AI: When a Robot Writes a Play

AI: When a Robot Writes a Play (in Czech: AI: Když robot píše hru) is a 2021 experimental theatre play, where 90% of its script was automatically generated by artificial intelligence (the GPT-2 language model). The play is in Czech language, but an English version of the script also exists. == Creation == The play is the first result of the THEaiTRE research project, aiming to commemorate the centenary of the R.U.R. play by Karel Čapek by investigating to what extent artificial intelligence could be used to create theatre play scripts. The script of the play was created using the THEaiTRobot tool, based on the GPT-2 language model. First, the play dramaturge, David Košťák, described the initial setting of each scene in a few sentences, and wrote the first line for each character. Next, THEaiTRobot suggested a continuation of the script, which the dramaturge could use, reject, or use part of it and let the tool generate a new continuation. Another option was to manually insert another line or a scenic remark. The script was generated in English and was automatically translated to Czech by the state-of-the-art CUBBITT machine translation tool. The resulting script was then further post-edited by the dramaturge. The resulting script was made freely available for non-commercial use both in English and in Czech, with marked manually inserted texts and manual edits. The analysis shows that 90% of the English script is automatically generated, with 10% manually written or manually post-edited. In the Czech script, a larger amount of edits were made, but the analysis claims that these additional edits are corrections of errors of the automated translation and stylistic corrections which do not change the meaning of the lines as represented by the English script, but rather bring the Czech script closer to the English one. == Characters == The play contains 9 characters. The Robot appears in all the scenes, while each of the other characters appears in only one scene. Robot – The lead character, a male humanoid robot. Master – An old man, the creator of the Robot. Boy – A schoolboy. Masseuse – A sex worker in a brothel. Stranger – An engineer. Man. Psychologist. Administrator – A female clerk at an employment agency. Actress – A film actress and a model in a robot-like costume. == Plot == The play is composed of 8 scenes. It tells the story of a humanoid robot, who encounters 8 other characters and engages into various typically human situations and activities, related to death, love, sex, violence, etc. The individual scenes are not tightly linked, but there are some linking points, such as the central character of the robot or some repeated and developing themes, such as the robot's search for love. The scenes often contain some absurd turns and it is often hard to find sense in them. It is therefore a very complicated piece interpretationally, requiring the director and the actors to invest a lot of effort and creativity in finding a meaningful interpretation which would not deviate from the script. In the interpretation by Švanda theatre, who premiered the play and who also participated on the creation of the script, the scenes typically contain non-verbally expressed content which can add a lot to the meaning of the scene compared to what is contained in the actual script (as the script only contains the lines said by the characters). === Scene 1: Death === The play opens by the Robot parting with his dying Master. The Master gives the Robot several last lessons and talks with him about death, soul, and love. === Scene 2: Sense of Humour === In the second scene, the Robot meets a sad and angry Boy, who complains that he wants to go to school, that his girlfriend is crazy, that he wants to buy a car, etc. The Robot tries to help the Boy by giving him advice, but the Boy's reactions are quite negative and irritated. The Boy then repeatedly asks the Robot to tell him a joke; the Robot keeps refusing, but ultimately tells the following joke: When you are dead. When your children are dead. When your grandchildren are dead, I will be still alive. === Scene 3: Nightclub === The Robot wants to feel pleasure, so he goes to a "night club" (a brothel), where he meets a "Masseuse" (a prostitute). The Robot is initially "a bit cold", but eventually manages to enjoy the experience and falls in love with the Masseuse. In the Švanda theatre performance, the Robot and the Masseuse seem to have a sort of virtual sex without touching each other, reminiscent of the sex scene in Demolition Man. === Scene 4: Fear of the Dark === It is the night. The Robot is standing under a lamp, unable to move away from the light as he finds that he is afraid of the dark. He meets a Stranger, an engineer who tells him that robots don't have feelings and that people cannot be trusted, and keeps hurting him. In the Švanda theatre performance, the Man repeatedly zaps the Robot with some kind of electric pulse. === Scene 5: Killer Robot === A Man approaches the Robot and repeatedly asks him to kill him. Instead, the Robot sticks a finger into the Man's anus, which leads to an argument between the Man and the Robot. === Scene 6: Burn Out === The Robot meets a Psychologist, who keeps asking him lots of questions regarding his life, burnout feeling, love, relationships, and emotions. They also talk about the Robot using a device called emotion machine which helps him to get rid of stress. === Scene 7: Search for Job === The Robot comes to an employment agency. He meets an Administrator and asks her to help him find a job. He expresses the wish to become an actor, and talks about his experience as a clown. He reveals his name to be Troy McClure, which is a character from The Simpsons who is an actor. In the Švanda theatre performance, the Administrator starts to seduce the Robot once his name is revealed, which he keeps ignoring; the Administrator then becomes irritated. === Scene 8: Love at First Sight === The Robot meets a human Actress in a robotic costume and falls in love with her immediately. The Actress is first reluctant, but the Robot manages to seduce her and she also falls in love with him. The Robot tells her about a binary world, in which he lives and where he will also take her. Ultimately, the Actress agrees, and the whole play concludes by the Robot and the Actress promising each to other to always be together. In the Švanda theatre performance, the Robot does not have a physical body in this scene, we can only hear his voice and see a pulsating light (based on the line in the script where the Robot says: "I have no body. So I don't need to wear clothes. You can't see me, you only hear me."), and the Actress eventually also agrees to lose her physical body so that she can be with the Robot forever. == Theatrical performances == The play premiered on 26 February 2021 in Švanda Theatre in Prague, Czech Republic, directed by Daniel Hrbek. Due to the COVID-19 pandemic, the play was not played in front of a live audience, but it was broadcast online, in Czech language with English subtitles. The play was followed by a panel discussion by the project members and experts on artificial intelligence. The premiere was viewed by 13,498 spectators worldwide. A short trailer of the premiere is available on YouTube. In 2021, after the opening of the theatres in the Czech Republic to spectators, the play can be viewed at Švanda Theatre. The performance takes approximately 60 minutes, and is followed by a discussion of the creators with the audience. The derniere is planned for 4 February 2023. == Reception == The play received a number of reviews, both in its country of origin as well as internationally. It is praised as first of its kind, although some reviewers note the similarity to previous works, such as the musical Beyond the Fence, the play Lifestyle of the Richard and Family, or the short movie Sunspring; however, these works used less advanced technology, and either were very short (Sunspring) or necessitated a larger amount of human interventions. The reviewers note that the script is far from perfect, with many inconsistencies and nonsensical parts, and conclude that the technology is definitely not yet ready to replace human authors; however, some find some parts of the script frighteningly human-like. The amount of human intervention is a somewhat controversial topic, with some reviewers finding the human influence too large (especially in interpreting the script and putting the play on scene), while others feel that a greater amount of human intervention would have been favorable as this could greatly improve the quality of the play. The reviews also frequently comment on the amount of sex, violence and strong language in the play; this can be attributed to the method used for creating the script, where the GPT-2 language model reflects topics and language common in the human-written articles on the internet that were used to train the model. Furthermore, some r

Noisy text analytics

Noisy text analytics is a process of information extraction whose goal is to automatically extract structured or semistructured information from noisy unstructured text data. While Text analytics is a growing and mature field that has great value because of the huge amounts of data being produced, processing of noisy text is gaining in importance because a lot of common applications produce noisy text data. Noisy unstructured text data is found in informal settings such as online chat, text messages, e-mails, message boards, newsgroups, blogs, wikis and web pages. Also, text produced by processing spontaneous speech using automatic speech recognition and printed or handwritten text using optical character recognition contains processing noise. Text produced under such circumstances is typically highly noisy containing spelling errors, abbreviations, non-standard words, false starts, repetitions, missing punctuations, missing letter case information, pause filling words such as “um” and “uh” and other texting and speech disfluencies. Such text can be seen in large amounts in contact centers, chat rooms, optical character recognition (OCR) of text documents, short message service (SMS) text, etc. Documents with historical language can also be considered noisy with respect to today's knowledge about the language. Such text contains important historical, religious, ancient medical knowledge that is useful. The nature of the noisy text produced in all these contexts warrants moving beyond traditional text analysis techniques. == Techniques for noisy text analysis == Missing punctuation and the use of non-standard words can often hinder standard natural language processing tools such as part-of-speech tagging and parsing. Techniques to both learn from the noisy data and then to be able to process the noisy data are only now being developed. == Possible source of noisy text == World Wide Web: Poorly written text is found in web pages, online chat, blogs, wikis, discussion forums, newsgroups. Most of these data are unstructured and the style of writing is very different from, say, well-written news articles. Analysis for the web data is important because they are sources for market buzz analysis, market review, trend estimation, etc. Also, because of the large amount of data, it is necessary to find efficient methods of information extraction, classification, automatic summarization and analysis of these data. Contact centers: This is a general term for help desks, information lines and customer service centers operating in domains ranging from computer sales and support to mobile phones to apparels. On an average a person in the developed world interacts at least once a week with a contact center agent. A typical contact center agent handles over a hundred calls per day. They operate in various modes such as voice, online chat and E-mail. The contact center industry produces gigabytes of data in the form of E-mails, chat logs, voice conversation transcriptions, customer feedback, etc. A bulk of the contact center data is voice conversations. Transcription of these using state of the art automatic speech recognition results in text with 30-40% word error rate. Further, even written modes of communication like online chat between customers and agents and even the interactions over email tend to be noisy. Analysis of contact center data is essential for customer relationship management, customer satisfaction analysis, call modeling, customer profiling, agent profiling, etc., and it requires sophisticated techniques to handle poorly written text. Printed Documents: Many libraries, government organizations and national defence organizations have vast repositories of hard copy documents. To retrieve and process the content from such documents, they need to be processed using Optical Character Recognition. In addition to printed text, these documents may also contain handwritten annotations. OCRed text can be highly noisy depending on the font size, quality of the print etc. It can range from 2-3% word error rates to as high as 50-60% word error rates. Handwritten annotations can be particularly hard to decipher, and error rates can be quite high in their presence. Short Messaging Service (SMS): Language usage over computer mediated discourses, like chats, emails and SMS texts, significantly differs from the standard form of the language. An urge towards shorter message length facilitating faster typing and the need for semantic clarity, shape the structure of this non-standard form known as the texting language.

Ground truth

Ground truth is information that is known to be real or true, provided by direct observation and measurement (i.e. empirical evidence) as opposed to information provided by inference. The term ground truth appeared in remote sensing literature as early as 1972, when NASA described it as essential "data about ... materials on the earth's surface" used to calibrate measurements. It was later adopted by the statistical modeling and machine learning communities. == Etymology == The Oxford English Dictionary (s.v. ground truth) records the use of the word Groundtruth in the sense of 'fundamental truth' from Henry Ellison's poem "The Siberian Exile's Tale", published in 1833. == Usage == The term "ground truth" can be used as a noun, adjective, and verb. Noun: "ground truth" (no hyphen). Example: "The ground truth is essential for training accurate models." Adjective: "ground-truth" (hyphenated compound adjective). Example: "We need to use ground-truth data to validate the model." Verb: "to ground-truth" or "to groundtruth" (compound verb,). Example: "We need to ground-truth the results to ensure their accuracy." == Statistics and machine learning == In statistics and machine learning, ground truth is the ideal expected result, used in statistical models to prove or disprove research hypotheses. "Ground truthing" is the process of gathering the good data for this test. Ground truth is typically included in labeled data. In machine learning, "ground truth" is not necessarily objectively correct or true. For example, in training AI models or relevance rankers, it may be a set of judgments made by people or inferred from user behavior, which may depend on context. For example, in Bayesian spam filtering, a supervised learning system is typically trained by examples labeled as spam and non-spam. Although these labels may be subjective or inaccurate, they are considered ground truth. True ground truth in machine learning is objective data. For example, suppose we are testing a stereo vision system to see how well it can estimate 3D positions. A calibrated laser rangefinder may provide accurate distances as ground truth. == Remote sensing == In remote sensing, "ground truth" refers to information collected at the imaged location. Ground truth allows image data to be related to real features and materials on the ground. The collection of ground truth data enables calibration of remote-sensing data, and aids in the interpretation and analysis of what is being sensed. Examples include cartography, meteorology, analysis of aerial photographs, satellite imagery and other techniques in which data are gathered at a distance. More specifically, ground truth may refer to a process in which "pixels" on a satellite image are compared to what is imaged (at the time of capture) in order to verify the contents of the "pixels" in the image (noting that the concept of "pixel" is imaging-system-dependent). In the case of a classified image, supervised classification can help to determine the accuracy of the classification by the remote sensing system which can minimize error in the classification. Ground truth is usually done on site, correlating what is known with surface observations and measurements of various properties of the features of the ground resolution cells under study in the remotely sensed digital image. The process also involves taking geographic coordinates of the ground resolution cell with GPS technology and comparing those with the coordinates of the "pixel" being studied provided by the remote sensing software to understand and analyze the location errors and how it may affect a particular study. Ground truth is important in the initial supervised classification of an image. When the identity and location of land cover types are known through a combination of field work, maps, and personal experience these areas are known as training sites. The spectral characteristics of these areas are used to train the remote sensing software using decision rules for classifying the rest of the image. These decision rules such as Maximum Likelihood Classification, Parallelopiped Classification, and Minimum Distance Classification offer different techniques to classify an image. Additional ground truth sites allow the remote sensor to establish an error matrix that validates the accuracy of the classification method used. Different classification methods may have different percentages of error for a given classification project. It is important that the remote sensor chooses a classification method that works best with the number of classifications used while providing the least amount of error. Ground truth also helps with atmospheric correction. Since images from satellites have to pass through the atmosphere, they can get distorted because of absorption in the atmosphere. So ground truth can help fully identify objects in satellite photos. === Errors of commission === An example of an error of commission is when a pixel reports the presence of a feature (such a tree) that, in reality, is absent (no tree is actually present). Ground truthing ensures that the error matrices have a higher accuracy percentage than would be the case if no pixels were ground-truthed. This value is the complement of the user's accuracy, i.e. Commission Error = 1 - user's accuracy. === Errors of omission === An example of an error of omission is when pixels of a certain type, for example, maple trees, are not classified as maple trees. The process of ground-truthing helps to ensure that the pixel is classified correctly and the error matrices are more accurate. This value is the complement of the producer's accuracy, i.e. Omission Error = 1 - producer's accuracy == Geographical information systems == In GIS the spatial data is modeled as field (like in remote sensing raster images) or as object (like in vectorial map representation). They are modeled from the real world (also named geographical reality), typically by a cartographic process (illustrated). Geographic information systems such as GIS, GPS, and GNSS, have become so widespread that the term "ground truth" has taken on special meaning in that context. If the location coordinates returned by a location method such as GPS are an estimate of a location, then the "ground truth" is the actual location on Earth. A smart phone might return a set of estimated location coordinates such as 43.87870, −103.45901. The ground truth being estimated by those coordinates is the tip of George Washington's nose on Mount Rushmore. The accuracy of the estimate is the maximum distance between the location coordinates and the ground truth. We could say in this case that the estimate accuracy is 10 meters, meaning that the point on Earth represented by the location coordinates is thought to be within 10 meters of George's nose—the ground truth. In slang, the coordinates indicate where we think George Washington's nose is located, and the ground truth is where it really is. In practice a smart phone or hand-held GPS unit is routinely able to estimate the ground truth within 6–10 meters. Specialized instruments can reduce GPS measurement error to under a centimeter. == Military usage == US military slang uses "ground truth" to refer to the facts comprising a tactical situation—as opposed to intelligence reports, mission plans, and other descriptions reflecting the conative or policy-based projections of the industrial·military complex. The term appears in the title of the Iraq War documentary film The Ground Truth (2006), and also in military publications, for example Stars and Stripes saying: "Stripes decided to figure out what the ground truth was in Iraq."

Softmax function

The softmax function, also known as softargmax or normalized exponential function, converts a tuple of K real numbers into a probability distribution over K possible outcomes. It is a generalization of the logistic function to multiple dimensions, and is used in multinomial logistic regression. The softmax function is often used as the last activation function of a neural network to normalize the output of a network to a probability distribution over predicted output classes. == Definition == The softmax function takes as input a tuple z of K real numbers, and normalizes it into a probability distribution consisting of K probabilities proportional to the exponentials of the input numbers. That is, prior to applying softmax, some tuple components could be negative, or greater than one; and might not sum to 1; but after applying softmax, each component will be in the interval ( 0 , 1 ) {\displaystyle (0,1)} , and the components will add up to 1, so that they can be interpreted as probabilities. Furthermore, the larger input components will correspond to larger probabilities. Formally, the standard (unit) softmax function σ : R K → ( 0 , 1 ) K {\displaystyle \sigma :\mathbb {R} ^{K}\to (0,1)^{K}} , where ⁠ K > 1 {\displaystyle K>1} ⁠, takes a tuple z = ( z 1 , … , z K ) ∈ R K {\displaystyle \mathbf {z} =(z_{1},\dotsc ,z_{K})\in \mathbb {R} ^{K}} and computes each component of vector σ ( z ) ∈ ( 0 , 1 ) K {\displaystyle \sigma (\mathbf {z} )\in (0,1)^{K}} with σ ( z ) i = e z i ∑ j = 1 K e z j . {\displaystyle \sigma (\mathbf {z} )_{i}={\frac {e^{z_{i}}}{\sum _{j=1}^{K}e^{z_{j}}}}\,.} In words, the softmax applies the standard exponential function to each element z i {\displaystyle z_{i}} of the input tuple z {\displaystyle \mathbf {z} } (consisting of K {\displaystyle K} real numbers), and normalizes these values by dividing by the sum of all these exponentials. The normalization ensures that the sum of the components of the output vector σ ( z ) {\displaystyle \sigma (\mathbf {z} )} is 1. The term "softmax" derives from the amplifying effects of the exponential on any maxima in the input tuple. For example, the standard softmax of ( 1 , 2 , 8 ) {\displaystyle (1,2,8)} is approximately ( 0.001 , 0.002 , 0.997 ) {\displaystyle (0.001,0.002,0.997)} , which amounts to assigning almost all of the total unit weight in the result to the position of the tuple's maximal element (of 8). In general, instead of e a different base b > 0 can be used. As above, if b > 1 then larger input components will result in larger output probabilities, and increasing the value of b will create probability distributions that are more concentrated around the positions of the largest input values. Conversely, if 0 < b < 1 then smaller input components will result in larger output probabilities, and decreasing the value of b will create probability distributions that are more concentrated around the positions of the smallest input values. Writing b = e β {\displaystyle b=e^{\beta }} or b = e − β {\displaystyle b=e^{-\beta }} (for real β) yields the expressions: σ ( z ) i = e β z i ∑ j = 1 K e β z j or σ ( z ) i = e − β z i ∑ j = 1 K e − β z j for i = 1 , … , K . {\displaystyle \sigma (\mathbf {z} )_{i}={\frac {e^{\beta z_{i}}}{\sum _{j=1}^{K}e^{\beta z_{j}}}}{\text{ or }}\sigma (\mathbf {z} )_{i}={\frac {e^{-\beta z_{i}}}{\sum _{j=1}^{K}e^{-\beta z_{j}}}}{\text{ for }}i=1,\dotsc ,K.} A value proportional to the reciprocal of β is sometimes referred to as the temperature: β = 1 / k T {\textstyle \beta =1/kT} , where k is typically 1 or the Boltzmann constant and T is the temperature. A higher temperature results in a more uniform output distribution (i.e. with higher entropy; it is "more random"), while a lower temperature results in a sharper output distribution, with one value dominating. In some fields, the base is fixed, corresponding to a fixed scale, while in others the parameter β (or T) is varied. The softmax function is a multiple-variable generalization of the logistic function. == Interpretations == === Smooth arg max === The Softmax function is a smooth approximation to the arg max function: the function whose value is the index of a tuple's largest element. The name "softmax" may be misleading. Softmax is not a smooth maximum (that is, a smooth approximation to the maximum function). The term "softmax" is also used for the closely related LogSumExp function, which is a smooth maximum. For this reason, some prefer the more accurate term "softargmax", though the term "softmax" is conventional in machine learning. This section uses the term "softargmax" for clarity. Formally, instead of considering the arg max as a function with categorical output 1 , … , n {\displaystyle 1,\dots ,n} (corresponding to the index), consider the arg max function with one-hot representation of the output (assuming there is a unique maximum arg): a r g m a x ⁡ ( z 1 , … , z n ) = ( y 1 , … , y n ) = ( 0 , … , 0 , 1 , 0 , … , 0 ) , {\displaystyle \operatorname {arg\,max} (z_{1},\,\dots ,\,z_{n})=(y_{1},\,\dots ,\,y_{n})=(0,\,\dots ,\,0,\,1,\,0,\,\dots ,\,0),} where the output coordinate y i = 1 {\displaystyle y_{i}=1} if and only if i {\displaystyle i} is the arg max of ( z 1 , … , z n ) {\displaystyle (z_{1},\dots ,z_{n})} , meaning z i {\displaystyle z_{i}} is the unique maximum value of ( z 1 , … , z n ) {\displaystyle (z_{1},\,\dots ,\,z_{n})} . For example, in this encoding a r g m a x ⁡ ( 1 , 5 , 10 ) = ( 0 , 0 , 1 ) , {\displaystyle \operatorname {arg\,max} (1,5,10)=(0,0,1),} since the third argument is the maximum. This can be generalized to multiple arg max values (multiple equal z i {\displaystyle z_{i}} being the maximum) by dividing the 1 between all max args; formally 1/k where k is the number of arguments assuming the maximum. For example, a r g m a x ⁡ ( 1 , 5 , 5 ) = ( 0 , 1 / 2 , 1 / 2 ) , {\displaystyle \operatorname {arg\,max} (1,\,5,\,5)=(0,\,1/2,\,1/2),} since the second and third argument are both the maximum. In case all arguments are equal, this is simply a r g m a x ⁡ ( z , … , z ) = ( 1 / n , … , 1 / n ) . {\displaystyle \operatorname {arg\,max} (z,\dots ,z)=(1/n,\dots ,1/n).} Points z with multiple arg max values are singular points (or singularities, and form the singular set) – these are the points where arg max is discontinuous (with a jump discontinuity) – while points with a single arg max are known as non-singular or regular points. With the last expression given in the introduction, softargmax is now a smooth approximation of arg max: as ⁠ β → ∞ {\displaystyle \beta \to \infty } ⁠, softargmax converges to arg max. There are various notions of convergence of a function; softargmax converges to arg max pointwise, meaning for each fixed input z as ⁠ β → ∞ {\displaystyle \beta \to \infty } ⁠, σ β ( z ) → a r g m a x ⁡ ( z ) . {\displaystyle \sigma _{\beta }(\mathbf {z} )\to \operatorname {arg\,max} (\mathbf {z} ).} However, softargmax does not converge uniformly to arg max, meaning intuitively that different points converge at different rates, and may converge arbitrarily slowly. In fact, softargmax is continuous, but arg max is not continuous at the singular set where two coordinates are equal, while the uniform limit of continuous functions is continuous. The reason it fails to converge uniformly is that for inputs where two coordinates are almost equal (and one is the maximum), the arg max is the index of one or the other, so a small change in input yields a large change in output. For example, σ β ( 1 , 1.0001 ) → ( 0 , 1 ) , {\displaystyle \sigma _{\beta }(1,\,1.0001)\to (0,1),} but σ β ( 1 , 0.9999 ) → ( 1 , 0 ) , {\displaystyle \sigma _{\beta }(1,\,0.9999)\to (1,\,0),} and σ β ( 1 , 1 ) = 1 / 2 {\displaystyle \sigma _{\beta }(1,\,1)=1/2} for all inputs: the closer the points are to the singular set ( x , x ) {\displaystyle (x,x)} , the slower they converge. However, softargmax does converge compactly on the non-singular set. Conversely, as ⁠ β → − ∞ {\displaystyle \beta \to -\infty } ⁠, softargmax converges to arg min in the same way, where here the singular set is points with two arg min values. In the language of tropical analysis, the softmax is a deformation or "quantization" of arg max and arg min, corresponding to using the log semiring instead of the max-plus semiring (respectively min-plus semiring), and recovering the arg max or arg min by taking the limit is called "tropicalization" or "dequantization". It is also the case that, for any fixed β, if one input ⁠ z i {\displaystyle z_{i}} ⁠ is much larger than the others relative to the temperature, T = 1 / β {\displaystyle T=1/\beta } , the output is approximately the arg max. For example, a difference of 10 is large relative to a temperature of 1: σ ( 0 , 10 ) := σ 1 ( 0 , 10 ) = ( 1 / ( 1 + e 10 ) , e 10 / ( 1 + e 10 ) ) ≈ ( 0.00005 , 0.99995 ) {\displaystyle \sigma (0,\,10):=\sigma _{1}(0,\,10)=\left(1/\left(1+e^{10}\right),\,e^{10}/\left(1+e^{10}\right)\right)\approx (0.00005

Oja's rule

Oja's learning rule, or simply Oja's rule, named after Finnish computer scientist Erkki Oja (Finnish pronunciation: [ˈojɑ], AW-yuh), is a model of how neurons in the brain or in artificial neural networks change connection strength, or learn, over time. It is a modification of the standard Hebb's Rule that, through multiplicative normalization, solves all stability problems and generates an algorithm for principal components analysis. This is a computational form of an effect which is believed to happen in biological neurons. == Theory == Oja's rule requires a number of simplifications to derive, but in its final form it is demonstrably stable, unlike Hebb's rule. It is a single-neuron special case of the Generalized Hebbian Algorithm. However, Oja's rule can also be generalized in other ways to varying degrees of stability and success. === Formula === Consider a simplified model of a neuron y {\displaystyle y} that returns a linear combination of its inputs x using presynaptic weights w: y ( x ) = ∑ j = 1 m x j w j {\displaystyle \,y(\mathbf {x} )~=~\sum _{j=1}^{m}x_{j}w_{j}} Oja's rule defines the change in presynaptic weights w given the output response y {\displaystyle y} of a neuron to its inputs x to be Δ w = w n + 1 − w n = η y n ( x n − y n w n ) , {\displaystyle \,\Delta \mathbf {w} ~=~\mathbf {w} _{n+1}-\mathbf {w} _{n}~=~\eta \,y_{n}(\mathbf {x} _{n}-y_{n}\mathbf {w} _{n}),} where η is the learning rate which can also change with time. Note that the bold symbols are vectors and n defines a discrete time iteration. The rule can also be made for continuous iterations as d w d t = η y ( t ) ( x ( t ) − y ( t ) w ( t ) ) . {\displaystyle \,{\frac {d\mathbf {w} }{dt}}~=~\eta \,y(t)(\mathbf {x} (t)-y(t)\mathbf {w} (t)).} === Derivation === The simplest learning rule known is Hebb's rule, which states in conceptual terms that neurons that fire together, wire together. In component form as a difference equation, it is written Δ w = η y ( x n ) x n {\displaystyle \,\Delta \mathbf {w} ~=~\eta \,y(\mathbf {x} _{n})\mathbf {x} _{n}} , or in scalar form with implicit n-dependence, w i ( n + 1 ) = w i ( n ) + η y ( x ) x i {\displaystyle \,w_{i}(n+1)~=~w_{i}(n)+\eta \,y(\mathbf {x} )x_{i}} , where y(xn) is again the output, this time explicitly dependent on its input vector x. Hebb's rule has synaptic weights approaching infinity with a positive learning rate. We can stop this by normalizing the weights so that each weight's magnitude is restricted between 0, corresponding to no weight, and 1, corresponding to being the only input neuron with any weight. We do this by normalizing the weight vector to be of length one: w i ( n + 1 ) = w i ( n ) + η y ( x ) x i ( ∑ j = 1 m [ w j ( n ) + η y ( x ) x j ] p ) 1 / p {\displaystyle \,w_{i}(n+1)~=~{\frac {w_{i}(n)+\eta \,y(\mathbf {x} )x_{i}}{\left(\sum _{j=1}^{m}[w_{j}(n)+\eta \,y(\mathbf {x} )x_{j}]^{p}\right)^{1/p}}}} . Note that in Oja's original paper, p=2, corresponding to quadrature (root sum of squares), which is the familiar Cartesian normalization rule. However, any type of normalization, even linear, will give the same result without loss of generality. For a small learning rate | η | ≪ 1 {\displaystyle |\eta |\ll 1} the equation can be expanded as a Power series in η {\displaystyle \eta } . w i ( n + 1 ) = w i ( n ) ( ∑ j w j p ( n ) ) 1 / p + η ( y x i ( ∑ j w j p ( n ) ) 1 / p − w i ( n ) ∑ j y x j w j p − 1 ( n ) ( ∑ j w j p ( n ) ) ( 1 + 1 / p ) ) + O ( η 2 ) {\displaystyle \,w_{i}(n+1)~=~{\frac {w_{i}(n)}{\left(\sum _{j}w_{j}^{p}(n)\right)^{1/p}}}~+~\eta \left({\frac {yx_{i}}{\left(\sum _{j}w_{j}^{p}(n)\right)^{1/p}}}-{\frac {w_{i}(n)\sum _{j}yx_{j}w_{j}^{p-1}(n)}{\left(\sum _{j}w_{j}^{p}(n)\right)^{(1+1/p)}}}\right)~+~O(\eta ^{2})} . For small η, our higher-order terms O(η2) go to zero. We again make the specification of a linear neuron, that is, the output of the neuron is equal to the sum of the product of each input and its synaptic weight to the power of p-1, which in the case of p=2 is synaptic weight itself, or y ( x ) = ∑ j = 1 m x j w j p − 1 {\displaystyle \,y(\mathbf {x} )~=~\sum _{j=1}^{m}x_{j}w_{j}^{p-1}} . We also specify that our weights normalize to 1, which will be a necessary condition for stability, so | w | = ( ∑ j = 1 m w j p ) 1 / p = 1 {\displaystyle \,|\mathbf {w} |~=~\left(\sum _{j=1}^{m}w_{j}^{p}\right)^{1/p}~=~1} , which, when substituted into our expansion, gives Oja's rule, or w i ( n + 1 ) = w i ( n ) + η y ( x i − w i ( n ) y ) {\displaystyle \,w_{i}(n+1)~=~w_{i}(n)+\eta \,y(x_{i}-w_{i}(n)y)} . === Stability and PCA === In analyzing the convergence of a single neuron evolving by Oja's rule, one extracts the first principal component, or feature, of a data set. Furthermore, with extensions using the Generalized Hebbian Algorithm, one can create a multi-Oja neural network that can extract as many features as desired, allowing for principal components analysis. A principal component aj is extracted from a dataset x through some associated vector qj, or aj = qj⋅x, and we can restore our original dataset by taking x = ∑ j a j q j {\displaystyle \mathbf {x} ~=~\sum _{j}a_{j}\mathbf {q} _{j}} . In the case of a single neuron trained by Oja's rule, we find the weight vector converges to q1, or the first principal component, as time or number of iterations approaches infinity. We can also define, given a set of input vectors Xi, that its correlation matrix Rij = XiXj has an associated eigenvector given by qj with eigenvalue λj. The variance of outputs of our Oja neuron σ2(n) = ⟨y2(n)⟩ then converges with time iterations to the principal eigenvalue, or lim n → ∞ σ 2 ( n ) = λ 1 {\displaystyle \lim _{n\rightarrow \infty }\sigma ^{2}(n)~=~\lambda _{1}} . These results are derived using Lyapunov function analysis, and they show that Oja's neuron necessarily converges on strictly the first principal component if certain conditions are met in our original learning rule. Most importantly, our learning rate η is allowed to vary with time, but only such that its sum is divergent but its power sum is convergent, that is ∑ n = 1 ∞ η ( n ) = ∞ , ∑ n = 1 ∞ η ( n ) p < ∞ , p > 1 {\displaystyle \sum _{n=1}^{\infty }\eta (n)=\infty ,~~~\sum _{n=1}^{\infty }\eta (n)^{p}<\infty ,~~~p>1} . Our output activation function y(x(n)) is also allowed to be nonlinear and nonstatic, but it must be continuously differentiable in both x and w and have derivatives bounded in time. == Applications == Oja's rule was originally described in Oja's 1982 paper, but the principle of self-organization to which it is applied is first attributed to Alan Turing in 1952. PCA has also had a long history of use before Oja's rule formalized its use in network computation in 1989. The model can thus be applied to any problem of self-organizing mapping, in particular those in which feature extraction is of primary interest. Therefore, Oja's rule has an important place in image and speech processing. It is also useful as it expands easily to higher dimensions of processing, thus being able to integrate multiple outputs quickly. A canonical example is its use in binocular vision. === Biology and Oja's subspace rule === There is clear evidence for both long-term potentiation and long-term depression in biological neural networks, along with a normalization effect in both input weights and neuron outputs. However, while there is no direct experimental evidence yet of Oja's rule active in a biological neural network, a biophysical derivation of a generalization of the rule is possible. Such a derivation requires retrograde signalling from the postsynaptic neuron, which is biologically plausible (see neural backpropagation), and takes the form of Δ w i j ∝ ⟨ x i y j ⟩ − ϵ ⟨ ( c p r e ∗ ∑ k w i k y k ) ⋅ ( c p o s t ∗ y j ) ⟩ , {\displaystyle \Delta w_{ij}~\propto ~\langle x_{i}y_{j}\rangle -\epsilon \left\langle \left(c_{\mathrm {pre} }\sum _{k}w_{ik}y_{k}\right)\cdot \left(c_{\mathrm {post} }y_{j}\right)\right\rangle ,} where as before wij is the synaptic weight between the ith input and jth output neurons, x is the input, y is the postsynaptic output, and we define ε to be a constant analogous the learning rate, and cpre and cpost are presynaptic and postsynaptic functions that model the weakening of signals over time. Note that the angle brackets denote the average and the ∗ operator is a convolution. By taking the pre- and post-synaptic functions into frequency space and combining integration terms with the convolution, we find that this gives an arbitrary-dimensional generalization of Oja's rule known as Oja's Subspace, namely Δ w = C x ⋅ w − w ⋅ C y . {\displaystyle \Delta w~=~Cx\cdot w-w\cdot Cy.}

Blackmagic Design

Blackmagic Design Pty Ltd is an Australian company that develops digital cinema technology and manufactures professional video production hardware and software. Headquartered in South Melbourne, it is known for producing high-end digital movie cameras and a range of broadcast and post-production equipment. The company also develops software applications, including the DaVinci Resolve application for non-linear video editing, color correction, color grading, visual effects, and audio post-production. == History == Blackmagic Design Pty Ltd was founded on 7 September 2001 by Grant Petty. Its first product, DeckLink, introduced in 2002, was a video capture card for macOS that supported uncompressed 10-bit video, marking a shift toward professional-grade yet affordable video workflows. Subsequent versions—including the DeckLink 2, Pro SDI, HD Plus, and Multibridge—added capabilities such as color correction, Windows support, and compatibility with major editing software like Adobe Premiere Pro, to broaden the product's appeal. At the 2012 NAB Show, Blackmagic announced its first Cinema Camera, a digital movie camera. Blackmagic made several acquisitions over the next decade. In 2009, it acquired da Vinci Systems, known for its color-grading tools. In 2010, it acquired Echolab's ATEM switcher line, in 2014, it added eyeon Software (developer of the Blackmagic Fusion compositing software) and London's Cintel (film scanning and restoration), and in 2016, it acquired Fairlight, an audio technology company known for its CMI synthesizers as well as mixing consoles. == Products == List of all products developed by the company. Editing, Color Correction and Audio Post Production DaVinci Resolve (free version) and DaVinci Resolve Studio (paid version), computer software for non-linear video editing, color correction, color grading, visual effects, and audio post-production. Audio/Video Controller Consoles: Editor Keyboard, Speed Editor, DaVinci Resolve Replay Editor, Micro Panel, Mini Panel, DaVinci Resolve Micro Color Panel, Advanced Panel, Fairlight Console Channel Fader, Fairlight Console Channel Control, Fairlight Console LCD Monitor, Fairlight Console Audio Editor, Fairlight Desktop Audio Editor, Fairlight Desktop Console, Fairlight Audio Interface Cintel Film Scanner (Generations 1-3) Live Production Home Streaming: ATEM Mini, ATEM Mini Pro/ISO, ATEM Mini Extreme, ATEM Mini Extreme ISO (The ATEM Mini series has both HDMI and SDI variants) Production Switchers: ATEM 1,2 & 4 M/E Constellation HD, ATEM 1,2 & 4 M/E Constellation 4K, ATEM Constellation 8K, ATEM 1,2 & 4 M/E Production Studio 4K, ATEM Television Studio HD8 & HD8 ISO Switcher & Camera Controllers: ATEM Camera Control Panel, ATEM 1 M/E Advanced Panel, ATEM 2 M/E Advanced Panel, ATEM 4 M/E Advanced Panel Chroma Keyers: Ultimatte 12 HD Mini, Ultimatte 12 HD, Ultimatte 12 4K, Ultimatte 12 8K Recording and Storage: HyperDeck Studio HD Mini, HyperDeck Studio HD Plus, HyperDeck Studio HD Plus, HyperDeck Studio 4K Pro, HyperDeck Extreme 8K HDR, HyperDeck Extreme 4K HDR, HyperDeck Extreme Control, HyperDeck Shuttle HD, Duplicator 4K, MultiDock 10G, Video Assist 7" 12G HDR, Video Assist 5" 12G HDR Capture and Playback UltraStudio: 3G, HD Mini, 4K Mini, 4K Extreme 3 DeckLink (PCIe cards): Mini Recorder, Mini Monitor, Mini Monitor 4K, Mini Recorder 4K, Duo 2 Mini, Duo 2, Quad 2, SDI 4K, Studio 4K, 4K Extreme 12G, 8K Pro, Quad HDMI Recorder Network Storage Cloud Store Cloud Pod Broadcast Converters Micro Converter: BiDirectional SDI/HDMI 3G wPSU, HDMI to SDI 3G wPSU, SDI to HDMI 3G wPSU, BiDirectional SDI/HDMI 3G, HDMI to SDI 3G, SDI to HDMI 3G Mini Converters: Audio to SDI, Optical Fiber 12G, SDI Multiplex 4K, Quad SDI to HDMI 4K, SDI Distribution 4K, SDI to Analog 4K, Audio to SDI 4K, SDI to Audio 4K, HDMI to SDI 6G, SDI to HDMI 6G Teranex Mini: SDI Distribution 12G, SDI to HDMI 12G, Audio to SDI 12G, SDI to Analog 12G, SDI to HDMI 8K HDR, SDI to DisplayPort 8K HDR 2110 IP Converters Routing and Distribution Videohub

Universal approximation theorem

In the field of machine learning, the universal approximation theorems (UATs) state that neural networks with a certain structure can, in principle, approximate any continuous function to any desired degree of accuracy. These theorems provide a mathematical justification for using neural networks, assuring researchers that a sufficiently large or deep network can model the complex, non-linear relationships often found in real-world data. The best-known version of the theorem applies to feedforward networks with a single hidden layer. It states that if the layer's activation function is non-polynomial (which is true for common choices like the sigmoid function or ReLU), then the network can act as a "universal approximator." Universality is achieved by increasing the number of neurons in the hidden layer, making the network "wider." Other versions of the theorem show that universality can also be achieved by keeping the network's width fixed but increasing its number of layers, making it "deeper." These are existence theorems. They guarantee that a network with the right structure exists, but they do not provide a method for finding the network's parameters (training it), nor do they specify exactly how large the network must be for a given function. Finding a suitable network remains a practical challenge that is typically addressed with optimization algorithms like backpropagation. == Setup == Artificial neural networks are combinations of multiple simple mathematical functions that implement more complicated functions from (typically) real-valued vectors to real-valued vectors. The spaces of multivariate functions that can be implemented by a network are determined by the structure of the network, the set of simple functions, and its multiplicative parameters. A great deal of theoretical work has gone into characterizing these function spaces. Most universal approximation theorems are in one of two classes. The first quantifies the approximation capabilities of neural networks with an arbitrary number of artificial neurons ("arbitrary width" case) and the second focuses on the case with an arbitrary number of hidden layers, each containing a limited number of artificial neurons ("arbitrary depth" case). In addition to these two classes, there are also universal approximation theorems for neural networks with bounded number of hidden layers and a limited number of neurons in each layer ("bounded depth and bounded width" case). == History == === Arbitrary width === The first results concerned the arbitrary width case. Ken-ichi Funahashi (May 1989) showed that Rumelhart–Hinton–Williams type backpropagation networks possess universal approximation capability with a class of sigmoidal activation functions, extending the result to multi-output mappings as well. Kurt Hornik, Maxwell Stinchcombe, and Halbert White (July 1989) showed that multilayer feed-forward networks with as few as one hidden layer are universal approximators, provided that the activation function satisfies certain conditions. George Cybenko (December 1989) independently established a related result for sigmoid activation functions using functional-analytic methods. Hornik also showed in 1991 that it is not the specific choice of the activation function but rather the multilayer feed-forward architecture itself that gives neural networks the potential of being universal approximators. Moshe Leshno et al in 1993 and later Allan Pinkus in 1999 showed that the universal approximation property is equivalent to having a nonpolynomial activation function. === Arbitrary depth === The arbitrary depth case was also studied by a number of authors such as Gustaf Gripenberg in 2003, Dmitry Yarotsky, Zhou Lu et al in 2017, Boris Hanin and Mark Sellke in 2018 who focused on neural networks with ReLU activation function. In 2020, Patrick Kidger and Terry Lyons extended those results to neural networks with general activation functions such, e.g. tanh or GeLU. One special case of arbitrary depth is that each composition component comes from a finite set of mappings. In 2024, Cai constructed a finite set of mappings, named a vocabulary, such that any continuous function can be approximated by compositing a sequence from the vocabulary. This is similar to the concept of compositionality in linguistics, which is the idea that a finite vocabulary of basic elements can be combined via grammar to express an infinite range of meanings. === Bounded depth and bounded width === The bounded depth and bounded width case was first studied by Maiorov and Pinkus in 1999. They showed that there exists an analytic sigmoidal activation function such that two hidden layer neural networks with bounded number of units in hidden layers are universal approximators. In 2018, Guliyev and Ismailov constructed a smooth sigmoidal activation function providing universal approximation property for two hidden layer feedforward neural networks with fewer units in hidden layers. In 2018, they also constructed single hidden layer networks with bounded width that are still universal approximators for univariate functions. However, this does not apply for multivariable functions. In 2022, Shen et al. obtained precise quantitative information on the depth and width required to approximate a target function by deep and wide ReLU neural networks. === Quantitative bounds === The question of minimal possible width for universality was first studied in 2021, Park et al obtained the minimum width required for the universal approximation of Lp functions using feed-forward neural networks with ReLU as activation functions. Similar results that can be directly applied to residual neural networks were also obtained in the same year by Paulo Tabuada and Bahman Gharesifard using control-theoretic arguments. In 2023, Cai obtained the optimal minimum width bound for the universal approximation. For the arbitrary depth case, Leonie Papon and Anastasis Kratsios derived explicit depth estimates depending on the regularity of the target function and of the activation function. === Kolmogorov network === The Kolmogorov–Arnold representation theorem is similar in spirit. Indeed, certain neural network families can directly apply the Kolmogorov–Arnold theorem to yield a universal approximation theorem. Robert Hecht-Nielsen showed that a three-layer neural network can approximate any continuous multivariate function. This was extended to the discontinuous case by Vugar Ismailov. In 2024, Ziming Liu and co-authors showed a practical application. === Reservoir computing and quantum reservoir computing === In reservoir computing a sparse recurrent neural network with fixed weights equipped of fading memory and echo state property is followed by a trainable output layer. Its universality has been demonstrated separately for what concerns networks of rate neurons and spiking neurons, respectively. In 2024, the framework has been generalized and extended to quantum reservoirs where the reservoir is based on qubits defined over Hilbert spaces. === Variants === Variants include discontinuous activation functions, noncompact domains, certifiable networks, random neural networks, and alternative network architectures and topologies. The universal approximation property of width-bounded networks has been studied as a dual of classical universal approximation results on depth-bounded networks. For input dimension d x {\displaystyle d_{x}} and output dimension d y {\displaystyle d_{y}} the minimum width required for the universal approximation of the Lp functions is exactly m a x { d x + 1 , d y } {\displaystyle max\{d_{x}+1,d_{y}\}} (for a ReLU network). More generally this also holds if both ReLU and a threshold activation function are used. Universal function approximation on graphs (or rather on graph isomorphism classes) by popular graph convolutional neural networks (GCNs or GNNs) can be made as discriminative as the Weisfeiler–Leman graph isomorphism test. In 2020, a universal approximation theorem result was established by Brüel-Gabrielsson, showing that graph representation with certain injective properties is sufficient for universal function approximation on bounded graphs and restricted universal function approximation on unbounded graphs, with an accompanying O ( | V | ⋅ | E | ) {\displaystyle {\mathcal {O}}(\left|V\right|\cdot \left|E\right|)} -runtime method that performed at state of the art on a collection of benchmarks (where V {\displaystyle V} and E {\displaystyle E} are the sets of nodes and edges of the graph respectively). There are also a variety of results between non-Euclidean spaces and other commonly used architectures and, more generally, algorithmically generated sets of functions, such as the convolutional neural network (CNN) architecture, radial basis functions, or neural networks with specific properties. == Arbitrary-width case == A universal approximation theorem formally states that a family of neural network funct