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Landmark point
In morphometrics, landmark point or shortly landmark is a point in a shape object in which correspondences between and within the populations of the object are preserved. In other disciplines, landmarks may be known as vertices, anchor points, control points, sites, profile points, 'sampling' points, nodes, markers, fiducial markers, etc. Landmarks can be defined either manually by experts or automatically by a computer program. There are three basic types of landmarks: anatomical landmarks, mathematical landmarks or pseudo-landmarks. An anatomical landmark is a biologically-meaningful point in an organism. Usually experts define anatomical points to ensure their correspondences within the same species. Examples of anatomical landmark in shape of a skull are the eye corner, tip of the nose, jaw, etc. Anatomical landmarks determine homologous parts of an organism, which share a common ancestry. Mathematical landmarks are points in a shape that are located according to some mathematical or geometrical property, for instance, a high curvature point or an extreme point. A computer program usually determines mathematical landmarks used for an automatic pattern recognition. Pseudo-landmarks are constructed points located between anatomical or mathematical landmarks. A typical example is an equally spaced set of points between two anatomical landmarks to get more sample points from a shape. Pseudo-landmarks are useful during shape matching, when the matching process requires a large number of points.
Determining the number of clusters in a data set
Determining the number of clusters in a data set, a quantity often labelled k as in the k-means algorithm, is a frequent problem in data clustering, and is a distinct issue from the process of actually solving the clustering problem. For a certain class of clustering algorithms (in particular k-means, k-medoids and expectation–maximization algorithm), there is a parameter commonly referred to as k that specifies the number of clusters to detect. Other algorithms such as DBSCAN and OPTICS algorithm do not require the specification of this parameter; hierarchical clustering avoids the problem altogether. The correct choice of k is often ambiguous, with interpretations depending on the shape and scale of the distribution of points in a data set and the desired clustering resolution of the user. In addition, increasing k without penalty will always reduce the amount of error in the resulting clustering, to the extreme case of zero error if each data point is considered its own cluster (i.e., when k equals the number of data points, n). Intuitively then, the optimal choice of k will strike a balance between maximum compression of the data using a single cluster, and maximum accuracy by assigning each data point to its own cluster. If an appropriate value of k is not apparent from prior knowledge of the properties of the data set, it must be chosen somehow. There are several categories of methods for making this decision. == Elbow method == The elbow method looks at the percentage of explained variance as a function of the number of clusters: One should choose a number of clusters so that adding another cluster does not give much better modeling of the data. More precisely, if one plots the percentage of variance explained by the clusters against the number of clusters, the first clusters will add much information (explain a lot of variance), but at some point the marginal gain will drop, giving an angle in the graph. The number of clusters is chosen at this point, hence the "elbow criterion". In most datasets, this "elbow" is ambiguous, making this method subjective and unreliable. Because the scale of the axes is arbitrary, the concept of an angle is not well-defined, and even on uniform random data, the curve produces an "elbow", making the method rather unreliable. Percentage of variance explained is the ratio of the between-group variance to the total variance, also known as an F-test. A slight variation of this method plots the curvature of the within group variance. The method can be traced to speculation by Robert L. Thorndike in 1953. While the idea of the elbow method sounds simple and straightforward, other methods (as detailed below) give better results. == X-means clustering == In statistics and data mining, X-means clustering is a variation of k-means clustering that refines cluster assignments by repeatedly attempting subdivision, and keeping the best resulting splits, until a criterion such as the Akaike information criterion (AIC) or Bayesian information criterion (BIC) is reached. == Information criterion approach == Another set of methods for determining the number of clusters are information criteria, such as the Akaike information criterion (AIC), Bayesian information criterion (BIC), or the deviance information criterion (DIC) — if it is possible to make a likelihood function for the clustering model. For example: The k-means model is "almost" a Gaussian mixture model and one can construct a likelihood for the Gaussian mixture model and thus also determine information criterion values. == Information–theoretic approach == Rate distortion theory has been applied to choosing k called the "jump" method, which determines the number of clusters that maximizes efficiency while minimizing error by information-theoretic standards. The strategy of the algorithm is to generate a distortion curve for the input data by running a standard clustering algorithm such as k-means for all values of k between 1 and n, and computing the distortion (described below) of the resulting clustering. The distortion curve is then transformed by a negative power chosen based on the dimensionality of the data. Jumps in the resulting values then signify reasonable choices for k, with the largest jump representing the best choice. The distortion of a clustering of some input data is formally defined as follows: Let the data set be modeled as a p-dimensional random variable, X, consisting of a mixture distribution of G components with common covariance, Γ. If we let c 1 … c K {\displaystyle c_{1}\ldots c_{K}} be a set of K cluster centers, with c X {\displaystyle c_{X}} the closest center to a given sample of X, then the minimum average distortion per dimension when fitting the K centers to the data is: d K = 1 p min c 1 … c K E [ ( X − c X ) T Γ − 1 ( X − c X ) ] {\displaystyle d_{K}={\frac {1}{p}}\min _{c_{1}\ldots c_{K}}{E[(X-c_{X})^{T}\Gamma ^{-1}(X-c_{X})]}} This is also the average Mahalanobis distance per dimension between X and the closest cluster center c X {\displaystyle c_{X}} . Because the minimization over all possible sets of cluster centers is prohibitively complex, the distortion is computed in practice by generating a set of cluster centers using a standard clustering algorithm and computing the distortion using the result. The pseudo-code for the jump method with an input set of p-dimensional data points X is: JumpMethod(X): Let Y = (p/2) Init a list D, of size n+1 Let D[0] = 0 For k = 1 ... n: Cluster X with k clusters (e.g., with k-means) Let d = Distortion of the resulting clustering D[k] = d^(-Y) Define J(i) = D[i] - D[i-1] Return the k between 1 and n that maximizes J(k) The choice of the transform power Y = ( p / 2 ) {\displaystyle Y=(p/2)} is motivated by asymptotic reasoning using results from rate distortion theory. Let the data X have a single, arbitrarily p-dimensional Gaussian distribution, and let fixed K = ⌊ α p ⌋ {\displaystyle K=\lfloor \alpha ^{p}\rfloor } , for some α greater than zero. Then the distortion of a clustering of K clusters in the limit as p goes to infinity is α − 2 {\displaystyle \alpha ^{-2}} . It can be seen that asymptotically, the distortion of a clustering to the power ( − p / 2 ) {\displaystyle (-p/2)} is proportional to α p {\displaystyle \alpha ^{p}} , which by definition is approximately the number of clusters K. In other words, for a single Gaussian distribution, increasing K beyond the true number of clusters, which should be one, causes a linear growth in distortion. This behavior is important in the general case of a mixture of multiple distribution components. Let X be a mixture of G p-dimensional Gaussian distributions with common covariance. Then for any fixed K less than G, the distortion of a clustering as p goes to infinity is infinite. Intuitively, this means that a clustering of less than the correct number of clusters is unable to describe asymptotically high-dimensional data, causing the distortion to increase without limit. If, as described above, K is made an increasing function of p, namely, K = ⌊ α p ⌋ {\displaystyle K=\lfloor \alpha ^{p}\rfloor } , the same result as above is achieved, with the value of the distortion in the limit as p goes to infinity being equal to α − 2 {\displaystyle \alpha ^{-2}} . Correspondingly, there is the same proportional relationship between the transformed distortion and the number of clusters, K. Putting the results above together, it can be seen that for sufficiently high values of p, the transformed distortion d K − p / 2 {\displaystyle d_{K}^{-p/2}} is approximately zero for K < G, then jumps suddenly and begins increasing linearly for K ≥ G. The jump algorithm for choosing K makes use of these behaviors to identify the most likely value for the true number of clusters. Although the mathematical support for the method is given in terms of asymptotic results, the algorithm has been empirically verified to work well in a variety of data sets with reasonable dimensionality. In addition to the localized jump method described above, there exists a second algorithm for choosing K using the same transformed distortion values known as the broken line method. The broken line method identifies the jump point in the graph of the transformed distortion by doing a simple least squares error line fit of two line segments, which in theory will fall along the x-axis for K < G, and along the linearly increasing phase of the transformed distortion plot for K ≥ G. The broken line method is more robust than the jump method in that its decision is global rather than local, but it also relies on the assumption of Gaussian mixture components, whereas the jump method is fully non-parametric and has been shown to be viable for general mixture distributions. == Silhouette method == The average silhouette of the data is another useful criterion for assessing the natural number of clusters. The silhouette of a data instance is a measure of how closely it is match
Language identification in the limit
Language identification in the limit is a formal model for inductive inference of formal languages, mainly by computers (see machine learning and induction of regular languages). It was introduced by E. Mark Gold in a technical report and a journal article with the same title. In this model, a teacher provides to a learner some presentation (i.e. a sequence of strings) of some formal language. The learning is seen as an infinite process. Each time the learner reads an element of the presentation, it should provide a representation (e.g. a formal grammar) for the language. Gold defines that a learner can identify in the limit a class of languages if, given any presentation of any language in the class, the learner will produce only a finite number of wrong representations, and then stick with the correct representation. However, the learner need not be able to announce its correctness; and the teacher might present a counterexample to any representation arbitrarily long after. Gold defined two types of presentations: Text (positive information): an enumeration of all strings the language consists of. Complete presentation (positive and negative information): an enumeration of all possible strings, each with a label indicating if the string belongs to the language or not. == Learnability == This model is an early attempt to formally capture the notion of learnability. Gold's journal article introduces for contrast the stronger models Finite identification (where the learner has to announce correctness after a finite number of steps), and Fixed-time identification (where correctness has to be reached after an apriori-specified number of steps). A weaker formal model of learnability is the Probably approximately correct learning (PAC) model, introduced by Leslie Valiant in 1984. == Examples == It is instructive to look at concrete examples (in the tables) of learning sessions the definition of identification in the limit speaks about. A fictitious session to learn a regular language L over the alphabet {a,b} from text presentation:In each step, the teacher gives a string belonging to L, and the learner answers a guess for L, encoded as a regular expression. In step 3, the learner's guess is not consistent with the strings seen so far; in step 4, the teacher gives a string repeatedly. After step 6, the learner sticks to the regular expression (ab+ba). If this happens to be a description of the language L the teacher has in mind, it is said that the learner has learned that language.If a computer program for the learner's role would exist that was able to successfully learn each regular language, that class of languages would be identifiable in the limit. Gold has shown that this is not the case. A particular learning algorithm always guessing L to be just the union of all strings seen so far:If L is a finite language, the learner will eventually guess it correctly, however, without being able to tell when. Although the guess didn't change during step 3 to 6, the learner couldn't be sure to be correct.Gold has shown that the class of finite languages is identifiable in the limit, however, this class is neither finitely nor fixed-time identifiable. Learning from complete presentation by telling:In each step, the teacher gives a string and tells whether it belongs to L (green) or not (red, struck-out). Each possible string is eventually classified in this way by the teacher. Learning from complete presentation by request:The learner gives a query string, the teacher tells whether it belongs to L (yes) or not (no); the learner then gives a guess for L, followed by the next query string. In this example, the learner happens to query in each step just the same string as given by the teacher in example 3.In general, Gold has shown that each language class identifiable in the request-presentation setting is also identifiable in the telling-presentation setting, since the learner, instead of querying a string, just needs to wait until it is eventually given by the teacher. == Gold's theorem == More formally, a language L {\displaystyle L} is a nonempty set, and its elements are called sentences. a language family is a set of languages. a language-learning environment E {\displaystyle E} for a language L {\displaystyle L} is a stream of sentences from L {\displaystyle L} , such that each sentence in L {\displaystyle L} appears at least once. a language learner is a function f {\displaystyle f} that sends a list of sentences to a language. This is interpreted as saying that, after seeing sentences a 1 , a 2 . . . , a n {\displaystyle a_{1},a_{2}...,a_{n}} in that order, the language learner guesses that the language that produces the sentences should be f ( a 1 , . . . , a n ) {\displaystyle f(a_{1},...,a_{n})} . Note that the learner is not obliged to be correct — it could very well guess a language that does not even contain a 1 , . . . , a n {\displaystyle a_{1},...,a_{n}} . a language learner f {\displaystyle f} learns a language L {\displaystyle L} in environment E = ( a 1 , a 2 , . . . ) {\displaystyle E=(a_{1},a_{2},...)} if the learner always guesses L {\displaystyle L} after seeing enough examples from the environment. a language learner f {\displaystyle f} learns a language L {\displaystyle L} if it learns L {\displaystyle L} in any environment E {\displaystyle E} for L {\displaystyle L} . a language family is learnable if there exists a language learner that can learn all languages in the family. Notes: In the context of Gold's theorem, sentences need only be distinguishable. They need not be anything in particular, such as finite strings (as usual in formal linguistics). Learnability is not a concept for individual languages. Any individual language L {\displaystyle L} could be learned by a trivial learner that always guesses L {\displaystyle L} . Learnability is not a concept for individual learners. A language family is learnable if, and only if, there exists some learner that can learn the family. It does not matter how well the learner performs for learning languages outside the family. Gold's theorem is easily bypassed if negative examples are allowed. In particular, the language family { L 1 , L 2 , . . . , L ∞ } {\displaystyle \{L_{1},L_{2},...,L_{\infty }\}} can be learned by a learner that always guesses L ∞ {\displaystyle L_{\infty }} until it receives the first negative example ¬ a n {\displaystyle \neg a_{n}} , where a n ∈ L n + 1 ∖ L n {\displaystyle a_{n}\in L_{n+1}\setminus L_{n}} , at which point it always guesses L n {\displaystyle L_{n}} . == Learnability characterization == Dana Angluin gave the characterizations of learnability from text (positive information) in a 1980 paper. If a learner is required to be effective, then an indexed class of recursive languages is learnable in the limit if there is an effective procedure that uniformly enumerates tell-tales for each language in the class (Condition 1). It is not hard to see that if an ideal learner (i.e., an arbitrary function) is allowed, then an indexed class of languages is learnable in the limit if each language in the class has a tell-tale (Condition 2). == Language classes learnable in the limit == The table shows which language classes are identifiable in the limit in which learning model. On the right-hand side, each language class is a superclass of all lower classes. Each learning model (i.e. type of presentation) can identify in the limit all classes below it. In particular, the class of finite languages is identifiable in the limit by text presentation (cf. Example 2 above), while the class of regular languages is not. Pattern Languages, introduced by Dana Angluin in another 1980 paper, are also identifiable by normal text presentation; they are omitted in the table, since they are above the singleton and below the primitive recursive language class, but incomparable to the classes in between. == Sufficient conditions for learnability == Condition 1 in Angluin's paper is not always easy to verify. Therefore, people come up with various sufficient conditions for the learnability of a language class. See also Induction of regular languages for learnable subclasses of regular languages. === Finite thickness === A class of languages has finite thickness if every non-empty set of strings is contained in at most finitely many languages of the class. This is exactly Condition 3 in Angluin's paper. Angluin showed that if a class of recursive languages has finite thickness, then it is learnable in the limit. A class with finite thickness certainly satisfies MEF-condition and MFF-condition; in other words, finite thickness implies M-finite thickness. === Finite elasticity === A class of languages is said to have finite elasticity if for every infinite sequence of strings s 0 , s 1 , . . . {\displaystyle s_{0},s_{1},...} and every infinite sequence of languages in the class L 1 , L 2 , . . . {\displaystyle L_{1},L_{2},...} , there exists a finite number n such
Ni1000
The Ni1000 is an artificial neural network chip developed by Nestor Corporation and Intel, developed in the 1990s. It is Intel's second-generation neural network chip, but the first all-digital chip. The chip is aimed at image analysis applications– containing more than 3 million transistors – and can analyze 40,000 patterns per second. Prototypes running Nestor's OCR software in 1994 were capable of recognizing around 100 handwritten characters per second. The development was funded with money from DARPA and Office of Naval Research.
Human-in-the-loop
Human-in-the-loop (HITL) is used in multiple contexts. It can be defined as a model requiring human interaction. HITL is associated with modeling and simulation (M&S) in the live, virtual, and constructive taxonomy. HITL, along with the related human-on-the-loop, are also used in relation to lethal autonomous weapons. Further, HITL is used in the context of machine learning.It is also used in conversational AI to manage complex interactions that require human empathy. == Machine learning == In machine learning, HITL is used in the sense of humans aiding the computer in making the correct decisions in building a model. HITL improves machine learning over random sampling by selecting the most critical data needed to refine the model. == Simulation == In simulation, HITL models may conform to human factors requirements as in the case of a mockup. In this type of simulation, a human is always part of the simulation and consequently influences the outcome in such a way that is difficult if not impossible to reproduce exactly. HITL also readily allows for the identification of problems and requirements that may not be easily identified by other means of simulation. HITL is often referred to as an interactive simulation, which is a special kind of physical simulation in which physical simulations include human operators, such as in a flight or a driving simulator. === Benefits === Human-in-the-loop allows the user to change the outcome of an event or process. The immersion effectively contributes to a positive transfer of acquired skills into the real world. This can be demonstrated by trainees utilizing flight simulators in preparation to become pilots. HITL also allows for the acquisition of knowledge regarding how a new process may affect a particular event. Utilizing HITL allows participants to interact with realistic models and attempt to perform as they would in an actual scenario. HITL simulations bring to the surface issues that would not otherwise be apparent until after a new process has been deployed. A real-world example of HITL simulation as an evaluation tool is its usage by the Federal Aviation Administration (FAA) to allow air traffic controllers to test new automation procedures by directing the activities of simulated air traffic while monitoring the effect of the newly implemented procedures. As with most processes, there is always the possibility of human error, which can only be reproduced using HITL simulation. Although much can be done to automate systems, humans typically still need to take the information provided by a system to determine the next course of action based on their judgment and experience. Intelligent systems can only go so far in certain circumstances to automate a process; only humans in the simulation can accurately judge the final design. Tabletop simulation may be useful in the very early stages of project development for the purpose of collecting data to set broad parameters, but the important decisions require human-in-the-loop simulation. HITL reflects scenarios where human input remains essential despite advances in automation. === Within the virtual simulation taxonomy === Virtual simulations inject HITL in a central role by exercising motor control skills (e.g. flying an airplane), decision making skills (e.g. committing fire control resources to action), or communication skills (e.g. as members of a C4I team). === Examples === Flight simulators Driving simulators Marine simulators Video games Supply chain management simulators Digital puppetry === Misconceptions === Although human-in-the-loop simulation can include a computer simulation in the form of a synthetic environment, computer simulation is not necessarily a form of human-in-the-loop simulation, and is often considered as human-out-of-the loop simulation. In this particular case, a computer model’s behavior is modified according to a set of initial parameters. The results of the model differ from the results stemming from a true human-in-the-loop simulation because the results can easily be replicated time and time again, by simply providing identical parameters. == Weapons == === Taxonomy === Three classifications of the degree of human control of autonomous weapon systems were laid out by Bonnie Docherty in a 2012 Human Rights Watch report. human-in-the-loop: a human must instigate the action of the weapon (in other words not fully autonomous) human-on-the-loop: a human may abort an action human-out-of-the-loop: no human action is involved === Positive human action === In discussions of autonomous weapons and nuclear command and control, the phrase positive human action has been used alongside "human-in-the-loop" to emphasize that a human operator must affirmatively authorize the use of force. Descriptions of the United States Navy's Aegis Combat System have used the phrase in characterizing a requirement for affirmative human action to initiate live firing. A survey of autonomous weapons systems described the Aegis "Auto SM" mode as one in which "the system fully develops the engagement process however engagement requires positive human action". The phrase entered United States federal law in the National Defense Authorization Act for Fiscal Year 2025, which stipulates that artificial intelligence systems not compromise "the principle of requiring positive human actions in execution of decisions by the President with respect to the employment of nuclear weapons".
Word2vec
Word2vec is a technique in natural language processing for obtaining vector representations of words. These vectors capture information about the meaning of the word based on the surrounding words. The word2vec algorithm estimates these representations by modeling text in a large corpus. Once trained, such a model can detect synonymous words or suggest additional words for a partial sentence. Word2vec was developed by Tomáš Mikolov, Kai Chen, Greg Corrado, Ilya Sutskever and Jeff Dean at Google, and published in 2013. Word2vec represents a word as a high-dimension vector of numbers which capture relationships between words. In particular, words which appear in similar contexts are mapped to vectors which are nearby as measured by cosine similarity. This indicates the level of semantic similarity between the words, so for example the vectors for walk and ran are nearby, as are those for "but" and "however", and "Berlin" and "Germany". == Approach == Word2vec is a group of related models that are used to produce word embeddings. These models are shallow, two-layer neural networks that are trained to reconstruct linguistic contexts of words. Word2vec takes as its input a large corpus of text and produces a mapping of the set of words to a vector space, typically of several hundred dimensions, with each unique word in the corpus being assigned a vector in the space. Word2vec can use either of two model architectures to produce these distributed representations of words: continuous bag of words (CBOW) or continuously sliding skip-gram. In both architectures, word2vec considers both individual words and a sliding context window as it iterates over the corpus. The CBOW can be viewed as a 'fill in the blank' task, where the word embedding represents the way the word influences the relative probabilities of other words in the context window. Words which are semantically similar should influence these probabilities in similar ways, because semantically similar words should be used in similar contexts. The order of context words does not influence prediction (bag of words assumption). In the continuous skip-gram architecture, the model uses the current word to predict the surrounding window of context words. The skip-gram architecture weighs nearby context words more heavily than more distant context words. According to the authors' note, CBOW is faster while skip-gram does a better job for infrequent words. After the model is trained, the learned word embeddings are positioned in the vector space such that words that share common contexts in the corpus — that is, words that are semantically and syntactically similar — are located close to one another in the space. More dissimilar words are located farther from one another in the space. == Mathematical details == This section is based on expositions. A corpus is a sequence of words. Both CBOW and skip-gram are methods to learn one vector per word appearing in the corpus. Let V {\displaystyle V} ("vocabulary") be the set of all words appearing in the corpus C {\displaystyle C} . Our goal is to learn one vector v w ∈ R d {\displaystyle v_{w}\in \mathbb {R} ^{d}} for each word w ∈ V {\displaystyle w\in V} . The idea of skip-gram is that the vector of a word should be close to the vector of each of its neighbors. The idea of CBOW is that the vector-sum of a word's neighbors should be close to the vector of the word. === Continuous bag-of-words (CBOW) === The idea of CBOW is to represent each word with a vector, such that it is possible to predict a word using the sum of the vectors of its neighbors. Specifically, for each word w i {\displaystyle w_{i}} in the corpus, the one-hot encoding of the word is used as the input to the neural network. The output of the neural network is a probability distribution over the dictionary, representing a prediction of individual words in the neighborhood of w i {\displaystyle w_{i}} . The objective of training is to maximize ∑ i ln Pr ( w i ∣ w i + j : j ∈ N ) {\displaystyle \sum _{i}\ln \Pr(w_{i}\mid w_{i+j}\colon j\in N)} where N {\displaystyle N} is a set of (non-zero) indices representing the relative locations of nearby words considered to be in w i {\displaystyle w_{i}} 's neighborhood. For example, if we want each word in the corpus to be predicted by every other word in a small span of 4 words. The set of relative indexes of neighbor words will be: N = { − 2 , − 1 , + 1 , + 2 } {\displaystyle N=\{-2,-1,+1,+2\}} , and the objective is to maximize ∑ i ln Pr ( w i ∣ w i − 2 , w i − 1 , w i + 1 , w i + 2 ) {\displaystyle \sum _{i}\ln \Pr(w_{i}\mid w_{i-2},w_{i-1},w_{i+1},w_{i+2})} . In standard bag-of-words, a word's context is represented by a word-count (aka a word histogram) of its neighboring words. For example, the "sat" in "the cat sat on the mat" is represented as {"the": 2, "cat": 1, "on": 1}. Note that the last word "mat" is not used to represent "sat", because it is outside the neighborhood N = { − 2 , − 1 , + 1 , + 2 } {\displaystyle N=\{-2,-1,+1,+2\}} . In continuous bag-of-words, the histogram is multiplied by a matrix V {\displaystyle V} to obtain a continuous representation of the word's context. The matrix V {\displaystyle V} is also called a dictionary. Its columns are the word vectors. It has D {\displaystyle D} columns, where D {\displaystyle D} is the size of the dictionary. Let d {\displaystyle d} be the length of each word vector. We have V ∈ R d × D {\displaystyle V\in \mathbb {R} ^{d\times D}} . For example, multiplying the word histogram {"the": 2, "cat": 1, "on": 1} with V {\displaystyle V} , we obtain 2 v the + v cat + v on {\displaystyle 2v_{\text{the}}+v_{\text{cat}}+v_{\text{on}}} . This is then multiplied with another matrix V ′ {\displaystyle V'} of shape R D × d {\displaystyle \mathbb {R} ^{D\times d}} . Each row of it is a word vector v ′ {\displaystyle v'} . This results in a vector of length D {\displaystyle D} , one entry per dictionary entry. Then, apply the softmax to obtain a probability distribution over the dictionary. This system can be visualized as a neural network, similar in spirit to an autoencoder, of architecture linear-linear-softmax, as depicted in the diagram. The system is trained by gradient descent to minimize the cross-entropy loss. In full formula, the cross-entropy loss is: − ∑ i ln e v w i ′ ⋅ ( ∑ j ∈ N v w j + i ) ∑ w ′ e v w ′ ′ ⋅ ( ∑ j ∈ N v w j + i ) {\displaystyle -\sum _{i}\ln {\frac {e^{v_{w_{i}}'\cdot (\sum _{j\in N}v_{w_{j+i}})}}{\sum _{w'}e^{v_{w'}'\cdot (\sum _{j\in N}v_{w_{j+i}})}}}} where the outer summation ∑ i {\displaystyle \sum _{i}} is over the words in a corpus, the quantity ∑ j ∈ N v w j + i {\displaystyle \sum _{j\in N}v_{w_{j+i}}} is the sum of a word's neighbors' vectors, etc. Once such a system is trained, we have two trained matrices V , V ′ {\displaystyle V,V'} . Either the column vectors of V {\displaystyle V} or the row vectors of V ′ {\displaystyle V'} can serve as the dictionary. For example, the word "sat" can be represented as either the "sat"-th column of V {\displaystyle V} or the "sat"-th row of V ′ {\displaystyle V'} . It is also possible to simply define V ′ = V ⊤ {\displaystyle V'=V^{\top }} , in which case there would no longer be a choice. === Skip-gram === The idea of skip-gram is to represent each word with a vector, such that it is possible to predict the vectors of its neighbors using the vector of a word. The architecture is still linear-linear-softmax, the same as CBOW, but the input and the output are switched. Specifically, for each word w i {\displaystyle w_{i}} in the corpus, the one-hot encoding of the word is used as the input to the neural network. The output of the neural network is a probability distribution over the dictionary, representing a prediction of individual words in the neighborhood of w i {\displaystyle w_{i}} . The objective of training is to maximize ∑ i ∑ j ∈ N ln Pr ( w j + i ∣ w i ) {\displaystyle \sum _{i}\sum _{j\in N}\ln \Pr(w_{j+i}\mid w_{i})} . In full formula, the loss function is − ∑ i ∑ j ∈ N ln e v w j + i ′ ⋅ v w i ∑ w ′ e v w ′ ′ ⋅ v w i {\displaystyle -\sum _{i}\sum _{j\in N}\ln {\frac {e^{v_{w_{j+i}}'\cdot v_{w_{i}}}}{\sum _{w'}e^{v_{w'}'\cdot v_{w_{i}}}}}} Same as CBOW, once such a system is trained, we have two trained matrices V , V ′ {\displaystyle V,V'} . Either the column vectors of V {\displaystyle V} or the row vectors of V ′ {\displaystyle V'} can serve as the dictionary. It is also possible to simply define V ′ = V ⊤ {\displaystyle V'=V^{\top }} , in which case there would no longer be a choice. Essentially, skip-gram and CBOW are exactly the same in architecture. They only differ in the objective function during training. == History == During the 1980s, there were some early attempts at using neural networks to represent words and concepts as vectors. In 2010, Tomáš Mikolov (then at Brno University of Technology) with co-authors applied a simple recurrent neural network with a single hidden