer capable of operating on a 200-word vocabulary. DTW processed speech by dividing it into short frames (e.g. 10 ms segments) and treating each frame as a unit. Speaker independence, however, remained unsolved. === 1970–1990 === 1971 – DARPA funded a five-year speech recognition research project, Speech Understanding Research, seeking a minimum vocabulary size of 1,000 words. The project considered speech understanding a key to achieving progress in speech recognition, which was later disproved. BBN, IBM, Carnegie Mellon (CMU), and Stanford Research Institute participated. 1972 – The IEEE Acoustics, Speech, and Signal Processing group held a conference in Newton, Massachusetts. 1976 – The first ICASSP was held in Philadelphia, which became a major venue for publishing on speech recognition. During the late 1960s, Leonard Baum developed the mathematics of Markov chains at the Institute for Defense Analysis. A decade later, at CMU, Raj Reddy's students James Baker and Janet M. Baker began using the hidden Markov model (HMM) for speech recognition. James Baker had learned about HMMs while at the Institute for Defense Analysis. HMMs enabled researchers to combine sources of knowledge, such as acoustics, language, and syntax, in a unified probabilistic model. By the mid-1980s, Fred Jelinek's team at IBM created a voice-activated typewriter called Tangora, which could handle a 20,000-word vocabulary. Jelinek's statistical approach placed less emphasis on emulating human brain processes in favor of statistical modelling. (Jelinek's group independently discovered the application of HMMs to speech.) This was controversial among linguists since HMMs are too simplistic to account for many features of human languages. However, the HMM proved to be a highly useful way for modelling speech and replaced dynamic time warping as the dominant speech recognition algorithm in the 1980s. 1982 – Dragon Systems, founded by James and Janet M. Baker, was one of IBM's few competitors. === Practical speech recognition === The 1980s also saw the introduction of the n-gram language model. 1987 – The back-off model enabled language models to use multiple-length n-grams, and CSELT used HMM to recognize languages (in software and hardware, e.g. RIPAC). At the end of the DARPA program in 1976, the best computer available to researchers was the PDP-10 with 4 MB of RAM. It could take up to 100 minutes to decode 30 seconds of speech. Practical products included: 1984 – the Apricot Portable was released with up to 4096 words support, of which only 64 could be held in RAM at a time. 1987 – a recognizer from Kurzweil Applied Intelligence 1990 – Dragon Dictate, a consumer product released in 1990. AT&T deployed the Voice Recognition Call Processing service in 1992 to route telephone calls without a human operator. The technology was developed by Lawrence Rabiner and others at Bell Labs. By the early 1990s, the vocabulary of the typical commercial speech recognition system had exceeded the average human vocabulary. Reddy's former student, Xuedong Huang, developed the Sphinx-II system at CMU. Sphinx-II was the first to do speaker-independent, large vocabulary, continuous speech recognition, and it won DARPA's 1992 evaluation. Handling continuous speech with a large vocabulary was a major milestone. Huang later founded the speech recognition group at Microsoft in 1993. Reddy's student Kai-Fu Lee joined Apple, where, in 1992, he helped develop the Casper speech interface prototype. Lernout & Hauspie, a Belgium-based speech recognition company, acquired other companies, including Kurzweil Applied Intelligence in 1997 and Dragon Systems in 2000. L&H was used in Windows XP. L&H was an industry leader until an accounting scandal destroyed it in 2001. L&H speech technology was bought by ScanSoft, which became Nuance in 2005. Apple licensed Nuance software for its digital assistant Siri. ==== 2000s ==== In the 2000s, DARPA sponsored two speech recognition programs: Effective Affordable Reusable Speech-to-Text (EARS) in 2002, followed by Global Autonomous Language Exploitation (GALE) in 2005. Four teams participated in EARS: IBM; a team led by BBN with LIMSI and the University of Pittsburgh; Cambridge University; and a team composed of ICSI, SRI, and the University of Washington. EARS funded the collection of the Switchboard telephone speech corpus, which contained 260 hours of recorded conversations from over 500 speakers. The GALE program focused on Arabic and Mandarin broadcast news. Google's first effort at speech recognition came in 2007 after recruiting Nuance researchers. Its first product, GOOG-411, was a telephone-based directory service. Since at least 2006, the U.S. National Security Agency has employed keyword spotting, allowing analysts to index large volumes of recorded conversations and identify speech containing "interesting" keywords. Other government research programs focused on intelligence applications, such as DARPA's EARS program and IARPA's Babel program. In the early 2000s, speech recognition was dominated by hidden Markov models combined with feed-forward artificial neural networks (ANN). Later, speech recognition was taken over by long short-term memory (LSTM), a recurrent neural network (RNN) published by Sepp Hochreiter & Jürgen Schmidhuber in 1997. LSTM RNNs avoid the vanishing gradient problem and can learn "Very Deep Learning" tasks that require memories of events that happened thousands of discrete time steps earlier, which is important for speech. Around 2007, LSTMs trained with Connectionist Temporal Classification (CTC) began to outperform. In 2015, Google reported a 49 percent error‑rate reduction in its speech recognition via CTC‑trained LSTM. Transformers, a type of neural network based solely on attention, were adopted in computer vision and language modelling, and then to speech recognition. Deep feed-forward (non-recurrent) networks for acoustic modelling were introduced in 2009 by Geoffrey Hinton and his students at the University of Toronto, and by Li Deng and colleagues at Microsoft Research. In contrast to the prioer incremental improvements, deep learning decreased error rates by 30%. Both shallow and deep forms (e.g., recurrent nets) of ANNs had been explored since the 1980s. Howev

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  Wasserstein GAN

  The Wasserstein Generative Adversarial Network (WGAN) is a variant of generative adversarial network (GAN) proposed in 2017 that aims to "improve the stability of learning, get rid of problems like mode collapse, and provide meaningful learning curves useful for debugging and hyperparameter searches". Compared with the original GAN discriminator, the Wasserstein GAN discriminator provides a better learning signal to the generator. This allows the training to be more stable when generator is learning distributions in very high dimensional spaces. == Motivation == === The GAN game === The original GAN method is based on the GAN game, a zero-sum game with 2 players: generator and discriminator. The game is defined over a probability space ( Ω , B , μ r e f ) {\displaystyle (\Omega ,{\mathcal {B}},\mu _{ref})} , The generator's strategy set is the set of all probability measures μ G {\displaystyle \mu _{G}} on ( Ω , B ) {\displaystyle (\Omega ,{\mathcal {B}})} , and the discriminator's strategy set is the set of measurable functions D : Ω → [ 0 , 1 ] {\displaystyle D:\Omega \to [0,1]} . The objective of the game is L ( μ G , D ) := E x ∼ μ r e f [ ln ⁡ D ( x ) ] + E x ∼ μ G [ ln ⁡ ( 1 − D ( x ) ) ] . {\displaystyle L(\mu _{G},D):=\mathbb {E} _{x\sim \mu _{ref}}[\ln D(x)]+\mathbb {E} _{x\sim \mu _{G}}[\ln(1-D(x))].} The generator aims to minimize it, and the discriminator aims to maximize it. A basic theorem of the GAN game states that Repeat the GAN game many times, each time with the generator moving first, and the discriminator moving second. Each time the generator μ G {\displaystyle \mu _{G}} changes, the discriminator must adapt by approaching the ideal D ∗ ( x ) = d μ r e f d ( μ r e f + μ G ) . {\displaystyle D^{}(x)={\frac {d\mu _{ref}}{d(\mu _{ref}+\mu _{G})}}.} Since we are really interested in μ r e f {\displaystyle \mu _{ref}} , the discriminator function D {\displaystyle D} is by itself rather uninteresting. It merely keeps track of the likelihood ratio between the generator distribution and the reference distribution. At equilibrium, the discriminator is just outputting 1 2 {\displaystyle {\frac {1}{2}}} constantly, having given up trying to perceive any difference. Concretely, in the GAN game, let us fix a generator μ G {\displaystyle \mu _{G}} , and improve the discriminator step-by-step, with μ D , t {\displaystyle \mu _{D,t}} being the discriminator at step t {\displaystyle t} . Then we (ideally) have L ( μ G , μ D , 1 ) ≤ L ( μ G , μ D , 2 ) ≤ ⋯ ≤ max μ D L ( μ G , μ D ) = 2 D J S ( μ r e f ‖ μ G ) − 2 ln ⁡ 2 , {\displaystyle L(\mu _{G},\mu _{D,1})\leq L(\mu _{G},\mu _{D,2})\leq \cdots \leq \max _{\mu _{D}}L(\mu _{G},\mu _{D})=2D_{JS}(\mu _{ref}\|\mu _{G})-2\ln 2,} so we see that the discriminator is actually lower-bounding D J S ( μ r e f ‖ μ G ) {\displaystyle D_{JS}(\mu _{ref}\|\mu _{G})} . === Wasserstein distance === Thus, we see that the point of the discriminator is mainly as a critic to provide feedback for the generator, about "how far it is from perfection", where "far" is defined as Jensen–Shannon divergence. Naturally, this brings the possibility of using a different criteria of farness. There are many possible divergences to choose from, such as the f-divergence family, which would give the f-GAN. The Wasserstein GAN is obtained by using the Wasserstein metric, which satisfies a "dual representation theorem" that renders it highly efficient to compute: A proof can be found in the main page on Wasserstein metric. == Definition == By the Kantorovich-Rubenstein duality, the definition of Wasserstein GAN is clear:A Wasserstein GAN game is defined by a probability space ( Ω , B , μ r e f ) {\displaystyle (\Omega ,{\mathcal {B}},\mu _{ref})} , where Ω {\displaystyle \Omega } is a metric space, and a constant K > 0 {\displaystyle K>0} . There are 2 players: generator and discriminator (also called "critic"). The generator's strategy set is the set of all probability measures μ G {\displaystyle \mu _{G}} on ( Ω , B ) {\displaystyle (\Omega ,{\mathcal {B}})} . The discriminator's strategy set is the set of measurable functions of type D : Ω → R {\displaystyle D:\Omega \to \mathbb {R} } with bounded Lipschitz-norm: ‖ D ‖ L ≤ K {\displaystyle \|D\|_{L}\leq K} . The Wasserstein GAN game is a zero-sum game, with objective function L W G A N ( μ G , D ) := E x ∼ μ G [ D ( x ) ] − E x ∼ μ r e f [ D ( x ) ] . {\displaystyle L_{WGAN}(\mu _{G},D):=\mathbb {E} _{x\sim \mu _{G}}[D(x)]-\mathbb {E} _{x\sim \mu _{ref}}[D(x)].} The generator goes first, and the discriminator goes second. The generator aims to minimize the objective, and the discriminator aims to maximize the objective: min μ G max D L W G A N ( μ G , D ) . {\displaystyle \min _{\mu _{G}}\max _{D}L_{WGAN}(\mu _{G},D).} By the Kantorovich-Rubenstein duality, for any generator strategy μ G {\displaystyle \mu _{G}} , the optimal reply by the discriminator is D ∗ {\displaystyle D^{}} , such that L W G A N ( μ G , D ∗ ) = K ⋅ W 1 ( μ G , μ r e f ) . {\displaystyle L_{WGAN}(\mu _{G},D^{})=K\cdot W_{1}(\mu _{G},\mu _{ref}).} Consequently, if the discriminator is good, the generator would be constantly pushed to minimize W 1 ( μ G , μ r e f ) {\displaystyle W_{1}(\mu _{G},\mu _{ref})} , and the optimal strategy for the generator is just μ G = μ r e f {\displaystyle \mu _{G}=\mu _{ref}} , as it should. == Comparison with GAN == In the Wasserstein GAN game, the discriminator provides a better gradient than in the GAN game. Consider for example a game on the real line where both μ G {\displaystyle \mu _{G}} and μ r e f {\displaystyle \mu _{ref}} are Gaussian. Then the optimal Wasserstein critic D W G A N {\displaystyle D_{WGAN}} and the optimal GAN discriminator D {\displaystyle D} are plotted as below: For fixed discriminator, the generator needs to minimize the following objectives: For GAN, E x ∼ μ G [ ln ⁡ ( 1 − D ( x ) ) ] {\displaystyle \mathbb {E} _{x\sim \mu _{G}}[\ln(1-D(x))]} . For Wasserstein GAN, E x ∼ μ G [ D W G A N ( x ) ] {\displaystyle \mathbb {E} _{x\sim \mu _{G}}[D_{WGAN}(x)]} . Let μ G {\displaystyle \mu _{G}} be parametrized by θ {\displaystyle \theta } , then we can perform stochastic gradient descent by using two unbiased estimators of the gradient: ∇ θ E x ∼ μ G [ ln ⁡ ( 1 − D ( x ) ) ] = E x ∼ μ G [ ln ⁡ ( 1 − D ( x ) ) ⋅ ∇ θ ln ⁡ ρ μ G ( x ) ] {\displaystyle \nabla _{\theta }\mathbb {E} _{x\sim \mu _{G}}[\ln(1-D(x))]=\mathbb {E} _{x\sim \mu _{G}}[\ln(1-D(x))\cdot \nabla _{\theta }\ln \rho _{\mu _{G}}(x)]} ∇ θ E x ∼ μ G [ D W G A N ( x ) ] = E x ∼ μ G [ D W G A N ( x ) ⋅ ∇ θ ln ⁡ ρ μ G ( x ) ] {\displaystyle \nabla _{\theta }\mathbb {E} _{x\sim \mu _{G}}[D_{WGAN}(x)]=\mathbb {E} _{x\sim \mu _{G}}[D_{WGAN}(x)\cdot \nabla _{\theta }\ln \rho _{\mu _{G}}(x)]} where we used the reparameterization trick. As shown, the generator in GAN is motivated to let its μ G {\displaystyle \mu _{G}} "slide down the peak" of ln ⁡ ( 1 − D ( x ) ) {\displaystyle \ln(1-D(x))} . Similarly for the generator in Wasserstein GAN. For Wasserstein GAN, D W G A N {\displaystyle D_{WGAN}} has gradient 1 almost everywhere, while for GAN, ln ⁡ ( 1 − D ) {\displaystyle \ln(1-D)} has flat gradient in the middle, and steep gradient elsewhere. As a result, the variance for the estimator in GAN is usually much larger than that in Wasserstein GAN. See also Figure 3 of. The problem with D J S {\displaystyle D_{JS}} is much more severe in actual machine learning situations. Consider training a GAN to generate ImageNet, a collection of photos of size 256-by-256. The space of all such photos is R 256 2 {\displaystyle \mathbb {R} ^{256^{2}}} , and the distribution of ImageNet pictures, μ r e f {\displaystyle \mu _{ref}} , concentrates on a manifold of much lower dimension in it. Consequently, any generator strategy μ G {\displaystyle \mu _{G}} would almost surely be entirely disjoint from μ r e f {\displaystyle \mu _{ref}} , making D J S ( μ G ‖ μ r e f ) = + ∞ {\displaystyle D_{JS}(\mu _{G}\|\mu _{ref})=+\infty } . Thus, a good discriminator can almost perfectly distinguish μ r e f {\displaystyle \mu _{ref}} from μ G {\displaystyle \mu _{G}} , as well as any μ G ′ {\displaystyle \mu _{G}'} close to μ G {\displaystyle \mu _{G}} . Thus, the gradient ∇ μ G L ( μ G , D ) ≈ 0 {\displaystyle \nabla _{\mu _{G}}L(\mu _{G},D)\approx 0} , creating no learning signal for the generator. Detailed theorems can be found in. == Training Wasserstein GANs == Training the generator in Wasserstein GAN is just gradient descent, the same as in GAN (or most deep learning methods), but training the discriminator is different, as the discriminator is now restricted to have bounded Lipschitz norm. There are several methods for this. === Upper-bounding the Lipschitz norm === Let the discriminator function D {\displaystyle D} to be implemented by a multilayer perceptron: D = D n ∘ D n − 1 ∘ ⋯ ∘ D 1 {\displaystyle D=D_{n}\circ D_{n-1}\circ \cdots \circ D_{1}} where D i ( x ) = h ( W i x ) {\displaystyle D_{i}(x)=h(W_

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  Cobham's theorem

  Cobham's theorem is a theorem in combinatorics on words that has important connections with number theory, notably transcendental numbers, and automata theory. Informally, the theorem gives the condition for the members of a set S of natural numbers written in bases b1 and base b2 to be recognised by finite automata. Specifically, consider bases b1 and b2 such that they are not powers of the same integer. Cobham's theorem states that S written in bases b1 and b2 is recognised by finite automata if and only if S differs by a finite set from a finite union of arithmetic progressions. The theorem was proved by Alan Cobham in 1969 and has since given rise to many extensions and generalisations. == Definitions == Let n > 0 {\displaystyle n>0} be an integer. The representation of a natural number n {\textstyle n} in base b {\textstyle b} is the sequence of digits n 0 n 1 ⋯ n h {\displaystyle n_{0}n_{1}\cdots n_{h}} such that n = n 0 + n 1 b + ⋯ + n h b h {\displaystyle n=n_{0}+n_{1}b+\cdots +n_{h}b^{h}} where 0 ≤ n 0 , n 1 , … , n h < b {\displaystyle 0\leq n_{0},n_{1},\ldots ,n_{h} 0 {\displaystyle n_{h}>0} . The word n 0 n 1 ⋯ n h {\displaystyle n_{0}n_{1}\cdots n_{h}} is often denoted ⟨ n ⟩ b {\displaystyle \langle n\rangle _{b}} , or more simply, n b {\displaystyle n_{b}} . A set of natural numbers S is recognisable in base b {\textstyle b} or more simply b {\textstyle b} -recognisable or b {\textstyle b} -automatic if the set { n b ∣ n ∈ S } {\displaystyle \{n_{b}\mid n\in S\}} of the representations of its elements in base b {\displaystyle b} is a language recognisable by a finite automaton on the alphabet { 0 , 1 , … , b − 1 } {\displaystyle \{0,1,\ldots ,b-1\}} . Two positive integers k {\displaystyle k} and ℓ {\displaystyle \ell } are multiplicatively independent if there are no non-negative integers p {\displaystyle p} and q {\displaystyle q} such that k p = ℓ q {\displaystyle k^{p}=\ell ^{q}} . For example, 2 and 3 are multiplicatively independent, but 8 and 16 are not since 8 4 = 16 3 {\displaystyle 8^{4}=16^{3}} . Two integers are multiplicatively dependent if and only if they are powers of a same third integer. == Problem statements == === Original problem statement === More equivalent statements of the theorem have been given. The original version by Cobham is the following: Another way to state the theorem is by using automatic sequences. Cobham himself calls them "uniform tag sequences." The following form is found in Allouche and Shallit's book:We can show that the characteristic sequence of a set of natural numbers S recognisable by finite automata in base k is a k-automatic sequence and that conversely, for all k-automatic sequences u {\displaystyle u} and all integers 0 ≤ i < k {\displaystyle 0\leq i 1 {\displaystyle \alpha >1} is the dominant eigenvalue of the matrix of morphism f {\displaystyle f} , namely, the matrix M ( f ) = ( m x , y ) x ∈ B , y ∈ A {\displaystyle M(f)=(m_{x,y})_{x\in B,y\in A}} , where m x , y {\displaystyle m_{x,y}} is the number of occurrences of the letter x {\displaystyle x} in the word f ( y ) {\displaystyle f(y)} . A set S of natural numbers is α {\displaystyle \alpha } -recognisable if its characteristic sequence s {\displaystyle s} is α {\displaystyle \alpha } -substitutive. A last definition: a Perron number is an algebraic number z > 1 {\displaystyle z>1} such that all its conjugates belong to the disc { z ′ ∈ C , | z ′ | < z } {\displaystyle \{z'\in \mathbb {C} ,|z'| Read more →