AlphaTensor is an artificial intelligence system developed by DeepMind for discovering efficient matrix multiplication algorithms using reinforcement learning. Introduced in 2022, the system was based on AlphaZero and formulated the search for matrix multiplication algorithms as a single-player game called TensorGame. AlphaTensor was designed to search for new ways to multiply matrices with fewer scalar multiplication operations. Matrix multiplication is a fundamental operation in linear algebra, numerical analysis, scientific computing, computer graphics, and machine learning. The system discovered thousands of matrix multiplication algorithms, including algorithms that rediscovered known human-designed methods and others that improved on previously known results for particular matrix sizes and mathematical settings. == Background == Matrix multiplication is one of the basic operations in numerical computing. The standard algorithm for multiplying two square matrices has cubic time complexity, while faster algorithms such as the Strassen algorithm reduce the number of multiplication operations by using more complex algebraic decompositions. Finding optimal matrix multiplication algorithms can be difficult because it involves searching through a large space of possible tensor decompositions. AlphaTensor approached this problem by representing algorithm discovery as TensorGame, in which each move corresponds to an operation that reduces a tensor representing matrix multiplication. The goal of the game is to find a low-rank decomposition of the matrix multiplication tensor, corresponding to an efficient multiplication algorithm. == Development == AlphaTensor was developed by DeepMind and described in a paper published in Nature in October 2022. The system built on the reinforcement-learning approach used in AlphaZero, which had previously been applied to games such as Go, chess, and shogi. Unlike those games, TensorGame involved a very large search space, requiring changes to the AlphaZero-style search method and neural network architecture. DeepMind released source code and discovered algorithms associated with the publication through a public GitHub repository. == Results == AlphaTensor discovered matrix multiplication algorithms over both standard arithmetic and finite fields. One widely reported result was a method for multiplying 4 × 4 matrices over the field with two elements using 47 multiplication operations, improving on the 49 operations required by applying Strassen's algorithm recursively in that setting. The system also found algorithms optimized for particular computer hardware, including algorithms designed for graphics processing units and Tensor Processing Units. DeepMind stated that some of the hardware-specific algorithms improved practical execution time compared with commonly used algorithms on the tested hardware. == Significance == AlphaTensor was described as an example of using machine learning not only to apply existing algorithms, but to assist in discovering new ones. The work was connected to broader research in algorithm discovery, automated machine learning, program synthesis, and computational complexity theory, especially the open problem of determining the optimal complexity of matrix multiplication. AlphaTensor later became part of a broader group of Google DeepMind systems for algorithm and mathematical discovery, alongside systems such as AlphaDev and AlphaEvolve.
Mean shift
Mean shift is a non-parametric feature-space mathematical analysis technique for locating the maxima of a density function, a so-called mode-seeking algorithm. Application domains include cluster analysis in computer vision and image processing. == History == The mean shift procedure is usually credited to work by Fukunaga and Hostetler in 1975. It is, however, reminiscent of earlier work by Schnell in 1964. == Overview == Mean shift is a procedure for locating the maxima—the modes—of a density function given discrete data sampled from that function. This is an iterative method, and we start with an initial estimate x {\displaystyle x} . Let a kernel function K ( x i − x ) {\displaystyle K(x_{i}-x)} be given. This function determines the weight of nearby points for re-estimation of the mean. Typically a Gaussian kernel on the distance to the current estimate is used, K ( x i − x ) = e − c | | x i − x | | 2 {\displaystyle K(x_{i}-x)=e^{-c||x_{i}-x||^{2}}} . The weighted mean of the density in the window determined by K {\displaystyle K} is m ( x ) = ∑ x i ∈ N ( x ) K ( x i − x ) x i ∑ x i ∈ N ( x ) K ( x i − x ) {\displaystyle m(x)={\frac {\sum _{x_{i}\in N(x)}K(x_{i}-x)x_{i}}{\sum _{x_{i}\in N(x)}K(x_{i}-x)}}} where N ( x ) {\displaystyle N(x)} is the neighborhood of x {\displaystyle x} , a set of points for which K ( x i − x ) ≠ 0 {\displaystyle K(x_{i}-x)\neq 0} . The difference m ( x ) − x {\displaystyle m(x)-x} is called mean shift in Fukunaga and Hostetler. The mean-shift algorithm now sets x ← m ( x ) {\displaystyle x\leftarrow m(x)} , and repeats the estimation until m ( x ) {\displaystyle m(x)} converges. Although the mean shift algorithm has been widely used in many applications, a rigid proof for the convergence of the algorithm using a general kernel in a high dimensional space is still not known. Aliyari Ghassabeh showed the convergence of the mean shift algorithm in one dimension with a differentiable, convex, and strictly decreasing profile function. However, the one-dimensional case has limited real world applications. Also, the convergence of the algorithm in higher dimensions with a finite number of the stationary (or isolated) points has been proved. However, sufficient conditions for a general kernel function to have finite stationary (or isolated) points have not been provided. Gaussian Mean-Shift is an Expectation–maximization algorithm. == Details == Let data be a finite set S {\displaystyle S} embedded in the n {\displaystyle n} -dimensional Euclidean space, X {\displaystyle X} . Let K {\displaystyle K} be a flat kernel that is the characteristic function of the λ {\displaystyle \lambda } -ball in X {\displaystyle X} , In each iteration of the algorithm, s ← m ( s ) {\displaystyle s\leftarrow m(s)} is performed for all s ∈ S {\displaystyle s\in S} simultaneously. The first question, then, is how to estimate the density function given a sparse set of samples. One of the simplest approaches is to just smooth the data, e.g., by convolving it with a fixed kernel of width h {\displaystyle h} , where x i {\displaystyle x_{i}} are the input samples and k ( r ) {\displaystyle k(r)} is the kernel function (or Parzen window). h {\displaystyle h} is the only parameter in the algorithm and is called the bandwidth. This approach is known as kernel density estimation or the Parzen window technique. Once we have computed f ( x ) {\displaystyle f(x)} from the equation above, we can find its local maxima using gradient ascent or some other optimization technique. The problem with this "brute force" approach is that, for higher dimensions, it becomes computationally prohibitive to evaluate f ( x ) {\displaystyle f(x)} over the complete search space. Instead, mean shift uses a variant of what is known in the optimization literature as multiple restart gradient descent. Starting at some guess for a local maximum, y k {\displaystyle y_{k}} , which can be a random input data point x 1 {\displaystyle x_{1}} , mean shift computes the gradient of the density estimate f ( x ) {\displaystyle f(x)} at y k {\displaystyle y_{k}} and takes an uphill step in that direction. == Types of kernels == Kernel definition: Let X {\displaystyle X} be the n {\displaystyle n} -dimensional Euclidean space, R n {\displaystyle \mathbb {R} ^{n}} . The norm of x {\displaystyle x} is a non-negative number, ‖ x ‖ 2 = x ⊤ x ≥ 0 {\displaystyle \|x\|^{2}=x^{\top }x\geq 0} . A function K : X → R {\displaystyle K:X\rightarrow \mathbb {R} } is said to be a kernel if there exists a profile, k : [ 0 , ∞ ] → R {\displaystyle k:[0,\infty ]\rightarrow \mathbb {R} } , such that K ( x ) = k ( ‖ x ‖ 2 ) {\displaystyle K(x)=k(\|x\|^{2})} and k is non-negative. k is non-increasing: k ( a ) ≥ k ( b ) {\displaystyle k(a)\geq k(b)} if a < b {\displaystyle a An image pipeline or video pipeline is the set of components commonly used between an image source (such as a camera, a scanner, or the rendering engine in a computer game), and an image renderer (such as a television set, a computer screen, a computer printer or cinema screen), or for performing any intermediate digital image processing consisting of two or more separate processing blocks. An image/video pipeline may be implemented as computer software, in a digital signal processor, on an FPGA, or as fixed-function ASIC. In addition, analog circuits can be used to do many of the same functions. Typical components include image sensor corrections (including debayering or applying a Bayer filter), noise reduction, image scaling, gamma correction, image enhancement, colorspace conversion (between formats such as RGB, YUV or YCbCr), chroma subsampling, framerate conversion, image compression/video compression (such as JPEG), and computer data storage/data transmission. Typical goals of an imaging pipeline may be perceptually pleasing end-results, colorimetric precision, a high degree of flexibility, low cost/low CPU utilization/long battery life, or reduction in bandwidth/file size. Some functions may be algorithmically linear. Mathematically, those elements can be connected in any order without changing the end-result. As digital computers use a finite approximation to numerical computing, this is in practice not true. Other elements may be non-linear or time-variant. For both cases, there is often one or a few sequences of components that makes sense for optimum precision and minimum hardware-cost/CPU-load. In computer programming, a data cube (or datacube) is a multi-dimensional array of values. Typically, the term "data cube" is applied in contexts where these arrays are massively larger than the hosting computer's main memory; examples include multi-terabyte/petabyte data warehouses and time series of image data. Even though it is called a cube, a data cube generally is a multi-dimensional concept which can be 1-dimensional, 2-dimensional, 3-dimensional, or higher-dimensional. The data cube is used to represent data (sometimes called facts) along some dimensions of interest. In satellite image timeseries, dimensions would be latitude and longitude coordinates and time; a fact (sometimes called measure) would be a pixel at a given space and time as taken by the satellite. For example, in online analytical processing, an OLAP cube about a company would have dimensions that could be the company subsidiaries, the company products, and time; in this setup, a fact would be a sales event where a particular product has been sold in a particular subsidiary at a particular time. In any case, every dimension divides data into groups of cells whereas each cell in the cube represents a single measure of interest. Sometimes cubes hold only a few values with the rest being empty, i.e. undefined, while sometimes most or all cube coordinates hold a cell value. In the first case such data are called sparse, and in the second case they are called dense, although there is no hard delineation between the two. Data cubes may be stored in database management systems (DBMS) as part of array DBMS. Spatio-temporal databases and geospatial databases may also be represented as coverage data. == History == Multi-dimensional arrays have long been familiar in programming languages. Fortran offers arbitrarily-indexed 1-D arrays and arrays of arrays, which allows the construction of higher-dimensional arrays, up to 15 dimensions. APL supports n-D arrays with a rich set of operations. All these have in common that arrays must fit into the main memory and are available only while the particular program maintaining them (such as image processing software) is running. A series of data exchange formats support storage and transmission of data cube-like data, often tailored towards particular application domains. Examples include MDX for statistical (in particular, business) data, Zarr and Hierarchical Data Format for general scientific data, and TIFF for imagery. In 1992, Peter Baumann introduced management of massive data cubes with high-level user functionality combined with an efficient software architecture. Datacube operations include subset extraction, processing, fusion, and in general queries in the spirit of data manipulation languages like SQL. Some years after, the data cube concept was applied to describe time-varying business data as data cubes by Jim Gray, et al., and by Venky Harinarayan, Anand Rajaraman and Jeff Ullman. Around that time, a working group on Multi-Dimensional Databases ("Arbeitskreis Multi-Dimensionale Datenbanken") was established at German Gesellschaft für Informatik. Datacube Inc. was an image processing company selling hardware and software applications for the PC market in 1996, however without addressing data cubes as such. The EarthServer initiative has established geo data cube service requirements. == Standardization == In 2018, the ISO SQL database language was extended with data cube functionality as "SQL – Part 15: Multi-dimensional arrays (SQL/MDA)". Web Coverage Processing Service is a geo data cube analytics language issued by the Open Geospatial Consortium in 2008. In addition to the common data cube operations, the language knows about the semantics of space and time and supports both regular and irregular grid data cubes, based on the concept of coverage data. An industry standard for querying business data cubes, originally developed by Microsoft, is MultiDimensional eXpressions. == Implementation == Many high-level computer languages treat data cubes and other large arrays as single entities distinct from their contents. These languages, of which Fortran, APL, IDL, NumPy, PDL, and S-Lang are examples, allow the programmer to manipulate complete film clips and other data en masse with simple expressions derived from linear algebra and vector mathematics. Some languages (such as PDL) distinguish between a list of images and a data cube, while many (such as IDL) do not. Array DBMSs (Database Management Systems) offer a data model which generically supports definition, management, retrieval, and manipulation of n-dimensional data cubes. This database category has been pioneered by the rasdaman system since 1994. == Applications == Multi-dimensional arrays can meaningfully represent spatio-temporal sensor, image, and simulation data, but also statistics data where the semantics of dimensions is not necessarily of spatial or temporal nature. Generally, any kind of axis can be combined with any other into a data cube. === Mathematics === In mathematics, a one-dimensional array corresponds to a vector, a two-dimensional array resembles a matrix; more generally, a tensor may be represented as an n-dimensional data cube. === Science and engineering === For a time sequence of color images, the array is generally four-dimensional, with the dimensions representing image X and Y coordinates, time, and RGB (or other color space) color plane. For example, the EarthServer initiative unites data centers from different continents offering 3-D x/y/t satellite image timeseries and 4-D x/y/z/t weather data for retrieval and server-side processing through the Open Geospatial Consortium WCPS geo data cube query language standard. A data cube is also used in the field of imaging spectroscopy, since a spectrally-resolved image is represented as a three-dimensional volume. Earth observation data cubes combine satellite imagery such as Landsat 8 and Sentinel-2 with Geographic information system analytics. === Business intelligence === In online analytical processing (OLAP), data cubes are a common arrangement of business data suitable for analysis from different perspectives through operations like slicing, dicing, pivoting, and aggregation. The Application Lifecycle Framework (ALF) was a project by the Eclipse Foundation that aimed to create a standardized, open-source system to allow different application lifecycle management (ALM) tools to work together more easily. The goal was to provide common protocols and integration services that would let software development tools from different vendors communicate and share data. However, the project failed to gain sufficient support from major industry players and was terminated in 2008. A bigram or digram is a sequence of two adjacent elements from a string of tokens, which are typically letters, syllables, or words. A bigram is an n-gram for n=2. The frequency distribution of every bigram in a string is commonly used for simple statistical analysis of text in many applications, including in computational linguistics, cryptography, and speech recognition. Gappy bigrams or skipping bigrams are word pairs which allow gaps (perhaps avoiding connecting words, or allowing some simulation of dependencies, as in a dependency grammar). == Applications == Bigrams, along with other n-grams, are used in most successful language models for speech recognition. Bigram frequency attacks can be used in cryptography to solve cryptograms. See frequency analysis. Bigram frequency is one approach to statistical language identification. Some activities in logology or recreational linguistics involve bigrams. These include attempts to find English words beginning with every possible bigram, or words containing a string of repeated bigrams, such as logogogue. == Bigram frequency in the English language == The frequency of the most common letter bigrams in a large English corpus is: th 3.56% of 1.17% io 0.83% he 3.07% ed 1.17% le 0.83% in 2.43% is 1.13% ve 0.83% er 2.05% it 1.12% co 0.79% an 1.99% al 1.09% me 0.79% re 1.85% ar 1.07% de 0.76% on 1.76% st 1.05% hi 0.76% at 1.49% to 1.05% ri 0.73% en 1.45% nt 1.04% ro 0.73% nd 1.35% ng 0.95% ic 0.70% ti 1.34% se 0.93% ne 0.69% es 1.34% ha 0.93% ea 0.69% or 1.28% as 0.87% ra 0.69% te 1.20% ou 0.87% ce 0.65% Yahoo! Mail (also written as Yahoo Mail) is a mailbox provider by Yahoo. It is one of the largest email services worldwide, with 225 million users. It is accessible via a web browser (webmail), mobile app, or through third-party email clients via the POP, SMTP, and IMAP protocols. Users can also connect non-Yahoo e-mail accounts to their Yahoo Mail inbox. The service was launched on October 8, 1997. The service is free for personal use, with an optional monthly fee for additional features. It is also available in several languages other than English. == History == === 1997–2002 === On October 8, 1997, Yahoo announced its acquisition of online communications company Four11 for $92 million in stock. As part of the purchase, Yahoo received Four11's RocketMail webmail service. Yahoo Mail, based on the RocketMail technology, launched at the same time. Yahoo! chose acquisition rather than internal platform development, because, as Healy said, "Hotmail was growing at thousands and thousands users per week. We did an analysis. For us to build, it would have taken four to six months, and by then, so many users would have taken an email account. The speed of the market was critical." On March 21, 2002, Yahoo! eliminated free software client access and introduced the $29.99 per year Mail Forwarding Service. Mary Osako, a Yahoo! Spokeswoman, told CNET, "For-pay services on Yahoo!, originally launched in February 1999, have experienced great acceptance from our base of active registered users, and we expect this adoption to continue to grow." === 2002–2010 === During 2002, the Yahoo network was gradually redesigned, including the company website, Yahoo Mail and other services. Along with the new design, new features were implemented, including drop-down menus in DHTML and keyboard shortcuts. On July 9, 2004, Yahoo! acquired Oddpost, a webmail service which simulated a desktop email client. Oddpost had features such as drag-and-drop support, right-click menus, RSS feeds, a preview pane, and increased speed using email caching to shorten response time. Many of the features were incorporated into an updated Yahoo! Mail service. ==== Competition ==== On April 1, 2004, Google announced its Gmail service with 1 GB of storage, although Gmail's invitation-only accounts kept the other webmail services at the forefront. Most major webmail providers, including Yahoo! Mail, increased their mailbox storage in response. Yahoo! first announced 100 MB of storage for basic accounts and 2 GB of storage for premium users. However, soon Yahoo Mail increased its free storage quota to 1 GB, before eventually allowing unlimited storage from March 27, 2007, until October 8, 2013. === 2011–2021 === In May 2011, Yahoo Mail rolled out a new interface. It included updated design, enhanced performance, and improved Facebook integration. In 2013, Yahoo! redesigned the site and removed several features, such as simultaneously opening multiple emails in tabs, sorting by sender name, and dragging mails to folders. The new email interface was geared to give an improved user-experience for mobile devices, but was criticized for having an inferior desktop interface. Many users objected to the unannounced nature of the changes through an online post asking Yahoo! to bring back mail tabs with one hundred thousand voting and nearly ten thousand commenting. The redesign produced a problem that caused an unknown number of users to lose access to their accounts for several weeks. In December 2013, Yahoo! Mail suffered a major outage where approximately one million users, one percent of the site's total users, could not access their emails for several days. Yahoo!'s then-CEO Marissa Mayer publicly apologized to the site's users. China Yahoo Mail announced in April 2013 that it would shut down that August as part of Yahoo ceasing services in China since acquiring a stake in Alibaba in 2005. Users with email address suffixes @yahoo.com.cn and @yahoo.cn could transfer their accounts to AliCloud to continue receiving messages through the end of 2014. In January 2014, an undisclosed number of usernames and passwords were released to hackers, following a security breach that Yahoo! believed had occurred through a third-party website. Yahoo! contacted affected users and requested that passwords be changed. In October 2015, Yahoo! updated the mail service with a "more subtle" redesign, as well as improved mobile features. The same release introduced the Yahoo! Account Key, a smartphone-based replacement for password logins. The app also added support for third-party mail accounts. In 2017, Yahoo! again redesigned the web interface with a "more minimal" look, and introduced the option to customize it with different color themes and layouts. In 2019, Yahoo released a redesigned Yahoo Mail app to organize user inboxes, introducing features including a one-tap unsubscribe tool, package tracking, and travel updates. In 2020, Yahoo Mail users were able to fill Walmart shopping carts directly from their inboxes, an industry first. Yahoo! also added a feature to view NFL matches. === 2022–present === In 2022, updates to the Yahoo Mail mobile app added tools to help manage receipts, gift cards, and subscriptions. AI-based additions in 2023 included a feature that automates tracking coupon codes and credits for online shopping, as well as updates to search suggestions, message summaries and AI writing assistance. In 2024, updates to the desktop interface added more AI-based features, including a "priority inbox" tab with automatically generated summaries of important messages and automated suggestions of next actions based on message contents. In February 2025, Yahoo aired its first Super Bowl ad since 2002, in which Bill Murray invited viewers to contact him at his Yahoo Mail email address ([email protected]). The address received nearly 150,000 emails in the first two hours after broadcast. In June 2025, Yahoo Mail introduced a "Catch Up" feature that provides AI-generated summaries and email previews and prompts users to choose to delete or retain each one. As part of the feature's launch, Yahoo Mail collaborated with streetwear brand Anti Social Social Club on an apparel release. == User interface == As many as three web interfaces were available at any given time. The traditional "Yahoo! Mail Classic" preserved the availability of their original 1997 interface until July 2013 in North America. A 2005 version included a new Ajax interface, drag-and-drop, improved search, keyboard shortcuts, address auto-completion, and tabs. However, other features were removed, such as column widths and one click delete-move-to-next. In October 2010, Yahoo! released a beta version of Yahoo! Mail, which included improvements to performance, search, and Facebook integration. In May 2011, this became the default interface. Their current Webmail interface was introduced in 2017. == Spam policy == Yahoo! Mail is often used by spammers to provide a "remove me" email address. Often, these addresses are used to verify the recipient's address, thus opening the door for more spam. Yahoo! does not tolerate this practice and terminates accounts connected with spam-related activities without warning, causing spammers to lose access to any other Yahoo! services connected with their ID under the Terms of Service. Additionally, Yahoo! stresses that its servers are based in California and any spam-related activity which uses its servers could potentially violate that state's anti-spam laws. In February 2006, Yahoo! announced its decision (along with AOL) to give some organizations the option to "certify" mail by paying up to one cent for each outgoing message, allowing the mail in question to bypass inbound spam filters. Few mailers used it and, Goodmail, the company running the certification process, shut down in 2011. === Filters === In order to prevent abuse, in 2002 Yahoo! Mail activated filters which changed certain words (that could trigger unwanted JavaScript events) and word fragments into other words. "mocha" was changed to "espresso", "expression" became "statement", and "eval" (short for "evaluation") became "review". This resulted in many unintended corrections, such as "prevent" (prevalent), "revalidation" (evaluation) and "media review" (medieval). When asked about these changes, Yahoo! explained that the changed words were common terms used in their privacy dashboard and were blacklisted to prevent hackers from sending damaging commands via the program's HTML function. Starting before February 7, 2006, Yahoo! Mail ended the practice, and began to add an underscore as a prefix to certain suspicious words and word fragments. === Greylisting === Incoming mail to Yahoo! addresses can be subjected to deferred delivery as part of Yahoo's incoming spam controls. This can delay delivery of mail sent to Yahoo! addresses without the sender or recipients being aware of it. The deferral is typically of short duration, but Color image pipeline
Data cube
Application Lifecycle Framework
Bigram
Yahoo Mail