Instapoetry is a style of poetry that emerged after the advent of social media, especially on Instagram. The term has been used to describe poems written specifically for being shared online, most commonly on Instagram, but also other platforms including Twitter, Tumblr, and TikTok. The style usually consists of short, direct lines in aesthetically pleasing fonts that are sometimes accompanied by an image or drawing, often without rhyme schemes or meter, and dealing with commonplace themes. Literary critics, poets, and writers have contended with Instapoetry's focus on brevity and plainness compared to traditional poetry, criticizing it for reproducing rather than subverting normative ideas on social media platforms that favor popularity and accessibility over craft and depth. == History == Instapoetry developed as a result of young, predominantly women, amateur poets sharing their output to expand their readership, who began using social media as their preferred method of distribution rather than traditional publishing methods. The term "Instapoetry" is a portmanteau of the words "Instagram" and "poetry," and was created by other writers trying to define and understand the new extension of "instant poetry" shared via social media, most prominently Instagram. In its most basic form, Instapoetry usually consists of bite-sized verses that consider political and social subjects such as immigration, domestic violence, sexual assault, love, culture, feminism, gun violence, war, racism, LGBTQ rights, and other social justice topics. All of these elements are usually made to fit social media feeds that are easily accessible through applications on smartphones. == Scholarship == Despite the diversity of poetry on Instagram, the Brazilian linguist Bruna Osaki Fazano found that shared "aspects of the compositional form, theme and style" mean that it can be understood as a specific genre. Camilla Holm Soelseth argues that taking on the platform-specific tasks of a social media creator is a prerequisite for being an Instapoet. Writing in Poetics Today, JuEunhae Knox combined quantitative and qualitative analysis to show that Instapoetry is a cohesive genre, in part because "the sheer volume and rapidity of content production in turn encourages posts that are not only visually appealing but also immediately recognizable as Instapoems". Instapoetry has been seen as a practice that serves as a form of self-staging for poets and "[crafts] authenticity". Eirik Vassenden describes the work of Norwegian poet Trygve Skaug as appearing to offer a "simple, almost direct access to the inner self". Vassenden writes that poems such as Rupi Kaur's "if you are not enough for yourself / you will never be enough / for someone else" are "authentic" to such an extent that they are not literary. Kiera Obbard describes how Rupi Kaur uses humour as a rhetorical device in her poetry performances to tell personal stories of trauma and challenge social inequalities. Scholars have also studied the work of specific Instapoets, such as Rupi Kaur, R.M. Drake, Aja Monet, Yrsa Daley-Ward, Nayyirah Waheed, Atticus, Nikita Gill and Trygve Skaug. == Overview == Academics have shown appreciation for the way in which Instapoetry has stimulated interest in poetry in general. Meanwhile, it has been argued that since Instapoets avoid critical evaluations, academics, and the publishing industry, Instapoets qualify more as online celebrities than literary figures. Additionally, although Instapoetry has been characterized as anti-establishment, Alyson Miller noted traditional or even conservative views in the online posts of Instapoets in contrast with the activist views the style is associated with, and that there is a contradiction between "the extra-textual commentary surrounding Instapoetry, particularly by way of interviews and artistic statements, and the content of works which repeatedly reinscribe conservative, patriarchal, and heteronormative worldviews". Thom Young, a poet and high school English teacher, created a parody Instagram page as a way to mock Instapoets and their work, describing it as "fidget-spinner poetry. Like they're just scrolling on their devices, to read something instantly, while the libraries are empty. I think people today don't want to read anything that causes a whole lot of critical thinking." According to Johnathan Ford's piece in the Financial Times, as Instagram's algorithms have limited prospective Instapoets' reach-per-post, it has pushed them to pay to promote their material. Popular Instagram accounts will be promoted to the front of users' feeds, with the app's algorithm, in the view of critics, favoring the spread of bland, inauthentic, or clichéd content while preventing disciplined poetry from reaching new audiences. == Writers described as Instapoets == Rupi Kaur Atticus Amanda Lovelace Tyler Knott Gregson Najwa Zebian Lang Leav Nikita Gill Upile Chisala Tendai M. Shaba Donna Ashworth Trista Mateer
GeoNetwork opensource
The GeoNetwork opensource (GNOS) project is a free and open source (FOSS) cataloging application for spatially referenced resources. It is a catalog of location-oriented information. == Outline == It is a standardized and decentralized spatial information management environment designed to enable access to geo-referenced databases, cartographic products and related metadata from a variety of sources, enhancing the spatial information exchange and sharing between organizations and their audience, using the capacities of the internet. Using the Z39.50 protocol it both accesses remote catalogs and makes its data available to other catalog services. As of 2007, OGC Web Catalog Service are being implemented. Maps, including those derived from satellite imagery, are effective communicational tools and play an important role in the work of decision makers (e.g., sustainable development planners and humanitarian and emergency managers) in need of quick, reliable and up-to-date user-friendly cartographic products as a basis for action and to better plan and monitor their activities; GIS experts in need of exchanging consistent and updated geographical data; and spatial analysts in need of multidisciplinary data to perform preliminary geographical analysis and make reliable forecasts. == Deployment == The software has been deployed to various organizations, the first being FAO GeoNetwork and WFP VAM-SIE-GeoNetwork, both at their headquarters in Rome, Italy. Furthermore, the WHO, CGIAR, BRGM, ESA, FGDC and the Global Change Information and Research Centre (GCIRC) of China are working on GeoNetwork opensource implementations as their spatial information management capacity. It is used for several risk information systems, in particular in the Gambia. Several related tools are packaged with GeoNetwork, including GeoServer. GeoServer stores geographical data, while GeoNetwork catalogs collections of such data.
NexDock
NexDock is a series of lapdock devices (containing a laptop screen, keyboard, trackpad, and battery connected to a phone or other device) sold by Nex Computer LLC. The product can be used with mobile desktop environments, including Samsung DeX and the former Windows Continuum. Critical reception for the series has been mixed, with reviewers praising the concept's utility for mobile productivity while noting hardware limitations and its niche appeal. == History == The first NexDock was introduced in 2016 through a successful Indiegogo campaign. Its development coincided with interest in smartphone-powered desktop interfaces, and it was marketed as a companion for Windows 10 Mobile's Continuum feature. Subsequent models, often launched via Kickstarter, added features like higher-resolution displays, touchscreens, and convertible hinges to adapt to the growing capabilities of smartphones. == Models == === NexDock (Original, 2016) === The first model featured a 14.1-inch 1366x768 display and connected primarily via a mini HDMI port. === NexDock 2 (2019) === This model introduced a 13.3-inch 1080p IPS display and a USB-C port, improvements aimed at better supporting platforms like Samsung DeX. === NexDock Touch (2020) === A touchscreen was added to the 13.3-inch display, allowing for more direct interaction with the connected device's operating system. === NexDock 360 (2021) === This version incorporated a 360-degree hinge, allowing the device to be used in laptop, tablet, tent, or stand modes. === NexDock Wireless (2023) === Wireless display connectivity was the key feature of this model, offering a cable-free connection to compatible phones and computers. === NexDock XL (2023) === The screen size was increased to 15.6 inches. It retained the 360-degree hinge and also offered a version with wireless charging for a connected phone. == Reception == Reviews of NexDock products have been mixed, generally praising the concept while pointing out execution flaws. The devices are often lauded for their utility with Samsung DeX, turning a high-end Samsung phone into a viable portable workstation. A review of the NexDock 2 from ZDNet concluded it was a "great companion for the modern road warrior," and Digital Trends called the original a "no-brainer shell" for expanding a phone's capability. However, reviewers have consistently highlighted hardware limitations. In its review of the NexDock Touch, TechRadar stated that while it was a "compelling package for a very specific niche," the "trackpad and keyboard are a bit of a letdown and the screen could be brighter." This sentiment was echoed in other reviews, with criticism often aimed at the trackpad's performance and feel. A review of the NexDock 2 from Android Authority described the experience as being "janky at times," concluding that the device "delivers on its promise — sort of." A common point across many reviews is that the overall performance is entirely dependent on the power of the connected phone, and the experience is often best suited for light productivity tasks rather than replacing a dedicated laptop.
Mass media use by the Islamic State
The Islamic State (IS) is known for its extensive and effective use of propaganda. It uses a version of the Muslim Black Standard flag and developed an emblem which has clear symbolic meaning in the Muslim world. The Islamic State targets younger audiences, such as teenagers and young adults, since they are more vulnerable to propaganda. It is known to exploit the internet to spread its propaganda by establishing websites, such as the Al Fustat domain. Videos by the Islamic State are commonly accompanied by nasheeds (chants), notable examples being the chant Dawlat al-Islam Qamat, which came to be viewed as an unofficial anthem of the Islamic State, and Salil al-Sawarim. Academic research has emphasized the scale and volume of Islamic State media production beyond its flagship magazines. A quantitative study cited in R. Malash’s academic work documented 1,373 distinct Islamic State media products released over a six-month period between 1 August 2017 and 28 February 2018, including magazines, newsletters, reports, photographic releases, audio recordings, and other media formats. Scholars have used such datasets to illustrate the breadth and intensity of the group’s media output, particularly during periods of territorial decline, when propaganda activity remained high despite military pressure. == Traditional media == === Al-Furqan Foundation for Media Production === In January 2006, shortly after the group's rebranding as the "Islamic State of Iraq", it established the Al-Furqan Foundation for Media Production (Arabic: مؤسسة الفرقان للإنتاج الإعلامي, romanized: Muasasat al-Furqān lil'īntāj al'ilāmī), which produces CDs, DVDs, posters, pamphlets, and web-related propaganda products and official statements. It is the primary media production house of the Islamic State and responsible for production of major media releases, including the statements of the spokesmen and leaders of the group. On January 10, 2006, Al-Furqan released its very first video, titled (Arabic: زحف الأنوار, romanized: Zahf al-Anwār) It was founded by the Iraqi man Dr Wa'il al-Fayad, known as Abu Muhammad al-Furqan. He got his name "Al-Furqan" from his role in founding this media house, which was named after the 25th surah of the Quran Al-Furqan. It is the oldest media production house for the Islamic State, being founded in November 2006 to release media for the Islamic State of Iraq. The earliest release indexed by the SITE Intelligence Group is on 21 November 2006, documenting the storming of a police station in the Iraqi town of Miqdadiyah. Al-Furqan is considered to be a considerable innovation in jihadist media, with Kavkaz Center describing it as "a milestone on the path of jihad, a distinguished media that takes the great care in the management of the conflict with the crusaders and their tails and to expose the lies in the crusader's media." In October 2007, the Long War Journal reported on United States Army raids targeting Al-Furqan media cell members across Iraq, including in Mosul and Samarra. Between August 2013 and March 2014 they released the 22 part series Messages from the Land of Epic Battles. On 2 September 2014 SITE Intelligence Group discovered the beheading video called A Second Message to America, about the death of Steven Sotloff. Since then, Al-Furqan has released videos of their operations across Iraq and Syria, as well as execution videos directed to governments around the world. In April 2019, Al-Furqan released a video Interviewing Abu Bakr al-Baghdadi. Al-Furqan also produces media in the form of audio, which consists mostly of recordings of IS leaders and spokesmen giving speeches, as well as producing a single nasheed under their name called "Ya Allah Al-Jannah" (O Allah, (we ask you for) Paradise), sung by now-dead member of IS, Uqab Al-Marzuqi. === Al-I'tisam Foundation for Media Production === The Islamic State of Iraq founded a second media foundation - Al-I'tisam Media Foundation - around 2011, marked by their first video release, titled "The Conqueror of the Murtaddin: Abu Ahmad Al-Ansari". The foundation has since released a few series of videos, 50 parts of "Windows on the Land of Battles", 9 parts of "Pictures from the Land of Battles", a 9-part series quoting leaders about the establishment of the Islamic State, and other series before their last release, "Deterring the Safavids in Salah ad-Din" in 2015. Since then, there were no further releases from their behalf. === Al-Hayat Media Center === In mid-2014, IS established the Al-Hayat Media Center, which targets Western audiences and produces material in English, German, Russian, Urdu, Indonesian, Turkish, Bengali, Chinese, Bosnian, Kurdish, Uyghur, and French. When IS announced its expansion to other countries in November 2014 it established media departments for the new branches, and its media apparatus ensured that the new branches follow the same models it uses in Iraq and Syria. Then FBI Director James Comey said that IS's "propaganda is unusually slick," noting that, "They are broadcasting... in something like 23 languages". In July 2014, Al-Hayat began publishing a digital magazine called Dabiq, in a number of different languages including English. According to the magazine, its name is taken from the town of Dabiq in northern Syria, which is mentioned in a hadith about Armageddon. Al-Hayat also began publishing other digital magazines, including the Turkish language Konstantiniyye, the Ottoman word for Istanbul, the French language Dar al-Islam, and the Russian language Istok (Russian: Исток). By late 2016, these magazines had apparently all been discontinued, with Al-Hayat's material being consolidated into a new magazine called Rumiyah (Arabic for Rome). === Al-Naba === While the group's glossy, foreign-language magazines like Dabiq and Rumiyah ceased publication as the group lost territory, the weekly Arabic newsletter Al-Naba (The News) has continued to publish regularly, becoming the central pillar of the group's "media jihad" in the post-territorial phase. Recent scholarship, including studies published in 2025, suggests that Al-Naba serves a dual purpose: maintaining internal cohesion among dispersed fighters and projecting a narrative of endurance to enemies. Unlike the earlier magazines which were designed for recruitment, Al-Naba focuses on bureaucratic reporting, military statistics, and religious instruction. These are then translated and disseminated by decentralized supporter networks ("media mujahideen") to reach non-Arabic speakers. === Furat Media Center === The Al-Furat Media Center is another media center established in around 2015 to cater towards non-Arab speaking audiences. However, unlike the other organizations, the production wasn't as professional as ones made by the other media centers. Instead, they partially relied on local media departments and foreign communities of the Mujahideen to produce short-form videos. However, some professional long-form videos were also made under their behalf. As of now, the media center is the only known active branch of all the media centers of the Islamic State, after heavy losses from past campaigns against them. Their last release was "The Resolve of Muwahhidin in Russia", where videos from the Surovikino penal colony hostage crisis were edited and released. === Ajnad Foundation for Media Production === Ajnad Foundation is one of the official media wings of Islamic State which produces nasheeds and Quran recitations. It was established in January 2014 and has released more than 150 nasheeds. === Asdaa Foundation === Like the Ajnad Foundation, the Asdaa Foundation (Arabic: مؤسسة أصداء) or Asedaa Foundation produces Anasheed (Islamic chants). The foundation is the closest counterpart to Ajnad in producing Islamic State nasheeds, only difference being Ajnad is directly linked to the Islamic State while Asdaa is only classified as a "supporter organization" (munaser/munasera). The foundation had humble beginnings possibly in Yemen, where low-quality nasheeds were produced at first by 2 munshids, Abu Layth Al-Iraqi and Abu Ya'qub Al-Yamani. After that, the quality had improved a bit (possibly with new equipment and increased recognition) and eventually had its nasheeds included in the Islamic State's official media releases. One of its munshids, Abu Hafs is a renowned munshid who sings around 70 nasheeds, who as well works with Ajnad Foundation in some instances. He is currently alive, and working under Ansar Production Center (مركز إنتاج الأنصار), another Munasir foundation and Asedaa. Another Yemeni munshid, Abu Musab al-Adani, worked temporarily with Asdaa Foundation before defecting back to AQAP, from which he previously defected from. Some of their anasheed is used in IS's execution videos, a popular one is their human slaughterhouse execution video released during the time of Eid Al-Adha in 2016. The background nasheed they used was "We Came To Fill The Horizons With Terror", produced by the Asd
Fan loyalty
Fan loyalty is the loyalty felt and expressed by a fan towards the object of their fanaticism. Fan loyalty is often used in the context of sports and the support of a specific team or institution. Fan loyalties can range from a passive support to radical allegiance and expressions of loyalty can take shape in many forms and be displayed across varying platforms. Fan loyalty can be threatened by team actions. The loyalties of sports fans in particular have been studied by psychologists, who have determined several factors that help to create such loyalties. == Underpinning psychology == Given the extensive costs involved in managing and operating a professional team sport, it is beneficial for sports marketers to be conscious of the elements that establish a strong brand and the effect they have on fan loyalty, so they can best cater to their current fans while acquiring new ones. This is because fans and spectators are considered key stakeholders of professional sports organisations. Fans directly and indirectly influence the production of operating revenue through purchasing merchandise, buying game tickets and improving the value that can be obtained from television broadcasting deals and sponsorship. Therefore, fans are a key factor to consider in determining the economic success of a sports club. Deep psychological connections with new teams can be built with individuals before a team has even played a match revealing insights can develop quickly in the mind of consumers without direct encounters or experiences e.g. watching a team compete. Brand management approaches are helping sport organisations to expand the sport experience, appeal to new fans and enable long term business to consumer relationships through multi faceted connection such as social media. To affect consumers’ loyalty with a team, they must develop a compelling, positive and distinctive brand in order to stand out amongst competitor and vie for fan support. Loyalty programmes positively shape fan attachment and behaviour as it connects teams and their fans, aside from a club's season ticketholder database. It not only provides marketers with essential information about consumers and their thinking, but also acts as a channel to promote attendance and an opportunity to add value to their game day experience. Bauer et al. concludes that non product related attributes such as contextual factors (other fans, the club history and tradition, logo, club colours and the stadium atmosphere) hold a higher place in fan experience than product related attributes such as the team's winning record. Therefore, to increase fan loyalty (customer retention) Bauer et al. suggests sports marketers focus on targeting non product related benefits and brand attributes. As a result of fostering this loyalty, sports organisations can afford to charge prices at premium. Fan loyalty also leads to dependable ratings in broadcast media which means broadcasters can also charge premiums for advertising time in team broadcasts with loyal followings. A flow on effect from fan loyalty is the ability to create additional revenue streams outside of the core product such as merchandise shops and food venues that are close to the location of the game if the team chooses to own and operate ventures or share licensing agreements. Fan loyalty, particularly with respect to team sports, is different from brand loyalty, in as much as if a consumer bought a product that was of lower quality than expected, he or she will usually abandon allegiance to the brand. However, fan loyalty continues even if the team that the fan supports continues to perform poorly year after year. Author Mark Conrad uses the Chicago Cubs as an example of a team with a loyal fan following, where fans spend their money in support of a poorly performing team that (until 2016) had not won a pennant since 1945 or a World Series since 1908. They attribute it to the following factors: Entertainment Value The entertainment value that a fan derives from spectating motivates him/her to remain a loyal fan. Entertainment value of team sports is also valuable to communities in general. Authenticity This is described by Passikoff as "the acceptance of the game as real and meaningful". Fan Bonding Fan bonding is where a fan bonds with the players, identifying with them as individuals, and bonds with the team. Team History and Tradition Shank gives the Cincinnati Reds, all-professional baseball's oldest team, as an example of a team where a long team history and tradition is a motivator for fans in the Cincinnati area. Group Affiliation Fans receive personal validation of their support for a team from being surrounded by a group of fans who also support the same team. Fair Weather Fans Fans that engage when a team is good, and lose interest when a team is bad. Bandwagon Fans Fans who support the winning team, instead of supporting the same team year after year. Diehard Fans Fans who follow their team no matter if they are winning or losing. == Factors influencing fan loyalty == === Community === Fan loyalty attachment is strengthened through communal ties that connect fans around a team, forming a community that results in regular fan interaction. This interaction is particularly important as fans may not develop solely an intra-psychic team identity but predominantly display behavioural loyalty through the group consumption of indirect sport experiences instead, such as wearing the team colours, singing, cheering, flags and interaction between the sport's team's fans (e.g. laughing, talking) Through indirect sport experiences, the stadium atmosphere can be heightened and as a result, the frequency of fan attendance can increase. Furthermore, by wearing team apparel, fans can visually identify with one another resulting an increased likelihood of opportunities to engage with others socially through this point of connection. For example, a study on NASCAR fans found that their personal identity was connected to the brand itself as they felt connected to the larger community of NASCAR revealing an emotional connection to the brand. This indicates that their fan loyalty will result in the notion that fans are naturally more resistant to the promotional efforts of competing brands (e.g. lower-price offers) as their emotional commitment to NASCAR is greatly embedded in their sense of identity. When they associate themselves with the sponsors because of the sponsor's relation to the brand, they are solidifying their relationship with NASCAR and are therefore reinforcing their identity. Consequently, their fan loyalty translates into brand loyalty so long as the sponsor remains attached to the subject of their fanaticism, NASCAR, meaning they are less price sensitive and more willing to pay premium prices for sponsor's products or services. Another aspect of consumer behaviour regarding fan loyalty is the existence of consumption communities where members feel a sense of unity when they participate in the group consumption of brand sponsors’ goods and services further strengthening their ties to a brand and its sponsors. However, a strategy sports marketers use to appeal to a wider range of fan identities is to sponsor more than one club in sports such as soccer. This is so they are careful not to come across as a singularly affiliated club brand, where the opinion or perceptions of opposing teams’ fans would be one of disfavour towards them. === Brand association === Any benefit or characteristic connected to a brand as perceived by a consumer is called a brand association. These hold significance over the thoughts and opinions a consumer holds about a brand and can therefore influence one's loyalty. These associations provide a reference point to gauge the salience of a brand which is the perceived favourability associated with it. Brand salience is vital because it ultimately effects the likelihood of brand selection and loyalty leading to steadier spectator numbers, and an increase in attention from the media such as advertisers and sponsors. However, loyalty is a developmental process. According to Bee & Havitz (2010), spectators who are highly involved in the participation of a sport and exhibit psychological commitment, possess the capability to display high levels of behavioural loyalty as they develop into committed fans. On the other hand, neutral or negative feelings towards a team are found to foster indifference or cause an individual to disidentify with a team altogether. A model of ‘escalating commitment’, put forward by Funk and James (2001), demonstrates an individual's movement from ‘awareness’ of team to a subsequent ‘allegiance’ but came to the conclusion that more research was required to find out the key influences that lead one to the highest state of commitment. However, brand association development is fostered under brand management within a sports organisation. It is important for sports management research to identify t
Proximal gradient methods for learning
Proximal gradient (forward backward splitting) methods for learning is an area of research in optimization and statistical learning theory which studies algorithms for a general class of convex regularization problems where the regularization penalty may not be differentiable. One such example is ℓ 1 {\displaystyle \ell _{1}} regularization (also known as Lasso) of the form min w ∈ R d 1 n ∑ i = 1 n ( y i − ⟨ w , x i ⟩ ) 2 + λ ‖ w ‖ 1 , where x i ∈ R d and y i ∈ R . {\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-\langle w,x_{i}\rangle )^{2}+\lambda \|w\|_{1},\quad {\text{ where }}x_{i}\in \mathbb {R} ^{d}{\text{ and }}y_{i}\in \mathbb {R} .} Proximal gradient methods offer a general framework for solving regularization problems from statistical learning theory with penalties that are tailored to a specific problem application. Such customized penalties can help to induce certain structure in problem solutions, such as sparsity (in the case of lasso) or group structure (in the case of group lasso). == Relevant background == Proximal gradient methods are applicable in a wide variety of scenarios for solving convex optimization problems of the form min x ∈ H F ( x ) + R ( x ) , {\displaystyle \min _{x\in {\mathcal {H}}}F(x)+R(x),} where F {\displaystyle F} is convex and differentiable with Lipschitz continuous gradient, R {\displaystyle R} is a convex, lower semicontinuous function which is possibly nondifferentiable, and H {\displaystyle {\mathcal {H}}} is some set, typically a Hilbert space. The usual criterion of x {\displaystyle x} minimizes F ( x ) + R ( x ) {\displaystyle F(x)+R(x)} if and only if ∇ ( F + R ) ( x ) = 0 {\displaystyle \nabla (F+R)(x)=0} in the convex, differentiable setting is now replaced by 0 ∈ ∂ ( F + R ) ( x ) , {\displaystyle 0\in \partial (F+R)(x),} where ∂ φ {\displaystyle \partial \varphi } denotes the subdifferential of a real-valued, convex function φ {\displaystyle \varphi } . Given a convex function φ : H → R {\displaystyle \varphi :{\mathcal {H}}\to \mathbb {R} } an important operator to consider is its proximal operator prox φ : H → H {\displaystyle \operatorname {prox} _{\varphi }:{\mathcal {H}}\to {\mathcal {H}}} defined by prox φ ( u ) = arg min x ∈ H φ ( x ) + 1 2 ‖ u − x ‖ 2 2 , {\displaystyle \operatorname {prox} _{\varphi }(u)=\operatorname {arg} \min _{x\in {\mathcal {H}}}\varphi (x)+{\frac {1}{2}}\|u-x\|_{2}^{2},} which is well-defined because of the strict convexity of the ℓ 2 {\displaystyle \ell _{2}} norm. The proximal operator can be seen as a generalization of a projection. We see that the proximity operator is important because x ∗ {\displaystyle x^{}} is a minimizer to the problem min x ∈ H F ( x ) + R ( x ) {\displaystyle \min _{x\in {\mathcal {H}}}F(x)+R(x)} if and only if x ∗ = prox γ R ( x ∗ − γ ∇ F ( x ∗ ) ) , {\displaystyle x^{}=\operatorname {prox} _{\gamma R}\left(x^{}-\gamma \nabla F(x^{})\right),} where γ > 0 {\displaystyle \gamma >0} is any positive real number. === Moreau decomposition === One important technique related to proximal gradient methods is the Moreau decomposition, which decomposes the identity operator as the sum of two proximity operators. Namely, let φ : X → R {\displaystyle \varphi :{\mathcal {X}}\to \mathbb {R} } be a lower semicontinuous, convex function on a vector space X {\displaystyle {\mathcal {X}}} . We define its Fenchel conjugate φ ∗ : X → R {\displaystyle \varphi ^{}:{\mathcal {X}}\to \mathbb {R} } to be the function φ ∗ ( u ) := sup x ∈ X ⟨ x , u ⟩ − φ ( x ) . {\displaystyle \varphi ^{}(u):=\sup _{x\in {\mathcal {X}}}\langle x,u\rangle -\varphi (x).} The general form of Moreau's decomposition states that for any x ∈ X {\displaystyle x\in {\mathcal {X}}} and any γ > 0 {\displaystyle \gamma >0} that x = prox γ φ ( x ) + γ prox φ ∗ / γ ( x / γ ) , {\displaystyle x=\operatorname {prox} _{\gamma \varphi }(x)+\gamma \operatorname {prox} _{\varphi ^{}/\gamma }(x/\gamma ),} which for γ = 1 {\displaystyle \gamma =1} implies that x = prox φ ( x ) + prox φ ∗ ( x ) {\displaystyle x=\operatorname {prox} _{\varphi }(x)+\operatorname {prox} _{\varphi ^{}}(x)} . The Moreau decomposition can be seen to be a generalization of the usual orthogonal decomposition of a vector space, analogous with the fact that proximity operators are generalizations of projections. In certain situations it may be easier to compute the proximity operator for the conjugate φ ∗ {\displaystyle \varphi ^{}} instead of the function φ {\displaystyle \varphi } , and therefore the Moreau decomposition can be applied. This is the case for group lasso. == Lasso regularization == Consider the regularized empirical risk minimization problem with square loss and with the ℓ 1 {\displaystyle \ell _{1}} norm as the regularization penalty: min w ∈ R d 1 n ∑ i = 1 n ( y i − ⟨ w , x i ⟩ ) 2 + λ ‖ w ‖ 1 , {\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-\langle w,x_{i}\rangle )^{2}+\lambda \|w\|_{1},} where x i ∈ R d and y i ∈ R . {\displaystyle x_{i}\in \mathbb {R} ^{d}{\text{ and }}y_{i}\in \mathbb {R} .} The ℓ 1 {\displaystyle \ell _{1}} regularization problem is sometimes referred to as lasso (least absolute shrinkage and selection operator). Such ℓ 1 {\displaystyle \ell _{1}} regularization problems are interesting because they induce sparse solutions, that is, solutions w {\displaystyle w} to the minimization problem have relatively few nonzero components. Lasso can be seen to be a convex relaxation of the non-convex problem min w ∈ R d 1 n ∑ i = 1 n ( y i − ⟨ w , x i ⟩ ) 2 + λ ‖ w ‖ 0 , {\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-\langle w,x_{i}\rangle )^{2}+\lambda \|w\|_{0},} where ‖ w ‖ 0 {\displaystyle \|w\|_{0}} denotes the ℓ 0 {\displaystyle \ell _{0}} "norm", which is the number of nonzero entries of the vector w {\displaystyle w} . Sparse solutions are of particular interest in learning theory for interpretability of results: a sparse solution can identify a small number of important factors. === Solving for L1 proximity operator === For simplicity we restrict our attention to the problem where λ = 1 {\displaystyle \lambda =1} . To solve the problem min w ∈ R d 1 n ∑ i = 1 n ( y i − ⟨ w , x i ⟩ ) 2 + ‖ w ‖ 1 , {\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-\langle w,x_{i}\rangle )^{2}+\|w\|_{1},} we consider our objective function in two parts: a convex, differentiable term F ( w ) = 1 n ∑ i = 1 n ( y i − ⟨ w , x i ⟩ ) 2 {\displaystyle F(w)={\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-\langle w,x_{i}\rangle )^{2}} and a convex function R ( w ) = ‖ w ‖ 1 {\displaystyle R(w)=\|w\|_{1}} . Note that R {\displaystyle R} is not strictly convex. Let us compute the proximity operator for R ( w ) {\displaystyle R(w)} . First we find an alternative characterization of the proximity operator prox R ( x ) {\displaystyle \operatorname {prox} _{R}(x)} as follows: u = prox R ( x ) ⟺ 0 ∈ ∂ ( R ( u ) + 1 2 ‖ u − x ‖ 2 2 ) ⟺ 0 ∈ ∂ R ( u ) + u − x ⟺ x − u ∈ ∂ R ( u ) . {\displaystyle {\begin{aligned}u=\operatorname {prox} _{R}(x)\iff &0\in \partial \left(R(u)+{\frac {1}{2}}\|u-x\|_{2}^{2}\right)\\\iff &0\in \partial R(u)+u-x\\\iff &x-u\in \partial R(u).\end{aligned}}} For R ( w ) = ‖ w ‖ 1 {\displaystyle R(w)=\|w\|_{1}} it is easy to compute ∂ R ( w ) {\displaystyle \partial R(w)} : the i {\displaystyle i} th entry of ∂ R ( w ) {\displaystyle \partial R(w)} is precisely ∂ | w i | = { 1 , w i > 0 − 1 , w i < 0 [ − 1 , 1 ] , w i = 0. {\displaystyle \partial |w_{i}|={\begin{cases}1,&w_{i}>0\\-1,&w_{i}<0\\\left[-1,1\right],&w_{i}=0.\end{cases}}} Using the recharacterization of the proximity operator given above, for the choice of R ( w ) = ‖ w ‖ 1 {\displaystyle R(w)=\|w\|_{1}} and γ > 0 {\displaystyle \gamma >0} we have that prox γ R ( x ) {\displaystyle \operatorname {prox} _{\gamma R}(x)} is defined entrywise by ( prox γ R ( x ) ) i = { x i − γ , x i > γ 0 , | x i | ≤ γ x i + γ , x i < − γ , {\displaystyle \left(\operatorname {prox} _{\gamma R}(x)\right)_{i}={\begin{cases}x_{i}-\gamma ,&x_{i}>\gamma \\0,&|x_{i}|\leq \gamma \\x_{i}+\gamma ,&x_{i}<-\gamma ,\end{cases}}} which is known as the soft thresholding operator S γ ( x ) = prox γ ‖ ⋅ ‖ 1 ( x ) {\displaystyle S_{\gamma }(x)=\operatorname {prox} _{\gamma \|\cdot \|_{1}}(x)} . === Fixed point iterative schemes === To finally solve the lasso problem we consider the fixed point equation shown earlier: x ∗ = prox γ R ( x ∗ − γ ∇ F ( x ∗ ) ) . {\displaystyle x^{}=\operatorname {prox} _{\gamma R}\left(x^{}-\gamma \nabla F(x^{})\right).} Given that we have computed the form of the proximity operator explicitly, then we can define a standard fixed point iteration procedure. Namely, fix some initial w 0 ∈ R d {\displaystyle w^{0}\in \mathbb {R} ^{d}} , and for k = 1 , 2 , … {\displaystyle k=1,2,\ldots } define w k + 1 = S γ ( w k − γ ∇ F ( w k ) ) . {\displaystyle w^{k+1}=S_{\gamma }\left(w^{k}-\gamma \nabla F\l
Robert Abel and Associates
Robert Abel and Associates (RA&A) was an American pioneering animation production company specializing in television commercials made with computer graphics. Founded by Robert Abel and Con Pederson in 1971, RA&A was especially known for their art direction and won many Clio Awards. Abel and his team created some of the most advanced and impressive computer-animated works of their time, including full ray-traced renders and fluid character animation at a time when such things were largely unknown. A variety of high-profile television advertisements, graphics sequences for motion pictures (including The Andromeda Strain and Tron), and work on laserdisc video games such as Cube Quest, put Abel and his team on the map in the early 1980s. The company was also originally commissioned to create the visual effects for Star Trek: The Motion Picture, but were subsequently taken off the project for mishandling funds. The company was also notable on its work for The Jacksons' 1981 music video "Can You Feel It." RA&A was on the southwest corner of Highland Avenue and Romaine in the heart of Hollywood, California. RA&A closed in 1987 following an ill-fated merger with now-defunct Omnibus Computer Graphics, Inc., a company which had been based in Toronto. Many people who worked at RA&A went on to other ground-breaking projects, including the founding of Wavefront Technologies, Rhythm & Hues and other studios. Many RA&A people went on to win Academy Awards.