Fuzzy mathematics is a branch of mathematics that extends classical set theory and logic to model reasoning under uncertainty. Initiated by Lotfi Asker Zadeh in 1965 with the introduction of fuzzy sets, the field has since evolved to include fuzzy set theory, fuzzy logic, and various fuzzy analogues of traditional mathematic structures. Unlike classical mathematics, which usually relies on binary membership (an element either belongs to a set or it does not), fuzzy mathematics allows elements to partially belong to a set, with degrees of membership represented by values in the interval [0, 1]. This framework enables more flexible modeling of imprecise or vague concepts. Fuzzy mathematics has found applications in numerous domains, including control theory, artificial intelligence, decision theory, pattern recognition, and linguistics, where the modeling of gradations and uncertainty is essential. == Definition == A fuzzy subset A of a set X is defined by a function A: X → L, where L is typically the interval [0, 1]. This function is called the membership function of the fuzzy subset and assigns to each element x in X a degree of membership A(x) in the fuzzy set A. In classical set theory, a subset of X can be represented by an indicator function (also known as a characteristic function), which maps elements to either 0 or 1, indicating non-membership or full membership, respectively. Fuzzy subsets generalize this concept by allowing any real value between 0 and 1, thereby enabling partial membership. More generally, the codomain L of the membership function can be replaced with any complete lattice, resulting in the broader framework of L-fuzzy sets. == Fuzzification == The development of fuzzification in mathematics can be broadly divided into three historical stages: Initial, straightforward fuzzifications (1960s–1970s), Expansion of generalization techniques (1980s), Standardization, axiomatization, and L-fuzzification (1990s). Fuzzification generally involves extending classical mathematical concepts from binary (crisp) logic, where membership is determined by characteristic functions, to fuzzy logic, where membership is expressed by values in the interval [0, 1] via membership functions. Let A and B be fuzzy subsets of a set X. The fuzzy versions of set-theoretic operations are commonly defined as: ( A ∩ B ) ( x ) = min ( A ( x ) , B ( x ) ) {\displaystyle (A\cap B)(x)=\min(A(x),B(x))} ( A ∪ B ) ( x ) = max ( A ( x ) , B ( x ) ) {\displaystyle (A\cup B)(x)=\max(A(x),B(x))} for all x ∈ X {\displaystyle x\in X} . These operations can be generalized using t-norms and t-conorms, respectively. For example, the minimum operation can be replaced by multiplication: ( A ∩ B ) ( x ) = A ( x ) ⋅ B ( x ) {\displaystyle (A\cap B)(x)=A(x)\cdot B(x)} Fuzzification of algebraic structures often relies on generalizing the closure property. Let ∗ {\displaystyle } be a binary operation on X, and let A be a fuzzy subset of X. Then A is said to satisfy fuzzy closure if: A ( x ∗ y ) ≥ min ( A ( x ) , A ( y ) ) {\displaystyle A(xy)\geq \min(A(x),A(y))} for all x , y ∈ X {\displaystyle x,y\in X} . If ( G , ∗ ) {\displaystyle (G,)} is a group, then a fuzzy subset A of G is a fuzzy subgroup if: A ( x ∗ y − 1 ) ≥ min ( A ( x ) , A ( y − 1 ) ) {\displaystyle A(xy^{-1})\geq \min(A(x),A(y^{-1}))} for all x , y ∈ G {\displaystyle x,y\in G} . Similar generalizations apply to relational properties. For example, for example, for fuzzification of the transitivity property, a fuzzy relation R {\displaystyle R} on X {\displaystyle X} (i.e., a fuzzy subset of X × X {\displaystyle X\times X} ) is said to be fuzzy transitive if: R ( x , z ) ≥ min ( R ( x , y ) , R ( y , z ) ) {\displaystyle R(x,z)\geq \min(R(x,y),R(y,z))} for all x , y , z ∈ X {\displaystyle x,y,z\in X} . == Fuzzy analogues == Fuzzy subgroupoids and fuzzy subgroups were introduced in 1971 by A. Rosenfeld. Analogues of other mathematical subjects have been translated to fuzzy mathematics, such as fuzzy field theory and fuzzy Galois theory, fuzzy topology, fuzzy geometry, fuzzy orderings, and fuzzy graphs.
Cloud computing
Cloud computing is defined by the International Organization for Standardization (ISO) as "a paradigm for enabling network access to a scalable and elastic pool of shareable physical or virtual resources with self-service provisioning and administration on demand". It is commonly referred to as "the cloud". == Characteristics == In 2011, the National Institute of Standards and Technology (NIST) identified five "essential characteristics" for cloud systems. Below are the exact definitions according to NIST: On-demand self-service: "A consumer can unilaterally provision computing capabilities, such as server time and network storage, as needed automatically without requiring human interaction with each service provider." Broad network access: "Capabilities are available over the network and accessed through standard mechanisms that promote use by heterogeneous thin or thick client platforms (e.g., mobile phones, tablets, laptops, and workstations)." Resource pooling: " The provider's computing resources are pooled to serve multiple consumers using a multi-tenant model, with different physical and virtual resources dynamically assigned and reassigned according to consumer demand." Rapid elasticity: "Capabilities can be elastically provisioned and released, in some cases automatically, to scale rapidly outward and inward commensurate with demand. To the consumer, the capabilities available for provisioning often appear unlimited and can be appropriated in any quantity at any time." Measured service: "Cloud systems automatically control and optimize resource use by leveraging a metering capability at some level of abstraction appropriate to the type of service (e.g., storage, processing, bandwidth, and active user accounts). Resource usage can be monitored, controlled, and reported, providing transparency for both the provider and consumer of the utilized service. By 2023, the International Organization for Standardization (ISO) had expanded and refined the list. == History == The history of cloud computing extends to the 1960s, with the initial concepts of time-sharing becoming popularized via remote job entry (RJE). The "data center" model, where users submitted jobs to operators to run on mainframes, was predominantly used during this era. This period saw broad experimentation with making large-scale computing power more accessible through time-sharing, while optimizing infrastructure, platforms, and applications to improve efficiency for end users. The "cloud" metaphor for virtualized services dates to 1994, when it was used by General Magic for the universe of "places" that mobile agents in the Telescript environment could "go". The metaphor is credited to David Hoffman, a General Magic communications specialist, based on its long-standing use in networking and telecom. The expression cloud computing became more widely known in 1996 when Compaq Computer Corporation drew up a business plan for future computing and the Internet. The company's ambition was to supercharge sales with "cloud computing-enabled applications". The business plan foresaw that online consumer file storage would likely be commercially successful. As a result, Compaq decided to sell server hardware to internet service providers. In the 2000s, the application of cloud computing began to take shape with the establishment of Amazon Web Services (AWS) in 2002, which allowed developers to build applications independently. In 2006 Amazon Simple Storage Service, known as Amazon S3, and the Amazon Elastic Compute Cloud (EC2) were released. In 2008 NASA's development of the first open-source software for deploying private and hybrid clouds. The following decade saw the launch of various cloud services. In 2010, Microsoft launched Microsoft Azure, and Rackspace Hosting and NASA initiated an open-source cloud-software project, OpenStack. IBM introduced the IBM SmartCloud framework in 2011, and Oracle announced the Oracle Cloud in 2012. In December 2019, Amazon launched AWS Outposts, a service that extends AWS infrastructure, services, APIs, and tools to customer data centers, co-location spaces, or on-premises facilities. == Value proposition == Cloud computing can shorten time to market by offering pre-configured tools, scalable resources, and managed services, allowing users to focus on core business value rather than maintaining infrastructure. Cloud platforms can enable organizations and individuals to reduce upfront capital expenditures on physical infrastructure by shifting to an operational expenditure model, where costs scale with usage. Cloud platforms also offer managed services and tools, such as artificial intelligence, data analytics, and machine learning, which might otherwise require significant in-house expertise and infrastructure investment. While cloud computing can offer cost advantages through effective resource optimization, organizations often face challenges such as unused resources, inefficient configurations, and hidden costs without proper oversight and governance. Many cloud platforms provide cost management tools, such as AWS Cost Explorer and Azure Cost Management, and frameworks like FinOps have emerged to standardize financial operations in the cloud. Cloud computing also facilitates collaboration, remote work, and global service delivery by enabling secure access to data and applications from any location with an internet connection. Cloud providers offer various redundancy options for core services, such as managed storage and managed databases, though redundancy configurations often vary by service tier. Advanced redundancy strategies, such as cross-region replication or failover systems, typically require explicit configuration and may incur additional costs or licensing fees. Cloud environments operate under a shared responsibility model, where providers are typically responsible for infrastructure security, physical hardware, and software updates, while customers are accountable for data encryption, identity and access management (IAM), and application-level security. These responsibilities vary depending on the cloud service model—Infrastructure as a Service (IaaS), Platform as a Service (PaaS), or Software as a Service (SaaS)—with customers typically having more control and responsibility in IaaS environments and progressively less in PaaS and SaaS models, often trading control for convenience and managed services. == Adoption and suitability == The decision to adopt cloud computing or maintain on-premises infrastructure depends on factors such as scalability, cost structure, latency requirements, regulatory constraints, and infrastructure customization. Organizations with variable or unpredictable workloads, limited capital for upfront investments, or a focus on rapid scalability benefit from cloud adoption. Startups, SaaS companies, and e-commerce platforms often prefer the pay-as-you-go operational expenditure (OpEx) model of cloud infrastructure. Additionally, companies prioritizing global accessibility, remote workforce enablement, disaster recovery, and leveraging advanced services such as AI/ML and analytics are well-suited for the cloud. In recent years, some cloud providers have started offering specialized services for high-performance computing and low-latency applications, addressing some use cases previously exclusive to on-premises setups. On the other hand, organizations with strict regulatory requirements, highly predictable workloads, or reliance on deeply integrated legacy systems may find cloud infrastructure less suitable. Businesses in industries like defense, government, or those handling highly sensitive data often favor on-premises setups for greater control and data sovereignty. Additionally, companies with ultra-low latency requirements, such as high-frequency trading (HFT) firms, rely on custom hardware (e.g., FPGAs) and physical proximity to exchanges, which most cloud providers cannot fully replicate despite recent advancements. Similarly, tech giants like Google, Meta, and Amazon build their own data centers due to economies of scale, predictable workloads, and the ability to customize hardware and network infrastructure for optimal efficiency. However, these companies also use cloud services selectively for certain workloads and applications where it aligns with their operational needs. In practice, many organizations are increasingly adopting hybrid cloud architectures, combining on-premises infrastructure with cloud services. This approach allows businesses to balance scalability, cost-effectiveness, and control, offering the benefits of both deployment models while mitigating their respective limitations. == Challenges and limitations == One of the primary challenges of cloud computing, compared with traditional on-premises systems, is maintaining data security and privacy. Cloud users entrust their sensitive data to third-party providers, who may not have adequate measures to protect it from unau
PCVC Speech Dataset
The PCVC (Persian Consonant Vowel Combination) Speech Dataset is a Modern Persian speech corpus for speech recognition and also speaker recognition. The dataset contains sound samples of Modern Persian combination of vowel and consonant phonemes from different speakers. Every sound sample contains just one consonant and one vowel So it is somehow labeled in phoneme level. This dataset consists of 23 Persian consonants and 6 vowels. The sound samples are all possible combinations of vowels and consonants (138 samples for each speaker). The sample rate of all speech samples is 48000 which means there are 48000 sound samples in every 1 second. Every sound sample starts with consonant then continues with vowel. In each sample, in average, 0.5 second of each sample is speech and the rest is silence. Each sound sample ends with silence. All of sound samples are denoised with "Adaptive noise reduction" algorithm. Compared to Farsdat speech dataset and Persian speech corpus it is more easy to use because it is prepared in .mat data files. Also it is more based on phoneme based separation and all samples are denoised. == Contents == The corpus is downloadable from its Kaggle web page, and contains the following: .mat data files of sound samples in a 23630000 matrix, in which 23 is number of consonants, 6 is the number of vowels and 30000 is the length of sound sample.
BigDog
BigDog is a dynamically stable quadruped military robot platform that was created in 2005 by Boston Dynamics with the Harvard University Concord Field Station. It was funded by the U.S. Defense Advanced Research Projects Agency (DARPA), but the project was shelved after the BigDog's gas engine was deemed too loud for combat. == History == BigDog was funded by the Defense Advanced Research Projects Agency (DARPA) in the hopes that it would be able to serve as a mechanic pack mule to accompany soldiers in terrain too rough for conventional vehicles. Instead of wheels or treads, BigDog uses four legs for movement, allowing it to move across surfaces that would be difficult for wheels. The legs contain a variety of sensors, including joint position and ground contact. BigDog also features a laser gyroscope and a stereo vision system. BigDog is 3 feet (0.91 m) long, stands 2.5 feet (0.76 m) tall, and weighs 240 pounds (110 kg), making it about the size of a small mule. It is capable of traversing difficult terrain, running at four miles per hour (6.4 km/h), carrying 340 pounds (150 kg), and climbing a 35 degree incline. Locomotion is controlled by an onboard computer that receives input from the robot's various sensors. Navigation and balance are also managed by the control system. BigDog's walking pattern is controlled through four legs, each equipped with four low-friction hydraulic cylinder actuators that power the joints. BigDog's locomotion behaviors can vary greatly. It can stand up, sit down, walk with a crawling gait that lifts one leg at a time, walk with a trotting gait lifting diagonal legs, or trot with a running gait. The travel speed of BigDog varies from a 0.62 mph (1 km/h) crawl to a 3.3 mph (5.3 km/h) trot. The BigDog project was headed by Dr. Martin Buehler, who received the Joseph Engelberger Award from the Robotics Industries Association in 2012 for the work. Dr. Buehler while previously a professor at McGill University, headed the robotics lab there, developing four-legged walking and running robots. Built onto the actuators are sensors for joint position and force, and movement is ultimately controlled through an onboard computer which manages the sensors. Approximately 50 sensors are located on BigDog. These measure the attitude and acceleration of the body, motion, and force of joint actuators as well as engine speed, temperature and hydraulic pressure inside the robot's internal engine. Low-level control, such as position and force of the joints, and high-level control such as velocity and altitude during locomotion, are both controlled through the onboard computer. BigDog was featured in episodes of Web Junk 20 and Hungry Beast, and in articles in New Scientist, Popular Science, Popular Mechanics, and The Wall Street Journal. In September 2011 Boston Dynamics released video footage of a new generation of BigDog known as AlphaDog. The footage shows AlphaDog's ability to walk on rough terrain and recover its balance when kicked from the side. The refined equivalent has been designed by Boston Dynamics to exceed the BigDog in terms of capabilities and use to dismounted soldiers. In February 2012, with further DARPA support, the militarized Legged Squad Support System (LS3) variant of BigDog demonstrated its capabilities during a hike over a rough terrain. Starting in the summer of 2012, DARPA planned to complete the overall development of the system and refine its key capabilities in 18 months, ensuring its worth to dismounted warfighters before it is rolled out to squads operating in-theatre. BigDog must be able to demonstrate its ability to complete a 20-mile (32 km) trail in 24 hours, without refuelling, while carrying a 325-pound (150 kg) load. A refinement of its vision sensors will also be conducted. At the end of February 2013, Boston Dynamics released video footage of a modified BigDog with an arm. The arm could pick up objects and throw them. The robot is relying on its legs and torso to help power the motions of the arm. It is believed that it can lift weights around 55 pounds (25 kg). This work was funded by the United States Army Research Laboratory and paved the way for integrating manipulators with quadrupeds as found on Spot, the spiritual successor of BigDog. === Discontinuation === At the end of December 2013, the BigDog project was discontinued. Despite hopes that it would one day work like a pack mule for US soldiers in the field, the gasoline-powered engine was deemed too noisy for use in combat, and it could be heard from hundreds of meters away. A similar project for an all-electric robot named Spot in 2016 was much quieter, but could only carry 45 pounds (20 kg). Both projects are no longer in progress, but the Spot was only released in 2020. == Hardware == BigDog is powered by a small two-stroke, one-cylinder, 15-brake-horsepower (11 kW) engine operating at 9,000 RPM. The engine drives a hydraulic pump, which in turn drives the hydraulic leg actuators. Each leg has four actuators (two for the hip joint, and two each for the knee and ankle joints), for a total of 16. Each actuator unit consists of a hydraulic cylinder, servo valve, position sensor, and force sensor. Onboard computing power is a ruggedized PC/104 board stack with two computers, one running a Pentium M processor running QNX (used for sensor data processing) and another running a Core Duo processor (used for visual data processing). == Gallery ==
Free Studio
Free Studio is a freeware set of multimedia computer programs developed by DVDVideoSoft. The programs are available in one integrated package and also as separate downloads (Free Studio Manager is included in both). == Overview == The Free Studio software bundle consists of about 48 programs, grouped into several sections: YouTube, MP3 & Audio, CD-DVD-BD, DVD & Video, Photo & Images, Mobiles, Apple Devices, and 3D. The largest group is the DVD & Video section containing 14 different applications. Mobiles section is the second largest group with 13 programs. However, the YouTube section, particularly YouTube downloading programs, has gained more popularity among users. The programs have been tested and endorsed by a dozen of software portals and have won awards from these sites. Free Studio is most popular in Germany, Greece, Italy, and the United States. It is also popular in Japan, France, and the United Kingdom. Some of the programs in the package are free and open-source software. == History == DVDVideoSoft project was launched in 2006 by company Digital Wave Ltd., for software development to produce multimedia application software. The founders distributed paid software as an affiliate at the start, later their own products appeared on the site. Free YouTube Download was the first successful program, then DVDVideoSoft created and launched several other 'Free YouTube' applications. Later on upon users' requests DVDVideoSoft started developing other kinds of applications including media converters etc. Today DVDVideoSoft offers up to 49 different programs for video, audio and image processing individually or integrated into the Free Studio package. == Features == DVDVideoSoft YouTube programs can be used to download YouTube videos in their original format and convert them to AVI, DVD, MP4, WMV etc. or different audio formats. YouTube section contains Free Video Call Recorder for Skype button, but the program itself is not included into FS installation (it has to be downloaded and installed separately). The "MP3 & Audio" section consists of the programs which convert audio files between different formats, convert audio files to Flash for web, extract audio from video files, edit audio files (Free Audio Dub), rip and burn CDs. Enclosed in the CD-DVD-BD section are the applications that enable users to burn files and folders to discs, to convert videos to a DVD format and vice versa, to burn CDs, and to copy music from audio CDs into files. The "DVD and Video" section contains several desktop video and DVD converters. Some of the programs can flip, rotate and cut (Free Video Dub) videos. One of the most popular programs from the section is Free Video Dub. Converted videos are now, contrary to previous versions, watermarked if no paid membership is present. Free Studio includes several applications for Apple phones, iPods and other devices. The Mobiles section contains a dozen video converters for various mobile devices such as cell phones, Tablets and Game consoles. They convert videos to play them on (BlackBerry, HTC, LG phones, Sony/Sony Ericsson, Nintendo, Xbox, Motorola phones, etc.) The "Photo & Images" section incorporates the programs for image conversion and resizing, extracting JPEG frames from videos (Free Video To JPEG Converter), recording screen activities, making screenshots (Free Screen Recorder). The 3D section is composed of the programs to make 3D videos and 3D images. There are several algorithms which allow to create different types of 3D images. == Supported formats == === Video formats === Input: .avi; .ivf; .div; .divx; .mpg; .mpeg; .mpe; .mp4; .m4v; .wmv; .asf; .webm; .mkv; .mov; .qt; .ts; .mts; .m2t; .m2ts; .mod; .tod; .vro; .dat; .3gp2; .3gpp; .3gp; .3g2; .dvr-ms; .flv; .f4v; .amv; .rm; .rmm; .rv; .rmvb; .ogv; DVD video Output: .mp4; .wmv; .avi; .mkv; .webm; .flv; .swf; .mov; .3gp; .m2ts; DVD video === Audio formats === Input: .mp3 .wav; .aac; .m4a; .m4b; .wma; .ogg; .flac; .ra; .ram; .amr; .ape; .mka; .tta; .aiff; .au; .mpc; .spx; .ac3; audio cd Output: .mp3; .m4a; .aac; .wav; .wma; .ogg; .flac; .ape; audio CD === Image formats === Input: .jpg, .png, .bmp, .gif, .tga Output: .jpg, .png, .bmp, .gif, .tga, .pdf == Reception == The programs have been tested and endorsed by Chip Online, Tucows, SnapFiles, Brothersoft, and Softonic and have won awards from these sites. Free Studio is most popular in Germany, United States and Italy. It is also popular in Japan, France and the United Kingdom. The most popular applications, according to CNET statistics, include Free YouTube to MP3 Converter, Free Video to MP3 Converter, Free MP4 Video Converter and Free YouTube Download. Other programs with high rank: Free AVI Video Converter, Free Video Editor, Free Audio Converter and Free Studio in a whole. == Criticism == Free Studio (as can be common for freeware packages) is criticized for toolbar and Web search engine installation. Older versions have also included OpenCandy, which is loaded automatically, with no request for user approval. There can be difficulties installing only the programs needed without installing bundled extra programs. In March 2017, DVDVideoSoft announced that it had stopped showing other products' ads during installation and removed all toolbars, search engines, and OpenCandy.
Weak supervision
Weak supervision (also known as semi-supervised learning) is a paradigm in machine learning, the relevance and notability of which increased with the advent of large language models due to the large amount of data required to train them. It is characterized by using a combination of a small amount of human-labeled data (exclusively used in more expensive and time-consuming supervised learning paradigm), followed by a large amount of unlabeled data (used exclusively in unsupervised learning paradigm). In other words, the desired output values are provided only for a subset of the training data. The remaining data is unlabeled or imprecisely labeled. Intuitively, it can be seen as an exam and labeled data as sample problems that the teacher solves for the class as an aid in solving another set of problems. In the transductive setting, these unsolved problems act as exam questions. In the inductive setting, they become practice problems of the sort that will make up the exam. == Problem == The acquisition of labeled data for a learning problem often requires a skilled human agent (e.g. to transcribe an audio segment) or a physical experiment (e.g. determining the 3D structure of a protein or determining whether there is oil at a particular location). The cost associated with the labeling process thus may render large, fully labeled training sets infeasible, whereas acquisition of unlabeled data is relatively inexpensive. In such situations, semi-supervised learning can be of great practical value. Semi-supervised learning is also of theoretical interest in machine learning and as a model for human learning. == Technique == More formally, semi-supervised learning assumes a set of l {\displaystyle l} independently identically distributed examples x 1 , … , x l ∈ X {\displaystyle x_{1},\dots ,x_{l}\in X} with corresponding labels y 1 , … , y l ∈ Y {\displaystyle y_{1},\dots ,y_{l}\in Y} and u {\displaystyle u} unlabeled examples x l + 1 , … , x l + u ∈ X {\displaystyle x_{l+1},\dots ,x_{l+u}\in X} are processed. Semi-supervised learning combines this information to surpass the classification performance that can be obtained either by discarding the unlabeled data and doing supervised learning or by discarding the labels and doing unsupervised learning. Semi-supervised learning may refer to either transductive learning or inductive learning. The goal of transductive learning is to infer the correct labels for the given unlabeled data x l + 1 , … , x l + u {\displaystyle x_{l+1},\dots ,x_{l+u}} only. The goal of inductive learning is to infer the correct mapping from X {\displaystyle X} to Y {\displaystyle Y} . It is unnecessary (and, according to Vapnik's principle, imprudent) to perform transductive learning by way of inferring a classification rule over the entire input space; however, in practice, algorithms formally designed for transduction or induction are often used interchangeably. == Assumptions == In order to make any use of unlabeled data, some relationship to the underlying distribution of data must exist. Semi-supervised learning algorithms make use of at least one of the following assumptions: === Continuity / smoothness assumption === Points that are close to each other are more likely to share a label. This is also generally assumed in supervised learning and yields a preference for geometrically simple decision boundaries. In the case of semi-supervised learning, the smoothness assumption additionally yields a preference for decision boundaries in low-density regions, so few points are close to each other but in different classes. === Cluster assumption === The data tend to form discrete clusters, and points in the same cluster are more likely to share a label (although data that shares a label may spread across multiple clusters). This is a special case of the smoothness assumption and gives rise to feature learning with clustering algorithms. === Manifold assumption === The data lie approximately on a manifold of much lower dimension than the input space. In this case learning the manifold using both the labeled and unlabeled data can avoid the curse of dimensionality. Then learning can proceed using distances and densities defined on the manifold. The manifold assumption is practical when high-dimensional data are generated by some process that may be hard to model directly, but which has only a few degrees of freedom. For instance, human voice is controlled by a few vocal folds, and images of various facial expressions are controlled by a few muscles. In these cases, it is better to consider distances and smoothness in the natural space of the generating problem, rather than in the space of all possible acoustic waves or images, respectively. == History == The heuristic approach of self-training (also known as self-learning or self-labeling) is historically the oldest approach to semi-supervised learning, with examples of applications starting in the 1960s. The transductive learning framework was formally introduced by Vladimir Vapnik in the 1970s. Interest in inductive learning using generative models also began in the 1970s. A probably approximately correct learning bound for semi-supervised learning of a Gaussian mixture was demonstrated by Ratsaby and Venkatesh in 1995. == Methods == === Generative models === Generative approaches to statistical learning first seek to estimate p ( x | y ) {\displaystyle p(x|y)} , the distribution of data points belonging to each class. The probability p ( y | x ) {\displaystyle p(y|x)} that a given point x {\displaystyle x} has label y {\displaystyle y} is then proportional to p ( x | y ) p ( y ) {\displaystyle p(x|y)p(y)} by Bayes' rule. Semi-supervised learning with generative models can be viewed either as an extension of supervised learning (classification plus information about p ( x ) {\displaystyle p(x)} ) or as an extension of unsupervised learning (clustering plus some labels). Generative models assume that the distributions take some particular form p ( x | y , θ ) {\displaystyle p(x|y,\theta )} parameterized by the vector θ {\displaystyle \theta } . If these assumptions are incorrect, the unlabeled data may actually decrease the accuracy of the solution relative to what would have been obtained from labeled data alone. However, if the assumptions are correct, then the unlabeled data necessarily improves performance. The unlabeled data are distributed according to a mixture of individual-class distributions. In order to learn the mixture distribution from the unlabeled data, it must be identifiable, that is, different parameters must yield different summed distributions. Gaussian mixture distributions are identifiable and commonly used for generative models. The parameterized joint distribution can be written as p ( x , y | θ ) = p ( y | θ ) p ( x | y , θ ) {\displaystyle p(x,y|\theta )=p(y|\theta )p(x|y,\theta )} by using the chain rule. Each parameter vector θ {\displaystyle \theta } is associated with a decision function f θ ( x ) = argmax y p ( y | x , θ ) {\displaystyle f_{\theta }(x)={\underset {y}{\operatorname {argmax} }}\ p(y|x,\theta )} . The parameter is then chosen based on fit to both the labeled and unlabeled data, weighted by λ {\displaystyle \lambda } : argmax Θ ( log p ( { x i , y i } i = 1 l | θ ) + λ log p ( { x i } i = l + 1 l + u | θ ) ) {\displaystyle {\underset {\Theta }{\operatorname {argmax} }}\left(\log p(\{x_{i},y_{i}\}_{i=1}^{l}|\theta )+\lambda \log p(\{x_{i}\}_{i=l+1}^{l+u}|\theta )\right)} === Low-density separation === Another major class of methods attempts to place boundaries in regions with few data points (labeled or unlabeled). One of the most commonly used algorithms is the transductive support vector machine, or TSVM (which, despite its name, may be used for inductive learning as well). Whereas support vector machines for supervised learning seek a decision boundary with maximal margin over the labeled data, the goal of TSVM is a labeling of the unlabeled data such that the decision boundary has maximal margin over all of the data. In addition to the standard hinge loss ( 1 − y f ( x ) ) + {\displaystyle (1-yf(x))_{+}} for labeled data, a loss function ( 1 − | f ( x ) | ) + {\displaystyle (1-|f(x)|)_{+}} is introduced over the unlabeled data by letting y = sign f ( x ) {\displaystyle y=\operatorname {sign} {f(x)}} . TSVM then selects f ∗ ( x ) = h ∗ ( x ) + b {\displaystyle f^{}(x)=h^{}(x)+b} from a reproducing kernel Hilbert space H {\displaystyle {\mathcal {H}}} by minimizing the regularized empirical risk: f ∗ = argmin f ( ∑ i = 1 l ( 1 − y i f ( x i ) ) + + λ 1 ‖ h ‖ H 2 + λ 2 ∑ i = l + 1 l + u ( 1 − | f ( x i ) | ) + ) {\displaystyle f^{}={\underset {f}{\operatorname {argmin} }}\left(\displaystyle \sum _{i=1}^{l}(1-y_{i}f(x_{i}))_{+}+\lambda _{1}\|h\|_{\mathcal {H}}^{2}+\lambda _{2}\sum _{i=l+1}^{l+u}(1-|f(x_{i})|)_{+}\right)} An exact solution is intractable due to the non-convex term ( 1 − | f ( x ) | ) + {\displayst
D/Vision Pro
D/Vision Pro was one of the earliest marketed non-linear editing systems. It was released by TouchVision Systems, Inc. in the mid-1990s. The program was DOS-based and worked on either Intel's 386 or 486 processor. The system used AVI compression and worked with the Action Media II board. The system allowed users to digitize video, audio, and timecode, create an edit decision list (EDL), instantly play back the edited program, and output the finished EDL in a wide variety of formats. These cost-effective editing systems were used by numerous independent filmmakers and in low-budget productions during the mid-late 1990s. D/Vision Pro's low-quality compression led TouchVision (later renamed D/Vision Systems) to abandon it in favor of D/Vision Online, which was purchased by Discreet Logic and renamed edit. In June 2002, Discreet discontinued edit, as they did not want it to interfere with smoke sales which were more profitable. Discreet was later purchased by Autodesk.