AI Ease Headshot Generator

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BabyCenter

BabyCenter is an online media company based in San Francisco, New York City, Chicago, and Los Angeles that provides information on conception, pregnancy, birth, and early childhood development for parents and expecting parents. BabyCenter operates 8 country and region specific properties including websites, apps, emails, print publications, and an online community where parents can connect on a variety of topics. The visitors of website and the users of the app can sign up for free weekly email newsletters that guide them through pregnancy and their child's development. In addition to publishing detailed, medically reviewed information about pregnancy and parenting, BabyCenter, under its Mission Motherhood initiative, ran numerous social programs and has participated in public health initiatives in partnership with hospitals, healthcare agencies, nonprofits, NGOs, and government agencies to provide pregnancy and parenting advice. It also annually publishes the most popular baby names. BabyCenter LLC is part of the Everyday Health Group, a division of Ziff Davis. == History == BabyCenter was founded in October 1997 by Stanford University MBA graduates Matt Glickman and Mark Selcow, who recognized a need for information about pregnancy and parenting on the internet. BabyCenter was initially funded through $13.5 million in startup capital funding from venture capital firms, including Bessemer Venture Partners, Intel, and Trinity Ventures. The funds were used to open the BabyCenter Store in October 1998. In the early years of its operation, BabyCenter offered multiple resources and services for parents, including a website that provided medically reviewed information and guidance to new and expectant parents on such topics as fertility, labor, and childcare; a weekly email for pregnant women tailored to their week of pregnancy (based on their pregnancy due date); and community groups and chat rooms for pregnant couples and parents to discuss pregnancy and child-rearing strategies. The site grew quickly, and by early 1999 had 175 employees and an annual revenue of $35 million. In April of that year, the two founders sold BabyCenter to another website, eToys.com, for $190 million in stock. Twenty-three months later, in 2001, shortly before declaring bankruptcy, eToys sold the site to Johnson & Johnson for $10 million. During the eToys ownership, BabyCenter launched its first international E-commerce site in the UK during the spring of 2000. Starting in 2005, BabyCenter launched an expansion plan, extending its global network to Australia, Canada and other countries, staffing each outpost with local editors. In 2007, BabyCenter debuted a Mandarin-language site in China, initiated operations in India, launched a Spanish language website, and introduced its first mobile site. BabyCenter released My Pregnancy Today, its first mobile app, to Apple's App Store in August 2010 and to the Android market in April 2011. The app provided daily information, nutrition tips, advice relevant to the user's week of pregnancy, and 3-D animated videos showcasing a baby's development in utero. The My Pregnancy app was joined by a My Baby Today app in October 2011. In 2015, BabyCenter released Mom Feed, its first mobile app for parents of toddlers and older children (ages 1 to 8). Mom Feed offered personalized, stage-based information as well as content from the BabyCenter Community and Blog in a real-time stream. In 2016, BabyCenter launched its web-based Baby Names Finder. In 2018, Mom Feed was discontinued and BabyCenter replaced that experience with a separate Child Health content area on its website. Also in 2018, BabyCenter launched its mobile baby name generator, the Baby Names app, which, like the web-based Baby Names Finder, leverages data from hundreds of thousands of parents that culminates in its annual most popular Baby Names Report. In 2019, Johnson & Johnson sold Baby Center to Everyday Health Group, a division of New York-based parent company of Ziff Davis, Inc. Neither side disclosed terms of the deal. == Popular research == BabyCenter's most popular baby names is released annually and often cited by the media. In March 2024, BabyCenter did a review of the app Temu and said that the website has found products that have been recalled, could be counterfeit or circumvent U.S. safety standards and features that are important in preventing issues like choking. In 2025, BabyCenter released a report about the cost of raising a newborn baby in the first year. == Content and products == === Websites === BabyCenter has 8 country and region-specific websites around the world, including sites for the United States, Canada, Australia, Brazil, India, Germany, the United Kingdom, and Latin America. Users can find parenting and pregnancy advice in seven languages: English, Spanish, Portuguese, Arabic, French, German, and Hindi BabyCenter content for each country- or region-specific site is written by an editorial team based in that country or region. Medical and health content for each site is reviewed by a medical advisory board based there and adheres to that country or region's medical standards. For example, the U.S. site works with and follows the recommendations of such U.S. medical authorities as the American Academy of Pediatrics, the American Congress of Obstetrics & Gynecology and the Society for Maternal-Fetal Medicine. BabyCenter regularly conducts research and provides thought leadership on pregnancy and parenting topics, popularly cited by major media outlets including The Wall Street Journal, Forbes, The Washington Post, BuzzFeed, Insider, MarketWatch, Axios. === Community, blogs and social === From its earliest days, BabyCenter has had a community area that allows people to join a group of parents with children born in the same month, known as a Birth Club. BabyCenter launched a blog called Momformation in 2007. Eventually, the name was changed to BabyCenter Blog. In April 2021, the BabyCenter Community was identified in a research article within the journal PLOS Computational Biology as facilitating "unobstructed communication" between parents, which avoids the "strong echo chamber phenomena" that can foster and perpetuate vaccine misinformation. === My Pregnancy and Baby Today App === The app is available in six languages, although not all features are supported for every market. Initially the apps only featured pregnancy articles that could be found on the BabyCenter website, but over the years the feature set has expanded to include a growing list of app-specific tools such as weekly fetal development information, a kick tracker, a birth plan worksheet, a contraction timer, a baby growth tracker, a photo journal for pregnant women to record their pregnancy bellies, and a photo journal for documenting a baby's first year. === Mission Motherhood™ === BabyCenter was a cofounder of the Mobile Alliance for Maternal Action (MAMA), a public-private partnership between USAID, Johnson & Johnson, the UN Foundation, and BabyCenter from 2011 to-to 2015. The MAMA program sparked the creation of MomConnect, an initiative of the South African Department of Health for which BabyCenter developed SMS messages with health information about pregnancy and a child's first year of life. BabyCenter helped develop similar messages for mMitra, a voice messaging program in India. A research article in the Maternal and Child Health Journal stated the mMitra program offered strong evidence "that tailored mobile phone voice messages can improve key infant care knowledge and practices that lead to improved infant health outcomes in low-resource settings. BabyCenter's Mission Motherhood Messages were available to qualifying organizations on the BabyCenter website. BabyCenter contributed websites for Free Basics. These websites featured age and stage-based pregnancy and baby articles targeted to low-income, lower-education women who would not otherwise have access to health information. Content developed for this program was also used to support a UNICEF SMS program during the 2016 Zika outbreak. == Awards and recognition == In 1998, BabyCenter won a Webby Award for Best Home Site. Since then, it has been nominated for a Webby Award 19 times and won either a Webby or a People's Choice Webby Award 12 times – including a People's Voice win in 2021 for Lifestyle websites and mobile sites. In 2002, it won Service Journalism award from Online Journalism Awards (OJA). In 2015, BabyCenter won five Digital Health Awards for content about autism in children. In 2016, BabyCenter won seven Digital Health Awards: four for videos about the aches and pains of pregnancy, baby sleep, and the walking milestone in child development; two for articles about baby sleep training and sleep apnea in babies; and one for the BabyCenter mobile app My Pregnancy & Baby Today. In 2021, Forbes Health chose My Pregnancy & Baby Today as the best pregnancy app of 2021, and Women's Health identified it
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Cellular evolutionary algorithm

A cellular evolutionary algorithm (cEA) is a kind of evolutionary algorithm (EA) in which individuals cannot mate arbitrarily, but every one interacts with its closer neighbors on which a basic EA is applied (selection, variation, replacement). The cellular model simulates natural evolution from the point of view of the individual, which encodes a tentative optimization, learning, or search problem solution. The essential idea of this model is to provide the EA population with a special structure defined as a connected graph, in which each vertex is an individual who communicates with his nearest neighbors. Particularly, individuals are conceptually set in a toroidal mesh, and are only allowed to recombine with close individuals. This leads to a kind of locality known as "isolation by distance". The set of potential mates of an individual is called its "neighborhood". It is known that, in this kind of algorithm, similar individuals tend to cluster creating niches, and these groups operate as if they were separate sub-populations (islands). There is no clear borderline between adjacent groups, and close niches could be easily colonized by competitive niches and potentially merge solution contents during the process. Simultaneously, farther niches can be affected more slowly. == Introduction == A cellular evolutionary algorithm (cEA) usually evolves a structured bidimensional grid of individuals, although other topologies are also possible. In this grid, clusters of similar individuals are naturally created during evolution, promoting exploration in their boundaries, while exploitation is mainly performed by direct competition and merging inside them. The grid is usually 2D toroidal structure, although the number of dimensions can be easily extended (to 3D) or reduced (to 1D, e.g. a ring). The neighborhood of a particular point of the grid (where an individual is placed) is defined in terms of the Manhattan distance from it to others in the population. Each point of the grid has a neighborhood that overlaps the neighborhoods of nearby individuals. In the basic algorithm, all the neighborhoods have the same size and identical shapes. The two most commonly used neighborhoods are L5, also called the Von Neumann or NEWS (North, East, West and South) neighborhood, and C9, also known as the Moore neighborhood. Here, L stands for "linear" while C stands for "compact". In cEAs, the individuals can only interact with their neighbors in the reproductive cycle where the variation operators are applied. This reproductive cycle is executed inside the neighborhood of each individual and, generally, consists in selecting two parents among its neighbors according to a certain criterion, applying the variation operators to them (recombination and mutation for example), and replacing the considered individual by the recently created offspring following a given criterion, for instance, replace if the offspring represents a better solution than the considered individual. == Synchronous versus asynchronous == In a regular synchronous cEA, the algorithm proceeds from the very first top left individual to the right and then to the several rows by using the information in the population to create a new temporary population. After finishing with the bottom-right last individual the temporary population is full with the newly computed individuals, and the replacement step starts. In it, the old population is completely and synchronously replaced with the newly computed one according to some criterion. Usually, the replacement keeps the best individual in the same position of both populations, that is, elitism is used. According to the update policy of the population used, an asynchronous cEA may also be defined and is a well-known issue in cellular automata. In asynchronous cEAs the order in which the individuals in the grid are update changes depending on the choice of criterion: line sweep, fixed random sweep, new random sweep, and uniform choice. All four proceed using the newly computed individual (or the original if better) for the computations of its neighbors. The overlap of the neighborhoods provides an implicit mechanism of solution migration to the cEA. Since the best solutions spread smoothly through the whole population, genetic diversity in the population is preserved longer than in non structured EAs. This soft dispersion of the best solutions through the population is one of the main issues of the good tradeoff between exploration and exploitation that cEAs perform during the search. This tradeoff can be tuned (and by extension the genetic diversity level along the evolution) by modifying (for instance) the size of the neighborhood used, as the overlap degree between the neighborhoods grows according to the size of the neighborhood. A cEA can be seen as a cellular automaton (CA) with probabilistic rewritable rules, where the alphabet of the CA is equivalent to the potential number of solutions of the problem. Hence, knowledge from research in CAs can be applied to cEAs. == Parallelism == Cellular EAs are very amenable to parallelism, thus usually found in the literature of parallel metaheuristics. In particular, fine grain parallelism can be used to assign independent threads of execution to every individual, thus allowing the whole cEA to run on a concurrent or actually parallel hardware platform. In this way, large time reductions can be obtained when running cEAs on FPGAs or GPUs. However, it is important to stress that cEAs are a model of search, in many senses different from traditional EAs. Also, they can be run in sequential and parallel platforms, reinforcing the fact that the model and the implementation are two different concepts. See here for a complete description on the fundamentals for the understanding, design, and application of cEAs.
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VIGRA

VIGRA is the abbreviation for "Vision with Generic Algorithms". It is a free open-source computer vision library which focuses on customizable algorithms and data structures. VIGRA component can be easily adapted to specific needs of target application without compromising execution speed, by using template techniques similar to those in the C++ Standard Template Library. == Features == VIGRA is cross-platform, with working builds on Microsoft Windows, Mac OS X, Linux, and OpenBSD. Since version 1.7.1, VIGRA provides Python bindings based on numpy framework. == History == VIGRA was originally designed and implemented by scientists at University of Hamburg faculty of computer science; its core maintainers are now working at Heidelberg Collaboratory for Image Processing (HCI) University of Heidelberg. In the meantime, many developers have contributed to the project. == Application == CellCognition and ilastik uses VIGRA computer vision library. OpenOffice.org uses VIGRA as part of its headless software rendering backend; LibreOffice does so until version 5.2.
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Homogeneity blockmodeling

In mathematics applied to analysis of social structures, homogeneity blockmodeling is an approach in blockmodeling, which is best suited for a preliminary or main approach to valued networks, when a prior knowledge about these networks is not available. This is because homogeneity blockmodeling emphasizes the similarity of link (tie) strengths within the blocks over the pattern of links. In this approach, tie (link) values (or statistical data computed on them) are assumed to be equal (homogenous) within blocks. This approach to the generalized blockmodeling of valued networks was first proposed by Aleš Žiberna in 2007 with the basic idea, "that the inconsistency of an empirical block with its ideal block can be measured by within block variability of appropriate values". The newly–formed ideal blocks, which are appropriate for blockmodeling of valued networks, are then presented together with the definitions of their block inconsistencies. Similar approach to the homogeneity blockmodeling, dealing with direct approach for structural equivalence, was previously suggested by Stephen P. Borgatti and Martin G. Everett (1992).
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Application Lifecycle Framework

The Application Lifecycle Framework (ALF) was a project by the Eclipse Foundation that aimed to create a standardized, open-source system to allow different application lifecycle management (ALM) tools to work together more easily. The goal was to provide common protocols and integration services that would let software development tools from different vendors communicate and share data. However, the project failed to gain sufficient support from major industry players and was terminated in 2008.
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Types of artificial neural networks

Types of neural networks (NN) include a family of techniques. The simplest types have static components, including number of units, number of layers, unit weights and topology. Dynamic NNs evolve via learning. Some types allow/require learning to be "supervised" by the operator, while others operate independently. Some types operate purely in hardware, while others are purely software and run on general purpose computers. The main types are: Transformers: these use attention to analyze every token in the input stream against every other token in the stream. That technique has enabled neural networks to reach the general public via chatbots, code generators and many other forms. Convolutional neural networks (CNN): a FNN that uses kernels and regularization to evade problems in prior generations of NNs. They are typically used to analyze visual and other two-dimensional data. Generative adversarial networks set networks (of varying structure) against each other, each trying to push the other(s) to produce better results such as winning a game or to deceive the opponent about the authenticity of an input. == Feedforward == In feedforward neural networks the information moves from the input to output directly in every layer. There can be hidden layers with or without cycles/loops to sequence inputs. Feedforward networks can be constructed with various types of units, such as binary McCulloch–Pitts neurons, the simplest of which is the perceptron. Continuous neurons, frequently with sigmoidal activation, are used in the context of backpropagation. == Group method of data handling == The Group Method of Data Handling (GMDH) features fully automatic structural and parametric model optimization. The node activation functions are Kolmogorov–Gabor polynomials that permit additions and multiplications. It uses a deep multilayer perceptron with eight layers. It is a supervised learning network that grows layer by layer, where each layer is trained by regression analysis. Useless items are detected using a validation set, and pruned through regularization. The size and depth of the resulting network depends on the task. == Autoencoder == An autoencoder, autoassociator or Diabolo network is similar to the multilayer perceptron (MLP) – with an input layer, an output layer and one or more hidden layers connecting them. However, the output layer has the same number of units as the input layer. Its purpose is to reconstruct its own inputs (instead of emitting a target value). Therefore, autoencoders are unsupervised learning models. An autoencoder is used for unsupervised learning of efficient codings, typically for the purpose of dimensionality reduction and for learning generative models of data. == Probabilistic == A probabilistic neural network (PNN) is a four-layer feedforward neural network. The layers are Input, hidden pattern, hidden summation, and output. In the PNN algorithm, the parent probability distribution function (PDF) of each class is approximated by a Parzen window and a non-parametric function. Then, using PDF of each class, the class probability of a new input is estimated and Bayes’ rule is employed to allocate it to the class with the highest posterior probability. It was derived from the Bayesian network and a statistical algorithm called Kernel Fisher discriminant analysis. It is used for classification and pattern recognition. == Time delay == A time delay neural network (TDNN) is a feedforward architecture for sequential data that recognizes features independent of sequence position. In order to achieve time-shift invariance, delays are added to the input so that multiple data points (points in time) are analyzed together. It usually forms part of a larger pattern recognition system. It has been implemented using a perceptron network whose connection weights were trained with back propagation (supervised learning). == Convolutional == A convolutional neural network (CNN, or ConvNet or shift invariant or space invariant) is a class of deep network, composed of one or more convolutional layers with fully connected layers (matching those in typical ANNs) on top. It uses tied weights and pooling layers. In particular, max-pooling. It is often structured via Fukushima's convolutional architecture. They are variations of multilayer perceptrons that use minimal preprocessing. This architecture allows CNNs to take advantage of the 2D structure of input data. Its unit connectivity pattern is inspired by the organization of the visual cortex. Units respond to stimuli in a restricted region of space known as the receptive field. Receptive fields partially overlap, over-covering the entire visual field. Unit response can be approximated mathematically by a convolution operation. CNNs are suitable for processing visual and other two-dimensional data. They have shown superior results in both image and speech applications. They can be trained with standard backpropagation. CNNs are easier to train than other regular, deep, feed-forward neural networks and have many fewer parameters to estimate. Capsule Neural Networks (CapsNet) add structures called capsules to a CNN and reuse output from several capsules to form more stable (with respect to various perturbations) representations. Examples of applications in computer vision include DeepDream and robot navigation. They have wide applications in image and video recognition, recommender systems and natural language processing. == Deep stacking network == A deep stacking network (DSN) (deep convex network) is based on a hierarchy of blocks of simplified neural network modules. It was introduced in 2011 by Deng and Yu. It formulates the learning as a convex optimization problem with a closed-form solution, emphasizing the mechanism's similarity to stacked generalization. Each DSN block is a simple module that is easy to train by itself in a supervised fashion without backpropagation for the entire blocks. Each block consists of a simplified multi-layer perceptron (MLP) with a single hidden layer. The hidden layer h has logistic sigmoidal units, and the output layer has linear units. Connections between these layers are represented by weight matrix U; input-to-hidden-layer connections have weight matrix W. Target vectors t form the columns of matrix T, and the input data vectors x form the columns of matrix X. The matrix of hidden units is H = σ ( W T X ) {\displaystyle {\boldsymbol {H}}=\sigma ({\boldsymbol {W}}^{T}{\boldsymbol {X}})} . Modules are trained in order, so lower-layer weights W are known at each stage. The function performs the element-wise logistic sigmoid operation. Each block estimates the same final label class y, and its estimate is concatenated with original input X to form the expanded input for the next block. Thus, the input to the first block contains the original data only, while downstream blocks' input adds the output of preceding blocks. Then learning the upper-layer weight matrix U given other weights in the network can be formulated as a convex optimization problem: min U T f = ‖ U T H − T ‖ F 2 , {\displaystyle \min _{U^{T}}f=\|{\boldsymbol {U}}^{T}{\boldsymbol {H}}-{\boldsymbol {T}}\|_{F}^{2},} which has a closed-form solution. Unlike other deep architectures, such as DBNs, the goal is not to discover the transformed feature representation. The structure of the hierarchy of this kind of architecture makes parallel learning straightforward, as a batch-mode optimization problem. In purely discriminative tasks, DSNs outperform conventional DBNs. === Tensor deep stacking networks === This architecture is a DSN extension. It offers two important improvements: it uses higher-order information from covariance statistics, and it transforms the non-convex problem of a lower-layer to a convex sub-problem of an upper-layer. TDSNs use covariance statistics in a bilinear mapping from each of two distinct sets of hidden units in the same layer to predictions, via a third-order tensor. While parallelization and scalability are not considered seriously in conventional DNNs, all learning for DSNs and TDSNs is done in batch mode, to allow parallelization. Parallelization allows scaling the design to larger (deeper) architectures and data sets. The basic architecture is suitable for diverse tasks such as classification and regression. == Physics-informed == Such a neural network is designed for the numerical solution of mathematical equations, such as differential, integral, delay, fractional and others. As input parameters, PINN accepts variables (spatial, temporal, and others), transmits them through the network block. At the output, it produces an approximate solution and substitutes it into the mathematical model, considering the initial and boundary conditions. If the solution does not satisfy the required accuracy, one uses the backpropagation and rectify the solution. Besides PINN, other architectures have been developed to produce surrogate models for scientific comput
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Memtransistor

The memtransistor (a blend word from Memory Transfer Resistor) is an experimental multi-terminal passive electronic component that might be used in the construction of artificial neural networks. It is a combination of the memristor and transistor technology. This technology is different from the 1T-1R approach since the devices are merged into one single entity. Multiple memristors can be embedded with a single transistor, enabling it to more accurately model a neuron with its multiple synaptic connections. A neural network produced from these would provide hardware-based artificial intelligence with a good foundation. == Applications == These types of devices would allow for a synapse model that could realise a learning rule, by which the synaptic efficacy is altered by voltages applied to the terminals of the device. An example of such a learning rule is spike-timing-dependant-plasticty by which the weight of the synapse, in this case the conductivity, could be modulated based on the timing of pre and post synaptic spikes arriving at each terminal. The advantage of this approach over two terminal memristive devices is that read and write protocols have the possibility to occur simultaneously and distinctly.
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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
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DABUS

DABUS (Device for the Autonomous Bootstrapping of Unified Sentience) is an artificial intelligence (AI) system created by Stephen Thaler. It reportedly conceived of two novel products — a food container constructed using fractal geometry, which enables rapid reheating, and a flashing beacon for attracting attention in an emergency. The filing of patent applications designating DABUS as inventor has led to decisions by patent offices and courts on whether a patent can be granted for an invention reportedly made by an AI system. == History in different jurisdictions == === Australia === On 17 September 2019, Thaler filed an application to patent a "Food container and devices and methods for attracting enhanced attention," naming DABUS as the inventor. On 21 September 2020, IP Australia found that section 15(1) of the Patents Act 1990 (Cth) is inconsistent with an artificial intelligence machine being treated as an inventor, and Thaler's application had lapsed. Thaler sought judicial review, and on 30 July 2021, the Federal Court set aside IP Australia's decision and ordered IP Australia to reconsider the application. On 13 April 2022, the Full Court of the Federal Court set aside that decision, holding that only a natural person can be an inventor for the purposes of the Patents Act 1990 (Cth) and the Patents Regulations 1991 (Cth), and that such an inventor must be identified for any person to be entitled to a grant of a patent. On 11 November 2022, Thaler was refused special leave to appeal to the High Court. === European Patent Office === On 17 October 2018 and 7 November 2018, Thaler filed two European patent applications with the European Patent Office. The first claimed invention was a "Food Container" and the second was "Devices and Methods for Attracting Enhanced Attention." On 27 January 2020, the EPO rejected the applications on the grounds that the application listed an AI system named DABUS, and not a human, as the inventor, based on Article 81 and Rule 19(1) of the European Patent Convention (EPC). On 21 December 2021, the Board of Appeal of the EPO dismissed Thaler's appeal from the EPO's primary decision. The Board of Appeal confirmed that "under the EPC the designated inventor has to be a person with legal capacity. This is not merely an assumption on which the EPC was drafted. It is the ordinary meaning of the term inventor." === United Kingdom === Similar applications were filed by Thaler to the United Kingdom Intellectual Property Office on 17 October and 7 November 2018. The Office asked Thaler to file statements of inventorship and of right of grant to a patent (Patent Form 7) in respect of each invention within 16 months of the filing date. Thaler filed those forms naming DABUS as the inventor and explaining in some detail why he believed that machines should be regarded as inventors in the circumstances. His application was rejected on the grounds that: (1) naming a machine as inventor did not meet the requirements of the Patents Act 1977; and (2) the IPO was not satisfied as to the manner in which Thaler had acquired rights that would otherwise vest in the inventor. Thaler was not satisfied with the decision and asked for a hearing before an official known as the "hearing officer". By a decision dated 4 December 2019 the hearing officer rejected Thaler's appeal. Thaler appealed against the hearing officer's decision to the Patents Court (a specialist court within the Chancery Division of the High Court of England and Wales that determines patent disputes). On 21 September 2020, Mr Justice Marcus Smith upheld the decision of the hearing officer. On 21 September 2021, Thaler's further appeal to the Court of Appeal was dismissed by Arnold LJ and Laing LJ (Birss LJ dissenting). On 20 December 2023, the UK Supreme Court dismissed a further appeal by Thaler. In its judgment, the court held that an "inventor" under the Patents Act 1977 must be a natural person. === United States === The patent applications on the inventions were refused by the USPTO, which held that only natural persons can be named as inventors in a patent application. Thaler first fought this result by filing a complaint under the Administrative Procedure Act alleging that the decision was "arbitrary, capricious, an abuse of discretion and not in accordance with the law; unsupported by substantial evidence, and in excess of Defendants’ statutory authority." A month later on August 19, 2019, Thaler filed a petition with the USPTO as allowed in 37 C.F.R. § 1.181 stating that DABUS should be the inventor. The judge and Thaler agreed in this case that Thaler himself is unable to receive the patent on behalf of DABUS. In their August 5, 2022, Thaler decision, the US Court of Appeals for the Federal Circuit affirmed that only a natural person could be an inventor, which means that the AI that invents any other type of invention is not addressed by the "who" mentioned in the legislation. === New Zealand === On January 31, 2022, the Intellectual Property Office of New Zealand (IPONZ) decided that a patent application (776029) filed by Stephen Thaler was void, on the basis that no inventor was identified on the patent application. IPONZ determined that DABUS could not be "an actual devisor of the invention" as required by the Patents Act 2013, and that this must be a natural person as held by the previous patent offices above. The High Court of New Zealand confirmed the decision in 2023. === South Africa === On 24 June 2021, the South African Companies and Intellectual Property Commission (CIPC) accepted Dr Thaler's Patent Cooperation Treaty, for a patent in respect of inventions generated by DABUS. In July 2021, the CIPC released a notice of issuance for the patent. It is the first patent granted for an AI invention. === Switzerland === On June 26, 2025, the Swiss Federal Administrative Court ruled that artificial intelligence systems such as DABUS cannot be listed as inventors in patent applications. The court upheld the existing practice of the Swiss Federal Institute of Intellectual Property (IPI), which requires that only natural persons can be recognized as inventors under Swiss patent law. The case concerned a patent application, which sought to designate DABUS as the sole inventor of a food container designed with a fractal geometry to enhance heat distribution. The IPI had rejected the application, arguing that both the absence of a human inventor and the attribution of inventorship to an AI system were inadmissible. While the court dismissed Thaler's main request, it accepted a subsidiary request: if a human applicant recognizes and files a patent based on an AI-generated invention, that person may be considered the inventor. As a result, the application may proceed with Thaler listed as the inventor. The decision (B-2532/2024) can still be appealed to the Swiss Federal Supreme Court.
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Correspondence analysis

Correspondence analysis (CA) is a multivariate statistical technique proposed by Herman Otto Hartley (Hirschfeld) and later developed by Jean-Paul Benzécri. It is conceptually similar to principal component analysis, but applies to categorical rather than continuous data. In a manner similar to principal component analysis, it provides a means of displaying or summarising a set of data in two-dimensional graphical form. Its aim is to display in a biplot any structure hidden in the multivariate setting of the data table. As such it is a technique from the field of multivariate ordination. Since the variant of CA described here can be applied either with a focus on the rows or on the columns it should in fact be called simple (symmetric) correspondence analysis. It is traditionally applied to the contingency table of a pair of nominal variables where each cell contains either a count or a zero value. If more than two categorical variables are to be summarized, a variant called multiple correspondence analysis should be chosen instead. CA may also be applied to binary data given the presence/absence coding represents simplified count data i.e. a 1 describes a positive count and 0 stands for a count of zero. Depending on the scores used CA preserves the chi-square distance between either the rows or the columns of the table. Because CA is a descriptive technique, it can be applied to tables regardless of a significant chi-squared test. Although the χ 2 {\displaystyle \chi ^{2}} statistic used in inferential statistics and the chi-square distance are computationally related they should not be confused since the latter works as a multivariate statistical distance measure in CA while the χ 2 {\displaystyle \chi ^{2}} statistic is in fact a scalar not a metric. == Details == Like principal components analysis, correspondence analysis creates orthogonal components (or axes) and, for each item in a table i.e. for each row, a set of scores (sometimes called factor scores, see Factor analysis). Correspondence analysis is performed on the data table, conceived as matrix C of size m × n where m is the number of rows and n is the number of columns. In the following mathematical description of the method capital letters in italics refer to a matrix while letters in italics refer to vectors. Understanding the following computations requires knowledge of matrix algebra. === Preprocessing === Before proceeding to the central computational step of the algorithm, the values in matrix C have to be transformed. First compute a set of weights for the columns and the rows (sometimes called masses), where row and column weights are given by the row and column vectors, respectively: w m = 1 n C C 1 , w n = 1 n C 1 T C . {\displaystyle w_{m}={\frac {1}{n_{C}}}C\mathbf {1} ,\quad w_{n}={\frac {1}{n_{C}}}\mathbf {1} ^{T}C.} Here n C = ∑ i = 1 n ∑ j = 1 m C i j {\displaystyle n_{C}=\sum _{i=1}^{n}\sum _{j=1}^{m}C_{ij}} is the sum of all cell values in matrix C, or short the sum of C, and 1 {\displaystyle \mathbf {1} } is a column vector of ones with the appropriate dimension. Put in simple words, w m {\displaystyle w_{m}} is just a vector whose elements are the row sums of C divided by the sum of C, and w n {\displaystyle w_{n}} is a vector whose elements are the column sums of C divided by the sum of C. The weights are transformed into diagonal matrices W m = diag ⁡ ( 1 / w m ) {\displaystyle W_{m}=\operatorname {diag} (1/{\sqrt {w_{m}}})} and W n = diag ⁡ ( 1 / w n ) {\displaystyle W_{n}=\operatorname {diag} (1/{\sqrt {w_{n}}})} where the diagonal elements of W n {\displaystyle W_{n}} are 1 / w n {\displaystyle 1/{\sqrt {w_{n}}}} and those of W m {\displaystyle W_{m}} are 1 / w m {\displaystyle 1/{\sqrt {w_{m}}}} respectively i.e. the vector elements are the inverses of the square roots of the masses. The off-diagonal elements are all 0. Next, compute matrix P {\displaystyle P} by dividing C {\displaystyle C} by its sum P = 1 n C C . {\displaystyle P={\frac {1}{n_{C}}}C.} In simple words, Matrix P {\displaystyle P} is just the data matrix (contingency table or binary table) transformed into portions i.e. each cell value is just the cell portion of the sum of the whole table. Finally, compute matrix S {\displaystyle S} , sometimes called the matrix of standardized residuals, by matrix multiplication as S = W m ( P − w m w n ) W n {\displaystyle S=W_{m}(P-w_{m}w_{n})W_{n}} Note, the vectors w m {\displaystyle w_{m}} and w n {\displaystyle w_{n}} are combined in an outer product resulting in a matrix of the same dimensions as P {\displaystyle P} . In words the formula reads: matrix outer ⁡ ( w m , w n ) {\displaystyle \operatorname {outer} (w_{m},w_{n})} is subtracted from matrix P {\displaystyle P} and the resulting matrix is scaled (weighted) by the diagonal matrices W m {\displaystyle W_{m}} and W n {\displaystyle W_{n}} . Multiplying the resulting matrix by the diagonal matrices is equivalent to multiply the i-th row (or column) of it by the i-th element of the diagonal of W m {\displaystyle W_{m}} or W n {\displaystyle W_{n}} , respectively. === Interpretation of preprocessing === The vectors w m {\displaystyle w_{m}} and w n {\displaystyle w_{n}} are the row and column masses or the marginal probabilities for the rows and columns, respectively. Subtracting matrix outer ⁡ ( w m , w n ) {\displaystyle \operatorname {outer} (w_{m},w_{n})} from matrix P {\displaystyle P} is the matrix algebra version of double centering the data. Multiplying this difference by the diagonal weighting matrices results in a matrix containing weighted deviations from the origin of a vector space. This origin is defined by matrix outer ⁡ ( w m , w n ) {\displaystyle \operatorname {outer} (w_{m},w_{n})} . In fact matrix outer ⁡ ( w m , w n ) {\displaystyle \operatorname {outer} (w_{m},w_{n})} is identical with the matrix of expected frequencies in the chi-squared test. Therefore S {\displaystyle S} is computationally related to the independence model used in that test. But since CA is not an inferential method the term independence model is inappropriate here. === Orthogonal components === The table S {\displaystyle S} is then decomposed by a singular value decomposition as S = U Σ V ∗ {\displaystyle S=U\Sigma V^{}\,} where U {\displaystyle U} and V {\displaystyle V} are the left and right singular vectors of S {\displaystyle S} and Σ {\displaystyle \Sigma } is a square diagonal matrix with the singular values σ i {\displaystyle \sigma _{i}} of S {\displaystyle S} on the diagonal. Σ {\displaystyle \Sigma } is of dimension p ≤ ( min ( m , n ) − 1 ) {\displaystyle p\leq (\min(m,n)-1)} hence U {\displaystyle U} is of dimension m×p and V {\displaystyle V} is of n×p. As orthonormal vectors U {\displaystyle U} and V {\displaystyle V} fulfill U ∗ U = V ∗ V = I {\displaystyle U^{}U=V^{}V=I} . In other words, the multivariate information that is contained in C {\displaystyle C} as well as in S {\displaystyle S} is now distributed across two (coordinate) matrices U {\displaystyle U} and V {\displaystyle V} and a diagonal (scaling) matrix Σ {\displaystyle \Sigma } . The vector space defined by them has as number of dimensions p, that is the smaller of the two values, number of rows and number of columns, minus 1. === Inertia === While a principal component analysis may be said to decompose the (co)variance, and hence its measure of success is the amount of (co-)variance covered by the first few PCA axes - measured in eigenvalue -, a CA works with a weighted (co-)variance which is called inertia. The sum of the squared singular values is the total inertia I {\displaystyle \mathrm {I} } of the data table, computed as I = ∑ i = 1 p σ i 2 . {\displaystyle \mathrm {I} =\sum _{i=1}^{p}\sigma _{i}^{2}.} The total inertia I {\displaystyle \mathrm {I} } of the data table can also computed directly from S {\displaystyle S} as I = ∑ i = 1 n ∑ j = 1 m s i j 2 . {\displaystyle \mathrm {I} =\sum _{i=1}^{n}\sum _{j=1}^{m}s_{ij}^{2}.} The amount of inertia covered by the i-th set of singular vectors is ι i {\displaystyle \iota _{i}} , the principal inertia. The higher the portion of inertia covered by the first few singular vectors i.e. the larger the sum of the principal inertiae in comparison to the total inertia, the more successful a CA is. Therefore, all principal inertia values are expressed as portion ϵ i {\displaystyle \epsilon _{i}} of the total inertia ϵ i = σ i 2 / ∑ i = 1 p σ i 2 {\displaystyle \epsilon _{i}=\sigma _{i}^{2}/\sum _{i=1}^{p}\sigma _{i}^{2}} and are presented in the form of a scree plot. In fact a scree plot is just a bar plot of all principal inertia portions ϵ i {\displaystyle \epsilon _{i}} . === Coordinates === To transform the singular vectors to coordinates which preserve the chi-square distances between rows or columns an additional weighting step is necessary. The resulting coordinates are called principal coordinates in CA text books. If principal coordinates are used for
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Modes of variation

In statistics, modes of variation are a continuously indexed set of vectors or functions that are centered at a mean and are used to depict the variation in a population or sample. Typically, variation patterns in the data can be decomposed in descending order of eigenvalues with the directions represented by the corresponding eigenvectors or eigenfunctions. Modes of variation provide a visualization of this decomposition and an efficient description of variation around the mean. Both in principal component analysis (PCA) and in functional principal component analysis (FPCA), modes of variation play an important role in visualizing and describing the variation in the data contributed by each eigencomponent. In real-world applications, the eigencomponents and associated modes of variation aid to interpret complex data, especially in exploratory data analysis (EDA). == Formulation == Modes of variation are a natural extension of PCA and FPCA. === Modes of variation in PCA === If a random vector X = ( X 1 , X 2 , ⋯ , X p ) T {\displaystyle \mathbf {X} =(X_{1},X_{2},\cdots ,X_{p})^{T}} has the mean vector μ p {\displaystyle {\boldsymbol {\mu }}_{p}} , and the covariance matrix Σ p × p {\displaystyle \mathbf {\Sigma } _{p\times p}} with eigenvalues λ 1 ≥ λ 2 ≥ ⋯ ≥ λ p ≥ 0 {\displaystyle \lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{p}\geq 0} and corresponding orthonormal eigenvectors e 1 , e 2 , ⋯ , e p {\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\cdots ,\mathbf {e} _{p}} , by eigendecomposition of a real symmetric matrix, the covariance matrix Σ {\displaystyle \mathbf {\Sigma } } can be decomposed as Σ = Q Λ Q T , {\displaystyle \mathbf {\Sigma } =\mathbf {Q} \mathbf {\Lambda } \mathbf {Q} ^{T},} where Q {\displaystyle \mathbf {Q} } is an orthogonal matrix whose columns are the eigenvectors of Σ {\displaystyle \mathbf {\Sigma } } , and Λ {\displaystyle \mathbf {\Lambda } } is a diagonal matrix whose entries are the eigenvalues of Σ {\displaystyle \mathbf {\Sigma } } . By the Karhunen–Loève expansion for random vectors, one can express the centered random vector in the eigenbasis X − μ = ∑ k = 1 p ξ k e k , {\displaystyle \mathbf {X} -{\boldsymbol {\mu }}=\sum _{k=1}^{p}\xi _{k}\mathbf {e} _{k},} where ξ k = e k T ( X − μ ) {\displaystyle \xi _{k}=\mathbf {e} _{k}^{T}(\mathbf {X} -{\boldsymbol {\mu }})} is the principal component associated with the k {\displaystyle k} -th eigenvector e k {\displaystyle \mathbf {e} _{k}} , with the properties E ⁡ ( ξ k ) = 0 , Var ⁡ ( ξ k ) = λ k , {\displaystyle \operatorname {E} (\xi _{k})=0,\operatorname {Var} (\xi _{k})=\lambda _{k},} and E ⁡ ( ξ k ξ l ) = 0 for l ≠ k . {\displaystyle \operatorname {E} (\xi _{k}\xi _{l})=0\ {\text{for}}\ l\neq k.} Then the k {\displaystyle k} -th mode of variation of X {\displaystyle \mathbf {X} } is the set of vectors, indexed by α {\displaystyle \alpha } , m k , α = μ ± α λ k e k , α ∈ [ − A , A ] , {\displaystyle \mathbf {m} _{k,\alpha }={\boldsymbol {\mu }}\pm \alpha {\sqrt {\lambda _{k}}}\mathbf {e} _{k},\alpha \in [-A,A],} where A {\displaystyle A} is typically selected as 2 or 3 {\displaystyle 2\ {\text{or}}\ 3} . === Modes of variation in FPCA === For a square-integrable random function X ( t ) , t ∈ T ⊂ R p {\displaystyle X(t),t\in {\mathcal {T}}\subset R^{p}} , where typically p = 1 {\displaystyle p=1} and T {\displaystyle {\mathcal {T}}} is an interval, denote the mean function by μ ( t ) = E ⁡ ( X ( t ) ) {\displaystyle \mu (t)=\operatorname {E} (X(t))} , and the covariance function by G ( s , t ) = Cov ⁡ ( X ( s ) , X ( t ) ) = ∑ k = 1 ∞ λ k φ k ( s ) φ k ( t ) , {\displaystyle G(s,t)=\operatorname {Cov} (X(s),X(t))=\sum _{k=1}^{\infty }\lambda _{k}\varphi _{k}(s)\varphi _{k}(t),} where λ 1 ≥ λ 2 ≥ ⋯ ≥ 0 {\displaystyle \lambda _{1}\geq \lambda _{2}\geq \cdots \geq 0} are the eigenvalues and { φ 1 , φ 2 , ⋯ } {\displaystyle \{\varphi _{1},\varphi _{2},\cdots \}} are the orthonormal eigenfunctions of the linear Hilbert–Schmidt operator G : L 2 ( T ) → L 2 ( T ) , G ( f ) = ∫ T G ( s , t ) f ( s ) d s . {\displaystyle G:L^{2}({\mathcal {T}})\rightarrow L^{2}({\mathcal {T}}),\,G(f)=\int _{\mathcal {T}}G(s,t)f(s)ds.} By the Karhunen–Loève theorem, one can express the centered function in the eigenbasis, X ( t ) − μ ( t ) = ∑ k = 1 ∞ ξ k φ k ( t ) , {\displaystyle X(t)-\mu (t)=\sum _{k=1}^{\infty }\xi _{k}\varphi _{k}(t),} where ξ k = ∫ T ( X ( t ) − μ ( t ) ) φ k ( t ) d t {\displaystyle \xi _{k}=\int _{\mathcal {T}}(X(t)-\mu (t))\varphi _{k}(t)dt} is the k {\displaystyle k} -th principal component with the properties E ⁡ ( ξ k ) = 0 , Var ⁡ ( ξ k ) = λ k , {\displaystyle \operatorname {E} (\xi _{k})=0,\operatorname {Var} (\xi _{k})=\lambda _{k},} and E ⁡ ( ξ k ξ l ) = 0 for l ≠ k . {\displaystyle \operatorname {E} (\xi _{k}\xi _{l})=0{\text{ for }}l\neq k.} Then the k {\displaystyle k} -th mode of variation of X ( t ) {\displaystyle X(t)} is the set of functions, indexed by α {\displaystyle \alpha } , m k , α ( t ) = μ ( t ) ± α λ k φ k ( t ) , t ∈ T , α ∈ [ − A , A ] {\displaystyle m_{k,\alpha }(t)=\mu (t)\pm \alpha {\sqrt {\lambda _{k}}}\varphi _{k}(t),\ t\in {\mathcal {T}},\ \alpha \in [-A,A]} that are viewed simultaneously over the range of α {\displaystyle \alpha } , usually for A = 2 or 3 {\displaystyle A=2\ {\text{or}}\ 3} . == Estimation == The formulation above is derived from properties of the population. Estimation is needed in real-world applications. The key idea is to estimate mean and covariance. === Modes of variation in PCA === Suppose the data x 1 , x 2 , ⋯ , x n {\displaystyle \mathbf {x} _{1},\mathbf {x} _{2},\cdots ,\mathbf {x} _{n}} represent n {\displaystyle n} independent drawings from some p {\displaystyle p} -dimensional population X {\displaystyle \mathbf {X} } with mean vector μ {\displaystyle {\boldsymbol {\mu }}} and covariance matrix Σ {\displaystyle \mathbf {\Sigma } } . These data yield the sample mean vector x ¯ {\displaystyle {\overline {\mathbf {x} }}} , and the sample covariance matrix S {\displaystyle \mathbf {S} } with eigenvalue-eigenvector pairs ( λ ^ 1 , e ^ 1 ) , ( λ ^ 2 , e ^ 2 ) , ⋯ , ( λ ^ p , e ^ p ) {\displaystyle ({\hat {\lambda }}_{1},{\hat {\mathbf {e} }}_{1}),({\hat {\lambda }}_{2},{\hat {\mathbf {e} }}_{2}),\cdots ,({\hat {\lambda }}_{p},{\hat {\mathbf {e} }}_{p})} . Then the k {\displaystyle k} -th mode of variation of X {\displaystyle \mathbf {X} } can be estimated by m ^ k , α = x ¯ ± α λ ^ k e ^ k , α ∈ [ − A , A ] . {\displaystyle {\hat {\mathbf {m} }}_{k,\alpha }={\overline {\mathbf {x} }}\pm \alpha {\sqrt {{\hat {\lambda }}_{k}}}{\hat {\mathbf {e} }}_{k},\alpha \in [-A,A].} === Modes of variation in FPCA === Consider n {\displaystyle n} realizations X 1 ( t ) , X 2 ( t ) , ⋯ , X n ( t ) {\displaystyle X_{1}(t),X_{2}(t),\cdots ,X_{n}(t)} of a square-integrable random function X ( t ) , t ∈ T {\displaystyle X(t),t\in {\mathcal {T}}} with the mean function μ ( t ) = E ⁡ ( X ( t ) ) {\displaystyle \mu (t)=\operatorname {E} (X(t))} and the covariance function G ( s , t ) = Cov ⁡ ( X ( s ) , X ( t ) ) {\displaystyle G(s,t)=\operatorname {Cov} (X(s),X(t))} . Functional principal component analysis provides methods for the estimation of μ ( t ) {\displaystyle \mu (t)} and G ( s , t ) {\displaystyle G(s,t)} in detail, often involving point wise estimate and interpolation. Substituting estimates for the unknown quantities, the k {\displaystyle k} -th mode of variation of X ( t ) {\displaystyle X(t)} can be estimated by m ^ k , α ( t ) = μ ^ ( t ) ± α λ ^ k φ ^ k ( t ) , t ∈ T , α ∈ [ − A , A ] . {\displaystyle {\hat {m}}_{k,\alpha }(t)={\hat {\mu }}(t)\pm \alpha {\sqrt {{\hat {\lambda }}_{k}}}{\hat {\varphi }}_{k}(t),t\in {\mathcal {T}},\alpha \in [-A,A].} == Applications == Modes of variation are useful to visualize and describe the variation patterns in the data sorted by the eigenvalues. In real-world applications, modes of variation associated with eigencomponents allow to interpret complex data, such as the evolution of function traits and other infinite-dimensional data. To illustrate how modes of variation work in practice, two examples are shown in the graphs to the right, which display the first two modes of variation. The solid curve represents the sample mean function. The dashed, dot-dashed, and dotted curves correspond to modes of variation with α = ± 1 , ± 2 , {\displaystyle \alpha =\pm 1,\pm 2,} and ± 3 {\displaystyle \pm 3} , respectively. The first graph displays the first two modes of variation of female mortality data from 41 countries in 2003. The object of interest is log hazard function between ages 0 and 100 years. The first mode of variation suggests that the variation of female mortality is smaller for ages around 0 or 100, and larger for ages around 25. An appropriate and intuitive interpretation is that mortality around 25 is driven by accidental death, while around 0 or 100, mortality is related to congenital disease or natural death. Compared to female mortality
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Pruning (artificial neural network)

In deep learning, pruning is the practice of removing parameters from an existing artificial neural network. The goal of this process is to reduce the size (parameter count) of the neural network (and therefore the computational resources required to run it) whilst maintaining accuracy. This can be compared to the biological process of synaptic pruning which takes place in mammalian brains during development. == Node (neuron) pruning == A basic algorithm for pruning is as follows: Evaluate the importance of each neuron. Rank the neurons according to their importance (assuming there is a clearly defined measure for "importance"). Remove the least important neuron. Check a termination condition (to be determined by the user) to see whether to continue pruning. == Edge (weight) pruning == Most work on neural network pruning does not remove full neurons or layers (structured pruning). Instead, it focuses on removing the most insignificant weights (unstructured pruning), namely, setting their values to zero. This can either be done globally by comparing weights from all layers in the network or locally by comparing weights in each layer separately. Different metrics can be used to measure the importance of each weight. Weight magnitude as well as combinations of weight and gradient information are commonly used metrics. Early work suggested also to change the values of non-pruned weights. == When to prune the neural network? == Pruning can be applied at three different stages: before training, during training, or after training. When pruning is performed during or after training, additional fine-tuning epochs are typically required. Each approach involves different trade-offs between accuracy and computational cost.
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Anna Becker

Anna Becker is an Israeli researcher known in the field of artificial intelligence and computer science within the financial field. == Early life and education == Becker was born in Russia and immigrated to Israel at 16 after graduating from a school in Moscow. At 17, she began her studies at Technion – Israel Institute of Technology. During her master's degree in computer science, she taught first-year students of the same course, and at 27, Becker completed her PhD in Computer Science and Artificial Intelligence. == Career == While pursuing her PhD, Becker resolved an NP-complete approximation algorithm that had been unresolved for over twenty years. This made her a recognized scholar in the field. After completing her PhD, she developed an approximation technique by a factor of two. This technique is widely used today in operating systems, database systems, and VLSI chip designs. She then founded and sold Strategy Runner, a fintech software. After this, she founded EndoTech, an algorithmic trading platform based on artificial intelligence and machine learning. EndoTech's trading strategies have been operating in live cryptocurrency markets since 2017. The platform's BTC Alpha strategy has reported an average annual return of 163% on fixed capital over eight years of live operation, with a maximum drawdown of 14% and a trade accuracy rate of approximately 83%. In 2026, EndoTech entered a partnership with Bit1 Exchange to make its BTC Alpha and ETH Alpha copy trading strategies accessible to retail investors with no minimum deposit requirement, through a full-custody model in which user funds remain in their own exchange wallets at all times.As of 2023, Becker is working on Fianchetto Fund, an AI-based investing analysis platform. Becker has also co-authored a book on Bayesian networks, which has been published widely in the field of computer science and artificial intelligence.
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Dimensionality reduction

Dimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the low-dimensional representation retains some meaningful properties of the original data, ideally close to its intrinsic dimension. Working in high-dimensional spaces can be undesirable for many reasons; raw data are often sparse as a consequence of the curse of dimensionality, and analyzing the data is usually computationally intractable. Dimensionality reduction is common in fields that deal with large numbers of observations and/or large numbers of variables, such as signal processing, speech recognition, neuroinformatics, and bioinformatics. Methods are commonly divided into linear and nonlinear approaches. Linear approaches can be further divided into feature selection and feature extraction. Dimensionality reduction can be used for noise reduction, data visualization, cluster analysis, or as an intermediate step to facilitate other analyses. == Feature selection == The process of feature selection aims to find a suitable subset of the input variables (features, or attributes) for the task at hand. The three strategies are: the filter strategy (e.g., information gain), the wrapper strategy (e.g., accuracy-guided search), and the embedded strategy (features are added or removed while building the model based on prediction errors). Data analysis such as regression or classification can be done in the reduced space more accurately than in the original space. == Feature projection == Feature projection (also called feature extraction) transforms the data from the high-dimensional space to a space of fewer dimensions. The data transformation may be linear, as in principal component analysis (PCA), but many nonlinear dimensionality reduction techniques also exist. For multidimensional data, tensor representation can be used in dimensionality reduction through multilinear subspace learning. === Principal component analysis (PCA) === The main linear technique for dimensionality reduction, principal component analysis, performs a linear mapping of the data to a lower-dimensional space in such a way that the variance of the data in the low-dimensional representation is maximized. In practice, the covariance (and sometimes the correlation) matrix of the data is constructed and the eigenvectors on this matrix are computed. The eigenvectors that correspond to the largest eigenvalues (the principal components) can now be used to reconstruct a large fraction of the variance of the original data. Moreover, the first few eigenvectors can often be interpreted in terms of the large-scale physical behavior of the system, because they often contribute the vast majority of the system's energy, especially in low-dimensional systems. Still, this must be proved on a case-by-case basis as not all systems exhibit this behavior. The original space (with dimension of the number of points) has been reduced (with data loss, but hopefully retaining the most important variance) to the space spanned by a few eigenvectors. === Non-negative matrix factorization (NMF) === NMF decomposes a non-negative matrix to the product of two non-negative ones, which has been a promising tool in fields where only non-negative signals exist, such as astronomy. NMF is well known since the multiplicative update rule by Lee & Seung, which has been continuously developed: the inclusion of uncertainties, the consideration of missing data and parallel computation, sequential construction which leads to the stability and linearity of NMF, as well as other updates including handling missing data in digital image processing. With a stable component basis during construction, and a linear modeling process, sequential NMF is able to preserve the flux in direct imaging of circumstellar structures in astronomy, as one of the methods of detecting exoplanets, especially for the direct imaging of circumstellar discs. In comparison with PCA, NMF does not remove the mean of the matrices, which leads to physical non-negative fluxes; therefore NMF is able to preserve more information than PCA as demonstrated by Ren et al. === Kernel PCA === Principal component analysis can be employed in a nonlinear way by means of the kernel trick. The resulting technique is capable of constructing nonlinear mappings that maximize the variance in the data. The resulting technique is called kernel PCA. === Graph-based kernel PCA === Other prominent nonlinear techniques include manifold learning techniques such as Isomap, locally linear embedding (LLE), Hessian LLE, Laplacian eigenmaps, and methods based on tangent space analysis. These techniques assume that the high-dimensional input data lies near a low-dimensional manifold embedded in the ambient space, and construct a low-dimensional representation using a cost function that retains local properties of the data; they can be viewed as defining a graph-based kernel for Kernel PCA. More recently, techniques have been proposed that, instead of defining a fixed kernel, try to learn the kernel using semidefinite programming. The most prominent example of such a technique is maximum variance unfolding (MVU). The central idea of MVU is to exactly preserve all pairwise distances between nearest neighbors (in the inner product space) while maximizing the distances between points that are not nearest neighbors. An alternative approach to neighborhood preservation is through the minimization of a cost function that measures differences between distances in the input and output spaces. Important examples of such techniques include: classical multidimensional scaling, which is identical to PCA; Isomap, which uses geodesic distances in the data space; diffusion maps, which use diffusion distances in the data space; t-distributed stochastic neighbor embedding (t-SNE), which minimizes the divergence between distributions over pairs of points; and curvilinear component analysis. A different approach to nonlinear dimensionality reduction is through the use of autoencoders, a special kind of feedforward neural networks with a bottleneck hidden layer. The training of deep encoders is typically performed using a greedy layer-wise pre-training (e.g., using a stack of restricted Boltzmann machines) that is followed by a finetuning stage based on backpropagation. === Linear discriminant analysis (LDA) === Linear discriminant analysis (LDA) is a generalization of Fisher's linear discriminant, a method used in statistics, pattern recognition, and machine learning to find a linear combination of features that characterizes or separates two or more classes of objects or events. === Generalized discriminant analysis (GDA) === GDA deals with nonlinear discriminant analysis using kernel function operator. The underlying theory is close to the support-vector machines (SVM) insofar as the GDA method provides a mapping of the input vectors into high-dimensional feature space. Similar to LDA, the objective of GDA is to find a projection for the features into a lower dimensional space by maximizing the ratio of between-class scatter to within-class scatter. === Autoencoder === Autoencoders can be used to learn nonlinear dimension reduction functions and codings together with an inverse function from the coding to the original representation. === t-SNE === T-distributed Stochastic Neighbor Embedding (t-SNE) is a nonlinear dimensionality reduction technique useful for the visualization of high-dimensional datasets. It is not recommended for use in analysis such as clustering or outlier detection since it does not necessarily preserve densities or distances well. === UMAP === Uniform manifold approximation and projection (UMAP) is a nonlinear dimensionality reduction technique. Visually, it is similar to t-SNE, but it assumes that the data is uniformly distributed on a locally connected Riemannian manifold and that the Riemannian metric is locally constant or approximately locally constant. == Dimension reduction == For high-dimensional datasets, dimension reduction is usually performed prior to applying a k-nearest neighbors (k-NN) algorithm in order to mitigate the curse of dimensionality. Feature extraction and dimension reduction can be combined in one step, using principal component analysis (PCA), linear discriminant analysis (LDA), canonical correlation analysis (CCA), or non-negative matrix factorization (NMF) techniques to pre-process the data, followed by clustering via k-NN on feature vectors in a reduced-dimension space. In machine learning, this process is also called low-dimensional embedding. For high-dimensional datasets (e.g., when performing similarity search on live video streams, DNA data, or high-dimensional time series), running a fast approximate k-NN search using locality-sensitive hashing, random projection, "sketches", or other high-dimensional similarity search techniques from the VLDB conference toolbox may be the only fe
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Vowpal Wabbit

Vowpal Wabbit (VW) is an open-source fast online interactive machine learning system library and program developed originally at Yahoo! Research, and currently at Microsoft Research. It was started and is led by John Langford. Vowpal Wabbit's interactive learning support is particularly notable including Contextual Bandits, Active Learning, and forms of guided Reinforcement Learning. Vowpal Wabbit provides an efficient scalable out-of-core implementation with support for a number of machine learning reductions, importance weighting, and a selection of different loss functions and optimization algorithms. == Notable features == The VW program supports: Multiple supervised (and semi-supervised) learning problems: Classification (both binary and multi-class) Regression Active learning (partially labeled data) for both regression and classification Multiple learning algorithms (model-types / representations) OLS regression Matrix factorization (sparse matrix SVD) Single layer neural net (with user specified hidden layer node count) Searn (Search and Learn) Latent Dirichlet Allocation (LDA) Stagewise polynomial approximation Recommend top-K out of N One-against-all (OAA) and cost-sensitive OAA reduction for multi-class Weighted all pairs Contextual-bandit (with multiple exploration/exploitation strategies) Multiple loss functions: squared error quantile hinge logistic poisson Multiple optimization algorithms Stochastic gradient descent (SGD) BFGS Conjugate gradient Regularization (L1 norm, L2 norm, & elastic net regularization) Flexible input - input features may be: Binary Numerical Categorical (via flexible feature-naming and the hash trick) Can deal with missing values/sparse-features Other features On the fly generation of feature interactions (quadratic and cubic) On the fly generation of N-grams with optional skips (useful for word/language data-sets) Automatic test-set holdout and early termination on multiple passes bootstrapping User settable online learning progress report + auditing of the model Hyperparameter optimization == Scalability == Vowpal wabbit has been used to learn a tera-feature (1012) data-set on 1000 nodes in one hour. Its scalability is aided by several factors: Out-of-core online learning: no need to load all data into memory The hashing trick: feature identities are converted to a weight index via a hash (uses 32-bit MurmurHash3) Exploiting multi-core CPUs: parsing of input and learning are done in separate threads. Compiled C++ code
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