AI Chat Image Generator Free

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  Learning rate

  In machine learning and statistics, the learning rate is a tuning parameter in an optimization algorithm that determines the step size at each iteration while moving toward a minimum of a loss function. Since it influences to what extent newly acquired information overrides old information, it metaphorically represents the speed at which a machine learning model "learns". In the adaptive control literature, the learning rate is commonly referred to as gain. In setting a learning rate, there is a trade-off between the rate of convergence and overshooting. While the descent direction is usually determined from the gradient of the loss function, the learning rate determines how big a step is taken in that direction. Too high a learning rate will make the learning jump over minima, but too low a learning rate will either take too long to converge or get stuck in an undesirable local minimum. In order to achieve faster convergence, prevent oscillations and getting stuck in undesirable local minima the learning rate is often varied during training either in accordance to a learning rate schedule or by using an adaptive learning rate. The learning rate and its adjustments may also differ per parameter, in which case it is a diagonal matrix that can be interpreted as an approximation to the inverse of the Hessian matrix in Newton's method. The learning rate is related to the step length determined by inexact line search in quasi-Newton methods and related optimization algorithms. == Learning rate schedule == Initial rate can be left as system default or can be selected using a range of techniques. A learning rate schedule changes the learning rate during learning and is most often changed between epochs/iterations. This is mainly done with two parameters: decay and momentum. There are many different learning rate schedules but the most common are time-based, step-based and exponential. Decay serves to settle the learning in a nice place and avoid oscillations, a situation that may arise when too high a constant learning rate makes the learning jump back and forth over a minimum, and is controlled by a hyperparameter. Momentum is analogous to a ball rolling down a hill; we want the ball to settle at the lowest point of the hill (corresponding to the lowest error). Momentum both speeds up the learning (increasing the learning rate) when the error cost gradient is heading in the same direction for a long time and also avoids local minima by 'rolling over' small bumps. Momentum is controlled by a hyperparameter analogous to a ball's mass which must be chosen manually—too high and the ball will roll over minima which we wish to find, too low and it will not fulfil its purpose. The formula for factoring in the momentum is more complex than for decay but is most often built in with deep learning libraries such as Keras. Time-based learning schedules alter the learning rate depending on the learning rate of the previous time iteration. Factoring in the decay the mathematical formula for the learning rate is: η n + 1 = η 0 1 + d n {\displaystyle \eta _{n+1}={\frac {\eta _{0}}{1+dn}}} where η {\displaystyle \eta } is the learning rate, η 0 {\displaystyle \eta _{0}} is the original learning rate, d {\displaystyle d} is a decay parameter and n {\displaystyle n} is the iteration step. Step-based learning schedules changes the learning rate according to some predefined steps. The decay application formula is here defined as: η n = η 0 d ⌊ 1 + n r ⌋ {\displaystyle \eta _{n}=\eta _{0}d^{\left\lfloor {\frac {1+n}{r}}\right\rfloor }} where η n {\displaystyle \eta _{n}} is the learning rate at iteration n {\displaystyle n} , η 0 {\displaystyle \eta _{0}} is the initial learning rate, d {\displaystyle d} is how much the learning rate should change at each drop (0.5 corresponds to a halving) and r {\displaystyle r} corresponds to the drop rate, or how often the rate should be dropped (10 corresponds to a drop every 10 iterations). The floor function ( ⌊ … ⌋ {\displaystyle \lfloor \dots \rfloor } ) here drops the value of its input to 0 for all values smaller than 1. Exponential learning schedules are similar to step-based, but instead of steps, a decreasing exponential function is used. The mathematical formula for factoring in the decay is: η n = η 0 e − d n {\displaystyle \eta _{n}=\eta _{0}e^{-dn}} where d {\displaystyle d} is a decay parameter. == Adaptive learning rate == The issue with learning rate schedules is that they all depend on hyperparameters that must be manually chosen for each given learning session and may vary greatly depending on the problem at hand or the model used. To combat this, there are many different types of adaptive gradient descent algorithms such as Adagrad, Adadelta, RMSprop, and Adam which are generally built into deep learning libraries such as Keras.

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  Vapnik–Chervonenkis dimension

  In Vapnik–Chervonenkis theory, the Vapnik–Chervonenkis (VC) dimension is a measure of the size (capacity, complexity, expressive power, richness, or flexibility) of a class of sets. The notion can be extended to classes of binary functions. It is defined as the cardinality of the largest set of points that the function class can shatter—that is, for which all possible binary labelings can be realized by some function in the class. It was originally defined by Vladimir Vapnik and Alexey Chervonenkis. Informally, the capacity of a classification model is related to how complicated it can be. For example, consider the thresholding of a high-degree polynomial: if the polynomial evaluates above zero, that point is classified as positive, otherwise as negative. A high-degree polynomial can be wiggly, so that it can fit a given set of training points well. Such a polynomial has a high capacity. A much simpler alternative is to threshold a linear function. This function may not fit the training set well, because it has a low capacity. This notion of capacity is made rigorous below. == Definitions == === VC dimension of a set-family === Let C = { C } C ∈ C {\displaystyle {\mathcal {C}}=\{C\}_{C\in {\mathcal {C}}}} be a family of sets (also called set family, collection of sets or set of sets) and X {\displaystyle X} a set. Their intersection is defined as the following set family: C ∩ X := { C ∩ X ∣ C ∈ C } . {\displaystyle {\mathcal {C}}\cap X:=\{C\cap X\mid C\in {\mathcal {C}}\}.} Here typically X {\displaystyle X} and each C ∈ C {\displaystyle C\in {\mathcal {C}}} are subsets of a big "universe" of possibilities U {\displaystyle U} where intersection takes place. We say that a set X {\displaystyle X} is shattered by C {\displaystyle {\mathcal {C}}} if P ( X ) = C ∩ X {\displaystyle {\mathcal {P}}(X)={\mathcal {C}}\cap X} i.e. the set of intersections contains (hence is equal to) all the subsets of X {\displaystyle X} . For finite sets X {\displaystyle X} this is equivalent to | C ∩ X | = 2 | X | . {\displaystyle |{\mathcal {C}}\cap X|=2^{|X|}.} The VC dimension D {\displaystyle D} of C {\displaystyle {\mathcal {C}}} is the cardinality of the largest set that is shattered by C {\displaystyle {\mathcal {C}}} . If arbitrarily large sets can be shattered, the VC dimension of C {\displaystyle {\mathcal {C}}} is ∞ {\displaystyle \infty } . === VC dimension of a classification model === A binary classification model f {\displaystyle f} with some parameter vector θ {\displaystyle \theta } is said to shatter a set of generally positioned data points ( x 1 , x 2 , … , x n ) {\displaystyle (x_{1},x_{2},\ldots ,x_{n})} if, for every assignment of labels to those points, there exists a θ {\displaystyle \theta } such that the model f {\displaystyle f} makes no errors when evaluating that set of data points. The VC dimension of a model f {\displaystyle f} is the maximum number of points that can be arranged so that f {\displaystyle f} shatters them. More formally, it is the maximum cardinal D {\displaystyle D} such that there exists a generally positioned data point set of cardinality D {\displaystyle D} that can be shattered by f {\displaystyle f} . == Examples == f {\displaystyle f} is a constant classifier (with no parameters); Its VC dimension is 0 since it cannot shatter even a single point. In general, the VC dimension of a finite classification model, which can return at most 2 d {\displaystyle 2^{d}} different classifiers, is at most d {\displaystyle d} (this is an upper bound on the VC dimension; the Sauer–Shelah lemma gives a lower bound on the dimension). f {\displaystyle f} is a single-parametric threshold classifier on real numbers; i.e., for a certain threshold θ {\displaystyle \theta } , the classifier f θ {\displaystyle f_{\theta }} returns 1 if the input number is larger than θ {\displaystyle \theta } and 0 otherwise. The VC dimension of f {\displaystyle f} is 1 because: (a) It can shatter a single point. For every point x {\displaystyle x} , a classifier f θ {\displaystyle f_{\theta }} labels it as 0 if θ > x {\displaystyle \theta >x} and labels it as 1 if θ < x {\displaystyle \theta x + 2 {\displaystyle \theta >x+2} , as (1,0) if θ ∈ [ x − 4 , x − 2 ) {\displaystyle \theta \in [x-4,x-2)} , as (1,1) if θ ∈ [ x − 2 , x ] {\displaystyle \theta \in [x-2,x]} , and as (0,1) if θ ∈ ( x , x + 2 ] {\displaystyle \theta \in (x,x+2]} . (b) It cannot shatter any set of three points. For every set of three numbers, if the smallest and the largest are labeled 1, then the middle one must also be labeled 1, so not all labelings are possible. f {\displaystyle f} is a straight line as a classification model on points in a two-dimensional plane (this is the model used by a perceptron). The line should separate positive data points from negative data points. There exist sets of 3 points that can indeed be shattered using this model (any 3 points that are not collinear can be shattered). However, no set of 4 points can be shattered: by Radon's theorem, any four points can be partitioned into two subsets with intersecting convex hulls, so it is not possible to separate one of these two subsets from the other. Thus, the VC dimension of this particular classifier is 3. It is important to remember that while one can choose any arrangement of points, the arrangement of those points cannot change when attempting to shatter for some label assignment. Note, only 3 of the 23 = 8 possible label assignments are shown for the three points. f {\displaystyle f} is a single-parametric sine classifier, i.e., for a certain parameter θ {\displaystyle \theta } , the classifier f θ {\displaystyle f_{\theta }} returns 1 if the input number x {\displaystyle x} has sin ⁡ ( θ x ) > 0 {\displaystyle \sin(\theta x)>0} and 0 otherwise. The VC dimension of f {\displaystyle f} is infinite, since it can shatter any finite subset of the set { 2 − m ∣ m ∈ N } {\displaystyle \{2^{-m}\mid m\in \mathbb {N} \}} . == Uses == === In statistical learning theory === The VC dimension can predict a probabilistic upper bound on the test error of a classification model. Vapnik proved that the probability of the test error (i.e., risk with 0–1 loss function) distancing from an upper bound (on data that is drawn i.i.d. from the same distribution as the training set) is given by: Pr ( test error ⩽ training error + 1 N [ D ( log ⁡ ( 2 N D ) + 1 ) − log ⁡ ( η 4 ) ] ) = 1 − η , {\displaystyle \Pr \left({\text{test error}}\leqslant {\text{training error}}+{\sqrt {{\frac {1}{N}}\left[D\left(\log \left({\tfrac {2N}{D}}\right)+1\right)-\log \left({\tfrac {\eta }{4}}\right)\right]}}\,\right)=1-\eta ,} where D {\displaystyle D} is the VC dimension of the classification model, 0 < η ⩽ 1 {\displaystyle 0<\eta \leqslant 1} , and N {\displaystyle N} is the size of the training set (restriction: this formula is valid when D ≪ N {\displaystyle D\ll N} . When D {\displaystyle D} is larger, the test-error may be much higher than the training-error. This is due to overfitting). The VC dimension also appears in sample-complexity bounds. A space of binary functions with VC dimension D {\displaystyle D} can be learned with: N = Θ ( D + ln ⁡ 1 δ ε 2 ) {\displaystyle N=\Theta \left({\frac {D+\ln {1 \over \delta }}{\varepsilon ^{2}}}\right)} samples, where ε {\displaystyle \varepsilon } is the learning error and δ {\displaystyle \delta } is the failure probability. Thus, the sample-complexity is a linear function of the VC dimension of the hypothesis space. === In computational geometry === The VC dimension is one of the critical parameters in the size of ε-nets, which determines the complexity of approximation algorithms based on them; range sets without finite VC dimension may not have finite ε-nets at all. == Bounds == The VC dimension of the dual set-family of C {\displaystyle {\mathcal {C}}} is strictly less than 2 vc ⁡ ( C ) + 1 {\displaystyle 2^{\operatorname {vc} ({\mathcal {C}})+1}} , and this is best possible. The VC dimension of a finite set-family C {\displaystyle {\mathcal {C}}} is at most log 2 ⁡ | C | {\displaystyle \log _{2}|{\mathcal {C}}|} . This is because | C ∩ X | ≤ | X | {\displaystyle |{\mathcal {C}}\cap X|\leq |X|} by definition. Given a set-fa

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  Analogical modeling

  Analogical modeling (AM) is a formal theory of exemplar based analogical reasoning, proposed by Royal Skousen, professor of Linguistics and English language at Brigham Young University in Provo, Utah. It is applicable to language modeling and other categorization tasks. Analogical modeling is related to connectionism and nearest neighbor approaches, in that it is data-based rather than abstraction-based; but it is distinguished by its ability to cope with imperfect datasets (such as caused by simulated short term memory limits) and to base predictions on all relevant segments of the dataset, whether near or far. In language modeling, AM has successfully predicted empirically valid forms for which no theoretical explanation was known (see the discussion of Finnish morphology in Skousen et al. 2002). == Implementation == === Overview === An exemplar-based model consists of a general-purpose modeling engine and a problem-specific dataset. Within the dataset, each exemplar (a case to be reasoned from, or an informative past experience) appears as a feature vector: a row of values for the set of parameters that define the problem. For example, in a spelling-to-sound task, the feature vector might consist of the letters of a word. Each exemplar in the dataset is stored with an outcome, such as a phoneme or phone to be generated. When the model is presented with a novel situation (in the form of an outcome-less feature vector), the engine algorithmically sorts the dataset to find exemplars that helpfully resemble it, and selects one, whose outcome is the model's prediction. The particulars of the algorithm distinguish one exemplar-based modeling system from another. In AM, we think of the feature values as characterizing a context, and the outcome as a behavior that occurs within that context. Accordingly, the novel situation is known as the given context. Given the known features of the context, the AM engine systematically generates all contexts that include it (all of its supracontexts), and extracts from the dataset the exemplars that belong to each. The engine then discards those supracontexts whose outcomes are inconsistent (this measure of consistency will be discussed further below), leaving an analogical set of supracontexts, and probabilistically selects an exemplar from the analogical set with a bias toward those in large supracontexts. This multilevel search exponentially magnifies the likelihood of a behavior's being predicted as it occurs reliably in settings that specifically resemble the given context. === Analogical modeling in detail === AM performs the same process for each case it is asked to evaluate. The given context, consisting of n variables, is used as a template to generate 2 n {\displaystyle 2^{n}} supracontexts. Each supracontext is a set of exemplars in which one or more variables have the same values that they do in the given context, and the other variables are ignored. In effect, each is a view of the data, created by filtering for some criteria of similarity to the given context, and the total set of supracontexts exhausts all such views. Alternatively, each supracontext is a theory of the task or a proposed rule whose predictive power needs to be evaluated. It is important to note that the supracontexts are not equal peers one with another; they are arranged by their distance from the given context, forming a hierarchy. If a supracontext specifies all of the variables that another one does and more, it is a subcontext of that other one, and it lies closer to the given context. (The hierarchy is not strictly branching; each supracontext can itself be a subcontext of several others, and can have several subcontexts.) This hierarchy becomes significant in the next step of the algorithm. The engine now chooses the analogical set from among the supracontexts. A supracontext may contain exemplars that only exhibit one behavior; it is deterministically homogeneous and is included. It is a view of the data that displays regularity, or a relevant theory that has never yet been disproven. A supracontext may exhibit several behaviors, but contain no exemplars that occur in any more specific supracontext (that is, in any of its subcontexts); in this case it is non-deterministically homogeneous and is included. Here there is no great evidence that a systematic behavior occurs, but also no counterargument. Finally, a supracontext may be heterogeneous, meaning that it exhibits behaviors that are found in a subcontext (closer to the given context), and also behaviors that are not. Where the ambiguous behavior of the nondeterministically homogeneous supracontext was accepted, this is rejected because the intervening subcontext demonstrates that there is a better theory to be found. The heterogeneous supracontext is therefore excluded. This guarantees that we see an increase in meaningfully consistent behavior in the analogical set as we approach the given context. With the analogical set chosen, each appearance of an exemplar (for a given exemplar may appear in several of the analogical supracontexts) is given a pointer to every other appearance of an exemplar within its supracontexts. One of these pointers is then selected at random and followed, and the exemplar to which it points provides the outcome. This gives each supracontext an importance proportional to the square of its size, and makes each exemplar likely to be selected in direct proportion to the sum of the sizes of all analogically consistent supracontexts in which it appears. Then, of course, the probability of predicting a particular outcome is proportional to the summed probabilities of all the exemplars that support it. (Skousen 2002, in Skousen et al. 2002, pp. 11–25, and Skousen 2003, both passim) === Formulas === Given a context with n {\displaystyle n} elements: total number of pairings: n 2 {\displaystyle n^{2}} number of agreements for outcome i: n i 2 {\displaystyle n_{i}^{2}} number of disagreements for outcome i: n i ( n − n i ) {\displaystyle n_{i}(n-n_{i})} total number of agreements: ∑ n i 2 {\displaystyle \sum {n_{i}^{2}}} total number of disagreements: ∑ n i ( n − n i ) = n 2 − ∑ n i 2 {\displaystyle \sum {n_{i}(n-n_{i})}=n^{2}-\sum {n_{i}^{2}}} === Example === This terminology is best understood through an example. In the example used in the second chapter of Skousen (1989), each context consists of three variables with potential values 0-3 Variable 1: 0,1,2,3 Variable 2: 0,1,2,3 Variable 3: 0,1,2,3 The two outcomes for the dataset are e and r, and the exemplars are: 3 1 0 e 0 3 2 r 2 1 0 r 2 1 2 r 3 1 1 r We define a network of pointers like so: The solid lines represent pointers between exemplars with matching outcomes; the dotted lines represent pointers between exemplars with non-matching outcomes. The statistics for this example are as follows: n = 5 {\displaystyle n=5} n r = 4 {\displaystyle n_{r}=4} n e = 1 {\displaystyle n_{e}=1} total number of pairings: n 2 = 25 {\displaystyle n^{2}=25} number of agreements for outcome r: n r 2 = 16 {\displaystyle n_{r}^{2}=16} number of agreements for outcome e: n e 2 = 1 {\displaystyle n_{e}^{2}=1} number of disagreements for outcome r: n r ( n − n r ) = 4 {\displaystyle n_{r}(n-n_{r})=4} number of disagreements for outcome e: n e ( n − n e ) = 4 {\displaystyle n_{e}(n-n_{e})=4} total number of agreements: n r 2 + n e 2 = 17 {\displaystyle n_{r}^{2}+n_{e}^{2}=17} total number of disagreements: n r ( n − n r ) + n e ( n − n e ) = n 2 − ( n r 2 + n e 2 ) = 8 {\displaystyle n_{r}(n-n_{r})+n_{e}(n-n_{e})=n^{2}-(n_{r}^{2}+n_{e}^{2})=8} uncertainty or fraction of disagreement: 8 / 25 = .32 {\displaystyle 8/25=.32} Behavior can only be predicted for a given context; in this example, let us predict the outcome for the context "3 1 2". To do this, we first find all of the contexts containing the given context; these contexts are called supracontexts. We find the supracontexts by systematically eliminating the variables in the given context; with m variables, there will generally be 2 m {\displaystyle 2^{m}} supracontexts. The following table lists each of the sub- and supracontexts; x means "not x", and - means "anything". These contexts are shown in the venn diagram below: The next step is to determine which exemplars belong to which contexts in order to determine which of the contexts are homogeneous. The table below shows each of the subcontexts, their behavior in terms of the given exemplars, and the number of disagreements within the behavior: Analyzing the subcontexts in the table above, we see that there is only 1 subcontext with any disagreements: "3 1 2", which in the dataset consists of "3 1 0 e" and "3 1 1 r". There are 2 disagreements in this subcontext; 1 pointing from each of the exemplars to the other (see the pointer network pictured above). Therefore, only supracontexts containing this subcontext will contain any disagreements. We use a simple rule to identify the homogeneous supraco

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  Sample exclusion dimension

  In computational learning theory, sample exclusion dimensions arise in the study of exact concept learning with queries. In algorithmic learning theory, a concept over a domain X is a Boolean function over X. Here we only consider finite domains. A partial approximation S of a concept c is a Boolean function over Y ⊆ X {\displaystyle Y\subseteq X} such that c is an extension to S. Let C be a class of concepts and c be a concept (not necessarily in C). Then a specifying set for c w.r.t. C, denoted by S is a partial approximation S of c such that C contains at most one extension to S. If we have observed a specifying set for some concept w.r.t. C, then we have enough information to verify a concept in C with at most one more mind change. The exclusion dimension, denoted by XD(C), of a concept class is the maximum of the size of the minimum specifying set of c' with respect to C, where c' is a concept not in C.

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  MSpy

  mSpy is a brand of mobile and computer parental control monitoring software for iOS, Android, Windows, and macOS. The app monitors and logs user activity on the client device and sends the data to a personalized dashboard. Data the users can monitor includes text messages, calls, GPS locations, social media chats, and more. It is owned by Virtuoso Holding. == History == mSpy was launched as a product for mobile monitoring by Altercon Group in 2010. In 2012, the application allowed parents to monitor not only smartphones but also computers running Windows and macOS. In 2013, mSpy became TopTenReviews cell phone monitoring software award winner. By 2014, the business grew nearly 400%, and the app's user numbers exceeded 1 million. In 2015, mSpy received the Parents Tested Parents Approved (PTPA) Winner’s Seal of Approval in the United States. In 2015 and 2018, mSpy was the victim of data breaches which released user data. In 2016, mLite, a light version of mSpy, became available from Google Play. The same year, it was awarded the kidSAFE Certified Seal in the United States. In 2017, mSpy collaborated with YouTuber and journalist Coby Persin to conduct a social experiment on the dangers of social media and online predators. A social experiment, conducted with parental consent, involved Coby Persin to befriend three children—aged 12, 13, and 14—via Snapchat and then invite them to meet personally. Each of the participants agreed to the meeting and arrived at the designated location. The video of the experiment received widespread attention and helped to raise awareness about the importance of online security and parental controls. In early 2021, mSpy released a new feature - Screenrecorder. The feature allows parents to take screenshots of the kid's screen when they are browsing certain apps. In 2024, mSpy's Zendesk was compromised by an unknown threat actor, revealing their customer list. As of 2025, mSpy is compatible with Android, iPhone, and iPad devices. It provides access to various types of data stored on the device, including contact information, calendar entries, emails, SMS messages, browser history, photos, videos, and installed applications. Functions also include GPS tracking, geofencing, keyword alerts etc. == Reception == It was noted that since MSpy runs inconspicuously, there is risk of the software being used illegally. mSpy was called "terrifying" by The Next Web and was featured in NPR coverage of spyware used against victims of stalking and other domestic violence. In response mSpy released security updates aimed at reducing the risk of misuse and stated that it "uses encryption protocols to protect user data and that access is restricted to the account holder". In May 2015, Brian Krebs reported that mSpy was hacked, leaking personal data for hundreds of thousands of users of devices with mSpy installed. mSpy claimed that there was no data leak, but that instead, it was the victim of blackmailers. In September 2018, Krebs claimed and demonstrated that anyone could easily gain access to the mSpy database containing data for millions of users. The company responded by stating that the exposed data consisted primarily of error logs and incorrect login attempts. Following the incident, mSpy implemented new security measures, changed encryption keys, and reset passwords for affected accounts. A 2024 Sky News story characterised mSpy as "stalkerware". Leaked customer support messages from mSpy reveal misuse of its app for illegally monitoring partners and children.

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  Quantum neural network

  Quantum neural networks are computational neural network models which are based on the principles of quantum mechanics. The first ideas on quantum neural computation were published independently in 1995 by Subhash Kak and Ron Chrisley, engaging with the theory of quantum mind, which posits that quantum effects play a role in cognitive function. However, typical research in quantum neural networks involves combining classical artificial neural network models (which are widely used in machine learning for the important task of pattern recognition) with the advantages of quantum information in order to develop more efficient algorithms. One important motivation for these investigations is the difficulty to train classical neural networks, especially in big data applications. The hope is that features of quantum computing such as quantum parallelism or the effects of interference and entanglement can be used as resources. Since the technological implementation of a quantum computer is still in a premature stage, such quantum neural network models are mostly theoretical proposals that await their full implementation in physical experiments. Most Quantum neural networks are developed as feed-forward networks. Similar to their classical counterparts, this structure intakes input from one layer of qubits, and passes that input onto another layer of qubits. This layer of qubits evaluates this information and passes on the output to the next layer. Eventually the path leads to the final layer of qubits. The layers do not have to be of the same width, meaning they don't have to have the same number of qubits as the layer before or after it. This structure is trained on which path to take similar to classical artificial neural networks. This is discussed in a lower section. Quantum neural networks refer to three different categories: Quantum computer with classical data, classical computer with quantum data, and quantum computer with quantum data. == Examples == Quantum neural network research is still in its infancy, and a conglomeration of proposals and ideas of varying scope and mathematical rigor have been put forward. Most of them are based on the idea of replacing classical binary or McCulloch-Pitts neurons with a qubit (which can be called a "quron"), resulting in neural units that can be in a superposition of the state 'firing' and 'resting'. === Quantum perceptrons === A lot of proposals attempt to find a quantum equivalent for the perceptron unit from which neural nets are constructed. A problem is that nonlinear activation functions do not immediately correspond to the mathematical structure of quantum theory, since a quantum evolution is described by linear operations and leads to probabilistic observation. Ideas to imitate the perceptron activation function with a quantum mechanical formalism reach from special measurements to postulating non-linear quantum operators (a mathematical framework that is disputed). A direct implementation of the activation function using the circuit-based model of quantum computation has recently been proposed by Schuld, Sinayskiy and Petruccione based on the quantum phase estimation algorithm. === Quantum networks === At a larger scale, researchers have attempted to generalize neural networks to the quantum setting. One way of constructing a quantum neuron is to first generalise classical neurons and then generalising them further to make unitary gates. Interactions between neurons can be controlled quantumly, with unitary gates, or classically, via measurement of the network states. This high-level theoretical technique can be applied broadly, by taking different types of networks and different implementations of quantum neurons, such as photonically implemented neurons and quantum reservoir processor (quantum version of reservoir computing). Most learning algorithms follow the classical model of training an artificial neural network to learn the input-output function of a given training set and use classical feedback loops to update parameters of the quantum system until they converge to an optimal configuration. Learning as a parameter optimisation problem has also been approached by adiabatic models of quantum computing. Quantum neural networks can be applied to algorithmic design: given qubits with tunable mutual interactions, one can attempt to learn interactions following the classical backpropagation rule from a training set of desired input-output relations, taken to be the desired output algorithm's behavior. The quantum network thus 'learns' an algorithm. === Quantum associative memory === The first quantum associative memory algorithm was introduced by Dan Ventura and Tony Martinez in 1999. The authors do not attempt to translate the structure of artificial neural network models into quantum theory, but propose an algorithm for a circuit-based quantum computer that simulates associative memory. The memory states (in Hopfield neural networks saved in the weights of the neural connections) are written into a superposition, and a Grover-like quantum search algorithm retrieves the memory state closest to a given input. As such, this is not a fully content-addressable memory, since only incomplete patterns can be retrieved. The first truly content-addressable quantum memory, which can retrieve patterns also from corrupted inputs, was proposed by Carlo A. Trugenberger. Both memories can store an exponential (in terms of n qubits) number of patterns but can be used only once due to the no-cloning theorem and their destruction upon measurement. Trugenberger, however, has shown that his probabilistic model of quantum associative memory can be efficiently implemented and re-used multiples times for any polynomial number of stored patterns, a large advantage with respect to classical associative memories. === Classical neural networks inspired by quantum theory === A substantial amount of interest has been given to a "quantum-inspired" model that uses ideas from quantum theory to implement a neural network based on fuzzy logic. == Training == Quantum Neural Networks can be theoretically trained similarly to training classical/artificial neural networks. A key difference lies in communication between the layers of a neural networks. For classical neural networks, at the end of a given operation, the current perceptron copies its output to the next layer of perceptron(s) in the network. However, in a quantum neural network, where each perceptron is a qubit, this would violate the no-cloning theorem. A proposed generalized solution to this is to replace the classical fan-out method with an arbitrary unitary that spreads out, but does not copy, the output of one qubit to the next layer of qubits. Using this fan-out Unitary ( U f {\displaystyle U_{f}} ) with a dummy state qubit in a known state (Ex. | 0 ⟩ {\displaystyle |0\rangle } in the computational basis), also known as an Ancilla bit, the information from the qubit can be transferred to the next layer of qubits. This process adheres to the quantum operation requirement of reversibility. Using this quantum feed-forward network, deep neural networks can be executed and trained efficiently. A deep neural network is essentially a network with many hidden-layers, as seen in the sample model neural network above. Since the Quantum neural network being discussed uses fan-out Unitary operators, and each operator only acts on its respective input, only two layers are used at any given time. In other words, no Unitary operator is acting on the entire network at any given time, meaning the number of qubits required for a given step depends on the number of inputs in a given layer. Since Quantum Computers are notorious for their ability to run multiple iterations in a short period of time, the efficiency of a quantum neural network is solely dependent on the number of qubits in any given layer, and not on the depth of the network. === Cost functions === To determine the effectiveness of a neural network, a cost function is used, which essentially measures the proximity of the network's output to the expected or desired output. In a Classical Neural Network, the weights ( w {\displaystyle w} ) and biases ( b {\displaystyle b} ) at each step determine the outcome of the cost function C ( w , b ) {\displaystyle C(w,b)} . When training a Classical Neural network, the weights and biases are adjusted after each iteration, and given equation 1 below, where y ( x ) {\displaystyle y(x)} is the desired output and a out ( x ) {\displaystyle a^{\text{out}}(x)} is the actual output, the cost function is optimized when C ( w , b ) {\displaystyle C(w,b)} = 0. For a quantum neural network, the cost function is determined by measuring the fidelity of the outcome state ( ρ out {\displaystyle \rho ^{\text{out}}} ) with the desired outcome state ( ϕ out {\displaystyle \phi ^{\text{out}}} ), seen in Equation 2 below. In this case, the Unitary operators are adjusted after each it

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  Grammatical evolution

  Grammatical evolution (GE) is a genetic programming (GP) technique (or approach) from evolutionary computation pioneered by Conor Ryan, JJ Collins and Michael O'Neill in 1998 at the BDS Group in the University of Limerick. As in any other GP approach, the objective is to find an executable program, program fragment, or function, which will achieve a good fitness value for a given objective function. In most published work on GP, a LISP-style tree-structured expression is directly manipulated, whereas GE applies genetic operators to an integer string, subsequently mapped to a program (or similar) through the use of a grammar, which is typically expressed in Backus–Naur form. One of the benefits of GE is that this mapping simplifies the application of search to different programming languages and other structures. == Problem addressed == In type-free, conventional Koza-style GP, the function set must meet the requirement of closure: all functions must be capable of accepting as their arguments the output of all other functions in the function set. Usually, this is implemented by dealing with a single data-type such as double-precision floating point. While modern Genetic Programming frameworks support typing, such type-systems have limitations that Grammatical Evolution does not suffer from. == GE's solution == GE offers a solution to the single-type limitation by evolving solutions according to a user-specified grammar (usually a grammar in Backus-Naur form). Therefore, the search space can be restricted, and domain knowledge of the problem can be incorporated. The inspiration for this approach comes from a desire to separate the "genotype" from the "phenotype": in GP, the objects the search algorithm operates on and what the fitness evaluation function interprets are one and the same. In contrast, GE's "genotypes" are ordered lists of integers which code for selecting rules from the provided context-free grammar. The phenotype, however, is the same as in Koza-style GP: a tree-like structure that is evaluated recursively. This model is more in line with how genetics work in nature, where there is a separation between an organism's genotype and the final expression of phenotype in proteins, etc. Separating genotype and phenotype allows a modular approach. In particular, the search portion of the GE paradigm needn't be carried out by any one particular algorithm or method. Observe that the objects GE performs search on are the same as those used in genetic algorithms. This means, in principle, that any existing genetic algorithm package, such as the popular GAlib, can be used to carry out the search, and a developer implementing a GE system need only worry about carrying out the mapping from list of integers to program tree. It is also in principle possible to perform the search using some other method, such as particle swarm optimization (see the remark below); the modular nature of GE creates many opportunities for hybrids as the problem of interest to be solved dictates. Brabazon and O'Neill have successfully applied GE to predicting corporate bankruptcy, forecasting stock indices, bond credit ratings, and other financial applications. GE has also been used with a classic predator-prey model to explore the impact of parameters such as predator efficiency, niche number, and random mutations on ecological stability. It is possible to structure a GE grammar that for a given function/terminal set is equivalent to genetic programming. == Criticism == Despite its successes, GE has been the subject of some criticism. One issue is that as a result of its mapping operation, GE's genetic operators do not achieve high locality which is a highly regarded property of genetic operators in evolutionary algorithms. == Variants == Although GE was originally described in terms of using an Evolutionary Algorithm, specifically, a Genetic Algorithm, other variants exist. For example, GE researchers have experimented with using particle swarm optimization to carry out the searching instead of genetic algorithms with results comparable to that of normal GE; this is referred to as a "grammatical swarm"; using only the basic PSO model it has been found that PSO is probably equally capable of carrying out the search process in GE as simple genetic algorithms are. (Although PSO is normally a floating-point search paradigm, it can be discretized, e.g., by simply rounding each vector to the nearest integer, for use with GE.) Yet another possible variation that has been experimented with in the literature is attempting to encode semantic information in the grammar in order to further bias the search process. Other work showed that, with biased grammars that leverage domain knowledge, even random search can be used to drive GE. == Related work == GE was originally a combination of the linear representation as used by the Genetic Algorithm for Developing Software (GADS) and Backus Naur Form grammars, which were originally used in tree-based GP by Wong and Leung in 1995 and Whigham in 1996. Other related work noted in the original GE paper was that of Frederic Gruau, who used a conceptually similar "embryonic" approach, as well as that of Keller and Banzhaf, which similarly used linear genomes. == Implementations == There are several implementations of GE. These include the following.

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  Multilinear principal component analysis

  Multilinear principal component analysis (MPCA) is a multilinear extension of principal component analysis (PCA) that is used to analyze M-way arrays, also informally referred to as "data tensors". M-way arrays may be modeled by linear tensor models, such as CANDECOMP/Parafac, or by multilinear tensor models, such as multilinear principal component analysis (MPCA) or multilinear (tensor) independent component analysis (MICA). In 2005, Vasilescu and Terzopoulos introduced the Multilinear PCA terminology as a way to better differentiate between multilinear data models that employed 2nd order statistics versus higher order statistics to compute a set of independent components for each mode, such as Multilinear ICA Multilinear PCA may be applied to compute the causal factors of data formation, or as signal processing tool on data tensors whose individual observation have either been vectorized, or whose observations are treated as a collection of column/row observations, an "observation as a matrix", and concatenated into a data tensor. The latter approach is suitable for compression and reducing redundancy in the rows, columns and fibers that are unrelated to the causal factors of data formation. Vasilescu and Terzopoulos in their paper "TensorFaces" introduced the M-mode SVD algorithm which are algorithms misidentified in the literature as the HOSVD or the Tucker which employ the power method or gradient descent, respectively. Vasilescu and Terzopoulos framed the data analysis, recognition and synthesis problems as multilinear tensor problems. Data is viewed as the compositional consequence of several causal factors, that are well suited for multi-modal tensor factor analysis. The power of the tensor framework was showcased by analyzing human motion joint angles, facial images or textures in the following papers: Human Motion Signatures (CVPR 2001, ICPR 2002), face recognition – TensorFaces, (ECCV 2002, CVPR 2003, etc.) and computer graphics – TensorTextures (Siggraph 2004). == The algorithm == The MPCA solution follows the alternating least square (ALS) approach. It is iterative in nature. As in PCA, MPCA works on centered data. Centering is a little more complicated for tensors, and it is problem dependent. == Feature selection == MPCA features: Supervised MPCA is employed in causal factor analysis that facilitates object recognition while a semi-supervised MPCA feature selection is employed in visualization tasks. == Extensions == Various extension of MPCA: Robust MPCA (RMPCA) Multi-Tensor Factorization, that also finds the number of components automatically (MTF)

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  NASA AI Assisted-Air Quality Monitoring Project

  The NASA Expert-System Ion Trap Mass Spectrometer (ES-ITMS) Project was a public-private partnership to develop an artificial intelligence assisted, air quality monitoring system and was qualified for use on the Space Shuttle. The partnership was also the first cost and intellectual property shared public-partnership implemented by NASA, which used the commercial Research and Development Limited Partnership (RDLP) model that had been adopted by the Reagan Administration for Department of Defense semiconductor development, and recommended for use by NASA for space commercialization. The project partners included NASA, the University of Florida and Finnigan MAT Corporation, was organized and administered by the NASA Joint Enterprise Institute (subsequently NASA Joint Sponsored Program) and ran from 1988 through 1990. The partnership concluded final testing in 1991, generating four patents, expert system software and application protocol reports. The system was space qualified for use on the Shuttle and elements of the ES-ITMS system were integrated into the product Improvements for Finnigan MAT corporation. The success of the partnership lead NASA to create a pilot program to develop partnership business models as an ongoing management practice. == Purpose and objectives == The need to monitor air quality in confined spaces represented an increasing challenge for NASA's planned space missions and private sector facility managers facing the increased scrutiny of possible air contaminants. Up to the early 1980's, air quality monitors generally required large spaces and human technicians to interpret readings. This created a need for miniaturized air quality monitors that could generate reliable and accurate analytic results without on-site technician presence. NASA initiated projects to develop..."mobile and/or portable mass spectrometers" that evaluated the "tradeoff between instrumentation capabilities and space, weight and power considerations." NASA selected a "commercial ITMS instrument capable of generating electron ionization, chemical ionization and mass spectrometry data", to develop a linked expert system to accomplish analysis without human intervention. The commercial instrumentation was from Finnigan MAT corporation while the scientific expertise to support expert system development was available at the University of Florida. The project managers at NASA Ames created a single, integrated project using the RDLP model with objectives to: Develop AI/expert system software for instrument control (NASA's role) Expand sensitivity, selectivity and speed of the spectrometer (Univ Florida role) Expand the spectrometer analytic capability and automate the screening (Finnigan role) == Membership == The partnership included seven specialists from five member organizations: Federal Government National Aeronautics and Space Administration (NASA) NASA Ames Research Center (ARC) NASA Kennedy Space Center (KSC) Commercial Finnigan MAT Corporation (Thermo-Fisher Scientific) TGS Technology, Inc. Research Management University of Florida == Organization, management and administration == The technical project was organized into two development teams, one located in at the NASA Ames Research Center covering expert systems and analytic capabilities and one in Florida covering improved sensitivity and testing. The partnership management and administration was provided by a non-profit, partnership support organization: the Joint Enterprise Institute operating through San Francisco State University Foundation (SFSUF) with a NASA employee liaison, Syed Shariq. == Public-private partnership == The partnership structure was as a prototype test of a pilot NASA program to develop public-private partnership business models. The pilot program was known as the NASA Joint Sponsored Research Program (JSRP), which operated as the NASA Joint Enterprise Institute between 1988 and 1991. The partnership was the first public-private, research and development partnership implemented by NASA in response to national policy shifts to increase technology transfer and space commercialization. The partnership structure included a two year technology development and testing plan that cost $610,000, of which NASA funded $310,000, Finnigan $175,000 and the University of Florida $95,000. == Results and commercialization == The project generated patents (4), software (2) and application protocol reports (8). NASA gained use of the patents and jointly development software while Finnigan received commercial utilization rights. The results were commercialized within eighteen months of project completion. == Recognition == NASA recognized the project as a space qualified instrument. Its achievements were reported to the NASA Administrator, directly leading to establishment of the agency-wide Joint Sponsored Research Program.

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  ID3 algorithm

  In decision tree learning, ID3 (Iterative Dichotomiser 3) is a greedy algorithm invented by Ross Quinlan used to generate a decision tree from a dataset. ID3 is the precursor to the C4.5 algorithm. The 3 in the name is meant to signify that this was Quinlan's third attempt at a model based on entropy-based splitting, and the term dichotimser is a misnomer as it implies a binary split, but the ID3 algorithm can split on multi-valued attributes. == Algorithm == The ID3 algorithm begins with the original set S {\displaystyle S} as the root node. On each iteration of the algorithm, it iterates through every unused attribute of the set S {\displaystyle S} and calculates the entropy H ( S ) {\displaystyle \mathrm {H} {(S)}} or the information gain I G ( S ) {\displaystyle IG(S)} of that attribute. It then selects the attribute which has the smallest entropy (or largest information gain) value. The set S {\displaystyle S} is then split or partitioned by the selected attribute to produce subsets of the data. (For example, a node can be split into child nodes based upon the subsets of the population whose ages are less than 50, between 50 and 100, and greater than 100.) The algorithm continues to recurse on each subset, considering only attributes never selected before. Recursion on a subset may stop in one of these cases: every element in the subset belongs to the same class; in which case the node is turned into a leaf node and labelled with the class of the examples. there are no more attributes to be selected, but the examples still do not belong to the same class. In this case, the node is made a leaf node and labelled with the most common class of the examples in the subset. there are no examples in the subset, which happens when no example in the parent set was found to match a specific value of the selected attribute. An example could be the absence of a person among the population with age over 100 years. Then a leaf node is created and labelled with the most common class of the examples in the parent node's set. Throughout the algorithm, the decision tree is constructed with each non-terminal node (internal node) representing the selected attribute on which the data was split, and terminal nodes (leaf nodes) representing the class label of the final subset of this branch. === Summary === Calculate the entropy of every attribute a {\displaystyle a} of the data set S {\displaystyle S} . Partition ("split") the set S {\displaystyle S} into subsets using the attribute for which the resulting entropy after splitting is minimized; or, equivalently, information gain is maximum. Make a decision tree node containing that attribute. Recurse on subsets using the remaining attributes. === Properties === ID3 does not guarantee an optimal solution. It can converge upon local optima. It uses a greedy strategy by selecting the locally best attribute to split the dataset on each iteration. The algorithm's optimality can be improved by using backtracking during the search for the optimal decision tree at the cost of possibly taking longer. ID3 can overfit the training data. To avoid overfitting, smaller decision trees should be preferred over larger ones. This algorithm usually produces small trees, but it does not always produce the smallest possible decision tree. ID3 is harder to use on continuous data than on factored data (factored data has a discrete number of possible values, thus reducing the possible branch points). If the values of any given attribute are continuous, then there are many more places to split the data on this attribute, and searching for the best value to split by can be time-consuming. === Usage === The ID3 algorithm is used by training on a data set S {\displaystyle S} to produce a decision tree which is stored in memory. At runtime, this decision tree is used to classify new test cases (feature vectors) by traversing the decision tree using the features of the datum to arrive at a leaf node. == The ID3 metrics == === Entropy === Entropy H ( S ) {\displaystyle \mathrm {H} {(S)}} is a measure of the amount of uncertainty in the (data) set S {\displaystyle S} (i.e. entropy characterizes the (data) set S {\displaystyle S} ). H ( S ) = ∑ x ∈ X − p ( x ) log 2 ⁡ p ( x ) {\displaystyle \mathrm {H} {(S)}=\sum _{x\in X}{-p(x)\log _{2}p(x)}} Where, S {\displaystyle S} – The current dataset for which entropy is being calculated This changes at each step of the ID3 algorithm, either to a subset of the previous set in the case of splitting on an attribute or to a "sibling" partition of the parent in case the recursion terminated previously. X {\displaystyle X} – The set of classes in S {\displaystyle S} p ( x ) {\displaystyle p(x)} – The proportion of the number of elements in class x {\displaystyle x} to the number of elements in set S {\displaystyle S} When H ( S ) = 0 {\displaystyle \mathrm {H} {(S)}=0} , the set S {\displaystyle S} is perfectly classified (i.e. all elements in S {\displaystyle S} are of the same class). In ID3, entropy is calculated for each remaining attribute. The attribute with the smallest entropy is used to split the set S {\displaystyle S} on this iteration. Entropy in information theory measures how much information is expected to be gained upon measuring a random variable; as such, it can also be used to quantify the amount to which the distribution of the quantity's values is unknown. A constant quantity has zero entropy, as its distribution is perfectly known. In contrast, a uniformly distributed random variable (discretely or continuously uniform) maximizes entropy. Therefore, the greater the entropy at a node, the less information is known about the classification of data at this stage of the tree; and therefore, the greater the potential to improve the classification here. As such, ID3 is a greedy heuristic performing a best-first search for locally optimal entropy values. Its accuracy can be improved by preprocessing the data. === Information gain === Information gain I G ( A ) {\displaystyle IG(A)} is the measure of the difference in entropy from before to after the set S {\displaystyle S} is split on an attribute A {\displaystyle A} . In other words, how much uncertainty in S {\displaystyle S} was reduced after splitting set S {\displaystyle S} on attribute A {\displaystyle A} . I G ( S , A ) = H ( S ) − ∑ t ∈ T p ( t ) H ( t ) = H ( S ) − H ( S | A ) . {\displaystyle IG(S,A)=\mathrm {H} {(S)}-\sum _{t\in T}p(t)\mathrm {H} {(t)}=\mathrm {H} {(S)}-\mathrm {H} {(S|A)}.} Where, H ( S ) {\displaystyle \mathrm {H} (S)} – Entropy of set S {\displaystyle S} T {\displaystyle T} – The subsets created from splitting set S {\displaystyle S} by attribute A {\displaystyle A} such that S = ⋃ t ∈ T t {\displaystyle S=\bigcup _{t\in T}t} p ( t ) {\displaystyle p(t)} – The proportion of the number of elements in t {\displaystyle t} to the number of elements in set S {\displaystyle S} H ( t ) {\displaystyle \mathrm {H} (t)} – Entropy of subset t {\displaystyle t} In ID3, information gain can be calculated (instead of entropy) for each remaining attribute. The attribute with the largest information gain is used to split the set S {\displaystyle S} on this iteration.

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  Mathematics of neural networks in machine learning

  An artificial neural network (ANN) or neural network combines biological principles with advanced statistics to solve problems in domains such as pattern recognition and game-play. ANNs adopt the basic model of neuron analogues connected to each other in a variety of ways. == Structure == === Neuron === A neuron with label j {\displaystyle j} receiving an input p j ( t ) {\displaystyle p_{j}(t)} from predecessor neurons consists of the following components: an activation a j ( t ) {\displaystyle a_{j}(t)} , the neuron's state, depending on a discrete time parameter, an optional threshold θ j {\displaystyle \theta _{j}} , which stays fixed unless changed by learning, an activation function f {\displaystyle f} that computes the new activation at a given time t + 1 {\displaystyle t+1} from a j ( t ) {\displaystyle a_{j}(t)} , θ j {\displaystyle \theta _{j}} and the net input p j ( t ) {\displaystyle p_{j}(t)} giving rise to the relation a j ( t + 1 ) = f ( a j ( t ) , p j ( t ) , θ j ) , {\displaystyle a_{j}(t+1)=f(a_{j}(t),p_{j}(t),\theta _{j}),} and an output function f out {\displaystyle f_{\text{out}}} computing the output from the activation o j ( t ) = f out ( a j ( t ) ) . {\displaystyle o_{j}(t)=f_{\text{out}}(a_{j}(t)).} Often the output function is simply the identity function. An input neuron has no predecessor but serves as input interface for the whole network. Similarly an output neuron has no successor and thus serves as output interface of the whole network. === Propagation function === The propagation function computes the input p j ( t ) {\displaystyle p_{j}(t)} to the neuron j {\displaystyle j} from the outputs o i ( t ) {\displaystyle o_{i}(t)} and typically has the form p j ( t ) = ∑ i o i ( t ) w i j . {\displaystyle p_{j}(t)=\sum _{i}o_{i}(t)w_{ij}.} === Bias === A bias term can be added, changing the form to the following: p j ( t ) = ∑ i o i ( t ) w i j + w 0 j , {\displaystyle p_{j}(t)=\sum _{i}o_{i}(t)w_{ij}+w_{0j},} where w 0 j {\displaystyle w_{0j}} is a bias. == Neural networks as functions == Neural network models can be viewed as defining a function that takes an input (observation) and produces an output (decision) f : X → Y {\displaystyle \textstyle f:X\rightarrow Y} or a distribution over X {\displaystyle \textstyle X} or both X {\displaystyle \textstyle X} and Y {\displaystyle \textstyle Y} . Sometimes models are intimately associated with a particular learning rule. A common use of the phrase "ANN model" is really the definition of a class of such functions (where members of the class are obtained by varying parameters, connection weights, or specifics of the architecture such as the number of neurons, number of layers or their connectivity). Mathematically, a neuron's network function f ( x ) {\displaystyle \textstyle f(x)} is defined as a composition of other functions g i ( x ) {\displaystyle \textstyle g_{i}(x)} , that can further be decomposed into other functions. This can be conveniently represented as a network structure, with arrows depicting the dependencies between functions. A widely used type of composition is the nonlinear weighted sum, where f ( x ) = K ( ∑ i w i g i ( x ) ) {\displaystyle \textstyle f(x)=K\left(\sum _{i}w_{i}g_{i}(x)\right)} , where K {\displaystyle \textstyle K} (commonly referred to as the activation function) is some predefined function, such as the hyperbolic tangent, sigmoid function, softmax function, or rectifier function. The important characteristic of the activation function is that it provides a smooth transition as input values change, i.e. a small change in input produces a small change in output. The following refers to a collection of functions g i {\displaystyle \textstyle g_{i}} as a vector g = ( g 1 , g 2 , … , g n ) {\displaystyle \textstyle g=(g_{1},g_{2},\ldots ,g_{n})} . This figure depicts such a decomposition of f {\displaystyle \textstyle f} , with dependencies between variables indicated by arrows. These can be interpreted in two ways. The first view is the functional view: the input x {\displaystyle \textstyle x} is transformed into a 3-dimensional vector h {\displaystyle \textstyle h} , which is then transformed into a 2-dimensional vector g {\displaystyle \textstyle g} , which is finally transformed into f {\displaystyle \textstyle f} . This view is most commonly encountered in the context of optimization. The second view is the probabilistic view: the random variable F = f ( G ) {\displaystyle \textstyle F=f(G)} depends upon the random variable G = g ( H ) {\displaystyle \textstyle G=g(H)} , which depends upon H = h ( X ) {\displaystyle \textstyle H=h(X)} , which depends upon the random variable X {\displaystyle \textstyle X} . This view is most commonly encountered in the context of graphical models. The two views are largely equivalent. In either case, for this particular architecture, the components of individual layers are independent of each other (e.g., the components of g {\displaystyle \textstyle g} are independent of each other given their input h {\displaystyle \textstyle h} ). This naturally enables a degree of parallelism in the implementation. Networks such as the previous one are commonly called feedforward, because their graph is a directed acyclic graph. Networks with cycles are commonly called recurrent. Such networks are commonly depicted in the manner shown at the top of the figure, where f {\displaystyle \textstyle f} is shown as dependent upon itself. However, an implied temporal dependence is not shown. == Backpropagation == Backpropagation training algorithms fall into three categories: steepest descent (with variable learning rate and momentum, resilient backpropagation); quasi-Newton (Broyden–Fletcher–Goldfarb–Shanno, one step secant); Levenberg–Marquardt and conjugate gradient (Fletcher–Reeves update, Polak–Ribiére update, Powell–Beale restart, scaled conjugate gradient). === Algorithm === Let N {\displaystyle N} be a network with e {\displaystyle e} connections, m {\displaystyle m} inputs and n {\displaystyle n} outputs. Below, x 1 , x 2 , … {\displaystyle x_{1},x_{2},\dots } denote vectors in R m {\displaystyle \mathbb {R} ^{m}} , y 1 , y 2 , … {\displaystyle y_{1},y_{2},\dots } vectors in R n {\displaystyle \mathbb {R} ^{n}} , and w 0 , w 1 , w 2 , … {\displaystyle w_{0},w_{1},w_{2},\ldots } vectors in R e {\displaystyle \mathbb {R} ^{e}} . These are called inputs, outputs and weights, respectively. The network corresponds to a function y = f N ( w , x ) {\displaystyle y=f_{N}(w,x)} which, given a weight w {\displaystyle w} , maps an input x {\displaystyle x} to an output y {\displaystyle y} . In supervised learning, a sequence of training examples ( x 1 , y 1 ) , … , ( x p , y p ) {\displaystyle (x_{1},y_{1}),\dots ,(x_{p},y_{p})} produces a sequence of weights w 0 , w 1 , … , w p {\displaystyle w_{0},w_{1},\dots ,w_{p}} starting from some initial weight w 0 {\displaystyle w_{0}} , usually chosen at random. These weights are computed in turn: first compute w i {\displaystyle w_{i}} using only ( x i , y i , w i − 1 ) {\displaystyle (x_{i},y_{i},w_{i-1})} for i = 1 , … , p {\displaystyle i=1,\dots ,p} . The output of the algorithm is then w p {\displaystyle w_{p}} , giving a new function x ↦ f N ( w p , x ) {\displaystyle x\mapsto f_{N}(w_{p},x)} . The computation is the same in each step, hence only the case i = 1 {\displaystyle i=1} is described. w 1 {\displaystyle w_{1}} is calculated from ( x 1 , y 1 , w 0 ) {\displaystyle (x_{1},y_{1},w_{0})} by considering a variable weight w {\displaystyle w} and applying gradient descent to the function w ↦ E ( f N ( w , x 1 ) , y 1 ) {\displaystyle w\mapsto E(f_{N}(w,x_{1}),y_{1})} to find a local minimum, starting at w = w 0 {\displaystyle w=w_{0}} . This makes w 1 {\displaystyle w_{1}} the minimizing weight found by gradient descent. == Learning pseudocode == To implement the algorithm above, explicit formulas are required for the gradient of the function w ↦ E ( f N ( w , x ) , y ) {\displaystyle w\mapsto E(f_{N}(w,x),y)} where the function is E ( y , y ′ ) = | y − y ′ | 2 {\displaystyle E(y,y')=|y-y'|^{2}} . The learning algorithm can be divided into two phases: propagation and weight update. === Propagation === Propagation involves the following steps: Propagation forward through the network to generate the output value(s) Calculation of the cost (error term) Propagation of the output activations back through the network using the training pattern target to generate the deltas (the difference between the targeted and actual output values) of all output and hidden neurons. === Weight update === For each weight: Multiply the weight's output delta and input activation to find the gradient of the weight. Subtract the ratio (percentage) of the weight's gradient from the weight. The learning rate is the ratio (percentage) that influences the speed and quality of learning. The greater the ratio, the faster the neuron trains, but the lower the ratio, the more accurat

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  Radial basis function kernel

  In machine learning, the radial basis function kernel, or RBF kernel, is a popular kernel function used in various kernelized learning algorithms. In particular, it is commonly used in support vector machine classification. The RBF kernel on two samples x , x ′ ∈ R k {\displaystyle \mathbf {x} ,\mathbf {x'} \in \mathbb {R} ^{k}} , represented as feature vectors in some input space, is defined as K ( x , x ′ ) = exp ⁡ ( − ‖ x − x ′ ‖ 2 2 σ 2 ) {\displaystyle K(\mathbf {x} ,\mathbf {x'} )=\exp \left(-{\frac {\|\mathbf {x} -\mathbf {x'} \|^{2}}{2\sigma ^{2}}}\right)} ‖ x − x ′ ‖ 2 {\displaystyle \textstyle \|\mathbf {x} -\mathbf {x'} \|^{2}} may be recognized as the squared Euclidean distance between the two feature vectors. σ {\displaystyle \sigma } is a free parameter. An equivalent definition involves a parameter γ = 1 2 σ 2 {\displaystyle \textstyle \gamma ={\tfrac {1}{2\sigma ^{2}}}} : K ( x , x ′ ) = exp ⁡ ( − γ ‖ x − x ′ ‖ 2 ) {\displaystyle K(\mathbf {x} ,\mathbf {x'} )=\exp(-\gamma \|\mathbf {x} -\mathbf {x'} \|^{2})} Since the value of the RBF kernel decreases with distance and ranges between zero (in the infinite-distance limit) and one (when x = x'), it has a ready interpretation as a similarity measure. The feature space of the kernel has an infinite number of dimensions; for σ = 1 {\displaystyle \sigma =1} , its expansion using the multinomial theorem is: exp ⁡ ( − 1 2 ‖ x − x ′ ‖ 2 ) = exp ⁡ ( 2 2 x ⊤ x ′ − 1 2 ‖ x ‖ 2 − 1 2 ‖ x ′ ‖ 2 ) = exp ⁡ ( x ⊤ x ′ ) exp ⁡ ( − 1 2 ‖ x ‖ 2 ) exp ⁡ ( − 1 2 ‖ x ′ ‖ 2 ) = ∑ j = 0 ∞ ( x ⊤ x ′ ) j j ! exp ⁡ ( − 1 2 ‖ x ‖ 2 ) exp ⁡ ( − 1 2 ‖ x ′ ‖ 2 ) = ∑ j = 0 ∞ ∑ n 1 + n 2 + ⋯ + n k = j exp ⁡ ( − 1 2 ‖ x ‖ 2 ) x 1 n 1 ⋯ x k n k n 1 ! ⋯ n k ! exp ⁡ ( − 1 2 ‖ x ′ ‖ 2 ) x ′ 1 n 1 ⋯ x ′ k n k n 1 ! ⋯ n k ! = ⟨ φ ( x ) , φ ( x ′ ) ⟩ {\displaystyle {\begin{alignedat}{2}\exp \left(-{\frac {1}{2}}\|\mathbf {x} -\mathbf {x'} \|^{2}\right)&=\exp \left({\frac {2}{2}}\mathbf {x} ^{\top }\mathbf {x'} -{\frac {1}{2}}\|\mathbf {x} \|^{2}-{\frac {1}{2}}\|\mathbf {x'} \|^{2}\right)\\[5pt]&=\exp \left(\mathbf {x} ^{\top }\mathbf {x'} \right)\exp \left(-{\frac {1}{2}}\|\mathbf {x} \|^{2}\right)\exp \left(-{\frac {1}{2}}\|\mathbf {x'} \|^{2}\right)\\[5pt]&=\sum _{j=0}^{\infty }{\frac {(\mathbf {x} ^{\top }\mathbf {x'} )^{j}}{j!}}\exp \left(-{\frac {1}{2}}\|\mathbf {x} \|^{2}\right)\exp \left(-{\frac {1}{2}}\|\mathbf {x'} \|^{2}\right)\\[5pt]&=\sum _{j=0}^{\infty }\quad \sum _{n_{1}+n_{2}+\dots +n_{k}=j}\exp \left(-{\frac {1}{2}}\|\mathbf {x} \|^{2}\right){\frac {x_{1}^{n_{1}}\cdots x_{k}^{n_{k}}}{\sqrt {n_{1}!\cdots n_{k}!}}}\exp \left(-{\frac {1}{2}}\|\mathbf {x'} \|^{2}\right){\frac {{x'}_{1}^{n_{1}}\cdots {x'}_{k}^{n_{k}}}{\sqrt {n_{1}!\cdots n_{k}!}}}\\[5pt]&=\langle \varphi (\mathbf {x} ),\varphi (\mathbf {x'} )\rangle \end{alignedat}}} φ ( x ) = exp ⁡ ( − 1 2 ‖ x ‖ 2 ) ( a ℓ 0 ( 0 ) , a 1 ( 1 ) , … , a ℓ 1 ( 1 ) , … , a 1 ( j ) , … , a ℓ j ( j ) , … ) {\displaystyle \varphi (\mathbf {x} )=\exp \left(-{\frac {1}{2}}\|\mathbf {x} \|^{2}\right)\left(a_{\ell _{0}}^{(0)},a_{1}^{(1)},\dots ,a_{\ell _{1}}^{(1)},\dots ,a_{1}^{(j)},\dots ,a_{\ell _{j}}^{(j)},\dots \right)} where ℓ j = ( k + j − 1 j ) {\displaystyle \ell _{j}={\tbinom {k+j-1}{j}}} , a ℓ ( j ) = x 1 n 1 ⋯ x k n k n 1 ! ⋯ n k ! | n 1 + n 2 + ⋯ + n k = j ∧ 1 ≤ ℓ ≤ ℓ j {\displaystyle a_{\ell }^{(j)}={\frac {x_{1}^{n_{1}}\cdots x_{k}^{n_{k}}}{\sqrt {n_{1}!\cdots n_{k}!}}}\quad |\quad n_{1}+n_{2}+\dots +n_{k}=j\wedge 1\leq \ell \leq \ell _{j}} == Approximations == Because support vector machines and other models employing the kernel trick do not scale well to large numbers of training samples or large numbers of features in the input space, several approximations to the RBF kernel (and similar kernels) have been introduced. Typically, these take the form of a function z that maps a single vector to a vector of higher dimensionality, approximating the kernel: ⟨ z ( x ) , z ( x ′ ) ⟩ ≈ ⟨ φ ( x ) , φ ( x ′ ) ⟩ = K ( x , x ′ ) {\displaystyle \langle z(\mathbf {x} ),z(\mathbf {x'} )\rangle \approx \langle \varphi (\mathbf {x} ),\varphi (\mathbf {x'} )\rangle =K(\mathbf {x} ,\mathbf {x'} )} where φ {\displaystyle \textstyle \varphi } is the implicit mapping embedded in the RBF kernel. === Fourier random features === One way to construct such a z is to randomly sample from the Fourier transformation of the kernel φ ( x ) = 1 D [ cos ⁡ ⟨ w 1 , x ⟩ , sin ⁡ ⟨ w 1 , x ⟩ , … , cos ⁡ ⟨ w D , x ⟩ , sin ⁡ ⟨ w D , x ⟩ ] T {\displaystyle \varphi (x)={\frac {1}{\sqrt {D}}}[\cos \langle w_{1},x\rangle ,\sin \langle w_{1},x\rangle ,\ldots ,\cos \langle w_{D},x\rangle ,\sin \langle w_{D},x\rangle ]^{T}} where w 1 , . . . , w D {\displaystyle w_{1},...,w_{D}} are independent samples from the normal distribution N ( 0 , σ − 2 I ) {\displaystyle N(0,\sigma ^{-2}I)} . Theorem: E ⁡ [ ⟨ φ ( x ) , φ ( y ) ⟩ ] = e ‖ x − y ‖ 2 / ( 2 σ 2 ) . {\displaystyle \operatorname {E} [\langle \varphi (x),\varphi (y)\rangle ]=e^{\|x-y\|^{2}/(2\sigma ^{2})}.} Proof: It suffices to prove the case of D = 1 {\displaystyle D=1} . Use the trigonometric identity cos ⁡ ( a − b ) = cos ⁡ ( a ) cos ⁡ ( b ) + sin ⁡ ( a ) sin ⁡ ( b ) {\displaystyle \cos(a-b)=\cos(a)\cos(b)+\sin(a)\sin(b)} , the spherical symmetry of Gaussian distribution, then evaluate the integral ∫ − ∞ ∞ cos ⁡ ( k x ) e − x 2 / 2 2 π d x = e − k 2 / 2 . {\displaystyle \int _{-\infty }^{\infty }{\frac {\cos(kx)e^{-x^{2}/2}}{\sqrt {2\pi }}}dx=e^{-k^{2}/2}.} Theorem: Var ⁡ [ ⟨ φ ( x ) , φ ( y ) ⟩ ] = O ( D − 1 ) {\displaystyle \operatorname {Var} [\langle \varphi (x),\varphi (y)\rangle ]=O(D^{-1})} . (Appendix A.2). === Nyström method === Another approach uses the Nyström method to approximate the eigendecomposition of the Gram matrix K, using only a random sample of the training set.

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  Conditional random field

  Conditional random fields (CRFs) are a class of statistical modeling methods often applied in pattern recognition and machine learning and used for structured prediction. Whereas a classifier predicts a label for a single sample without considering "neighbouring" samples, a CRF can take context into account. To do so, the predictions are modelled as a graphical model, which represents the presence of dependencies between the predictions. The kind of graph used depends on the application. For example, in natural language processing, "linear chain" CRFs are popular, for which each prediction is dependent only on its immediate neighbours. In image processing, the graph typically connects locations to nearby and/or similar locations to enforce that they receive similar predictions. Other examples where CRFs are used are: labeling or parsing of sequential data for natural language processing or biological sequences, part-of-speech tagging, shallow parsing, named entity recognition, gene finding, peptide critical functional region finding, and object recognition and image segmentation in computer vision. == Description == CRFs are a type of discriminative undirected probabilistic graphical model. Lafferty, McCallum and Pereira define a CRF on observations X {\displaystyle {\boldsymbol {X}}} and random variables Y {\displaystyle {\boldsymbol {Y}}} as follows: Let G = ( V , E ) {\displaystyle G=(V,E)} be a graph such that Y = ( Y v ) v ∈ V {\displaystyle {\boldsymbol {Y}}=({\boldsymbol {Y}}_{v})_{v\in V}} , so that Y {\displaystyle {\boldsymbol {Y}}} is indexed by the vertices of G {\displaystyle G} . Then ( X , Y ) {\displaystyle ({\boldsymbol {X}},{\boldsymbol {Y}})} is a conditional random field when each random variable Y v {\displaystyle {\boldsymbol {Y}}_{v}} , conditioned on X {\displaystyle {\boldsymbol {X}}} , obeys the Markov property with respect to the graph; that is, its probability is dependent only on its neighbours in G and not its past states: P ( Y v | X , { Y w : w ≠ v } ) = P ( Y v | X , { Y w : w ∼ v } ) {\displaystyle P({\boldsymbol {Y}}_{v}|{\boldsymbol {X}},\{{\boldsymbol {Y}}_{w}:w\neq v\})=P({\boldsymbol {Y}}_{v}|{\boldsymbol {X}},\{{\boldsymbol {Y}}_{w}:w\sim v\})} , where w ∼ v {\displaystyle {\mathit {w}}\sim v} means that w {\displaystyle w} and v {\displaystyle v} are neighbors in G {\displaystyle G} . What this means is that a CRF is an undirected graphical model whose nodes can be divided into exactly two disjoint sets X {\displaystyle {\boldsymbol {X}}} and Y {\displaystyle {\boldsymbol {Y}}} , the observed and output variables, respectively; the conditional distribution p ( Y | X ) {\displaystyle p({\boldsymbol {Y}}|{\boldsymbol {X}})} is then modeled. === Inference === For general graphs, the problem of exact inference in CRFs is intractable. The inference problem for a CRF is basically the same as for an MRF and the same arguments hold. However, there exist special cases for which exact inference is feasible: If the graph is a chain or a tree, message passing algorithms yield exact solutions. The algorithms used in these cases are analogous to the forward-backward and Viterbi algorithm for the case of HMMs. If the CRF only contains pair-wise potentials and the energy is submodular, combinatorial min cut/max flow algorithms yield exact solutions. If exact inference is impossible, several algorithms can be used to obtain approximate solutions. These include: Loopy belief propagation Alpha expansion Mean field inference Linear programming relaxations === Parameter learning === Learning the parameters θ {\displaystyle \theta } is usually done by maximum likelihood learning for p ( Y i | X i ; θ ) {\displaystyle p(Y_{i}|X_{i};\theta )} . If all nodes have exponential family distributions and all nodes are observed during training, this optimization is convex. It can be solved for example using gradient descent algorithms, or Quasi-Newton methods such as the L-BFGS algorithm. On the other hand, if some variables are unobserved, the inference problem has to be solved for these variables. Exact inference is intractable in general graphs, so approximations have to be used. === Examples === In sequence modeling, the graph of interest is usually a chain graph. An input sequence of observed variables X {\displaystyle X} represents a sequence of observations and Y {\displaystyle Y} represents a hidden (or unknown) state variable that needs to be inferred given the observations. The Y i {\displaystyle Y_{i}} are structured to form a chain, with an edge between each Y i − 1 {\displaystyle Y_{i-1}} and Y i {\displaystyle Y_{i}} . As well as having a simple interpretation of the Y i {\displaystyle Y_{i}} as "labels" for each element in the input sequence, this layout admits efficient algorithms for: model training, learning the conditional distributions between the Y i {\displaystyle Y_{i}} and feature functions from some corpus of training data. decoding, determining the probability of a given label sequence Y {\displaystyle Y} given X {\displaystyle X} . inference, determining the most likely label sequence Y {\displaystyle Y} given X {\displaystyle X} . The conditional dependency of each Y i {\displaystyle Y_{i}} on X {\displaystyle X} is defined through a fixed set of feature functions of the form f ( i , Y i − 1 , Y i , X ) {\displaystyle f(i,Y_{i-1},Y_{i},X)} , which can be thought of as measurements on the input sequence that partially determine the likelihood of each possible value for Y i {\displaystyle Y_{i}} . The model assigns each feature a numerical weight and combines them to determine the probability of a certain value for Y i {\displaystyle Y_{i}} . Linear-chain CRFs have many of the same applications as conceptually simpler hidden Markov models (HMMs), but relax certain assumptions about the input and output sequence distributions. An HMM can loosely be understood as a CRF with very specific feature functions that use constant probabilities to model state transitions and emissions. Conversely, a CRF can loosely be understood as a generalization of an HMM that makes the constant transition probabilities into arbitrary functions that vary across the positions in the sequence of hidden states, depending on the input sequence. Notably, in contrast to HMMs, CRFs can contain any number of feature functions, the feature functions can inspect the entire input sequence X {\displaystyle X} at any point during inference, and the range of the feature functions need not have a probabilistic interpretation. == Variants == === Higher-order CRFs and semi-Markov CRFs === CRFs can be extended into higher order models by making each Y i {\displaystyle Y_{i}} dependent on a fixed number k {\displaystyle k} of previous variables Y i − k , . . . , Y i − 1 {\displaystyle Y_{i-k},...,Y_{i-1}} . In conventional formulations of higher order CRFs, training and inference are only practical for small values of k {\displaystyle k} (such as k ≤ 5), since their computational cost increases exponentially with k {\displaystyle k} . However, another recent advance has managed to ameliorate these issues by leveraging concepts and tools from the field of Bayesian nonparametrics. Specifically, the CRF-infinity approach constitutes a CRF-type model that is capable of learning infinitely-long temporal dynamics in a scalable fashion. This is effected by introducing a novel potential function for CRFs that is based on the Sequence Memoizer (SM), a nonparametric Bayesian model for learning infinitely-long dynamics in sequential observations. To render such a model computationally tractable, CRF-infinity employs a mean-field approximation of the postulated novel potential functions (which are driven by an SM). This allows for devising efficient approximate training and inference algorithms for the model, without undermining its capability to capture and model temporal dependencies of arbitrary length. There exists another generalization of CRFs, the semi-Markov conditional random field (semi-CRF), which models variable-length segmentations of the label sequence Y {\displaystyle Y} . This provides much of the power of higher-order CRFs to model long-range dependencies of the Y i {\displaystyle Y_{i}} , at a reasonable computational cost. Finally, large-margin models for structured prediction, such as the structured Support Vector Machine can be seen as an alternative training procedure to CRFs. === Latent-dynamic conditional random field === Latent-dynamic conditional random fields (LDCRF) or discriminative probabilistic latent variable models (DPLVM) are a type of CRFs for sequence tagging tasks. They are latent variable models that are trained discriminatively. In an LDCRF, like in any sequence tagging task, given a sequence of observations x = x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} , the main problem the model must solve is how to assign a sequence of labels y = y 1 , … , y n {\displaystyle y_{1},\dots ,y_{n}} from one finite set

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  Local tangent space alignment

  Local tangent space alignment (LTSA) is a method for manifold learning, which can efficiently learn a nonlinear embedding into low-dimensional coordinates from high-dimensional data, and can also reconstruct high-dimensional coordinates from embedding coordinates. It is based on the intuition that when a manifold is correctly unfolded, all of the tangent hyperplanes to the manifold will become aligned. It begins by computing the k-nearest neighbors of every point. It computes the tangent space at every point by computing the d-first principal components in each local neighborhood. It then optimizes to find an embedding that aligns the tangent spaces, but it ignores the label information conveyed by data samples, and thus can not be used for classification directly.

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  Local tangent space alignment

  Local tangent space alignment (LTSA) is a method for manifold learning, which can efficiently learn a nonlinear embedding into low-dimensional coordinates from high-dimensional data, and can also reconstruct high-dimensional coordinates from embedding coordinates. It is based on the intuition that when a manifold is correctly unfolded, all of the tangent hyperplanes to the manifold will become aligned. It begins by computing the k-nearest neighbors of every point. It computes the tangent space at every point by computing the d-first principal components in each local neighborhood. It then optimizes to find an embedding that aligns the tangent spaces, but it ignores the label information conveyed by data samples, and thus can not be used for classification directly.

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