Scanimate is an analog computer animation (video synthesizer) system created by Lee Harrison III of Denver, Colorado. Harrison had developed its predecessor, ANIMAC, which generated used a motion capture system, based on a body suit with potentiometers. In contrast, Scanimate included TV technology. Scanimate's successor was called Caesar, and used a digital computer to control the analog system. The eight Scanimate systems were used to produce much of the video-based animation seen on television between most of the 1970s and early 1980s in commercials, promotions, and show openings. One of the major advantages the Scanimate system had over film-based animation and computer animation was the ability to create animations in real time. The speed with which animation could be produced on the system because of this, as well as its range of possible effects, helped it to supersede film-based animation techniques for television graphics. By the mid-1980s, it was superseded by digital computer animation, which produced sharper images and more sophisticated 3D imagery. Animations created on Scanimate and similar analog computer animation systems have a number of characteristic features that distinguish them from film-based animation: the motion is extremely fluid, using all 60 fields per second (in NTSC format video) or 50 fields (in PAL format video) rather than the 24 frames per second that film uses; the colors are much brighter and more saturated; and the images have a very "electronic" look that results from the direct manipulation of video signals through which the Scanimate produces the images. == How it works == A special high-resolution (around 945 lines) monochrome camera records high-contrast artwork. The image is then displayed on a high-resolution screen. Unlike a normal monitor, its deflection signals are passed through a special analog computer that enables the operator to bend the image in a variety of ways. The image is then shot from the screen by either a film camera or a video camera. In the case of a video camera, this signal is then fed into a colorizer, a device that takes certain shades of grey and turns it into color as well as transparency. The idea behind this is that the output of the Scanimate itself is always monochrome. Another advantage of the colorizer is that it gives the operator the ability to continuously add layers of graphics. This makes possible the creation of very complex graphics. This is done by using two video recorders. The background is played by one recorder and then recorded by another one. This process is repeated for every layer. This requires very high-quality video recorders (such as both the Ampex VR-2000 or IVC's IVC-9000 of Scanimate's era, the IVC-9000 being used quite frequently for Scanimate composition due to its very high generational quality between re-recordings). == Current usage == Two of the Scanimates are still in use at ZFx studios in Asheville, North Carolina. The original "Black Swan" R&D machine has been updated with more modern power supplies and can produce material in standard or 1080P high definition video. The "white Pearl" machine is the last one produced and is being kept in its original configuration for historical purposes by David Sieg at ZFx inc. The machines are installed in a working production environment with Grass Valley switchers, Kaleidoscope digital video effects systems, and Accom digital disk recorders for layering. == Use in television, music and films == === Music videos === Let's Groove by Earth, Wind & Fire Get Down on It by Kool & the Gang Blame It on the Boogie by The Jacksons Knock on Wood by Amii Stewart Popcorn Love by New Edition === TV programs/movies === === TV channels/home video/TV productions ===
Uncertain data
In computer science, uncertain data is data that contains noise that makes it deviate from the correct, intended or original values. In the age of big data, uncertainty or data veracity is one of the defining characteristics of data. Data is constantly growing in volume, variety, velocity and uncertainty (1/veracity). Uncertain data is found in abundance today on the web, in sensor networks, within enterprises both in their structured and unstructured sources. For example, there may be uncertainty regarding the address of a customer in an enterprise dataset, or the temperature readings captured by a sensor due to aging of the sensor. In 2012 IBM called out managing uncertain data at scale in its global technology outlook report that presents a comprehensive analysis looking three to ten years into the future seeking to identify significant, disruptive technologies that will change the world. In order to make confident business decisions based on real-world data, analyses must necessarily account for many different kinds of uncertainty present in very large amounts of data. Analyses based on uncertain data will have an effect on the quality of subsequent decisions, so the degree and types of inaccuracies in this uncertain data cannot be ignored. Uncertain data is found in the area of sensor networks; text where noisy text is found in abundance on social media, web and within enterprises where the structured and unstructured data may be old, outdated, or plain incorrect; in modeling where the mathematical model may only be an approximation of the actual process. When representing such data in a database, an appropriate uncertain database model needs to be selected. == Example data model for uncertain data == One way to represent uncertain data is through probability distributions. Let us take the example of a relational database. There are three main ways to do represent uncertainty as probability distributions in such a database model. In attribute uncertainty, each uncertain attribute in a tuple is subject to its own independent probability distribution. For example, if readings are taken of temperature and wind speed, each would be described by its own probability distribution, as knowing the reading for one measurement would not provide any information about the other. In correlated uncertainty, multiple attributes may be described by a joint probability distribution. For example, if readings are taken of the position of an object, and the x- and y-coordinates stored, the probability of different values may depend on the distance from the recorded coordinates. As distance depends on both coordinates, it may be appropriate to use a joint distribution for these coordinates, as they are not independent. In tuple uncertainty, all the attributes of a tuple are subject to a joint probability distribution. This covers the case of correlated uncertainty, but also includes the case where there is a probability of a tuple not belonging in the relevant relation, which is indicated by all the probabilities not summing to one. For example, assume we have the following tuple from a probabilistic database: Then, the tuple has 10% chance of not existing in the database.
Are We Dating The Same Guy?
Are We Dating The Same Guy?, also abbreviated AWDTSG is a series of over 200 individual Facebook groups where women share dating profiles of men they matched with on dating networks to seek the opinion of other women who may have dated the same man in the past. The first group was created by Paola Sanchez and aimed at women living in the New York City environs. The groups have over 3.5 million members as of January 2024. The group's function is to post screenshots of a man's dating profile to that city's designated Facebook group, after which the poster asks "any tea?". Other users in the group will then share information about the man and share warnings. The groups are moderated by volunteers, and have been described as a feminist group. The groups have rules saying that personal information such as addresses must not be included in the Facebook posts. Users attempting to join the group are also examined to prevent fake profiles. The group is mainly for straight women. According to Vice, the men being posted about have no way to defend against accusations made about them, and on the other hand, posters cannot prove their stories unless backed up by others. Often times, members post pictures alongside personal information such as names, which may infringe on subjects' legal right to privacy. Lawyers have said these issues can lead to defamation lawsuits, and members can make false allegations and create fabricated stories. If members tell a man that he's been talked about on the group, the "snitch" will be banned and be "exposed to the whole group". == History == The first Are We Dating The Same Guy group was created by Paola Sanchez. The first group was created in March 2022 in New York City. A male counterpart, named "Are We Dating the Same Girl NYC" was created for New York, with mostly the same guidelines and rules to the original. When the original Are We Dating The Same Guy group found it, they denounced the new group. == Operations == Administrators are told not to respond to men asking to have posts about them removed, and to not remove said posts. The people being posted about have reported being questioned by their employers about things they have not done. Members of the groups sometimes criticise the physical appearance of the men being posted about. According to the Evening Standard, the groups "frequent[ly] mock" the appearance or dating profiles of the men who are posted about, despite being against the rules. For this behaviour, women are sometimes kicked out, or the group is disciplined en masse by admins. The groups have rules against hate towards men, but the rules can be difficult to enforce in large groups, with some having over 100,000 members. Some men have also been able to join the groups without being noticed. == Reception == In October 2023, Sera Bozza of Body+Soul wrote that consistently using Are We Dating The Same Guy can "affect your real-world view". She wrote that "A few stories of cheating may persuade you to believe that all men are unfaithful". Some lawyers and commentators have expressed concern that the groups fail to acknowledge the legal right to privacy and users can create false allegations and fabricated stories, and cyberbully men without them being able to defend themselves. This may lead to civil lawsuits against the author for defamation, harassment, and other related privacy torts. Netsafe, an online safety organisation in New Zealand, advises users of a similar group to familiarise themselves with the Harmful Digital Communications Act to ensure that posts do not lead to "harmful consequences". The Independent reported that men who have been posted on the dating groups have felt violated, and that even if reviewed positively by potentially thousands of strangers, the men being discussed about may have their reputation slightly decreased due to the association with being on the groups. The Independent also reported that some men believe that the groups are created to spread lies or mock them. Mashable reported that the growth of AWDTSG in recent years has led to the rise of a small industry of online reputation and content removal services, as increasing numbers of men seek assistance. A co-founder of Maximatic Media, one such agency offering these removal services, stated that many of the men contacting the firm do so in a state of panic after learning that allegations about them have circulated among tens of thousands of participants without their knowledge. Mashable similarly reported that the growing visibility of AWDTSG and similar platforms has contributed to what commentators describe as a "public trial" dynamic, where subjective accounts about dating behavior are interpreted as factual assessments and can influence a person's reputation among large audiences within their locale. The Oklahoman reported that anonymous, unverified claims in these groups have led some men to experience social and dating repercussions, although legal analysts argue that the benefits of community-based safety networks still outweigh these concerns in modern, app based dating environments. UTV/ITV News reportedly spoke to a man who was posted who alleged he attempted suicide, was clinically dead for three minutes, and spent three weeks in a psychiatric hospital as a result of the posts made about him. Many other men have talked about malicious false claims made about them. Self-described men’s rights activists have taken a dislike to these groups and have gotten multiple North American groups shut down by running campaigns, threatening lawsuits, and mass Facebook reporting. They also have Reddit communities dedicated to getting rid of such groups. Women who have posted in the groups have felt that they have put their safety at risk, with some having been confronted by the men they posted about. The group has been noted for exposing men who use dating apps while already in a relationship, misrepresent their ages, or repeatedly stand up the women they meet through apps, among other bad dating behaviors. For example, some members of the group had matched on a dating site with a man who had, several years prior, killed a stranger while having a mental break. After this information came to light, members of the group were warned. The group has also been noted to be complimentary of some men. == Lawsuits == In 2023, a 41-year-old man sued the administrators of the London group for $35,000 under defamation, alleging that the group "called names, accused of sending lewd photos and of being a bad parent". In January 2024 a man sued Meta, the owner of Facebook, along with Patreon, GoFundMe, and the AWDTSG website, as well as almost 30 group members due to alleged defamation, emotional distress, and invasion of privacy. Claiming that the groups violate anti-doxxing laws and do not fact check, seeking $75,000 in damages. He claims that the group shared fake images of him sending women texts containing harassment, his name and photo. His attorneys claim that if the images were real, they would fall under free speech in the First Amendment. By February, groups had raised $80,000. The Washington Post said that this case caused AWDTSG to "explode into public view". The case was dismissed in 2025 by the United States District Court for the Northern District of Illinois. On May 15, 2026, the United States Court of Appeals for the Seventh Circuit declined to renew the case in D'Ambrosio v. Meta Platforms Inc., et al. The plaintiff and his attorneys, Marc Trent and Aaron Walner of Trent Law Firm, were sanctioned "for frivolously appealing the dismissal of the claims," "misrepresentations of law," in connection with falsified citations included in the plaintiff's brief, and " disputing at oral argument without any evidentiary basis that [the plaintiff] client sent the text message she attributed to him." == By country == === Australia === In Australia, there are groups for multiple cities including Sydney, Melbourne, Adelaide, Perth, Brisbane and Rockhampton with many having several thousand members. The Sydney group has 30,000 members. In March 2023, the Adelaide version of the group, which had 7,000 members, was shut down. In 2024, groups titled "Sis, Are We Dating The Same Guy" stopped accepting new posts after an admin was sued for defamation and had to pay over AU$20,000 in legal fees. The case was settled out of court. The administrator announcing these closures cited a 2021 defamation High Court case involving detainee Dylan Voller, which led to the High Court saying that owners of Facebook groups can be held liable for defamatory comments, even if they did not know the comments had been made. === Canada === In 2023, a group was started for Ottawa. The founder previously was in a relationship full of "cheating and lies", which prompted her to creating the Facebook community. In 2023, the group for Vancouver and British Columbia was shut down after concerns about men being unable to protect themselves against fa
Elonis v. United States
Elonis v. United States, 575 U.S. 723 (2015), was a United States Supreme Court case concerning whether conviction of threatening another person over interstate lines (under 18 U.S.C. § 875(c)) requires proof of subjective intent to threaten or whether it is enough to show that a "reasonable person" would regard the statement as threatening. In controversy were the purported threats of violent rap lyrics written by Anthony Douglas Elonis and posted to Facebook under a pseudonym. The ACLU filed an amicus brief in support of the petitioner. It was the first time the Court has heard a case considering true threats and the limits of speech on social media. == Background == In May 2010, Elonis was in the process of divorce and made a number of public Facebook posts. Prior to his postings, he had lost his job at an amusement park. He "posted the script of a sketch" by The Whitest Kids U' Know, which originally referenced saying "I want to kill the President of the United States" and replaced the president with his wife: Elonis ended the post with this statement: "Art is about pushing limits. I'm willing to go to jail for my constitutional rights. Are you?" A week later, Elonis posted about local law enforcement and a kindergarten class, which caught the attention of the Federal Bureau of Investigation. Then, he wrote a post on Facebook about one of the agents who visited him: He concluded: == Arrest and Conviction == These actions led to Elonis's arrest on December 8, 2010. He was indicted by a grand jury on five counts of threats to his estranged ex-wife, park employees and visitors, local law enforcement, an FBI agent, and a kindergarten class that had been relayed through interstate communication. At the district court, Elonis moved to dismiss the indictment for failing to allege that he had intended to threaten anyone, claiming his Facebook post was not were not intended as a threat. He argued that, as an aspiring rap artist, his posts were intended to be a form of artistic expression to help him cope with his recent loses. According to him, he did not mean anything said in his posts in a literal sense. His motion was denied. He requested a jury instruction that "the government must prove that he intended to communicate a true threat", which was also denied. He was convicted on the last four of the five counts, and was sentenced to 44 months in prison and three years on supervised release. He appealed unsuccessfully to the Third Circuit, renewing his challenge to the jury instructions. He then appealed to the U.S. Supreme Court based on lack of any attempt to show intent to threaten and on First Amendment rights. == Decision == On June 1, 2015, the U.S. Supreme Court reversed Elonis's conviction in an 8–1 decision. Chief Justice John Roberts wrote for a seven-justice majority, Samuel Alito authored an opinion concurring in part and dissenting in part, and Clarence Thomas authored a dissenting opinion. The finding of the circuit court was reversed and the matter remanded. === Majority opinion === The majority opinion, written by Roberts, did not rule on First Amendment matters or on the question of whether recklessness was sufficient mens rea to show intent. It ruled that mens rea was required to prove the commission of a crime under §875(c). Importantly, the mens rea issue had been preserved for review, since Elonis had raised that objection at every stage of the previous proceedings. The government contended that the presence of the words "intent to extort" in §875(b) and §875(d) implied that the absence in §875(c) was constructive. The court disagreed, holding that the absence of the language in §875(c) was because the section was intended to have a broader scope than threats relating to extortion. The opinion drew on many Supreme Court cases holding that in criminal law, mens rea was required though it had not been mentioned explicitly in statute. Consequently, the Supreme Court ruled in favor of Elonis. === Alito's concurrence === Justice Samuel Alito, concurring in part and dissenting in part, opined that while agreeing that mens rea was required and specifically that showing negligence was not sufficient, the court should have ruled on the question of recklessness. He further opined that recklessness was sufficient to show a crime under that provision on the basis that going further would amount to amending the statute, rather than interpreting it. Since Elonis explicitly argued that recklessness was not sufficient, Alito said: I would therefore remand for the Third Circuit to determine if Elonis’s failure (indeed, refusal) to argue for recklessness prevents reversal of his conviction. The Third Circuit should also have the opportunity to consider whether the conviction could be upheld on harmless error grounds. Alito also addressed the First Amendment question, elided by the majority opinion. He held that "lyrics in songs that are performed for an audience or sold in recorded form are unlikely to be interpreted as a real threat to a real person. ... Statements on social media that are pointedly directed at their victims, by contrast, are much more likely to be taken seriously." === Thomas's dissent === Justice Clarence Thomas, dissenting, wrote against discarding the "general intent" standard without replacing it with a clearer standard. Thomas argued that "there is no historical practice requiring more than general intent when a statute regulates speech." Thomas cited Rosen v. United States, arguing that general intent was sufficient in this case. However, the majority opinion offers refutation in that Rosen turned on ignorance of the law: knowledge as to whether material was legally obscene, not on whether it was intended to be obscene. Thomas also supported the government's claim that the presence of "intent to extort" language in the adjacent §875(b) and did not address the majority's reasoning on that language. Thomas used precedent, notably from the states and 18th-century England based on other but similar and, arguably, influencing legislation to support his "general intent" claim. Thomas also drew a parallel with general intent in tort. While he sought to address the First Amendment issues, he never strayed far from "general intent". == Aftermath == On remand, the Third Circuit reaffirmed the conviction "concluding beyond a reasonable doubt that Elonis would have been convicted if the jury had been properly instructed" and therefore was harmless error. In 2022, Elonis was once again arrested and indicted on three counts of cyberstalking involving three people. It was discovered that between 2018 and 2021, Elonis had sent numerous threatening messages over email, text, voice mail, and social media platforms like Twitter to a former prosecutor of the Eastern District of Pennsylvania, his ex-girlfriend, and ex-wife. On August 5, after a five-day trial, Elonis was found guilty on all three counts, and on March 23, 2023, he was sentenced by U.S. District Court Judge Edward G. Smith of Easton, Pennsylvania to twelve years and seven months in prison.
Facebook Platform
The Facebook Platform is the set of services, tools, and products provided by the social networking service Facebook for third-party developers to create their own applications and services that access data in Facebook. The current Facebook Platform was launched in 2010. The platform offers a set of programming interfaces and tools which enable developers to integrate with the open "social graph" of personal relations and other things like songs, places, and Facebook pages. Applications on facebook.com, external websites, and devices are all allowed to access the graph. == History == Facebook launched the Facebook Platform on May 24, 2007, providing a framework for software developers to create applications that interact with core Facebook features. A markup language called Facebook Markup Language was introduced simultaneously; it is used to customize the "look and feel" of applications that developers create. Prior to the Facebook platform, Facebook had built many applications themselves within the Facebook website, including Gifts, allowing users to send virtual gifts to each other, Marketplace, allowing users to post free classified ads, Facebook events, giving users a method of informing their friends about upcoming events, Video, letting users share homemade videos with one another, and social network game, where users can use their connections to friends to help them advance in games they are playing. The Facebook Platform made it possible for outside partners to build similar applications. Many of the popular early social network games would combine capabilities. For instance, one of the early games to reach the top application spot, (Lil) Green Patch, combined virtual Gifts with Event notifications to friends and contributions to charities through Causes. Third-party companies provide application metrics, and several blogs arose in response to the clamor for Facebook applications. On July 4, 2007, Altura Ventures announced the "Altura 1 Facebook Investment Fund," becoming the world's first Facebook-only venture capital firm. On August 29, 2007, Facebook changed the way in which the popularity of applications is measured, to give attention to the more engaging applications, following criticism that ranking applications only by the number of people who had installed the application was giving an advantage to the highly viral, yet useless applications. Tech blog Valleywag has criticized Facebook Applications, labeling them a "cornucopia of uselessness." Others have called for limiting third-party applications so the Facebook user experience is not degraded. Applications that have been created on the Platform include chess, which both allow users to play games with their friends. In such games, a user's moves are saved on the website, allowing the next move to be made at any time rather than immediately after the previous move. By November 3, 2007, seven thousand applications had been developed on the Facebook Platform, with another hundred created every day. By the second annual f8 developers conference on July 23, 2008, the number of applications had grown to 33,000, and the number of registered developers had exceeded 400,000. Within a few months of launching the Facebook Platform, issues arose regarding "application spam", which involves Facebook applications "spamming" users to request it be installed. Facebook integration was announced for the Xbox 360 and Nintendo DSi on June 1, 2009 at E3. On November 18, 2009, Sony announced an integration with Facebook to deliver the first phase of a variety of new features to further connect and enhance the online social experiences of PlayStation 3. On February 2, 2010, Facebook announced the release of HipHop for PHP as an opensource project. Mark Zuckerberg said that his team from Facebook is developing a Facebook search engine. “Facebook is pretty well placed to respond to people’s questions. At some point, we will. We have a team that is working on it", said Mark Zuckerberg. For him, the traditional search engines return too many results that do not necessarily respond to questions. “The search engines really need to evolve a set of answers: 'I have a specific question, answer this question for me.'" On June 10, 2014, Facebook announced Haxl, a Haskell library that simplified the access to remote data, such as databases or web-based services. === Partnerships with device manufacturers === Starting in 2007, Facebook formed data sharing partnerships with at least 60 handset manufacturers, including Apple, Amazon, BlackBerry, Microsoft and Samsung. Those manufacturers were provided with Facebook user data without the users' consent. Most of the partnerships remained in place as of 2018, when the partnerships were first publicly reported. == High-level Platform components == === Graph API === The Graph API is the core of Facebook Platform, enabling developers to read from and write data into Facebook. The Graph API presents a simple, consistent view of the Facebook social graph, uniformly representing objects in the graph (e.g., people, photos, events, and pages) and the connections between them (e.g., friend relationships, shared content, and photo tags). On April 30, 2015, Facebook shut down friends' data API prior to the v2.0 release. === Authentication === Facebook authentication enables developers’ applications to interact with the Graph API on behalf of Facebook users, and it provides a single-sign on mechanism across web, mobile, and desktop apps. ==== Facebook Connect ==== Facebook Connect, also called Log in with Facebook, like OpenID, is a set of authentication APIs from Facebook that developers can use to help their users connect and share with such users' Facebook friends (on and off Facebook) and increase engagement for their website or application. When so used, Facebook members can log on to third-party websites, applications, mobile devices and gaming systems with their Facebook identity and, while logged in, can connect with friends via these media and post information and updates to their Facebook profile. Originally unveiled during Facebook's developer conference, F8, in July 2008, Log in with Facebook became generally available in December 2008. According to an article from The New York Times, "Some say the services are representative of surprising new thinking in Silicon Valley. Instead of trying to hoard information about their users, the Internet companies (including Facebook, Google, MySpace and Twitter) all share at least some of that data so people do not have to enter the same identifying information again and again on different sites." Log in with Facebook cannot be used by users in locations that cannot access Facebook, even if the third-party site is otherwise accessible from that location. According to Facebook, users who logged into The Huffington Post with Facebook spent more time on the site than the average user. === Social plugins === Social plugins – including the Like Button, Recommendations, and Activity Feed – enable developers to provide social experiences to their users with just a few lines of HTML. All social plugins are extensions of Facebook and are designed so that no user data is shared with the sites on which they appear. On the other hand, the social plugins let Facebook track its users’ browsing habits through any sites that feature the plugins. === Open Graph protocol === The Open Graph protocol enables developers to integrate their pages into Facebook's global mapping/tracking tool Social Graph. These pages gain the functionality of other graph objects including profile links and stream updates for connected users. OpenGraph tags in HTML5 might look like this: === iframes === Facebook uses iframes to allow third-party developers to create applications that are hosted separately from Facebook, but operate within a Facebook session and are accessed through a user's profile. Since iframes essentially nest independent websites within a Facebook session, their content is distinct from Facebook formatting. Facebook originally used 'Facebook Markup Language (FBML)' to allow Facebook Application developers to customize the "look and feel" of their applications, to a limited extent. FBML is a specification of how to encode content so that Facebook's servers can read and publish it, which is needed in the Facebook-specific feed so that Facebook's system can properly parse content and publish it as specified. FBML set by any application is cached by Facebook until a subsequent API call replaces it. Facebook also offers a specialized Facebook JavaScript (FBJS) library. Facebook stopped accepting new FBML applications on March 18, 2011, but continued to support existing FBML tabs and applications. Since January 1, 2012 FBML was no longer supported, and FBML no longer functioned as of June 1, 2012. === Microformats === In February 2011, Facebook began to use the hCalendar microformat to mark up events, and the hCard for the events' venues,
Rademacher complexity
In computational learning theory (machine learning and theory of computation), Rademacher complexity, named after Hans Rademacher, measures richness of a class of sets with respect to a probability distribution. The concept can also be extended to real valued functions. == Definitions == === Rademacher complexity of a set === Given a set A ⊆ R m {\displaystyle A\subseteq \mathbb {R} ^{m}} , the Rademacher complexity of A is defined as follows: Rad ( A ) := 1 m E σ [ sup a ∈ A ∑ i = 1 m σ i a i ] {\displaystyle \operatorname {Rad} (A):={\frac {1}{m}}\mathbb {E} _{\sigma }\left[\sup _{a\in A}\sum _{i=1}^{m}\sigma _{i}a_{i}\right]} where σ 1 , σ 2 , … , σ m {\displaystyle \sigma _{1},\sigma _{2},\dots ,\sigma _{m}} are independent random variables drawn from the Rademacher distribution i.e. Pr ( σ i = + 1 ) = Pr ( σ i = − 1 ) = 1 / 2 {\displaystyle \Pr(\sigma _{i}=+1)=\Pr(\sigma _{i}=-1)=1/2} for i ∈ { 1 , 2 , … , m } {\displaystyle i\in \{1,2,\dots ,m\}} , and a = ( a 1 , … , a m ) ∈ A {\displaystyle a=(a_{1},\ldots ,a_{m})\in A} . Some authors take the absolute value of the sum before taking the supremum, but if A {\displaystyle A} is symmetric this makes no difference. === Rademacher complexity of a function class === Let S = { z 1 , z 2 , … , z m } ⊆ Z {\displaystyle S=\{z_{1},z_{2},\dots ,z_{m}\}\subseteq Z} be a sample of points and consider a function class F {\displaystyle {\mathcal {F}}} of real-valued functions over Z {\displaystyle Z} . Then, the empirical Rademacher complexity of F {\displaystyle {\mathcal {F}}} given S {\displaystyle S} is defined as: Rad S ( F ) = 1 m E σ [ sup f ∈ F | ∑ i = 1 m σ i f ( z i ) | ] {\displaystyle \operatorname {Rad} _{S}({\mathcal {F}})={\frac {1}{m}}\mathbb {E} _{\sigma }\left[\sup _{f\in {\mathcal {F}}}\left|\sum _{i=1}^{m}\sigma _{i}f(z_{i})\right|\right]} This can also be written using the previous definition: Rad S ( F ) = Rad ( F ∘ S ) {\displaystyle \operatorname {Rad} _{S}({\mathcal {F}})=\operatorname {Rad} ({\mathcal {F}}\circ S)} where F ∘ S {\displaystyle {\mathcal {F}}\circ S} denotes function composition, i.e.: F ∘ S := { ( f ( z 1 ) , … , f ( z m ) ) ∣ f ∈ F } {\displaystyle {\mathcal {F}}\circ S:=\{(f(z_{1}),\ldots ,f(z_{m}))\mid f\in {\mathcal {F}}\}} The worst case empirical Rademacher complexity is Rad ¯ m ( F ) = sup S = { z 1 , … , z m } Rad S ( F ) {\displaystyle {\overline {\operatorname {Rad} }}_{m}({\mathcal {F}})=\sup _{S=\{z_{1},\dots ,z_{m}\}}\operatorname {Rad} _{S}({\mathcal {F}})} Let P {\displaystyle P} be a probability distribution over Z {\displaystyle Z} . The Rademacher complexity of the function class F {\displaystyle {\mathcal {F}}} with respect to P {\displaystyle P} for sample size m {\displaystyle m} is: Rad P , m ( F ) := E S ∼ P m [ Rad S ( F ) ] {\displaystyle \operatorname {Rad} _{P,m}({\mathcal {F}}):=\mathbb {E} _{S\sim P^{m}}\left[\operatorname {Rad} _{S}({\mathcal {F}})\right]} where the above expectation is taken over an identically independently distributed (i.i.d.) sample S = ( z 1 , z 2 , … , z m ) {\displaystyle S=(z_{1},z_{2},\dots ,z_{m})} generated according to P {\displaystyle P} . == Intuition == The Rademacher complexity is typically applied on a function class of models that are used for classification, with the goal of measuring their ability to classify points drawn from a probability space under arbitrary labellings. When the function class is rich enough, it contains functions that can appropriately adapt for each arrangement of labels, simulated by the random draw of σ i {\displaystyle \sigma _{i}} under the expectation, so that this quantity in the sum is maximized. The Rademacher complexity of a set A {\displaystyle A} can be rewritten as Rad ( A ) := 1 m E σ [ sup a ∈ A ∑ i = 1 m σ i a i ] = 1 m 2 m ∑ σ ∈ { − 1 / m , + 1 / m } m [ sup a ∈ A ⟨ σ , a ⟩ ] . {\displaystyle \operatorname {Rad} (A):={\frac {1}{m}}\mathbb {E} _{\sigma }\left[\sup _{a\in A}\sum _{i=1}^{m}\sigma _{i}a_{i}\right]={\frac {1}{{\sqrt {m}}2^{m}}}\sum _{\sigma \in \{-1/{\sqrt {m}},+1/{\sqrt {m}}\}^{m}}\left[\sup _{a\in A}\langle \sigma ,a\rangle \right].} Each term in the summation is the farthest distance of the set A {\displaystyle A} from the origin, along a unit-length direction σ {\displaystyle \sigma } . The directions are along the vertices of a hypercube. Thus, we can also write it as Rad ( A ) = 1 2 m 1 2 m − 1 ∑ σ ∈ { − 1 / m , + 1 / m } m / { − 1 , + 1 } [ sup a ∈ A ⟨ σ , a ⟩ − inf a ∈ A ⟨ σ , a ⟩ ] {\displaystyle \operatorname {Rad} (A)={\frac {1}{2{\sqrt {m}}}}{\frac {1}{2^{m-1}}}\sum _{\sigma \in \{-1/{\sqrt {m}},+1/{\sqrt {m}}\}^{m}/\{-1,+1\}}\left[\sup _{a\in A}\langle \sigma ,a\rangle -\inf _{a\in A}\langle \sigma ,a\rangle \right]} Here, the set { − 1 / m , + 1 / m } m / { − 1 , + 1 } {\displaystyle \{-1/{\sqrt {m}},+1/{\sqrt {m}}\}^{m}/\{-1,+1\}} denotes half of the vertices of a hypercube, selected so that each diagonal has exactly one vertex selected. In words, this states that 2 m Rad ( A ) {\displaystyle 2{\sqrt {m}}\operatorname {Rad} (A)} is precisely the average width of the set A {\displaystyle A} along all diagonal directions of a hypercube. == Examples == A singleton set has 0 width in any direction, so it has Rademacher complexity 0. The set A = { ( 1 , 1 ) , ( 1 , 2 ) } ⊆ R 2 {\displaystyle A=\{(1,1),(1,2)\}\subseteq \mathbb {R} ^{2}} has average width 1 / 2 {\displaystyle 1/{\sqrt {2}}} along the two diagonal directions of the square, so it has Rademacher complexity 1 / 4 {\displaystyle 1/4} . The unit cube [ 0 , 1 ] m {\displaystyle [0,1]^{m}} has constant width m {\displaystyle {\sqrt {m}}} along the diagonal directions, so it has Rademacher complexity 1 / 2 {\displaystyle 1/2} . Similarly, the unit cross-polytope { x ∈ R m : ‖ x ‖ 1 ≤ 1 } {\displaystyle \{x\in \mathbb {R} ^{m}:\|x\|_{1}\leq 1\}} has constant width 2 / m {\displaystyle 2/{\sqrt {m}}} along the diagonal directions, so it has Rademacher complexity 1 / m {\displaystyle 1/m} . == Using the Rademacher complexity == The Rademacher complexity can be used to derive data-dependent upper-bounds on the learnability of function classes. Intuitively, a function-class with smaller Rademacher complexity is easier to learn. === Bounding the representativeness === In machine learning, it is desired to have a training set that represents the true distribution of some sample data S {\displaystyle S} . This can be quantified using the notion of representativeness. Denote by P {\displaystyle P} the probability distribution from which the samples are drawn. Denote by H {\displaystyle H} the set of hypotheses (potential classifiers) and denote by F {\displaystyle {\mathcal {F}}} the corresponding set of error functions, i.e., for every hypothesis h ∈ H {\displaystyle h\in H} , there is a function f h ∈ F {\displaystyle f_{h}\in F} , that maps each training sample (features,label) to the error of the classifier h {\displaystyle h} (note in this case hypothesis and classifier are used interchangeably). For example, in the case that h {\displaystyle h} represents a binary classifier, the error function is a 0–1 loss function, i.e. the error function f h {\displaystyle f_{h}} returns 0 if h {\displaystyle h} correctly classifies a sample and 1 else. We omit the index and write f {\displaystyle f} instead of f h {\displaystyle f_{h}} when the underlying hypothesis is irrelevant. Define: L P ( f ) := E z ∼ P [ f ( z ) ] {\displaystyle L_{P}(f):=\mathbb {E} _{z\sim P}[f(z)]} – the expected error of some error function f ∈ F {\displaystyle f\in {\mathcal {F}}} on the real distribution P {\displaystyle P} ; L S ( f ) := 1 m ∑ i = 1 m f ( z i ) {\displaystyle L_{S}(f):={1 \over m}\sum _{i=1}^{m}f(z_{i})} – the estimated error of some error function f ∈ F {\displaystyle f\in {\mathcal {F}}} on the sample S {\displaystyle S} . The representativeness of the sample S {\displaystyle S} , with respect to P {\displaystyle P} and F {\displaystyle {\mathcal {F}}} , is defined as: Rep P ( F , S ) := sup f ∈ F ( L P ( f ) − L S ( f ) ) {\displaystyle \operatorname {Rep} _{P}({\mathcal {F}},S):=\sup _{f\in F}(L_{P}(f)-L_{S}(f))} Smaller representativeness is better, since it provides a way to avoid overfitting: it means that the true error of a classifier is not much higher than its estimated error, and so selecting a classifier that has low estimated error will ensure that the true error is also low. Note however that the concept of representativeness is relative and hence can not be compared across distinct samples. The expected representativeness of a sample can be bounded above by the Rademacher complexity of the function class: If F {\displaystyle {\mathcal {F}}} is a set of functions with range within [ 0 , 1 ] {\displaystyle [0,1]} , then Rad P , m ( F ) − ln 2 2 m ≤ E S ∼ P m [ Rep P ( F , S ) ] ≤ 2 Rad P , m ( F ) {\displaystyle \operatorname {Rad} _{P,m}({\mathcal {F}})-{\sqrt {\frac {\ln 2}{2m}}}\leq \mathbb {E} _{S\sim P^{m}}[\operatorname {Rep} _{P}({\
The Morning After (web series)
The Morning After is a Hulu original web series that premiered on January 17, 2011, and ended April 24, 2014. It was produced by Hulu and Jace Hall's HDFilms, streaming Monday through Friday. The show originally featured Brian Kimmet and Ginger Gonzaga as hosts. Later shows used a rotation of hosts including Alison Haislip, Dave Holmes, Damien Fahey, Bradley Hasemeyer, Haley Mancini, Paul Nyhart, and Rachel Perry. The series advertises itself as "a smart, daily shot of pop culture to help Hulu users stay up to date" and typically highlights notable moments from television shows and current news in an entertaining fashion. In keeping with its focus on pop culture, The Morning After will sometimes stream an episode featuring past pop culture titled "From the Archives," such as its April Fools' Day episode. == History == While not the first original series to appear exclusively on Hulu, The Morning After is the company's first self-branded production. It was preceded by If I Can Dream, a reality series co-produced with 19 Entertainment and created by Simon Fuller. Hulu originated the idea in house, based on user feedback and observations from discussion boards hosted by the website. The concept was modeled after The Big Show with Olbermann and Patrick. The company sought out a production partner and ultimately chose Jace Hall and his team at HDFilms to executive produce. Initial stream of the series was held on January 17, 2011, and featured coverage of Piers Morgan, the Golden Globes, and The Bachelor. Senior VP of Content and Distribution Andy Forssell made the announcement for the show the same day. The show aired its last episode April 24, 2014. == Format == A typical episode usually begins with a cold open shared by the varying hosts listing the highlights to be covered. The topics focus on TV and Pop Culture Highlights from the previous night, with the intention of helping Hulu users digest hours of content in a matter of moments. The show has the hosts trade humorous remarks regarding the news and each other, taking turns reviewing the night's TV and injecting their own personality. The Morning After was named as an honoree by the Webbys on April 10, 2012, in the variety section of its online video category.