A variable- (also changeable-, electronic-, or dynamic-) message sign or message board, often abbreviated VMS, VMB, CMS, or DMS, and in the UK known as a matrix sign, is an electronic traffic sign often used on roadways to give travelers information about special events. Such signs warn of traffic congestion, accidents, incidents such as terrorist attacks, Amber/Silver/Blue Alerts, roadwork zones, or speed limits on a specific highway segment. In urban areas, VMS are used within parking guidance and information systems to guide drivers to available car parking spaces. They may also ask vehicles to take alternative routes, limit travel speed, warn of duration and location of the incidents, inform of the traffic conditions, or display general public safety messages. == History == VMS systems were deployed at least as early as the 1950s on the New Jersey Turnpike. The road's signs of that period, and up to around 2012, were capable of displaying a few messages in neon, all oriented around warning drivers to slow down: "REDUCE SPEED", followed by a warning of either construction, accident, congestion, ice, snow, or fog at a certain distance ahead. The New Jersey Turnpike Authority replaced those signs (along with 1990s-vintage dot-matrix VMS systems along the Garden State Parkway) with more flexible electronic signs between 2010 and 2016. The current VMS systems are largely deployed on freeways, trunk highways, or in work zones. On the interchange of I-5 and SR 120 in San Joaquin County, California, an automated visibility and speed warning system was installed in 1996 to warn traffic of reduced visibility due to fog (where tule fog is a common problem in the winter), and of slow or stopped traffic. Message Signs were deployed in Ontario during the 1990s and are now being upgraded on 400-series highways as well as two pilot secondary highways in northeastern Ontario. == Technologies and types == Early variable message signs included static signs with words that would illuminate (often using neon tubing) indicating the type of incident that occurred, or signs that used rotating prisms (trilons) to change the message being displayed. These were later replaced by dot matrix displays typically using eggcrate, fiber optic, or flip-disc technology, which were capable of displaying a much wider range of messages than earlier static variable message signs. Since the late 1990s, the most common technology used in new installations for variable message signs are LED displays. In recent years, some newer LED variable message signs have the ability to display colored text and graphics. Dot-matrix variable message signs are divided into three subgroups: character matrix, row matrix, and full matrix. In a character matrix VMS, each character is given its own matrix with equal horizontal spacing between them, typically with two or three rows of characters. In a full matrix VMS, the entire sign is a single large dot matrix display, allowing the display of different fonts and graphics. A row matrix VMS is a hybrid of the two types, divided into two or three rows like a character matrix display, except each row is a single long dot matrix display instead of being split per character horizontally. Overhead variable message signs are today available in three form factors: front access, rear access, and walk-in. In a front access variable message sign, maintenance is performed by lifting the sign open from the front. Most smaller VMS are of the front access form factor, and are typically installed today on major arterials. The rear access form factor is similar to the front access form factor, except that maintenance is performed from the rear of the sign, and are commonly used for medium-sized dynamic message signs installed along the roadside of freeways (instead of overhead). The walk-in form factor is a more recent introduction, where maintenance on the sign is performed from the inside of the sign. A key advantage of the walk-in form factor is that lane closures are generally not required to perform maintenance on the sign. Most of the largest VMS units installed today are walk-in units, and are typically installed overhead on freeways. The NJ Turnpike Authority counts five unique types of variable message signs under its jurisdiction, at least one of which has been replaced by newer signs. They are: "REDUCE SPEED" neon signs (1950s-2010, obsolete, have now been replaced). "Changeable message signs" (trilon/ rotating-drum signs that can be used for closing roads or moving traffic to other roadways). Electronic VMS: signs with remotely controlled messages displayed on them; the messages are sent from the State Traffic Management Center, updating the signs automatically. Variable speed limit signs - used for varying the posted speed limits within work zones and in emergencies. Portable VMS: movable "electronic VMS". A portable VMS has much the same characteristics as a fixed electronic VMS, but can be moved from location to location as the need dictates. == Usage == Early models required an operator to be physically present when programming a message, whereas newer models may be reprogrammed remotely via a wired or wireless network or cellphone connection. A complete message on a panel generally includes a problem statement indicating incident, roadwork, stalled vehicle etc.; a location statement indicating where the incident is located; an effect statement indicating lane closure, delay, etc. and an action statement giving suggestion what to do traffic conditions ahead. These signs are also used for Amber alert messages, and in some states, Silver and Blue Alert messages. In some places, VMSes are set up with permanent, semi-static displays indicating predicted travel times to important traffic destinations such as major cities or interchanges along the route of a highway. Typical messages provide the following information: Promotional messages about services provided by a road authority during non-critical hours, such as carpooling efforts, travelers' information stations and 5-1-1 lines Crashes, including vehicle spin-out or rollover Road Works Incidents affecting normal traffic flow in a lane or on shoulders Non-recurring congestion, often a residual effect of cleared crash Closures of an entire road, e.g. over a mountain pass in winter. Exit ramp closures Debris on roadway Vehicle fires Wildfires Short-term maintenance or construction lasting less than three days Pavement failure alerts AMBER, Silver, and Blue Alerts, as well as weather warnings via the warning infrastructure of NOAA Weather Radio's SAME system Travel times Variable speed limits Car park occupancy levels speed sign, for recommending a speed to approach the next traffic light in its green phase. The information comes from a variety of traffic monitoring and surveillance systems. It is expected that by providing real-time information on special events on the oncoming road, VMS can improve motorists' route selection, reduce travel time, mitigate the severity and duration of incidents and improve the performance of the transportation network. === United Kingdom === Do not enter the motorway when the red lamps are flashing in pairs from side to side. On 27 March 1972, the first motorway computer-controlled warning lights in the UK, with 59 miles on the M6 from Broughton, Lancashire to Barthomley, on the Cheshire boundary, and 26 miles on the M62 east of Whitefield, was switched on by Michael Heseltine and Charles Legh Shuldham Cornwall-Legh, 5th Baron Grey of Codnor at the headquarters of Cheshire Constabulary on Nuns Road. It was centred at a police computer centre at Westhoughton, that connected to police stations in Preston and Chester. The Chester site was soon be connected to the M53 and M57. Four other regional computer centres would be opened at Perry Barr near the M6, Scratchwood near the M1, at Hook near the M3, and at Almondsbury near the M4. Most British motorways would be covered by 1975. The system was designed by GEC and had taken five years to design. == Safety messages for drivers == Increasingly, signs have been used to remind drivers to buckle seat belts ("Click It or Ticket"), obey the speed limit, and stay off the road if impaired ("Drive sober or get pulled over"). In a federal study, a slight majority of drivers reported that public safety messages on dynamic message signs impacted their driving behaviors. The Ohio Department of Transportation began using humorous dynamic message signs in 2015, perplexing some drivers. Examples of humorous signs seen in New Jersey, Arizona, Texas, Pennsylvania, Delaware, Iowa, New York, Minnesota and Ohio include: "Hold on to your butts. Help prevent forest fires." "We'll be blunt. Don't drive high." "Visiting in-laws? Slow down, get there late." "Only sparklers should be lit." and “Don’t drive Star Spangled hammered." (for Fourth of July) "Hocus pocus – drive with focus." and "Slow down in work zones - my mummy works here." (f
Elasticity (data store)
The elasticity of a data store relates to the flexibility of its data model and clustering capabilities. The greater the number of data model changes that can be tolerated, and the more easily the clustering can be managed, the more elastic the data store is considered to be. == Types == === Clustering elasticity === Clustering elasticity is the ease of adding or removing nodes from the distributed data store. Usually, this is a difficult and delicate task to be done by an expert in a relational database system. Some NoSQL data stores, like Apache Cassandra have an easy solution, and a node can be added/removed with a few changes in the properties and by adding specifying at least one seed. === Data-modelling elasticity === Relational databases are most often very inelastic, as they have a predefined data model that can only be adapted through redesign. Most NoSQL data stores, however, do not have a fixed schema. Each row can have a different number and even different type of columns. Concerning the data store, modifications in the schema are no problem. This makes this kind of data stores more elastic concerning the data model. The drawback is that the programmer has to take into account that the data model may change over time.
Loss function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its opposite (in specific domains, variously called a reward function, a profit function, a utility function, a fitness function, etc.), in which case it is to be maximized. The loss function could include terms from several levels of the hierarchy. In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as Laplace, was reintroduced in statistics by Abraham Wald in the middle of the 20th century. In the context of economics, for example, this is usually economic cost or regret. In classification, it is the penalty for an incorrect classification of an example. In actuarial science, it is used in an insurance context to model benefits paid over premiums, particularly since the works of Harald Cramér in the 1920s. In optimal control, the loss is the penalty for failing to achieve a desired value. In financial risk management, the function is mapped to a monetary loss. == Examples == === Regret === Leonard J. Savage argued that using non-Bayesian methods such as minimax, the loss function should be based on the idea of regret, i.e., the loss associated with a decision should be the difference between the consequences of the best decision that could have been made under circumstances will be known and the decision that was in fact taken before they were known. === Quadratic loss function === The use of a quadratic loss function is common, for example when using least squares techniques. It is often more mathematically tractable than other loss functions because of the properties of variances, as well as being symmetric: an error above the target causes the same loss as the same magnitude of error below the target. If the target is t {\displaystyle t} , then a quadratic loss function is λ ( x ) = C ( t − x ) 2 {\displaystyle \lambda (x)=C(t-x)^{2}\;} for some constant C {\displaystyle C} ; the value of the constant makes no difference to a decision, and can be ignored by setting it equal to 1. This is also known as the squared error loss (SEL). Many common statistics, including t-tests, regression models, design of experiments, and much else, use least squares methods applied using linear regression theory, which is based on the quadratic loss function. The quadratic loss function is also used in linear-quadratic optimal control problems. In these problems, even in the absence of uncertainty, it may not be possible to achieve the desired values of all target variables. Often loss is expressed as a quadratic form in the deviations of the variables of interest from their desired values; this approach is tractable because it results in linear first-order conditions. In the context of stochastic control, the expected value of the quadratic form is used. The quadratic loss assigns more importance to outliers than to the true data due to its square nature, so alternatives like the Huber, log-cosh and SMAE losses are used when the data has many large outliers. === 0-1 loss function === In statistics and decision theory, a frequently used loss function is the 0-1 loss function L ( y ^ , y ) = { 0 if y = y ^ 1 if y ≠ y ^ {\displaystyle L({\hat {y}},y)={\begin{cases}0&{\text{if }}y={\hat {y}}\\1&{\text{if }}y\neq {\hat {y}}\end{cases}}} In information theory, this loss function is known as Hamming distortion. == Constructing loss and objective functions == In many applications, objective functions, including loss functions as a particular case, are determined by the problem formulation. In other situations, the decision maker’s preference must be elicited and represented by a scalar-valued function (called also utility function) in a form suitable for optimization — the problem that Ragnar Frisch has highlighted in his Nobel Prize lecture. The existing methods for constructing objective functions are collected in the proceedings of two dedicated conferences. In particular, Andranik Tangian showed that the most usable objective functions — quadratic and additive — are determined by a few indifference points. He used this property in the models for constructing these objective functions from either ordinal or cardinal data that were elicited through computer-assisted interviews with decision makers. Among other things, he constructed objective functions to optimally distribute budgets for 16 Westfalian universities and the European subsidies for equalizing unemployment rates among 271 German regions. == Expected loss == In some contexts, the value of the loss function itself is a random quantity because it depends on the outcome of a random variable X {\displaystyle X} . === Statistics === Both frequentist and Bayesian statistical theory involve making a decision based on the expected value of the loss function; however, this quantity is defined differently under the two paradigms. ==== Frequentist expected loss ==== We first define the expected loss in the frequentist context. It is obtained by taking the expected value with respect to the probability distribution, P θ {\displaystyle P_{\theta }} , of the observed data, X {\displaystyle X} . This is also referred to as the risk function of the decision rule δ {\displaystyle \delta } and the parameter θ {\displaystyle \theta } . Here the decision rule depends on the outcome of X {\displaystyle X} . The risk function is given by: R ( θ , δ ) = E θ L ( θ , δ ( X ) ) = ∫ X L ( θ , δ ( x ) ) d P θ ( x ) . {\displaystyle R(\theta ,\delta )=\operatorname {E} _{\theta }L{\big (}\theta ,\delta (X){\big )}=\int _{X}L{\big (}\theta ,\delta (x){\big )}\,\mathrm {d} P_{\theta }(x).} Here, θ {\displaystyle \theta } is a fixed but possibly unknown state of nature, X {\displaystyle X} is a vector of observations stochastically drawn from a population, E θ {\displaystyle \operatorname {E} _{\theta }} is the expectation over all population values of X {\displaystyle X} , d P θ {\displaystyle \mathrm {d} P_{\theta }} is a probability measure over the event space of X {\displaystyle X} (parametrized by θ {\displaystyle \theta } ) and the integral is evaluated over the entire support of X {\displaystyle X} . ==== Bayes Risk ==== In a Bayesian approach, the expectation is calculated using the prior distribution π ∗ {\displaystyle \pi ^{}} of the parameter θ {\displaystyle \theta } : ρ ( π ∗ , a ) = ∫ Θ ∫ X L ( θ , a ( x ) ) d P ( x | θ ) d π ∗ ( θ ) = ∫ X ∫ Θ L ( θ , a ( x ) ) d π ∗ ( θ | x ) d M ( x ) {\displaystyle \rho (\pi ^{},a)=\int _{\Theta }\int _{\mathbf {X}}L(\theta ,a({\mathbf {x}}))\,\mathrm {d} P({\mathbf {x}}\vert \theta )\,\mathrm {d} \pi ^{}(\theta )=\int _{\mathbf {X}}\int _{\Theta }L(\theta ,a({\mathbf {x}}))\,\mathrm {d} \pi ^{}(\theta \vert {\mathbf {x}})\,\mathrm {d} M({\mathbf {x}})} where M ( x ) {\displaystyle M(\mathbf {x} )} is known as the predictive likelihood wherein θ {\displaystyle \theta } has been "integrated out," π ∗ ( θ | x ) {\displaystyle \pi ^{}(\theta |\mathbf {x} )} is the posterior distribution, and the order of integration has been changed. One then should choose the action a ∗ {\displaystyle a^{}} which minimises this expected loss, which is referred to as Bayes Risk. In the latter equation, the integrand inside d x {\displaystyle \mathrm {d} x} is known as the Posterior Risk, and minimising it with respect to decision a {\displaystyle a} also minimizes the overall Bayes Risk. This optimal decision, a ∗ {\displaystyle a^{}} is known as the Bayes (decision) Rule - it minimises the average loss over all possible states of nature θ {\displaystyle \theta } , over all possible (probability-weighted) data outcomes. One advantage of the Bayesian approach is to that one need only choose the optimal action under the actual observed data to obtain a uniformly optimal one, whereas choosing the actual frequentist optimal decision rule as a function of all possible observations, is a much more difficult problem. Of equal importance though, the Bayes Rule reflects consideration of loss outcomes under different states of nature, θ {\displaystyle \theta } . ==== Examples in statistics ==== For a scalar parameter θ {\displaystyle \theta } , a decision function whose output θ ^ {\displaystyle {\hat {\theta }}} is an estimate of θ {\displaystyle \theta } , and a quadratic loss function (squared error loss) L ( θ , θ ^ ) = ( θ − θ ^ ) 2 , {\displaystyle L(\theta ,{\hat {\theta }})=(\theta -{\hat {\theta }})^{2},} the risk function becomes the mean squared error of the estimate, R ( θ , θ ^ ) = E θ [ ( θ − θ ^ ) 2 ] . {\displaystyle R(\theta ,{\hat {\thet
L-1 Identity Solutions
L-1 Identity Solutions, Inc. was an American biometric technology company headquartered in Stamford, Connecticut, specializing in identity management products and services including facial recognition systems, fingerprint readers, and secure credentialing solutions for governments and commercial enterprises. The company's shares traded on the New York Stock Exchange under the ticker symbol "ID." == History == L-1 Identity Solutions was formed on August 29, 2006, from a merger of Viisage Technology, Inc. and Identix Incorporated. Prior to the Safran acquisition, L-1 divested its Intelligence Services Group (ISG) comprising SpecTal LLC, Advanced Concepts Inc., and McClendon LLC to BAE Systems, Inc. for approximately $297 million. The transaction, initially announced in September 2010, closed on February 15, 2011, with more than 1,000 ISG employees joining BAE Systems' Intelligence & Security sector. It specializes in selling face recognition systems, electronic passports, such as Fly Clear, and other biometric technology to governments such as the United States and Saudi Arabia. It also licenses technology to other companies internationally, including China. On July 26, 2011, Safran (NYSE Euronext Paris: SAF) acquired L-1 Identity Solutions, Inc. for a total cash amount of USD 1.09 billion. L-1 was part of Morpho's MorphoTrust department which rebranded to Idemia in 2017. Bioscrypt is a biometrics research, development and manufacturing company purchased by L-1 Identity Solutions. It provides fingerprint IP readers for physical access control systems, Facial recognition system readers for contactless access control authentication and OEM fingerprint modules for embedded applications. According to IMS Research, Bioscrypt has been the world market leader in biometric access control for enterprises (since 2006) with a worldwide market share of over 13%. In 2011, Bioscrypt was sold to Safran Morpho.
Locality-sensitive hashing
In computer science, locality-sensitive hashing (LSH) is a fuzzy hashing technique that hashes similar input items into the same "buckets" with high probability. The number of buckets is much smaller than the universe of possible input items. Since similar items end up in the same buckets, this technique can be used for data clustering and nearest neighbor search. It differs from conventional hashing techniques in that hash collisions are maximized, not minimized. Alternatively, the technique can be seen as a way to reduce the dimensionality of high-dimensional data; high-dimensional input items can be reduced to low-dimensional versions while preserving relative distances between items. Hashing-based approximate nearest-neighbor search algorithms generally use one of two main categories of hashing methods: either data-independent methods, such as locality-sensitive hashing (LSH); or data-dependent methods, such as locality-preserving hashing (LPH). Locality-preserving hashing was initially devised as a way to facilitate data pipelining in implementations of massively parallel algorithms that use randomized routing and universal hashing to reduce memory contention and network congestion. == Definitions == A finite family F {\displaystyle {\mathcal {F}}} of functions h : M → S {\displaystyle h\colon M\to S} is defined to be an LSH family for a metric space M = ( M , d ) {\displaystyle {\mathcal {M}}=(M,d)} , a threshold r > 0 {\displaystyle r>0} , an approximation factor c > 1 {\displaystyle c>1} , and probabilities p 1 > p 2 {\displaystyle p_{1}>p_{2}} if it satisfies the following condition. For any two points a , b ∈ M {\displaystyle a,b\in M} and a hash function h {\displaystyle h} chosen uniformly at random from F {\displaystyle {\mathcal {F}}} : If d ( a , b ) ≤ r {\displaystyle d(a,b)\leq r} , then h ( a ) = h ( b ) {\displaystyle h(a)=h(b)} (i.e., a and b collide) with probability at least p 1 {\displaystyle p_{1}} , If d ( a , b ) ≥ c r {\displaystyle d(a,b)\geq cr} , then h ( a ) = h ( b ) {\displaystyle h(a)=h(b)} with probability at most p 2 {\displaystyle p_{2}} . Such a family F {\displaystyle {\mathcal {F}}} is called ( r , c r , p 1 , p 2 ) {\displaystyle (r,cr,p_{1},p_{2})} -sensitive. === LSH with respect to a similarity measure === Alternatively it is possible to define an LSH family on a universe of items U endowed with a similarity function ϕ : U × U → [ 0 , 1 ] {\displaystyle \phi \colon U\times U\to [0,1]} . In this setting, a LSH scheme is a family of hash functions H coupled with a probability distribution D over H such that a function h ∈ H {\displaystyle h\in H} chosen according to D satisfies P r [ h ( a ) = h ( b ) ] = ϕ ( a , b ) {\displaystyle Pr[h(a)=h(b)]=\phi (a,b)} for each a , b ∈ U {\displaystyle a,b\in U} . === Amplification === Given a ( d 1 , d 2 , p 1 , p 2 ) {\displaystyle (d_{1},d_{2},p_{1},p_{2})} -sensitive family F {\displaystyle {\mathcal {F}}} , we can construct new families G {\displaystyle {\mathcal {G}}} by either the AND-construction or OR-construction of F {\displaystyle {\mathcal {F}}} . To create an AND-construction, we define a new family G {\displaystyle {\mathcal {G}}} of hash functions g, where each function g is constructed from k random functions h 1 , … , h k {\displaystyle h_{1},\ldots ,h_{k}} from F {\displaystyle {\mathcal {F}}} . We then say that for a hash function g ∈ G {\displaystyle g\in {\mathcal {G}}} , g ( x ) = g ( y ) {\displaystyle g(x)=g(y)} if and only if all h i ( x ) = h i ( y ) {\displaystyle h_{i}(x)=h_{i}(y)} for i = 1 , 2 , … , k {\displaystyle i=1,2,\ldots ,k} . Since the members of F {\displaystyle {\mathcal {F}}} are independently chosen for any g ∈ G {\displaystyle g\in {\mathcal {G}}} , G {\displaystyle {\mathcal {G}}} is a ( d 1 , d 2 , p 1 k , p 2 k ) {\displaystyle (d_{1},d_{2},p_{1}^{k},p_{2}^{k})} -sensitive family. To create an OR-construction, we define a new family G {\displaystyle {\mathcal {G}}} of hash functions g, where each function g is constructed from k random functions h 1 , … , h k {\displaystyle h_{1},\ldots ,h_{k}} from F {\displaystyle {\mathcal {F}}} . We then say that for a hash function g ∈ G {\displaystyle g\in {\mathcal {G}}} , g ( x ) = g ( y ) {\displaystyle g(x)=g(y)} if and only if h i ( x ) = h i ( y ) {\displaystyle h_{i}(x)=h_{i}(y)} for one or more values of i. Since the members of F {\displaystyle {\mathcal {F}}} are independently chosen for any g ∈ G {\displaystyle g\in {\mathcal {G}}} , G {\displaystyle {\mathcal {G}}} is a ( d 1 , d 2 , 1 − ( 1 − p 1 ) k , 1 − ( 1 − p 2 ) k ) {\displaystyle (d_{1},d_{2},1-(1-p_{1})^{k},1-(1-p_{2})^{k})} -sensitive family. == Applications == LSH has been applied to several problem domains, including: Near-duplicate detection Hierarchical clustering Genome-wide association study Image similarity identification VisualRank Gene expression similarity identification Audio similarity identification Nearest neighbor search Audio fingerprint Digital video fingerprinting Shared memory organization in parallel computing Physical data organization in database management systems Training fully connected neural networks Computer security Machine learning == Methods == === Bit sampling for Hamming distance === One of the easiest ways to construct an LSH family is by bit sampling. This approach works for the Hamming distance over d-dimensional vectors { 0 , 1 } d {\displaystyle \{0,1\}^{d}} . Here, the family F {\displaystyle {\mathcal {F}}} of hash functions is simply the family of all the projections of points on one of the d {\displaystyle d} coordinates, i.e., F = { h : { 0 , 1 } d → { 0 , 1 } ∣ h ( x ) = x i for some i ∈ { 1 , … , d } } {\displaystyle {\mathcal {F}}=\{h\colon \{0,1\}^{d}\to \{0,1\}\mid h(x)=x_{i}{\text{ for some }}i\in \{1,\ldots ,d\}\}} , where x i {\displaystyle x_{i}} is the i {\displaystyle i} th coordinate of x {\displaystyle x} . A random function h {\displaystyle h} from F {\displaystyle {\mathcal {F}}} simply selects a random bit from the input point. This family has the following parameters: P 1 = 1 − R / d {\displaystyle P_{1}=1-R/d} , P 2 = 1 − c R / d {\displaystyle P_{2}=1-cR/d} . That is, any two vectors x , y {\displaystyle x,y} with Hamming distance at most R {\displaystyle R} collide under a random h {\displaystyle h} with probability at least P 1 {\displaystyle P_{1}} . Any x , y {\displaystyle x,y} with Hamming distance at least c R {\displaystyle cR} collide with probability at most P 2 {\displaystyle P_{2}} . === Min-wise independent permutations === Suppose U is composed of subsets of some ground set of enumerable items S and the similarity function of interest is the Jaccard index J. If π is a permutation on the indices of S, for A ⊆ S {\displaystyle A\subseteq S} let h ( A ) = min a ∈ A { π ( a ) } {\displaystyle h(A)=\min _{a\in A}\{\pi (a)\}} . Each possible choice of π defines a single hash function h mapping input sets to elements of S. Define the function family H to be the set of all such functions and let D be the uniform distribution. Given two sets A , B ⊆ S {\displaystyle A,B\subseteq S} the event that h ( A ) = h ( B ) {\displaystyle h(A)=h(B)} corresponds exactly to the event that the minimizer of π over A ∪ B {\displaystyle A\cup B} lies inside A ∩ B {\displaystyle A\cap B} . As h was chosen uniformly at random, P r [ h ( A ) = h ( B ) ] = J ( A , B ) {\displaystyle Pr[h(A)=h(B)]=J(A,B)\,} and ( H , D ) {\displaystyle (H,D)\,} define an LSH scheme for the Jaccard index. Because the symmetric group on n elements has size n!, choosing a truly random permutation from the full symmetric group is infeasible for even moderately sized n. Because of this fact, there has been significant work on finding a family of permutations that is "min-wise independent" — a permutation family for which each element of the domain has equal probability of being the minimum under a randomly chosen π. It has been established that a min-wise independent family of permutations is at least of size lcm { 1 , 2 , … , n } ≥ e n − o ( n ) {\displaystyle \operatorname {lcm} \{\,1,2,\ldots ,n\,\}\geq e^{n-o(n)}} , and that this bound is tight. Because min-wise independent families are too big for practical applications, two variant notions of min-wise independence are introduced: restricted min-wise independent permutations families, and approximate min-wise independent families. Restricted min-wise independence is the min-wise independence property restricted to certain sets of cardinality at most k. Approximate min-wise independence differs from the property by at most a fixed ε. === Open source methods === ==== Nilsimsa Hash ==== Nilsimsa is a locality-sensitive hashing algorithm used in anti-spam efforts. The goal of Nilsimsa is to generate a hash digest of an email message such that the digests of two similar messages are similar to each other. The paper suggests that the Nilsimsa satisfies three requirements: The digest identifying each message should not
Arattai
Arattai Messenger (or simply Arattai) is an encrypted messaging service for instant messaging, voice calls, and video calls, developed by Zoho Corporation. The name Arattai means "chat" or "conversation" in Tamil. The app was soft-launched in January 2021. The app saw a sharp surge in downloads in September 2025, partially fueled by endorsements from Indian government officials. However, the app dropped from the top rankings in October 2025. == History == Arattai was initially tested internally among Zoho employees before being released publicly in early 2021. The launch coincided with a surge in interest for privacy-focused and messaging services, triggered by concerns over WhatsApp's updated terms of service. In September 2025, Arattai experienced a major surge in adoption, with daily sign-ups reportedly increasing 100-fold, from around 3,000 to more than 350,000 in three days. The surge in downloads was attributed to Zoho products being promoted by Indian government officials as part of their Make in India push for homegrown alternatives to foreign‐owned apps, amid deteriorating India–US relations. The growth temporarily strained Zoho's infrastructure, prompting rapid scaling of servers and capacity expansion. During the same period, the app reached the top position in Apple's App Store charts for the "Social Networking" category in India. The app dropped from the top ranking in late October 2025. == Reception == At launch, Arattai was positioned as a potential domestic rival to WhatsApp in India, but analysts noted that it faced challenges with encryption, ecosystem, and network effect. Critics pointed to occasional sync delays.
Stress majorization
Stress majorization is an optimization strategy used in multidimensional scaling (MDS) where, for a set of n {\displaystyle n} m {\displaystyle m} -dimensional data items, a configuration X {\displaystyle X} of n {\displaystyle n} points in r {\displaystyle r} ( ≪ m ) {\displaystyle (\ll m)} -dimensional space is sought that minimizes the so-called stress function σ ( X ) {\displaystyle \sigma (X)} . Usually r {\displaystyle r} is 2 {\displaystyle 2} or 3 {\displaystyle 3} , i.e. the ( n × r ) {\displaystyle (n\times r)} matrix X {\displaystyle X} lists points in 2 − {\displaystyle 2-} or 3 − {\displaystyle 3-} dimensional Euclidean space so that the result may be visualised (i.e. an MDS plot). The function σ {\displaystyle \sigma } is a cost or loss function that measures the squared differences between ideal ( m {\displaystyle m} -dimensional) distances and actual distances in r-dimensional space. It is defined as: σ ( X ) = ∑ i < j ≤ n w i j ( d i j ( X ) − δ i j ) 2 {\displaystyle \sigma (X)=\sum _{i