ObjectVision was a forms-based programming language and environment for Windows 3.x developed by Borland. The latest version, 2.1, was released in 1992. An ObjectVision application is composed by forms designed in a graphic way that contains objects and events to provide interactivity. Forms are connected together with logic in the form of decision trees. ObjectVision applications also can interact with databases using multiple engines, like Paradox and dBase. A finished project is saved as an OVD file, that is executed by an interpreted runtime that can be freely distributed. ObjectVision was not used broadly except in some niche segments, but the visual programming ideas were the basis for Borland Delphi.
Computer Law & Security Review
The Computer Law & Security Review is an international peer-reviewed journal published by Elsevier. It has been published six times a year since 1985 and is indexed in Scopus and SSCI. It is accessible to a wide range of professional legal and IT practitioners, businesses, academics, researchers, libraries and organisations in both the public and private sectors. The journal regularly covers: CLSR Briefing with special emphasis on UK/US developments European Union update National news from 10 European jurisdictions Pacific rim news column Refereed practitioner and academic papers on topics such as Web 2.0, IT security, Identity management, ID cards, RFID, interference with privacy, Internet law, telecoms regulation, online broadcasting, intellectual property, software law, e-commerce, outsourcing, data protection and freedom of information and many other topics. The Journal's Correspondent Panel includes more than 40 specialists in IT law and security. Each issue contains articles, case law analysis and current news on information and communications technology. Special Features High quality peer reviewed papers from internationally renowned practitioner and academic experts Latest developments reported in situ by more than 20 leading law firms from around the world Highly experienced and respected editor and correspondents panel Online access to all 23 volumes of CLSR with embedded web links to primary sources Contact details of all authors A pool of expertise that can collectively identify the key topics that need to be examined.
Weka (software)
Waikato Environment for Knowledge Analysis (Weka) is a collection of machine learning and data analysis free software licensed under the GNU General Public License. It was developed at the University of Waikato, New Zealand, and is the companion software to the book "Data Mining: Practical Machine Learning Tools and Techniques". == Description == Weka contains a collection of visualization tools and algorithms for data analysis and predictive modeling, together with graphical user interfaces for easy access to these functions. The original non-Java version of Weka was a Tcl/Tk front-end to (mostly third-party) modeling algorithms implemented in other programming languages, plus data preprocessing utilities in C, and a makefile-based system for running machine learning experiments. This original version was primarily designed as a tool for analyzing data from agricultural domains, but the more recent fully Java-based version (Weka 3), for which development started in 1997, is now used in many different application areas, in particular for educational purposes and research. Advantages of Weka include: Free availability under the GNU General Public License. Portability, since it is fully implemented in the Java programming language and thus runs on almost any modern computing platform. A comprehensive collection of data preprocessing and modeling techniques. Ease of use due to its graphical user interfaces. Weka supports several standard data mining tasks, more specifically, data preprocessing, clustering, classification, regression, visualization, and feature selection. Input to Weka is expected to be formatted according the Attribute-Relational File Format and with the filename bearing the .arff extension. All of Weka's techniques are predicated on the assumption that the data is available as one flat file or relation, where each data point is described by a fixed number of attributes (normally, numeric or nominal attributes, but some other attribute types are also supported). Weka provides access to SQL databases using Java Database Connectivity and can process the result returned by a database query. Weka provides access to deep learning with Deeplearning4j. It is not capable of multi-relational data mining, but there is separate software for converting a collection of linked database tables into a single table that is suitable for processing using Weka. Another important area that is currently not covered by the algorithms included in the Weka distribution is sequence modeling. == Extension packages == In version 3.7.2, a package manager was added to allow the easier installation of extension packages. Some functionality that used to be included with Weka prior to this version has since been moved into such extension packages, but this change also makes it easier for others to contribute extensions to Weka and to maintain the software, as this modular architecture allows independent updates of the Weka core and individual extensions. == History == In 1993, the University of Waikato in New Zealand began development of the original version of Weka, which became a mix of Tcl/Tk, C, and makefiles. In 1997, the decision was made to redevelop Weka from scratch in Java, including implementations of modeling algorithms. In 2005, Weka received the SIGKDD Data Mining and Knowledge Discovery Service Award. In 2006, Pentaho Corporation acquired an exclusive licence to use Weka for business intelligence. It forms the data mining and predictive analytics component of the Pentaho business intelligence suite. Pentaho has since been acquired by Hitachi Vantara, and Weka now underpins the PMI (Plugin for Machine Intelligence) open source component. == Related tools == Auto-WEKA is an automated machine learning system for Weka. Environment for DeveLoping KDD-Applications Supported by Index-Structures (ELKI) is a similar project to Weka with a focus on cluster analysis, i.e., unsupervised methods. H2O.ai is an open-source data science and machine learning platform KNIME is a machine learning and data mining software implemented in Java. Massive Online Analysis (MOA) is an open-source project for large scale mining of data streams, also developed at the University of Waikato in New Zealand. Neural Designer is a data mining software based on deep learning techniques written in C++. Orange is a similar open-source project for data mining, machine learning and visualization based on scikit-learn. RapidMiner is a commercial machine learning framework implemented in Java which integrates Weka. scikit-learn is a popular machine learning library in Python.
Liquid state machine
A liquid state machine (LSM) is a type of reservoir computer that uses a spiking neural network. An LSM consists of a large collection of units (called nodes, or neurons). Each node receives time varying input from external sources (the inputs) as well as from other nodes. Nodes are randomly connected to each other. The recurrent nature of the connections turns the time varying input into a spatio-temporal pattern of activations in the network nodes. The spatio-temporal patterns of activation are read out by linear discriminant units. The soup of recurrently connected nodes will end up computing a large variety of nonlinear functions on the input. Given a large enough variety of such nonlinear functions, it is theoretically possible to obtain linear combinations (using the read out units) to perform whatever mathematical operation is needed to perform a certain task, such as speech recognition or computer vision. The word liquid in the name comes from the analogy drawn to dropping a stone into a still body of water or other liquid. The falling stone will generate ripples in the liquid. The input (motion of the falling stone) has been converted into a spatio-temporal pattern of liquid displacement (ripples). LSMs have been put forward as a way to explain the operation of brains. LSMs are argued to be an improvement over the theory of artificial neural networks because: Circuits are not hard coded to perform a specific task. Continuous time inputs are handled "naturally". Computations on various time scales can be done using the same network. The same network can perform multiple computations. Criticisms of LSMs as used in computational neuroscience are that LSMs don't actually explain how the brain functions. At best they can replicate some parts of brain functionality. There is no guaranteed way to dissect a working network and figure out how or what computations are being performed. There is very little control over the process. == Universal function approximation == If a reservoir has fading memory and input separability, with help of a readout, it can be proven the liquid state machine is a universal function approximator using Stone–Weierstrass theorem.
Evolutionary attractor
An evolutionary attractor is a point in an evolutionary space where a selection process will always drive trait values towards that point from the region around it. Because of the importance of evolution through natural selection, often such an evolutionary space will be defined by genetic or phenotypic traits, or possibly both. In this case the selection process will be a form of natural selection. The existence of an evolutionary attractor in a biological evolutionary space does not always imply that it can be reached from all points in that evolutionary space, nor does it identify what will happen when the evolutionary attractor is reached. While an evolutionary attractor may represent a point in evolutionary space that is resistant to further selection, such as an evolutionarily stable strategy, other possibilities are available. Because identification of an evolutionary attractor on its own does not describe everything about the evolutionary space in which it lies, this has led to interest in the evolutionary dynamics surrounding evolutionary attractors and in evolutionary spaces in general. (Theoretical biologists and mathematicians working in the area may prefer the terms adaptive dynamics or evolutionary invasion analysis to evolutionary dynamics.) These fields use differential equations which allows a more complete understanding of the dynamics in evolutionary spaces including the existence or otherwise of evolutionary attractors. Advances in the study of molecular evolution have also led to the identification of evolutionary attractors at a molecular level. Because biological evolutionary processes have been studied using evolutionary game theory, a technique inspired by game theory originally derived to address economic problems, not only can evolutionary attractors be found in biology but economists studying evolutionary economic models have also identified evolutionary attractors. Evolution in biology has also inspired evolutionary computation in computer science. Many algorithms in this field use a form of selection inspired by natural selection to generate results through evolutionary algorithms. This is therefore another area in which evolutionary attractors have been identified. == Evolutionary attractors in biology == It is not probably not surprising that biology is the field where most examples of evolutionary attractors have been identified, given the importance of evolution through natural selection. === Evolutionary attractors in adaptive landscapes === An evolutionary attractor is a point in genetic and/or phenotypic trait space, that evolution will always drive trait values towards via a selection process. The concept of an evolutionary attractor arose in population genetics following the origin of the adaptive landscape originally proposed by Sewall Wright in 1932. The height of a point in an adaptive landscape is a measure of evolutionary fitness. If a point in an adaptive landscape is a peak, then selection will always drive traits towards it and it will be an evolutionary attractor. While population genetics deals with discrete genetic traits, quantitative genetics extended such concepts to deal with continuous genetic traits, where the concept of evolutionary attractor is also valid. === Evolutionary attractors in evolutionary game models === Evolutionary game theory introduced into evolutionary biology concepts originally used in economics, with the advantage that evolution could be studied in relation to strategic choices made in animal conflicts. This is of particular interest because of the concept of the evolutionarily stable strategy or ESS, a strategy that once established is resistant to invasion by other strategies. ESSs will not always be evolutionary attractors, but if they are they will persist over evolutionary time. === Dynamics around evolutionary attractors in biology === Evolutionary attractors in biology do not exist in isolation. By definition they must exist in an evolutionary trait space where selection drives all traits towards them from a region immediately around them. That is, they must be convergence stable. Eshel (1983) modified the definition of an ESS by considering individually advantageous reduction from a majority deviation: he created the term continuous stability. A continuously stable ESS can be shown to be convergence stable, therefore it will act as an evolutionary attractor. But the nature of evolutionary trait spaces in biology means that it is not possible to guarantee that the region of convergence to the evolutionary attractor covers the whole of the trait space, nor that there is only one evolutionary attractor in a particular trait space. These issues have led to the emergence of the related fields of evolutionary dynamics, adaptive dynamics and evolutionary invasion analysis, all of which use differential equations to understand the dynamics in evolutionary trait spaces. Hence, if one or more evolutionary attractor exists in an evolutionary trait space, they provide techniques to understand the dynamics in that trait space around the evolutionary attractor. === Evolutionary attractors in an ecological context === Evolution in biology does not take place in single species in isolation. Ecological interaction of species leads to coevolution. Important examples of this are host-parasite or host-pathogen interaction, which can make both the dynamics around evolutionary attractors more complex, and the occurrence and number of evolutionary attractors more diverse. Evolutionary attractors have been identified in the analysis of evolutionary epidemiology of plant pathogens. In the above study working on plant populations the authors were able to identify evolutionary attractors using methods from adaptive dynamics. A model applied to the analysis of a maize (Zea mays L.) virus identified convergence stable equilibria through simulation modelling. A related model identified evolutionary attractors in the interaction of plants with fungal pathogens. === Evolutionary attractors in molecular genetics === As mentioned above much of the consideration of evolutionary attractors in biology has been through investigation of selection at a genetic or phenotypic level or both, in a single species or in coevolving species. Advances in the study of molecular genetics now allow the study of evolutionary attractors to be taken to a molecular genetic level. Wilson et. al (2019) studied the evolution of gene regulatory networks and identified the emergence of evolutionary attractors. == Evolutionary attractors in economics == Evolutionary game theory as applied in biology was inspired by game theory originally devised for applications in economics. Game theory remains an active field of research outside of biology, and thus it is not surprising that researchers in evolutionary economics use evolutionary game theory. Evolutionary attractors have been demonstrated by economists studying the evolutionary dynamics of market entry with market dynamics based on the replicator dynamics of biological evolutionary games. == Evolutionary attractors in computing == Evolutionary computation is a branch of computer science inspired by biological evolution. Many algorithms in evolutionary computation use a form of selection. Thus evolutionary attractors have been identified in computer science as well as in biology and economics. Evolutionary algorithms have generated evolutionary attractors, probably because of the similarity between adaptive hill-climbing in evolutionary heuristics and the adaptive landscape originated to explain evolution through natural selection.
Color quantization
In computer graphics, color quantization or color image quantization is quantization applied to color spaces; it is a process that reduces the number of distinct colors used in an image, usually with the intention that the new image should be as visually similar as possible to the original image. Computer algorithms to perform color quantization on bitmaps have been studied since the 1970s. Color quantization is critical for displaying images with many colors on devices that can only display a limited number of colors, usually due to memory limitations, and enables efficient compression of certain types of images. The name "color quantization" is primarily used in computer graphics research literature; in applications, terms such as optimized palette generation, optimal palette generation, or decreasing color depth are used. Some of these are misleading, as the palettes generated by standard algorithms are not necessarily the best possible. == Algorithms == Most standard techniques treat color quantization as a problem of clustering points in three-dimensional space, where the points represent colors found in the original image and the three axes represent the three color channels. Almost any three-dimensional clustering algorithm can be applied to color quantization, and vice versa. After the clusters are located, typically the points in each cluster are averaged to obtain the representative color that all colors in that cluster are mapped to. The three color channels are usually red, green, and blue, but another popular choice is the Lab color space, in which Euclidean distance is more consistent with perceptual difference. The most popular algorithm by far for color quantization, invented by Paul Heckbert in 1979, is the median cut algorithm. Many variations on this scheme are in use. Before this time, most color quantization was done using the population algorithm or population method, which essentially constructs a histogram of equal-sized ranges and assigns colors to the ranges containing the most points. A more modern popular method is clustering using octrees, first conceived by Gervautz and Purgathofer and improved by Xerox PARC researcher Dan Bloomberg. If the palette is fixed, as is often the case in real-time color quantization systems such as those used in operating systems, color quantization is usually done using the "straight-line distance" or "nearest color" algorithm, which simply takes each color in the original image and finds the closest palette entry, where distance is determined by the distance between the two corresponding points in three-dimensional space. In other words, if the colors are ( r 1 , g 1 , b 1 ) {\displaystyle (r_{1},g_{1},b_{1})} and ( r 2 , g 2 , b 2 ) {\displaystyle (r_{2},g_{2},b_{2})} , we want to minimize the Euclidean distance: ( r 1 − r 2 ) 2 + ( g 1 − g 2 ) 2 + ( b 1 − b 2 ) 2 . {\displaystyle {\sqrt {(r_{1}-r_{2})^{2}+(g_{1}-g_{2})^{2}+(b_{1}-b_{2})^{2}}}.} This effectively decomposes the color cube into a Voronoi diagram, where the palette entries are the points and a cell contains all colors mapping to a single palette entry. There are efficient algorithms from computational geometry for computing Voronoi diagrams and determining which region a given point falls in; in practice, indexed palettes are so small that these are usually overkill. Color quantization is frequently combined with dithering, which can eliminate unpleasant artifacts such as banding that appear when quantizing smooth gradients and give the appearance of a larger number of colors. Some modern schemes for color quantization attempt to combine palette selection with dithering in one stage, rather than perform them independently. A number of other much less frequently used methods have been invented that use entirely different approaches. The Local K-means algorithm, conceived by Oleg Verevka in 1995, is designed for use in windowing systems where a core set of "reserved colors" is fixed for use by the system and many images with different color schemes might be displayed simultaneously. It is a post-clustering scheme that makes an initial guess at the palette and then iteratively refines it. In the early days of color quantization, the k-means clustering algorithm was deemed unsuitable because of its high computational requirements and sensitivity to initialization. In 2011, M. Emre Celebi reinvestigated the performance of k-means as a color quantizer. He demonstrated that an efficient implementation of k-means outperforms a large number of color quantization methods. The high-quality but slow NeuQuant algorithm reduces images to 256 colors by training a Kohonen neural network "which self-organises through learning to match the distribution of colours in an input image. Taking the position in RGB-space of each neuron gives a high-quality colour map in which adjacent colours are similar." It is particularly advantageous for images with gradients. Finally, one of the newer methods is spatial color quantization, conceived by Puzicha, Held, Ketterer, Buhmann, and Fellner of the University of Bonn, which combines dithering with palette generation and a simplified model of human perception to produce visually impressive results even for very small numbers of colors. It does not treat palette selection strictly as a clustering problem, in that the colors of nearby pixels in the original image also affect the color of a pixel. See sample images. == History and applications == In the early days of PCs, it was common for video adapters to support only 2, 4, 16, or (eventually) 256 colors due to video memory limitations; they preferred to dedicate the video memory to having more pixels (higher resolution) rather than more colors. Color quantization helped to justify this tradeoff by making it possible to display many high color images in 16- and 256-color modes with limited visual degradation. Many operating systems automatically perform quantization and dithering when viewing high color images in a 256 color video mode, which was important when video devices limited to 256 color modes were dominant. Modern computers can now display millions of colors at once, far more than can be distinguished by the human eye, limiting this application primarily to mobile devices and legacy hardware. Nowadays, color quantization is mainly used in GIF and PNG images. GIF, for a long time the most popular lossless and animated bitmap format on the World Wide Web, only supports up to 256 colors, necessitating quantization for many images. Some early web browsers constrained images to use a specific palette known as the web colors, leading to severe degradation in quality compared to optimized palettes. PNG images support 24-bit color, but can often be made much smaller in filesize without much visual degradation by application of color quantization, since PNG files use fewer bits per pixel for palettized images. The infinite number of colors available through the lens of a camera is impossible to display on a computer screen; thus converting any photograph to a digital representation necessarily involves some quantization. Practically speaking, 24-bit color is sufficiently rich to represent almost all colors perceivable by humans with sufficiently small error as to be visually identical (if presented faithfully), within the available color space. However, the digitization of color, either in a camera detector or on a screen, necessarily limits the available color space. Consequently there are many colors that may be impossible to reproduce, regardless of how many bits are used to represent the color. For example, it is impossible in typical RGB color spaces (common on computer monitors) to reproduce the full range of green colors that the human eye is capable of perceiving. With the few colors available on early computers, different quantization algorithms produced very different-looking output images. As a result, a lot of time was spent on writing sophisticated algorithms to be more lifelike. === Quantization for image compression === Many image file formats support indexed color. A whole-image palette typically selects 256 "representative" colors for the entire image, where each pixel references any one of the colors in the palette, as in the GIF and PNG file formats. A block palette typically selects 2 or 4 colors for each block of 4x4 pixels, used in BTC, CCC, S2TC, and S3TC. === Editor support === Many bitmap graphics editors contain built-in support for color quantization, and will automatically perform it when converting an image with many colors to an image format with fewer colors. Most of these implementations allow the user to set exactly the number of desired colors. Examples of such support include: Photoshop's Mode→Indexed Color function supplies a number of quantization algorithms ranging from the fixed Windows system and Web palettes to the proprietary Local and Global algorithms for generating palettes suited to a particu
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