In the field of machine learning, the universal approximation theorems (UATs) state that neural networks with a certain structure can, in principle, approximate any continuous function to any desired degree of accuracy. These theorems provide a mathematical justification for using neural networks, assuring researchers that a sufficiently large or deep network can model the complex, non-linear relationships often found in real-world data. The best-known version of the theorem applies to feedforward networks with a single hidden layer. It states that if the layer's activation function is non-polynomial (which is true for common choices like the sigmoid function or ReLU), then the network can act as a "universal approximator." Universality is achieved by increasing the number of neurons in the hidden layer, making the network "wider." Other versions of the theorem show that universality can also be achieved by keeping the network's width fixed but increasing its number of layers, making it "deeper." These are existence theorems. They guarantee that a network with the right structure exists, but they do not provide a method for finding the network's parameters (training it), nor do they specify exactly how large the network must be for a given function. Finding a suitable network remains a practical challenge that is typically addressed with optimization algorithms like backpropagation. == Setup == Artificial neural networks are combinations of multiple simple mathematical functions that implement more complicated functions from (typically) real-valued vectors to real-valued vectors. The spaces of multivariate functions that can be implemented by a network are determined by the structure of the network, the set of simple functions, and its multiplicative parameters. A great deal of theoretical work has gone into characterizing these function spaces. Most universal approximation theorems are in one of two classes. The first quantifies the approximation capabilities of neural networks with an arbitrary number of artificial neurons ("arbitrary width" case) and the second focuses on the case with an arbitrary number of hidden layers, each containing a limited number of artificial neurons ("arbitrary depth" case). In addition to these two classes, there are also universal approximation theorems for neural networks with bounded number of hidden layers and a limited number of neurons in each layer ("bounded depth and bounded width" case). == History == === Arbitrary width === The first results concerned the arbitrary width case. Ken-ichi Funahashi (May 1989) showed that Rumelhart–Hinton–Williams type backpropagation networks possess universal approximation capability with a class of sigmoidal activation functions, extending the result to multi-output mappings as well. Kurt Hornik, Maxwell Stinchcombe, and Halbert White (July 1989) showed that multilayer feed-forward networks with as few as one hidden layer are universal approximators, provided that the activation function satisfies certain conditions. George Cybenko (December 1989) independently established a related result for sigmoid activation functions using functional-analytic methods. Hornik also showed in 1991 that it is not the specific choice of the activation function but rather the multilayer feed-forward architecture itself that gives neural networks the potential of being universal approximators. Moshe Leshno et al in 1993 and later Allan Pinkus in 1999 showed that the universal approximation property is equivalent to having a nonpolynomial activation function. === Arbitrary depth === The arbitrary depth case was also studied by a number of authors such as Gustaf Gripenberg in 2003, Dmitry Yarotsky, Zhou Lu et al in 2017, Boris Hanin and Mark Sellke in 2018 who focused on neural networks with ReLU activation function. In 2020, Patrick Kidger and Terry Lyons extended those results to neural networks with general activation functions such, e.g. tanh or GeLU. One special case of arbitrary depth is that each composition component comes from a finite set of mappings. In 2024, Cai constructed a finite set of mappings, named a vocabulary, such that any continuous function can be approximated by compositing a sequence from the vocabulary. This is similar to the concept of compositionality in linguistics, which is the idea that a finite vocabulary of basic elements can be combined via grammar to express an infinite range of meanings. === Bounded depth and bounded width === The bounded depth and bounded width case was first studied by Maiorov and Pinkus in 1999. They showed that there exists an analytic sigmoidal activation function such that two hidden layer neural networks with bounded number of units in hidden layers are universal approximators. In 2018, Guliyev and Ismailov constructed a smooth sigmoidal activation function providing universal approximation property for two hidden layer feedforward neural networks with fewer units in hidden layers. In 2018, they also constructed single hidden layer networks with bounded width that are still universal approximators for univariate functions. However, this does not apply for multivariable functions. In 2022, Shen et al. obtained precise quantitative information on the depth and width required to approximate a target function by deep and wide ReLU neural networks. === Quantitative bounds === The question of minimal possible width for universality was first studied in 2021, Park et al obtained the minimum width required for the universal approximation of Lp functions using feed-forward neural networks with ReLU as activation functions. Similar results that can be directly applied to residual neural networks were also obtained in the same year by Paulo Tabuada and Bahman Gharesifard using control-theoretic arguments. In 2023, Cai obtained the optimal minimum width bound for the universal approximation. For the arbitrary depth case, Leonie Papon and Anastasis Kratsios derived explicit depth estimates depending on the regularity of the target function and of the activation function. === Kolmogorov network === The Kolmogorov–Arnold representation theorem is similar in spirit. Indeed, certain neural network families can directly apply the Kolmogorov–Arnold theorem to yield a universal approximation theorem. Robert Hecht-Nielsen showed that a three-layer neural network can approximate any continuous multivariate function. This was extended to the discontinuous case by Vugar Ismailov. In 2024, Ziming Liu and co-authors showed a practical application. === Reservoir computing and quantum reservoir computing === In reservoir computing a sparse recurrent neural network with fixed weights equipped of fading memory and echo state property is followed by a trainable output layer. Its universality has been demonstrated separately for what concerns networks of rate neurons and spiking neurons, respectively. In 2024, the framework has been generalized and extended to quantum reservoirs where the reservoir is based on qubits defined over Hilbert spaces. === Variants === Variants include discontinuous activation functions, noncompact domains, certifiable networks, random neural networks, and alternative network architectures and topologies. The universal approximation property of width-bounded networks has been studied as a dual of classical universal approximation results on depth-bounded networks. For input dimension d x {\displaystyle d_{x}} and output dimension d y {\displaystyle d_{y}} the minimum width required for the universal approximation of the Lp functions is exactly m a x { d x + 1 , d y } {\displaystyle max\{d_{x}+1,d_{y}\}} (for a ReLU network). More generally this also holds if both ReLU and a threshold activation function are used. Universal function approximation on graphs (or rather on graph isomorphism classes) by popular graph convolutional neural networks (GCNs or GNNs) can be made as discriminative as the Weisfeiler–Leman graph isomorphism test. In 2020, a universal approximation theorem result was established by Brüel-Gabrielsson, showing that graph representation with certain injective properties is sufficient for universal function approximation on bounded graphs and restricted universal function approximation on unbounded graphs, with an accompanying O ( | V | ⋅ | E | ) {\displaystyle {\mathcal {O}}(\left|V\right|\cdot \left|E\right|)} -runtime method that performed at state of the art on a collection of benchmarks (where V {\displaystyle V} and E {\displaystyle E} are the sets of nodes and edges of the graph respectively). There are also a variety of results between non-Euclidean spaces and other commonly used architectures and, more generally, algorithmically generated sets of functions, such as the convolutional neural network (CNN) architecture, radial basis functions, or neural networks with specific properties. == Arbitrary-width case == A universal approximation theorem formally states that a family of neural network funct
Scrolling
In computer displays, filmmaking, television production, video games and other kinetic displays, scrolling is sliding text, images or video across a monitor or display, vertically or horizontally. "Scrolling," as such, does not change the layout of the text or pictures but moves (pans or tilts) the user's view across what is apparently a larger image that is not wholly seen. A common television and movie special effect is to scroll credits, while leaving the background stationary. Scrolling may take place completely without user intervention (as in film credits) or, on an interactive device, be triggered by touchscreen or a keypress and continue without further intervention until a further user action, or be entirely controlled by input devices. Scrolling may take place in discrete increments (perhaps one or a few lines of text at a time), or continuously (smooth scrolling). Frame rate is the speed at which an entire image is redisplayed. It is related to scrolling in that changes to text and image position can only happen as often as the image can be redisplayed. When frame rate is a limiting factor, one smooth scrolling technique is to blur images during movement that would otherwise appear to "jump". == Computing == === Implementation === Scrolling is often carried out on a computer by the CPU (software scrolling) or by a graphics processor. Some systems feature hardware scrolling, where an image may be offset as it is displayed, without any frame buffer manipulation (see also hardware windowing). This was especially common in 8 and 16bit video game consoles. === UI paradigms === In a WIMP-style graphical user interface (GUI), user-controlled scrolling is carried out by manipulating a scrollbar with a mouse, or using keyboard shortcuts, often the arrow keys. Scrolling is often supported by text user interfaces and command line interfaces. Older computer terminals changed the entire contents of the display one screenful ("page") at a time; this paging mode requires fewer resources than scrolling. Scrolling displays often also support page mode. Typically certain keys or key combinations page up or down; on PC-compatible keyboards the page up and page down keys or the space bar are used; earlier computers often used control key combinations. Some computer mice have a scroll wheel, which scrolls the display, often vertically, when rolled; others have scroll balls or tilt wheels which allow both vertical and horizontal scrolling. Some software supports other ways of scrolling. Adobe Reader has a mode identified by a small hand icon ("hand tool") on the document, which can then be dragged by clicking on it and moving the mouse as if sliding a large sheet of paper. When this feature is implemented on a touchscreen it is called kinetic scrolling. Touch-screens often use inertial scrolling, in which the scrolling motion of an object continues in a decaying fashion after release of the touch, simulating the appearance of an object with inertia. An early implementation of such behavior was in the "Star7" PDA of Sun Microsystems ca. 1991–1992. Scrolling can be controlled in other software-dependent ways by a PC mouse. Some scroll wheels can be pressed down, functioning like a button. Depending on the software, this allows both horizontal and vertical scrolling by dragging in the direction desired; when the mouse is moved to the original position, scrolling stops. A few scroll wheels can also be tilted, scrolling horizontally in one direction until released. On touchscreen devices, scrolling is a multi-touch gesture, done by swiping a finger on the screen vertically in the direction opposite to where the user wants to scroll to. If any content is too wide to fit on a display, horizontal scrolling is required to view all of it. In applications such as graphics and spreadsheets there is often more content than can fit either the width or the height of the screen at a comfortable scale, and scrolling in both directions is necessary. === Infinite scrolling === In contrast to material divided into discrete pages, the web design approach of infinite scrolling dynamically adds new material to the user display, leading to a continuous, apparently bottomless or endless scrolling experience. === Text === In languages written horizontally, such as most Western languages, text documents longer than will fit on the screen are often displayed wrapped and sized to fit the screen width, and scrolled vertically to bring desired content into view. It is possible to display lines too long to fit the display without wrapping, scrolling horizontally to view each entire line. However, this requires inconvenient constant line-by-line scrolling, while vertical scrolling is only needed after reading a full screenful. Software such as word processors and web browsers normally uses word-wrapping to display as many words in a single line as will fit the width of the screen or window or, for text organised in columns, each column. === Demos === Scrolling texts, also referred to as scrolltexts or scrollers, played an important part in the birth of the computer demo culture. The software crackers often used their deep knowledge of computer platforms to transform the information that accompanied their releases into crack intros. The sole role of these intros was to scroll the text on the screen in an impressive way. == Film and television == Scrolling is commonly used to display the credits at the end of films and television programs. Scrolling is often used in the form of a news ticker towards the bottom of the picture for content such as television news, scrolling sideways across the screen, delivering short-form content. In the dynamic layout of kinetic typography, scrolling typography can scroll across the flat screen, or can appear to recede or advance. An iconic example is the Star Wars opening crawl inspired by the Flash Gordon serials. == Video games == In computer and video games, scrolling of a playing field allows the player to control an object in a large contiguous area. Early examples of this method include Taito's 1974 vertical-scrolling racing video game Speed Race, Sega's 1976 forward-scrolling racing games Moto-Cross (Fonz) and Road Race, and Super Bug. Previously the flip-screen method was used to indicate moving backgrounds. The Namco Galaxian arcade system board introduced with Galaxian in 1979 pioneered a sprite system that animated pre-loaded sprites over a scrolling background, which became the basis for Nintendo's Radar Scope and Donkey Kong arcade hardware and home consoles such as the Nintendo Entertainment System. Parallax scrolling, which was first featured in Moon Patrol, involves several semi-transparent layers (called playfields), which scroll on top of each other at varying rates in order to give an early pseudo-3D illusion of depth. Belt scrolling is a method used in side-scrolling beat 'em up games with a downward camera angle where players can move up and down in addition to left and right. == Studies == A 1993 article by George Fitzmaurice studied spatially aware palmtop computers. These devices had a 3D sensor, and moving the device caused the contents to move as if the contents were fixed in place. This interaction could be referred to as “moving to scroll.” Also, if the user moved the device away from their body, they would zoom in; conversely, the device would zoom out if the user pulled the device closer to them. Smartphone cameras and “optical flow” image analysis utilize this technique nowadays. A 1996 research paper by Jun Rekimoto analyzed tilting operations as scrolling techniques on small screen interfaces. Users could not only tilt to scroll, but also tilt to select menu items. These techniques proved especially useful for field workers, since they only needed to hold and control the device with one hand. A study from 2013 by Selina Sharmin, Oleg Špakov, and Kari-Jouko Räihä explored the action of reading text on a screen while the text auto-scrolls based on the user's eye tracking patterns. The control group simply read text on a screen and manually scrolled. The study found that participants preferred to read primarily at the top of the screen, so the screen scrolled down whenever participants’ eyes began to look toward the bottom of the screen. This auto-scrolling caused no statistically significant difference in reading speed or performance. An undated study occurring during or after 2010 by Dede Frederick, James Mohler, Mihaela Vorvoreanu, and Ronald Glotzbach noted that parallax scrolling "may cause certain people to experience nausea."
Dave's Redistricting
Dave's Redistricting App (DRA) is an online web app originally created by Dave Bradlee that allows anyone to simulate redistricting a U.S. state's congressional and legislative districts. == Purpose == According to Bradlee, the software was designed to "put power in people's hands," and so that they "can see how the process works, so it's a little less mysterious than it was 10 years ago." Bradlee has noticed that many citizens are taking this process seriously and using his app to create legitimate redistricting maps that could be put in place. Some websites have called Bradlee the pioneer and cause of the rise of do-it-yourself redistricting. States such as Montana in 2021 allowed the general population to use it to submit redistricting proposals following the 2020 United States Census. Dave's Redistricting has frequently been mentioned as a resource that can be used to combat gerrymandering, given that the public has free access to it. Political science firms such as FiveThirtyEight have used the website to draw examples of gerrymandered districts, including on their famous Atlas of Redistricting. Dave Bradlee built the first generation of DRA. DRA 2020 is built by a small team of volunteers—Dave Bradlee, Terry Crowley, Alec Ramsay, and David Rinn—all with a shared passion for technology & democracy and all Microsoft veterans. Their mission is to empower civic organizations and citizen activists to advocate for fair congressional and legislative districts and increased transparency in the redistricting process. == Functions == Users can redraw the congressional and state legislative districts for all 50 states, the District of Columbia, and Puerto Rico using a variety of census and election datasets including Cook PVI. Maps can be optimized for different criteria. DRA 2020 added several major features to the first generation app: Sharing & collaborative editing of maps, like Google Docs Multiple statewide elections for all 50 states including the ability to import your own data Comprehensive analytics for evaluating and comparing maps Custom overlays, and Block-level editing DRA remains free to use. == Versions == 2.2: This uses Bing Maps, an outdated software that projects the districts of a single state onto a map of the United States. 2.5: After Bing Maps announced that it would no longer be updating for the foreseen future, the U.S. Map feature was removed. DRA 2020: At the end of 2018, a beta version of 2020 was released. This version that did not require Microsoft Silverlight and could be used in any web browser. DRA 2020 has been under continuous development since and is built using React (JavaScript library), Mapbox, OpenStreetMap, TypeScript, Node.js, Amazon Web Services, as well as many open source components, tools, and icons.
Bring your own encryption
Bring your own encryption (BYOE), also known as bring your own key (BYOK), is a cloud computing security model that allows cloud service customers to use their own encryption software and manage their own encryption keys. == Overview == BYOE enables cloud service customers to utilize a virtual instance of their encryption software alongside their cloud-hosted business applications to encrypt their data. In this model, hosted business applications are configured to process all data through the encryption software. This software then writes the ciphertext version of the data to the cloud service provider's physical data store and decrypts ciphertext data upon retrieval requests. This approach provides enterprises with control over their keys and the ability to generate their own master key using internal hardware security modules (HSM), which are then transmitted to the cloud provider's HSM. When the data is no longer needed, such as when users discontinue the cloud service, the keys can be deleted, rendering the encrypted data permanently inaccessible. This practice is known as crypto-shredding. == Potential Advantages == Organizations can store data with unique encryption that only they can access. Multiple organizations can share the same hardware infrastructure via cloud services like Amazon Web Services (AWS) or Google Cloud while maintaining encryption to comply with regulations such as HIPAA. == Potential Challenges == Resource utilization may be higher compared to traditional encryption practices when multiple users share the same hardware and use their own encryption. Efforts to minimize resource utilization issues may potentially impact security benefits.
ZygoteBody
ZygoteBody, formerly Google Body, is a web application by Zygote Media Group that renders manipulable 3D anatomical models of the human body. Several layers, from muscle tissues down to blood vessels, can be removed or made transparent to allow better study of individual body parts. Most of the body parts are labelled and are searchable. == Technology == The human models are based on data from the Zygote Media Group. The website uses JavaScript and WebGL technology to display 3D images inside the web browser without requiring the installation of external browser plug-ins. == History == ZygoteBody was launched as Google Body on December 15, 2010. On April Fools' Day 2011, users were greeted with the anatomy of a cow on the home page. The cow model is still available as part of the open-3d-viewer open source project. As part of the wind down on Google Labs, it was announced that Google Body will be shut down but will continue to be maintained by Zygote as ZygoteBody. On October 13, 2011, the Google Body site was shut down. Then, on January 9, 2012, ZygoteBody was launched and core code base (with the Google Cow model as a demo) was made available as an open source project called open-3d-viewer.
FloodAlerts
FloodAlerts is a software application, developed by software specialists Shoothill, which takes real-time flooding information, and displays the data on an interactive Bing map, updating and warning its users when they, their premises or the routes they need to travel could be at risk of flooding. == History == FloodAlerts was launched in 2012, originally as the world's first Facebook flood warning app. == Operation == FloodAlerts is made available free of charge to individuals. Users are able to set up their own monitored locations and receive alerts via the application or their Facebook wall if the locations they are monitoring are at imminent risk of flooding. Hosted in the Cloud, using the Microsoft Windows Azure platform, the FloodAlerts application processes the data received from the Environment Agency, automatically creates the required map tiles, pins and alerts and displays them on an interactive Bing map, updating the content every 15 minutes. Users are able to see the latest information on the map without having to refresh their browser. FloodAlerts can also be provided as a customised risk management solution to businesses that require infrastructure or asset safety monitoring in areas where water levels are rising or receding. == Awards and recognition == FloodAlerts has received The Guardian and Virgin Media Business's 2012 Innovation Nation Awards and was shortlisted as a finalist for a further two national awards: the UK IT Industry Awards for Innovation and Entrepreneurship and The Institution of Engineering and Technology Innovation Awards for Information Technology. == In the press == The FloodAlerts application was reviewed on the BBC website. It was also reviewed on BBC Click.
Box blur
A box blur (also known as a box linear filter) is a spatial domain linear filter in which each pixel in the resulting image has a value equal to the average value of its neighboring pixels in the input image. It is a form of low-pass ("blurring") filter. A 3 by 3 box blur ("radius 1") can be written as matrix 1 9 [ 1 1 1 1 1 1 1 1 1 ] . {\displaystyle {\frac {1}{9}}{\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\end{bmatrix}}.} Due to its property of using equal weights, it can be implemented using a much simpler accumulation algorithm, which is significantly faster than using a sliding-window algorithm. Box blurs are frequently used to approximate a Gaussian blur. By the central limit theorem, repeated application of a box blur will approximate a Gaussian blur. In the frequency domain, a box blur has zeros and negative components. That is, a sine wave with a period equal to the size of the box will be blurred away entirely, and wavelengths shorter than the size of the box may be phase-reversed, as seen when two bokeh circles touch to form a bright spot where there would be a dark spot between two bright spots in the original image. == Extensions == Gwosdek, et al. has extended Box blur to take a fractional radius: the edges of the 1-D filter are expanded with a fraction. It makes slightly better gaussian approximation possible due to the elimination of integer-rounding error. Mario Klingemann has a "stack blur" that tries to better emulate gaussian's look in one pass by stacking weights: 1 9 [ 1 2 3 2 1 ] {\displaystyle {\frac {1}{9}}{\begin{bmatrix}1&2&3&2&1\end{bmatrix}}} The triangular impulse response it forms decomposes to two rounds of box blur. Stacked Integral Image by Bhatia et al. takes the weighted average of a few box blurs to fit the gaussian response curve. == Implementation == The following pseudocode implements a 3x3 box blur. The example does not handle the edges of the image, which would not fit inside the kernel, so that these areas remain unblurred. In practice, the issue is better handled by: Introducing an alpha channel to represent the absence of colors; Extending the boundary by filling in values, ranked by quality: Fill in a mirrored image at the border Fill in a constant color extending from the last pixel Pad in a fixed color A number of optimizations can be applied when implementing the box blur of a radius r and N pixels: The box blur is a separable filter, so that only two 1D passes of averaging 2 r + 1 pixels will be needed, one horizontal and one vertical, for each pixel. This lowers the complexity from O(Nr2) to O(Nr). In digital signal processing terminology, each pass is a moving-average filter. Accumulation. Instead of discarding the sum for each pixel, the algorithm re-uses the previous sum, and updates it by subtracting away the old pixel and adding the new pixel in the blurring range. A summed-area table can be used similarly. This lowers the complexity from O(Nr) to O(N). When being used in multiple passes to approximate a Gaussian blur, the cascaded integrator–comb filter construction allows for doing the equivalent operation in a single pass.