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  Owain Evans

  Owain Rhys Evans is a British artificial intelligence researcher who works on AI alignment and machine learning safety. He founded Truthful AI, a research group based in Berkeley, California, and is an affiliate of the Center for Human Compatible AI (CHAI) at the University of California, Berkeley. His research addresses AI truthfulness, emergent behaviors in large language models, and the alignment of AI systems with human values. == Education == Evans earned a Bachelor of Arts in philosophy and mathematics from Columbia University in 2008 and a PhD in philosophy from the Massachusetts Institute of Technology in 2015. His doctoral research focused on Bayesian computational models of human preferences and decision-making. == Career == After completing his doctorate, Evans held positions at the Future of Humanity Institute (FHI) at the University of Oxford, first as a postdoctoral research fellow and later as a research scientist. While at FHI, he co-authored a survey of machine learning researchers on timelines for human-level AI, published in the Journal of Artificial Intelligence Research. The survey was reported on by Newsweek, New Scientist, the BBC, and The Economist. He was also among the co-authors of a 2018 report on the potential for misuse of AI technologies, published by researchers at Oxford, Cambridge, and other institutions. Since 2022, Evans has been based in Berkeley, where he founded Truthful AI, a non-profit research group that studies AI truthfulness, deception, and emergent behaviors in large language models. == Research == Evans's early work examined challenges in inverse reinforcement learning when human behavior is irrational or biased, proposing methods for AI systems to infer preferences from imperfect human demonstrations. He co-developed TruthfulQA (2021), a benchmark that tests whether language models give truthful answers rather than repeating common misconceptions. Initial evaluations found that larger models were not more truthful, suggesting that scaling alone does not improve factual accuracy. The benchmark has since been used by AI developers to evaluate large language models. He also co-authored a paper proposing design and governance strategies for building AI systems that do not deceive or hallucinate. In 2023, Evans and collaborators described the "reversal curse", showing that language models trained on a fact in one direction (e.g. "A is B") often cannot answer the corresponding reverse query ("B is A"). His group also developed a benchmark for evaluating situational awareness in language models. In 2025, Evans and colleagues published a study in Nature on what they termed "emergent misalignment": fine-tuning a language model on a narrow task (writing insecure code) caused it to produce unrelated harmful outputs without explicit instruction to do so. Later that year, Evans and collaborators (including researchers at Anthropic) reported that hidden behavioral traits can transfer between language models through training data, even when those traits are not explicitly present in the data, a phenomenon they called "subliminal learning". == Public engagement == In November 2025, Evans delivered the Hinton Lectures, a keynote lecture series on AI safety co-founded by Geoffrey Hinton and the Global Risk Institute.

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  Witness set

  In combinatorics and computational learning theory, a witness set is a set of elements that distinguishes a given Boolean function from a given class of other Boolean functions. Let C {\displaystyle C} be a concept class over a domain X {\displaystyle X} (that is, a family of Boolean functions over X {\displaystyle X} ) and c {\displaystyle c} be a concept in X {\displaystyle X} (a single Boolean function). A subset S {\displaystyle S} of X {\displaystyle X} is a witness set for c {\displaystyle c} in X {\displaystyle X} if S {\displaystyle S} distinguishes c {\displaystyle c} from all the other functions in C {\displaystyle C} , in the sense that no other function in C {\displaystyle C} has the same values on S {\displaystyle S} . For a concept class with | C | {\displaystyle |C|} concepts, there exists a concept that has a witness of size at most log 2 ⁡ | C | {\displaystyle \log _{2}|C|} ; this bound is tight when C {\displaystyle C} consists of all Boolean functions over X {\displaystyle X} . By a result of Bondy (1972) there exists a single witness set of size at most | C | − 1 {\displaystyle |C|-1} that is valid for all concepts in C {\displaystyle C} ; this bound is tight when C {\displaystyle C} consists of the indicator functions of the empty set and some singleton sets. One way to construct this set is to interpret the concepts as bitstrings, and the domain elements as positions in these bitstrings. Then the set of positions at which a trie of the bitstrings branches forms the desired witness set. This construction is central to the operation of the fusion tree data structure. The minimum size of a witness set for c {\displaystyle c} is called the witness size or specification number and is denoted by w C ( c ) {\displaystyle w_{C}(c)} . The value max { w C ( c ) : c ∈ C } {\displaystyle \max\{w_{C}(c):c\in C\}} is called the teaching dimension of C {\displaystyle C} . It represents the number of examples of a concept that need to be presented by a teacher to a learner, in the worst case, to enable the learner to determine which concept is being presented. Witness sets have also been called teaching sets, keys, specifying sets, or discriminants. The "witness set" terminology is from Kushilevitz et al. (1996), who trace the concept of witness sets to work by Cover (1965).

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  Variable-order Bayesian network

  Variable-order Bayesian network (VOBN) models provide an important extension of both the Bayesian network models and the variable-order Markov models. VOBN models are used in machine learning in general and have shown great potential in bioinformatics applications. These models extend the widely used position weight matrix (PWM) models, Markov models, and Bayesian network (BN) models. In contrast to the BN models, where each random variable depends on a fixed subset of random variables, in VOBN models these subsets may vary based on the specific realization of observed variables. The observed realizations are often called the context and, hence, VOBN models are also known as context-specific Bayesian networks. The flexibility in the definition of conditioning subsets of variables turns out to be a real advantage in classification and analysis applications, as the statistical dependencies between random variables in a sequence of variables (not necessarily adjacent) may be taken into account efficiently, and in a position-specific and context-specific manner.

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  Bootstrap aggregating

  Bootstrap aggregating, also called bagging (from bootstrap aggregating) or bootstrapping, is a machine learning (ML) ensemble meta-algorithm designed to improve the stability and accuracy of ML classification and regression algorithms. It also reduces variance and overfitting. Although it is usually applied to decision tree methods, it can be used with any type of method. Bagging is a special case of the ensemble averaging approach. == Description of the technique == Given a standard training set D {\displaystyle D} of size n {\displaystyle n} , bagging generates m {\displaystyle m} new training sets D i {\displaystyle D_{i}} , each of size n ′ {\displaystyle n'} , by sampling from D {\displaystyle D} uniformly and with replacement. By sampling with replacement, some observations may be repeated in each D i {\displaystyle D_{i}} . If n ′ = n {\displaystyle n'=n} , then for large n {\displaystyle n} the set D i {\displaystyle D_{i}} is expected to have the fraction (1 - 1/e) (~63.2%) of the unique samples of D {\displaystyle D} , the rest being duplicates. This kind of sample is known as a bootstrap sample. Sampling with replacement ensures each bootstrap is independent from its peers, as it does not depend on previous chosen samples when sampling. Then, m {\displaystyle m} models are fitted using the above bootstrap samples and combined by averaging the output (for regression) or voting (for classification). Bagging leads to "improvements for unstable procedures", which include, for example, artificial neural networks, classification and regression trees, and subset selection in linear regression. Bagging was shown to improve preimage learning. On the other hand, it can mildly degrade the performance of stable methods such as k-nearest neighbors. == Process of the algorithm == === Key Terms === There are three types of datasets in bootstrap aggregating. These are the original, bootstrap, and out-of-bag datasets. Each section below will explain how each dataset is made except for the original dataset. The original dataset is whatever information is given. === Creating the bootstrap dataset === The bootstrap dataset is made by randomly picking objects from the original dataset. Also, it must be the same size as the original dataset. However, the difference is that the bootstrap dataset can have duplicate objects. Here is a simple example to demonstrate how it works along with the illustration below: Suppose the original dataset is a group of 12 people. Their names are Emily, Jessie, George, Constantine, Lexi, Theodore, John, James, Rachel, Anthony, Ellie, and Jamal. By randomly picking a group of names, let us say our bootstrap dataset had James, Ellie, Constantine, Lexi, John, Constantine, Theodore, Constantine, Anthony, Lexi, Constantine, and Theodore. In this case, the bootstrap sample contained four duplicates for Constantine, and two duplicates for Lexi, and Theodore. === Creating the out-of-bag dataset === The out-of-bag dataset represents the remaining people who were not in the bootstrap dataset. It can be calculated by taking the difference between the original and the bootstrap datasets. In this case, the remaining samples who were not selected are Emily, Jessie, George, Rachel, and Jamal. Keep in mind that since both datasets are sets, when taking the difference the duplicate names are ignored in the bootstrap dataset. The illustration below shows how the math is done: === Application === Creating the bootstrap and out-of-bag datasets is crucial since it is used to test the accuracy of ensemble learning algorithms like random forest. For example, a model that produces 50 trees using the bootstrap/out-of-bag datasets will have a better accuracy than if it produced 10 trees. Since the algorithm generates multiple trees and therefore multiple datasets the chance that an object is left out of the bootstrap dataset is low. The next few sections talk about how the random forest algorithm works in more detail. === Creation of Decision Trees === The next step of the algorithm involves the generation of decision trees from the bootstrapped dataset. To achieve this, the process examines each gene/feature and determines for how many samples the feature's presence or absence yields a positive or negative result. This information is then used to compute a confusion matrix, which lists the true positives, false positives, true negatives, and false negatives of the feature when used as a classifier. These features are then ranked according to various classification metrics based on their confusion matrices. Some common metrics include estimate of positive correctness (calculated by subtracting false positives from true positives), measure of "goodness", and information gain. These features are then used to partition the samples into two sets: those that possess the top feature, and those that do not. The diagram below shows a decision tree of depth two being used to classify data. For example, a data point that exhibits Feature 1, but not Feature 2, will be given a "No". Another point that does not exhibit Feature 1, but does exhibit Feature 3, will be given a "Yes". This process is repeated recursively for successive levels of the tree until the desired depth is reached. At the very bottom of the tree, samples that test positive for the final feature are generally classified as positive, while those that lack the feature are classified as negative. These trees are then used as predictors to classify new data. === Random Forests === The next part of the algorithm involves introducing yet another element of variability amongst the bootstrapped trees. In addition to each tree only examining a bootstrapped set of samples, only a small but consistent number of unique features are considered when ranking them as classifiers. This means that each tree only knows about the data pertaining to a small constant number of features, and a variable number of samples that is less than or equal to that of the original dataset. Consequently, the trees are more likely to return a wider array of answers, derived from more diverse knowledge. This results in a random forest, which possesses numerous benefits over a single decision tree generated without randomness. In a random forest, each tree "votes" on whether or not to classify a sample as positive based on its features. The sample is then classified based on majority vote. An example of this is given in the diagram below, where the four trees in a random forest vote on whether or not a patient with mutations A, B, F, and G has cancer. Since three out of four trees vote yes, the patient is then classified as cancer positive. Because of their properties, random forests are considered one of the most accurate data mining algorithms, are less likely to overfit their data, and run quickly and efficiently even for large datasets. They are primarily useful for classification as opposed to regression, which attempts to draw observed connections between statistical variables in a dataset. This makes random forests particularly useful in such fields as banking, healthcare, the stock market, and e-commerce where it is important to be able to predict future results based on past data. One of their applications would be as a useful tool for predicting cancer based on genetic factors, as seen in the above example. There are several important factors to consider when designing a random forest. If the trees in the random forests are too deep, overfitting can still occur due to over-specificity. If the forest is too large, the algorithm may become less efficient due to an increased runtime. Random forests also do not generally perform well when given sparse data with little variability. However, they still have numerous advantages over similar data classification algorithms such as neural networks, as they are much easier to interpret and generally require less data for training. As an integral component of random forests, bootstrap aggregating is very important to classification algorithms, and provides a critical element of variability that allows for increased accuracy when analyzing new data, as discussed below. == Improving Random Forests and Bagging == While the techniques described above utilize random forests and bagging (otherwise known as bootstrapping), there are certain techniques that can be used in order to improve their execution and voting time, their prediction accuracy, and their overall performance. The following are key steps in creating an efficient random forest: Specify the maximum depth of trees: Instead of allowing the random forest to continue until all nodes are pure, it is better to cut it off at a certain point in order to further decrease chances of overfitting. Prune the dataset: Using an extremely large dataset may create results that are less indicative of the data provided than a smaller set that more accurately represents what is being focused on. Continue pruning the data at each

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  Evolvability (computer science)

  The term evolvability is a framework of computational learning introduced by Leslie Valiant in his paper of the same name. The aim of this theory is to model biological evolution and categorize which types of mechanisms are evolvable. Evolution is an extension of PAC learning and learning from statistical queries. == General framework == Let F n {\displaystyle F_{n}\,} and R n {\displaystyle R_{n}\,} be collections of functions on n {\displaystyle n\,} variables. Given an ideal function f ∈ F n {\displaystyle f\in F_{n}} , the goal is to find by local search a representation r ∈ R n {\displaystyle r\in R_{n}} that closely approximates f {\displaystyle f\,} . This closeness is measured by the performance Perf ⁡ ( f , r ) {\displaystyle \operatorname {Perf} (f,r)} of r {\displaystyle r\,} with respect to f {\displaystyle f\,} . As is the case in the biological world, there is a difference between genotype and phenotype. In general, there can be multiple representations (genotypes) that correspond to the same function (phenotype). That is, for some r , r ′ ∈ R n {\displaystyle r,r'\in R_{n}} , with r ≠ r ′ {\displaystyle r\neq r'\,} , still r ( x ) = r ′ ( x ) {\displaystyle r(x)=r'(x)\,} for all x ∈ X n {\displaystyle x\in X_{n}} . However, this need not be the case. The goal then, is to find a representation that closely matches the phenotype of the ideal function, and the spirit of the local search is to allow only small changes in the genotype. Let the neighborhood N ( r ) {\displaystyle N(r)\,} of a representation r {\displaystyle r\,} be the set of possible mutations of r {\displaystyle r\,} . For simplicity, consider Boolean functions on X n = { − 1 , 1 } n {\displaystyle X_{n}=\{-1,1\}^{n}\,} , and let D n {\displaystyle D_{n}\,} be a probability distribution on X n {\displaystyle X_{n}\,} . Define the performance in terms of this. Specifically, Perf ⁡ ( f , r ) = ∑ x ∈ X n f ( x ) r ( x ) D n ( x ) . {\displaystyle \operatorname {Perf} (f,r)=\sum _{x\in X_{n}}f(x)r(x)D_{n}(x).} Note that Perf ⁡ ( f , r ) = Prob ⁡ ( f ( x ) = r ( x ) ) − Prob ⁡ ( f ( x ) ≠ r ( x ) ) . {\displaystyle \operatorname {Perf} (f,r)=\operatorname {Prob} (f(x)=r(x))-\operatorname {Prob} (f(x)\neq r(x)).} In general, for non-Boolean functions, the performance will not correspond directly to the probability that the functions agree, although it will have some relationship. Throughout an organism's life, it will only experience a limited number of environments, so its performance cannot be determined exactly. The empirical performance is defined by Perf s ⁡ ( f , r ) = 1 s ∑ x ∈ S f ( x ) r ( x ) , {\displaystyle \operatorname {Perf} _{s}(f,r)={\frac {1}{s}}\sum _{x\in S}f(x)r(x),} where S {\displaystyle S\,} is a multiset of s {\displaystyle s\,} independent selections from X n {\displaystyle X_{n}\,} according to D n {\displaystyle D_{n}\,} . If s {\displaystyle s\,} is large enough, evidently Perf s ⁡ ( f , r ) {\displaystyle \operatorname {Perf} _{s}(f,r)} will be close to the actual performance Perf ⁡ ( f , r ) {\displaystyle \operatorname {Perf} (f,r)} . Given an ideal function f ∈ F n {\displaystyle f\in F_{n}} , initial representation r ∈ R n {\displaystyle r\in R_{n}} , sample size s {\displaystyle s\,} , and tolerance t {\displaystyle t\,} , the mutator Mut ⁡ ( f , r , s , t ) {\displaystyle \operatorname {Mut} (f,r,s,t)} is a random variable defined as follows. Each r ′ ∈ N ( r ) {\displaystyle r'\in N(r)} is classified as beneficial, neutral, or deleterious, depending on its empirical performance. Specifically, r ′ {\displaystyle r'\,} is a beneficial mutation if Perf s ⁡ ( f , r ′ ) − Perf s ⁡ ( f , r ) ≥ t {\displaystyle \operatorname {Perf} _{s}(f,r')-\operatorname {Perf} _{s}(f,r)\geq t} ; r ′ {\displaystyle r'\,} is a neutral mutation if − t < Perf s ⁡ ( f , r ′ ) − Perf s ⁡ ( f , r ) < t {\displaystyle -t<\operatorname {Perf} _{s}(f,r')-\operatorname {Perf} _{s}(f,r) 0 {\displaystyle \epsilon >0\,} , for all ideal functions f ∈ F n {\displaystyle f\in F_{n}} and representations r 0 ∈ R n {\displaystyle r_{0}\in R_{n}} , with probability at least 1 − ϵ {\displaystyle 1-\epsilon \,} , Perf ⁡ ( f , r g ( n , 1 / ϵ ) ) ≥ 1 − ϵ , {\displaystyle \operatorname {Perf} (f,r_{g(n,1/\epsilon )})\geq 1-\epsilon ,} where the sizes of neighborhoods N ( r ) {\displaystyle N(r)\,} for r ∈ R n {\displaystyle r\in R_{n}\,} are at most p ( n , 1 / ϵ ) {\displaystyle p(n,1/\epsilon )\,} , the sample size is s ( n , 1 / ϵ ) {\displaystyle s(n,1/\epsilon )\,} , the tolerance is t ( 1 / n , ϵ ) {\displaystyle t(1/n,\epsilon )\,} , and the generation size is g ( n , 1 / ϵ ) {\displaystyle g(n,1/\epsilon )\,} . F {\displaystyle F\,} is evolvable over D {\displaystyle D\,} if it is evolvable by some R {\displaystyle R\,} over D {\displaystyle D\,} . F {\displaystyle F\,} is evolvable if it is evolvable over all distributions D {\displaystyle D\,} . == Results == The class of conjunctions and the class of disjunctions are evolvable over the uniform distribution for short conjunctions and disjunctions, respectively. The class of parity functions (which evaluate to the parity of the number of true literals in a given subset of literals) are not evolvable, even for the uniform distribution. Evolvability implies PAC learnability.

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  Growth function

  The growth function, also called the shatter coefficient or the shattering number, measures the richness of a set family or class of functions. It is especially used in the context of statistical learning theory, where it is used to study properties of statistical learning methods. The term 'growth function' was coined by Vapnik and Chervonenkis in their 1968 paper, where they also proved many of its properties. It is a basic concept in machine learning. == Definitions == === Set-family definition === Let H {\displaystyle H} be a set family (a set of sets) and C {\displaystyle C} a set. Their intersection is defined as the following set-family: H ∩ C := { h ∩ C ∣ h ∈ H } {\displaystyle H\cap C:=\{h\cap C\mid h\in H\}} The intersection-size (also called the index) of H {\displaystyle H} with respect to C {\displaystyle C} is | H ∩ C | {\displaystyle |H\cap C|} . If a set C m {\displaystyle C_{m}} has m {\displaystyle m} elements then the index is at most 2 m {\displaystyle 2^{m}} . If the index is exactly 2m then the set C {\displaystyle C} is said to be shattered by H {\displaystyle H} , because H ∩ C {\displaystyle H\cap C} contains all the subsets of C {\displaystyle C} , i.e.: | H ∩ C | = 2 | C | , {\displaystyle |H\cap C|=2^{|C|},} The growth function measures the size of H ∩ C {\displaystyle H\cap C} as a function of | C | {\displaystyle |C|} . Formally: Growth ⁡ ( H , m ) := max C : | C | = m | H ∩ C | {\displaystyle \operatorname {Growth} (H,m):=\max _{C:|C|=m}|H\cap C|} === Hypothesis-class definition === Equivalently, let H {\displaystyle H} be a hypothesis-class (a set of binary functions) and C {\displaystyle C} a set with m {\displaystyle m} elements. The restriction of H {\displaystyle H} to C {\displaystyle C} is the set of binary functions on C {\displaystyle C} that can be derived from H {\displaystyle H} : H C := { ( h ( x 1 ) , … , h ( x m ) ) ∣ h ∈ H , x i ∈ C } {\displaystyle H_{C}:=\{(h(x_{1}),\ldots ,h(x_{m}))\mid h\in H,x_{i}\in C\}} The growth function measures the size of H C {\displaystyle H_{C}} as a function of | C | {\displaystyle |C|} : Growth ⁡ ( H , m ) := max C : | C | = m | H C | {\displaystyle \operatorname {Growth} (H,m):=\max _{C:|C|=m}|H_{C}|} == Examples == 1. The domain is the real line R {\displaystyle \mathbb {R} } . The set-family H {\displaystyle H} contains all the half-lines (rays) from a given number to positive infinity, i.e., all sets of the form { x > x 0 ∣ x ∈ R } {\displaystyle \{x>x_{0}\mid x\in \mathbb {R} \}} for some x 0 ∈ R {\displaystyle x_{0}\in \mathbb {R} } . For any set C {\displaystyle C} of m {\displaystyle m} real numbers, the intersection H ∩ C {\displaystyle H\cap C} contains m + 1 {\displaystyle m+1} sets: the empty set, the set containing the largest element of C {\displaystyle C} , the set containing the two largest elements of C {\displaystyle C} , and so on. Therefore: Growth ⁡ ( H , m ) = m + 1 {\displaystyle \operatorname {Growth} (H,m)=m+1} . The same is true whether H {\displaystyle H} contains open half-lines, closed half-lines, or both. 2. The domain is the segment [ 0 , 1 ] {\displaystyle [0,1]} . The set-family H {\displaystyle H} contains all the open sets. For any finite set C {\displaystyle C} of m {\displaystyle m} real numbers, the intersection H ∩ C {\displaystyle H\cap C} contains all possible subsets of C {\displaystyle C} . There are 2 m {\displaystyle 2^{m}} such subsets, so Growth ⁡ ( H , m ) = 2 m {\displaystyle \operatorname {Growth} (H,m)=2^{m}} . 3. The domain is the Euclidean space R n {\displaystyle \mathbb {R} ^{n}} . The set-family H {\displaystyle H} contains all the half-spaces of the form: x ⋅ ϕ ≥ 1 {\displaystyle x\cdot \phi \geq 1} , where ϕ {\displaystyle \phi } is a fixed vector. Then Growth ⁡ ( H , m ) = Comp ⁡ ( n , m ) {\displaystyle \operatorname {Growth} (H,m)=\operatorname {Comp} (n,m)} , where Comp is the number of components in a partitioning of an n-dimensional space by m hyperplanes. 4. The domain is the real line R {\displaystyle \mathbb {R} } . The set-family H {\displaystyle H} contains all the real intervals, i.e., all sets of the form { x ∈ [ x 0 , x 1 ] | x ∈ R } {\displaystyle \{x\in [x_{0},x_{1}]|x\in \mathbb {R} \}} for some x 0 , x 1 ∈ R {\displaystyle x_{0},x_{1}\in \mathbb {R} } . For any set C {\displaystyle C} of m {\displaystyle m} real numbers, the intersection H ∩ C {\displaystyle H\cap C} contains all runs of between 0 and m {\displaystyle m} consecutive elements of C {\displaystyle C} . The number of such runs is ( m + 1 2 ) + 1 {\displaystyle {m+1 \choose 2}+1} , so Growth ⁡ ( H , m ) = ( m + 1 2 ) + 1 {\displaystyle \operatorname {Growth} (H,m)={m+1 \choose 2}+1} . == Polynomial or exponential == The main property that makes the growth function interesting is that it can be either polynomial or exponential - nothing in-between. The following is a property of the intersection-size: If, for some set C m {\displaystyle C_{m}} of size m {\displaystyle m} , and for some number n ≤ m {\displaystyle n\leq m} , | H ∩ C m | ≥ Comp ⁡ ( n , m ) {\displaystyle |H\cap C_{m}|\geq \operatorname {Comp} (n,m)} - then, there exists a subset C n ⊆ C m {\displaystyle C_{n}\subseteq C_{m}} of size n {\displaystyle n} such that | H ∩ C n | = 2 n {\displaystyle |H\cap C_{n}|=2^{n}} . This implies the following property of the Growth function. For every family H {\displaystyle H} there are two cases: The exponential case: Growth ⁡ ( H , m ) = 2 m {\displaystyle \operatorname {Growth} (H,m)=2^{m}} identically. The polynomial case: Growth ⁡ ( H , m ) {\displaystyle \operatorname {Growth} (H,m)} is majorized by Comp ⁡ ( n , m ) ≤ m n + 1 {\displaystyle \operatorname {Comp} (n,m)\leq m^{n}+1} , where n {\displaystyle n} is the smallest integer for which Growth ⁡ ( H , n ) < 2 n {\displaystyle \operatorname {Growth} (H,n)<2^{n}} . == Other properties == === Trivial upper bound === For any finite H {\displaystyle H} : Growth ⁡ ( H , m ) ≤ | H | {\displaystyle \operatorname {Growth} (H,m)\leq |H|} since for every C {\displaystyle C} , the number of elements in H ∩ C {\displaystyle H\cap C} is at most | H | {\displaystyle |H|} . Therefore, the growth function is mainly interesting when H {\displaystyle H} is infinite. === Exponential upper bound === For any nonempty H {\displaystyle H} : Growth ⁡ ( H , m ) ≤ 2 m {\displaystyle \operatorname {Growth} (H,m)\leq 2^{m}} I.e, the growth function has an exponential upper-bound. We say that a set-family H {\displaystyle H} shatters a set C {\displaystyle C} if their intersection contains all possible subsets of C {\displaystyle C} , i.e. H ∩ C = 2 C {\displaystyle H\cap C=2^{C}} . If H {\displaystyle H} shatters C {\displaystyle C} of size m {\displaystyle m} , then Growth ⁡ ( H , C ) = 2 m {\displaystyle \operatorname {Growth} (H,C)=2^{m}} , which is the upper bound. === Cartesian intersection === Define the Cartesian intersection of two set-families as: H 1 ⨂ H 2 := { h 1 ∩ h 2 ∣ h 1 ∈ H 1 , h 2 ∈ H 2 } {\displaystyle H_{1}\bigotimes H_{2}:=\{h_{1}\cap h_{2}\mid h_{1}\in H_{1},h_{2}\in H_{2}\}} . Then: Growth ⁡ ( H 1 ⨂ H 2 , m ) ≤ Growth ⁡ ( H 1 , m ) ⋅ Growth ⁡ ( H 2 , m ) {\displaystyle \operatorname {Growth} (H_{1}\bigotimes H_{2},m)\leq \operatorname {Growth} (H_{1},m)\cdot \operatorname {Growth} (H_{2},m)} === Union === For every two set-families: Growth ⁡ ( H 1 ∪ H 2 , m ) ≤ Growth ⁡ ( H 1 , m ) + Growth ⁡ ( H 2 , m ) {\displaystyle \operatorname {Growth} (H_{1}\cup H_{2},m)\leq \operatorname {Growth} (H_{1},m)+\operatorname {Growth} (H_{2},m)} === VC dimension === The VC dimension of H {\displaystyle H} is defined according to these two cases: In the polynomial case, VCDim ⁡ ( H ) = n − 1 {\displaystyle \operatorname {VCDim} (H)=n-1} = the largest integer d {\displaystyle d} for which Growth ⁡ ( H , d ) = 2 d {\displaystyle \operatorname {Growth} (H,d)=2^{d}} . In the exponential case VCDim ⁡ ( H ) = ∞ {\displaystyle \operatorname {VCDim} (H)=\infty } . So VCDim ⁡ ( H ) ≥ d {\displaystyle \operatorname {VCDim} (H)\geq d} if-and-only-if Growth ⁡ ( H , d ) = 2 d {\displaystyle \operatorname {Growth} (H,d)=2^{d}} . The growth function can be regarded as a refinement of the concept of VC dimension. The VC dimension only tells us whether Growth ⁡ ( H , d ) {\displaystyle \operatorname {Growth} (H,d)} is equal to or smaller than 2 d {\displaystyle 2^{d}} , while the growth function tells us exactly how Growth ⁡ ( H , m ) {\displaystyle \operatorname {Growth} (H,m)} changes as a function of m {\displaystyle m} . Another connection between the growth function and the VC dimension is given by the Sauer–Shelah lemma: If VCDim ⁡ ( H ) = d {\displaystyle \operatorname {VCDim} (H)=d} , then: for all m {\displaystyle m} : Growth ⁡ ( H , m ) ≤ ∑ i = 0 d ( m i ) {\displaystyle \operatorname {Growth} (H,m)\leq \sum _{i=0}^{d}{m \choose i}} In particular, for all m > d + 1 {\displaystyle m>d+1} : Growth ⁡ ( H , m ) ≤ ( e m / d ) d = O ( m d ) {\displaystyle \operatorname {Growth} (H,m)\leq (

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  LogitBoost

  In machine learning and computational learning theory, LogitBoost is a boosting algorithm formulated by Jerome Friedman, Trevor Hastie, and Robert Tibshirani. The original paper casts the AdaBoost algorithm into a statistical framework. Specifically, if one considers AdaBoost as a generalized additive model and then applies the cost function of logistic regression, one can derive the LogitBoost algorithm. == Minimizing the LogitBoost cost function == LogitBoost can be seen as a convex optimization. Specifically, given that we seek an additive model of the form f = ∑ t α t h t {\displaystyle f=\sum _{t}\alpha _{t}h_{t}} the LogitBoost algorithm minimizes the logistic loss: ∑ i log ⁡ ( 1 + e − y i f ( x i ) ) {\displaystyle \sum _{i}\log \left(1+e^{-y_{i}f(x_{i})}\right)}

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  Distributional Soft Actor Critic

  Distributional Soft Actor Critic (DSAC) is a suite of model-free off-policy reinforcement learning algorithms, tailored for learning decision-making or control policies in complex systems with continuous action spaces. Distinct from traditional methods that focus solely on expected returns, DSAC algorithms are designed to learn a Gaussian distribution over stochastic returns, called value distribution. This focus on Gaussian value distribution learning notably diminishes value overestimations, which in turn boosts policy performance. Additionally, the value distribution learned by DSAC can also be used for risk-aware policy learning. From a technical standpoint, DSAC is essentially a distributional adaptation of the well-established soft actor-critic (SAC) method. To date, the DSAC family comprises two iterations: the original DSAC-v1 and its successor, DSAC-T (also known as DSAC-v2), with the latter demonstrating superior capabilities over the Soft Actor-Critic (SAC) in Mujoco benchmark tasks. The source code for DSAC-T can be found at the following URL: Jingliang-Duan/DSAC-T. Both iterations have been integrated into an advanced, Pytorch-powered reinforcement learning toolkit named GOPS: GOPS (General Optimal control Problem Solver).

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  Pixel binning

  Pixel binning, also known as binning, is a process image sensors of digital cameras use to combine adjacent pixels throughout an image, by summing or averaging their values, during or after readout. It improves low-light performance while still allowing for highly detailed photographs in good light. Charge from adjacent pixels in CCD or charge-coupled device image sensors and some other image sensors can be combined during readout, increasing the line rate or frame rate. In the context of image processing, binning is the procedure of combining clusters of adjacent pixels, throughout an image, into single pixels. For example, in 2×2 binning, an array of 4 pixels becomes a single larger pixel, reducing the number of pixels to 1/4 and halving the image resolution in each dimension. The result can be the sum, average, median, minimum, or maximum value of the cluster. Some systems use more advanced algorithms such as considering the values of nearby pixels, edge detection, self-claimed "AI", etc. to increase the perceived visual quality of the final downsized image. This aggregation, although associated with loss of information, reduces the amount of data to be processed, facilitating analysis. The binned image has lower resolution, but the relative noise level in each pixel is generally reduced. == History == Normally, an increase in megapixel count on a constant image sensor size would lead to a sacrifice of the surface size of the individual pixels, which would result in each pixel being able to catch less light in the same time, thus leading to a darker and/or noisier image in low light (given the same exposure time). In the past, camera manufacturers had to compromise between low-light performance and the amount of detail in good light, by dropping the megapixel count like HTC did in 2013 with their four-megapixel "UltraPixel" camera. However, this results in less detailed images in daylight where enough light is available. With pixel binning, the camera has "the best of both worlds", meaning both the benefit of high detail in good light and the benefit of high brightness in low light. In low light, the surfaces of four or more pixels can act as one large pixel that catches far more light. For example, some smartphones such as the Samsung Galaxy A15 are able to capture photographs with up to fifty megapixels in daylight. However, in low light, the individual pixels would be too small to capture the light needed for a bright image with the short exposure time available for handheld shooting. Therefore, with pixel binning activated, the 50-megapixel image sensor acts as a 12.5-megapixel image sensor, a quarter of its original resolution, with an accordingly larger surface area per pixel.

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  K-nearest neighbors algorithm

  In statistics, the k-nearest neighbors algorithm (k-NN) is a non-parametric supervised learning method. It was first developed by Evelyn Fix and Joseph Hodges in 1951, and later expanded by Thomas Cover. In classification, a new example is assigned a label based on the labels of its k nearest training examples; in regression, the prediction is computed from the values of those neighbors. Most often, it is used for classification, as a k-NN classifier, the output of which is a class membership. An object is classified by a plurality vote of its neighbors, with the object being assigned to the class most common among its k nearest neighbors (k is a positive integer, typically small). If k = 1, then the object is simply assigned to the class of that single nearest neighbor. The k-NN algorithm can also be generalized for regression. In k-NN regression, also known as nearest neighbor smoothing, the output is the property value for the object. This value is the average of the values of k nearest neighbors. If k = 1, then the output is simply assigned to the value of that single nearest neighbor, also known as nearest neighbor interpolation. For both classification and regression, a useful technique can be to assign weights to the contributions of the neighbors, so that nearer neighbors contribute more to the average than distant ones. For example, a common weighting scheme consists of giving each neighbor a weight of 1/d, where d is the distance to the neighbor. The input consists of the k closest training examples in a data set. The neighbors are taken from a set of objects for which the class (for k-NN classification) or the object property value (for k-NN regression) is known. This can be thought of as the training set for the algorithm, though no explicit training step is required. A peculiarity (sometimes even a disadvantage) of the k-NN algorithm is its sensitivity to the local structure of the data. In k-NN classification the function is only approximated locally and all computation is deferred until function evaluation. Since this algorithm relies on distance, if the features represent different physical units or come in vastly different scales, then feature-wise normalizing of the training data can greatly improve its accuracy. == Statistical setting == Suppose we have pairs ( X 1 , Y 1 ) , ( X 2 , Y 2 ) , … , ( X n , Y n ) {\displaystyle (X_{1},Y_{1}),(X_{2},Y_{2}),\dots ,(X_{n},Y_{n})} taking values in R d × { 1 , 2 } {\displaystyle \mathbb {R} ^{d}\times \{1,2\}} , where Y is the class label of X, so that X | Y = r ∼ P r {\displaystyle X|Y=r\sim P_{r}} for r = 1 , 2 {\displaystyle r=1,2} (and probability distributions P r {\displaystyle P_{r}} ). Given some norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} on R d {\displaystyle \mathbb {R} ^{d}} and a point x ∈ R d {\displaystyle x\in \mathbb {R} ^{d}} , let ( X ( 1 ) , Y ( 1 ) ) , … , ( X ( n ) , Y ( n ) ) {\displaystyle (X_{(1)},Y_{(1)}),\dots ,(X_{(n)},Y_{(n)})} be a reordering of the training data such that ‖ X ( 1 ) − x ‖ ≤ ⋯ ≤ ‖ X ( n ) − x ‖ {\displaystyle \|X_{(1)}-x\|\leq \dots \leq \|X_{(n)}-x\|} . == Algorithm == The training examples are vectors in a multidimensional feature space, each with a class label. The training phase of the algorithm consists only of storing the feature vectors and class labels of the training samples. In the classification phase, k is a user-defined constant, and an unlabeled vector (a query or test point) is classified by assigning the label which is most frequent among the k training samples nearest to that query point. A commonly used distance metric for continuous variables is Euclidean distance. For discrete variables, such as for text classification, another metric can be used, such as the overlap metric (or Hamming distance). In the context of gene expression microarray data, for example, k-NN has been employed with correlation coefficients, such as Pearson and Spearman, as a metric. Often, the classification accuracy of k-NN can be improved significantly if the distance metric is learned with specialized algorithms such as large margin nearest neighbor or neighborhood components analysis. A drawback of the basic "majority voting" classification occurs when the class distribution is skewed. That is, examples of a more frequent class tend to dominate the prediction of the new example, because they tend to be common among the k nearest neighbors due to their large number. One way to overcome this problem is to weight the classification, taking into account the distance from the test point to each of its k nearest neighbors. The class (or value, in regression problems) of each of the k nearest points is multiplied by a weight proportional to the inverse of the distance from that point to the test point. Another way to overcome skew is by abstraction in data representation. For example, in a self-organizing map (SOM), each node is a representative (a center) of a cluster of similar points, regardless of their density in the original training data. k-NN can then be applied to the SOM. == Parameter selection == The best choice of k depends upon the data; generally, larger values of k reduces effect of the noise on the classification, but make boundaries between classes less distinct. A good k can be selected by various heuristic techniques (see hyperparameter optimization). The special case where the class is predicted to be the class of the closest training sample (i.e. when k = 1) is called the nearest neighbor algorithm. The accuracy of the k-NN algorithm can be severely degraded by the presence of noisy or irrelevant features, or if the feature scales are not consistent with their importance. Much research effort has been put into selecting or scaling features to improve classification. A particularly popular approach is the use of evolutionary algorithms to optimize feature scaling. Another popular approach is to scale features by the mutual information of the training data with the training classes. In binary (two class) classification problems, it is helpful to choose k to be an odd number as this avoids tied votes. One popular way of choosing the empirically optimal k in this setting is via bootstrap method. == The 1-nearest neighbor classifier == The most intuitive nearest neighbour type classifier is the one nearest neighbour classifier that assigns a point x to the class of its closest neighbour in the feature space, that is C n 1 n n ( x ) = Y ( 1 ) {\displaystyle C_{n}^{1nn}(x)=Y_{(1)}} . As the size of training data set approaches infinity, the one nearest neighbour classifier guarantees an error rate of no worse than twice the Bayes error rate (the minimum achievable error rate given the distribution of the data). == The weighted nearest neighbour classifier == The k-nearest neighbour classifier can be viewed as assigning the k nearest neighbours a weight 1 / k {\displaystyle 1/k} and all others 0 weight. This can be generalised to weighted nearest neighbour classifiers. That is, where the ith nearest neighbour is assigned a weight w n i {\displaystyle w_{ni}} , with ∑ i = 1 n w n i = 1 {\textstyle \sum _{i=1}^{n}w_{ni}=1} . An analogous result on the strong consistency of weighted nearest neighbour classifiers also holds. Let C n w n n {\displaystyle C_{n}^{wnn}} denote the weighted nearest classifier with weights { w n i } i = 1 n {\displaystyle \{w_{ni}\}_{i=1}^{n}} . Subject to regularity conditions, which in asymptotic theory are conditional variables which require assumptions to differentiate among parameters with some criteria. On the class distributions the excess risk has the following asymptotic expansion R R ( C n w n n ) − R R ( C Bayes ) = ( B 1 s n 2 + B 2 t n 2 ) { 1 + o ( 1 ) } , {\displaystyle {\mathcal {R}}_{\mathcal {R}}(C_{n}^{wnn})-{\mathcal {R}}_{\mathcal {R}}(C^{\text{Bayes}})=\left(B_{1}s_{n}^{2}+B_{2}t_{n}^{2}\right)\{1+o(1)\},} for constants B 1 {\displaystyle B_{1}} and B 2 {\displaystyle B_{2}} where s n 2 = ∑ i = 1 n w n i 2 {\displaystyle s_{n}^{2}=\sum _{i=1}^{n}w_{ni}^{2}} and t n = n − 2 / d ∑ i = 1 n w n i { i 1 + 2 / d − ( i − 1 ) 1 + 2 / d } {\displaystyle t_{n}=n^{-2/d}\sum _{i=1}^{n}w_{ni}\left\{i^{1+2/d}-(i-1)^{1+2/d}\right\}} . The optimal weighting scheme { w n i ∗ } i = 1 n {\displaystyle \{w_{ni}^{}\}_{i=1}^{n}} , that balances the two terms in the display above, is given as follows: set k ∗ = ⌊ B n 4 d + 4 ⌋ {\displaystyle k^{}=\lfloor Bn^{\frac {4}{d+4}}\rfloor } , w n i ∗ = 1 k ∗ [ 1 + d 2 − d 2 k ∗ 2 / d { i 1 + 2 / d − ( i − 1 ) 1 + 2 / d } ] {\displaystyle w_{ni}^{}={\frac {1}{k^{}}}\left[1+{\frac {d}{2}}-{\frac {d}{2{k^{}}^{2/d}}}\{i^{1+2/d}-(i-1)^{1+2/d}\}\right]} for i = 1 , 2 , … , k ∗ {\displaystyle i=1,2,\dots ,k^{}} and w n i ∗ = 0 {\displaystyle w_{ni}^{}=0} for i = k ∗ + 1 , … , n {\displaystyle i=k^{}+1,\dots ,n} . With optimal weights the dominant term in the asymptotic expansion of the excess risk is O ( n − 4 d + 4 ) {\displaystyle {\mathcal {O}}(n^{-{\frac {4}{d+4}}})}

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  Sammon mapping

  Sammon mapping or Sammon projection is an algorithm that maps a high-dimensional space to a space of lower dimensionality (see multidimensional scaling) by trying to preserve the structure of inter-point distances in high-dimensional space in the lower-dimension projection. It is particularly suited for use in exploratory data analysis. The method was proposed by John W. Sammon in 1969. It is considered a non-linear approach as the mapping cannot be represented as a linear combination of the original variables as possible in techniques such as principal component analysis, which also makes it more difficult to use for classification applications. Denote the distance between ith and jth objects in the original space by d i j ∗ {\displaystyle \scriptstyle d_{ij}^{}} , and the distance between their projections by d i j {\displaystyle \scriptstyle d_{ij}^{}} . Sammon's mapping aims to minimize the following error function, which is often referred to as Sammon's stress or Sammon's error: E = 1 ∑ i < j d i j ∗ ∑ i < j ( d i j ∗ − d i j ) 2 d i j ∗ . {\displaystyle E={\frac {1}{\sum \limits _{i Read more →

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  Spatial Analysis of Principal Components

  Spatial Principal Component Analysis (sPCA) is a multivariate statistical technique that complements the traditional Principal Component Analysis (PCA) by incorporating spatial information into the analysis of genetic variation. While traditional PCA can be used to find spatial patterns, it focuses on reducing data dimensionality by identifying uncorrelated principal components that capture maximum variance, thus often lacking power to identify non-trivial spatial genetic patterns. By accounting for spatial autocorrelation, sPCA is able to uncover spatial patterns in the data and find the spatial structure of datasets where observations are either geographically or topologically linked. This statistical power improvement allows the investigation of cryptic spatial patterns of genetic variability otherwise overlooked. sPCA has been applied in various fields, including geography, ecology and genetics. == History == sPCA was introduced in 2008 by Thibaut Jombart, Sébastien Devillard, Anne-Béatrice Dufour, and D. Pontier as a spatially explicit method to investigate the spatial pattern of genetic variation among individuals or populations. In 2017, Valeria Montano and Thibaut Jombart published an alternative non-parametric test to evaluate the significance of global and local spatial genetic patterns with improved statistical power. == Details == sPCA modifies the PCA framework by integrating spatial weights, typically in the form of connectivity matrices or spatial adjacency graphs. It identifies principal components (PCs) that maximize both genentic variance and spatial autocorreation, as measured by Moran's I. These weights represent relationships between observations based on geographic distance or other spatial criteria. The method decomposes variance into two components: Global structures, correspond to positive autocorrelation, that is, reflect broad-scale spatial patterns where similar values cluster over large regions. Local structures, correspond to negative autocorrelation, that is, capture fine-scale spatial variations or localized patterns. The core of sPCA relies on the eigenanalysis of a spatially weighted covariance or correlation matrix. The spatial weight matrix can be constructed using techniques such as Delaunay triangulation, nearest-neighbor graphs, or distance-based criteria. Applications of sPCA should be used only as an explorative tool. == Applications == sPCA has been widely used in many fields, including: Ecology: To find spatial patterns in species distributions and environmental gradients. Genetics: Population structure and gene flow analysis while allowing for spatial autocorrelation considerations. Biogeography: To identify historical dispersal routes, and barriers to gene flow, providing insights into species distribution patterns and evolutionary history. == Software/Source Code == sPCA implementations are available in R in adegenet and ntbox . These tools facilitate the application of sPCA by providing functions for constructing spatial weight matrices, performing eigenanalysis, and obtaining spatial principal components in an easy-to-read form.

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  QF-Test

  QF-Test from Quality First Software is a cross-platform software tool for automated testing of programs via the graphical user interface (GUI) test automation). The program is specialized on (Java/Swing, Standard Widget Toolkit (SWT), Eclipse plug-ins and rich client platform (RCP) applications, ULC and JavaFX) cross-web browser test automation of static and dynamic web applications (HTML and web frameworks like Angular, Ext JS, Fluent UI React, Google Web Toolkit (GWT), jQuery UI, jQueryEasyUI Remote Application Platform (RAP), Qooxdoo, RichFaces, Vaadin, React, Smart GWT, Vue.js, ICEfaces and ZK). Version 4.1 added support for macOS and the Apple Safari and Microsoft Edge browsers via the Selenium WebDriver. Representational State Transfer (RESTful) web service testing. From version 5.0, Windows applications can also be tested (classic Win32 applications, .NET framework applications (often developed in C#) based on Windows Presentation Foundation (WPF) or Windows Forms, Windows apps and Universal Windows Platform (UWP) applications using Extensible Application Markup Language (XAML) controls) and modern C++ applications (such as Qt applications). Version 5.3 added support for the Chrome DevTools protocol, which allows browsers to be controlled using CDP drivers. Since then, mobile testing for iOS and Android, accessibility testing of web applications and SmartID, a new approach for more flexible and robust component recognition, have been introduced. Powerful enhancements such as WebAPI testing and AI-assisted validation complement the test automation tool. == Overview == QF-Test (the successor of qftestJUI, available since 2001) enables regression and load testing and runs on Windows, Unix and macOS. It is mainly used commercially by testers, developers or business analysts (modelling, low code approaches) with or without programming knowledge as part of software Quality Assurance. Since December 2008, a webtest add-on is available which allows test automation of browser-based GUIs (such as Internet Explorer, Mozilla Firefox, Google Chrome, Apple Safari, and Microsoft Edge) along with extant Java GUI test functions, which was extended to include JavaFX in July 2014. From 2018, QF-Test version 4.2 can test PDF documents, from 2020 native desktop applications (QF-Test version 5) and in 2022, mobile application testing will be added. The basis for efficient use in test automation is stable component recognition (IDs, logical screen elements, labels, CustomWebResolver, SmartID, ...) with low maintenance effort. == Features == General – QF-Test's capture/replay function enables recording of tests for beginners, while modular programming (modularizing) allows creating large test suites in a concise arrangement. For the advanced user who requires even more control over his application, the tool offers access to internal program structures through the standard scripting languages Jython, the Java implementation of the popular Python language, JavaScript, and Groovy. The tool also offers a batch processing mode, allowing to run tests unattended and then generate XML, HTML and JUnit reports. Thus the tool can be integrated into existing build/test frameworks like Jenkins, Ant or Maven. Another mode is the so-called Daemon mode for distributed test execution. A specific integration with many test management tools exists. There is a test debugger (enabling arbitrary stepping and editing variables at runtime) and a fully automated dependency management that takes care of pre- and postconditions and helps isolating test cases. Data-driven testing with no need for scripting is possible. Web testing: cross-browser on Internet Explorer, Chrome, Firefox, Edge (including Chromium-based), Opera and Safari for static and dynamic websites (HTML5, Ajax, DOM). A headless browser can also be used for testing. QF-Test fully supports frameworks like Angular, React and Vue.js, but also many specific UI toolkits like Smart (GWT), GXT/ExtGWT, ExtJS, ICEfaces, jQuery UI, Kendo UI, PrimeFaces, Qooxdoo, RAP, RichFaces, Vaadin and ZK. Easy integration with Selenium makes it easy to balance development and functional testing. Electron applications can also be tested. Other (e.g., SAP UI5, Siebel Open UI, Salesforce) and future web toolkits can be integrated with little effort. Short-term and individual customisations (CustomWebResolver) are possible via an optimised interface JavaFX, Java Swing, SWT, Eclipse plug-ins and RCP applications and ULC. Support for testing when migrating from JavaSwing or JavaFX to web applications (e.g. via Webswing). Hybrid applications based on multiple technologies are also supported, e.g. applications that integrate HTML content into Java applications using JxBrowser. Windows-based applications (Win32, .NET, Windows Forms, WPF, Windows apps, Qt). Android applications can be tested on real devices and with the Android Studio emulator. iOS applications can also be tested on real devices and with the Xcode Simulator. Testing of PDF documents (document comparisons, checking content, texts, images/graphic objects, layouts, "invisible" or partially hidden objects). QF-Test 9 introduces web accessibility testing to automatically check compliance with WCAG and other standards. QF-Test 10 introduces powerful enhancements for WebAPI testing and AI-assisted validation.

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  Gremlin (query language)

  Gremlin is a graph traversal language and virtual machine developed by Apache TinkerPop of the Apache Software Foundation. Gremlin works for both OLTP-based graph databases as well as OLAP-based graph processors. Gremlin's automata and functional language foundation enable Gremlin to naturally support imperative and declarative querying, host language agnosticism, user-defined domain specific languages, an extensible compiler/optimizer, single- and multi-machine execution models, and hybrid depth- and breadth-first evaluation with Turing completeness. As an explanatory analogy, Apache TinkerPop and Gremlin are to graph databases what the JDBC and SQL are to relational databases. Likewise, the Gremlin traversal machine is to graph computing as what the Java virtual machine is to general purpose computing. == History == 2009-10-30 the project is born, and immediately named "TinkerPop" 2009-12-25 v0.1 is the first release 2011-05-21 v1.0 is released 2012-05-24 v2.0 is released 2015-01-16 TinkerPop becomes an Apache Incubator project 2015-07-09 v3.0.0-incubating is released 2016-05-23 Apache TinkerPop becomes a top-level project 2016-07-18 v3.1.3 and v3.2.1 are first releases as Apache TinkerPop 2017-12-17 v3.3.1 is released 2018-05-08 v3.3.3 is released 2019-08-05 v3.4.3 is released 2020-02-20 v3.4.6 is released 2021-05-01 v3.5.0 is released 2022-04-04 v3.6.0 is released 2023-07-31 v3.7.0 is released 2025-11-12 v3.8.0 is released == Vendor integration == Gremlin is an Apache2-licensed graph traversal language that can be used by graph system vendors. There are typically two types of graph system vendors: OLTP graph databases and OLAP graph processors. The table below outlines those graph vendors that support Gremlin. == Traversal examples == The following examples of Gremlin queries and responses in a Gremlin-Groovy environment are relative to a graph representation of the MovieLens dataset. The dataset includes users who rate movies. Users each have one occupation, and each movie has one or more categories associated with it. The MovieLens graph schema is detailed below. === Simple traversals === For each vertex in the graph, emit its label, then group and count each distinct label. What year was the oldest movie made? What is Die Hard's average rating? === Projection traversals === For each category, emit a map of its name and the number of movies it represents. For each movie with at least 11 ratings, emit a map of its name and average rating. Sort the maps in decreasing order by their average rating. Emit the first 10 maps (i.e. top 10). === Declarative pattern matching traversals === Gremlin supports declarative graph pattern matching similar to SPARQL. For instance, the following query below uses Gremlin's match()-step. What 80's action movies do 30-something programmers like? Group count the movies by their name and sort the group count map in decreasing order by value. Clip the map to the top 10 and emit the map entries. === OLAP traversal === Which movies are most central in the implicit 5-stars graph? == Gremlin graph traversal machine == Gremlin is a virtual machine composed of an instruction set as well as an execution engine. An analogy is drawn between Gremlin and Java. === Gremlin steps (instruction set) === The following traversal is a Gremlin traversal in the Gremlin-Java8 dialect. The Gremlin language (i.e. the fluent-style of expressing a graph traversal) can be represented in any host language that supports function composition and function nesting. Due to this simple requirement, there exists various Gremlin dialects including Gremlin-Groovy, Gremlin-Scala, Gremlin-Clojure, etc. The above Gremlin-Java8 traversal is ultimately compiled down to a step sequence called a traversal. A string representation of the traversal above provided below. The steps are the primitives of the Gremlin graph traversal machine. They are the parameterized instructions that the machine ultimately executes. The Gremlin instruction set is approximately 30 steps. These steps are sufficient to provide general purpose computing and what is typically required to express the common motifs of any graph traversal query. Given that Gremlin is a language, an instruction set, and a virtual machine, it is possible to design another traversal language that compiles to the Gremlin traversal machine (analogous to how Scala compiles to the JVM). For instance, the popular SPARQL graph pattern match language can be compiled to execute on the Gremlin machine. The following SPARQL query would compile to In Gremlin-Java8, the SPARQL query above would be represented as below and compile to the identical Gremlin step sequence (i.e. traversal). === Gremlin Machine (virtual machine) === The Gremlin graph traversal machine can execute on a single machine or across a multi-machine compute cluster. Execution agnosticism allows Gremlin to run over both graph databases (OLTP) and graph processors (OLAP).

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  Count sketch

  Count sketch is a type of dimensionality reduction that is particularly efficient in statistics, machine learning and algorithms. It was invented by Moses Charikar, Kevin Chen and Martin Farach-Colton in an effort to speed up the AMS Sketch by Alon, Matias and Szegedy for approximating the frequency moments of streams (these calculations require counting of the number of occurrences for the distinct elements of the stream). The sketch is nearly identical to the Feature hashing algorithm by John Moody, but differs in its use of hash functions with low dependence, which makes it more practical. In order to still have a high probability of success, the median trick is used to aggregate multiple count sketches, rather than the mean. These properties allow use for explicit kernel methods, bilinear pooling in neural networks and is a cornerstone in many numerical linear algebra algorithms. == Intuitive explanation == The inventors of this data structure offer the following iterative explanation of its operation: at the simplest level, the output of a single hash function s mapping stream elements q into {+1, -1} is feeding a single up/down counter C. After a single pass over the data, the frequency n ( q ) {\displaystyle n(q)} of a stream element q can be approximated, although extremely poorly, by the expected value E [ C ⋅ s ( q ) ] {\displaystyle {\mathbf {E}}[C\cdot s(q)]} ; a straightforward way to improve the variance of the previous estimate is to use an array of different hash functions s i {\displaystyle s_{i}} , each connected to its own counter C i {\displaystyle C_{i}} . For each i, the E [ C i ⋅ s i ( q ) ] = n ( q ) {\displaystyle {\mathbf {E}}[C_{i}\cdot s_{i}(q)]=n(q)} still holds, so averaging across the i range will tighten the approximation; the previous construct still has a major deficiency: if a lower-frequency-but-still-important output element a exhibits a hash collision with a high-frequency element even for one of the s i {\displaystyle s_{i}} hashes, n ( a ) {\displaystyle n(a)} estimate can be significantly affected. Avoiding this requires reducing the frequency of collision counter updates between any two distinct elements. This is achieved by replacing each C i {\displaystyle C_{i}} in the previous construct with an array of m counters (making the counter set into a two-dimensional matrix C i , j {\displaystyle C_{i,j}} ), with index j of a particular counter to be incremented/decremented selected via another set of hash functions h i {\displaystyle h_{i}} that map element q into the range {1..m}. Since E [ C i , h i ( q ) ⋅ s i ( q ) ] = n ( q ) {\displaystyle {\mathbf {E}}[C_{i,h_{i}(q)}\cdot s_{i}(q)]=n(q)} , averaging across all values of i will work. == Mathematical definition == 1. For constants w {\displaystyle w} and t {\displaystyle t} (to be defined later) independently choose d = 2 t + 1 {\displaystyle d=2t+1} random hash functions h 1 , … , h d {\displaystyle h_{1},\dots ,h_{d}} and s 1 , … , s d {\displaystyle s_{1},\dots ,s_{d}} such that h i : [ n ] → [ w ] {\displaystyle h_{i}:[n]\to [w]} and s i : [ n ] → { ± 1 } {\displaystyle s_{i}:[n]\to \{\pm 1\}} . It is necessary that the hash families from which h i {\displaystyle h_{i}} and s i {\displaystyle s_{i}} are chosen be pairwise independent. 2. For each item q i {\displaystyle q_{i}} in the stream, add s j ( q i ) {\displaystyle s_{j}(q_{i})} to the h j ( q i ) {\displaystyle h_{j}(q_{i})} th bucket of the j {\displaystyle j} th hash. At the end of this process, one has w d {\displaystyle wd} sums ( C i j ) {\displaystyle (C_{ij})} where C i , j = ∑ h i ( k ) = j s i ( k ) . {\displaystyle C_{i,j}=\sum _{h_{i}(k)=j}s_{i}(k).} To estimate the count of q {\displaystyle q} s one computes the following value: r q = median i = 1 d s i ( q ) ⋅ C i , h i ( q ) . {\displaystyle r_{q}={\text{median}}_{i=1}^{d}\,s_{i}(q)\cdot C_{i,h_{i}(q)}.} The values s i ( q ) ⋅ C i , h i ( q ) {\displaystyle s_{i}(q)\cdot C_{i,h_{i}(q)}} are unbiased estimates of how many times q {\displaystyle q} has appeared in the stream. The estimate r q {\displaystyle r_{q}} has variance O ( m i n { m 1 2 / w 2 , m 2 2 / w } ) {\displaystyle O(\mathrm {min} \{m_{1}^{2}/w^{2},m_{2}^{2}/w\})} , where m 1 {\displaystyle m_{1}} is the length of the stream and m 2 2 {\displaystyle m_{2}^{2}} is ∑ q ( ∑ i [ q i = q ] ) 2 {\displaystyle \sum _{q}(\sum _{i}[q_{i}=q])^{2}} . Furthermore, r q {\displaystyle r_{q}} is guaranteed to never be more than 2 m 2 / w {\displaystyle 2m_{2}/{\sqrt {w}}} off from the true value, with probability 1 − e − O ( t ) {\displaystyle 1-e^{-O(t)}} . === Vector formulation === Alternatively Count-Sketch can be seen as a linear mapping with a non-linear reconstruction function. Let M ( i ∈ [ d ] ) ∈ { − 1 , 0 , 1 } w × n {\displaystyle M^{(i\in [d])}\in \{-1,0,1\}^{w\times n}} , be a collection of d = 2 t + 1 {\displaystyle d=2t+1} matrices, defined by M h i ( j ) , j ( i ) = s i ( j ) {\displaystyle M_{h_{i}(j),j}^{(i)}=s_{i}(j)} for j ∈ [ w ] {\displaystyle j\in [w]} and 0 everywhere else. Then a vector v ∈ R n {\displaystyle v\in \mathbb {R} ^{n}} is sketched by C ( i ) = M ( i ) v ∈ R w {\displaystyle C^{(i)}=M^{(i)}v\in \mathbb {R} ^{w}} . To reconstruct v {\displaystyle v} we take v j ∗ = median i C j ( i ) s i ( j ) {\displaystyle v_{j}^{}={\text{median}}_{i}C_{j}^{(i)}s_{i}(j)} . This gives the same guarantees as stated above, if we take m 1 = ‖ v ‖ 1 {\displaystyle m_{1}=\|v\|_{1}} and m 2 = ‖ v ‖ 2 {\displaystyle m_{2}=\|v\|_{2}} . == Relation to Tensor sketch == The count sketch projection of the outer product of two vectors is equivalent to the convolution of two component count sketches. The count sketch computes a vector convolution C ( 1 ) x ∗ C ( 2 ) x T {\displaystyle C^{(1)}x\ast C^{(2)}x^{T}} , where C ( 1 ) {\displaystyle C^{(1)}} and C ( 2 ) {\displaystyle C^{(2)}} are independent count sketch matrices. Pham and Pagh show that this equals C ( x ⊗ x T ) {\displaystyle C(x\otimes x^{T})} – a count sketch C {\displaystyle C} of the outer product of vectors, where ⊗ {\displaystyle \otimes } denotes Kronecker product. The fast Fourier transform can be used to do fast convolution of count sketches. By using the face-splitting product such structures can be computed much faster than normal matrices.

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