FastICA

FastICA is an efficient and popular algorithm for independent component analysis invented by Aapo Hyvärinen at Helsinki University of Technology. Like most ICA algorithms, FastICA seeks an orthogonal rotation of prewhitened data, through a fixed-point iteration scheme, that maximizes a measure of non-Gaussianity of the rotated components. Non-gaussianity serves as a proxy for statistical independence, which is a very strong condition and requires infinite data to verify. FastICA can also be alternatively derived as an approximative Newton iteration. == Algorithm == === Prewhitening the data === Let the X := ( x i j ) ∈ R N × M {\displaystyle \mathbf {X} :=(x_{ij})\in \mathbb {R} ^{N\times M}} denote the input data matrix, M {\displaystyle M} the number of columns corresponding with the number of samples of mixed signals and N {\displaystyle N} the number of rows corresponding with the number of independent source signals. The input data matrix X {\displaystyle \mathbf {X} } must be prewhitened, or centered and whitened, before applying the FastICA algorithm to it. Centering the data entails demeaning each component of the input data X {\displaystyle \mathbf {X} } , that is, for each i = 1 , … , N {\displaystyle i=1,\ldots ,N} and j = 1 , … , M {\displaystyle j=1,\ldots ,M} . After centering, each row of X {\displaystyle \mathbf {X} } has an expected value of 0 {\displaystyle 0} . Whitening the data requires a linear transformation L : R N × M → R N × M {\displaystyle \mathbf {L} :\mathbb {R} ^{N\times M}\to \mathbb {R} ^{N\times M}} of the centered data so that the components of L ( X ) {\displaystyle \mathbf {L} (\mathbf {X} )} are uncorrelated and have variance one. More precisely, if X {\displaystyle \mathbf {X} } is a centered data matrix, the covariance of L x := L ( X ) {\displaystyle \mathbf {L} _{\mathbf {x} }:=\mathbf {L} (\mathbf {X} )} is the ( N × N ) {\displaystyle (N\times N)} -dimensional identity matrix, that is, A common method for whitening is by performing an eigenvalue decomposition on the covariance matrix of the centered data X {\displaystyle \mathbf {X} } , E { X X T } = E D E T {\displaystyle E\left\{\mathbf {X} \mathbf {X} ^{T}\right\}=\mathbf {E} \mathbf {D} \mathbf {E} ^{T}} , where E {\displaystyle \mathbf {E} } is the matrix of eigenvectors and D {\displaystyle \mathbf {D} } is the diagonal matrix of eigenvalues. The whitened data matrix is defined thus by === Single component extraction === The iterative algorithm finds the direction for the weight vector w ∈ R N {\displaystyle \mathbf {w} \in \mathbb {R} ^{N}} that maximizes a measure of non-Gaussianity of the projection w T X {\displaystyle \mathbf {w} ^{T}\mathbf {X} } , with X ∈ R N × M {\displaystyle \mathbf {X} \in \mathbb {R} ^{N\times M}} denoting a prewhitened data matrix as described above. Note that w {\displaystyle \mathbf {w} } is a column vector. To measure non-Gaussianity, FastICA relies on a nonquadratic nonlinear function f ( u ) {\displaystyle f(u)} , its first derivative g ( u ) {\displaystyle g(u)} , and its second derivative g ′ ( u ) {\displaystyle g^{\prime }(u)} . Hyvärinen states that the functions are useful for general purposes, while may be highly robust. The steps for extracting the weight vector w {\displaystyle \mathbf {w} } for single component in FastICA are the following: Randomize the initial weight vector w {\displaystyle \mathbf {w} } Let w + ← E { X g ( w T X ) T } − E { g ′ ( w T X ) } w {\displaystyle \mathbf {w} ^{+}\leftarrow E\left\{\mathbf {X} g(\mathbf {w} ^{T}\mathbf {X} )^{T}\right\}-E\left\{g'(\mathbf {w} ^{T}\mathbf {X} )\right\}\mathbf {w} } , where E { . . . } {\displaystyle E\left\{...\right\}} means averaging over all column-vectors of matrix X {\displaystyle \mathbf {X} } Let w ← w + / ‖ w + ‖ {\displaystyle \mathbf {w} \leftarrow \mathbf {w} ^{+}/\|\mathbf {w} ^{+}\|} If not converged, go back to 2 === Multiple component extraction === The single unit iterative algorithm estimates only one weight vector which extracts a single component. Estimating additional components that are mutually "independent" requires repeating the algorithm to obtain linearly independent projection vectors - note that the notion of independence here refers to maximizing non-Gaussianity in the estimated components. Hyvärinen provides several ways of extracting multiple components with the simplest being the following. Here, 1 M {\displaystyle \mathbf {1_{M}} } is a column vector of 1's of dimension M {\displaystyle M} . Algorithm FastICA Input: C {\displaystyle C} Number of desired components Input: X ∈ R N × M {\displaystyle \mathbf {X} \in \mathbb {R} ^{N\times M}} Prewhitened matrix, where each column represents an N {\displaystyle N} -dimensional sample, where C <= N {\displaystyle C<=N} Output: W ∈ R N × C {\displaystyle \mathbf {W} \in \mathbb {R} ^{N\times C}} Un-mixing matrix where each column projects X {\displaystyle \mathbf {X} } onto independent component. Output: S ∈ R C × M {\displaystyle \mathbf {S} \in \mathbb {R} ^{C\times M}} Independent components matrix, with M {\displaystyle M} columns representing a sample with C {\displaystyle C} dimensions. for p in 1 to C: w p ← {\displaystyle \mathbf {w_{p}} \leftarrow } Random vector of length N while w p {\displaystyle \mathbf {w_{p}} } changes w p ← 1 M X g ( w p T X ) T − 1 M g ′ ( w p T X ) 1 M w p {\displaystyle \mathbf {w_{p}} \leftarrow {\frac {1}{M}}\mathbf {X} g(\mathbf {w_{p}} ^{T}\mathbf {X} )^{T}-{\frac {1}{M}}g'(\mathbf {w_{p}} ^{T}\mathbf {X} )\mathbf {1_{M}} \mathbf {w_{p}} } w p ← w p − ∑ j = 1 p − 1 ( w p T w j ) w j {\displaystyle \mathbf {w_{p}} \leftarrow \mathbf {w_{p}} -\sum _{j=1}^{p-1}(\mathbf {w_{p}} ^{T}\mathbf {w_{j}} )\mathbf {w_{j}} } w p ← w p ‖ w p ‖ {\displaystyle \mathbf {w_{p}} \leftarrow {\frac {\mathbf {w_{p}} }{\|\mathbf {w_{p}} \|}}} output W ← [ w 1 , … , w C ] {\displaystyle \mathbf {W} \leftarrow {\begin{bmatrix}\mathbf {w_{1}} ,\dots ,\mathbf {w_{C}} \end{bmatrix}}} output S ← W T X {\displaystyle \mathbf {S} \leftarrow \mathbf {W^{T}} \mathbf {X} }

Elastic cloud storage

An elastic cloud is a cloud computing offering that provides variable service levels based on changing needs. Elasticity is an attribute that can be applied to most cloud services. It states that the capacity and performance of any given cloud service can expand or contract according to a customer's requirements and that this can potentially be changed automatically as a consequence of some software-driven event or, at worst, can be reconfigured quickly by the customer's infrastructure management team. Elasticity has been described as one of the five main principles of cloud computing by Rosenburg and Mateos in The Cloud at Your Service - Manning 2011. == History == Cloud computing was first described by Gillet and Kapor in 1996; however, the first practical implementation was a consequence of a strategy to leverage Amazon's excess data center capacity. Amazon and other pioneers of the commercial use of this technology were primarily interested in providing a “public” cloud service, whereby they could offer customers the benefits of using the cloud, particularly the utility-based pricing model benefit. Other suppliers followed suit with a range of cloud-based models all offering elasticity as a core component, but these suppliers were only offering this service as an element of their public cloud service. Due to perceived weaknesses in security, or at least a lack of proven compliance, many organizations, particularly in the financial and public sectors, have been slow adopters of cloud technologies. These wary organizations can achieve some of the benefits of cloud computing by adopting private cloud technologies. An alternative form of the elastic cloud has been offered by vendors such as EMC and IBM, whereby the service is based around an enterprise's own infrastructure but still retains elements of elasticity and the potential to bill by consumption. == Description == Elasticity in cloud computing is the ability for the organization to adjust its storage requirements in terms of capacity and processing with respect to operational requirements. This has the following benefits: Operational Benefits - Services can be acquired quickly, meaning that the evolving requirements of the business can be addressed almost immediately, giving an organization a potential agility advantage. A properly implemented elastic system will provision/de-provision according to application demands, so if a particular business has activity spikes then the provision can be enabled to match the demand and the capacity can be re-allocated. Research and Development (R&D) Projects - R&D activities are no longer hindered by a requirement to secure a capex budget prior to a project starting. Capability can simply be provisioned from the cloud and released at the end of the exercise. Testing and Deployment - With most large-scale projects a size test needs to be performed prior to final rollout. By taking advantage of the elasticity of the cloud and creating a full-scale avatar of the proposed production system, realistic data and traffic volumes can be provisioned and released as needed. Expensive Resources Allocated - This will normally apply only in the context where a customer is applying at least some of their own servers as part of a cloud infrastructure, specifically where a business (for performance reasons) has decided to invest in solid-state storage as opposed to spinning platters. There are instances when, due to activity spikes, a less critical process may need to be moved from the high-performance resources to more traditional storage. Server Specification - When a customer has elected to own/lease hardware, they can select and specify servers that are specifically tuned to meet the likely needs of their operation (i.e., directly controlling the cost/benefit equation). Utility Based Payments - There is, of course, a key cost driver in this process, and the notion that you should pay for what you consume is acceptable for many organizations. When hardware capacity is sourced internally, organizations need to over-provision. This applies just as much to traditional outsourcing as it does to capex-related expenditure on in-house servers. Cloud Platform – At the heart of any cloud storage system is the ability to manage hyperscale object storage and a Hadoop Distributed Files System (HDFS). Elastic storage capability is particularly well suited to hyperscale and Hadoop environments, where its capability to rapidly respond to changing circumstances and priorities is essential

Harvest now, decrypt later

Harvest now, decrypt later (HNDL) is a surveillance strategy that relies on the acquisition and long-term storage of currently unreadable encrypted data awaiting possible breakthroughs in decryption technology that would render it readable in the future—a hypothetical date referred to as Y2Q (a reference to Y2K), or Q-Day. The most common concern is the prospect of developments in quantum computing which would allow current strong encryption algorithms to be broken at some time in the future, making it possible to decrypt any stored material that had been encrypted using those algorithms. However, the improvement in decryption technology need not be due to a quantum-cryptographic advance; any other form of attack capable of enabling decryption would be sufficient. The existence of this strategy has led to concerns about the need to urgently deploy post-quantum cryptography; even though no practical quantum attacks yet exist, some data stored now may still remain sensitive even decades into the future. As of 2022, the U.S. federal government has proposed a roadmap for organizations to start migrating toward quantum-cryptography-resistant algorithms to mitigate these threats. This new version of Commercial National Security Algorithm Suite uses publicly-available algorithms and is allowed for government use up to the TOP SECRET level. == Terminology and scope == The term “harvest now, decrypt later” encompasses various surveillance or espionage operations in which ciphertext or encrypted communications are collected today with the view that they may one day be decrypted, given sufficient advances in computing power or cryptanalysis. The abbreviation HNDL is sometimes used in technical and policy documents. The “Y2Q” (or “Q-Day”) label draws an analogy to the Y2K date-change issue, emphasising a potential future point at which current cryptography may collapse. The strategy is particularly relevant for data with long confidentiality lifetimes, such as diplomatic communications, personal health records, critical infrastructure logs, or intellectual property. == Mitigation strategies == The primary defense against HNDL attacks is the transition to post-quantum cryptography (PQC), which utilizes algorithms believed to be secure against quantum computer attacks. However, because PQC protects the data payload digitally, rather than the transmission itself, the encrypted data can still be harvested and stored. A complementary approach involves physical layer security (also known as optical layer encryption or photonic shielding). Unlike algorithmic encryption, this method modifies the optical waveform itself—often by burying the signal within optical noise or using spectral phase encoding—to render the transmission unrecordable by standard receivers. By preventing the attacker from capturing a valid signal in the first place, this approach aims to eliminate the "harvest" phase of the threat. Commercial implementations of harvest-proof optical encryption have been developed by firms such as CyberRidge to secure long-haul fiber networks. Field trials have demonstrated 100 Gbps throughput over legacy DWDM networks using this method.

Social television

Social television is the union of television and social media. Millions of people now share their TV experience with other viewers on social media such as Twitter and Facebook using smartphones and tablets. TV networks and rights holders are increasingly sharing video clips on social platforms to monetise engagement and drive tune-in. The social TV market covers the technologies that support communication and social interaction around TV as well as companies that study television-related social behavior and measure social media activities tied to specific TV broadcasts – many of which have attracted significant investment from established media and technology companies. The market is also seeing numerous tie-ups between broadcasters and social networking players such as Twitter and Facebook. The market is expected to be worth $256bn by 2017. Social TV was named one of the 10 most important emerging technologies by the MIT Technology Review on Social TV in 2010. And in 2011, David Rowan, the editor of Wired magazine, named Social TV at number three of six in his peek into 2011 and what tech trends to expect to get traction. Ynon Kreiz, CEO of the Endemol Group told the audience at the Digital Life Design (DLD) conference in January 2011: "Everyone says that social television will be big. I think it's not going to be big—it's going to be huge". Much of the investment in the earlier years of social TV went into standalone social TV apps. The industry believed these apps would provide an appealing and complimentary consumer experience which could then be monetized with ads. These apps featured TV listings, check-ins, stickers and synchronised second-screen content but struggled to attract users away from Twitter and Facebook. Most of these companies have since gone out of business or been acquired amid a wave of consolidation and the market has instead focused on the activities of the social media channels themselves – such as Twitter Amplify, Facebook Suggested Videos and Snapchat Discover – and the technologies that support them. == Twitter == Twitter and Facebook are both helping users connect around media, which can provoke strong debate and engagement. Both social platforms want to be the 'digital watercooler' and host conversation around TV because the engagement and data about what media people consume can then be used to generate advertising revenue. As an open platform, conversation on Twitter is closely aligned with real-time events. In May 2013, it launched Twitter Amplify – an advertising product for media and consumer brands. With Amplify, Twitter runs video highlights from major live broadcasts, with advertisers' names and messages playing before the clip. By February 2014, all four major U.S. TV networks had signed up to the Amplify program, bringing a variety of premium TV content onto the social platform in the form of in-tweet real-time video clips. In June 2014, Twitter acquired its Twitter Amplify partner in the U.S. SnappyTV, a company that was helping broadcasters and rights holders to share video content both organically across social and via Twitter's Amplify program. Twitter continues to rely on Grabyo, which has also struck numerous deals with some of the largest broadcasters and rights holders in Europe and North America to share video content across Facebook and Twitter. == Facebook == Facebook made significant changes to its platform in 2014 including updates to its algorithm to enhance how it serves video in users' feeds. It also launched video autoplay to get users to watch the videos in their feeds. It rapidly surpassed Twitter and by the end of 2014 it was enjoying three billion video views a day on its platform and had announced a partnership with the NFL, one of Twitter's most active Twitter Amplify partners. In April 2015, at its F8 Developer Conference, it revealed it was working with Grabyo among other technology partners to bring video onto its platform. Then in July it announced it would be launching Facebook Suggested Videos, bringing related videos and ads to anyone that clicks on a video – a move that not only competed with Twitter's commercial video offering but also put it in direct competition with YouTube. == TV Time == TV Time is a television dedicated social network that allows users to keep track of the television series they watch, as well as films. It also allows them to express their reaction to the media they have seen with episode specific voting for favorite characters and emotional reaction to episodes, as well as commenting in episode restrictive pages. This way users are able to avoid spoilers while also finding a precise audience and community for each of their interactions, as opposed to bigger, non-television dedicated social medias such as Facebook and Twitter where the likelihood of unintentionally reading spoilers is much higher. TV Time offers an analytics service called "TVLytics" where the votes and reactions collected from users can be studied for research and television production purposes. == Advertising == According to Businessinsider.com, there are variety of applications for social TV, including support for TV ad sales, optimizing TV ad buys, making ad buys more efficient, as a complement to audience measurement, and eventually, audience forecasting and real-time optimization. Social TV data can ease access to focus groups and may create a positive feedback loop for generating ultra-sticky TV programming and multi-screen ad campaigns. == In numbers == Viewers share their TV experience on social media in real-time as events unfold: between 88-100m Facebook users login to the platform during the primetime hours of 8pm – 11pm in the US. The volume of social media engagement in TV is also rising – according to Nielsen SocialGuide, there was a 38% increase in tweets about TV in 2013 to 263m. For the 2014 Super Bowl, Twitter reported that a record 24.9 million tweets about the game were sent during the telecast, peaking at 381,605 tweets per minute. Facebook reported that 50 million people discussed the Super Bowl, generating 185 million interactions. The 2014 Oscars generated 5m tweets, viewed by an audience of 37m unique Twitter users and delivering 3.3bn impressions globally as conversation and key moments were shared virally across the platform. In 2014 the All England Lawn Tennis Club (AELTC), hosts of Wimbledon, used Grabyo to share video content across social. The videos were viewed 3.5 million times across Facebook and Twitter. In partnered with Grabyo again in 2015 and the videos generated over 48 million views across Facebook and Twitter. == Television shows with social integration == Here are some examples of how TV executives are integrating social elements with TV shows: C-SPAN streamed tweets from US Senators and Representatives during the quorum call The Voice had the judges of the program tweet during the show and the posts scrolls on the bottom of the screen. The use of Twitter also led to an increase in viewers. "Glee" Entertainment Weekly created a second screen viewing platform for the Glee season 3 premiere. == Related publications == Erika Jonietz. "Making TV Social, Virtually" MIT Technology Review. (January 11, 2010) AmigoTV (Alcatel-Lucent; Coppens et al.) – 2004 www.ist-ipmedianet.org/Alcatel_EuroiTV2004_AmigoTV_short_paper_S4-2.pdf Nextream (MIT Media Lab, Martin et al.) – 2010 Social Interactive Television: Immersive Shared Experiences and Perspectives (P. Cesar, D. Geerts, and K. Chorianopoulos (eds.)) – 2009 Social TV and the Emergence of Interactive TV – Multimedia Research Group – November 2010 Interactive Social TV on Service Oriented Environments: Challenges and Enablers (May 2011) == Systems == Boxee – acquired by Samsung GetGlue – acquired by i.TV Grabyo KIT digital Miso TV Tank Top TV WiO Xbox Live

Social network hosting service

A social network hosting service is a web hosting service that specifically hosts the user creation of web-based social networking services, alongside related applications. Such services are also known as vertical social networks due to the creation of SNSes which cater to specific user interests and niches; like larger, interest-agnostic SNSes, such niche networking services may also possess the ability to create increasingly niche groups of users. == List of social network hosting services == Federated Media Publishing's BigTent BroadVision Clearvale Ning Wall.fm

Stability (learning theory)

Stability, also known as algorithmic stability, is a notion in computational learning theory of how a machine learning algorithm output is changed with small perturbations to its inputs. A stable learning algorithm is one for which the prediction does not change much when the training data is modified slightly. For instance, consider a machine learning algorithm that is being trained to recognize handwritten letters of the alphabet, using 1000 examples of handwritten letters and their labels ("A" to "Z") as a training set. One way to modify this training set is to leave out an example, so that only 999 examples of handwritten letters and their labels are available. A stable learning algorithm would produce a similar classifier with both the 1000-element and 999-element training sets. Stability can be studied for many types of learning problems, from language learning to inverse problems in physics and engineering, as it is a property of the learning process rather than the type of information being learned. The study of stability gained importance in computational learning theory in the 2000s when it was shown to have a connection with generalization. It was shown that for large classes of learning algorithms, notably empirical risk minimization algorithms, certain types of stability ensure good generalization. == History == A central goal in designing a machine learning system is to guarantee that the learning algorithm will generalize, or perform accurately on new examples after being trained on a finite number of them. In the 1990s, milestones were reached in obtaining generalization bounds for supervised learning algorithms. The technique historically used to prove generalization was to show that an algorithm was consistent, using the uniform convergence properties of empirical quantities to their means. This technique was used to obtain generalization bounds for the large class of empirical risk minimization (ERM) algorithms. An ERM algorithm is one that selects a solution from a hypothesis space H {\displaystyle H} in such a way to minimize the empirical error on a training set S {\displaystyle S} . A general result, proved by Vladimir Vapnik for an ERM binary classification algorithms, is that for any target function and input distribution, any hypothesis space H {\displaystyle H} with VC-dimension d {\displaystyle d} , and n {\displaystyle n} training examples, the algorithm is consistent and will produce a training error that is at most O ( d n ) {\displaystyle O\left({\sqrt {\frac {d}{n}}}\right)} (plus logarithmic factors) from the true error. The result was later extended to almost-ERM algorithms with function classes that do not have unique minimizers. Vapnik's work, using what became known as VC theory, established a relationship between generalization of a learning algorithm and properties of the hypothesis space H {\displaystyle H} of functions being learned. However, these results could not be applied to algorithms with hypothesis spaces of unbounded VC-dimension. Put another way, these results could not be applied when the information being learned had a complexity that was too large to measure. Some of the simplest machine learning algorithms—for instance, for regression—have hypothesis spaces with unbounded VC-dimension. Another example is language learning algorithms that can produce sentences of arbitrary length. Stability analysis was developed in the 2000s for computational learning theory and is an alternative method for obtaining generalization bounds. The stability of an algorithm is a property of the learning process, rather than a direct property of the hypothesis space H {\displaystyle H} , and it can be assessed in algorithms that have hypothesis spaces with unbounded or undefined VC-dimension such as nearest neighbor. A stable learning algorithm is one for which the learned function does not change much when the training set is slightly modified, for instance by leaving out an example. A measure of Leave one out error is used in a Cross Validation Leave One Out (CVloo) algorithm to evaluate a learning algorithm's stability with respect to the loss function. As such, stability analysis is the application of sensitivity analysis to machine learning. == Summary of classic results == Early 1900s - Stability in learning theory was earliest described in terms of continuity of the learning map L {\displaystyle L} , traced to Andrey Nikolayevich Tikhonov. 1979 - Devroye and Wagner observed that the leave-one-out behavior of an algorithm is related to its sensitivity to small changes in the sample. 1999 - Kearns and Ron discovered a connection between finite VC-dimension and stability. 2002 - In a landmark paper, Bousquet and Elisseeff proposed the notion of uniform hypothesis stability of a learning algorithm and showed that it implies low generalization error. Uniform hypothesis stability, however, is a strong condition that does not apply to large classes of algorithms, including ERM algorithms with a hypothesis space of only two functions. 2002 - Kutin and Niyogi extended Bousquet and Elisseeff's results by providing generalization bounds for several weaker forms of stability which they called almost-everywhere stability. Furthermore, they took an initial step in establishing the relationship between stability and consistency in ERM algorithms in the Probably Approximately Correct (PAC) setting. 2004 - Poggio et al. proved a general relationship between stability and ERM consistency. They proposed a statistical form of leave-one-out-stability which they called CVEEEloo stability, and showed that it is a) sufficient for generalization in bounded loss classes, and b) necessary and sufficient for consistency (and thus generalization) of ERM algorithms for certain loss functions such as the square loss, the absolute value and the binary classification loss. 2010 - Shalev Shwartz et al. noticed problems with the original results of Vapnik due to the complex relations between hypothesis space and loss class. They discuss stability notions that capture different loss classes and different types of learning, supervised and unsupervised. 2016 - Moritz Hardt et al. proved stability of gradient descent given certain assumption on the hypothesis and number of times each instance is used to update the model. == Preliminary definitions == We define several terms related to learning algorithms training sets, so that we can then define stability in multiple ways and present theorems from the field. A machine learning algorithm, also known as a learning map L {\displaystyle L} , maps a training data set, which is a set of labeled examples ( x , y ) {\displaystyle (x,y)} , onto a function f {\displaystyle f} from X {\displaystyle X} to Y {\displaystyle Y} , where X {\displaystyle X} and Y {\displaystyle Y} are in the same space of the training examples. The functions f {\displaystyle f} are selected from a hypothesis space of functions called H {\displaystyle H} . The training set from which an algorithm learns is defined as S = { z 1 = ( x 1 , y 1 ) , . . , z m = ( x m , y m ) } {\displaystyle S=\{z_{1}=(x_{1},\ y_{1})\ ,..,\ z_{m}=(x_{m},\ y_{m})\}} and is of size m {\displaystyle m} in Z = X × Y {\displaystyle Z=X\times Y} drawn i.i.d. from an unknown distribution D. Thus, the learning map L {\displaystyle L} is defined as a mapping from Z m {\displaystyle Z_{m}} into H {\displaystyle H} , mapping a training set S {\displaystyle S} onto a function f S {\displaystyle f_{S}} from X {\displaystyle X} to Y {\displaystyle Y} . Here, we consider only deterministic algorithms where L {\displaystyle L} is symmetric with respect to S {\displaystyle S} , i.e. it does not depend on the order of the elements in the training set. Furthermore, we assume that all functions are measurable and all sets are countable. The loss V {\displaystyle V} of a hypothesis f {\displaystyle f} with respect to an example z = ( x , y ) {\displaystyle z=(x,y)} is then defined as V ( f , z ) = V ( f ( x ) , y ) {\displaystyle V(f,z)=V(f(x),y)} . The empirical error of f {\displaystyle f} is I S [ f ] = 1 n ∑ V ( f , z i ) {\displaystyle I_{S}[f]={\frac {1}{n}}\sum V(f,z_{i})} . The true error of f {\displaystyle f} is I [ f ] = E z V ( f , z ) {\displaystyle I[f]=\mathbb {E} _{z}V(f,z)} Given a training set S of size m, we will build, for all i = 1....,m, modified training sets as follows: By removing the i-th element S | i = { z 1 , . . . , z i − 1 , z i + 1 , . . . , z m } {\displaystyle S^{|i}=\{z_{1},...,\ z_{i-1},\ z_{i+1},...,\ z_{m}\}} By replacing the i-th element S i = { z 1 , . . . , z i − 1 , z i ′ , z i + 1 , . . . , z m } {\displaystyle S^{i}=\{z_{1},...,\ z_{i-1},\ z_{i}',\ z_{i+1},...,\ z_{m}\}} == Definitions of stability == === Hypothesis Stability === An algorithm L {\displaystyle L} has hypothesis stability β with respect to the loss function V if the following holds: ∀ i ∈ { 1 , . . . , m } , E S , z [ | V ( f S , z ) − V ( f S |

Social media and psychology

Social media began in the form of generalized online communities. These online communities formed on websites like Geocities.com in 1994, Theglobe.com in 1995, and Tripod.com in 1995. Many of these early communities focused on social interaction by bringing people together through the use of chat rooms. The chat rooms encouraged users to share personal information, ideas, or even personal web pages. Later the social networking community Classmates took a different approach by simply having people link to each other by using their personal email addresses. By the late 1990s, social networking websites began to develop more advanced features to help users find and manage friends. These newer generation of social networking websites began to flourish with the emergence of SixDegrees.com in 1997, Makeoutclub in 2000, Hub Culture in 2002, and Friendster in 2002. However, the first profitable mass social networking website was the South Korean service, Cyworld. Cyworld initially launched as a blog-based website in 1999 and social networking features were added to the website in 2001. Other social networking websites emerged like Myspace in 2002, LinkedIn in 2003, and Bebo in 2005. In 2009, the social networking website Facebook (launched in 2004) became the largest social networking website in the world. Both Instagram and Kik were launched in October 2010. Active users of Facebook increased from just a million in 2004 to over 750 million by the year 2011. Making internet-based social networking both a cultural and financial phenomenon. In September 2011, Snapchat was launched and reported over 300 million users in 2021. == Psychology of social networking == A social network is a social structure made up of individuals or organizations who communicate and interact with each other. Social networking sites – such as Facebook, Twitter, Instagram, Pinterest and LinkedIn – are defined as technology-enabled tools that assist users with creating and maintaining their relationships. A study found that middle schoolers reported using social media to see what their friends are doing, to post pictures, and to connect with friends. Human behavior related to social networking is influenced by major individual differences, meaning that people differ quite systematically in the quantity and quality of their social relationships. Two of the main personality traits that are responsible for this variability are the traits of extraversion and introversion. Extraversion refers to the tendency to be socially dominant, exert leadership, and influence on others. In contrast, introversion reflects a tendency towards shyness, social phobia, or even avoid social situations altogether, which could potentially reduce the number of social contacts a person may have. These individual differences may result in different social networking outcomes. Other psychology factors related to social media and Media psychology are depression, anxiety, attachment, self-identity, well-being, and the need to belong. === Neuroscience === The three domains that neural systems rely on to be strengthened to support social media use are social cognition, self-referential cognition, and social rewarding. When someone posts something on social media, they think of how their audience will react, while the audience thinks of the motivations behind posting the information. Both parties are analyzing the other's thoughts and feelings, which coherently rely on multiple network systems of the brain including the dorsomedial prefrontal cortex, bilateral temporoparietal junction, anterior temporal lobes, inferior frontal gyri, and posterior cingulate cortex. All of these systems work to help us process social behaviors and thoughts drawn out on social media. Social media requires a great deal of self-referential thought. People use social media as a platform to express their opinions and show off their past and present selves. In other words, as Bailey Parnell said in her Ted Talk, we're showing off our "highlight reel" (4). When one receives feedback from others, the individual obtains more reflected self-appraisal which leads to comparisons of their social behaviors or "highlights" to other users. Self-referential thought involves activity in the medial prefrontal cortex and the posterior cingulate cortex. The brain uses these systems when thinking of oneself. A 2021 umbrella review found that most associations between adolescent social media use and mental health were characterized as weak or inconsistent, though certain studies identified 'substantial' negative impacts, particularly linked to passive consumption and problematic use. Social media also provides a constant supply of rewards that keeps users coming back for more. Whenever users receive a like or a new follower, it activates the brain's social reward system which includes the ventromedial prefrontal cortex, ventral striatum, and ventral tegmental area. This system has been found to activate in response to positive feedback from peers, suggesting that users experience online acceptance in a similar manner to other material rewards or positive experiences, further acting as a potential reward. While these areas of the brain become strengthened, other parts of the brain start to weaken. Technology is encouraging multi-tasking, especially because of how easy it is to switch from one task to another by opening another tab or using two devices at once. The brain's hippocampus is mainly associated with long-term memory. In a study done by Russell Poldark, a professor at UCLA, they found that "for the task learned without distraction, the hippocampus was involved. However, for the task learned with the distraction of the beeps, the hippocampus was not involved; but the striatum was, which is the brain system that underlies our ability to learn new skills." The study concludes that multitasking can cause reliance on the striatum more than the hippocampus, which can change the way we learn. The striatum is known to be connected to mainly the brain's reward system. The brain will strengthen the neurons to the striatum while it weakens the neurons to the hippocampus to make the brain more efficient. Because our brain starts to rely on the striatum more than the hippocampus, it becomes harder for us to process new information. Nicholas Carr, author of The Shallows: How The Internet Is Changing Our Brains, agrees: "What psychologists and brain scientists tell us about interruptions is that they have a fairly profound effect on the way we think. It becomes much harder to sustain attention, to think about one thing for a long period of time, and to think deeply when new stimuli are pouring at you all day long. I argue that the price we pay for being constantly inundated with information is a loss of our ability to be contemplative and to engage in the kind of deep thinking that requires you to concentrate on one thing." === Well-Being === How does well-being relate to social media? In an article titled Social Impact of Psychological Research on Well-Being Shared in Social Media, Pulido et al. found a 15.7% social impact in their results. These new results were compared to a previous study conducted by Pulido et al., which had a high of 4.98% compared to 27.5% in the new study. These results show the ESISM, which is evidence of social impact present. In a two-year span, the difference between social impact rose 22.52% according to these studies. When taking into consideration that an increasingly large number of teens report either being active on, or having used, some form of social media, ranging from apps such as Facebook to TikTok, researching the effects of social media on the well-being of teens and young adults has become more of a topic of focus in recent years. === Depression === Especially in today's society, social media has gained a new perspective on younger generations. It is what younger generations are born into and are growing up to use, particularly what is running today's society. Social Media has its downfalls regarding depression and mental health. Many users often compare their lives regarding what they see on these platforms. In an article Does Social Media Cause Depression? by the Child Mind Institute, Miller states that "several studies, teenage and young adult users who spend the most time on Instagram, Facebook and other platforms for have shown to have substantially (from 13 to 66 percent) higher rates of reported depression than those who spent the least time", what the study shows how Facebook and Instagram, platforms showcasing daily lives and or lifestyles, or less fulfilling or less satisfied or more flaunting base or superficial. Instead of social community, there has become a perception of individuals striving for a life that is not real, whether that is editing photos or making life seem perfect when it is not. This causes a sense of depression by the weight of a comparing game. In "How Social Media Affects Y