Lernmatrix (German for "learning matrix") is a special type of artificial neural network (ANN) architecture, similar to associative memory, invented around 1960 by Karl Steinbuch, a pioneer in computer science and ANNs. This model for learning systems could establish complex associations between certain sets of characteristics (e.g., letters of an alphabet) and their meanings. == Function == The Lernmatrix generally consists of n "characteristic lines" and m "meaning lines," where each characteristic line is connected to each meaning line, similar to how neurons in the brain are connected by synapses. (This can be realized in various ways – according to Steinbuch, this could be done by hardware or software). To train a Lernmatrix, values are specified on the corresponding characteristic and meaning lines (binary or real); then the connections between all pairs of characteristic and meaning lines are strengthened by the Hebb rule. A trained Lernmatrix, when given a specific input on the characteristic lines, activates the corresponding meaning lines. In modern language, it is a linear projection module. By appropriately interconnecting several Lernmatrices, a switching system can be built that, after completing certain training phases, is ultimately able to automatically determine the most probable associated meaning for an input sequence of features.
Clip Studio Paint
Clip Studio Paint (previously marketed as Manga Studio in North America), informally known in Japan as Kurisuta (クリスタ), is a family of software applications developed by Japanese graphics software company Celsys. It is used for the digital creation of comics, general illustration, and 2D animation. The software is available in versions for macOS, Windows, iOS, iPadOS, Android, and ChromeOS. The program is widely used by amateur and professional comics creators, and animation studios. The application is sold in editions with varying feature sets. The full-featured edition is a page-based, layered drawing program, with support for bitmap and vector art, text, imported 3D models, and frame-by-frame animation. It is designed for use with a stylus and a graphics tablet or tablet computer. It has drawing tools which emulate natural media such as pencils, ink pens, and brushes, as well as patterns and decorations. It is distinguished from similar programs by features designed for creating comics: tools for creating panel layouts, perspective rulers, sketching, inking, applying tones and textures, coloring, and creating word balloons and captions. == History == The application has it origins in a program for macOS and Windows, released in Japan in 2001 as "Comic Studio". It was sold as "Manga Studio" in the Western market by E Frontier America until 2007, then by Smith Micro Software. Early versions were designed for creating black and white art with only spot color (a typical format for Japanese manga), with version 4 adding support for full-color art. Celsys developed Clip Studio Paint as a replacement for this product, based on the company's Illust Studio application, and it was released on May 31, 2012. It was initially distributed in Western markets as "Manga Studio 5", but in 2016, the branding was unified worldwide as "Clip Studio Paint". At this time, version 1.5.4 introduced a new file format (extension .clip) and frame-by-frame animation. In late 2017, Celsys took over direct support for the software worldwide, and ceased its relationship with Smith Micro. In July 2018, Celsys began a partnership with Graphixly for distribution in North America, South America, and Europe. Clip Studio Paint for the Apple iPad was introduced in November 2017, and for the iPhone in December 2019. Clip Studio Paint for Samsung Galaxy tablets and smartphones was released in August 2020 on the Galaxy Store, with versions for other Android devices and Chromebooks released in December. The Windows and macOS versions of the software have been sold and distributed either from the developer's web site or on DVD, and purchased either with a perpetual license or an ongoing subscription. The versions for iPhone, iPad, and Android-based devices are distributed through the corresponding app stores free of charge, but require a subscription – which includes cloud storage – for unrestricted use. Without a subscription, the tablet versions can be used only for a specified number of months, and the phone versions can be used only for 30 hours per month. From 2013 to 2023, regular updates for version 1 were distributed free of additional charge to both perpetual and subscription users. Since the release of version 2 in 2023, feature updates are included only in subscription plans and are available to perpetual licenses at an additional cost. Perpetual licenses can be upgraded permanently or with an annual "update pass". The "update pass" provides early access to features to be included in subsequent perpetual licenses for 12 months, after which the software reverts to the original license if not renewed. In March 2024, version 3 was released, and version 4 introduced additional features in March 2025. == Editions == Clip Studio Paint is available in three editions, with differing feature sets and prices: Debut (bundle-only grade), Pro (adding support for vector-based drawing, custom textures, and comics-focused features), and EX (adding support for multi-page documents, book exporting, and 2D animation). Companion programs include Clip Studio (for managing and sharing digital assets distributed through the Clip Studio web site, managing licenses, and getting updates and support) and Clip Studio Modeler (for setting up 3D materials to use in Clip Studio Paint).
Transkribus
Transkribus is a platform for the text recognition, image analysis and structure recognition of historical documents. The platform was created in the context of the two EU projects "tranScriptorium" (2013–2015) and "READ" (Recognition and Enrichment of Archival Documents – 2016–2019). It was developed by the University of Innsbruck. Since July 1, 2019 the platform has been directed and further developed by the READ-COOP, a non-profit cooperative. The platform integrates tools developed by research groups throughout Europe, including the Pattern Recognition and Human Language Technology (PRHLT) group of the Technical University of Valencia and the Computational Intelligence Technology Lab (CITlab) group of University of Rostock. Comparable programs that offer similar functions are eScriptorium and OCR4All.
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
Facial recognition system
A facial recognition system is a technology potentially capable of matching a human face from a digital image or a video frame against a database of faces. Such a system is typically employed to authenticate users through ID verification services, and works by pinpointing and measuring facial features from a given image. Development on similar systems began in the 1960s as a form of computer application. Since their inception, facial recognition systems have seen wider uses in recent times on smartphones and in other forms of technology, such as robotics. Because computerized facial recognition involves the measurement of a human's physiological characteristics, facial recognition systems are categorized as biometrics. Although the accuracy of facial recognition systems as a biometric technology is lower than iris recognition, fingerprint image acquisition, palm recognition or voice recognition, it is widely adopted due to its contactless process. Facial recognition systems have been deployed in advanced human–computer interaction, video surveillance, law enforcement, passenger screening, decisions on employment and housing, and automatic indexing of images. Facial recognition systems are employed throughout the world today by governments and private companies. Their effectiveness varies, and some systems have previously been scrapped because of their ineffectiveness. The use of facial recognition systems has also raised controversy, with claims that the systems violate citizens' privacy, commonly make incorrect identifications, encourage gender norms and racial profiling, and do not protect important biometric data. The appearance of synthetic media such as deepfakes has also raised concerns about its security. These claims have led to the ban of facial recognition systems in several cities in the United States. Growing societal concerns led social networking company Meta Platforms to shut down its Facebook facial recognition system in 2021, deleting the face-scan data of more than one billion users. The change represented one of the largest shifts in facial recognition usage in the technology's history. IBM also stopped offering facial recognition technology due to similar concerns. == History of facial recognition technology == Automated facial recognition was pioneered in the 1960s by Woody Bledsoe, Helen Chan Wolf, and Charles Bisson, whose work focused on teaching computers to recognize human faces. Their early facial recognition project was dubbed "man-machine" because a human first needed to establish the coordinates of facial features in a photograph before they could be used by a computer for recognition. Using a graphics tablet, a human would pinpoint facial features coordinates, such as the pupil centers, the inside and outside corners of eyes, and the widows peak in the hairline. The coordinates were used to calculate 20 individual distances, including the width of the mouth and of the eyes. A human could process about 40 pictures an hour, building a database of these computed distances. A computer would then automatically compare the distances for each photograph, calculate the difference between the distances, and return the closed records as a possible match. In 1970, Takeo Kanade publicly demonstrated a face-matching system that located anatomical features such as the chin and calculated the distance ratio between facial features without human intervention. Later tests revealed that the system could not always reliably identify facial features. Nonetheless, interest in the subject grew and in 1977 Kanade published the first detailed book on facial recognition technology. In 1993, the Defense Advanced Research Project Agency (DARPA) and the Army Research Laboratory (ARL) established the face recognition technology program FERET to develop "automatic face recognition capabilities" that could be employed in a productive real life environment "to assist security, intelligence, and law enforcement personnel in the performance of their duties." Face recognition systems that had been trialled in research labs were evaluated. The FERET tests found that while the performance of existing automated facial recognition systems varied, a handful of existing methods could viably be used to recognize faces in still images taken in a controlled environment. The FERET tests spawned three US companies that sold automated facial recognition systems. Vision Corporation and Miros Inc were founded in 1994, by researchers who used the results of the FERET tests as a selling point. Viisage Technology was established by an identification card defense contractor in 1996 to commercially exploit the rights to the facial recognition algorithm developed by Alex Pentland at MIT. Following the 1993 FERET face-recognition vendor test, the Department of Motor Vehicles (DMV) offices in West Virginia and New Mexico became the first DMV offices to use automated facial recognition systems to prevent people from obtaining multiple driving licenses using different names. Driver's licenses in the United States were at that point a commonly accepted form of photo identification. DMV offices across the United States were undergoing a technological upgrade and were in the process of establishing databases of digital ID photographs. This enabled DMV offices to deploy the facial recognition systems on the market to search photographs for new driving licenses against the existing DMV database. DMV offices became one of the first major markets for automated facial recognition technology and introduced US citizens to facial recognition as a standard method of identification. The increase of the US prison population in the 1990s prompted U.S. states to established connected and automated identification systems that incorporated digital biometric databases, in some instances this included facial recognition. In 1999, Minnesota incorporated the facial recognition system FaceIT by Visionics into a mug shot booking system that allowed police, judges and court officers to track criminals across the state. Until the 1990s, facial recognition systems were developed primarily by using photographic portraits of human faces. Research on face recognition to reliably locate a face in an image that contains other objects gained traction in the early 1990s with the principal component analysis (PCA). The PCA method of face detection is also known as Eigenface and was developed by Matthew Turk and Alex Pentland. Turk and Pentland combined the conceptual approach of the Karhunen–Loève theorem and factor analysis, to develop a linear model. Eigenfaces are determined based on global and orthogonal features in human faces. A human face is calculated as a weighted combination of a number of Eigenfaces. Because few Eigenfaces were used to encode human faces of a given population, Turk and Pentland's PCA face detection method greatly reduced the amount of data that had to be processed to detect a face. Pentland in 1994 defined Eigenface features, including eigen eyes, eigen mouths and eigen noses, to advance the use of PCA in facial recognition. In 1997, the PCA Eigenface method of face recognition was improved upon using linear discriminant analysis (LDA) to produce Fisherfaces. LDA Fisherfaces became dominantly used in PCA feature based face recognition. While Eigenfaces were also used for face reconstruction. In these approaches no global structure of the face is calculated which links the facial features or parts. Purely feature based approaches to facial recognition were overtaken in the late 1990s by the Bochum system, which used Gabor filter to record the face features and computed a grid of the face structure to link the features. Christoph von der Malsburg and his research team at the University of Bochum developed Elastic Bunch Graph Matching in the mid-1990s to extract a face out of an image using skin segmentation. By 1997, the face detection method developed by Malsburg outperformed most other facial detection systems on the market. The so-called "Bochum system" of face detection was sold commercially on the market as ZN-Face to operators of airports and other busy locations. The software was "robust enough to make identifications from less-than-perfect face views. It can also often see through such impediments to identification as mustaches, beards, changed hairstyles and glasses—even sunglasses". Real-time face detection in video footage became possible in 2001 with the Viola–Jones object detection framework for faces. Paul Viola and Michael Jones combined their face detection method with the Haar-like feature approach to object recognition in digital images to launch AdaBoost, the first real-time frontal-view face detector. By 2015, the Viola–Jones algorithm had been implemented using small low power detectors on handheld devices and embedded systems. Therefore, the Viola–Jones algorithm has not only broadened the practical application of face recognition systems but
Pyramid (image processing)
Pyramid, or pyramid representation, is a type of multi-scale signal representation developed by the computer vision, image processing and signal processing communities, in which a signal or an image is subject to repeated smoothing and subsampling. Pyramid representation is a predecessor to scale-space representation and multiresolution analysis. == Pyramid generation == There are two main types of pyramids: lowpass and bandpass. A lowpass pyramid is made by smoothing the image with an appropriate smoothing filter and then subsampling the smoothed image, usually by a factor of 2 along each coordinate direction. The resulting image is then subjected to the same procedure, and the cycle is repeated multiple times. Each cycle of this process results in a smaller image with increased smoothing, but with decreased spatial sampling density (that is, decreased image resolution). If illustrated graphically, the entire multi-scale representation will look like a pyramid, with the original image on the bottom and each cycle's resulting smaller image stacked one atop the other. A bandpass pyramid is made by forming the difference between images at adjacent levels in the pyramid and performing image interpolation between adjacent levels of resolution, to enable computation of pixelwise differences. == Pyramid generation kernels == A variety of different smoothing kernels have been proposed for generating pyramids. Among the suggestions that have been given, the binomial kernels arising from the binomial coefficients stand out as a particularly useful and theoretically well-founded class. Thus, given a two-dimensional image, we may apply the (normalized) binomial filter (1/4, 1/2, 1/4) typically twice or more along each spatial dimension and then subsample the image by a factor of two. This operation may then proceed as many times as desired, leading to a compact and efficient multi-scale representation. If motivated by specific requirements, intermediate scale levels may also be generated where the subsampling stage is sometimes left out, leading to an oversampled or hybrid pyramid. With the increasing computational efficiency of CPUs available today, it is in some situations also feasible to use wider supported Gaussian filters as smoothing kernels in the pyramid generation steps. === Gaussian pyramid === In a Gaussian pyramid, subsequent images are weighted down using a Gaussian average (Gaussian blur) and scaled down. Each pixel containing a local average corresponds to a neighborhood pixel on a lower level of the pyramid. This technique is used especially in texture synthesis. === Laplacian pyramid === A Laplacian pyramid is very similar to a Gaussian pyramid but saves the difference image of the blurred versions between each levels. Only the smallest level is not a difference image to enable reconstruction of the high resolution image using the difference images on higher levels. This technique can be used in image compression. === Steerable pyramid === A steerable pyramid, developed by Simoncelli and others, is an implementation of a multi-scale, multi-orientation band-pass filter bank used for applications including image compression, texture synthesis, and object recognition. It can be thought of as an orientation selective version of a Laplacian pyramid, in which a bank of steerable filters are used at each level of the pyramid instead of a single Laplacian or Gaussian filter. == Applications of pyramids == === Alternative representation === In the early days of computer vision, pyramids were used as the main type of multi-scale representation for computing multi-scale image features from real-world image data. More recent techniques include scale-space representation, which has been popular among some researchers due to its theoretical foundation, the ability to decouple the subsampling stage from the multi-scale representation, the more powerful tools for theoretical analysis as well as the ability to compute a representation at any desired scale, thus avoiding the algorithmic problems of relating image representations at different resolution. Nevertheless, pyramids are still frequently used for expressing computationally efficient approximations to scale-space representation. === Detail manipulation === Levels of a Laplacian pyramid can be added to or removed from the original image to amplify or reduce detail at different scales. However, detail manipulation of this form is known to produce halo artifacts in many cases, leading to the development of alternatives such as the bilateral filter. Some image compression file formats use the Adam7 algorithm or some other interlacing technique. These can be seen as a kind of image pyramid. Because those file format store the "large-scale" features first, and fine-grain details later in the file, a particular viewer displaying a small "thumbnail" or on a small screen can quickly download just enough of the image to display it in the available pixels—so one file can support many viewer resolutions, rather than having to store or generate a different file for each resolution.
Discrete diffusion model
In machine learning, discrete diffusion models are a class of diffusion models, which themselves are a class of latent variable generative models. Each discrete diffusion model consists of two major components: the forward jump diffusion process, and the reverse jump diffusion process. The goal of diffusion modeling is, given a given dataset and a forward process, to learn a model for the reverse process, such that the reverse process can generate new elements that are distributed similarly as the original dataset. A trained discrete diffusion model can be sampled in many ways, which trades off computational efficiency and sample quality. In general, higher quality data can be obtained, but at the price of higher computational cost. In standard diffusion modeling, the diffusion process takes place over a state space that is continuous space of R n {\displaystyle \mathbb {R} ^{n}} , but over a discrete set S {\displaystyle S} . A discrete set is simply a set where one cannot speak of "infinitesimally close" points. Points can be more or less separated from each other, but the separation is always a finite number. This in particular means the standard framework of continuous diffusion does not apply, since it uses gaussian noise, which is continuous. Nevertheless, an analogous theory can be produced. Discrete diffusion is usually used for language modeling. In practice, the state space S {\displaystyle S} is not only discrete, but finite, so this is what we will assume from now on. == Continuous time Markov process == In the case of continuous state space, during the forward discrete diffusion process, at each step t → t + d t {\displaystyle t\to t+dt} , we mix in an infinitesimal amount of gaussian noise d x t = − 1 2 β ( t ) x t d t + β ( t ) d W t {\displaystyle dx_{t}=-{\frac {1}{2}}\beta (t)x_{t}dt+{\sqrt {\beta (t)}}dW_{t}} . This changes the probability density function, by first a convolution with the density of a gaussian, followed by a scaling. In the case of discrete state space, the gaussian noise must be replaced by a noise that takes values over a finite set. For example, if the noise is the uniform distribution over S {\displaystyle S} , then the probability distribution at time t + d t {\displaystyle t+dt} satisfies q t + d t ( x ) = ( 1 − d t ) q t ( x ) + d t ( 1 | S | ∑ y ∈ S q t ( y ) ) {\displaystyle q_{t+dt}(x)=(1-dt)q_{t}(x)+dt\left({\frac {1}{|S|}}\sum _{y\in S}q_{t}(y)\right)} More succinctly, ∂ t q t ( x ) = − ( 1 − 1 | S | ) q t ( x ) + ∑ y ∈ S , y ≠ x 1 | S | q t ( y ) {\displaystyle \partial _{t}q_{t}(x)=-\left(1-{\frac {1}{|S|}}\right)q_{t}(x)+\sum _{y\in S,y\neq x}{\frac {1}{|S|}}q_{t}(y)} In general, we do not need to convolve with a uniformly distributed noise, but with an arbitrary noise process. That is, we use an arbitrary matrix Q t {\displaystyle Q_{t}} such that ∂ t q t ( y ) = ∑ x ∈ S Q t ( y , x ) q t ( x ) {\displaystyle \partial _{t}q_{t}(y)=\sum _{x\in S}Q_{t}(y,x)q_{t}(x)} where Q t {\displaystyle Q_{t}} is called the rate matrix. Any matrix may be used as a rate matrix if it has non-negative off-diagonals, and each column sums to 0: Q t ( y , x ) ≥ 0 ∀ y ≠ x , ∑ y ∈ S Q t ( y , x ) = 0 ∀ x {\displaystyle Q_{t}(y,x)\geq 0\quad \forall y\neq x,\quad \sum _{y\in S}Q_{t}(y,x)=0\quad \forall x} A continuous time Markov chain (CTMC) is defined by a continuous function Q {\displaystyle Q} that maps any time t ∈ [ 0 , T ) {\displaystyle t\in [0,T)} to a rate matrix Q t {\displaystyle Q_{t}} . Given the function Q {\displaystyle Q} , time-evolution under the CTMC is done as follows: Given state x t {\displaystyle x_{t}} at time t {\displaystyle t} , and given an infinitesimal d t {\displaystyle dt} , the state at t + d t {\displaystyle t+dt} is x t + d t {\displaystyle x_{t+dt}} , such that Pr ( x t + d t | x t ) = { 1 + Q t ( x t + d t , x t ) d t if x t + d t = x t Q t ( x t + d t , x t ) d t else {\displaystyle \Pr(x_{t+dt}|x_{t})={\begin{cases}1+Q_{t}(x_{t+dt},x_{t})dt&{\text{if }}x_{t+dt}=x_{t}\\Q_{t}(x_{t+dt},x_{t})dt&{\text{else}}\end{cases}}} This implies that the probability distribution function evolves according to ∂ t q t ( y ) = ∑ x ∈ S Q t ( y , x ) q t ( x ) {\displaystyle \partial _{t}q_{t}(y)=\sum _{x\in S}Q_{t}(y,x)q_{t}(x)} which is what we previously specified. === Backward process === Similarly to the case of continuous diffusion, in discrete diffusion, there exists a backward diffusion process Q ¯ t {\displaystyle {\bar {Q}}_{t}} : s ( x , t ) y := q t ( y ) q t ( x ) , Q ¯ t ( y , x ) := { s ( x , t ) y Q t ( x , y ) if y ≠ x − ∑ y : y ≠ x Q ¯ t ( y , x ) if y = x {\displaystyle s(x,t)_{y}:={\frac {q_{t}(y)}{q_{t}(x)}},\quad {\bar {Q}}_{t}(y,x):={\begin{cases}s(x,t)_{y}Q_{t}(x,y)&{\text{if }}y\neq x\\-\sum _{y:y\neq x}{\bar {Q}}_{t}(y,x)&{\text{if }}y=x\end{cases}}} where s ( x , t ) y {\displaystyle s(x,t)_{y}} should be interpreted as the discrete score or concrete score, since, abusing notation a bit, the score function is ∇ ln ρ t ( x ) = 1 d x ( ρ t ( x + d x ) ρ t ( x ) − 1 ) {\displaystyle \nabla \ln \rho _{t}(x)={\frac {1}{dx}}\left({\frac {\rho _{t}(x+dx)}{\rho _{t}(x)}}-1\right)} . If we picture the distribution q t {\displaystyle q_{t}} as a bunch of point-masses, one per state x ∈ S {\displaystyle x\in S} , then the forward diffusion from time t {\displaystyle t} to t + d t {\displaystyle t+dt} is performed by removing Q t ( x , y ) q t ( y ) d t {\displaystyle Q_{t}(x,y)q_{t}(y)dt} from the mass at y {\displaystyle y} and moving it to the mass at x {\displaystyle x} , for each pair x ≠ y {\displaystyle x\neq y} . Thus, the process is reversed in detail by the CTMC defined by Q ¯ {\displaystyle {\bar {Q}}} , since Q ¯ t ( y , x ) q t ( x ) = Q t ( x , y ) q t ( y ) {\displaystyle {\bar {Q}}_{t}(y,x)q_{t}(x)=Q_{t}(x,y)q_{t}(y)} . Given Q ¯ t {\displaystyle {\bar {Q}}_{t}} , if we have a way to sample from q t {\displaystyle q_{t}} , then we can sample from q t − d t {\displaystyle q_{t-dt}} by first sampling x t ∼ q t {\displaystyle x_{t}\sim q_{t}} , then sampling x t − d t {\displaystyle x_{t-dt}} according to Pr ( x t − d t | x t ) = { 1 + Q ¯ t ( x t − d t , x t ) d t if x t − d t = x t Q ¯ t ( x t − d t , x t ) d t else {\displaystyle \Pr(x_{t-dt}|x_{t})={\begin{cases}1+{\bar {Q}}_{t}(x_{t-dt},x_{t})dt&{\text{if }}x_{t-dt}=x_{t}\\{\bar {Q}}_{t}(x_{t-dt},x_{t})dt&{\text{else}}\end{cases}}} === Overall plan of score-matching discrete diffusion modeling === Similar to score-matching continuous diffusion, score-matching discrete diffusion is a method to sample an initial distribution. If we have a certain function s θ {\displaystyle s_{\theta }} that approximates the true score function s θ ( x , t ) y ≈ s ( x , t ) y {\displaystyle s_{\theta }(x,t)_{y}\approx s(x,t)_{y}} , then it allows a corresponding Q ¯ θ {\displaystyle {\bar {Q}}^{\theta }} to be defined in the same way. If we also have a base distribution q base {\displaystyle q_{\text{base}}} such that it is easy to sample from, and approximately equal to the true terminal distribution q base ≈ q T {\displaystyle q_{\text{base}}\approx q_{T}} , then we can perform the backward CTMC with Q ¯ θ {\displaystyle {\bar {Q}}^{\theta }} and q T θ := q terminal {\displaystyle q_{T}^{\theta }:=q_{\text{terminal}}} . When both approximations are good, the backward CTMC would give q 0 θ ≈ q 0 {\displaystyle q_{0}^{\theta }\approx q_{0}} . This is the idea of score-matching discrete diffusion modeling. If q data {\displaystyle q_{\text{data}}} is sharp, in the sense that for some x , x ′ {\displaystyle x,x'} , we have q data ( x ) ≫ q data ( x ′ ) {\displaystyle q_{\text{data}}(x)\gg q_{\text{data}}(x')} , then the score function would diverge as 1 / t {\displaystyle 1/t} at the t → 0 {\displaystyle t\to 0} limit. To avoid this in practice, it is common to use early stopping, which is to stop the backward process at some time δ > 0 {\displaystyle \delta >0} , and sample from q δ θ {\displaystyle q_{\delta }^{\theta }} instead of q 0 θ {\displaystyle q_{0}^{\theta }} . === Tractable forward processes === The theory of CTMC works for any continuous choice of rate matrices Q {\displaystyle Q} . However, most choices are computationally expensive and cannot be used in practice. In the case of continuous diffusion, the gaussian noise is used for the simple reason that the sum of any number of gaussians is still a gaussian. This allows one to sample any x t ∼ ρ t {\displaystyle x_{t}\sim \rho _{t}} by sampling a single x 0 ∼ ρ 0 {\displaystyle x_{0}\sim \rho _{0}} , followed by a single gaussian noise z ∼ N ( 0 , I ) {\displaystyle z\sim {\mathcal {N}}(0,I)} , and let x t = α ¯ t x 0 + σ t z {\displaystyle x_{t}={\sqrt {{\bar {\alpha }}_{t}}}x_{0}+\sigma _{t}z} , without needing any x s {\displaystyle x_{s}} for any 0 < s < t {\displaystyle 0