AI Face Paint

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  Spatial embedding

  Spatial embedding is one of feature learning techniques used in spatial analysis where points, lines, polygons or other spatial data types. representing geographic locations are mapped to vectors of real numbers. Conceptually it involves a mathematical embedding from a space with many dimensions per geographic object to a continuous vector space with a much lower dimension. Such embedding methods allow complex spatial data to be used in neural networks and have been shown to improve performance in spatial analysis tasks == Embedded data types == Geographic data can take many forms: text, images, graphs, trajectories, polygons. Depending on the task, there may be a need to combine multimodal data from different sources. The next section describes examples of different types of data and their uses. === Text === Geolocated posts on social media can be used to acquire a library of documents bound to a given place that can be later transformed to embedded vectors using word embedding techniques. === Image === Satellites and aircraft collect digital spatial data acquired from remotely sensed images which can be used in machine learning. They are sometimes hard to analyse using basic image analysis methods and convolutional neural networks can be used to acquire an embedding of images bound to a given geographical object or a region. === Point === A single point of interest (POI) can be assigned multiple features that can be used in machine learning. These could be demographic, transportation, meteorological, or economic data, for example. When embedding single points, it is common to consider the entire set of available points as nodes in a graph. === Line / multiline === Among other things, motion trajectories are represented as lines (multilines). Individual trajectories are embedded taking into account travel time, distances and also features of points visited along the way. Embedding of trajectories allows to improve performance of such tasks as clustering and also categorization. === Polygon === The geographic areas analyzed in machine learning are defined by both administrative boundaries and top-down division into grids of regular shapes such as rectangles, for example. Both types are represented as polygons and, like points, can be assigned different demographic, transportation, or economic features. A polygon can also have features related to the size of the area or shape it represents. === Graph === An example domain where graph representation is used is the street layout in a city, where vertices can be intersections and edges can be roads. The vertices can also be destination points like public transport stops or important points in the city, and the edges represent the flow between them. Embedding graphs or single vertices allows to improve accuracy of analysis methods in which the treated geographical domain can be represented as a network. == Usage == POI recommendation - generating personalized point of interest recommendations based on user preferences. Next/future location prediction - prediction of the next location a person will go to based on their historical trajectory. Zone functions classification - based on different mobility of people or POI distribution a function of a given area in a city can be predicted. Crime prediction - estimation of crime rate in different regions of a city. Local event detection - studying spatio-temporal changes in embeddings can provide valuable information in detection of local event occurring in specific location. Regional mobility popularity prediction - analysis of mobility can show patterns in popularity of different regions in a city. Shape matching - finding a similar shape of given polygon, for example finding building with the same shape as input building. Travel time estimation - predicting estimated travel time given current traffic conditions and special occurring events. Time estimation for on-demand food delivery - estimation of delivery time when placing an order through the website. == Temporal aspect == Some of the data analyzed has a timestamp associated with it. In some cases of data analysis this information is omitted and in others it is used to divide the set into groups. The most common division is the separation of weekdays from weekends or division into hours of the day. This is particularly important in the analysis of mobility data, because the characteristics of mobility during the week and at different times of the day are very different from each other. Another area in which time division into, for example, individual months can be used is in the analysis of tourism of a given region. In order to take such a split into account, embedding methods treat the time stamp specifically or separate versions of the model are developed for different subgroups of the analyzed set.

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  Nearest neighbor search

  Nearest neighbor search (NNS), as a form of proximity search, is the optimization problem of finding the point in a given set that is closest (or most similar) to a given point. Closeness is typically expressed in terms of a dissimilarity function: the less similar the objects, the larger the function values. Formally, the nearest neighbor (NN) search problem is defined as follows: given a set S of points in a space M and a query point q ∈ M {\displaystyle q\in M} , find the closest point in S to q. Donald Knuth in volume 3 of The Art of Computer Programming (1973) called it the post-office problem, referring to an application of assigning to a residence the nearest post office. A direct generalization of this problem is a k-NN search, where we need to find the k closest points. Most commonly M is a metric space and dissimilarity is expressed as a distance metric, which is symmetric and satisfies the triangle inequality. Even more common, M is taken to be the d-dimensional vector space where dissimilarity is measured using the Euclidean distance, Manhattan distance or other distance metric. However, the dissimilarity function can be arbitrary. One example is asymmetric Bregman divergence, for which the triangle inequality does not hold. == Applications == The nearest neighbor search problem arises in numerous fields of application, including: Pattern recognition – in particular for optical character recognition Statistical classification – see k-nearest neighbor algorithm Computer vision – for point cloud registration Computational geometry – see Closest pair of points problem Cryptanalysis – for lattice problem Databases – e.g. content-based image retrieval Coding theory – see maximum likelihood decoding Semantic search Vector databases, where nearest-neighbor lookup over embeddings is used to retrieve semantically similar records Retrieval-augmented generation systems, where nearest-neighbor retrieval over embeddings is used to fetch candidate passages or documents before generation Data compression – see MPEG-2 standard Robotic sensing Recommendation systems, e.g. see Collaborative filtering Internet marketing – see contextual advertising and behavioral targeting DNA sequencing Spell checking – suggesting correct spelling Plagiarism detection Similarity scores for predicting career paths of professional athletes. Cluster analysis – assignment of a set of observations into subsets (called clusters) so that observations in the same cluster are similar in some sense, usually based on Euclidean distance Chemical similarity Sampling-based motion planning == Methods == Various solutions to the NNS problem have been proposed. The quality and usefulness of the algorithms are determined by the time complexity of queries as well as the space complexity of any search data structures that must be maintained. The informal observation usually referred to as the curse of dimensionality states that there is no general-purpose exact solution for NNS in high-dimensional Euclidean space using polynomial preprocessing and polylogarithmic search time. === Exact methods === ==== Linear search ==== The simplest solution to the NNS problem is to compute the distance from the query point to every other point in the database, keeping track of the "best so far". This algorithm, sometimes referred to as the naive approach, has a running time of O(dN), where N is the cardinality of S and d is the dimensionality of S. There are no search data structures to maintain, so the linear search has no space complexity beyond the storage of the database. Naive search can, on average, outperform space partitioning approaches on higher dimensional spaces. The absolute distance is not required for distance comparison, only the relative distance. In geometric coordinate systems the distance calculation can be sped up considerably by omitting the square root calculation from the distance calculation between two coordinates. The distance comparison will still yield identical results. ==== Space partitioning ==== Since the 1970s, the branch and bound methodology has been applied to the problem. In the case of Euclidean space, this approach encompasses spatial index or spatial access methods. Several space-partitioning methods have been developed for solving the NNS problem. Perhaps the simplest is the k-d tree, which iteratively bisects the search space into two regions containing half of the points of the parent region. Queries are performed via traversal of the tree from the root to a leaf by evaluating the query point at each split. Depending on the distance specified in the query, neighboring branches that might contain hits may also need to be evaluated. For constant dimension query time, average complexity is O(log N) in the case of randomly distributed points, worst case complexity is O(kN^(1-1/k)) Alternatively the R-tree data structure was designed to support nearest neighbor search in dynamic context, as it has efficient algorithms for insertions and deletions such as the R tree. R-trees can yield nearest neighbors not only for Euclidean distance, but can also be used with other distances. In the case of general metric space, the branch-and-bound approach is known as the metric tree approach. Particular examples include vp-tree and BK-tree methods. Using a set of points taken from a 3-dimensional space and put into a BSP tree, and given a query point taken from the same space, a possible solution to the problem of finding the nearest point-cloud point to the query point is given in the following description of an algorithm. (Strictly speaking, no such point may exist, because it may not be unique. But in practice, usually we only care about finding any one of the subset of all point-cloud points that exist at the shortest distance to a given query point.) The idea is, for each branching of the tree, guess that the closest point in the cloud resides in the half-space containing the query point. This may not be the case, but it is a good heuristic. After having recursively gone through all the trouble of solving the problem for the guessed half-space, now compare the distance returned by this result with the shortest distance from the query point to the partitioning plane. This latter distance is that between the query point and the closest possible point that could exist in the half-space not searched. If this distance is greater than that returned in the earlier result, then clearly there is no need to search the other half-space. If there is such a need, then you must go through the trouble of solving the problem for the other half space, and then compare its result to the former result, and then return the proper result. The performance of this algorithm is nearer to logarithmic time than linear time when the query point is near the cloud, because as the distance between the query point and the closest point-cloud point nears zero, the algorithm needs only perform a look-up using the query point as a key to get the correct result. === Approximation methods === An approximate nearest neighbor search algorithm is allowed to return points whose distance from the query is at most c {\displaystyle c} times the distance from the query to its nearest points. The appeal of this approach is that, in many cases, an approximate nearest neighbor is almost as good as the exact one. In particular, if the distance measure accurately captures the notion of user quality, then small differences in the distance should not matter. ==== Greedy search in proximity neighborhood graphs ==== Proximity graph methods (such as navigable small world graphs and HNSW) are considered the current state-of-the-art for the approximate nearest neighbors search. The methods are based on greedy traversing in proximity neighborhood graphs G ( V , E ) {\displaystyle G(V,E)} in which every point x i ∈ S {\displaystyle x_{i}\in S} is uniquely associated with vertex v i ∈ V {\displaystyle v_{i}\in V} . The search for the nearest neighbors to a query q in the set S takes the form of searching for the vertex in the graph G ( V , E ) {\displaystyle G(V,E)} . The basic algorithm – greedy search – works as follows: search starts from an enter-point vertex v i ∈ V {\displaystyle v_{i}\in V} by computing the distances from the query q to each vertex of its neighborhood { v j : ( v i , v j ) ∈ E } {\displaystyle \{v_{j}:(v_{i},v_{j})\in E\}} , and then finds a vertex with the minimal distance value. If the distance value between the query and the selected vertex is smaller than the one between the query and the current element, then the algorithm moves to the selected vertex, and it becomes new enter-point. The algorithm stops when it reaches a local minimum: a vertex whose neighborhood does not contain a vertex that is closer to the query than the vertex itself. The idea of proximity neighborhood graphs was exploited in multiple publications, including the seminal paper by Arya and Mount, in the VoroNet syst

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  Q-learning

  Q-learning is a reinforcement learning algorithm that trains an agent to assign values to its possible actions based on its current state, without requiring a model of the environment (model-free). It can handle problems with stochastic transitions and rewards without requiring adaptations. For example, in a grid maze, an agent learns to reach an exit worth 10 points. At a junction, Q-learning might assign a higher value to moving right than left if right gets to the exit faster, improving this choice by trying both directions over time. For any finite Markov decision process, Q-learning finds an optimal policy in the sense of maximizing the expected value of the total reward over any and all successive steps, starting from the current state. Q-learning can identify an optimal action-selection policy for any given finite Markov decision process, given infinite exploration time and a partly random policy. "Q" refers to the function that the algorithm computes: the expected reward—that is, the quality—of an action taken in a given state. == Reinforcement learning == Reinforcement learning involves an agent, a set of states S {\displaystyle {\mathcal {S}}} , and a set A {\displaystyle {\mathcal {A}}} of actions per state. By performing an action a ∈ A {\displaystyle a\in {\mathcal {A}}} , the agent transitions from state to state. Executing an action in a specific state provides the agent with a reward (a numerical score). The goal of the agent is to maximize its total reward. It does this by adding the maximum reward attainable from future states to the reward for achieving its current state, effectively influencing the current action by the potential future reward. This potential reward is a weighted sum of expected values of the rewards of all future steps starting from the current state. As an example, consider the process of boarding a train, in which the reward is measured by the negative of the total time spent boarding (alternatively, the cost of boarding the train is equal to the boarding time). One strategy is to enter the train door as soon as they open, minimizing the initial wait time for yourself. If the train is crowded, however, then you will have a slow entry after the initial action of entering the door as people are fighting you to depart the train as you attempt to board. The total boarding time, or cost, is then: 0 seconds wait time + 15 seconds fight time On the next day, by random chance (exploration), you decide to wait and let other people depart first. This initially results in a longer wait time. However, less time is spent fighting the departing passengers. Overall, this path has a higher reward than that of the previous day, since the total boarding time is now: 5 second wait time + 0 second fight time Through exploration, despite the initial (patient) action resulting in a larger cost (or negative reward) than in the forceful strategy, the overall cost is lower, thus revealing a more rewarding strategy. == Algorithm == After Δ t {\displaystyle \Delta t} steps into the future the agent will decide some next step. The weight for this step is calculated as γ Δ t {\displaystyle \gamma ^{\Delta t}} , where γ {\displaystyle \gamma } (the discount factor) is a number between 0 and 1 ( 0 ≤ γ ≤ 1 {\displaystyle 0\leq \gamma \leq 1} ). Assuming γ < 1 {\displaystyle \gamma <1} , it has the effect of valuing rewards received earlier higher than those received later (reflecting the value of a "good start"). γ {\displaystyle \gamma } may also be interpreted as the probability to succeed (or survive) at every step Δ t {\displaystyle \Delta t} . The algorithm, therefore, has a function that calculates the quality of a state–action combination: Q : S × A → R {\displaystyle Q:{\mathcal {S}}\times {\mathcal {A}}\to \mathbb {R} } . Before learning begins, ⁠ Q {\displaystyle Q} ⁠ is initialized to a possibly arbitrary fixed value (chosen by the programmer). Then, at each time t {\displaystyle t} the agent selects an action A t {\displaystyle A_{t}} , observes a reward R t + 1 {\displaystyle R_{t+1}} , enters a new state S t + 1 {\displaystyle S_{t+1}} (that may depend on both the previous state S t {\displaystyle S_{t}} and the selected action), and Q {\displaystyle Q} is updated. The core of the algorithm is a Bellman equation as a simple value iteration update, using the weighted average of the current value and the new information: Q n e w ( S t , A t ) ← ( 1 − α ⏟ learning rate ) ⋅ Q ( S t , A t ) ⏟ current value + α ⏟ learning rate ⋅ ( R t + 1 ⏟ reward + γ ⏟ discount factor ⋅ max a Q ( S t + 1 , a ) ⏟ estimate of optimal future value ⏟ new value (temporal difference target) ) {\displaystyle Q^{new}(S_{t},A_{t})\leftarrow (1-\underbrace {\alpha } _{\text{learning rate}})\cdot \underbrace {Q(S_{t},A_{t})} _{\text{current value}}+\underbrace {\alpha } _{\text{learning rate}}\cdot {\bigg (}\underbrace {\underbrace {R_{t+1}} _{\text{reward}}+\underbrace {\gamma } _{\text{discount factor}}\cdot \underbrace {\max _{a}Q(S_{t+1},a)} _{\text{estimate of optimal future value}}} _{\text{new value (temporal difference target)}}{\bigg )}} where R t + 1 {\displaystyle R_{t+1}} is the reward received when moving from the state S t {\displaystyle S_{t}} to the state S t + 1 {\displaystyle S_{t+1}} , and α {\displaystyle \alpha } is the learning rate ( 0 < α ≤ 1 ) {\displaystyle (0<\alpha \leq 1)} . Note that Q n e w ( S t , A t ) {\displaystyle Q^{new}(S_{t},A_{t})} is the sum of three terms: ( 1 − α ) Q ( S t , A t ) {\displaystyle (1-\alpha )Q(S_{t},A_{t})} : the current value (weighted by one minus the learning rate) α R t + 1 {\displaystyle \alpha \,R_{t+1}} : the reward R t + 1 {\displaystyle R_{t+1}} to obtain if action A t {\displaystyle A_{t}} is taken when in state S t {\displaystyle S_{t}} (weighted by learning rate) α γ max a Q ( S t + 1 , a ) {\displaystyle \alpha \gamma \max _{a}Q(S_{t+1},a)} : the maximum reward that can be obtained from state S t + 1 {\displaystyle S_{t+1}} (weighted by learning rate and discount factor) An episode of the algorithm ends when state S t + 1 {\displaystyle S_{t+1}} is a final or terminal state. However, Q-learning can also learn in non-episodic tasks (as a result of the property of convergent infinite series). If the discount factor is lower than 1, the action values are finite even if the problem can contain infinite loops or paths. For all final states s f {\displaystyle s_{f}} , Q ( s f , a ) {\displaystyle Q(s_{f},a)} is never updated, but is set to the reward value r {\displaystyle r} observed for state s f {\displaystyle s_{f}} . In most cases, Q ( s f , a ) {\displaystyle Q(s_{f},a)} can be taken to equal zero. == Influence of variables == === Learning rate === The learning rate or step size determines to what extent newly acquired information overrides old information. A factor of 0 makes the agent learn nothing (exclusively exploiting prior knowledge), while a factor of 1 makes the agent consider only the most recent information (ignoring prior knowledge to explore possibilities). In fully deterministic environments, a learning rate of α t = 1 {\displaystyle \alpha _{t}=1} is optimal. When the problem is stochastic, the algorithm converges under some technical conditions on the learning rate that require it to decrease to zero. In practice, often a constant learning rate is used, such as α t = 0.1 {\displaystyle \alpha _{t}=0.1} for all t {\displaystyle t} . === Discount factor === The discount factor ⁠ γ {\displaystyle \gamma } ⁠ determines the importance of future rewards. A factor of 0 will make the agent "myopic" (or short-sighted) by only considering current rewards, i.e. r t {\displaystyle r_{t}} (in the update rule above), while a factor approaching 1 will make it strive for a long-term high reward. If the discount factor meets or exceeds 1, the action values may diverge. For ⁠ γ = 1 {\displaystyle \gamma =1} ⁠, without a terminal state, or if the agent never reaches one, all environment histories become infinitely long, and utilities with additive, undiscounted rewards generally become infinite. Even with a discount factor only slightly lower than 1, Q-function learning leads to propagation of errors and instabilities when the value function is approximated with an artificial neural network. In that case, starting with a lower discount factor and increasing it towards its final value accelerates learning. === Initial conditions (Q0) === Since Q-learning is an iterative algorithm, it implicitly assumes an initial condition before the first update occurs. High initial values, also known as "optimistic initial conditions", can encourage exploration: no matter what action is selected, the update rule will cause it to have lower values than the other alternative, thus increasing their choice probability. The first reward r {\displaystyle r} can be used to reset the initial conditions. According to this idea, the first time an action is taken the reward is used to set the value

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  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

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  List of chatbots

  A chatbot is a software application or web interface that is designed to mimic human conversation through text or voice interactions. Modern chatbots are typically online and use generative artificial intelligence systems that are capable of maintaining a conversation with a user in natural language and simulating the way a human would behave as a conversational partner. Such chatbots often use large language models (LLMs) and natural language processing, but simpler chatbots have existed for decades. == LLM chatbots == == General chatbots == == Historical chatbots ==

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  Genetic programming

  Genetic programming (GP) is an evolutionary algorithm, an artificial intelligence technique mimicking natural evolution, which operates on a population of programs. It applies the genetic operators selection according to a predefined fitness measure, mutation and crossover. The crossover operation involves swapping specified parts of selected pairs (parents) to produce new and different offspring that become part of the new generation of programs. Some programs not selected for reproduction are copied from the current generation to the new generation. Mutation involves substitution of some random part of a program with some other random part of a program. Then the selection and other operations are recursively applied to the new generation of programs. Typically, members of each new generation are on average more fit than the members of the previous generation, and the best-of-generation program is often better than the best-of-generation programs from previous generations. Termination of the evolution usually occurs when some individual program reaches a predefined proficiency or fitness level. It may and often does happen that a particular run of the algorithm results in premature convergence to some local maximum that is not a globally optimal or even good solution. Multiple runs (dozens to hundreds) are usually necessary to produce a very good result. It may also be necessary to have a large starting population size and variability of the individuals to avoid pathologies. == History == The first record of the proposal to evolve programs is probably that of Alan Turing in 1950 in "Computing Machinery and Intelligence". There was a gap of 25 years before the publication of John Holland's 'Adaptation in Natural and Artificial Systems' laid out the theoretical and empirical foundations of the science. In 1981, Richard Forsyth demonstrated the successful evolution of small programs, represented as trees, to perform classification of crime scene evidence for the UK Home Office. Although the idea of evolving programs, initially in the computer language Lisp, was current amongst John Holland's students, it was not until they organised the first Genetic Algorithms (GA) conference in Pittsburgh that Nichael Cramer published evolved programs in two specially designed languages, which included the first statement of modern "tree-based" genetic programming (that is, procedural languages organized in tree-based structures and operated on by suitably defined GA-operators). In 1988, John Koza (also a PhD student of John Holland) patented his invention of a GA for program evolution. This was followed by publication in the International Joint Conference on Artificial Intelligence IJCAI-89. Koza followed this with 205 publications on "genetic programming", a term coined by David Goldberg, also a PhD student of John Holland. However, it is the series of 4 books by Koza, starting in 1992 with accompanying videos, that really established GP. Subsequently, there was an enormous expansion of the number of publications with the Genetic Programming Bibliography, surpassing 10,000 entries. In 2010, Koza listed 77 results where genetic programming was human competitive. The departure of GP from the rigid, fixed-length representations typical of early GA models was not entirely without precedent. Early work on variable-length representations laid the groundwork. One notable example is messy genetic algorithms, which introduced irregular, variable-length chromosomes to address building block disruption and positional bias in standard GAs. Another precursor was robot trajectory programming, where genome representations encoded program instructions for robotic movements—structures inherently variable in length. Even earlier, unfixed-length representations were proposed in a doctoral dissertation by Cavicchio, who explored adaptive search using simulated evolution. His work provided foundational ideas for flexible program structures. In 1996, Koza started the annual Genetic Programming conference, which was followed in 1998 by the annual EuroGP conference, and the first book in a GP series edited by Koza. 1998 also saw the first GP textbook. GP continued to flourish, leading to the first specialist GP journal and three years later (2003) the annual Genetic Programming Theory and Practice (GPTP) workshop was established by Rick Riolo. Genetic programming papers continue to be published at a diversity of conferences and associated journals. Today there are nineteen GP books including several for students. === Foundational work in GP === Early work that set the stage for current genetic programming research topics and applications is diverse, and includes software synthesis and repair, predictive modeling, data mining, financial modeling, soft sensors, design, and image processing. Applications in some areas, such as design, often make use of intermediate representations, such as Fred Gruau's cellular encoding. Industrial uptake has been significant in several areas including finance, the chemical industry, bioinformatics and the steel industry. == Methods == === Program representation === GP evolves computer programs, traditionally represented in memory as tree structures. Trees can be easily evaluated in a recursive manner. Every internal node has an operator function and every terminal node has an operand, making mathematical expressions easy to evolve and evaluate. Thus traditionally GP favors the use of programming languages that naturally embody tree structures (for example, Lisp; other functional programming languages are also suitable). Non-tree representations have been suggested and successfully implemented, such as linear genetic programming, which perhaps suits the more traditional imperative languages. The commercial GP software Discipulus uses automatic induction of binary machine code ("AIM") to achieve better performance. μGP uses directed multigraphs to generate programs that fully exploit the syntax of a given assembly language. Multi expression programming uses three-address code for encoding solutions. Other program representations on which significant research and development have been conducted include programs for stack-based virtual machines, and sequences of integers that are mapped to arbitrary programming languages via grammars. Cartesian genetic programming is another form of GP, which uses a graph representation instead of the usual tree based representation to encode computer programs. Most representations have structurally noneffective code (introns). Such non-coding genes may seem to be useless because they have no effect on the performance of any one individual. However, they alter the probabilities of generating different offspring under the variation operators, and thus alter the individual's variational properties. Experiments seem to show faster convergence when using program representations that allow such non-coding genes, compared to program representations that do not have any non-coding genes. Instantiations may have both trees with introns and those without; the latter are called canonical trees. Special canonical crossover operators are introduced that maintain the canonical structure of parents in their children. === Initialisation === The methods for creation of the initial population include: Grow creates the individuals sequentially. Every GP tree is created starting from the root, creating functional nodes with children as well as terminal nodes up to a certain depth. Full is similar to the Grow. The difference is that all brunches in a tree are of same predetermined depth. Ramped half-and-half creates a population consisting of m d − 1 {\displaystyle md-1} parts and a maximum depth of m d {\displaystyle md} for its trees. The first part has a maximum depth of 2, second of 3 and so on up to the m d − 1 {\displaystyle md-1} -th part with maximum depth m d {\displaystyle md} . Half of every part is created by Grow, while the other part is created by Full. === Selection === Selection is a process whereby certain individuals are selected from the current generation that would serve as parents for the next generation. The individuals are selected probabilistically such that the better performing individuals have a higher chance of getting selected. The most commonly used selection method in GP is tournament selection, although other methods such as fitness proportionate selection, lexicase selection, and others have been demonstrated to perform better for many GP problems. Elitism, which involves seeding the next generation with the best individual (or best n individuals) from the current generation, is a technique sometimes employed to avoid regression. === Crossover === In genetic programming two fit individuals are chosen from the population to be parents for one or two children. In tree genetic programming, these parents are represented as inverted lisp like trees, with their root nodes at the top. In subtree cro

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  Neural gas

  Neural gas is an artificial neural network, inspired by the self-organizing map and introduced in 1991 by Thomas Martinetz and Klaus Schulten. The neural gas is a simple algorithm for finding optimal data representations based on feature vectors. The algorithm was coined "neural gas" because of the dynamics of the feature vectors during the adaptation process, which distribute themselves like a gas within the data space. It is applied where data compression or vector quantization is an issue, for example speech recognition, image processing or pattern recognition. As a robustly converging alternative to the k-means clustering it is also used for cluster analysis. == Algorithm == Suppose we want to model a probability distribution P ( x ) {\displaystyle P(x)} of data vectors x {\displaystyle x} using a finite number of feature vectors w i {\displaystyle w_{i}} , where i = 1 , ⋯ , N {\displaystyle i=1,\cdots ,N} . For each time step t {\displaystyle t} Sample data vector x {\displaystyle x} from P ( x ) {\displaystyle P(x)} Compute the distance between x {\displaystyle x} and each feature vector. Rank the distances. Let i 0 {\displaystyle i_{0}} be the index of the closest feature vector, i 1 {\displaystyle i_{1}} the index of the second closest feature vector, and so on. Update each feature vector by: w i k t + 1 = w i k t + ε ⋅ e − k / λ ⋅ ( x − w i k t ) , k = 0 , ⋯ , N − 1 {\displaystyle w_{i_{k}}^{t+1}=w_{i_{k}}^{t}+\varepsilon \cdot e^{-k/\lambda }\cdot (x-w_{i_{k}}^{t}),k=0,\cdots ,N-1} In the algorithm, ε {\displaystyle \varepsilon } can be understood as the learning rate, and λ {\displaystyle \lambda } as the neighborhood range. ε {\displaystyle \varepsilon } and λ {\displaystyle \lambda } are reduced with increasing t {\displaystyle t} so that the algorithm converges after many adaptation steps. The adaptation step of the neural gas can be interpreted as gradient descent on a cost function. By adapting not only the closest feature vector but all of them with a step size decreasing with increasing distance order, compared to (online) k-means clustering a much more robust convergence of the algorithm can be achieved. The neural gas model does not delete a node and also does not create new nodes. === Comparison with SOM === Compared to self-organized map, the neural gas model does not assume that some vectors are neighbors. If two vectors happen to be close together, they would tend to move together, and if two vectors happen to be apart, they would tend to not move together. In contrast, in an SOM, if two vectors are neighbors in the underlying graph, then they will always tend to move together, no matter whether the two vectors happen to be neighbors in the Euclidean space. The name "neural gas" is because one can imagine it to be what an SOM would be like if there is no underlying graph, and all points are free to move without the bonds that bind them together. == Variants == A number of variants of the neural gas algorithm exists in the literature so as to mitigate some of its shortcomings. More notable is perhaps Bernd Fritzke's growing neural gas, but also one should mention further elaborations such as the Growing When Required network and also the incremental growing neural gas. A performance-oriented approach that avoids the risk of overfitting is the Plastic Neural gas model. === Growing neural gas === Fritzke describes the growing neural gas (GNG) as an incremental network model that learns topological relations by using a "Hebb-like learning rule", only, unlike the neural gas, it has no parameters that change over time and it is capable of continuous learning, i.e. learning on data streams. GNG has been widely used in several domains, demonstrating its capabilities for clustering data incrementally. The GNG is initialized with two randomly positioned nodes which are initially connected with a zero age edge and whose errors are set to 0. Since in the GNG input data is presented sequentially one by one, the following steps are followed at each iteration: It is calculating the errors (distances) between the two closest nodes to the current input data. The error of the winner node (only the closest one) is respectively accumulated. The winner node and its topological neighbors (connected by an edge) are moving towards the current input by different fractions of their respective errors. The age of all edges connected to the winner node are incremented. If the winner node and the second-winner are connected by an edge, such an edge is set to 0. Else, an edge is created between them. If there are edges with an age larger than a threshold, they are removed. Nodes without connections are eliminated. If the current iteration is an integer multiple of a predefined frequency-creation threshold, a new node is inserted between the node with the largest error (among all) and its topological neighbor presenting the highest error. The link between the former and the latter nodes is eliminated (their errors are decreased by a given factor) and the new node is connected to both of them. The error of the new node is initialized as the updated error of the node which had the largest error (among all). The accumulated error of all nodes is decreased by a given factor. If the stopping criterion is not met, the algorithm takes a following input. The criterion might be a given number of epochs, i.e., a pre-set number of times where all data is presented, or the reach of a maximum number of nodes. === Incremental growing neural gas === Another neural gas variant inspired by the GNG algorithm is the incremental growing neural gas (IGNG). The authors propose the main advantage of this algorithm to be "learning new data (plasticity) without degrading the previously trained network and forgetting the old input data (stability)." === Growing when required === Having a network with a growing set of nodes, like the one implemented by the GNG algorithm was seen as a great advantage, however some limitation on the learning was seen by the introduction of the parameter λ, in which the network would only be able to grow when iterations were a multiple of this parameter. The proposal to mitigate this problem was a new algorithm, the Growing When Required network (GWR), which would have the network grow more quickly, by adding nodes as quickly as possible whenever the network identified that the existing nodes would not describe the input well enough. === Plastic neural gas === The ability to only grow a network may quickly introduce overfitting; on the other hand, removing nodes on the basis of age only, as in the GNG model, does not ensure that the removed nodes are actually useless, because removal depends on a model parameter that should be carefully tuned to the "memory length" of the stream of input data. The "Plastic Neural Gas" model solves this problem by making decisions to add or remove nodes using an unsupervised version of cross-validation, which controls an equivalent notion of "generalization ability" for the unsupervised setting. While growing-only methods only cater for the incremental learning scenario, the ability to grow and shrink is suited to the more general streaming data problem. == Implementations == To find the ranking i 0 , i 1 , … , i N − 1 {\displaystyle i_{0},i_{1},\ldots ,i_{N-1}} of the feature vectors, the neural gas algorithm involves sorting, which is a procedure that does not lend itself easily to parallelization or implementation in analog hardware. However, implementations in both parallel software and analog hardware were actually designed.

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  Concept class

  In computational learning theory in mathematics, a concept over a domain X is a total Boolean function over X. A concept class is a class of concepts. Concept classes are a subject of computational learning theory. Concept class terminology frequently appears in model theory associated with probably approximately correct (PAC) learning. In this setting, if one takes a set Y as a set of (classifier output) labels, and X is a set of examples, the map c : X → Y {\displaystyle c:X\to Y} , i.e. from examples to classifier labels (where Y = { 0 , 1 } {\displaystyle Y=\{0,1\}} and where c is a subset of X), c is then said to be a concept. A concept class C {\displaystyle C} is then a collection of such concepts. Given a class of concepts C, a subclass D is reachable if there exists a sample s such that D contains exactly those concepts in C that are extensions to s. Not every subclass is reachable. == Background == A sample s {\displaystyle s} is a partial function from X {\displaystyle X} to { 0 , 1 } {\displaystyle \{0,1\}} . Identifying a concept with its characteristic function mapping X {\displaystyle X} to { 0 , 1 } {\displaystyle \{0,1\}} , it is a special case of a sample. Two samples are consistent if they agree on the intersection of their domains. A sample s ′ {\displaystyle s'} extends another sample s {\displaystyle s} if the two are consistent and the domain of s {\displaystyle s} is contained in the domain of s ′ {\displaystyle s'} . == Examples == Suppose that C = S + ( X ) {\displaystyle C=S^{+}(X)} . Then: the subclass { { x } } {\displaystyle \{\{x\}\}} is reachable with the sample s = { ( x , 1 ) } {\displaystyle s=\{(x,1)\}} ; the subclass S + ( Y ) {\displaystyle S^{+}(Y)} for Y ⊆ X {\displaystyle Y\subseteq X} are reachable with a sample that maps the elements of X − Y {\displaystyle X-Y} to zero; the subclass S ( X ) {\displaystyle S(X)} , which consists of the singleton sets, is not reachable. == Applications == Let C {\displaystyle C} be some concept class. For any concept c ∈ C {\displaystyle c\in C} , we call this concept 1 / d {\displaystyle 1/d} -good for a positive integer d {\displaystyle d} if, for all x ∈ X {\displaystyle x\in X} , at least 1 / d {\displaystyle 1/d} of the concepts in C {\displaystyle C} agree with c {\displaystyle c} on the classification of x {\displaystyle x} . The fingerprint dimension F D ( C ) {\displaystyle FD(C)} of the entire concept class C {\displaystyle C} is the least positive integer d {\displaystyle d} such that every reachable subclass C ′ ⊆ C {\displaystyle C'\subseteq C} contains a concept that is 1 / d {\displaystyle 1/d} -good for it. This quantity can be used to bound the minimum number of equivalence queries needed to learn a class of concepts according to the following inequality: F D ( C ) − 1 ≤ # E Q ( C ) ≤ ⌈ F D ( C ) ln ⁡ ( | C | ) ⌉ {\textstyle FD(C)-1\leq \#EQ(C)\leq \lceil FD(C)\ln(|C|)\rceil } .

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  Discovery system (artificial intelligence)

  A discovery system is an artificial intelligence system that attempts to discover new scientific concepts or laws. The aim of discovery systems is to automate scientific data analysis and the scientific discovery process. Ideally, an artificial intelligence system should be able to search systematically through the space of all possible hypotheses and yield the hypothesis - or set of equally likely hypotheses - that best describes the complex patterns in data. During the era known as the second AI summer (approximately 1978–1987), various systems akin to the era's dominant expert systems were developed to tackle the problem of extracting scientific hypotheses from data, with or without interacting with a human scientist. These systems included Autoclass, Automated Mathematician, Eurisko, which aimed at general-purpose hypothesis discovery, and more specific systems such as Dalton, which uncovers molecular properties from data. The dream of building systems that discover scientific hypotheses was pushed to the background with the second AI winter and the subsequent resurgence of subsymbolic methods such as neural networks. Subsymbolic methods emphasize prediction over explanation, and yield models which works well but are difficult or impossible to explain which has earned them the name black box AI. A black-box model cannot be considered a scientific hypothesis, and this development has even led some researchers to suggest that the traditional aim of science - to uncover hypotheses and theories about the structure of reality - is obsolete. Other researchers disagree and argue that subsymbolic methods are useful in many cases, just not for generating scientific theories. == Discovery systems from the 1970s and 1980s == Autoclass was a Bayesian Classification System written in 1986 Automated Mathematician was one of the earliest successful discovery systems. It was written in 1977 and worked by generating a modifying small Lisp programs Eurisko was a Sequel to Automated Mathematician written in 1984 Dalton is a still maintained program capable of calculating various molecular properties initially launched in 1983 and available in open source since 2017 Glauber is a scientific discovery method written in the context of computational philosophy of science launched in 1983 == Modern discovery systems (2009–present) == After a couple of decades with little interest in discovery systems, the interest in using AI to uncover natural laws and scientific explanations was renewed by the work of Michael Schmidt, then a PhD student in Computational Biology at Cornell University. Schmidt and his advisor, Hod Lipson, invented Eureqa, which they described as a symbolic regression approach to "distilling free-form natural laws from experimental data". This work effectively demonstrated that symbolic regression was a promising way forward for AI-driven scientific discovery. Since 2009, symbolic regression has matured further, and today, various commercial and open source systems are actively used in scientific research. Notable examples include Eureqa, now a part of DataRobot AI Cloud Platform, AI Feynman, and QLattice.

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  Prefrontal cortex basal ganglia working memory

  Prefrontal cortex basal ganglia working memory (PBWM) is an algorithm that models working memory in the prefrontal cortex and the basal ganglia. It can be compared to long short-term memory (LSTM) in functionality, but is more biologically explainable. It uses the primary value learned value model to train prefrontal cortex working-memory updating system, based on the biology of the prefrontal cortex and basal ganglia. It is used as part of the Leabra framework and was implemented in Emergent in 2019. == Abstract == The prefrontal cortex has long been thought to subserve both working memory (the holding of information online for processing) and "executive" functions (deciding how to manipulate working memory and perform processing). Although many computational models of working memory have been developed, the mechanistic basis of executive function remains elusive. PBWM is a computational model of the prefrontal cortex to control both itself and other brain areas in a strategic, task-appropriate manner. These learning mechanisms are based on subcortical structures in the midbrain, basal ganglia and amygdala, which together form an actor/critic architecture. The critic system learns which prefrontal representations are task-relevant and trains the actor, which in turn provides a dynamic gating mechanism for controlling working memory updating. Computationally, the learning mechanism is designed to simultaneously solve the temporal and structural credit assignment problems. The model's performance compares favorably with standard backpropagation-based temporal learning mechanisms on the challenging 1-2-AX working memory task, and other benchmark working memory tasks. == Model == First, there are multiple separate stripes (groups of units) in the prefrontal cortex and striatum layers. Each stripe can be independently updated, such that this system can remember several different things at the same time, each with a different "updating policy" of when memories are updated and maintained. The active maintenance of the memory is in prefrontal cortex (PFC), and the updating signals (and updating policy more generally) come from the striatum units (a subset of basal ganglia units). PVLV provides reinforcement learning signals to train up the dynamic gating system in the basal ganglia. === Sensory input and motor output === The sensory input is connected to the posterior cortex which is connected to the motor output. The sensory input is also linked to the PVLV system. === Posterior cortex === The posterior cortex form the hidden layers of the input/output mapping. The PFC is connected with the posterior cortex to contextualize this input/output mapping. === PFC === The PFC (for output gating) has a localist one-to-one representation of the input units for every stripe. Thus, you can look at these PFC representations and see directly what the network is maintaining. The PFC maintains the working memory needed to perform the task. === Striatum === This is the dynamic gating system representing the striatum units of the basal ganglia. Every even-index unit within a stripe represents "Go", while the odd-index units represent "NoGo." The Go units cause updating of the PFC, while the NoGo units cause the PFC to maintain its existing memory representation. There are groups of units for every stripe. In the PBWM model in Emergent, the matrices represent the striatum. === PVLV === All of these layers are part of PVLV system. The PVLV system controls the dopaminergic modulation of the basal ganglia (BG). Thus, BG/PVLV form an actor-critic architecture where the PVLV system learns when to update. ==== SNrThal ==== SNrThal represents the substantia nigra pars reticulata (SNr) and the associated area of the thalamus, which produce a competition among the Go/NoGo units within a given stripe and mediates competition using k-winners-take-all dynamics. If there is more overall Go activity in a given stripe, then the associated SNrThal unit gets activated, and it drives updating in PFC. For every stripe, there is one unit in SNrThal. ==== VTA and SNc ==== Ventral tegmental area (VTA) and substantia nigra pars compacta (SNc) are part of the dopamine layer. This layer models midbrain dopamine neurons. They control the dopaminergic modulation of the basal ganglia.

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  Elastic net regularization

  In statistics and, in particular, in the fitting of linear or logistic regression models, the elastic net is a regularized regression method that linearly combines the L1 and L2 penalties of the lasso and ridge methods. Nevertheless, elastic net regularization is typically more accurate than both methods with regard to reconstruction. == Specification == The elastic net method overcomes the limitations of the LASSO (least absolute shrinkage and selection operator) method which uses a penalty function based on ‖ β ‖ 1 = ∑ j = 1 p | β j | . {\displaystyle \|\beta \|_{1}=\textstyle \sum _{j=1}^{p}|\beta _{j}|.} Use of this penalty function has several limitations. For example, in the "large p, small n" case (high-dimensional data with few examples), the LASSO selects at most n variables before it saturates. Also if there is a group of highly correlated variables, then the LASSO tends to select one variable from a group and ignore the others. To overcome these limitations, the elastic net adds a quadratic part ( ‖ β ‖ 2 {\displaystyle \|\beta \|^{2}} ) to the penalty, which when used alone is ridge regression (known also as Tikhonov regularization). The estimates from the elastic net method are defined by β ^ ≡ argmin β ( ‖ y − X β ‖ 2 + λ 2 ‖ β ‖ 2 + λ 1 ‖ β ‖ 1 ) . {\displaystyle {\hat {\beta }}\equiv {\underset {\beta }{\operatorname {argmin} }}(\|y-X\beta \|^{2}+\lambda _{2}\|\beta \|^{2}+\lambda _{1}\|\beta \|_{1}).} The quadratic penalty term makes the loss function strongly convex, and it therefore has a unique minimum. The elastic net method includes the LASSO and ridge regression: in other words, each of them is a special case where λ 1 = λ , λ 2 = 0 {\displaystyle \lambda _{1}=\lambda ,\lambda _{2}=0} or λ 1 = 0 , λ 2 = λ {\displaystyle \lambda _{1}=0,\lambda _{2}=\lambda } . Meanwhile, the naive version of elastic net method finds an estimator in a two-stage procedure : first for each fixed λ 2 {\displaystyle \lambda _{2}} it finds the ridge regression coefficients, and then does a LASSO type shrinkage. This kind of estimation incurs a double amount of shrinkage, which leads to increased bias and poor predictions. To improve the prediction performance, sometimes the coefficients of the naive version of elastic net is rescaled by multiplying the estimated coefficients by ( 1 + λ 2 ) {\displaystyle (1+\lambda _{2})} . Examples of where the elastic net method has been applied are: Support vector machine Metric learning Portfolio optimization Cancer prognosis == Reduction to support vector machine == It was proven in 2014 that the elastic net can be reduced to the linear support vector machine. A similar reduction was previously proven for the LASSO in 2014. The authors showed that for every instance of the elastic net, an artificial binary classification problem can be constructed such that the hyper-plane solution of a linear support vector machine (SVM) is identical to the solution β {\displaystyle \beta } (after re-scaling). The reduction immediately enables the use of highly optimized SVM solvers for elastic net problems. It also enables the use of GPU acceleration, which is often already used for large-scale SVM solvers. The reduction is a simple transformation of the original data and regularization constants X ∈ R n × p , y ∈ R n , λ 1 ≥ 0 , λ 2 ≥ 0 {\displaystyle X\in {\mathbb {R} }^{n\times p},y\in {\mathbb {R} }^{n},\lambda _{1}\geq 0,\lambda _{2}\geq 0} into new artificial data instances and a regularization constant that specify a binary classification problem and the SVM regularization constant X 2 ∈ R 2 p × n , y 2 ∈ { − 1 , 1 } 2 p , C ≥ 0. {\displaystyle X_{2}\in {\mathbb {R} }^{2p\times n},y_{2}\in \{-1,1\}^{2p},C\geq 0.} Here, y 2 {\displaystyle y_{2}} consists of binary labels − 1 , 1 {\displaystyle {-1,1}} . When 2 p > n {\displaystyle 2p>n} it is typically faster to solve the linear SVM in the primal, whereas otherwise the dual formulation is faster. Some authors have referred to the transformation as Support Vector Elastic Net (SVEN), and provided the following MATLAB pseudo-code: == Software == "Glmnet: Lasso and elastic-net regularized generalized linear models" is a software which is implemented as an R source package and as a MATLAB toolbox. This includes fast algorithms for estimation of generalized linear models with ℓ1 (the lasso), ℓ2 (ridge regression) and mixtures of the two penalties (the elastic net) using cyclical coordinate descent, computed along a regularization path. JMP Pro 11 includes elastic net regularization, using the Generalized Regression personality with Fit Model. "pensim: Simulation of high-dimensional data and parallelized repeated penalized regression" implements an alternate, parallelised "2D" tuning method of the ℓ parameters, a method claimed to result in improved prediction accuracy. scikit-learn includes linear regression and logistic regression with elastic net regularization. SVEN, a Matlab implementation of Support Vector Elastic Net. This solver reduces the Elastic Net problem to an instance of SVM binary classification and uses a Matlab SVM solver to find the solution. Because SVM is easily parallelizable, the code can be faster than Glmnet on modern hardware. SpaSM, a Matlab implementation of sparse regression, classification and principal component analysis, including elastic net regularized regression. Apache Spark provides support for Elastic Net Regression in its MLlib machine learning library. The method is available as a parameter of the more general LinearRegression class. SAS (software) The SAS procedure Glmselect and SAS Viya procedure Regselect support the use of elastic net regularization for model selection.

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  Robust principal component analysis

  Robust Principal Component Analysis (RPCA) is a modification of the widely used statistical procedure of principal component analysis (PCA) which works well with respect to grossly corrupted observations. A number of different approaches exist for Robust PCA, including an idealized version of Robust PCA, which aims to recover a low-rank matrix L0 from highly corrupted measurements M = L0 +S0. This decomposition in low-rank and sparse matrices can be achieved by techniques such as Principal Component Pursuit method (PCP), Stable PCP, Quantized PCP, Block based PCP, and Local PCP. Then, optimization methods are used such as the Augmented Lagrange Multiplier Method (ALM), Alternating Direction Method (ADM), Fast Alternating Minimization (FAM), Iteratively Reweighted Least Squares (IRLS ) or alternating projections (AP). == Algorithms == === Non-convex method === The 2014 guaranteed algorithm for the robust PCA problem (with the input matrix being M = L + S {\displaystyle M=L+S} ) is an alternating minimization type algorithm. The computational complexity is O ( m n r 2 log ⁡ 1 ϵ ) {\displaystyle O\left(mnr^{2}\log {\frac {1}{\epsilon }}\right)} where the input is the superposition of a low-rank (of rank r {\displaystyle r} ) and a sparse matrix of dimension m × n {\displaystyle m\times n} and ϵ {\displaystyle \epsilon } is the desired accuracy of the recovered solution, i.e., ‖ L ^ − L ‖ F ≤ ϵ {\displaystyle \|{\widehat {L}}-L\|_{F}\leq \epsilon } where L {\displaystyle L} is the true low-rank component and L ^ {\displaystyle {\widehat {L}}} is the estimated or recovered low-rank component. Intuitively, this algorithm performs projections of the residual onto the set of low-rank matrices (via the SVD operation) and sparse matrices (via entry-wise hard thresholding) in an alternating manner - that is, low-rank projection of the difference the input matrix and the sparse matrix obtained at a given iteration followed by sparse projection of the difference of the input matrix and the low-rank matrix obtained in the previous step, and iterating the two steps until convergence. This alternating projections algorithm is later improved by an accelerated version, coined AccAltProj. The acceleration is achieved by applying a tangent space projection before projecting the residue onto the set of low-rank matrices. This trick improves the computational complexity to O ( m n r log ⁡ 1 ϵ ) {\displaystyle O\left(mnr\log {\frac {1}{\epsilon }}\right)} with a much smaller constant in front while it maintains the theoretically guaranteed linear convergence. Another fast version of accelerated alternating projections algorithm is IRCUR. It uses the structure of CUR decomposition in alternating projections framework to dramatically reduces the computational complexity of RPCA to O ( max { m , n } r 2 log ⁡ ( m ) log ⁡ ( n ) log ⁡ 1 ϵ ) {\displaystyle O\left(\max\{m,n\}r^{2}\log(m)\log(n)\log {\frac {1}{\epsilon }}\right)} === Convex relaxation === This method consists of relaxing the rank constraint r a n k ( L ) {\displaystyle rank(L)} in the optimization problem to the nuclear norm ‖ L ‖ ∗ {\displaystyle \|L\|_{}} and the sparsity constraint ‖ S ‖ 0 {\displaystyle \|S\|_{0}} to ℓ 1 {\displaystyle \ell _{1}} -norm ‖ S ‖ 1 {\displaystyle \|S\|_{1}} . The resulting program can be solved using methods such as the method of Augmented Lagrange Multipliers. === Deep-learning augmented method === Some recent works propose RPCA algorithms with learnable/training parameters. Such a learnable/trainable algorithm can be unfolded as a deep neural network whose parameters can be learned via machine learning techniques from a given dataset or problem distribution. The learned algorithm will have superior performance on the corresponding problem distribution. == Applications == RPCA has many real life important applications particularly when the data under study can naturally be modeled as a low-rank plus a sparse contribution. Following examples are inspired by contemporary challenges in computer science, and depending on the applications, either the low-rank component or the sparse component could be the object of interest: === Video surveillance === Given a sequence of surveillance video frames, it is often required to identify the activities that stand out from the background. If we stack the video frames as columns of a matrix M, then the low-rank component L0 naturally corresponds to the stationary background and the sparse component S0 captures the moving objects in the foreground. === Face recognition === Images of a convex, Lambertian surface under varying illuminations span a low-dimensional subspace. This is one of the reasons for effectiveness of low-dimensional models for imagery data. In particular, it is easy to approximate images of a human's face by a low-dimensional subspace. To be able to correctly retrieve this subspace is crucial in many applications such as face recognition and alignment. It turns out that RPCA can be applied successfully to this problem to exactly recover the face.

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  Legendre moment

  In mathematics, Legendre moments are a type of image moment and are achieved by using the Legendre polynomial. Legendre moments are used in areas of image processing including: pattern and object recognition, image indexing, line fitting, feature extraction, edge detection, and texture analysis. Legendre moments have been studied as a means to reduce image moment calculation complexity by limiting the amount of information redundancy through approximation. == Legendre moments == Source: With order of m + n, and object intensity function f(x,y): L m n = ( 2 m + 1 ) ( 2 n + 1 ) 4 ∫ − 1 1 ∫ − 1 1 P m ( x ) P n ( y ) f ( x , y ) d x d y {\displaystyle L_{mn}={\frac {(2m+1)(2n+1)}{4}}\int \limits _{-1}^{1}\int \limits _{-1}^{1}P_{m}(x)P_{n}(y)f(x,y)\,dx\,dy} where m,n = 1, 2, 3, ...∞ with the nth-order Legendre polynomials being: P n ( x ) = ∑ k = 0 n a k , n x k = ( − 1 ) n 2 n n ! ( d d x ) [ ( 1 − x 2 ) n ] {\displaystyle P_{n}(x)=\sum _{k=0}^{n}a_{k,n}x^{k}={\frac {(-1)^{n}}{2^{n}n!}}\left({\frac {d}{dx}}\right)[(1-x^{2})^{n}]} which can also be written: P n ( x ) = ∑ k = 0 D ( n ) ( − 1 ) k ( 2 n − 2 k ) ! 2 n k ! ( n − k ) ! ( n − 2 k ) ! x n − 2 k = ( 2 n ) ! 2 n ( n ! ) 2 x n − ( 2 n − 2 ) ! 2 n 1 ! ( n − 1 ) ! ( n − 2 ) ! x n − 2 + ⋯ {\displaystyle {\begin{aligned}P_{n}(x)&=\sum _{k=0}^{D(n)}(-1)^{k}{\frac {(2n-2k)!}{2^{n}k!(n-k)!(n-2k)!}}x^{n-2k}\\[5pt]&={\frac {(2n)!}{2^{n}(n!)^{2}}}x^{n}-{\frac {(2n-2)!}{2^{n}1!(n-1)!(n-2)!}}x^{n-2}+\cdots \end{aligned}}} where D(n) = floor(n/2). The set of Legendre polynomials {Pn(x)} form an orthogonal set on the interval [−1,1]: ∫ − 1 1 P n ( x ) P m ( x ) d x = 2 2 n + 1 δ n m {\displaystyle \int _{-1}^{1}P_{n}(x)P_{m}(x)\,dx={\frac {2}{2n+1}}\delta _{nm}} A recurrence relation can be used to compute the Legendre polynomial: ( n + 1 ) P n + 1 ( x ) − ( 2 n + 1 ) x P n ( x ) + n P n − 1 ( x ) = 0 {\displaystyle (n+1)P_{n+1}(x)-(2n+1)xP_{n}(x)+nP_{n-1}(x)=0} f(x,y) can be written as an infinite series expansion in terms of Legendre polynomials [−1 ≤ x,y ≤ 1.]: f ( x , y ) = ∑ m = 0 ∞ ∑ n = 0 ∞ λ m n P m ( x ) P n ( y ) {\displaystyle f(x,y)=\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }\lambda _{mn}P_{m}(x)P_{n}(y)}

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  Teaching dimension

  In computational learning theory, the teaching dimension of a concept class C is defined to be max c ∈ C { w C ( c ) } {\displaystyle \max _{c\in C}\{w_{C}(c)\}} , where w C ( c ) {\displaystyle {w_{C}(c)}} is the minimum size of a witness set for c in C. Intuitively, this measures the number of instances that are needed to identify a concept in the class, using supervised learning with examples provided by a helpful teacher who is trying to convey the concept as succinctly as possible. This definition was formulated in 1995 by Sally Goldman and Michael Kearns, based on earlier work by Goldman, Ron Rivest, and Robert Schapire. The teaching dimension of a finite concept class can be used to give a lower and an upper bound on the membership query cost of the concept class. In Stasys Jukna's book "Extremal Combinatorics", a lower bound is given for the teaching dimension in general: Let C be a concept class over a finite domain X. If the size of C is greater than 2 k ( | X | k ) , {\displaystyle 2^{k}{|X| \choose k},} then the teaching dimension of C is greater than k. However, there are more specific teaching models that make assumptions about teacher or learner, and can get lower values for the teaching dimension. For instance, several models are the classical teaching (CT) model, the optimal teacher (OT) model, recursive teaching (RT), preference-based teaching (PBT), and non-clashing teaching (NCT).

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  Ordinal regression

  In statistics, ordinal regression, also called ordinal classification, is a type of regression analysis used for predicting an ordinal variable, i.e. a variable whose value exists on an arbitrary scale where only the relative ordering between different values is significant. It can be considered an intermediate problem between regression and classification. Examples of ordinal regression are ordered logit and ordered probit. Ordinal regression turns up often in the social sciences, for example in the modeling of human levels of preference (on a scale from, say, 1–5 for "very poor" through "excellent"), as well as in information retrieval. In machine learning, ordinal regression may also be called ranking learning. == Linear models for ordinal regression == Ordinal regression can be performed using a generalized linear model (GLM) that fits both a coefficient vector and a set of thresholds to a dataset. Suppose one has a set of observations, represented by length-p vectors x1 through xn, with associated responses y1 through yn, where each yi is an ordinal variable on a scale 1, ..., K. For simplicity, and without loss of generality, we assume y is a non-decreasing vector, that is, yi ≤ {\displaystyle \leq } yi+1. To this data, one fits a length-p coefficient vector w and a set of thresholds θ1, ..., θK−1 with the property that θ1 < θ2 < ... < θK−1. This set of thresholds divides the real number line into K disjoint segments, corresponding to the K response levels. The model can now be formulated as Pr ( y ≤ i ∣ x ) = σ ( θ i − w ⋅ x ) {\displaystyle \Pr(y\leq i\mid \mathbf {x} )=\sigma (\theta _{i}-\mathbf {w} \cdot \mathbf {x} )} or, the cumulative probability of the response y being at most i is given by a function σ (the inverse link function) applied to a linear function of x. Several choices exist for σ; the logistic function σ ( θ i − w ⋅ x ) = 1 1 + e − ( θ i − w ⋅ x ) {\displaystyle \sigma (\theta _{i}-\mathbf {w} \cdot \mathbf {x} )={\frac {1}{1+e^{-(\theta _{i}-\mathbf {w} \cdot \mathbf {x} )}}}} gives the ordered logit model, while using the CDF of the standard normal distribution gives the ordered probit model. A third option is to use an exponential function σ ( θ i − w ⋅ x ) = 1 − exp ⁡ ( − exp ⁡ ( θ i − w ⋅ x ) ) {\displaystyle \sigma (\theta _{i}-\mathbf {w} \cdot \mathbf {x} )=1-\exp(-\exp(\theta _{i}-\mathbf {w} \cdot \mathbf {x} ))} which gives the proportional hazards model. === Latent variable model === The probit version of the above model can be justified by assuming the existence of a real-valued latent variable (unobserved quantity) y, determined by y ∗ = w ⋅ x + ε {\displaystyle y^{}=\mathbf {w} \cdot \mathbf {x} +\varepsilon } where ε is normally distributed with zero mean and unit variance, conditioned on x. The response variable y results from an "incomplete measurement" of y, where one only determines the interval into which y falls: y = { 1 if y ∗ ≤ θ 1 , 2 if θ 1 < y ∗ ≤ θ 2 , 3 if θ 2 < y ∗ ≤ θ 3 ⋮ K if θ K − 1 < y ∗ . {\displaystyle y={\begin{cases}1&{\text{if}}~~y^{}\leq \theta _{1},\\2&{\text{if}}~~\theta _{1} Read more →