RadioVIS is a protocol for sideband signalling of images and text messages for a broadcast audio service to provide a richer visual experience. It is an application and sub-project of RadioDNS, which allows radio consumption devices to look up an IP-based service based on the parameters of the currently tuned broadcast station. In January 2015, the functionality of RadioVIS was integrated to Visual Slideshow (ETSI TS 101 499 v3.1.1). The original RVIS01 document is now deprecated. == Details == The protocol enables either Streaming Text Oriented Messaging Protocol (STOMP) or Comet to deliver text and image URLs to a client, with the images being acquired over a HTTP connection. The technology is currently implemented by a number of broadcasters across the world, including Global Radio, Bauer Radio in the UK, RTÉ in the Republic Of Ireland, Südwestrundfunk in Germany and a number of Australian media groups amongst others. A number of software clients exist to show the protocol, as well as hardware devices such as the Pure Sensia from Pure Digital, and the Colourstream from Roberts Radio.
NER model
NER is one of several formulas for accessing live subtitles in television broadcasts and events that are produced using speech recognition. The three letters stand for number, edit error and recognition error. It has been promoted as an alternative to Word error rate (Word Error Rate) which is a more objective measure. The overall score is calculated as follows: Firstly, the number of edit and recognition errors is deducted from the total number of words in the live subtitles. This number is then divided by the total number of words in the live subtitles and finally multiplied by one hundred. N E R v a l u e = N − E − R N ∗ 100 {\displaystyle NERvalue={\frac {N-E-R}{N}}100} . The acronyms stand for the following: N (number) = total number of words in the live subtitles E (Edit error) = edit error R (Recognition error) = recognition error This measurement process has been used for public television broadcasts in European countries like Italy and Switzerland. One major drawback with NER is that it requires a human assessor to rate errors as either: 1 Minor edition or recognition errors 2 Normal edition or recognition errors 3 Serious errors which are then weighted in the assessment process. This is both subjective, time consuming and costly. Also, NER fails to account for words left out subtitles which is something that does not take account of the D/deaf audience who want verbatim subtitles. As a result, NER cannot accurately reflect the audience's experience of subtitles. Another problem is the inconsistency of human evaluation of subtitles, particularly with live subtitles, where there are differing opinions of the importance of subtitle errors. By way of contrast, Word error rate is an objective measure of subtitle errors, since it measures the textual discrepancy between the subtitles and the speech.
WordNet
WordNet is a lexical database of semantic relations between words that links words into semantic relations including synonyms, hyponyms, and meronyms. The synonyms are grouped into synsets with short definitions and usage examples. It can thus be seen as a combination and extension of a dictionary and thesaurus. Its primary use is in automatic text analysis and artificial intelligence applications. It was first created in the English language and the English WordNet database and software tools have been released under a BSD style license and are freely available for download. The latest official release from Princeton was released in 2011. Princeton currently has no plans to release any new versions due to staffing and funding issues. New versions are still being released annually through the Open English WordNet website. Until about 2024 an online version was previously available through wordnet.princeton.edu. That version of WordNet has been deprecated, but a new online version is available at en-word.net. There are now WordNets in more than 200 languages. == History and team members == WordNet was first created in 1985, in English only, in the Cognitive Science Laboratory of Princeton University under the direction of psychology professor George Armitage Miller. It was later directed by Christiane Fellbaum. The project was initially funded by the U.S. Office of Naval Research, and later also by other U.S. government agencies including the DARPA, the National Science Foundation, the Disruptive Technology Office (formerly the Advanced Research and Development Activity) and REFLEX. George Miller and Christiane Fellbaum received the 2006 Antonio Zampolli Prize for their work with WordNet. The Global WordNet Association is a non-commercial organization that provides a platform for discussing, sharing and connecting WordNets for all languages in the world. Christiane Fellbaum and Piek Th.J.M. Vossen are its co-presidents. == Database contents == The database contains 155,327 words organized in 175,979 synsets for a total of 207,016 word-sense pairs; in compressed form, it is about 12 megabytes in size. It includes the lexical categories nouns, verbs, adjectives and adverbs but ignores prepositions, determiners and other function words. Words from the same lexical category that are roughly synonymous are grouped into synsets, which include simplex words as well as collocations like "eat out" and "car pool." The different senses of a polysemous word form are assigned to different synsets. A synset's meaning is further clarified with a short defining gloss and one or more usage examples. An example adjective synset is: good, right, ripe – (most suitable or right for a particular purpose; "a good time to plant tomatoes"; "the right time to act"; "the time is ripe for great sociological changes") All synsets are connected by means of semantic relations. These relations, which are not all shared by all lexical categories, include: Nouns hypernym: Y is a hypernym of X if every X is a (kind of) Y (canine is a hypernym of dog) hyponym: Y is a hyponym of X if every Y is a (kind of) X (dog is a hyponym of canine) coordinate term: Y is a coordinate term of X if X and Y share a hypernym (wolf is a coordinate term of dog, and dog is a coordinate term of wolf) holonym: Y is a holonym of X if X is a part of Y (building is a holonym of window) meronym: Y is a meronym of X if Y is a part of X (window is a meronym of building) Verbs hypernym: the verb Y is a hypernym of the verb X if the activity X is a (kind of) Y (to perceive is an hypernym of to listen) troponym: the verb Y is a troponym of the verb X if the activity Y is doing X in some manner (to lisp is a troponym of to talk) entailment: the verb Y is entailed by the verb X if by doing X you must be doing Y (to sleep is entailed by to snore) coordinate term: the verb Y is a coordinate term of the verb X if X and Y share a hypernym (to lisp is a coordinate term of to yell, and to yell is a coordinate term of to lisp) These semantic relations hold among all members of the linked synsets. Individual synset members (words) can also be connected with lexical relations. For example, (one sense of) the noun "director" is linked to (one sense of) the verb "direct" from which it is derived via a "morphosemantic" link. The morphology functions of the software distributed with the database try to deduce the lemma or stem form of a word from the user's input. Irregular forms are stored in a list, and looking up "ate" will return "eat," for example. == Knowledge structure == Both nouns and verbs are organized into hierarchies, defined by hypernym or IS A relationships. For instance, one sense of the word dog is found following hypernym hierarchy; the words at the same level represent synset members. Each set of synonyms has a unique index. At the top level, these hierarchies are organized into 25 beginner "trees" for nouns and 15 for verbs (called lexicographic files at a maintenance level). All are linked to a unique beginner synset, "entity". Noun hierarchies are far deeper than verb hierarchies. Adjectives are not organized into hierarchical trees. Instead, two "central" antonyms such as "hot" and "cold" form binary poles, while 'satellite' synonyms such as "steaming" and "chilly" connect to their respective poles via a "similarity" relations. The adjectives can be visualized in this way as "dumbbells" rather than as "trees". == Psycholinguistic aspects == The initial goal of the WordNet project was to build a lexical database that would be consistent with theories of human semantic memory developed in the late 1960s. Psychological experiments indicated that speakers organized their knowledge of concepts in an economic, hierarchical fashion. Retrieval time required to access conceptual knowledge seemed to be directly related to the number of hierarchies the speaker needed to "traverse" to access the knowledge. Thus, speakers could more quickly verify that canaries can sing because a canary is a songbird, but required slightly more time to verify that canaries can fly (where they had to access the concept "bird" on the superordinate level) and even more time to verify canaries have skin (requiring look-up across multiple levels of hyponymy, up to "animal"). While such psycholinguistic experiments and the underlying theories have been subject to criticism, some of WordNet's organization is consistent with experimental evidence. For example, anomic aphasia selectively affects speakers' ability to produce words from a specific semantic category, a WordNet hierarchy. Antonymous adjectives (WordNet's central adjectives in the dumbbell structure) are found to co-occur far more frequently than chance, a fact that has been found to hold for many languages. == As a lexical ontology == WordNet is sometimes called an ontology, a persistent claim that its creators do not make. The hypernym/hyponym relationships among the noun synsets can be interpreted as specialization relations among conceptual categories. In other words, WordNet can be interpreted and used as a lexical ontology in the computer science sense. However, such an ontology should be corrected before being used, because it contains hundreds of basic semantic inconsistencies; for example there are, (i) common specializations for exclusive categories and (ii) redundancies in the specialization hierarchy. Furthermore, transforming WordNet into a lexical ontology usable for knowledge representation should normally also involve (i) distinguishing the specialization relations into subtypeOf and instanceOf relations, and (ii) associating intuitive unique identifiers to each category. Although such corrections and transformations have been performed and documented as part of the integration of WordNet 1.7 into the cooperatively updatable knowledge base of WebKB-2, most projects claiming to reuse WordNet for knowledge-based applications (typically, knowledge-oriented information retrieval) simply reuse it directly. WordNet has also been converted to a formal specification, by means of a hybrid bottom-up top-down methodology to automatically extract association relations from it and interpret these associations in terms of a set of conceptual relations, formally defined in the DOLCE foundational ontology. In most works that claim to have integrated WordNet into ontologies, the content of WordNet has not simply been corrected when it seemed necessary; instead, it has been heavily reinterpreted and updated whenever suitable. This was the case when, for example, the top-level ontology of WordNet was restructured according to the OntoClean-based approach, or when it was used as a primary source for constructing the lower classes of the SENSUS ontology. == Limitations == The most widely discussed limitation of WordNet (and related resources like ImageNet) is that some of the semantic relations are more suited to concrete concepts than to abstract concepts. For example,
General Problem Solver
General Problem Solver (GPS) is a computer program created in 1957 by Herbert A. Simon, J. C. Shaw, and Allen Newell (RAND Corporation) intended to work as a universal problem solver machine. In contrast to the former Logic Theorist project, the GPS works with means–ends analysis. == Overview == Any problem that can be expressed as a set of well-formed formulas (WFFs) or Horn clauses, and that constitutes a directed graph with one or more sources (that is, hypotheses) and sinks (that is, desired conclusions), can be solved, in principle, by GPS. Proofs in the predicate logic and Euclidean geometry problem spaces are prime examples of the domain of applicability of GPS. It was based on Simon and Newell's theoretical work on logic machines. GPS was the first computer program that separated its knowledge of problems (rules represented as input data) from its strategy of how to solve problems (a generic solver engine). GPS was implemented in the third-order programming language, IPL. While GPS solved simple problems such as the Towers of Hanoi that could be sufficiently formalized, it could not solve any real-world problems because the search was easily lost in the combinatorial explosion. Put another way, the number of "walks" through the inferential digraph became computationally untenable. (In practice, even a straightforward state space search such as the Towers of Hanoi can become computationally infeasible, albeit judicious prunings of the state space can be achieved by such elementary AI techniques as A and IDA). The user defined objects and operations that could be done on the objects, and GPS generated heuristics by means–ends analysis in order to solve problems. It focused on the available operations, finding what inputs were acceptable and what outputs were generated. It then created subgoals to get closer and closer to the goal. The GPS paradigm eventually evolved into the Soar architecture for artificial intelligence.
DREAM Challenges
DREAM Challenges (Dialogue for Reverse Engineering Assessment and Methods) is a non-profit initiative for advancing biomedical and systems biology research via crowd-sourced competitions. Started in 2006, DREAM challenges collaborate with Sage Bionetworks to provide a platform for competitions run on the Synapse platform. Over 60 DREAM challenges have been conducted over the span of over 15 years. == Overview == DREAM Challenges were founded in 2006 by Gustavo Stolovizky from IBM Research and Andrea Califano from Columbia University. Current chair of the DREAM organization is Paul Boutros from University of California. Further organization spans emeritus chairs Justin Guinney and Gustavo Stolovizky, and multiple DREAM directors. Individual challenges focus on tackling a specific biomedical research question, typically narrowed down to a specific disease. A prominent disease focus has been on oncology, with multiple past challenges focused on breast cancer, acute myeloid leukemia, and prostate cancer or similar diseases. The data involved in an individual challenge reflects the disease context; while cancers typically involve data such as mutations in the human genome, gene expression and gene networks in transcriptomics, and large scale proteomics, newer challenges have shifted towards single cell sequencing technologies as well as emerging gut microbiome related research questions, thus reflecting trends in the wider research community. Motivation for DREAM Challenges is that via crowd-sourcing data to a larger audience via competitions, better models and insight is gained than if the analysis was conducted by a single entity. Past competitions have been published in such scientific venues as the flagship journals of the Nature Portfolio and PLOS publishing groups. Results of DREAM challenges are announced via web platforms, and the top performing participants are invited to present their results in the annual RECOMB/ISCB Conferences with RSG/DREAM organized by the ISCB. While DREAM Challenges have emphasized open science and data, in order to mitigate issues rising from highly sensitive data such as genomics in patient cohorts, "model to data" approaches have been adopted. In such challenges participants submit their models via containers such as Docker or Singularity. This allows retaining confidentiality of the original data as these containers are then run by the organizers on the confidential data. This differs from the more traditional open data model, where participants submit predictions directly based on the provided open data. == Challenge organization == DREAM challenge comprises a core DREAM/Sage Bionetworks organization group as well as an extended scientific expert group, who may have contributed to creation and conception of the challenge or by providing key data. Additionally, new DREAM challenges may be proposed by the wider research community. Pharmaceutical companies or other private entities may also be involved in DREAM challenges, for example in providing data. == Challenge structure == Timelines for key stages (such as introduction webinars, model submission deadlines, and final deadline for participation) are provided in advance. After the winners are announced, organizers start collaborating with the top performing participants to conduct post hoc analyses for a publication describing key findings from the competition. Challenges may be split into sub-challenges, each addressing a different subtopic within the research question. For example, regarding cancer treatment efficacy predictions, these may be separate predictions for progression-free survival, overall survival, best overall response according to RECIST, or exact time until event (progression or death). == Participation == During DREAM challenges, participants typically build models on provided data, and submit predictions or models that are then validated on held-out data by the organizers. While DREAM challenges avoid leaking validation data to participants, there are typically mid-challenge submission leaderboards available to assist participants in evaluating their performance on a sub-sampled or scrambled dataset. DREAM challenges are free for participants. During the open phase anybody can register via Synapse to participate either individually or as a team. A person may only register once and may not use any aliases. There are some exceptions, which disqualify an individual from participating, for example: Person has privileged access to the data for the particular challenge, thus providing them with an unfair advantage. Person has been caught or is under suspicion of cheating or abusing previous DREAM Challenges. Person is a minor (under age 18 or the age of majority in jurisdiction of residence). This may be alleviated via parental consent.
Similarity learning
Similarity learning is an area of supervised machine learning in artificial intelligence. It is closely related to regression and classification, but the goal is to learn a similarity function that measures how similar or related two objects are. It has applications in ranking, in recommendation systems, visual identity tracking, face verification, and speaker verification. == Learning setup == There are four common setups for similarity and metric distance learning. Regression similarity learning In this setup, pairs of objects are given ( x i 1 , x i 2 ) {\displaystyle (x_{i}^{1},x_{i}^{2})} together with a measure of their similarity y i ∈ R {\displaystyle y_{i}\in R} . The goal is to learn a function that approximates f ( x i 1 , x i 2 ) ∼ y i {\displaystyle f(x_{i}^{1},x_{i}^{2})\sim y_{i}} for every new labeled triplet example ( x i 1 , x i 2 , y i ) {\displaystyle (x_{i}^{1},x_{i}^{2},y_{i})} . This is typically achieved by minimizing a regularized loss min W ∑ i l o s s ( w ; x i 1 , x i 2 , y i ) + r e g ( w ) {\displaystyle \min _{W}\sum _{i}loss(w;x_{i}^{1},x_{i}^{2},y_{i})+reg(w)} . Classification similarity learning Given are pairs of similar objects ( x i , x i + ) {\displaystyle (x_{i},x_{i}^{+})} and non similar objects ( x i , x i − ) {\displaystyle (x_{i},x_{i}^{-})} . An equivalent formulation is that every pair ( x i 1 , x i 2 ) {\displaystyle (x_{i}^{1},x_{i}^{2})} is given together with a binary label y i ∈ { 0 , 1 } {\displaystyle y_{i}\in \{0,1\}} that determines if the two objects are similar or not. The goal is again to learn a classifier that can decide if a new pair of objects is similar or not. Ranking similarity learning Given are triplets of objects ( x i , x i + , x i − ) {\displaystyle (x_{i},x_{i}^{+},x_{i}^{-})} whose relative similarity obey a predefined order: x i {\displaystyle x_{i}} is known to be more similar to x i + {\displaystyle x_{i}^{+}} than to x i − {\displaystyle x_{i}^{-}} . The goal is to learn a function f {\displaystyle f} such that for any new triplet of objects ( x , x + , x − ) {\displaystyle (x,x^{+},x^{-})} , it obeys f ( x , x + ) > f ( x , x − ) {\displaystyle f(x,x^{+})>f(x,x^{-})} (contrastive learning). This setup assumes a weaker form of supervision than in regression, because instead of providing an exact measure of similarity, one only has to provide the relative order of similarity. For this reason, ranking-based similarity learning is easier to apply in real large-scale applications. Locality sensitive hashing (LSH) Hashes input items so that similar items map to the same "buckets" in memory with high probability (the number of buckets being much smaller than the universe of possible input items). It is often applied in nearest neighbor search on large-scale high-dimensional data, e.g., image databases, document collections, time-series databases, and genome databases. A common approach for learning similarity is to model the similarity function as a bilinear form. For example, in the case of ranking similarity learning, one aims to learn a matrix W that parametrizes the similarity function f W ( x , z ) = x T W z {\displaystyle f_{W}(x,z)=x^{T}Wz} . When data is abundant, a common approach is to learn a siamese network – a deep network model with parameter sharing. == Metric learning == Similarity learning is closely related to distance metric learning. Metric learning is the task of learning a distance function over objects. A metric or distance function has to obey four axioms: non-negativity, identity of indiscernibles, symmetry and subadditivity (or the triangle inequality). In practice, metric learning algorithms ignore the condition of identity of indiscernibles and learn a pseudo-metric. When the objects x i {\displaystyle x_{i}} are vectors in R d {\displaystyle R^{d}} , then any matrix W {\displaystyle W} in the symmetric positive semi-definite cone S + d {\displaystyle S_{+}^{d}} defines a distance pseudo-metric of the space of x through the form D W ( x 1 , x 2 ) 2 = ( x 1 − x 2 ) ⊤ W ( x 1 − x 2 ) {\displaystyle D_{W}(x_{1},x_{2})^{2}=(x_{1}-x_{2})^{\top }W(x_{1}-x_{2})} . When W {\displaystyle W} is a symmetric positive definite matrix, D W {\displaystyle D_{W}} is a metric. Moreover, as any symmetric positive semi-definite matrix W ∈ S + d {\displaystyle W\in S_{+}^{d}} can be decomposed as W = L ⊤ L {\displaystyle W=L^{\top }L} where L ∈ R e × d {\displaystyle L\in R^{e\times d}} and e ≥ r a n k ( W ) {\displaystyle e\geq rank(W)} , the distance function D W {\displaystyle D_{W}} can be rewritten equivalently D W ( x 1 , x 2 ) 2 = ( x 1 − x 2 ) ⊤ L ⊤ L ( x 1 − x 2 ) = ‖ L ( x 1 − x 2 ) ‖ 2 2 {\displaystyle D_{W}(x_{1},x_{2})^{2}=(x_{1}-x_{2})^{\top }L^{\top }L(x_{1}-x_{2})=\|L(x_{1}-x_{2})\|_{2}^{2}} . The distance D W ( x 1 , x 2 ) 2 = ‖ x 1 ′ − x 2 ′ ‖ 2 2 {\displaystyle D_{W}(x_{1},x_{2})^{2}=\|x_{1}'-x_{2}'\|_{2}^{2}} corresponds to the Euclidean distance between the transformed feature vectors x 1 ′ = L x 1 {\displaystyle x_{1}'=Lx_{1}} and x 2 ′ = L x 2 {\displaystyle x_{2}'=Lx_{2}} . Many formulations for metric learning have been proposed. Some well-known approaches for metric learning include learning from relative comparisons, which is based on the triplet loss, large margin nearest neighbor, and information theoretic metric learning (ITML). In statistics, the covariance matrix of the data is sometimes used to define a distance metric called Mahalanobis distance. == Applications == Similarity learning is used in information retrieval for learning to rank, in face verification or face identification, and in recommendation systems. Also, many machine learning approaches rely on some metric. This includes unsupervised learning such as clustering, which groups together close or similar objects. It also includes supervised approaches like K-nearest neighbor algorithm which rely on labels of nearby objects to decide on the label of a new object. Metric learning has been proposed as a preprocessing step for many of these approaches. == Scalability == Metric and similarity learning scale quadratically with the dimension of the input space, as can easily see when the learned metric has a bilinear form f W ( x , z ) = x T W z {\displaystyle f_{W}(x,z)=x^{T}Wz} . Scaling to higher dimensions can be achieved by enforcing a sparseness structure over the matrix model, as done with HDSL, and with COMET. == Software == metric-learn is a free software Python library which offers efficient implementations of several supervised and weakly-supervised similarity and metric learning algorithms. The API of metric-learn is compatible with scikit-learn. OpenMetricLearning is a Python framework to train and validate the models producing high-quality embeddings. == Further information == For further information on this topic, see the surveys on metric and similarity learning by Bellet et al. and Kulis.
Learning vector quantization
In computer science, learning vector quantization (LVQ) is a prototype-based supervised classification algorithm. LVQ is the supervised counterpart of vector quantization systems. LVQ can be understood as a special case of an artificial neural network, more precisely, it applies a winner-take-all Hebbian learning-based approach. It is a precursor to self-organizing maps (SOM) and related to neural gas and the k-nearest neighbor algorithm (k-NN). LVQ was invented by Teuvo Kohonen. == Definition == An LVQ system is represented by prototypes W = ( w ( i ) , . . . , w ( n ) ) {\displaystyle W=(w(i),...,w(n))} which are defined in the feature space of observed data. In winner-take-all training algorithms one determines, for each data point, the prototype which is closest to the input according to a given distance measure. The position of this so-called winner prototype is then adapted, i.e. the winner is moved closer if it correctly classifies the data point or moved away if it classifies the data point incorrectly. An advantage of LVQ is that it creates prototypes that are easy to interpret for experts in the respective application domain. LVQ systems can be applied to multi-class classification problems in a natural way. A key issue in LVQ is the choice of an appropriate measure of distance or similarity for training and classification. Recently, techniques have been developed which adapt a parameterized distance measure in the course of training the system, see e.g. (Schneider, Biehl, and Hammer, 2009) and references therein. LVQ can be a valuable aid in classifying text documents. == Algorithm == The algorithms are presented as in. Set up: Let the data be denoted by x i ∈ R D {\displaystyle x_{i}\in \mathbb {R} ^{D}} , and their corresponding labels by y i ∈ { 1 , 2 , … , C } {\displaystyle y_{i}\in \{1,2,\dots ,C\}} . The complete dataset is { ( x i , y i ) } i = 1 N {\displaystyle \{(x_{i},y_{i})\}_{i=1}^{N}} . The set of code vectors is w j ∈ R D {\displaystyle w_{j}\in \mathbb {R} ^{D}} . The learning rate at iteration step t {\displaystyle t} is denoted by α t {\displaystyle \alpha _{t}} . The hyperparameters w {\displaystyle w} and ϵ {\displaystyle \epsilon } are used by LVQ2 and LVQ3. The original paper suggests ϵ ∈ [ 0.1 , 0.5 ] {\displaystyle \epsilon \in [0.1,0.5]} and w ∈ [ 0.2 , 0.3 ] {\displaystyle w\in [0.2,0.3]} . === LVQ1 === Initialize several code vectors per label. Iterate until convergence criteria is reached. Sample a datum x i {\displaystyle x_{i}} , and find out the code vector w j {\displaystyle w_{j}} , such that x i {\displaystyle x_{i}} falls within the Voronoi cell of w j {\displaystyle w_{j}} . If its label y i {\displaystyle y_{i}} is the same as that of w j {\displaystyle w_{j}} , then w j ← w j + α t ( x i − w j ) {\displaystyle w_{j}\leftarrow w_{j}+\alpha _{t}(x_{i}-w_{j})} , otherwise, w j ← w j − α t ( x i − w j ) {\displaystyle w_{j}\leftarrow w_{j}-\alpha _{t}(x_{i}-w_{j})} . === LVQ2 === LVQ2 is the same as LVQ3, but with this sentence removed: "If w j {\displaystyle w_{j}} and w k {\displaystyle w_{k}} and x i {\displaystyle x_{i}} have the same class, then w j ← w j − α t ( x i − w j ) {\displaystyle w_{j}\leftarrow w_{j}-\alpha _{t}(x_{i}-w_{j})} and w k ← w k + α t ( x i − w k ) {\displaystyle w_{k}\leftarrow w_{k}+\alpha _{t}(x_{i}-w_{k})} .". If w j {\displaystyle w_{j}} and w k {\displaystyle w_{k}} and x i {\displaystyle x_{i}} have the same class, then nothing happens. === LVQ3 === Initialize several code vectors per label. Iterate until convergence criteria is reached. Sample a datum x i {\displaystyle x_{i}} , and find out two code vectors w j , w k {\displaystyle w_{j},w_{k}} closest to it. Let d j := ‖ x i − w j ‖ , d k := ‖ x i − w k ‖ {\displaystyle d_{j}:=\|x_{i}-w_{j}\|,d_{k}:=\|x_{i}-w_{k}\|} . If min ( d j d k , d k d j ) > s {\displaystyle \min \left({\frac {d_{j}}{d_{k}}},{\frac {d_{k}}{d_{j}}}\right)>s} , where s = 1 − w 1 + w {\displaystyle s={\frac {1-w}{1+w}}} , then If w j {\displaystyle w_{j}} and x i {\displaystyle x_{i}} have the same class, and w k {\displaystyle w_{k}} and x i {\displaystyle x_{i}} have different classes, then w j ← w j + α t ( x i − w j ) {\displaystyle w_{j}\leftarrow w_{j}+\alpha _{t}(x_{i}-w_{j})} and w k ← w k − α t ( x i − w k ) {\displaystyle w_{k}\leftarrow w_{k}-\alpha _{t}(x_{i}-w_{k})} . If w k {\displaystyle w_{k}} and x i {\displaystyle x_{i}} have the same class, and w j {\displaystyle w_{j}} and x i {\displaystyle x_{i}} have different classes, then w j ← w j − α t ( x i − w j ) {\displaystyle w_{j}\leftarrow w_{j}-\alpha _{t}(x_{i}-w_{j})} and w k ← w k + α t ( x i − w k ) {\displaystyle w_{k}\leftarrow w_{k}+\alpha _{t}(x_{i}-w_{k})} . If w j {\displaystyle w_{j}} and w k {\displaystyle w_{k}} and x i {\displaystyle x_{i}} have the same class, then w j ← w j − ϵ α t ( x i − w j ) {\displaystyle w_{j}\leftarrow w_{j}-\epsilon \alpha _{t}(x_{i}-w_{j})} and w k ← w k + ϵ α t ( x i − w k ) {\displaystyle w_{k}\leftarrow w_{k}+\epsilon \alpha _{t}(x_{i}-w_{k})} . If w k {\displaystyle w_{k}} and x i {\displaystyle x_{i}} have different classes, and w j {\displaystyle w_{j}} and x i {\displaystyle x_{i}} have different classes, then the original paper simply does not explain what happens in this case, but presumably nothing happens in this case. Otherwise, skip. Note that condition min ( d j d k , d k d j ) > s {\displaystyle \min \left({\frac {d_{j}}{d_{k}}},{\frac {d_{k}}{d_{j}}}\right)>s} , where s = 1 − w 1 + w {\displaystyle s={\frac {1-w}{1+w}}} , precisely means that the point x i {\displaystyle x_{i}} falls between two Apollonian spheres.