Willy's Chocolate Experience was an unlicensed event based on Charlie and the Chocolate Factory that took place in Glasgow, Scotland, in February 2024. The event was promoted as an immersive and interactive family experience, illustrated on a promotional website with "dreamlike" AI-generated images. Once it was discovered that the event was held in a sparsely decorated warehouse, many customers complained, and the police were called to the venue. The event went viral on the Internet and attracted worldwide media attention. The event drew comparisons to the 2008 Lapland New Forest controversy, the 2014 Tumblr fan convention DashCon, and Billy McFarland's 2017 Fyre Festival. == Background and advertising == The event was stated to take place over the weekend of 24–25 February 2024. Promotional material advertised "stunning and intricately designed settings inspired by Roald Dahl's timeless tale" and "an array of delectable treats scattered throughout the experience". Both the website and promotional material used poor-quality AI-generated images, which included several spelling errors such as "cartchy tuns" and "a pasadise of sweet teats" and nonsensical words such as "catgacating" and "exarserdray". Tickets cost up to £35 per person. While the event was being promoted in early February, a Reddit user who saw Facebook advertisements suspected it to be a scam and was surprised that people were apparently buying tickets based solely on AI-generated images. The event was organised by House of Illuminati, a company registered to Billy Coull which claimed to offer "unparalleled immersive experiences". An investigation by Third Force News conducted after the event described Coull's previous "murky involvement in the charity sector." Coull had previously registered several other companies and claimed to work as a "consultant" for the now-defunct brand Empowerity, formerly known as the charity Gowanbank Community Hub. In 2021, Gowanbank was forced to remove claims of a £95-per-ticket fundraising "gala" at DoubleTree Glasgow which had been falsely advertised to feature TV personalities and performers including Gok Wan and Joe Black. Coull had claimed to be a doctor with a fake degree from a false university that provided "metaphysical degrees", and had attempted to use the charity to win the 2022 Glasgow City Council election in the seat of Greater Pollok, though he never registered for the election. In the summer of 2023, he independently published 17 AI-generated books on various topics, including vaccine conspiracy theories. Rolling Stone concluded that House of Illuminati's websites and event descriptions were likely written by an AI chatbot, such as ChatGPT. Three actors were hired to portray "Willy McDuff", a character based on Willy Wonka. One of them, Paul Connell, said that the cast were given one day to learn the script. Another actor playing Willy McDuff was 18-year-old Michael Archibald; the experience was his first ever acting job, and he was given the script at 6 pm on Friday before the event began on Saturday. Kirsty Paterson, an actress who played one of the Oompa-Loompas (called "Wonkidoodles" in the script), said that the job offer had been posted on Indeed.com and offered £500 for two days of work. The day before the event, the actors attended a dress rehearsal at the sparsely decorated venue. They were told that others would be working through the night on the production. When they returned on the day of the event, the venue was in the same condition. Paterson was given her costume an hour before the event opened, saying that "We were just handed an Amazon box that probably arrived that morning." == Script == The script for the event is titled Wonkidoodles at McDuff's Chocolate Factory: A Script, and describes Willy McDuff leading an audience through the Garden of Enchantment and the Twilight Tunnel. Once there, they are confronted by a character called The Unknown, described as "an evil chocolate maker who lives in the walls" who seeks to steal the magical "Anti-Graffiti Gobstopper" from McDuff's Imagination Lab. The gobstopper is "a sweet so powerful, it can make any room sparkle without lifting a finger". McDuff defeats The Unknown by amplifying the power of the gobstopper and causing his enemy to be "gently swept up by a robotic vacuum, humorously ending the confrontation". The script was unusual in that it included stage directions for the audience, and descriptions of their reactions. Connell described it as "15 pages of AI-generated gibberish of me just monologuing these mad things", and compared the vacuum cleaner plot point to that of the Nintendo video game Luigi's Mansion. Interviewed after the event, Coull claimed to have written the script himself, using AI only to "check spelling, grammar, and continuity" as he said he had dyslexia. == Event == The event was held at the Box Hub Warehouse event space in Whiteinch, an industrial area of Glasgow. Customers described the venue as "little more than an abandoned, empty warehouse", with set dressings including a small bouncy castle, AI-generated backdrop images pinned to some of the walls, and props which were "strewn about on bare concrete floors". The venue's windows were dirty and its air conditioning systems were left exposed. Paterson has stated that by the time she saw the venue, she had already signed her contract and "didn't want to disappoint the kids", and thus chose to proceed with the work. The Unknown was played by a 16-year-old actress named Felicia Dawkins, who wore a silver mask and a black cloak. Young children were frightened by the character, who appeared from behind a large rectangular mirror. Despite the script calling for The Unknown to be defeated with a vacuum cleaner, no such prop was provided, and actors were instead asked to improvise. Connell said that he and other employees were told to give each child "two jelly beans and a quarter of a cup of lemonade", although the limited supply of jelly beans quickly ran out. Paterson and another "Wonkidoodle" actress, Jenny Fogarty, said that after the first three 45-minute performances, the cast were told to abandon the script and instead let guests walk through the venue, a process that Paterson said took "about two minutes". The character of The Unknown, previously introduced as the main antagonist, was now "scaring children for no reason". One of the actors playing McDuff improvised the idea that children should pull a "silly face" at The Unknown to scare them away, but Dawkins said that, in other cases, she "just had to awkwardly walk back to my corner". Connell was told he would be given a 15-minute break every 45 minutes, but on the day of the event, he played Willy McDuff for three and a half hours without a break. After returning from a lunch break, Connell encountered a crowd of customers demanding refunds from Coull, and the other actors were unsure what to do next. After being told that the event was now cancelled halfway through its opening day, the actors left and went to a pub. Upon returning to the venue some time later, Connell said that he felt "the threat of violence had become quite high" and that there were two police vans and two squad cars at the scene. == Customer reviews and response == Willy's Chocolate Experience was widely criticised by those who attended it, many of whom demanded refunds. One customer, who had driven with his children for two hours to reach the event, described it as an "absolute con". Other visitors who arrived after the event was closed and were not informed of its cancellation requested compensation for wasted rail fares. Following the event's cancellation, Coull offered to refund 850 people, a statement repeated by the event's Facebook page. Some Facebook users stated that they had received their money back. Paterson and Fogarty stated that they only received half of their paycheque. Box Hub, the organisation that had rented the warehouse to House of Illuminati, issued an apology on House of Illuminati's behalf, stating that they "either have no regards for the families and young children they have disappointed or are too embarrassed to comment", and offered to provide a venue free of charge for those who attended the event. House of Illuminati later stated that they would not host any future events. Coull deleted his LinkedIn profile, his YouTube channel, and his personal website in response to the controversy. A few days after the event, Connell said he felt that Coull was "probably one of the most disliked people in Glasgow right now". In an interview with The Sunday Times, Coull apologised for how the event turned out, saying he would accept responsibility. == Fundraising == In an interview with Wired magazine, Connell stated that he and the other actors were working with parents to provide a free show for the children who attended. Some items from the event were later auctioned for charity. The venue auctioned the leftover hand-written "even
Autognostics
Autognostics is a new paradigm that describes the capacity for computer networks to be self-aware. It is considered one of the major components of Autonomic Networking. == Introduction == One of the most important characteristics of today's Internet that has contributed to its success is its basic design principle: a simple and transparent core with intelligence at the edges (the so-called "end-to-end principle"). Based on this principle, the network carries data without knowing the characteristics of that data (e.g., voice, video, etc.) - only the end-points have application-specific knowledge. If something goes wrong with the data, only the edge may be able to recognize that since it knows about the application and what the expected behavior is. The core has no information about what should happen with that data - it only forwards packets. Although an effective and beneficial attribute, this design principle has also led to many of today's problems, limitations, and frustrations. Currently, it is almost impossible for most end-users to know why certain network-based applications do not work well and what they need to do to make it better. Also, network operators who interact with the core in low-level terms such as router configuration have problems expressing their high-level goals into low-level actions. In high-level terms, this may be summarized as a weak coupling between the network and application layers of the overall system. As a consequence of the Internet end-to-end principle, the network performance experienced by a particular application is difficult to attribute based on the behavior of the individual elements. At any given moment, the measure of performance between any two points is typically unknown and applications must operate blindly. As a further consequence, changes to the configuration of given element, or changes in the end-to-end path, cannot easily be validated. Optimization and provisioning cannot then be automated except against only the simplest design specifications. There is an increasing interest in Autonomic Networking research, and a strong conviction that an evolution from the current networking status quo is necessary. Although to date there have not been any practical implementations demonstrating the benefits of an effective autonomic networking paradigm, there seems to be a consensus as to the characteristics which such implementations would need to demonstrate. These specifically include continuous monitoring, identifying, diagnosing and fixing problems based on high-level policies and objectives. Autognostics, as a major part of the autonomic networking concept, intends to bring networks to a new level of awareness and eliminate the lack of visibility which currently exists in today's networks. == Definition == Autognostics is a new paradigm that describes the capacity for computer networks to be self-aware, in part and as a whole, and dynamically adapt to the applications running on them by autonomously monitoring, identifying, diagnosing, resolving issues, subsequently verifying that any remediation was successful, and reporting the impact with respect to the application's use (i.e., providing visibility into the changes to networks and their effects). Although similar to the concept of network awareness, i.e., the capability of network devices and applications to be aware of network characteristics (see References section below), it is noteworthy that autognostics takes that concept one step further. The main difference is the auto part of autognostics, which entails that network devices are self-aware of network characteristics, and have the capability to adapt themselves as a result of continuous monitoring and diagnostics. == Path to autognostics == Autognostics, or in other words deep self-knowledge, can be best described as the ability of a network to know itself and the applications that run on it. This knowledge is used to autonomously adapt to dynamic network and application conditions such as utilization, capacity, quality of service/application/user experience, etc. In order to achieve autognosis, networks need a means to: Continuously monitor/test the network for application-specific performance Analyze the monitoring/test data to detect problems (e.g., performance degradation) Diagnose, identify and localize sources of degradation Automatically take actions to resolve problems via remediation/provisioning Verify the problems have been resolved (potentially rolling back changes if ineffective) Subsequently, continue to monitor/test for performance
One-class classification
In machine learning, one-class classification (OCC), also known as unary classification or class-modelling, is an approach to the training of binary classifiers in which only examples of one of the two classes are used. Examples include the monitoring of helicopter gearboxes, motor failure prediction, or assessing the operational status of a nuclear plant as 'normal': In such scenarios, there are few, if any, examples of the catastrophic system states – rare outliers – that comprise the second class. Alternatively, the class that is being focused on may cover a small, coherent subset of the data and the training may rely on an information bottleneck approach. In practice, counter-examples from the second class may be used in later rounds of training to further refine the algorithm. == Overview == The term one-class classification (OCC) was coined by Moya & Hush (1996) and many applications can be found in scientific literature, for example outlier detection, anomaly detection, novelty detection. A feature of OCC is that it uses only sample points from the assigned class, so that a representative sampling is not strictly required for non-target classes. == Introduction == SVM based one-class classification (OCC) relies on identifying the smallest hypersphere (with radius r, and center c) consisting of all the data points. This method is called Support Vector Data Description (SVDD). Formally, the problem can be defined in the following constrained optimization form, min r , c r 2 subject to, | | Φ ( x i ) − c | | 2 ≤ r 2 ∀ i = 1 , 2 , . . . , n {\displaystyle \min _{r,c}r^{2}{\text{ subject to, }}||\Phi (x_{i})-c||^{2}\leq r^{2}\;\;\forall i=1,2,...,n} However, the above formulation is highly restrictive, and is sensitive to the presence of outliers. Therefore, a flexible formulation, that allow for the presence of outliers is formulated as shown below, min r , c , ζ r 2 + 1 ν n ∑ i = 1 n ζ i {\displaystyle \min _{r,c,\zeta }r^{2}+{\frac {1}{\nu n}}\sum _{i=1}^{n}\zeta _{i}} subject to, | | Φ ( x i ) − c | | 2 ≤ r 2 + ζ i ∀ i = 1 , 2 , . . . , n {\displaystyle {\text{subject to, }}||\Phi (x_{i})-c||^{2}\leq r^{2}+\zeta _{i}\;\;\forall i=1,2,...,n} From the Karush–Kuhn–Tucker conditions for optimality, we get c = ∑ i = 1 n α i Φ ( x i ) , {\displaystyle c=\sum _{i=1}^{n}\alpha _{i}\Phi (x_{i}),} where the α i {\displaystyle \alpha _{i}} 's are the solution to the following optimization problem: max α ∑ i = 1 n α i κ ( x i , x i ) − ∑ i , j = 1 n α i α j κ ( x i , x j ) {\displaystyle \max _{\alpha }\sum _{i=1}^{n}\alpha _{i}\kappa (x_{i},x_{i})-\sum _{i,j=1}^{n}\alpha _{i}\alpha _{j}\kappa (x_{i},x_{j})} subject to, ∑ i = 1 n α i = 1 and 0 ≤ α i ≤ 1 ν n for all i = 1 , 2 , . . . , n . {\displaystyle \sum _{i=1}^{n}\alpha _{i}=1{\text{ and }}0\leq \alpha _{i}\leq {\frac {1}{\nu n}}{\text{for all }}i=1,2,...,n.} The introduction of kernel function provide additional flexibility to the One-class SVM (OSVM) algorithm. === PU (Positive Unlabeled) learning === A similar problem is PU learning, in which a binary classifier is constructed by semi-supervised learning from only positive and unlabeled sample points. In PU learning, two sets of examples are assumed to be available for training: the positive set P {\displaystyle P} and a mixed set U {\displaystyle U} , which is assumed to contain both positive and negative samples, but without these being labeled as such. This contrasts with other forms of semisupervised learning, where it is assumed that a labeled set containing examples of both classes is available in addition to unlabeled samples. A variety of techniques exist to adapt supervised classifiers to the PU learning setting, including variants of the EM algorithm. PU learning has been successfully applied to text, time series, bioinformatics tasks, and remote sensing data. == Approaches == Several approaches have been proposed to solve one-class classification (OCC). The approaches can be distinguished into three main categories, density estimation, boundary methods, and reconstruction methods. === Density estimation methods === Density estimation methods rely on estimating the density of the data points, and set the threshold. These methods rely on assuming distributions, such as Gaussian, or a Poisson distribution. Following which discordancy tests can be used to test the new objects. These methods are robust to scale variance. Gaussian model is one of the simplest methods to create one-class classifiers. Due to Central Limit Theorem (CLT), these methods work best when large number of samples are present, and they are perturbed by small independent error values. The probability distribution for a d-dimensional object is given by: p N ( z ; μ ; Σ ) = 1 ( 2 π ) d 2 | Σ | 1 2 exp { − 1 2 ( z − μ ) T Σ − 1 ( z − μ ) } {\displaystyle p_{\mathcal {N}}(z;\mu ;\Sigma )={\frac {1}{(2\pi )^{\frac {d}{2}}|\Sigma |^{\frac {1}{2}}}}\exp \left\{-{\frac {1}{2}}(z-\mu )^{T}\Sigma ^{-1}(z-\mu )\right\}} Where, μ {\displaystyle \mu } is the mean and Σ {\displaystyle \Sigma } is the covariance matrix. Computing the inverse of covariance matrix ( Σ − 1 {\displaystyle \Sigma ^{-1}} ) is the costliest operation, and in the cases where the data is not scaled properly, or data has singular directions pseudo-inverse Σ + {\displaystyle \Sigma ^{+}} is used to approximate the inverse, and is calculated as Σ T ( Σ Σ T ) − 1 {\displaystyle \Sigma ^{T}(\Sigma \Sigma ^{T})^{-1}} . === Boundary methods === Boundary methods focus on setting boundaries around a few set of points, called target points. These methods attempt to optimize the volume. Boundary methods rely on distances, and hence are not robust to scale variance. K-centers method, NN-d, and SVDD are some of the key examples. K-centers In K-center algorithm, k {\displaystyle k} small balls with equal radius are placed to minimize the maximum distance of all minimum distances between training objects and the centers. Formally, the following error is minimized, ε k − c e n t e r = max i ( min k | | x i − μ k | | 2 ) {\displaystyle \varepsilon _{k-center}=\max _{i}(\min _{k}||x_{i}-\mu _{k}||^{2})} The algorithm uses forward search method with random initialization, where the radius is determined by the maximum distance of the object, any given ball should capture. After the centers are determined, for any given test object z {\displaystyle z} the distance can be calculated as, d k − c e n t r ( z ) = min k | | z − μ k | | 2 {\displaystyle d_{k-centr}(z)=\min _{k}||z-\mu _{k}||^{2}} === Reconstruction methods === Reconstruction methods use prior knowledge and generating process to build a generating model that best fits the data. New objects can be described in terms of a state of the generating model. Some examples of reconstruction methods for OCC are, k-means clustering, learning vector quantization, self-organizing maps, etc. == Applications == === Document classification === The basic Support Vector Machine (SVM) paradigm is trained using both positive and negative examples, however studies have shown there are many valid reasons for using only positive examples. When the SVM algorithm is modified to only use positive examples, the process is considered one-class classification. One situation where this type of classification might prove useful to the SVM paradigm is in trying to identify a web browser's sites of interest based only off of the user's browsing history. === Biomedical studies === One-class classification can be particularly useful in biomedical studies where often data from other classes can be difficult or impossible to obtain. In studying biomedical data it can be difficult and/or expensive to obtain the set of labeled data from the second class that would be necessary to perform a two-class classification. A study from The Scientific World Journal found that the typicality approach is the most useful in analysing biomedical data because it can be applied to any type of dataset (continuous, discrete, or nominal). The typicality approach is based on the clustering of data by examining data and placing it into new or existing clusters. To apply typicality to one-class classification for biomedical studies, each new observation, y 0 {\displaystyle y_{0}} , is compared to the target class, C {\displaystyle C} , and identified as an outlier or a member of the target class. === Unsupervised Concept Drift Detection === One-class classification has similarities with unsupervised concept drift detection, where both aim to identify whether the unseen data share similar characteristics to the initial data. A concept is referred to as the fixed probability distribution which data is drawn from. In unsupervised concept drift detection, the goal is to detect if the data distribution changes without utilizing class labels. In one-class classification, the flow of data is not important. Unseen data is classified as typical or outlier depending on its characteristics, whether it is from the initi
Variational autoencoder
In machine learning, a variational autoencoder (VAE) is an artificial neural network architecture introduced by Diederik P. Kingma and Max Welling in 2013. It is part of the families of probabilistic graphical models and variational Bayesian methods. In addition to being seen as an autoencoder neural network architecture, variational autoencoders can also be studied within the mathematical formulation of variational Bayesian methods, connecting a neural encoder network to its decoder through a probabilistic latent space (for example, as a multivariate Gaussian distribution) that corresponds to the parameters of a variational distribution. Thus, the encoder maps each point (such as an image) from a large complex dataset into a distribution within the latent space, rather than to a single point in that space. The decoder has the opposite function, which is to map from the latent space to the input space, again according to a distribution (although in practice, noise is rarely added during the decoding stage). By mapping a point to a distribution instead of a single point, the network can avoid overfitting the training data. Both networks are typically trained together with the usage of the reparameterization trick, although the variance of the noise model can be learned separately. Although this type of model was initially designed for unsupervised learning, its effectiveness has been proven for semi-supervised learning and supervised learning. == Overview of architecture and operation == A variational autoencoder is a generative model with a prior and noise distribution respectively. Usually such models are trained using the expectation-maximization meta-algorithm (e.g. probabilistic PCA, (spike & slab) sparse coding). Such a scheme optimizes a lower bound of the data likelihood, which is usually computationally intractable, and in doing so requires the discovery of q-distributions, or variational posteriors. These q-distributions are normally parameterized for each individual data point in a separate optimization process. However, variational autoencoders use a neural network as an amortized approach to jointly optimize across data points. In that way, the same parameters are reused for multiple data points, which can result in massive memory savings. The first neural network takes as input the data points themselves, and outputs parameters for the variational distribution. As it maps from a known input space to the low-dimensional latent space, it is called the encoder. The decoder is the second neural network of this model. It is a function that maps from the latent space to the input space, e.g. as the means of the noise distribution. It is possible to use another neural network that maps to the variance, however this can be omitted for simplicity. In such a case, the variance can be optimized with gradient descent. To optimize this model, one needs to know two terms: the "reconstruction error", and the Kullback–Leibler divergence (KL-D). Both terms are derived from the free energy expression of the probabilistic model, and therefore differ depending on the noise distribution and the assumed prior of the data, here referred to as p-distribution. For example, a standard VAE task such as IMAGENET is typically assumed to have a gaussianly distributed noise; however, tasks such as binarized MNIST require a Bernoulli noise. The KL-D from the free energy expression maximizes the probability mass of the q-distribution that overlaps with the p-distribution, which unfortunately can result in mode-seeking behaviour. The "reconstruction" term is the remainder of the free energy expression, and requires a sampling approximation to compute its expectation value. More recent approaches replace Kullback–Leibler divergence (KL-D) with various statistical distances, see "Statistical distance VAE variants" below. == Formulation == From the point of view of probabilistic modeling, one wants to maximize the likelihood of the data x {\displaystyle x} by their chosen parameterized probability distribution p θ ( x ) = p ( x | θ ) {\displaystyle p_{\theta }(x)=p(x|\theta )} . This distribution is usually chosen to be a Gaussian N ( x | μ , σ ) {\displaystyle N(x|\mu ,\sigma )} which is parameterized by μ {\displaystyle \mu } and σ {\displaystyle \sigma } respectively, and as a member of the exponential family it is easy to work with as a noise distribution. Simple distributions are easy enough to maximize, however distributions where a prior is assumed over the latents z {\displaystyle z} results in intractable integrals. Let us find p θ ( x ) {\displaystyle p_{\theta }(x)} via marginalizing over z {\displaystyle z} . p θ ( x ) = ∫ z p θ ( x , z ) d z , {\displaystyle p_{\theta }(x)=\int _{z}p_{\theta }({x,z})\,dz,} where p θ ( x , z ) {\displaystyle p_{\theta }({x,z})} represents the joint distribution under p θ {\displaystyle p_{\theta }} of the observable data x {\displaystyle x} and its latent representation or encoding z {\displaystyle z} . According to the chain rule, the equation can be rewritten as p θ ( x ) = ∫ z p θ ( x | z ) p θ ( z ) d z {\displaystyle p_{\theta }(x)=\int _{z}p_{\theta }({x|z})p_{\theta }(z)\,dz} In the vanilla variational autoencoder, z {\displaystyle z} is usually taken to be a finite-dimensional vector of real numbers, and p θ ( x | z ) {\displaystyle p_{\theta }({x|z})} to be a Gaussian distribution. Then p θ ( x ) {\displaystyle p_{\theta }(x)} is a mixture of Gaussian distributions. It is now possible to define the set of the relationships between the input data and its latent representation as Prior p θ ( z ) {\displaystyle p_{\theta }(z)} Likelihood p θ ( x | z ) {\displaystyle p_{\theta }(x|z)} Posterior p θ ( z | x ) {\displaystyle p_{\theta }(z|x)} Unfortunately, the computation of p θ ( z | x ) {\displaystyle p_{\theta }(z|x)} is expensive and in most cases intractable. To speed up the calculus to make it feasible, it is necessary to introduce a further function to approximate the posterior distribution as q ϕ ( z | x ) ≈ p θ ( z | x ) {\displaystyle q_{\phi }({z|x})\approx p_{\theta }({z|x})} with ϕ {\displaystyle \phi } defined as the set of real values that parametrize q {\displaystyle q} . This is sometimes called amortized inference, since by "investing" in finding a good q ϕ {\displaystyle q_{\phi }} , one can later infer z {\displaystyle z} from x {\displaystyle x} quickly without doing any integrals. In this way, the problem is to find a good probabilistic autoencoder, in which the conditional likelihood distribution p θ ( x | z ) {\displaystyle p_{\theta }(x|z)} is computed by the probabilistic decoder, and the approximated posterior distribution q ϕ ( z | x ) {\displaystyle q_{\phi }(z|x)} is computed by the probabilistic encoder. Parametrize the encoder as E ϕ {\displaystyle E_{\phi }} , and the decoder as D θ {\displaystyle D_{\theta }} . == Evidence lower bound (ELBO) == Like many deep learning approaches that use gradient-based optimization, VAEs require a differentiable loss function to update the network weights through backpropagation. For variational autoencoders, the idea is to jointly optimize the generative model parameters θ {\displaystyle \theta } to reduce the reconstruction error between the input and the output, and ϕ {\displaystyle \phi } to make q ϕ ( z | x ) {\displaystyle q_{\phi }({z|x})} as close as possible to p θ ( z | x ) {\displaystyle p_{\theta }(z|x)} . As reconstruction loss, mean squared error and cross entropy are often used. The Kullback–Leibler divergence D K L ( q ϕ ( z | x ) ∥ p θ ( z | x ) ) {\displaystyle D_{KL}(q_{\phi }({z|x})\parallel p_{\theta }({z|x}))} can be used as a loss function to squeeze q ϕ ( z | x ) {\displaystyle q_{\phi }({z|x})} under p θ ( z | x ) {\displaystyle p_{\theta }(z|x)} . This divergence loss expands to D K L ( q ϕ ( z | x ) ∥ p θ ( z | x ) ) = E z ∼ q ϕ ( ⋅ | x ) [ ln q ϕ ( z | x ) p θ ( z | x ) ] = E z ∼ q ϕ ( ⋅ | x ) [ ln q ϕ ( z | x ) p θ ( x ) p θ ( x , z ) ] = ln p θ ( x ) + E z ∼ q ϕ ( ⋅ | x ) [ ln q ϕ ( z | x ) p θ ( x , z ) ] . {\displaystyle {\begin{aligned}D_{KL}(q_{\phi }({z|x})\parallel p_{\theta }({z|x}))&=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {q_{\phi }(z|x)}{p_{\theta }(z|x)}}\right]\\&=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {q_{\phi }({z|x})p_{\theta }(x)}{p_{\theta }(x,z)}}\right]\\&=\ln p_{\theta }(x)+\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {q_{\phi }({z|x})}{p_{\theta }(x,z)}}\right].\end{aligned}}} Now, define the evidence lower bound (ELBO): L θ , ϕ ( x ) := E z ∼ q ϕ ( ⋅ | x ) [ ln p θ ( x , z ) q ϕ ( z | x ) ] = ln p θ ( x ) − D K L ( q ϕ ( ⋅ | x ) ∥ p θ ( ⋅ | x ) ) {\displaystyle L_{\theta ,\phi }(x):=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {p_{\theta }(x,z)}{q_{\phi }({z|x})}}\right]=\ln p_{\theta }(x)-D_{KL}(q_{\phi }({\cdot |x})\parallel p_{\theta }({\cdot |x}))} Maximizing the ELBO θ ∗ , ϕ ∗ = argmax θ , ϕ L θ , ϕ ( x ) {\dis
Radial basis function network
In the field of mathematical modeling, a radial basis function network is an artificial neural network that uses radial basis functions as activation functions. The output of the network is a linear combination of radial basis functions of the inputs and neuron parameters. Radial basis function networks have many uses, including function approximation, time series prediction, classification, and system control. They were first formulated in a 1988 paper by Broomhead and Lowe, both researchers at the Royal Signals and Radar Establishment. == Network architecture == Radial basis function (RBF) networks typically have three layers: an input layer, a hidden layer with a non-linear RBF activation function and a linear output layer. The input can be modeled as a vector of real numbers x ∈ R n {\displaystyle \mathbf {x} \in \mathbb {R} ^{n}} . The output of the network is then a scalar function of the input vector, φ : R n → R {\displaystyle \varphi :\mathbb {R} ^{n}\to \mathbb {R} } , and is given by φ ( x ) = ∑ i = 1 N a i ρ ( | | x − c i | | ) {\displaystyle \varphi (\mathbf {x} )=\sum _{i=1}^{N}a_{i}\rho (||\mathbf {x} -\mathbf {c} _{i}||)} where N {\displaystyle N} is the number of neurons in the hidden layer, c i {\displaystyle \mathbf {c} _{i}} is the center vector for neuron i {\displaystyle i} , and a i {\displaystyle a_{i}} is the weight of neuron i {\displaystyle i} in the linear output neuron. Functions that depend only on the distance from a center vector are radially symmetric about that vector, hence the name radial basis function. In the basic form, all inputs are connected to each hidden neuron. The norm is typically taken to be the Euclidean distance (although the Mahalanobis distance appears to perform better with pattern recognition) and the radial basis function is commonly taken to be Gaussian ρ ( ‖ x − c i ‖ ) = exp [ − β i ‖ x − c i ‖ 2 ] {\displaystyle \rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}=\exp \left[-\beta _{i}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert ^{2}\right]} . The Gaussian basis functions are local to the center vector in the sense that lim | | x | | → ∞ ρ ( ‖ x − c i ‖ ) = 0 {\displaystyle \lim _{||x||\to \infty }\rho (\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert )=0} i.e. changing parameters of one neuron has only a small effect for input values that are far away from the center of that neuron. Given certain mild conditions on the shape of the activation function, RBF networks are universal approximators on a compact subset of R n {\displaystyle \mathbb {R} ^{n}} . This means that an RBF network with enough hidden neurons can approximate any continuous function on a closed, bounded set with arbitrary precision. The parameters a i {\displaystyle a_{i}} , c i {\displaystyle \mathbf {c} _{i}} , and β i {\displaystyle \beta _{i}} are determined in a manner that optimizes the fit between φ {\displaystyle \varphi } and the data. === Normalization === ==== Normalized architecture ==== In addition to the above unnormalized architecture, RBF networks can be normalized. In this case the mapping is φ ( x ) = d e f ∑ i = 1 N a i ρ ( ‖ x − c i ‖ ) ∑ i = 1 N ρ ( ‖ x − c i ‖ ) = ∑ i = 1 N a i u ( ‖ x − c i ‖ ) {\displaystyle \varphi (\mathbf {x} )\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\sum _{i=1}^{N}a_{i}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}{\sum _{i=1}^{N}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}}=\sum _{i=1}^{N}a_{i}u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}} where u ( ‖ x − c i ‖ ) = d e f ρ ( ‖ x − c i ‖ ) ∑ j = 1 N ρ ( ‖ x − c j ‖ ) {\displaystyle u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}{\sum _{j=1}^{N}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{j}\right\Vert {\big )}}}} is known as a normalized radial basis function. ==== Theoretical motivation for normalization ==== There is theoretical justification for this architecture in the case of stochastic data flow. Assume a stochastic kernel approximation for the joint probability density P ( x ∧ y ) = 1 N ∑ i = 1 N ρ ( ‖ x − c i ‖ ) σ ( | y − e i | ) {\displaystyle P\left(\mathbf {x} \land y\right)={1 \over N}\sum _{i=1}^{N}\,\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}\,\sigma {\big (}\left\vert y-e_{i}\right\vert {\big )}} where the weights c i {\displaystyle \mathbf {c} _{i}} and e i {\displaystyle e_{i}} are exemplars from the data and we require the kernels to be normalized ∫ ρ ( ‖ x − c i ‖ ) d n x = 1 {\displaystyle \int \rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}\,d^{n}\mathbf {x} =1} and ∫ σ ( | y − e i | ) d y = 1 {\displaystyle \int \sigma {\big (}\left\vert y-e_{i}\right\vert {\big )}\,dy=1} . The probability densities in the input and output spaces are P ( x ) = ∫ P ( x ∧ y ) d y = 1 N ∑ i = 1 N ρ ( ‖ x − c i ‖ ) {\displaystyle P\left(\mathbf {x} \right)=\int P\left(\mathbf {x} \land y\right)\,dy={1 \over N}\sum _{i=1}^{N}\,\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}} and The expectation of y given an input x {\displaystyle \mathbf {x} } is φ ( x ) = d e f E ( y ∣ x ) = ∫ y P ( y ∣ x ) d y {\displaystyle \varphi \left(\mathbf {x} \right)\ {\stackrel {\mathrm {def} }{=}}\ E\left(y\mid \mathbf {x} \right)=\int y\,P\left(y\mid \mathbf {x} \right)dy} where P ( y ∣ x ) {\displaystyle P\left(y\mid \mathbf {x} \right)} is the conditional probability of y given x {\displaystyle \mathbf {x} } . The conditional probability is related to the joint probability through Bayes' theorem P ( y ∣ x ) = P ( x ∧ y ) P ( x ) {\displaystyle P\left(y\mid \mathbf {x} \right)={\frac {P\left(\mathbf {x} \land y\right)}{P\left(\mathbf {x} \right)}}} which yields φ ( x ) = ∫ y P ( x ∧ y ) P ( x ) d y {\displaystyle \varphi \left(\mathbf {x} \right)=\int y\,{\frac {P\left(\mathbf {x} \land y\right)}{P\left(\mathbf {x} \right)}}\,dy} . This becomes φ ( x ) = ∑ i = 1 N e i ρ ( ‖ x − c i ‖ ) ∑ i = 1 N ρ ( ‖ x − c i ‖ ) = ∑ i = 1 N e i u ( ‖ x − c i ‖ ) {\displaystyle \varphi \left(\mathbf {x} \right)={\frac {\sum _{i=1}^{N}e_{i}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}{\sum _{i=1}^{N}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}}=\sum _{i=1}^{N}e_{i}u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}} when the integrations are performed. === Local linear models === It is sometimes convenient to expand the architecture to include local linear models. In that case the architectures become, to first order, φ ( x ) = ∑ i = 1 N ( a i + b i ⋅ ( x − c i ) ) ρ ( ‖ x − c i ‖ ) {\displaystyle \varphi \left(\mathbf {x} \right)=\sum _{i=1}^{N}\left(a_{i}+\mathbf {b} _{i}\cdot \left(\mathbf {x} -\mathbf {c} _{i}\right)\right)\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}} and φ ( x ) = ∑ i = 1 N ( a i + b i ⋅ ( x − c i ) ) u ( ‖ x − c i ‖ ) {\displaystyle \varphi \left(\mathbf {x} \right)=\sum _{i=1}^{N}\left(a_{i}+\mathbf {b} _{i}\cdot \left(\mathbf {x} -\mathbf {c} _{i}\right)\right)u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}} in the unnormalized and normalized cases, respectively. Here b i {\displaystyle \mathbf {b} _{i}} are weights to be determined. Higher order linear terms are also possible. This result can be written φ ( x ) = ∑ i = 1 2 N ∑ j = 1 n e i j v i j ( x − c i ) {\displaystyle \varphi \left(\mathbf {x} \right)=\sum _{i=1}^{2N}\sum _{j=1}^{n}e_{ij}v_{ij}{\big (}\mathbf {x} -\mathbf {c} _{i}{\big )}} where e i j = { a i , if i ∈ [ 1 , N ] b i j , if i ∈ [ N + 1 , 2 N ] {\displaystyle e_{ij}={\begin{cases}a_{i},&{\mbox{if }}i\in [1,N]\\b_{ij},&{\mbox{if }}i\in [N+1,2N]\end{cases}}} and v i j ( x − c i ) = d e f { δ i j ρ ( ‖ x − c i ‖ ) , if i ∈ [ 1 , N ] ( x i j − c i j ) ρ ( ‖ x − c i ‖ ) , if i ∈ [ N + 1 , 2 N ] {\displaystyle v_{ij}{\big (}\mathbf {x} -\mathbf {c} _{i}{\big )}\ {\stackrel {\mathrm {def} }{=}}\ {\begin{cases}\delta _{ij}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )},&{\mbox{if }}i\in [1,N]\\\left(x_{ij}-c_{ij}\right)\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )},&{\mbox{if }}i\in [N+1,2N]\end{cases}}} in the unnormalized case and in the normalized case. Here δ i j {\displaystyle \delta _{ij}} is a Kronecker delta function defined as δ i j = { 1 , if i = j 0 , if i ≠ j {\displaystyle \delta _{ij}={\begin{cases}1,&{\mbox{if }}i=j\\0,&{\mbox{if }}i\neq j\end{cases}}} . == Training == RBF networks are typically trained from pairs of input and target values x ( t ) , y ( t ) {\displaystyle \mathbf {x} (t),y(t)} , t = 1 , … , T {\displaystyle t=1,\dots ,T} by a two-step algorithm. In the first step, the center vectors c i {\displaystyle \mathbf {c} _{i}} of the RBF functions in the hidden layer
Three-factor learning
In neuroscience and machine learning, three-factor learning is the combination of Hebbian plasticity with a third modulatory factor to stabilise and enhance synaptic learning. This third factor can represent various signals such as reward, punishment, error, surprise, or novelty, often implemented through neuromodulators. == Description == Three-factor learning introduces the concept of eligibility traces, which flag synapses for potential modification pending the arrival of the third factor, and helps temporal credit assignement by bridging the gap between rapid neuronal firing and slower behavioral timescales, from which learning can be done. Biological basis for Three-factor learning rules have been supported by experimental evidence. This approach addresses the instability of classical Hebbian learning by minimizing autocorrelation and maximizing cross-correlation between inputs.
Radial basis function
In mathematics a radial basis function (RBF) is a real-valued function φ {\textstyle \varphi } whose value depends only on the distance between the input and some fixed point, either the origin, so that φ ( x ) = φ ^ ( ‖ x ‖ ) {\textstyle \varphi (\mathbf {x} )={\hat {\varphi }}(\left\|\mathbf {x} \right\|)} , or some other fixed point c {\textstyle \mathbf {c} } , called a center, so that φ ( x ) = φ ^ ( ‖ x − c ‖ ) {\textstyle \varphi (\mathbf {x} )={\hat {\varphi }}(\left\|\mathbf {x} -\mathbf {c} \right\|)} . Any function φ {\textstyle \varphi } that satisfies the property φ ( x ) = φ ^ ( ‖ x ‖ ) {\textstyle \varphi (\mathbf {x} )={\hat {\varphi }}(\left\|\mathbf {x} \right\|)} is a radial function. The distance is usually Euclidean distance, although other metrics are sometimes used. They are often used as a collection { φ k } k {\displaystyle \{\varphi _{k}\}_{k}} which forms a basis for some function space of interest, hence the name. Sums of radial basis functions are typically used to approximate given functions. This approximation process can also be interpreted as a simple kind of neural network; this was the context in which they were originally applied to machine learning, in work by David Broomhead and David Lowe in 1988, which stemmed from Michael J. D. Powell's seminal research from 1977. RBFs are also used as a kernel in support vector classification. The technique has proven effective and flexible enough that radial basis functions are now applied in a variety of engineering applications. == Definition == A radial function is a function φ : [ 0 , ∞ ) → R {\textstyle \varphi :[0,\infty )\to \mathbb {R} } . When paired with a norm ‖ ⋅ ‖ : V → [ 0 , ∞ ) {\textstyle \|\cdot \|:V\to [0,\infty )} on a vector space, a function of the form φ c = φ ( ‖ x − c ‖ ) {\textstyle \varphi _{\mathbf {c} }=\varphi (\|\mathbf {x} -\mathbf {c} \|)} is said to be a radial kernel centered at c ∈ V {\textstyle \mathbf {c} \in V} . A radial function and the associated radial kernels are said to be radial basis functions if, for any finite set of nodes { x k } k = 1 n ⊆ V {\displaystyle \{\mathbf {x} _{k}\}_{k=1}^{n}\subseteq V} , all of the following conditions are true: === Examples === Commonly used types of radial basis functions include (writing r = ‖ x − x i ‖ {\textstyle r=\left\|\mathbf {x} -\mathbf {x} _{i}\right\|} and using ε {\textstyle \varepsilon } to indicate a shape parameter that can be used to scale the input of the radial kernel): == Approximation == Radial basis functions are typically used to build up function approximations of the form where the approximating function y ( x ) {\textstyle y(\mathbf {x} )} is represented as a sum of N {\displaystyle N} radial basis functions, each associated with a different center x i {\textstyle \mathbf {x} _{i}} , and weighted by an appropriate coefficient w i . {\textstyle w_{i}.} The weights w i {\textstyle w_{i}} can be estimated using the matrix methods of linear least squares, because the approximating function is linear in the weights w i {\textstyle w_{i}} . Approximation schemes of this kind have been particularly used in time series prediction and control of nonlinear systems exhibiting sufficiently simple chaotic behaviour and 3D reconstruction in computer graphics (for example, hierarchical RBF and Pose Space Deformation). == RBF Network == The sum can also be interpreted as a rather simple single-layer type of artificial neural network called a radial basis function network, with the radial basis functions taking on the role of the activation functions of the network. It can be shown that any continuous function on a compact interval can in principle be interpolated with arbitrary accuracy by a sum of this form, if a sufficiently large number N {\textstyle N} of radial basis functions is used. The approximant y ( x ) {\textstyle y(\mathbf {x} )} is differentiable with respect to the weights w i {\textstyle w_{i}} . The weights could thus be learned using any of the standard iterative methods for neural networks. Using radial basis functions in this manner yields a reasonable interpolation approach provided that the fitting set has been chosen such that it covers the entire range systematically (equidistant data points are ideal). However, without a polynomial term that is orthogonal to the radial basis functions, estimates outside the fitting set tend to perform poorly. == RBFs for PDEs == Radial basis functions are used to approximate functions and so can be used to discretize and numerically solve Partial Differential Equations (PDEs). This was first done in 1990 by E. J. Kansa who developed the first RBF based numerical method. It is called the Kansa method and was used to solve the elliptic Poisson equation and the linear advection-diffusion equation. The function values at points x {\displaystyle \mathbf {x} } in the domain are approximated by the linear combination of RBFs: The derivatives are approximated as such: where N {\displaystyle N} are the number of points in the discretized domain, d {\displaystyle d} the dimension of the domain and λ {\displaystyle \lambda } the scalar coefficients that are unchanged by the differential operator. Different numerical methods based on Radial Basis Functions were developed thereafter. Some methods are the RBF-FD method, the RBF-QR method and the RBF-PUM method.