Toggl Track (formerly Toggl) is a time tracking software developed by Toggl OÜ which is headquartered in Tallinn, Estonia. The company offers online time tracking and reporting services through their website along with mobile and desktop applications. Time can be tracked through a start/stop button, manual entry, or dragging and resizing time blocks in a calendar view. == History == According to Alari Aho, Toggl's CEO and founder, the application has been fully self-funded from the start. The name was created using a random name generator.
SemEval
SemEval (Semantic Evaluation) is an ongoing series of evaluations of computational semantic analysis systems; it evolved from the Senseval word sense evaluation series. The evaluations are intended to explore the nature of meaning in language. While meaning is intuitive to humans, transferring those intuitions to computational analysis has proved elusive. This series of evaluations provides a mechanism to characterize in more precise terms exactly what is necessary to compute in meaning. As such, the evaluations provide an emergent mechanism to identify the problems and solutions for computations with meaning. These exercises have evolved to articulate more of the dimensions that are involved in our use of language. They began with apparently simple attempts to identify word senses computationally. They have evolved to investigate the interrelationships among the elements in a sentence (e.g., semantic role labeling), relations between sentences (e.g., coreference), and the nature of what we are saying (semantic relations and sentiment analysis). The purpose of the SemEval and Senseval exercises is to evaluate semantic analysis systems. "Semantic Analysis" refers to a formal analysis of meaning, and "computational" refer to approaches that in principle support effective implementation. The first three evaluations, Senseval-1 through Senseval-3, were focused on word sense disambiguation (WSD), each time growing in the number of languages offered in the tasks and in the number of participating teams. Beginning with the fourth workshop, SemEval-2007 (SemEval-1), the nature of the tasks evolved to include semantic analysis tasks outside of word sense disambiguation. Triggered by the conception of the SEM conference, the SemEval community had decided to hold the evaluation workshops yearly in association with the SEM conference. It was also the decision that not every evaluation task will be run every year, e.g. none of the WSD tasks were included in the SemEval-2012 workshop. == History == === Early evaluation of algorithms for word sense disambiguation === From the earliest days, assessing the quality of word sense disambiguation algorithms had been primarily a matter of intrinsic evaluation, and “almost no attempts had been made to evaluate embedded WSD components”. Only very recently (2006) had extrinsic evaluations begun to provide some evidence for the value of WSD in end-user applications. Until 1990 or so, discussions of the sense disambiguation task focused mainly on illustrative examples rather than comprehensive evaluation. The early 1990s saw the beginnings of more systematic and rigorous intrinsic evaluations, including more formal experimentation on small sets of ambiguous words. === Senseval to SemEval === In April 1997, Martha Palmer and Marc Light organized a workshop entitled Tagging with Lexical Semantics: Why, What, and How? in conjunction with the Conference on Applied Natural Language Processing. At the time, there was a clear recognition that manually annotated corpora had revolutionized other areas of NLP, such as part-of-speech tagging and parsing, and that corpus-driven approaches had the potential to revolutionize automatic semantic analysis as well. Kilgarriff recalled that there was "a high degree of consensus that the field needed evaluation", and several practical proposals by Resnik and Yarowsky kicked off a discussion that led to the creation of the Senseval evaluation exercises. === SemEval's 3, 2 or 1 year(s) cycle === After SemEval-2010, many participants feel that the 3-year cycle is a long wait. Many other shared tasks such as Conference on Natural Language Learning (CoNLL) and Recognizing Textual Entailments (RTE) run annually. For this reason, the SemEval coordinators gave the opportunity for task organizers to choose between a 2-year or a 3-year cycle. The SemEval community favored the 3-year cycle. Although the votes within the SemEval community favored a 3-year cycle, organizers and coordinators had settled to split the SemEval task into 2 evaluation workshops. This was triggered by the introduction of the new SEM conference. The SemEval organizers thought it would be appropriate to associate our event with the SEM conference and collocate the SemEval workshop with the SEM conference. The organizers got very positive responses (from the task coordinators/organizers and participants) about the association with the yearly SEM, and 8 tasks were willing to switch to 2012. Thus was born SemEval-2012 and SemEval-2013. The current plan is to switch to a yearly SemEval schedule to associate it with the SEM conference but not every task needs to run every year. ==== List of Senseval and SemEval Workshops ==== Senseval-1 took place in the summer of 1998 for English, French, and Italian, culminating in a workshop held at Herstmonceux Castle, Sussex, England on September 2–4. Senseval-2 took place in the summer of 2001, and was followed by a workshop held in July 2001 in Toulouse, in conjunction with ACL 2001. Senseval-2 included tasks for Basque, Chinese, Czech, Danish, Dutch, English, Estonian, Italian, Japanese, Korean, Spanish and Swedish. Senseval-3 took place in March–April 2004, followed by a workshop held in July 2004 in Barcelona, in conjunction with ACL 2004. Senseval-3 included 14 different tasks for core word sense disambiguation, as well as identification of semantic roles, multilingual annotations, logic forms, subcategorization acquisition. SemEval-2007 (Senseval-4) took place in 2007, followed by a workshop held in conjunction with ACL in Prague. SemEval-2007 included 18 different tasks targeting the evaluation of systems for the semantic analysis of text. A special issue of Language Resources and Evaluation is devoted to the result. SemEval-2010 took place in 2010, followed by a workshop held in conjunction with ACL in Uppsala. SemEval-2010 included 18 different tasks targeting the evaluation of semantic analysis systems. SemEval-2012 took place in 2012; it was associated with the new SEM, First Joint Conference on Lexical and Computational Semantics, and co-located with NAACL, Montreal, Canada. SemEval-2012 included 8 different tasks targeting at evaluating computational semantic systems. However, there was no WSD task involved in SemEval-2012, the WSD related tasks were scheduled in the upcoming SemEval-2013. SemEval-2013 was associated with NAACL 2013, North American Association of Computational Linguistics, Georgia, USA and took place in 2013. It included 13 different tasks targeting at evaluating computational semantic systems. SemEval-2014 took place in 2014. It was co-located with COLING 2014, 25th International Conference on Computational Linguistics and SEM 2014, Second Joint Conference on Lexical and Computational Semantics, Dublin, Ireland. There were 10 different tasks in SemEval-2014 evaluating various computational semantic systems. SemEval-2015 took place in 2015. It was co-located with NAACL-HLT 2015, 2015 Conference of the North American Chapter of the Association for Computational Linguistics – Human Language Technologies and SEM 2015, Third Joint Conference on Lexical and Computational Semantics, Denver, USA. There were 17 different tasks in SemEval-2015 evaluating various computational semantic systems. == SemEval Workshop framework == The framework of the SemEval/Senseval evaluation workshops emulates the Message Understanding Conferences (MUCs) and other evaluation workshops ran by ARPA (Advanced Research Projects Agency, renamed the Defense Advanced Research Projects Agency (DARPA)). Stages of SemEval/Senseval evaluation workshops Firstly, all likely participants were invited to express their interest and participate in the exercise design. A timetable towards a final workshop was worked out. A plan for selecting evaluation materials was agreed. 'Gold standards' for the individual tasks were acquired, often human annotators were considered as a gold standard to measure precision and recall scores of computer systems. These 'gold standards' are what the computational systems strive towards. In WSD tasks, human annotators were set on the task of generating a set of correct WSD answers (i.e. the correct sense for a given word in a given context) The gold standard materials, without answers, were released to participants, who then had a short time to run their programs over them and return their sets of answers to the organizers. The organizers then scored the answers and the scores were announced and discussed at a workshop. == Semantic evaluation tasks == Senseval-1 & Senseval-2 focused on evaluation WSD systems on major languages that were available corpus and computerized dictionary. Senseval-3 looked beyond the lexemes and started to evaluate systems that looked into wider areas of semantics, such as Semantic Roles (technically known as Theta roles in formal semantics), Logic Form Transformation (commonly semantics of phrases, clauses or sentences were represented
Smart data capture
Smart data capture (SDC), also known as 'intelligent data capture' or 'automated data capture', describes the branch of technology concerned with using computer vision techniques like optical character recognition (OCR), barcode scanning, object recognition and other similar technologies to extract and process information from semi-structured and unstructured data sources. IDC characterize smart data capture as an integrated hardware, software, and connectivity strategy to help organizations enable the capture of data in an efficient, repeatable, scalable, and future-proof way. Data is captured visually from barcodes, text, IDs and other objects - often from many sources simultaneously - before being converted and prepared for digital use, typically by artificial intelligence-powered software. An important feature of SDC is that it focuses not just on capturing data more efficiently but serving up easy-to-access, actionable insights at the instant of data collection to both frontline and desk-based workers, aiding decision-making and making it a two-way process. Smart data capture automates and accelerates capture, applying insights in real time and automating processes based on extracted input. Smart data capture is designed to be repeatable and scalable to reduce low-level manual tasks and eliminate human error. To achieve this goal, smart data capture solutions are often made available using specialist software installed on commodity hardware such as smartphones. However, some solutions may rely on specialized hardware such as dedicated scanning devices, wearables or shop floor robots. == Differences from OCR == Optical character recognition applications are typically concerned with the actual data capture process; they are intended to faithfully reproduce text, words, letters and symbols from a printed document. Smart data capture is multimodal, capable of extracting data from a wider range of semi-structured and unstructured sources, going beyond basic text recognition to offer a wider scope of applications. By extending functionality to provide actionable insights at the point of capture, SDC is also a two-way process (capture-display), while OCR is more commonly one-way (capture only), primarily used for data input. Smart data capture solutions typically have two parts: Data capture (which includes OCR, barcode scanning, object recognition) Functionality that then uses this data to provide actionable insights at the point of capture. == Applications == Smart data capture can be applied to almost any industry and application that requires visual information capture and interpretation. This may include: Retail Warehouse inventory control Logistics, handling and shipping Manufacturing Field service Healthcare Transport and travel Fraud detection
Pyramid (image processing)
Pyramid, or pyramid representation, is a type of multi-scale signal representation developed by the computer vision, image processing and signal processing communities, in which a signal or an image is subject to repeated smoothing and subsampling. Pyramid representation is a predecessor to scale-space representation and multiresolution analysis. == Pyramid generation == There are two main types of pyramids: lowpass and bandpass. A lowpass pyramid is made by smoothing the image with an appropriate smoothing filter and then subsampling the smoothed image, usually by a factor of 2 along each coordinate direction. The resulting image is then subjected to the same procedure, and the cycle is repeated multiple times. Each cycle of this process results in a smaller image with increased smoothing, but with decreased spatial sampling density (that is, decreased image resolution). If illustrated graphically, the entire multi-scale representation will look like a pyramid, with the original image on the bottom and each cycle's resulting smaller image stacked one atop the other. A bandpass pyramid is made by forming the difference between images at adjacent levels in the pyramid and performing image interpolation between adjacent levels of resolution, to enable computation of pixelwise differences. == Pyramid generation kernels == A variety of different smoothing kernels have been proposed for generating pyramids. Among the suggestions that have been given, the binomial kernels arising from the binomial coefficients stand out as a particularly useful and theoretically well-founded class. Thus, given a two-dimensional image, we may apply the (normalized) binomial filter (1/4, 1/2, 1/4) typically twice or more along each spatial dimension and then subsample the image by a factor of two. This operation may then proceed as many times as desired, leading to a compact and efficient multi-scale representation. If motivated by specific requirements, intermediate scale levels may also be generated where the subsampling stage is sometimes left out, leading to an oversampled or hybrid pyramid. With the increasing computational efficiency of CPUs available today, it is in some situations also feasible to use wider supported Gaussian filters as smoothing kernels in the pyramid generation steps. === Gaussian pyramid === In a Gaussian pyramid, subsequent images are weighted down using a Gaussian average (Gaussian blur) and scaled down. Each pixel containing a local average corresponds to a neighborhood pixel on a lower level of the pyramid. This technique is used especially in texture synthesis. === Laplacian pyramid === A Laplacian pyramid is very similar to a Gaussian pyramid but saves the difference image of the blurred versions between each levels. Only the smallest level is not a difference image to enable reconstruction of the high resolution image using the difference images on higher levels. This technique can be used in image compression. === Steerable pyramid === A steerable pyramid, developed by Simoncelli and others, is an implementation of a multi-scale, multi-orientation band-pass filter bank used for applications including image compression, texture synthesis, and object recognition. It can be thought of as an orientation selective version of a Laplacian pyramid, in which a bank of steerable filters are used at each level of the pyramid instead of a single Laplacian or Gaussian filter. == Applications of pyramids == === Alternative representation === In the early days of computer vision, pyramids were used as the main type of multi-scale representation for computing multi-scale image features from real-world image data. More recent techniques include scale-space representation, which has been popular among some researchers due to its theoretical foundation, the ability to decouple the subsampling stage from the multi-scale representation, the more powerful tools for theoretical analysis as well as the ability to compute a representation at any desired scale, thus avoiding the algorithmic problems of relating image representations at different resolution. Nevertheless, pyramids are still frequently used for expressing computationally efficient approximations to scale-space representation. === Detail manipulation === Levels of a Laplacian pyramid can be added to or removed from the original image to amplify or reduce detail at different scales. However, detail manipulation of this form is known to produce halo artifacts in many cases, leading to the development of alternatives such as the bilateral filter. Some image compression file formats use the Adam7 algorithm or some other interlacing technique. These can be seen as a kind of image pyramid. Because those file format store the "large-scale" features first, and fine-grain details later in the file, a particular viewer displaying a small "thumbnail" or on a small screen can quickly download just enough of the image to display it in the available pixels—so one file can support many viewer resolutions, rather than having to store or generate a different file for each resolution.
Phase congruency
Phase congruency is a measure of feature significance in computer images, a method of edge detection that is particularly robust against changes in illumination and contrast. == Foundations == Phase congruency reflects the behaviour of the image in the frequency domain. It has been noted that edgelike features have many of their frequency components in the same phase. The concept is similar to coherence, except that it applies to functions of different wavelength. For example, the Fourier decomposition of a square wave consists of sine functions, whose frequencies are odd multiples of the fundamental frequency. At the rising edges of the square wave, each sinusoidal component has a rising phase; the phases have maximal congruency at the edges. This corresponds to the human-perceived edges in an image where there are sharp changes between light and dark. == Definition == Phase congruency compares the weighted alignment of the Fourier components of a signal A n {\displaystyle A_{\rm {n}}} with the sum of the Fourier components. P C ( t ) = max ϕ ¯ ∑ n A n cos ( ϕ n ( t ) − ϕ ¯ ) ∑ n A n {\displaystyle PC(t)=\max _{\bar {\phi }}{\frac {\sum _{\rm {n}}A_{\rm {n}}\cos(\phi _{\rm {n}}(t)-{\bar {\phi }})}{\sum _{\rm {n}}A_{n}}}} where ϕ n {\displaystyle \phi _{\rm {n}}} is the local or instantaneous phase as can be calculated using the Hilbert transform and A n {\displaystyle A_{\rm {n}}} are the local amplitude, or energy, of the signal. When all the phases are aligned, this is equal to 1. Several ways of implementing phase congruency have been developed, of which two versions are available in open source, one written for MATLAB and the other written in Java as a plugin for the ImageJ software. Given the different notations used for its formulation, a unified version has been recently presented, where a methodology for the parameter tuning is also presented. == Advantages == The square-wave example is naive in that most edge detection methods deal with it equally well. For example, the first derivative has a maximal magnitude at the edges. However, there are cases where the perceived edge does not have a sharp step or a large derivative. The method of phase congruency applies to many cases where other methods fail. A notable example is an image feature consisting of a single line, such as the letter "l". Many edge-detection algorithms will pick up two adjacent edges: the transitions from white to black, and black to white. On the other hand, the phase congruency map has a single line. A simple Fourier analogy of this case is a triangle wave. In each of its crests there is a congruency of crests from different sinusoidal functions. == Disadvantages == Calculating the phase congruency map of an image is very computationally intensive, and sensitive to image noise. Techniques of noise reduction are usually applied prior to the calculation.
Automated decision-making
Automated decision-making (ADM) is the use of data, machines and algorithms to make decisions in a range of contexts, including public administration, business, health, education, law, employment, transport, media and entertainment, with varying degrees of human oversight or intervention. ADM may involve large-scale data from a range of sources, such as databases, text, social media, sensors, images or speech, that is processed using various technologies including computer software, algorithms, machine learning, natural language processing, artificial intelligence, augmented intelligence and robotics. The increasing use of automated decision-making systems (ADMS) across a range of contexts presents many benefits and challenges to human society requiring consideration of the technical, legal, ethical, societal, educational, economic and health consequences. == Overview == There are different definitions of ADM based on the level of automation involved. Some definitions suggests ADM involves decisions made through purely technological means without human input, such as the EU's General Data Protection Regulation (Article 22). However, ADM technologies and applications can take many forms ranging from decision-support systems that make recommendations for human decision-makers to act on, sometimes known as augmented intelligence or 'shared decision-making', to fully automated decision-making processes that make decisions on behalf of individuals or organizations without human involvement. Models used in automated decision-making systems can be as simple as checklists and decision trees through to artificial intelligence and deep neural networks (DNN). Since the 1950s computers have gone from being able to do basic processing to having the capacity to undertake complex, ambiguous and highly skilled tasks such as image and speech recognition, gameplay, scientific and medical analysis and inferencing across multiple data sources. ADM is now being increasingly deployed across all sectors of society and many diverse domains from entertainment to transport. An ADM system (ADMS) may involve multiple decision points, data sets, and technologies (ADMT) and may sit within a larger administrative or technical system such as a criminal justice system or business process. == Data == Automated decision-making involves using data as input to be analyzed within a process, model, or algorithm or for learning and generating new models. ADM systems may use and connect a wide range of data types and sources depending on the goals and contexts of the system, for example, sensor data for self-driving cars and robotics, identity data for security systems, demographic and financial data for public administration, medical records in health, criminal records in law. This can sometimes involve vast amounts of data and computing power. === Data quality === The quality of the available data and its ability to be used in ADM systems is fundamental to the outcomes. It is often highly problematic for many reasons. Datasets are often highly variable; corporations or governments may control large-scale data, restricted for privacy or security reasons, incomplete, biased, limited in terms of time or coverage, measuring and describing terms in different ways, and many other issues. For machines to learn from data, large corpora are often required, which can be challenging to obtain or compute; however, where available, they have provided significant breakthroughs, for example, in diagnosing chest X-rays. == ADM technologies == Automated decision-making technologies (ADMT) are software-coded digital tools that automate the translation of input data to output data, contributing to the function of automated decision-making systems. There are a wide range of technologies in use across ADM applications and systems. ADMTs involving basic computational operations Search (includes 1-2-1, 1-2-many, data matching/merge) Matching (two different things) Mathematical Calculation (formula) ADMTs for assessment and grouping: User profiling Recommender systems Clustering Classification Feature learning Predictive analytics (includes forecasting) ADMTs relating to space and flows: Social network analysis (includes link prediction) Mapping Routing ADMTs for processing of complex data formats Image processing Audio processing Natural Language Processing (NLP) Other ADMT Business rules management systems Time series analysis Anomaly detection Modelling/Simulation === Machine learning === Machine learning (ML) involves training computer programs through exposure to large data sets and examples to learn from experience and solve problems. Machine learning can be used to generate and analyse data as well as make algorithmic calculations and has been applied to image and speech recognition, translations, text, data and simulations. While machine learning has been around for some time, it is becoming increasingly powerful due to recent breakthroughs in training deep neural networks (DNNs), and dramatic increases in data storage capacity and computational power with GPU coprocessors and cloud computing. Machine learning systems based on foundation models run on deep neural networks and use pattern matching to train a single huge system on large amounts of general data such as text and images. Early models tended to start from scratch for each new problem however since the early 2020s many are able to be adapted to new problems. Examples of these technologies include Open AI's DALL-E (an image creation program) and their various GPT language models, and Google's PaLM language model program. == Applications == ADM is being used to replace or augment human decision-making by both public and private-sector organisations for a range of reasons including to help increase consistency, improve efficiency, reduce costs and enable new solutions to complex problems. === Debate === Research and development are underway into uses of technology to assess argument quality, assess argumentative essays and judge debates. Potential applications of these argument technologies span education and society. Scenarios to consider, in these regards, include those involving the assessment and evaluation of conversational, mathematical, scientific, interpretive, legal, and political argumentation and debate. === Law === In legal systems around the world, algorithmic tools such as risk assessment instruments (RAI), are being used to supplement or replace the human judgment of judges, civil servants and police officers in many contexts. In the United States RAI are being used to generate scores to predict the risk of recidivism in pre-trial detention and sentencing decisions, evaluate parole for prisoners and to predict "hot spots" for future crime. These scores may result in automatic effects or may be used to inform decisions made by officials within the justice system. In Canada ADM has been used since 2014 to automate certain activities conducted by immigration officials and to support the evaluation of some immigrant and visitor applications. === Economics === Automated decision-making systems are used in certain computer programs to create buy and sell orders related to specific financial transactions and automatically submit the orders in the international markets. Computer programs can automatically generate orders based on predefined set of rules using trading strategies which are based on technical analyses, advanced statistical and mathematical computations, or inputs from other electronic sources. === Business === ==== Continuous auditing ==== Continuous auditing uses advanced analytical tools to automate auditing processes. It can be utilized in the private sector by business enterprises and in the public sector by governmental organizations and municipalities. As artificial intelligence and machine learning continue to advance, accountants and auditors may make use of increasingly sophisticated algorithms which make decisions such as those involving determining what is anomalous, whether to notify personnel, and how to prioritize those tasks assigned to personnel. === Media and entertainment === Digital media, entertainment platforms, and information services increasingly provide content to audiences via automated recommender systems based on demographic information, previous selections, collaborative filtering or content-based filtering. This includes music and video platforms, publishing, health information, product databases and search engines. Many recommender systems also provide some agency to users in accepting recommendations and incorporate data-driven algorithmic feedback loops based on the actions of the system user. Large-scale machine learning language models and image creation programs being developed by companies such as OpenAI and Google in the 2020s have restricted access however they are likely to have widespread application in fields such as advertising, copywriting, stock imagery and gra
Neural network Gaussian process
A Neural Network Gaussian Process (NNGP) is a Gaussian process (GP) obtained as the limit of a certain type of sequence of neural networks. Specifically, a wide variety of network architectures converges to a GP in the infinitely wide limit, in the sense of distribution. The concept constitutes an intensional definition, i.e., a NNGP is just a GP, but distinguished by how it is obtained. == Motivation == Bayesian networks are a modeling tool for assigning probabilities to events, and thereby characterizing the uncertainty in a model's predictions. Deep learning and artificial neural networks are approaches used in machine learning to build computational models which learn from training examples. Bayesian neural networks merge these fields. They are a type of neural network whose parameters and predictions are both probabilistic. While standard neural networks often assign high confidence even to incorrect predictions, Bayesian neural networks can more accurately evaluate how likely their predictions are to be correct. Computation in artificial neural networks is usually organized into sequential layers of artificial neurons. The number of neurons in a layer is called the layer width. When we consider a sequence of Bayesian neural networks with increasingly wide layers (see figure), they converge in distribution to a NNGP. This large width limit is of practical interest, since the networks often improve as layers get wider. And the process may give a closed form way to evaluate networks. NNGPs also appears in several other contexts: It describes the distribution over predictions made by wide non-Bayesian artificial neural networks after random initialization of their parameters, but before training; it appears as a term in neural tangent kernel prediction equations; it is used in deep information propagation to characterize whether hyperparameters and architectures will be trainable. It is related to other large width limits of neural networks. === Scope === The first correspondence result had been established in the 1995 PhD thesis of Radford M. Neal, then supervised by Geoffrey Hinton at University of Toronto. Neal cites David J. C. MacKay as inspiration, who worked in Bayesian learning. Today the correspondence is proven for: Single hidden layer Bayesian neural networks; deep fully connected networks as the number of units per layer is taken to infinity; convolutional neural networks as the number of channels is taken to infinity; transformer networks as the number of attention heads is taken to infinity; recurrent networks as the number of units is taken to infinity. In fact, this NNGP correspondence holds for almost any architecture: Generally, if an architecture can be expressed solely via matrix multiplication and coordinatewise nonlinearities (i.e., a tensor program), then it has an infinite-width GP. This in particular includes all feedforward or recurrent neural networks composed of multilayer perceptron, recurrent neural networks (e.g., LSTMs, GRUs), (nD or graph) convolution, pooling, skip connection, attention, batch normalization, and/or layer normalization. === Illustration === Every setting of a neural network's parameters θ {\displaystyle \theta } corresponds to a specific function computed by the neural network. A prior distribution p ( θ ) {\displaystyle p(\theta )} over neural network parameters therefore corresponds to a prior distribution over functions computed by the network. As neural networks are made infinitely wide, this distribution over functions converges to a Gaussian process for many architectures. The notation used in this section is the same as the notation used below to derive the correspondence between NNGPs and fully connected networks, and more details can be found there. The figure to the right plots the one-dimensional outputs z L ( ⋅ ; θ ) {\displaystyle z^{L}(\cdot ;\theta )} of a neural network for two inputs x {\displaystyle x} and x ∗ {\displaystyle x^{}} against each other. The black dots show the function computed by the neural network on these inputs for random draws of the parameters from p ( θ ) {\displaystyle p(\theta )} . The red lines are iso-probability contours for the joint distribution over network outputs z L ( x ; θ ) {\displaystyle z^{L}(x;\theta )} and z L ( x ∗ ; θ ) {\displaystyle z^{L}(x^{};\theta )} induced by p ( θ ) {\displaystyle p(\theta )} . This is the distribution in function space corresponding to the distribution p ( θ ) {\displaystyle p(\theta )} in parameter space, and the black dots are samples from this distribution. For infinitely wide neural networks, since the distribution over functions computed by the neural network is a Gaussian process, the joint distribution over network outputs is a multivariate Gaussian for any finite set of network inputs. == Discussion == === Infinitely wide fully connected network === This section expands on the correspondence between infinitely wide neural networks and Gaussian processes for the specific case of a fully connected architecture. It provides a proof sketch outlining why the correspondence holds, and introduces the specific functional form of the NNGP for fully connected networks. The proof sketch closely follows the approach by Novak and coauthors. ==== Network architecture specification ==== Consider a fully connected artificial neural network with inputs x {\displaystyle x} , parameters θ {\displaystyle \theta } consisting of weights W l {\displaystyle W^{l}} and biases b l {\displaystyle b^{l}} for each layer l {\displaystyle l} in the network, pre-activations (pre-nonlinearity) z l {\displaystyle z^{l}} , activations (post-nonlinearity) y l {\displaystyle y^{l}} , pointwise nonlinearity ϕ ( ⋅ ) {\displaystyle \phi (\cdot )} , and layer widths n l {\displaystyle n^{l}} . For simplicity, the width n L + 1 {\displaystyle n^{L+1}} of the readout vector z L {\displaystyle z^{L}} is taken to be 1. The parameters of this network have a prior distribution p ( θ ) {\displaystyle p(\theta )} , which consists of an isotropic Gaussian for each weight and bias, with the variance of the weights scaled inversely with layer width. This network is illustrated in the figure to the right, and described by the following set of equations: x ≡ input y l ( x ) = { x l = 0 ϕ ( z l − 1 ( x ) ) l > 0 z i l ( x ) = ∑ j W i j l y j l ( x ) + b i l W i j l ∼ N ( 0 , σ w 2 n l ) b i l ∼ N ( 0 , σ b 2 ) ϕ ( ⋅ ) ≡ nonlinearity y l ( x ) , z l − 1 ( x ) ∈ R n l × 1 n L + 1 = 1 θ = { W 0 , b 0 , … , W L , b L } {\displaystyle {\begin{aligned}x&\equiv {\text{input}}\\y^{l}(x)&=\left\{{\begin{array}{lcl}x&&l=0\\\phi \left(z^{l-1}(x)\right)&&l>0\end{array}}\right.\\z_{i}^{l}(x)&=\sum _{j}W_{ij}^{l}y_{j}^{l}(x)+b_{i}^{l}\\W_{ij}^{l}&\sim {\mathcal {N}}\left(0,{\frac {\sigma _{w}^{2}}{n^{l}}}\right)\\b_{i}^{l}&\sim {\mathcal {N}}\left(0,\sigma _{b}^{2}\right)\\\phi (\cdot )&\equiv {\text{nonlinearity}}\\y^{l}(x),z^{l-1}(x)&\in \mathbb {R} ^{n^{l}\times 1}\\n^{L+1}&=1\\\theta &=\left\{W^{0},b^{0},\dots ,W^{L},b^{L}\right\}\end{aligned}}} ==== ==== z l | y l {\displaystyle z^{l}|y^{l}} is a Gaussian process We first observe that the pre-activations z l {\displaystyle z^{l}} are described by a Gaussian process conditioned on the preceding activations y l {\displaystyle y^{l}} . This result holds even at finite width. Each pre-activation z i l {\displaystyle z_{i}^{l}} is a weighted sum of Gaussian random variables, corresponding to the weights W i j l {\displaystyle W_{ij}^{l}} and biases b i l {\displaystyle b_{i}^{l}} , where the coefficients for each of those Gaussian variables are the preceding activations y j l {\displaystyle y_{j}^{l}} . Because they are a weighted sum of zero-mean Gaussians, the z i l {\displaystyle z_{i}^{l}} are themselves zero-mean Gaussians (conditioned on the coefficients y j l {\displaystyle y_{j}^{l}} ). Since the z l {\displaystyle z^{l}} are jointly Gaussian for any set of y l {\displaystyle y^{l}} , they are described by a Gaussian process conditioned on the preceding activations y l {\displaystyle y^{l}} . The covariance or kernel of this Gaussian process depends on the weight and bias variances σ w 2 {\displaystyle \sigma _{w}^{2}} and σ b 2 {\displaystyle \sigma _{b}^{2}} , as well as the second moment matrix K l {\displaystyle K^{l}} of the preceding activations y l {\displaystyle y^{l}} , z i l ∣ y l ∼ G P ( 0 , σ w 2 K l + σ b 2 ) K l ( x , x ′ ) = 1 n l ∑ i y i l ( x ) y i l ( x ′ ) {\displaystyle {\begin{aligned}z_{i}^{l}\mid y^{l}&\sim {\mathcal {GP}}\left(0,\sigma _{w}^{2}K^{l}+\sigma _{b}^{2}\right)\\K^{l}(x,x')&={\frac {1}{n^{l}}}\sum _{i}y_{i}^{l}(x)y_{i}^{l}(x')\end{aligned}}} The effect of the weight scale σ w 2 {\displaystyle \sigma _{w}^{2}} is to rescale the contribution to the covariance matrix from K l {\displaystyle K^{l}} , while the bias is shared for all inputs, and so σ b 2 {\displaystyle \sigma _{b}^{2}} makes the z i l {\displaystyle z_{i}^{l}} for different datapoints more similar and