Coreu

COREU (French: Correspondance Européenne – Telex network of European correspondents, also EUKOR-Netzwerk in Austria) is a communication network of the European Union for the communication of the Council of the European Union, the European correspondents of the foreign ministries of the EU member states, permanent representatives of member states in Brussels, the European Commission, and the General Secretariat of the Council of the European Union. The European Parliament is not among the participants. COREU is the European equivalent of the American Secret Internet Protocol Router Network (SIPRNet, also known as Intelink-S). COREU's official aim is fast communication in case of crisis. The network enables a closer cooperation in matters regarding foreign affairs. In actuality the system's function exceeds that of mere communication, it also enables decision-making. COREU's first goal is to enable the exchange of information before and after decisions. Relaying upfront negotiations in preparation of meetings is the second goal. In addition, the system also allows the editing of documents and the decision-making, especially if there is little time. While the first two goals are preparatory measures for a shared foreign policy, the third is a methodical variant marked by practise that is defining for the image of the Common Foreign and Security Policy. == Members == (The following information dates from 2013): There is one representative in each of the capital cities in the EU.(since 1973) In Germany for example, this is the European correspondent (EU-KOR) from the Foreign Office. In Austria it is the European correspondent from the Referat II.1.a in the Federal Ministry for Europe, Integration and Foreign Affairs They are the correspondents (since 1982) for the European Commission They comprise the secretariat for the European Council They also make up the European External Action Service (EEAS) (responsible for foreign policy issues, since 1987) == Data volume and technical details == COREU functions as a spoke-hub distribution paradigm system with the hub in Brussels. The network is operated by the European Union Intelligence and Situation Centre (formerly Joint Situation Center, JSC). The technical infrastructure is located in a building of the European Council. COREU may be described as an advanced telex system with encrypted messages via dedicated terminals. Once a message has reached the destination, it is then redistributed via the local media. In contrast, messages of governments are transmitted via local media to the correspondents and from there delivered point-to-point to Brussels via COREU. In 2010, approximately 8500 communications had been distributed over this network. == History == A telex-based communication system under the name COREU was established in 1973. Originally, only the ministries of Foreign Affairs in the European capitals were connected to it. This telex system was replaced in 1997 by the mail system CORTESY (COREU Terminal Equipment System). The name was retained despite the technical innovation. COREU was reportedly compromised by hackers working for the People's Liberation Army Strategic Support Force, allowing for the theft of thousands of low-classified documents and diplomatic cables.

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

Qloo

Qloo (pronounced "clue") is a company that uses artificial intelligence (AI) to understand taste and cultural correlations. It provides companies with an application programming interface (API). It received funding from Leonardo DiCaprio, Elton John, Barry Sternlicht, Pierre Lagrange and others. Qloo establishes consumer preference correlations via machine learning across data spanning cultural domains including music, film, television, dining, nightlife, fashion, books, and travel. The recommender system uses AI to predict correlations for further applications. == History == Qloo was founded in 2012 by chief executive officer Alex Elias and chief operating officer Jay Alger. Qloo initially launched an app designed for consumers, allowing them to understand their own tastes and receive personalized recommendations. The company amassed several million users and built a large catalog of cultural entities and corresponding user sentiment. In 2012, Qloo raised $1.4 million in seed funding from investors including Cedric the Entertainer, and venture capital firm Kindler Capital. Qloo had a public beta release in November 2012 after its initial funding. In 2013, the company raised an additional $1.6 million from Cross Creek Pictures founding partner Tommy Thompson, and Samih Toukan and Hussam Khoury, founders of Maktoob, an Internet services company purchased by Yahoo! for $164 million in 2009. On November 14, 2013, a website and an iPhone app were announced. The company later released an Android app, and tablet versions, in mid-2014. In 2015, Twitter approached Qloo about powering personalized social feeds and targeted eCommerce ads on the platform based on what users were posting. Qloo developed an enterprise-grade API to support Twitter’s needs. Twitter ended up pivoting to enable brands to use the social platform for customer service and support, but Qloo was able to sell access to its cultural intelligence via API to many other enterprise clients, marking the official transition from a B2C company to a B2B company. In 2016, Qloo secured $4.5 million in venture capital investment. The $4.5 million was split between a number of investors, including Barry Sternlicht, Pierre Lagrange, and Leonardo DiCaprio. In July 2017, Qloo raised $6.5 million in funding rounds from AXA Strategic Ventures, and Elton John. Following the investment, the founders stated in an interview with Tech Crunch that they would use the investment to expand Qloo's database. They hoped the move would secure larger contracts with corporate clients. At the time, clients already included Fortune 500 companies such as Twitter, PepsiCo, and BMW. In 2019, the company announced that it had acquired cultural recommendation service TasteDive, with Alex Elias becoming chairman of TasteDive. In September 2019, Qloo was named among the Top 14 Artificial Intelligence APIs by ProgrammableWeb. In 2022, Qloo raised $15M in Series B funding from Eldridge and AXA Venture Partners, enabling the privacy-centric AI leader to expand its team of world-class data scientists, enrich its technology, and build on its sales channels in order to continue to offer premier insights into global consumer taste for Fortune 500 companies across the globe. Qloo was recognized as the "Best Decision Intelligence Company" at the 2023 AI Breakthrough Awards. Also in 2023, the company was awarded a Top Performer Award by SourceForge. As of 2024, Qloo is a three-time Inc. 5000 honoree: No. 360 (2022), No. 344 (2021), No. 187 (2020). Qloo raised $25 million Series C round on February 21, 2024. The round was led by AI Ventures with participation from AXA Venture Partners, Eldridge, and Moderne Ventures, allowing Qloo to address new commercial surface areas for Taste AI, including on-device learning and foundational models leveraging Qloo, as well as introduce self-service platform to make consumer and taste analytics available to small and mid-sized enterprises and individuals. Qloo also announced pursuing opportunistic M&A using its balance sheet along the lines of the TasteDive acquisition completed, which expanded Qloo's first-party data moat and corpus of cultural learning. This latest financing brought the total amount raised since the company's founding in 2012 to over $56 million. == Services and features == Qloo calls itself a cultural AI platform to provide real-time correlation data across domains of culture and entertainment including: film, music, television, dining, nightlife, fashion, books, and travel. Each category contains subcategories. Qloo’s knowledge of a user's taste in one category can be utilized to offer suggestions in other categories. Users then rate the suggestions, providing it with feedback for future suggestions. Qloo has partnerships with companies such as Expedia and iTunes. == Technology == Qloo’s Taste AI technology uses machine learning to decode and predict consumers’ interests, maintaining user anonymity. It is powered by 3.7 billion lifestyle entities (brands, music, film, TV, dining, nightlife, fashion, books, travel, and more) and trillions of anonymized consumer behavioral signals. Through AI, Qloo identifies patterns in these data signals, making predictions about how much interest a person or group has in a concept or thing. Central to Qloo’s technology are algorithms designed to detect and mitigate biases within datasets and models, allowing Qloo to assess the fairness of its AI systems with a focus on attributes such as age, gender, and race, enabling the company to fine-tune its AI models to align with their ethical standards. They also use visualization tools to probe the behavior of their AI models for conducting counterfactual analyses and for comparing the performances of the AI models across diverse demographic segments. Qloo’s Taste AI doesn’t collect or use any Personally Identifiable Information (PII). Instead, it derives recommendations for audience segments based on co-occurrences between lifestyle entities and anonymized behavioral signals. == Applications == Starbucks uses Qloo to create in-store music playlists tailored to specific neighborhoods. Hershey’s uses Qloo to customize the content of assorted candy bags. Michelin uses Qloo to serve recommendations in its Michelin Guide app. Netflix leverages Qloo’s technology to enhance merchandising by identifying actors who resonate with certain demographics. Qloo also works with PepsiCo, Samsung, The New York Mets, BuzzFeed, and Ticketmaster, Universal Music Group, and OOH advertising company JCDecaux.

Smart speaker industry in South Korea

Smart speakers, or AI speakers, have been developed by multiple domestic electronics and telecommunications firms in South Korea. Since their introduction to the local market in 2016, they have been used by millions of people in the country. == Brands == === Google === In September 2018, Google Home (including the Google Home Mini) launched in South Korea. Running Google Assistant, it featured simultaneous recognition of two languages among a total of seven, including Korean. At launch, it could play music from Bugs!, in addition to YouTube. === Kakao === In November 2017, Kakao launched the Kakao Mini, featuring integrated KakaoTalk functionality. === KT === KT launched the GiGA Genie smart speaker in January 2017, using a Harman Kardon speaker. In November 2017, KT announced GiGA Genie LTE, a portable AI speaker with LTE support. They also released a mini speaker called GiGA Genie Buddy. In 2018, KT created a special version of GiGa Genie with a screen for use in hotels. On 29 April 2019, KT announced the GiGA Genie Table TV, a consumer-oriented smart speaker with a display. It featured paid TV access through Wi-Fi. Based on usage data from the hotel model, KT decided not to add a touchscreen. The Table TV also featured a limited-access "personalized-text-to-speech technology" which could use parents' voice recording inputs to read children books. In February 2022, KT began rolling out Amazon Alexa integration into its speakers for English support. === Naver === In August 2017, Naver announced the Wave smart speaker, operating on Clova. In October 2017, Naver launched the Friends smart speaker, which were designed based on Line characters. ==== LG Uplus ==== In December 2017, LG Uplus launched the Friends+ speaker with Naver, operating on U+ Home AI. === Samsung === In August 2018, Samsung announced the Samsung Galaxy Home in partnership with Spotify. The original size was delayed, while the Galaxy Home Mini appeared briefly as a bonus for Samsung Galaxy S20 preorders in South Korea in February 2020. === SK Telecom === SK Telecom launched the Nugu smart speaker in September 2016, using an Astell & Kern audio system. In August 2017, SKT released a portable speaker named Nugu mini. In July 2018, SKT launched the Nugu Candle, featuring expanded mood lighting. The first-generation Nugu was subsequently discontinued. On 18 April 2019, SKT released the NUGU Nemo AI, which featured a display and JBL stereo speaker. In August 2019, SKT collaborated with SM Entertainment, incorporating functions related to the agency's artists into Nugu. In January 2022, SKT showcased the NUGU Candle SE, introducing Alexa support. == Usage == In 2018, approximately 3 million people in South Korea used smart speakers. According to data from KT in 2018, the most common commands to its speakers were for controlling televisions. Based on a broader survey in 2017, music was selected as the most frequent use case. By 2018, smart speaker companies were partnering with reading and other education services, adding potential use-cases for children. By 2022, smart speakers were being utilized by the South Korean government. SKT, in partnership with 70 regional governments, distributed smart speakers to 12,000 senior citizens living alone. The government paid for monthly subscriptions to help seniors stay mentally engaged. Naver made an agreement with the Seoul Metropolitan Government to provide Clova CareCall, an automated health checkup program to hundreds of senior citizens living alone. KT's AI care service included an emergency dispatch call function and medication notifications. == Criticism == === Communication === In a survey of 300 users in 2017, approximately half reported having some type of communication issue with their smart speakers. === Privacy === South Korean smart speakers sparked privacy concerns when they were found to be collecting and documenting user audio data in 2019. The speaker companies responded that only a minority of data was collected and that it was anonymized. They stated that such recordings were collected for performance improvements.

Grammar systems theory

Grammar systems theory is a field of theoretical computer science that studies systems of finite collections of formal grammars generating a formal language. Each grammar works on a string, a so-called sequential form that represents an environment. Grammar systems can thus be used as a formalization of decentralized or distributed systems of agents in artificial intelligence. Let A {\displaystyle \mathbb {A} } be a simple reactive agent moving on the table and trying not to fall down from the table with two reactions, t for turning and ƒ for moving forward. The set of possible behaviors of A {\displaystyle \mathbb {A} } can then be described as formal language L A = { ( f m t n f r ) + : 1 ≤ m ≤ k ; 1 ≤ n ≤ ℓ ; 1 ≤ r ≤ k } , {\displaystyle \mathbb {L_{A}} =\{(f^{m}t^{n}f^{r})^{+}:1\leq m\leq k;1\leq n\leq \ell ;1\leq r\leq k\},} where ƒ can be done maximally k times and t can be done maximally ℓ times considering the dimensions of the table. Let G A {\displaystyle \mathbb {G_{A}} } be a formal grammar which generates language L A {\displaystyle \mathbb {L_{A}} } . The behavior of A {\displaystyle \mathbb {A} } is then described by this grammar. Suppose the A {\displaystyle \mathbb {A} } has a subsumption architecture; each component of this architecture can be then represented as a formal grammar, too, and the final behavior of the agent is then described by this system of grammars. The schema on the right describes such a system of grammars which shares a common string representing an environment. The shared sequential form is sequentially rewritten by each grammar, which can represent either a component or generally an agent. If grammars communicate together and work on a shared sequential form, it is called a Cooperating Distributed (DC) grammar system. Shared sequential form is a similar concept to the blackboard approach in AI, which is inspired by an idea of experts solving some problem together while they share their proposals and ideas on a shared blackboard. Each grammar in a grammar system can also work on its own string and communicate with other grammars in a system by sending their sequential forms on request. Such a grammar system is then called a Parallel Communicating (PC) grammar system. PC and DC are inspired by distributed AI. If there is no communication between grammars, the system is close to the decentralized approaches in AI. These kinds of grammar systems are sometimes called colonies or Eco-Grammar systems, depending (besides others) on whether the environment is changing on its own (Eco-Grammar system) or not (colonies).

Keith Youngin George II

Keith "Youngin" George II is a former mixtape DJ, music executive, manager, producer, and technology app director. He has collaborated with Maino, T-Pain, Nas and Soulja Boy, among others. He was instrumental in the launch of social media app and website, Kandiid in 2021 and served as Fliiks App Director of Regional Development. == Career == Keith Anthony George II was born in Upper Heyford, Oxfordshire, England. His father was in the Air Force which exposed him to different cultures and music. He graduated from Allen High School and attended San Antonio College. George's music career began in 2006 as a mixtape DJ working as DJ Youngin Beatz. He performed at various shows and worked with a variety of artists, managers, and music executives. In 2007, George released the mixtape, Untapped market Vol. 1 (Da Underdogz), which featured tracks from artists including Kanye West, Lil Wayne, 50 Cent, Yung Berg, and Nelly. In 2008, he began working with Def Jam executive Sarah Alminawi who was managing Maino at the time. George played a key role in the marketing and promotional success of Maino's single, Hi Hater, which peaked at #8 on Billboard's US Bubbling Under Hot 100 chart. In 2021, George was an advisor and infrastructure head at Kandiid, a social media app which won a W3 Award in 2022. In 2023, he became involved with Fliiks App as Director of Regional Development which earned a Telly Award, two Muse Awards, and a W3 Award in 2025. In 2025, George was a composer and producer on two singles on Sekou Andrews's album, Koumami; The Chosen One: ACT 1 (featuring Lion Babe) and Love Don't Care (featuring Jordin Sparks and Omari Hardwick). In 2025, he was awarded an Atlanta City Proclamation for Philanthropy and Community Leadership for his partnership with Women's International Grail, a nonprofit organization that assists women, single mothers, and low-income families. He also collaborates with local youth programs, creative networks, and minority-owned startups, providing access to mentorship and industry knowledge. == Awards ==

Matrix regularization

In the field of statistical learning theory, matrix regularization generalizes notions of vector regularization to cases where the object to be learned is a matrix. The purpose of regularization is to enforce conditions, for example sparsity or smoothness, that can produce stable predictive functions. For example, in the more common vector framework, Tikhonov regularization optimizes over min x ‖ A x − y ‖ 2 + λ ‖ x ‖ 2 {\displaystyle \min _{x}\left\|Ax-y\right\|^{2}+\lambda \left\|x\right\|^{2}} to find a vector x {\displaystyle x} that is a stable solution to the regression problem. When the system is described by a matrix rather than a vector, this problem can be written as min X ‖ A X − Y ‖ 2 + λ ‖ X ‖ 2 , {\displaystyle \min _{X}\left\|AX-Y\right\|^{2}+\lambda \left\|X\right\|^{2},} where the vector norm enforcing a regularization penalty on x {\displaystyle x} has been extended to a matrix norm on X {\displaystyle X} . Matrix regularization has applications in matrix completion, multivariate regression, and multi-task learning. Ideas of feature and group selection can also be extended to matrices, and these can be generalized to the nonparametric case of multiple kernel learning. == Basic definition == Consider a matrix W {\displaystyle W} to be learned from a set of examples, S = ( X i t , y i t ) {\displaystyle S=(X_{i}^{t},y_{i}^{t})} , where i {\displaystyle i} goes from 1 {\displaystyle 1} to n {\displaystyle n} , and t {\displaystyle t} goes from 1 {\displaystyle 1} to T {\displaystyle T} . Let each input matrix X i {\displaystyle X_{i}} be ∈ R D T {\displaystyle \in \mathbb {R} ^{DT}} , and let W {\displaystyle W} be of size D × T {\displaystyle D\times T} . A general model for the output y {\displaystyle y} can be posed as y i t = ⟨ W , X i t ⟩ F , {\displaystyle y_{i}^{t}=\left\langle W,X_{i}^{t}\right\rangle _{F},} where the inner product is the Frobenius inner product. For different applications the matrices X i {\displaystyle X_{i}} will have different forms, but for each of these the optimization problem to infer W {\displaystyle W} can be written as min W ∈ H E ( W ) + R ( W ) , {\displaystyle \min _{W\in {\mathcal {H}}}E(W)+R(W),} where E {\displaystyle E} defines the empirical error for a given W {\displaystyle W} , and R ( W ) {\displaystyle R(W)} is a matrix regularization penalty. The function R ( W ) {\displaystyle R(W)} is typically chosen to be convex and is often selected to enforce sparsity (using ℓ 1 {\displaystyle \ell ^{1}} -norms) and/or smoothness (using ℓ 2 {\displaystyle \ell ^{2}} -norms). Finally, W {\displaystyle W} is in the space of matrices H {\displaystyle {\mathcal {H}}} with Frobenius inner product ⟨ … ⟩ F {\displaystyle \langle \dots \rangle _{F}} . == General applications == === Matrix completion === In the problem of matrix completion, the matrix X i t {\displaystyle X_{i}^{t}} takes the form X i t = e t ⊗ e i ′ , {\displaystyle X_{i}^{t}=e_{t}\otimes e_{i}',} where ( e t ) t {\displaystyle (e_{t})_{t}} and ( e i ′ ) i {\displaystyle (e_{i}')_{i}} are the canonical basis in R T {\displaystyle \mathbb {R} ^{T}} and R D {\displaystyle \mathbb {R} ^{D}} . In this case the role of the Frobenius inner product is to select individual elements w i t {\displaystyle w_{i}^{t}} from the matrix W {\displaystyle W} . Thus, the output y {\displaystyle y} is a sampling of entries from the matrix W {\displaystyle W} . The problem of reconstructing W {\displaystyle W} from a small set of sampled entries is possible only under certain restrictions on the matrix, and these restrictions can be enforced by a regularization function. For example, it might be assumed that W {\displaystyle W} is low-rank, in which case the regularization penalty can take the form of a nuclear norm. R ( W ) = λ ‖ W ‖ ∗ = λ ∑ i | σ i | , {\displaystyle R(W)=\lambda \left\|W\right\|_{}=\lambda \sum _{i}\left|\sigma _{i}\right|,} where σ i {\displaystyle \sigma _{i}} , with i {\displaystyle i} from 1 {\displaystyle 1} to min D , T {\displaystyle \min D,T} , are the singular values of W {\displaystyle W} . === Multivariate regression === Models used in multivariate regression are parameterized by a matrix of coefficients. In the Frobenius inner product above, each matrix X {\displaystyle X} is X i t = e t ⊗ x i {\displaystyle X_{i}^{t}=e_{t}\otimes x_{i}} such that the output of the inner product is the dot product of one row of the input with one column of the coefficient matrix. The familiar form of such models is Y = X W + b {\displaystyle Y=XW+b} Many of the vector norms used in single variable regression can be extended to the multivariate case. One example is the squared Frobenius norm, which can be viewed as an ℓ 2 {\displaystyle \ell ^{2}} -norm acting either entrywise, or on the singular values of the matrix: R ( W ) = λ ‖ W ‖ F 2 = λ ∑ i ∑ j | w i j | 2 = λ Tr ⁡ ( W ∗ W ) = λ ∑ i σ i 2 . {\displaystyle R(W)=\lambda \left\|W\right\|_{F}^{2}=\lambda \sum _{i}\sum _{j}\left|w_{ij}\right|^{2}=\lambda \operatorname {Tr} \left(W^{}W\right)=\lambda \sum _{i}\sigma _{i}^{2}.} In the multivariate case the effect of regularizing with the Frobenius norm is the same as the vector case; very complex models will have larger norms, and, thus, will be penalized more. === Multi-task learning === The setup for multi-task learning is almost the same as the setup for multivariate regression. The primary difference is that the input variables are also indexed by task (columns of Y {\displaystyle Y} ). The representation with the Frobenius inner product is then X i t = e t ⊗ x i t . {\displaystyle X_{i}^{t}=e_{t}\otimes x_{i}^{t}.} The role of matrix regularization in this setting can be the same as in multivariate regression, but matrix norms can also be used to couple learning problems across tasks. In particular, note that for the optimization problem min W ‖ X W − Y ‖ 2 2 + λ ‖ W ‖ 2 2 {\displaystyle \min _{W}\left\|XW-Y\right\|_{2}^{2}+\lambda \left\|W\right\|_{2}^{2}} the solutions corresponding to each column of Y {\displaystyle Y} are decoupled. That is, the same solution can be found by solving the joint problem, or by solving an isolated regression problem for each column. The problems can be coupled by adding an additional regularization penalty on the covariance of solutions min W , Ω ‖ X W − Y ‖ 2 2 + λ 1 ‖ W ‖ 2 2 + λ 2 Tr ⁡ ( W T Ω − 1 W ) {\displaystyle \min _{W,\Omega }\left\|XW-Y\right\|_{2}^{2}+\lambda _{1}\left\|W\right\|_{2}^{2}+\lambda _{2}\operatorname {Tr} \left(W^{T}\Omega ^{-1}W\right)} where Ω {\displaystyle \Omega } models the relationship between tasks. This scheme can be used to both enforce similarity of solutions across tasks, and to learn the specific structure of task similarity by alternating between optimizations of W {\displaystyle W} and Ω {\displaystyle \Omega } . When the relationship between tasks is known to lie on a graph, the Laplacian matrix of the graph can be used to couple the learning problems. == Spectral regularization == Regularization by spectral filtering has been used to find stable solutions to problems such as those discussed above by addressing ill-posed matrix inversions (see for example Filter function for Tikhonov regularization). In many cases the regularization function acts on the input (or kernel) to ensure a bounded inverse by eliminating small singular values, but it can also be useful to have spectral norms that act on the matrix that is to be learned. There are a number of matrix norms that act on the singular values of the matrix. Frequently used examples include the Schatten p-norms, with p = 1 or 2. For example, matrix regularization with a Schatten 1-norm, also called the nuclear norm, can be used to enforce sparsity in the spectrum of a matrix. This has been used in the context of matrix completion when the matrix in question is believed to have a restricted rank. In this case the optimization problem becomes: min ‖ W ‖ ∗ subject to W i , j = Y i j . {\displaystyle \min \left\|W\right\|_{}~~{\text{ subject to }}~~W_{i,j}=Y_{ij}.} Spectral Regularization is also used to enforce a reduced rank coefficient matrix in multivariate regression. In this setting, a reduced rank coefficient matrix can be found by keeping just the top n {\displaystyle n} singular values, but this can be extended to keep any reduced set of singular values and vectors. == Structured sparsity == Sparse optimization has become the focus of much research interest as a way to find solutions that depend on a small number of variables (see e.g. the Lasso method). In principle, entry-wise sparsity can be enforced by penalizing the entry-wise ℓ 0 {\displaystyle \ell ^{0}} -norm of the matrix, but the ℓ 0 {\displaystyle \ell ^{0}} -norm is not convex. In practice this can be implemented by convex relaxation to the ℓ 1 {\displaystyle \ell ^{1}} -norm. While entry-wise regularization with an ℓ 1 {\displaystyle \ell ^{1}} -norm will find solutions with a small number of nonzero elements, applying an ℓ 1 {