Broadcast is a collective communication primitive in parallel programming to distribute programming instructions or data to nodes in a cluster. It is the reverse operation of reduction. The broadcast operation is widely used in parallel algorithms, such as matrix-vector multiplication, Gaussian elimination and shortest paths. The Message Passing Interface implements broadcast in MPI_Bcast. == Definition == A message M [ 1.. m ] {\displaystyle M[1..m]} of length m {\displaystyle m} should be distributed from one node to all other p − 1 {\displaystyle p-1} nodes. T byte {\displaystyle T_{\text{byte}}} is the time it takes to send one byte. T start {\displaystyle T_{\text{start}}} is the time it takes for a message to travel to another node, independent of its length. Therefore, the time to send a package from one node to another is t = s i z e × T byte + T start {\displaystyle t=\mathrm {size} \times T_{\text{byte}}+T_{\text{start}}} . p {\displaystyle p} is the number of nodes and the number of processors. == Binomial Tree Broadcast == With Binomial Tree Broadcast the whole message is sent at once. Each node that has already received the message sends it on further. This grows exponentially as each time step the amount of sending nodes is doubled. The algorithm is ideal for short messages but falls short with longer ones as during the time when the first transfer happens only one node is busy. Sending a message to all nodes takes log 2 ( p ) t {\displaystyle \log _{2}(p)t} time which results in a runtime of log 2 ( p ) ( m T byte + T start ) {\displaystyle \log _{2}(p)(mT_{\text{byte}}+T_{\text{start}})} == Linear Pipeline Broadcast == The message is split up into k {\displaystyle k} packages and sent piecewise from node n {\displaystyle n} to node n + 1 {\displaystyle n+1} . The time needed to distribute the first message piece is p t = m k T byte + T start {\textstyle pt={\frac {m}{k}}T_{\text{byte}}+T_{\text{start}}} whereby t {\displaystyle t} is the time needed to send a package from one processor to another. Sending a whole message takes ( p + k ) ( m T byte k + T start ) = ( p + k ) t = p t + k t {\displaystyle (p+k)\left({\frac {mT_{\text{byte}}}{k}}+T_{\text{start}}\right)=(p+k)t=pt+kt} . Optimal is to choose k = m ( p − 2 ) T byte T start {\displaystyle k={\sqrt {\frac {m(p-2)T_{\text{byte}}}{T_{\text{start}}}}}} resulting in a runtime of approximately m T byte + p T start + m p T start T byte {\displaystyle mT_{\text{byte}}+pT_{\text{start}}+{\sqrt {mpT_{\text{start}}T_{\text{byte}}}}} The run time is dependent on not only message length but also the number of processors that play roles. This approach shines when the length of the message is much larger than the amount of processors. == Pipelined Binary Tree Broadcast == This algorithm combines Binomial Tree Broadcast and Linear Pipeline Broadcast, which makes the algorithm work well for both short and long messages. The aim is to have as many nodes work as possible while maintaining the ability to send short messages quickly. A good approach is to use Fibonacci trees for splitting up the tree, which are a good choice as a message cannot be sent to both children at the same time. This results in a binary tree structure. We will assume in the following that communication is full-duplex. The Fibonacci tree structure has a depth of about d ≈ log Φ ( p ) {\displaystyle d\approx \log _{\Phi }(p)} whereby Φ = 1 + 5 2 {\displaystyle \Phi ={\frac {1+{\sqrt {5}}}{2}}} the golden ratio. The resulting runtime is ( m k T byte + T start ) ( d + 2 k − 2 ) {\textstyle ({\frac {m}{k}}T_{\text{byte}}+T_{\text{start}})(d+2k-2)} . Optimal is k = n ( d − 2 ) T byte 3 T start {\displaystyle k={\sqrt {\frac {n(d-2)T_{\text{byte}}}{3T_{\text{start}}}}}} . This results in a runtime of 2 m T byte + T start log Φ ( p ) + 2 m log Φ ( p ) T start T byte {\displaystyle 2mT_{\text{byte}}+T_{\text{start}}\log _{\Phi }(p)+{\sqrt {2m\log _{\Phi }(p)T_{\text{start}}T_{\text{byte}}}}} . == Two Tree Broadcast (23-Broadcast) == === Definition === This algorithm aims to improve on some disadvantages of tree structure models with pipelines. Normally in tree structure models with pipelines (see above methods), leaves receive just their data and cannot contribute to send and spread data. The algorithm concurrently uses two binary trees to communicate over. Those trees will be called tree A and B. Structurally in binary trees there are relatively more leave nodes than inner nodes. Basic Idea of this algorithm is to make a leaf node of tree A be an inner node of tree B. It has also the same technical function in opposite side from B to A tree. This means, two packets are sent and received by inner nodes and leaves in different steps. === Tree construction === The number of steps needed to construct two parallel-working binary trees is dependent on the amount of processors. Like with other structures one processor can is the root node who sends messages to two trees. It is not necessary to set a root node, because it is not hard to recognize that the direction of sending messages in binary tree is normally top to bottom. There is no limitation on the number of processors to build two binary trees. Let the height of the combined tree be h = ⌈log(p + 2)⌉. Tree A and B can have a height of h − 1 {\displaystyle h-1} . Especially, if the number of processors correspond to p = 2 h − 1 {\displaystyle p=2^{h}-1} , we can make both sides trees and a root node. To construct this model efficiently and easily with a fully built tree, we can use two methods called "Shifting" and "Mirroring" to get second tree. Let assume tree A is already modeled and tree B is supposed to be constructed based on tree A. We assume that we have p {\displaystyle p} processors ordered from 0 to p − 1 {\displaystyle p-1} . ==== Shifting ==== The "Shifting" method, first copies tree A and moves every node one position to the left to get tree B. The node, which will be located on -1, becomes a child of processor p − 2 {\displaystyle p-2} . ==== Mirroring ==== "Mirroring" is ideal for an even number of processors. With this method tree B can be more easily constructed by tree A, because there are no structural transformations in order to create the new tree. In addition, a symmetric process makes this approach simple. This method can also handle an odd number of processors, in this case, we can set processor p − 1 {\displaystyle p-1} as root node for both trees. For the remaining processors "Mirroring" can be used. === Coloring === We need to find a schedule in order to make sure that no processor has to send or receive two messages from two trees in a step. The edge, is a communication connection to connect two nodes, and can be labelled as either 0 or 1 to make sure that every processor can alternate between 0 and 1-labelled edges. The edges of A and B can be colored with two colors (0 and 1) such that no processor is connected to its parent nodes in A and B using edges of the same color- no processor is connected to its children nodes in A or B using edges of the same color. In every even step the edges with 0 are activated and edges with 1 are activated in every odd step. === Time complexity === In this case the number of packet k is divided in half for each tree. Both trees are working together the total number of packets k = k / 2 + k / 2 {\displaystyle k=k/2+k/2} (upper tree + bottom tree) In each binary tree sending a message to another nodes takes 2 i {\displaystyle 2i} steps until a processor has at least a packet in step i {\displaystyle i} . Therefore, we can calculate all steps as d := log 2 ( p + 1 ) ⇒ log 2 ( p + 1 ) ≈ log 2 ( p ) {\displaystyle d:=\log _{2}(p+1)\Rightarrow \log _{2}(p+1)\approx \log _{2}(p)} . The resulting run time is T ( m , p , k ) ≈ ( m k T byte + T start ) ( 2 d + k − 1 ) {\textstyle T(m,p,k)\approx ({\frac {m}{k}}T_{\text{byte}}+T_{\text{start}})(2d+k-1)} . (Optimal k = m ( 2 d − 1 ) T byte / T start {\textstyle k={\sqrt {{m(2d-1)T_{\text{byte}}}/{T_{\text{start}}}}}} ) This results in a run time of T ( m , p ) ≈ m T byte + T start ⋅ 2 log 2 ( p ) + m ⋅ 2 log 2 ( p ) T start T byte {\displaystyle T(m,p)\approx mT_{\text{byte}}+T_{\text{start}}\cdot 2\log _{2}(p)+{\sqrt {m\cdot 2\log _{2}(p)T_{\text{start}}T_{\text{byte}}}}} . == ESBT-Broadcasting (Edge-disjoint Spanning Binomial Trees) == In this section, another broadcasting algorithm with an underlying telephone communication model will be introduced. A Hypercube creates network system with p = 2 d ( d = 0 , 1 , 2 , 3 , . . . ) {\displaystyle p=2^{d}(d=0,1,2,3,...)} . Every node is represented by binary 0 , 1 {\displaystyle {0,1}} depending on the number of dimensions. Fundamentally ESBT(Edge-disjoint Spanning Binomial Trees) is based on hypercube graphs, pipelining( m {\displaystyle m} messages are divided by k {\displaystyle k} packets) and binomial trees. The Processor 0 d {\displaystyle 0^{d}} cyclically spreads packets to roots of ESB
Crackme
A crackme is a small computer program designed to test a programmer's reverse engineering skills. Crackmes are made as a legal way to crack software, since no intellectual property is being infringed. == Description == Crackmes often incorporate protection schemes and algorithms similar to those used in proprietary software. However, they can sometimes be more challenging because they may use advanced packing or protection techniques, making the underlying algorithm harder to analyze and modify. == Keygenme == A keygenme is specifically designed for the reverser to not only identify the protection algorithm used in the application but also create a small key generator (keygen) in the programming language of their choice. Most keygenmes, when properly manipulated, can be made self-keygenning. For example, during validation, they might generate the correct key internally and compare it to the user's input. This allows the key generation algorithm to be easily replicated. Anti-debugging and anti-disassembly routines are often used to confuse debuggers or render disassembly output useless. Code obfuscation is also used to further complicate reverse engineering.
Bulletin (service)
Bulletin was an online newsletter platform launched by Facebook on July 6, 2021, that allows notable writers to make announcements directly to their subscribers. Its competitors included Substack, of which Bulletin was called a "near-clone." Writers participating in the platform's launch included Malcolm Gladwell, Mitch Albom, Tan France, Jessica Yellin, Jane Wells, Erin Andrews and Dorie Greenspan. Facebook CEO Mark Zuckerberg stated that Bulletin represented the first time that the company had "built a project that is directly for journalists and individual writers." In October 2022 Meta announced the shutdown of Bulletin. The platform went into read only mode in January 2023 and became unavailable in April 2023. == History == Facebook announced Bulletin as its online newsletter platform on June 29, 2021. and launched by the company on July 6, 2021. Facebook CEO Mark Zuckerberg touted the service by saying that Bulletin represented the first time that the company had "built a project that is directly for journalists and individual writers." Writers participating in the platform's launch included Malcolm Gladwell, Mitch Albom, Tan France, Jessica Yellin, Jane Wells, Erin Andrews and Dorie Greenspan. == Reception == Unlike competitor such as Substack, Facebook indicated upon service's launch that it would not take a cut of subscription fees of writers using that platform. According to Washington Post technology writer Will Oremus, the move was criticized by those who viewed it as a form of predatory pricing intended by Facebook to force those competitors out of business. Sandeep Vaheesan, legal director of the think tank Open Markets, called for the government to reexamine predatory pricing as a violation of antitrust law, saying, "We want companies to compete by making better products, investing in new equipment and tech — not purely relying on their financial advantages to capture market share."
Quality of experience
Quality of experience (QoE) is a measure of the delight or annoyance of a customer's experiences with a service (e.g., web browsing, phone call, TV broadcast). QoE focuses on the entire service experience; it is a holistic concept, similar to the field of user experience, but with its roots in telecommunication. QoE is an emerging multidisciplinary field based on social psychology, cognitive science, economics, and engineering science, focused on understanding overall human quality requirements. == Definition and concepts == In 2013, within the context of the COST Action QUALINET, QoE has been defined as:The degree of delight or annoyance of the user of an application or service. It results from the fulfillment of his or her expectations with respect to the utility and / or enjoyment of the application or service in the light of the user’s personality and current state.This definition has been adopted in 2016 by the International Telecommunication Union in Recommendation ITU-T P.10/G.100. Before, various definitions of QoE had existed in the domain, with the above-mentioned definition now finding wide acceptance in the community. QoE has historically emerged from Quality of Service (QoS), which attempts to objectively measure service parameters (such as packet loss rates or average throughput). QoS measurement is most of the time not related to a customer, but to the media or network itself. QoE however is a purely subjective measure from the user's perspective of the overall quality of the service provided, by capturing people's aesthetic and hedonic needs. QoE looks at a vendor's or purveyor's offering from the standpoint of the customer or end user, and asks, "What mix of goods, services, and support, do you think will provide you with the perception that the total product is providing you with the experience you desired and/or expected?" It then asks, "Is this what the vendor/purveyor has actually provided?" If not, "What changes need to be made to enhance your total experience?" In short, QoE provides an assessment of human expectations, feelings, perceptions, cognition and satisfaction with respect to a particular product, service or application. QoE is a blueprint of all human subjective and objective quality needs and experiences arising from the interaction of a person with technology and with business entities in a particular context. Although QoE is perceived as subjective, it is an important measure that counts for customers of a service. Being able to measure it in a controlled manner helps operators understand what may be wrong with their services and how to improve them. == QoE factors == QoE aims at taking into consideration every factor that contributes to a user's perceived quality of a system or service. This includes system, human and contextual factors. The following so-called "influence factors" have been identified and classified by Reiter et al.: Human Influence Factors Low-level processing (visual and auditory acuity, gender, age, mood, …) Higher-level processing (cognitive processes, socio-cultural and economic background, expectations, needs and goals, other personality traits…) System Influence Factors Content-related Media-related (encoding, resolution, sample rate, …) Network-related (bandwidth, delay, jitter, …) Device-related (screen resolution, display size, …) Context Influence Factors Physical context (location and space) Temporal context (time of day, frequency of use, …) Social context (inter-personal relations during experience) Economic context Task context (multitasking, interruptions, task type) Technical and information context (relationship between systems) Studies in the field of QoE have typically focused on system factors, primarily due to its origin in the QoS and network engineering domains. Through the use of dedicated test laboratories, the context is often sought to be kept constant. == QoE versus User Experience == QoE is strongly related to but different from the field of User Experience (UX), which also focuses on users' experiences with services. Historically, QoE has emerged from telecommunication research, while UX has its roots in Human–Computer Interaction. Both fields can be considered multi-disciplinary. In contrast to UX, the goal of improving QoE for users was more strongly motivated by economic needs. Wechsung and De Moor identify the following key differences between the fields: == QoE measurement == As a measure of the end-to-end performance at the service level from the user's perspective, QoE is an important metric for the design of systems and engineering processes. This is particularly relevant for video services because – due to their high traffic demands –, bad network performance may highly affect the user's experience. So, when designing systems, the expected output, i.e. the expected QoE, is often taken into account – also as a system output metric and optimization goal. To measure this level of QoE, human ratings can be used. The mean opinion score (MOS) is a widely used measure for assessing the quality of media signals. It is a limited form of QoE measurement, relating to a specific media type, in a controlled environment and without explicitly taking into account user expectations. The MOS as an indicator of experienced quality has been used for audio and speech communication, as well as for the assessment of quality of Internet video, television and other multimedia signals, and web browsing. Due to inherent limitations in measuring QoE in a single scalar value, the usefulness of the MOS is often debated. Subjective quality evaluation requires a lot of human resources, establishing it as a time-consuming process. Objective evaluation methods can provide quality results faster, but require dedicated computing resources. Since such instrumental video quality algorithms are often developed based on a limited set of subjective data, their QoE prediction accuracy may be low when compared to human ratings. QoE metrics are often measured at the end devices and can conceptually be seen as the remaining quality after the distortion introduced during the preparation of the content and the delivery through the network, until it reaches the decoder at the end device. There are several elements in the media preparation and delivery chain, and some of them may introduce distortion. This causes degradation of the content, and several elements in this chain can be considered as "QoE-relevant" for the offered services. The causes of degradation are applicable for any multimedia service, that is, not exclusive to video or speech. Typical degradations occur at the encoding system (compression degradation), transport network, access network (e.g., packet loss or packet delay), home network (e.g. WiFi performance) and end device (e.g. decoding performance). == QoE management == Several QoE-centric network management and bandwidth management solutions have been proposed, which aim to improve the QoE delivered to the end-users. When managing a network, QoE fairness may be taken into account in order to keep the users sufficiently satisfied (i.e., high QoE) in a fair manner. From a QoE perspective, network resources and multimedia services should be managed in order to guarantee specific QoE levels instead of classical QoS parameters, which are unable to reflect the actual delivered QoE. A pure QoE-centric management is challenged by the nature of the Internet itself, as the Internet protocols and architecture were not originally designed to support today's complex and high demanding multimedia services. As an example for an implementation of QoE management, network nodes can become QoE-aware by estimating the status of the multimedia service as perceived by the end-users. This information can then be used to improve the delivery of the multimedia service over the network and proactively improve the users' QoE. This can be achieved, for example, via traffic shaping. QoE management gives the service provider and network operator the capability to minimize storage and network resources by allocating only the resources that are sufficient to maintain a specific level of user satisfaction. As it may involve limiting resources for some users or services in order to increase the overall network performance and QoE, the practice of QoE management requires that net neutrality regulations are considered.
CSS HTML Validator
CSS HTML Validator (previously named CSE HTML Validator) is an HTML editor and CSS editor for Microsoft Windows (and Linux and other Unix-like operating systems when used with Wine) that helps web developers create syntactically correct and accessible HTML/HTML5, XHTML, and CSS documents by locating errors, potential problems like browser compatibility issues, and common mistakes. It is also able to check links, check spelling, suggest improvements, alert developers to deprecated, obsolete, or proprietary tags, attributes, and CSS properties, and find issues that can affect search engine optimization. CSS HTML Validator is developed, marketed, and sold by AI Internet Solutions LLC located in the United States. The first version of CSS HTML Validator was released in 1997 for Windows 95. The current version is 2026/v26.02 (as of January 9, 2026) and is for Windows 10 and above, including Windows 11. A native macOS and Linux command-line console tool (called htmlval) became available with version 23. There are currently three main editions of CSS HTML Validator — Pro/Professional, Home/Standard, and Lite. The Enterprise edition was discontinued in 2025/v25. While the application is generally a commercial product (except for the Lite edition), a free version of the Home edition is available for personal/educational, non-commercial use. A free limited version of the htmlval command-line console tool for macOS and Linux is also available. == Features == CSS HTML Validator includes an HTML editor, validator for HTML, XHTML, htmx, polyglot markup, CSS, PHP and JavaScript (using JSLint or JSHint), link checker (to find dead and broken links), spell checker, accessibility checker, and search engine optimization (SEO) checker. An integrated web browser allows developers to browse the web while the pages are automatically validated. Because documents are checked locally and not uploaded over the Internet to a server in order to be checked, validations are performed relatively quickly, and security and privacy are increased. A custom scripting language called TNPL, included in the Pro and Enterprise editions, can be used to customize validations by adding, eliminating, or changing validator messages. TNPL can also be used to integrate customized validation checks to meet the unique requirements of an individual or entity. A Batch Wizard tool, included in the Pro and Enterprise editions, can check entire Web sites, parts of Web sites, or a list of local web documents. The Batch Wizard generates reports in standard HTML or XML format. The reports can be viewed using a normal web browser. The accessibility checker includes support for Section 508 Amendment to the Rehabilitation Act of 1973 and Web Content Accessibility Guidelines (both WCAG 1.0 and WCAG 2.0/2.1/2.2). Using a version of HTML Tidy with HTML5 support and the Pretty Print & Fix Tool, CSS HTML Validator can automatically fix some common problems with HTML and XHTML documents. However, some problems cannot be fixed (or fixed correctly) with automated tools and require manual review and repair. == Version history == Validation of polyglot markup was added in version 12, and mobile development support (for HTML and CSS) was added in version 14 and improved in version 15. Version 15 added built-in syntax checking for JSON and HTML5 cache manifest files. Version 16 added JavaScript linting using JSHint, a static code analysis tool for checking JavaScript, but also continues to support JSLint. Version 17 added support for the Accelerated Mobile Pages Project, which is a type of HTML optimized for mobile web browsing, and support for live DOM validation using Google Chrome CSS HTML Validator 2018/v18 renames the software from CSE HTML Validator to CSS HTML Validator and includes updated HTML5 and CSS support. Version 18 also added a new "By Message" report in the Batch Wizard and dropped support for Windows Vista and below. CSS HTML Validator 2019/v19 includes updated HTML and CSS support, adds WCAG 2.1 support, improves support when running under Wine (software), and is a native 64-bit application (previously releases were 32-bit). CSS HTML Validator 2020/v20, first released in January 2020, includes HTML, CSS, accessibility, and other updates, including improved support for the Accelerated Mobile Pages Project. Also, beginning with version 20, the Standard edition was renamed to the Home edition. CSS HTML Validator 2021/v21, first released in January 2021, includes further HTML, CSS, accessibility, and other updates. CSS HTML Validator 2022/v22, released in January 2022, includes improvements and updates to keep the program up-to-date, a new Microsoft Edge WebView2 rendering engine for the integrated web browser, and three new dark themes. Later updates to version 22 added support for checking JSON Lines and NDJSON documents. CSS HTML Validator 2023/v23, released in January 2023, includes more improvements and updates to keep the program up-to-date. The new release also introduced new command-line macOS and Linux ports of the core validation engine, called htmlval for Mac and Linux. Official support for Windows 7, 8, and 8.1 was dropped in the 2023/v23 version. CSS HTML Validator 2024/v24, released in January 2024, includes updates and improvements. It also adds support for htmx. CSS HTML Validator 2025/v25, released in December 2024, includes further updates and improvements for 2025. Version 25 discontinues the Enterprise edition, moving Enterprise functionality to the Pro edition. CSS HTML Validator 2026/v26, released in January 2026, includes updated support for HTML and CSS. An online edition based on CSS HTML Validator Pro that can check documents via file upload, URL, or snippets (direct text input) was discontinued May 2017 in favor of the desktop version for Microsoft Windows. == Purpose of validation == The purpose of validation and computerized checking of HTML, XHTML, and CSS documents is to help make sure that the documents are syntactically correct and problem-free. Checked HTML, XHTML, and CSS documents are more likely to: be more accessible for people with disabilities (such as blindness), as well as all users in general render faster (user agents don't have to "figure out" and decipher bad syntax) render as intended and with fewer problems on a variety of user agents, including mobile devices cause browsers and user agents to build a more consistent Document Object Model, which is important for CSS and JavaScript be forward-compatible with future versions of user agents and browsers ("future-proof") be compatible with current and future HTML, XHTML, and CSS specifications cause fewer problems for visitors and web indexing not contain dead, broken, or rotting links While automated checking tools are helpful for website development and continued maintenance, they cannot guarantee that a document will display (render) and behave as intended in all browsers. Developers should always test documents in a variety of browsers (including mobile browsers) to locate problems that cannot be detected with a computerized checking tool. == Differences from other HTML validators == CSS HTML Validator is an offline desktop app for Microsoft Windows and a native macOS and Linux command-line console tool that does not require an Internet connection. The offline nature of CSS HTML Validator is in contrast to online web-based services. CSS HTML Validator primarily works offline (except for link checking when it must go online), which has speed and privacy benefits compared to web-based solutions and services like the W3C Markup Validation Service. However, the user must keep the software updated unlike web-based solutions which are typically kept updated by the solution provider. CSS HTML Validator checks HTML/XHTML syntax, CSS, links, spelling, accessibility, JavaScript, SEO, and PHP with one pass, while DTD-based validators are more limited and cannot check HTML5. CSS HTML Validator includes a built-in scripting language (called TNPL) which allows for a high degree of customization via scripting and "user functions". This allows developers to add custom (specialized) validation checks and messages. CSS HTML Validator includes a DTD-based validator which can optionally be used for checking DTD-based versions of HTML (versions prior to HTML5), however one of CSS HTML Validator's primary differences is that its custom validation engine can perform more checks on a document than a DTD-based validator can. This is because DTD-based validators are limited to checking only what can be specified in a Document Type Definition. == Integration == CSS HTML Validator integrates with other third-party software like those listed below. This allows validation using CSS HTML Validator from within the third-party program. EmEditor - includes a special Lite edition build of CSS HTML Validator for built-in checking of HTML and CSS Blumentals Software - several Blumentals software products integrate with CSS H
Pooling layer
In neural networks, a pooling layer is a kind of network layer that downsamples and aggregates information that is dispersed among many vectors into fewer vectors. It has several uses. It removes redundant information, thus reducing the amount of computation and memory required, which makes the model more robust to small variations in the input; and it increases the receptive field of neurons in later layers in the network. == Convolutional neural network pooling == Pooling is most commonly used in convolutional neural networks (CNN). Below is a description of pooling in 2-dimensional CNNs. The generalization to n-dimensions is immediate. As notation, we consider a tensor x ∈ R H × W × C {\displaystyle x\in \mathbb {R} ^{H\times W\times C}} , where H {\displaystyle H} is height, W {\displaystyle W} is width, and C {\displaystyle C} is the number of channels. A pooling layer outputs a tensor y ∈ R H ′ × W ′ × C ′ {\displaystyle y\in \mathbb {R} ^{H'\times W'\times C'}} . We define two variables f , s {\displaystyle f,s} called "filter size" (aka "kernel size") and "stride". Sometimes, it is necessary to use a different filter size and stride for horizontal and vertical directions. In such cases, we define 4 variables: f H , f W , s H , s W {\displaystyle f_{H},f_{W},s_{H},s_{W}} . The receptive field of an entry in the output tensor, y {\displaystyle y} , are all the entries in x {\displaystyle x} that can affect that entry. === Max pooling === Max Pooling (MaxPool) is commonly used in CNNs to reduce the spatial dimensions of feature maps. Define M a x P o o l ( x | f , s ) 0 , 0 , 0 = max ( x 0 : f − 1 , 0 : f − 1 , 0 ) {\displaystyle \mathrm {MaxPool} (x|f,s)_{0,0,0}=\max(x_{0:f-1,0:f-1,0})} where 0 : f − 1 {\displaystyle 0:f-1} means the range 0 , 1 , … , f − 1 {\displaystyle 0,1,\dots ,f-1} . Note that we need to avoid the off-by-one error. The next input is M a x P o o l ( x | f , s ) 1 , 0 , 0 = max ( x s : s + f − 1 , 0 : f − 1 , 0 ) {\displaystyle \mathrm {MaxPool} (x|f,s)_{1,0,0}=\max(x_{s:s+f-1,0:f-1,0})} and so on. The receptive field of y i , j , c {\displaystyle y_{i,j,c}} is x i s + f − 1 , j s + f − 1 , c {\displaystyle x_{is+f-1,js+f-1,c}} , so in general, M a x P o o l ( x | f , s ) i , j , c = m a x ( x i s : i s + f − 1 , j s : j s + f − 1 , c ) {\displaystyle \mathrm {MaxPool} (x|f,s)_{i,j,c}=\mathrm {max} (x_{is:is+f-1,js:js+f-1,c})} If the horizontal and vertical filter size and strides differ, then in general, M a x P o o l ( x | f , s ) i , j , c = m a x ( x i s H : i s H + f H − 1 , j s W : j s W + f W − 1 , c ) {\displaystyle \mathrm {MaxPool} (x|f,s)_{i,j,c}=\mathrm {max} (x_{is_{H}:is_{H}+f_{H}-1,js_{W}:js_{W}+f_{W}-1,c})} More succinctly, we can write y k = max ( { x k ′ | k ′ in the receptive field of k } ) {\displaystyle y_{k}=\max(\{x_{k'}|k'{\text{ in the receptive field of }}k\})} . If H {\displaystyle H} is not expressible as k s + f {\displaystyle ks+f} where k {\displaystyle k} is an integer, then for computing the entries of the output tensor on the boundaries, max pooling would attempt to take as inputs variables off the tensor. In this case, how those non-existent variables are handled depends on the padding conditions, illustrated on the right. Global Max Pooling (GMP) is a specific kind of max pooling where the output tensor has shape R C {\displaystyle \mathbb {R} ^{C}} and the receptive field of y c {\displaystyle y_{c}} is all of x 0 : H , 0 : W , c {\displaystyle x_{0:H,0:W,c}} . That is, it takes the maximum over each entire channel. It is often used just before the final fully connected layers in a CNN classification head. === Average pooling === Average pooling (AvgPool) is similarly defined A v g P o o l ( x | f , s ) i , j , c = a v e r a g e ( x i s : i s + f − 1 , j s : j s + f − 1 , c ) = 1 f 2 ∑ k ∈ i s : i s + f − 1 ∑ l ∈ j s : j s + f − 1 x k , l , c {\displaystyle \mathrm {AvgPool} (x|f,s)_{i,j,c}=\mathrm {average} (x_{is:is+f-1,js:js+f-1,c})={\frac {1}{f^{2}}}\sum _{k\in is:is+f-1}\sum _{l\in js:js+f-1}x_{k,l,c}} Global Average Pooling (GAP) is defined similarly to GMP. It was first proposed in Network-in-Network. Similarly to GMP, it is often used just before the final fully connected layers in a CNN classification head. === Interpolations === There are some interpolations of max pooling and average pooling. Mixed Pooling is a linear sum of max pooling and average pooling. That is, M i x e d P o o l ( x | f , s , w ) = w M a x P o o l ( x | f , s ) + ( 1 − w ) A v g P o o l ( x | f , s ) {\displaystyle \mathrm {MixedPool} (x|f,s,w)=w\mathrm {MaxPool} (x|f,s)+(1-w)\mathrm {AvgPool} (x|f,s)} where w ∈ [ 0 , 1 ] {\displaystyle w\in [0,1]} is either a hyperparameter, a learnable parameter, or randomly sampled anew every time. Lp Pooling is similar to average pooling, but uses Lp norm average instead of average: y k = ( 1 N ∑ k ′ in the receptive field of k | x k ′ | p ) 1 / p {\displaystyle y_{k}=\left({\frac {1}{N}}\sum _{k'{\text{ in the receptive field of }}k}|x_{k'}|^{p}\right)^{1/p}} where N {\displaystyle N} is the size of receptive field, and p ≥ 1 {\displaystyle p\geq 1} is a hyperparameter. If all activations are non-negative, then average pooling is the case of p = 1 {\displaystyle p=1} , and max pooling is the case of p → ∞ {\displaystyle p\to \infty } . Square-root pooling is the case of p = 2 {\displaystyle p=2} . Stochastic pooling samples a random activation x k ′ {\displaystyle x_{k'}} from the receptive field with probability x k ′ ∑ k ″ x k ″ {\displaystyle {\frac {x_{k'}}{\sum _{k''}x_{k''}}}} . It is the same as average pooling in expectation. Softmax pooling is like max pooling, but uses softmax, i.e. ∑ k ′ e β x k ′ x k ′ ∑ k ″ e β x k ″ {\displaystyle {\frac {\sum _{k'}e^{\beta x_{k'}}x_{k'}}{\sum _{k''}e^{\beta x_{k''}}}}} where β > 0 {\displaystyle \beta >0} . Average pooling is the case of β ↓ 0 {\displaystyle \beta \downarrow 0} , and max pooling is the case of β ↑ ∞ {\displaystyle \beta \uparrow \infty } Local Importance-based Pooling generalizes softmax pooling by ∑ k ′ e g ( x k ′ ) x k ′ ∑ k ″ e g ( x k ″ ) {\displaystyle {\frac {\sum _{k'}e^{g(x_{k'})}x_{k'}}{\sum _{k''}e^{g(x_{k''})}}}} where g {\displaystyle g} is a learnable function. === Other poolings === Spatial pyramidal pooling applies max pooling (or any other form of pooling) in a pyramid structure. That is, it applies global max pooling, then applies max pooling to the image divided into 4 equal parts, then 16, etc. The results are then concatenated. It is a hierarchical form of global pooling, and similar to global pooling, it is often used just before a classification head. Region of Interest Pooling (also known as RoI pooling) is a variant of max pooling used in R-CNNs for object detection. It is designed to take an arbitrarily-sized input matrix, and output a fixed-sized output matrix. Covariance pooling computes the covariance matrix of the vectors { x k , l , 0 : C − 1 } k ∈ i s : i s + f − 1 , l ∈ j s : j s + f − 1 {\displaystyle \{x_{k,l,0:C-1}\}_{k\in is:is+f-1,l\in js:js+f-1}} which is then flattened to a C 2 {\displaystyle C^{2}} -dimensional vector y i , j , 0 : C 2 − 1 {\displaystyle y_{i,j,0:C^{2}-1}} . Global covariance pooling is used similarly to global max pooling. As average pooling computes the average, which is a first-degree statistic, and covariance is a second-degree statistic, covariance pooling is also called "second-order pooling". It can be generalized to higher-order poolings. Blur Pooling means applying a blurring method before downsampling. For example, the Rect-2 blur pooling means taking an average pooling at f = 2 , s = 1 {\displaystyle f=2,s=1} , then taking every second pixel (identity with s = 2 {\displaystyle s=2} ). == Vision Transformer pooling == In Vision Transformers (ViT), there are the following common kinds of poolings. BERT-like pooling uses a dummy [CLS] token, "classification". For classification, the output at [CLS] is the classification token, which is then processed by a LayerNorm-feedforward-softmax module into a probability distribution, which is the network's prediction of class probability distribution. This is the one used by the original ViT and Masked Autoencoder. Global average pooling (GAP) does not use the dummy token, but simply takes the average of all output tokens as the classification token. It was mentioned in the original ViT as being equally good. Multihead attention pooling (MAP) applies a multi headed attention block to pooling. Specifically, it takes as input a list of vectors x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\dots ,x_{n}} , which might be thought of as the output vectors of a layer of a ViT. It then applies a feedforward layer F F N {\displaystyle \mathrm {FFN} } on each vector, resulting in a matrix V = [ F F N ( v 1 ) , … , F F N ( v n ) ] {\displaystyle V=[\mathrm {FFN} (v_{1}),\dots ,\mathrm {FFN} (v_{n})]} . This is then sent to a multi-headed attention, resulting in M u l t i h e a d e d A
Temporal resolution
Temporal resolution (TR) refers to the discrete resolution of a measurement with respect to time. It is defined as the amount of time needed to revisit and acquire data for the same location. When applied to remote sensing, this amount of time is influenced by the sensor platform's orbital characteristics and the features of the sensor itself. The temporal resolution is low when the revisiting delay is high and vice versa. Temporal resolution is typically expressed in days. == Physics == Often there is a trade-off between the temporal resolution of a measurement and its spatial resolution, due to Heisenberg's uncertainty principle. In some contexts, such as particle physics, this trade-off can be attributed to the finite speed of light and the fact that it takes a certain period of time for the photons carrying information to reach the observer. In this time, the system might have undergone changes itself. Thus, the longer the light has to travel, the lower the temporal resolution. == Technology == === Computing === In another context, there is often a tradeoff between temporal resolution and computer storage. A transducer may be able to record data every millisecond, but available storage may not allow this, and in the case of 4D PET imaging the resolution may be limited to several minutes. === Electronic displays === In some applications, temporal resolution may instead be equated to the sampling period, or its inverse, the refresh rate, or update frequency in Hertz, of a TV, for example. The temporal resolution is distinct from temporal uncertainty. This would be analogous to conflating image resolution with optical resolution. One is discrete, the other, continuous. The temporal resolution is a resolution somewhat the 'time' dual to the 'space' resolution of an image. In a similar way, the sample rate is equivalent to the pixel pitch on a display screen, whereas the optical resolution of a display screen is equivalent to temporal uncertainty. Note that both this form of image space and time resolutions are orthogonal to measurement resolution, even though space and time are also orthogonal to each other. Both an image or an oscilloscope capture can have a signal-to-noise ratio, since both also have measurement resolution. === Oscilloscopy === An oscilloscope is the temporal equivalent of a microscope, and it is limited by temporal uncertainty the same way a microscope is limited by optical resolution. A digital sampling oscilloscope has also a limitation analogous to image resolution, which is the sample rate. A non-digital non-sampling oscilloscope is still limited by temporal uncertainty. The temporal uncertainty can be related to the maximum frequency of continuous signal the oscilloscope could respond to, called the bandwidth and given in Hertz. But for oscilloscopes, this figure is not the temporal resolution. To reduce confusion, oscilloscope manufacturers use 'Sa/s' instead of 'Hz' to specify the temporal resolution. Two cases for oscilloscopes exist: either the probe settling time is much shorter than the real time sampling rate, or it is much larger. The case where the settling time is the same as the sampling time is usually undesirable in an oscilloscope. It is more typical to prefer a larger ratio either way, or if not, to be somewhat longer than two sample periods. In the case where it is much longer, the most typical case, it dominates the temporal resolution. The shape of the response during the settling time also has as strong effect on the temporal resolution. For this reason probe leads usually offer an arrangement to 'compensate' the leads to alter the trade off between minimal settling time, and minimal overshoot. If it is much shorter, the oscilloscope may be prone to aliasing from radio frequency interference, but this can be removed by repeatedly sampling a repetitive signal and averaging the results together. If the relationship between the 'trigger' time and the sample clock can be controlled with greater accuracy than the sampling time, then it is possible to make a measurement of a repetitive waveform with much higher temporal resolution than the sample period by upsampling each record before averaging. In this case the temporal uncertainty may be limited by clock jitter.