Virtual Print Fee (VPF) is a subsidy paid by a film distributor towards the purchase of digital cinema projection equipment for use by a film exhibitor in the presentation of first release motion pictures. The subsidy is paid in the form of a fee per booking of a movie, intended to match the savings that occurs by not shipping a film print. The model is designed to help redistribute the savings realized by studios when using digital distribution instead of film print distribution and is intended to vanish when the transition phase is over when the vast majority of cinemas screens are equipped. == History == The first public demonstration of digital projection for cinema took place at ShoWest in 1999, and it was readily apparent that the technology was further ahead than the business model. Early technology presentations attempted to claim that the technology would pay for itself through new revenues generated by new forms of content. But exhibitors knew their audience, and could see that digital projection was only a replacement technology, creating new financial liabilities, and not new revenue. It wasn’t until the rollout of digital 3-D years later in 2005 that digital projection demonstrated that it could be used to generate additional revenue. The economics were challenging. Film projectors and platters cost in the neighborhood of US$30,000, while early digital projectors cost up to US$150,000. Further, film projectors had a lifetime of 30 years with relatively small annual expenditures in maintenance and replacement parts. On the other hand, exhibitors felt they would be lucky to get 10 years of service from a digital projector, after which there would have to be a refresh in capital expenditure. Meanwhile, distributors would realize significant savings by eliminating the high cost of film prints with corresponding shipping costs, and instead distributing digital files either by satellite or hard drive. The Virtual Print Fee was designed to better balance savings and expenditures for both exhibitors and distributors. It is intended to primarily assist in the replacement of film projectors, and not assist in the purchase of new projection equipment for new construction. To give confidence to financial institutions that digital cinema technology was stable and worthy of investment, Digital Cinema Initiatives was created in 2002, resulting in the release of the first version of the DCI Digital Cinema System Specification in 2005. The DCI Specification continues to be the core specification for digital cinema, establishing the baseline technology and system requirements for which studios will release digital movies. The first set of VPF agreements executed with four major studios were announced by Christie/AIX in November 2005. Christie/AIX at that time was a subsidiary of Access Integrated Technology, now renamed Cinedigm Digital Cinema Corp. The agreements were for the rollout of digital cinema technology to 4000 screens. Since that time, numerous other Digital Cinema Deployment Agreements have been executed around the world, allowing exhibitors in nearly every territory to benefit from VPF subsidies in the conversion from film projection to digital projection.
Artificial intelligence in fraud detection
Artificial intelligence is used by many different businesses and organizations. It is widely used in the financial sector, especially by accounting firms, to help detect fraud. In 2022, PricewaterhouseCoopers reported that fraud has impacted 46% of all businesses in the world. The shift from working in person to working from home has brought increased access to data. According to an FTC (Federal Trade Commission) study from 2022, customers reported fraud of approximately $5.8 billion in 2021, an increase of 70% from the year before. The majority of these scams were imposter scams and online shopping frauds. Furthermore, artificial intelligence plays a crucial role in developing advanced algorithms and machine learning models that enhance fraud detection systems, enabling businesses to stay ahead of evolving fraudulent tactics in an increasingly digital landscape. == Tools == === Expert systems === Expert systems were first designed in the 1970s as an expansion into artificial intelligence technologies. Their design is based on the premise of decreasing potential user error in decision-making and emulating mental reasoning used by experts in a particular field. They differentiate themselves from traditional linear reasoning models by separating identified points in data and processing them individually at the same time. Though, these systems do not rely purely on machine-learned intelligence. Information regarding rules, practices, and procedures in the form of "if-then" statements are implemented into the programming of the system. Users interact with the system by feeding information into the system either through direct entry or import of external data. An inference system compares the information provided by the user with corresponding rules that are believed to specifically apply to the situation. Using this information and the corresponding rules will be used to create a solution to the user's query. Expert systems will generally not operate properly when the common procedures for a specified situation are ambiguous due to the need for well-defined rules. Implementation of expert systems in accounting procedures is feasible in areas where professional judgment is required. Situations where expert systems are applicable include investigations into transactions that involve potential fraudulent entries, instances of going concern, and the evaluation of risk in the planning stages of an audit. === Continuous auditing === Continuous auditing is a set of processes that assess various aspects of information gathered in an audit to classify areas of risk and potential weaknesses in financial Internal controls at a more frequent rate than traditional methods. Instead of analyzing recorded transactions and journal entries periodically, continuous auditing focuses on interpreting the character of these actions more frequently. The frequency of these processes being undertaken as well as highlighting areas of importance is up to the discretion of their implementer, who commonly makes such decisions based on the level of risk in the accounts being evaluated and the goals of implementing the system. Performance of these processes can occur as frequently as being nearly instantaneous with an entry being posted. The processes involved with analyzing financial data in continuous auditing can include the creation of spreadsheets to allow for interactive information gathering, calculation of financial ratios for comparison with previously created models, and detection of errors in entered figures. A primary goal of this practice is to allow for quicker and easier detection of instances of faulty controls, errors, and instances of fraud. === Machine learning and deep learning === The ability of machine learning and deep learning to swiftly and effectively sort through vast volumes of data in the forms of various documents relevant to companies and documents being audited makes them applicable to the domains of audit and fraud detection. Examples of this include recognizing key language in contracts, identifying levels of risk of fraud in transactions, and assessing journal entries for misstatement. == Applications == === 'Big 4' Accounting Firms === Deloitte created an Al-enabled document-reviewing system in 2014. The system automates the method of reviewing and extracting relevant information from different business documents. Deloitte claims that this innovation has made a difference by reducing time spent going through lawful contract documents, invoices, money-related articulations, and board minutes by up to 50%. Working with IBM's Watson, Deloitte is developing cognitive-technology-enhanced commerce arrangements for its clients. LeasePoint is fueled by IBM TRIRIGA (this product evolved into IBM Maximo Real Estate and Facilities) and uses Deloitte's industrial information to create an end-to-end leasing portfolio. Automated Cognitive Resource Assessment employs IBM's Maximo innovation to progress the proficiency of asset inspection. Ernst and Young (EY) connected Al to the investigation of lease contracts. EY (Australia) has also received Al-enabled auditing technology. Collaborating with H20.ai, PwC developed an Al-enabled framework (GL.ai) capable of analyzing reports and preparing reports. PwC claims to have made a significant investment in normal dialect processing (NLP), an Al-enabled innovation to process unstructured information efficiently. KPMG built a portfolio of Al instruments, called KPMG Ignite, to upgrade trade decisions and forms. Working with Microsoft and IBM Watson, KPMG is creating instruments to coordinate Al, data analytics, Cognitive Technologies, and RPA. == Advantages == === Efficiency === The process of auditing an entity in an attempt to detect fraudulent activity requires the repeating of investigatory processes until an error or misstatement may be identified. Under traditional methods, these processes would be carried out by a human being. Proponents of artificial intelligence in fraud detection have stated that these traditional methods are inefficient and can be more quickly accomplished with the aid of an intelligent computing system. A survey of 400 chief executive officers created by KPMG in 2016 found that approximately 58% believed that artificial intelligence would play a key role in making audits more efficient in the future. === Data interpretation === Higher levels of fraud detection entail the use of professional judgement to interpret data. Supporters of artificial intelligence being used in financial audits have claimed that increased risks from instances of higher data interpretation can be minimized through such technologies. One necessary element of an audit of financial statements that requires professional judgement is the implementation of thresholds for materiality. Materiality entails the distinction between errors and transactions in financial statements that would impact decisions made by users of those financial statements. The threshold for materiality in an audit is set by the auditor based on various factors. Artificial intelligence has been used to interpret data and suggest materiality thresholds to be implemented through the use of expert systems. === Decreased costs === Those in favor of using artificial intelligence to complete investigations of fraud have stated that such technologies decrease the amount of time required to complete tasks that are repetitive. The claim further states that such efficiencies allow for lowered resource requirements, which can then be further spent on tasks that have not been fully automated. The audit firm Ernst & Young has posited these claims by declaring that their deep learning systems have been used to reduce time spent on administrative tasks by analyzing relevant audit documents. According to the firm, this has allowed their employees to focus more on judgement and analysis. == Disadvantages == === Job Displacement === The inescapable reception of computer based intelligence and robotization advancements might prompt critical work relocation across different enterprises. As artificial intelligence frameworks become more equipped for performing undertakings customarily completed by people, there is a worry that specific work jobs could become out of date, prompting joblessness and financial imbalance. === Initial investment requirement === Along with a knowledge of coding and building systems through computer programs, we are seeing the advantages of these systems, but since they are so new, they require a large investment to start building such a system. Any firm that is planning on implementing an AI system to detect fraud must hire a team of data scientists, along with upgrading their cloud system and data storage. The system must be consistently monitored and updated to be the most efficient form of itself, otherwise the likelihood of fraud being involved in those transactions increases. If one does not initially invest in such a syst
Geographical cluster
A geographical cluster is a localized anomaly, usually an excess of something given the distribution or variation of something else. Often it is considered as an incidence rate that is unusual in that there is more of some variable than might be expected. Examples would include: a local excess disease rate, a crime hot spot, areas of high unemployment, accident blackspots, unusually high positive residuals from a model, high concentrations of flora or fauna, physical features or events like earthquake epicenters etc... Identifying these extreme regions may be useful in that there could be implicit geographical associations with other variables that can be identified and would be of interest. Pattern detection via the identification of such geographical clusters is a very simple and generic form of geographical analysis that has many applications in many different contexts. The emphasis is on localized clustering or patterning because this may well contain the most useful information. A geographical cluster is different from a high concentration as it is generally second order, involving the factoring in of the distribution of something else. == Geographical cluster detection == Identifying geographical clusters can be an important stage in a geographical analysis. Mapping the locations of unusual concentrations may help identify causes of these. Some techniques include the Geographical Analysis Machine and Besag and Newell's cluster detection method.
Neocognitron
The neocognitron is a hierarchical, multilayered artificial neural network proposed by Kunihiko Fukushima in 1979. It has been used for Japanese handwritten character recognition and other pattern recognition tasks, and served as the inspiration for convolutional neural networks. Previously in 1969, he published a similar architecture, but with hand-designed kernels inspired by convolutions in mammalian vision. In 1975 he improved it to the Cognitron, and in 1979 he improved it to the neocognitron, which learns all convolutional kernels by unsupervised learning (in his terminology, "self-organized by 'learning without a teacher'"). The neocognitron was inspired by the model proposed by Hubel & Wiesel in 1959. They found two types of cells in the visual primary cortex called simple cell and complex cell, and also proposed a cascading model of these two types of cells for use in pattern recognition tasks. The neocognitron is a natural extension of these cascading models. The neocognitron consists of multiple types of cells, the most important of which are called S-cells and C-cells. The local features are extracted by S-cells, and these features' deformation, such as local shifts, are tolerated by C-cells. Local features in the input are integrated gradually and classified in the higher layers. The idea of local feature integration is found in several other models, such as the Convolutional Neural Network model, the SIFT method, and the HoG method. There are various kinds of neocognitron. For example, some types of neocognitron can detect multiple patterns in the same input by using backward signals to achieve selective attention.
Latent class model
In statistics, a latent class model (LCM) is a model for clustering multivariate discrete data. It assumes that the data arise from a mixture of discrete distributions, within each of which the variables are independent. It is called a latent class model because the class to which each data point belongs is unobserved (or latent). Latent class analysis (LCA) is a subset of structural equation modeling used to find groups or subtypes of cases in multivariate categorical data. These groups or subtypes of cases are called "latent classes". When faced with the following situation, a researcher might opt to use LCA to better understand the data: Symptoms a, b, c, and d have been recorded in a variety of patients diagnosed with diseases X, Y, and Z. Disease X is associated with symptoms a, b, and c; disease Y is linked to symptoms b, c, and d; and disease Z is connected to symptoms a, c, and d. In this context, the LCA would attempt to detect the presence of latent classes (i.e., the disease entities), thus creating patterns of association in the symptoms. As in factor analysis, LCA can also be used to classify cases according to their maximum likelihood class membership probability. The key criterion for resolving the LCA is identifying latent classes in which the observed symptom associations are effectively rendered null. This is because within each class, the diseases responsible for the symptoms create a structure of dependencies. As a result, the symptoms become conditionally independent, meaning that, given the class a case belongs to, the symptoms are no longer related to one another. == Model == Within each latent class, the observed variables are statistically independent—an essential aspect of latent class modeling. Usually, the observed variables are statistically dependent. By introducing the latent variable, independence is restored in the sense that within classes, variables are independent (local independence). Therefore, the association between the observed variables is explained by the classes of the latent variable (McCutcheon, 1987). In one form, the LCM is written as p i 1 , i 2 , … , i N ≈ ∑ t T p t ∏ n N p i n , t n , {\displaystyle p_{i_{1},i_{2},\ldots ,i_{N}}\approx \sum _{t}^{T}p_{t}\,\prod _{n}^{N}p_{i_{n},t}^{n},} where T {\displaystyle T} is the number of latent classes and p t {\displaystyle p_{t}} are the so-called recruitment or unconditional probabilities that should sum to one. p i n , t n {\displaystyle p_{i_{n},t}^{n}} are the marginal or conditional probabilities. For a two-way latent class model, the form is p i j ≈ ∑ t T p t p i t p j t . {\displaystyle p_{ij}\approx \sum _{t}^{T}p_{t}\,p_{it}\,p_{jt}.} This two-way model is related to probabilistic latent semantic analysis and non-negative matrix factorization. The probability model used in LCA is closely related to the Naive Bayes classifier. The main difference is that in LCA, the class membership of an individual is a latent variable, whereas in Naive Bayes classifiers, the class membership is an observed label. == Related methods == There are a number of methods with distinct names and uses that share a common relationship. Cluster analysis is, like LCA, used to discover taxon-like groups of cases in data. Multivariate mixture estimation (MME) is applicable to continuous data and assumes that such data arise from a mixture of distributions, such as a set of heights arising from a mixture of men and women. If a multivariate mixture estimation is constrained so that measures must be uncorrelated within each distribution, it is termed latent profile analysis. Modified to handle discrete data, this constrained analysis is known as LCA. Discrete latent trait models further constrain the classes to form from segments of a single dimension, allocating members to classes based on that dimension. An example would be assigning cases to social classes based on ability or merit. In a practical instance, the variables could be multiple choice items of a political questionnaire. In this case, the data consists of an N-way contingency table with answers to the items for a number of respondents. In this example, the latent variable refers to political opinion, and the latent classes to political groups. Given group membership, the conditional probabilities specify the chance that certain answers are chosen. == Application == LCA may be used in many fields, such as: collaborative filtering, Behavior Genetics and Evaluation of diagnostic tests.
RockMyRun
Rock My Run (stylized as RockMyRun; trademarked slogan: "The Best Running Music in the World™") is a mobile running/fitness app founded in 2011 that provides running and workout music in the form of DJ mixes. It is owned by Rock My World, Inc., a health and fitness technology company based in San Diego, California. The app allows users to listen to these professional DJ mixes on their smartphone while running or working out to enhance and motivate their performance. Rock My World, Inc. also developed the app Jolt.ai for the software Slack. == History == During the early stages of the company, Rock My World, Inc. raised more than $2 million in funding generated by the Irvine Company's The Vine SD and from institutional investors including Skullcandy, ZTE and Lighter Capital and were admitted to the Plug and Play Tech Center in Sunnyvale and to the tech incubator EvoNexus in San Diego. In an interview with co-founder and ex-Qualcomm staff Adam Riggs-Zeigen, he said that "from the beginning [their] big goal is to help people live healthier lives." == Features == The RockMyRun app contains thousands of mixes or "stations" produced by its professional DJs intended to increase enjoyment and performance during exercise. DJs who have provided mixes for the app include David Guetta, Zedd, Steve Aoki, Major Lazer and Afrojack. All of the music can be personalized based on the user's steps per minute, heart rate or ideal cadence allowing the user to "always hear the right music at the right time at the right tempo". All RockMyRun mixes are organized into stations to help users discover music that suits their needs. RockMyRun contains mixes of all genres and each station is categorized into their respective genres and displays tags to let users know the type of music contained in the mix. RockMyRun has two membership types; it is free as a standard member, but for uninterrupted listening and additional features, users can upgrade to a paid "Rockstar" membership. Since March 2023, couples can now be on the same RockMyRun playlists and "share" earbuds. This allows people to train together, easier. A group of DJs curate playlists for specific training needs and different energy levels. == Reception == RockMyRun has been featured on television programs such as The Today Show on two occasions and on The Rachael Ray Show, and in positive reviews by many publications and websites including The New York Times on four separate occasions, TIME, The Huffington Post, The Denver Post, Men's Fitness, Real Simple, The Vulcan Post, The L.A. Times, Glamour, Paste magazine, PCMag, Dubai Week, BetaNews, CNET, CNBC, Reuters, Insider, Tom's Guide and Yahoo! Tech. RockMyRun has also been mentioned/recommended in books/publications such as A Practical Guide to Teacher Wellbeing by Elizabeth Holmes and Applying Music in Exercise and Sport by Dr. Costas Karageorghis. Ultimate Ears placed RockMyRun at the top of their list at No. 1 on their "5 Favorite Workout Music Apps". In a positive review by David Strausser for AndroidGuys in 2015, he praised the app in a detailed review, saying "The mixes are incredible and the rates are reasonable. The app is quick, beautiful." In 2015, Jill Duffy of PC Magazine gave a review of the app, pointing out its key features, and stating that the app is great if you enjoy listening to different, or new music, that can match your tempo while running. Also in 2015, Digital Trends listed RockMyRun, as one of the best exercise music apps in the article "No need to make exercise playlists with these music apps". In 2018, Redbull.com recommended RockMyRun in preparation for the Wings for Life World Run in their article "10 essential hacks for running to work to get you in World Run shape". In 2019, The Fashion Spot included RockMyRun in their list of "The Best Workout Apps for People Who Hate to Work Out", saying: "RockMyRun matches music to the tempo of your running pace – the music literally follows your steps/heart rate. The app has thousands of mixes/music options along with tracking capabilities." Also in 2019, MakeUseOf.com included RockMyRun in their list of "The 7 Best Running and Workout Music Apps". In September 2022, VeryWellFit listed RockMyRun as the first of three "Other Playlist Options" in the article "How to Create a Running Playlist, According to Running Coaches". Tech Grapple recommended the app in "The best workout free music apps for iPhone and Android" saying that "RockMyRun is the best application that you can use during workout. It comes with amazing DJs to craft mixes that will keep you moving." == Partners == RockMyRun is partnered with the following brands/companies: C25K Del Taco JLab Audio iFit Active Network, LLC Night Nation Run (the world's first running music festival) Lady Foot Locker Mayweather Boxing + Fitness Mio Global Orangetheory Fitness Red Rock Apps Tapout Fitness
Random projection
In mathematics and statistics, random projection is a technique used to reduce the dimensionality of a set of points which lie in Euclidean space. According to theoretical results, random projection preserves distances well, but empirical results are sparse. They have been applied to many natural language tasks under the name random indexing. == Dimensionality reduction == Dimensionality reduction, as the name suggests, is reducing the number of random variables using various mathematical methods from statistics and machine learning. Dimensionality reduction is often used to reduce the problem of managing and manipulating large data sets. Dimensionality reduction techniques generally use linear transformations in determining the intrinsic dimensionality of the manifold as well as extracting its principal directions. For this purpose there are various related techniques, including: principal component analysis, linear discriminant analysis, canonical correlation analysis, discrete cosine transform, random projection, etc. Random projection is a simple and computationally efficient way to reduce the dimensionality of data by trading a controlled amount of error for faster processing times and smaller model sizes. The dimensions and distribution of random projection matrices are controlled so as to approximately preserve the pairwise distances between any two samples of the dataset. == Method == The core idea behind random projection is given in the Johnson-Lindenstrauss lemma, which states that if points in a vector space are of sufficiently high dimension, then they may be projected into a suitable lower-dimensional space in a way which approximately preserves pairwise distances between the points with high probability. In random projection, the original d {\displaystyle d} -dimensional data is projected to a k {\displaystyle k} -dimensional subspace, by multiplying on the left by a random matrix R ∈ R k × d {\displaystyle R\in \mathbb {R} ^{k\times d}} . Using matrix notation: If X d × N {\displaystyle X_{d\times N}} is the original set of N d-dimensional observations, then X k × N R P = R k × d X d × N {\displaystyle X_{k\times N}^{RP}=R_{k\times d}X_{d\times N}} is the projection of the data onto a lower k-dimensional subspace. Random projection is computationally simple: form the random matrix "R" and project the d × N {\displaystyle d\times N} data matrix X onto K dimensions of order O ( d k N ) {\displaystyle O(dkN)} . If the data matrix X is sparse with about c nonzero entries per column, then the complexity of this operation is of order O ( c k N ) {\displaystyle O(ckN)} . === Orthogonal random projection === A unit vector can be orthogonally projected to a random subspace. Let u {\displaystyle u} be the original unit vector, and let v {\displaystyle v} be its projection. The norm-squared ‖ v ‖ 2 2 {\displaystyle \|v\|_{2}^{2}} has the same distribution as projecting a random point, uniformly sampled on the unit sphere, to its first k {\displaystyle k} coordinates. This is equivalent to sampling a random point in the multivariate gaussian distribution x ∼ N ( 0 , I d × d ) {\displaystyle x\sim {\mathcal {N}}(0,I_{d\times d})} , then normalizing it. Therefore, ‖ v ‖ 2 2 {\displaystyle \|v\|_{2}^{2}} has the same distribution as ∑ i = 1 k x i 2 ∑ i = 1 k x i 2 + ∑ i = k + 1 d x i 2 {\displaystyle {\frac {\sum _{i=1}^{k}x_{i}^{2}}{\sum _{i=1}^{k}x_{i}^{2}+\sum _{i=k+1}^{d}x_{i}^{2}}}} , which by the chi-squared construction of the Beta distribution, has distribution Beta ( k / 2 , ( d − k ) / 2 ) {\displaystyle \operatorname {Beta} (k/2,(d-k)/2)} , with mean k / d {\displaystyle k/d} . We have a concentration inequality P r [ | ‖ v ‖ 2 − k d | ≥ ϵ k d ] ≤ 3 exp ( − k ϵ 2 / 64 ) {\displaystyle Pr\left[\left|\|v\|_{2}-{\frac {k}{d}}\right|\geq \epsilon {\sqrt {\frac {k}{d}}}\right]\leq 3\exp \left(-k\epsilon ^{2}/64\right)} for any ϵ ∈ ( 0 , 1 ) {\displaystyle \epsilon \in (0,1)} . === Gaussian random projection === The random matrix R can be generated using a Gaussian distribution. The first row is a random unit vector uniformly chosen from S d − 1 {\displaystyle S^{d-1}} . The second row is a random unit vector from the space orthogonal to the first row, the third row is a random unit vector from the space orthogonal to the first two rows, and so on. In this way of choosing R, and the following properties are satisfied: Spherical symmetry: For any orthogonal matrix A ∈ O ( d ) {\displaystyle A\in O(d)} , RA and R have the same distribution. Orthogonality: The rows of R are orthogonal to each other. Normality: The rows of R are unit-length vectors. === More computationally efficient random projections === Achlioptas has shown that the random matrix can be sampled more efficiently. Either the full matrix can be sampled IID according to R i , j = 3 / k × { + 1 with probability 1 6 0 with probability 2 3 − 1 with probability 1 6 {\displaystyle R_{i,j}={\sqrt {3/k}}\times {\begin{cases}+1&{\text{with probability }}{\frac {1}{6}}\\0&{\text{with probability }}{\frac {2}{3}}\\-1&{\text{with probability }}{\frac {1}{6}}\end{cases}}} or the full matrix can be sampled IID according to R i , j = 1 / k × { + 1 with probability 1 2 − 1 with probability 1 2 {\displaystyle R_{i,j}={\sqrt {1/k}}\times {\begin{cases}+1&{\text{with probability }}{\frac {1}{2}}\\-1&{\text{with probability }}{\frac {1}{2}}\end{cases}}} Both are efficient for database applications because the computations can be performed using integer arithmetic. More related study is conducted in. It was later shown how to use integer arithmetic while making the distribution even sparser, having very few nonzeroes per column, in work on the Sparse JL Transform. This is advantageous since a sparse embedding matrix means being able to project the data to lower dimension even faster. === Random Projection with Quantization === Random projection can be further condensed by quantization (discretization), with 1-bit (sign random projection) or multi-bits. It is the building block of SimHash, RP tree, and other memory efficient estimation and learning methods. == Large quasiorthogonal bases == The Johnson-Lindenstrauss lemma states that large sets of vectors in a high-dimensional space can be linearly mapped in a space of much lower (but still high) dimension n with approximate preservation of distances. One of the explanations of this effect is the exponentially high quasiorthogonal dimension of n-dimensional Euclidean space. There are exponentially large (in dimension n) sets of almost orthogonal vectors (with small value of inner products) in n–dimensional Euclidean space. This observation is useful in indexing of high-dimensional data. Quasiorthogonality of large random sets is important for methods of random approximation in machine learning. In high dimensions, exponentially large numbers of randomly and independently chosen vectors from equidistribution on a sphere (and from many other distributions) are almost orthogonal with probability close to one. This implies that in order to represent an element of such a high-dimensional space by linear combinations of randomly and independently chosen vectors, it may often be necessary to generate samples of exponentially large length if we use bounded coefficients in linear combinations. On the other hand, if coefficients with arbitrarily large values are allowed, the number of randomly generated elements that are sufficient for approximation is even less than dimension of the data space. == Implementations == RandPro - An R package for random projection sklearn.random_projection - A module for random projection from the scikit-learn Python library Weka implementation [1]