AI Data Operations

AI Data Operations — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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  Microsoft Teams

  Microsoft Teams is a team collaboration platform developed by Microsoft as part of the Microsoft 365 suite. It offers features such as workspace chat, video conferencing, file storage, and integration with both Microsoft and third-party applications and services. Teams gradually replaced earlier Microsoft messaging and collaboration platforms, including Skype for Business, Skype, Flip, and Microsoft Classroom. The platform saw significant growth during the COVID-19 pandemic, alongside competitors such as Zoom, Slack, and Google Meet, as organizations shifted to remote work and virtual meetings. As of January 2023, Microsoft reported approximately 280 million monthly active users. == History == On August 29, 2007, Microsoft acquired Parlano, the developer of the persistent group chat tool MindAlign. Years later, on March 4, 2016, Microsoft considered acquiring Slack for $8 billion. However, the proposal was reportedly opposed by Bill Gates, who advocated for focusing on enhancing Skype for Business instead. Lu Qi, then executive vice president of Applications and Services, had led the initiative to pursue the Slack acquisition. Following Lu's departure later that year, Microsoft announced Microsoft Teams on November 2, 2016, at an event in New York City, positioning it as a direct competitor to Slack. Teams launched worldwide on March 14, 2017. The service was initially led by corporate vice president Brian MacDonald. In response to the launch, Slack published a full-page advertisement in The New York Times welcoming the competition and outlining its product philosophy. Although Slack was used by 28 companies in the Fortune 100, The Verge wrote that executives would question paying for the service if Teams provides a similar function in their company's existing Office 365 subscription. However, ZDNET noted that the platforms initially served different markets, as Teams did not support external users, making it less appealing to small businesses and freelancers, a limitation Microsoft later addressed. In response to Teams' announcement, Slack deepened in-product integration with Google services. In May 2017, Microsoft announced that Teams would replace Microsoft Classroom in Office 365 Education. A free version of Teams was released on July 12, 2018, offering most core features at no cost, albeit with limits on users and storage. In January 2019, Microsoft introduced updates targeting "Firstline Workers" to improve Teams’ performance across shared or limited-access devices. In September 2019, Microsoft announced the retirement of Skype for Business in favor of Teams, which took effect on July 31, 2021. In early 2020, Microsoft introduced a push-to-talk "Walkie Talkie" feature aimed at firstline workers using smartphones and tablets over Wi-Fi or cellular networks. The COVID-19 pandemic significantly boosted usage of Teams. On March 19, 2020, Microsoft reported 44 million daily active users. In April, the platform logged 4.1 billion meeting minutes in a single day. A public preview of Microsoft Teams for Linux was released in December 2019, but the Linux client was discontinued in 2022. In July 2020, Microsoft shut down its video game livestreaming platform Mixer, and announced that some of its technologies would be repurposed for use in Teams. On February 28, 2025, Microsoft announced that Skype would be fully retired on May 5, 2025, with users given options to export their data or transition to Microsoft Teams. In October 2025, together with other Microsoft 365 suite apps, Teams had its logo updated. == Usage == == Underlying software == Microsoft Teams, as part of the Microsoft 365 suite, utilizes SharePoint and Exchange Online. Each Team, Shared Channel, and Private Channel has its own Microsoft 365 Group and SharePoint Site used for file storage. Messages are stored in Cosmos DB and are journaled to Exchange Online mailboxes. Private messages, including messages in Private Channels, are journaled to the sender and recipients' mailboxes. Public Channel messages are journaled to their corresponding Team's group mailbox, whereas, messages from Shared Channels are journaled to their own mailboxes. Contacts and voicemail are stored in Exchange Online. Microsoft Teams client is a web-based desktop app, originally developed on top of the Electron framework which combines the Chromium rendering engine and the Node.js JavaScript platform. Version 2.0 client was rebuilt using the Evergreen version of Microsoft Edge WebView2 in place of Electron. == Features == === Chats === Teams allows users to communicate in two-way persistent chats with one or multiple participants. Participants can message using text, emojis, stickers and gifs, as well as sharing links and files. In August 2022, the chat feature was updated for "chat with yourself"; allowing for the organization of files, notes, comments, images, and videos within a private chat tab. === Teams === Teams allows communities, groups, or teams to contribute in a shared workspace where messages and digital content on a specific topic are shared. Team members can join through an invitation sent by a team administrator or owner or sharing of a specific URL. Teams for Education allows admins and teachers to set up groups for classes, professional learning communities (PLCs), staff members, and everyone. === Channels === Channels allow team members to communicate without the use of email or group SMS (texting). Users can reply to posts with text, images, GIFs, and image macros. Direct messages send private messages to designated users rather than the entire channel. Connectors can be used within a channel to submit information contacted through a third-party service. Connectors include Mailchimp, Facebook Pages, Twitter, Power BI and Bing News. === Group conversations === Ad-hoc groups can be created to share instant messaging, audio calls (VoIP), and video calls inside the client software. === Telephone replacement === A feature on one of the higher cost licencing tiers allows connectivity to the public switched telephone network (PSTN) telephone system. This allows users to use Teams as if it were a telephone, making and receiving calls over the PSTN, including the ability to host "conference calls" with multiple participants. === Meeting === Meetings can be scheduled with multiple participants able to share audio, video, chat and presented content with all participants. Multiple users can connect via a meeting link. Automated minutes are possible using the recording and transcript features. Teams has a plugin for Microsoft Outlook to schedule a Teams Meeting in Outlook for a specific date and time and invite others to attend. If a meeting is scheduled within a channel, users visiting the channel are able to see if a meeting is in progress. ==== Teams Live Events ==== Teams Live Events replaces Skype Meeting Broadcast for users to broadcast to 10,000 participants on Teams, Yammer, or Microsoft Stream. ==== Breakout Rooms ==== Breakout rooms split a meeting into small groups. This is often utilized for collaboration during trainings or any environment where having all participants speak at once could be disruptive or unfeasible. Breakout rooms can be set by the hosts to a certain length of time, after which all participants will automatically rejoin the main meeting room. ==== Front Row ==== Front Row adjusts the layout of the viewer's screen, placing the speaker or content in the center of the gallery with other meeting participant's video feeds reduced in size and located below the speaker. === Education === Microsoft Teams for Education allows teachers to distribute, provide feedback, and grade student assignments turned in via Teams using the Assignments tab through Office 365 for Education subscribers. Quizzes can also be assigned to students through an integration with Office Forms. === Protocols === Microsoft Teams is based on a number of Microsoft-specific protocols. Video conferences are realized over the protocol MNP24, known from the Skype consumer version. VoIP and video conference clients based on SIP and H.323 need special gateways to connect to Microsoft Teams servers. With the help of Interactive Connectivity Establishment (ICE), clients behind Network address translation routers and restrictive firewalls are also able to connect, if peer-to-peer is not possible. === Integrations === Microsoft Teams has integrations through Microsoft AppSource, its integration marketplace. In 2020, Microsoft partnered with KUDO, a cloud-based solution with language interpretation, to allow integrated language meeting controls. In June 2022, an update was released using AI to improve call audio through the elimination of background feedback loops and cancelling non-vocal audio. == Anti-trust controversy == In July 2023, the European Commission opened an anti-trust investigation into the possibility that Microsoft unfairly used its office suite market power to increase sales of Teams and hurt

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  Elastic net regularization

  In statistics and, in particular, in the fitting of linear or logistic regression models, the elastic net is a regularized regression method that linearly combines the L1 and L2 penalties of the lasso and ridge methods. Nevertheless, elastic net regularization is typically more accurate than both methods with regard to reconstruction. == Specification == The elastic net method overcomes the limitations of the LASSO (least absolute shrinkage and selection operator) method which uses a penalty function based on ‖ β ‖ 1 = ∑ j = 1 p | β j | . {\displaystyle \|\beta \|_{1}=\textstyle \sum _{j=1}^{p}|\beta _{j}|.} Use of this penalty function has several limitations. For example, in the "large p, small n" case (high-dimensional data with few examples), the LASSO selects at most n variables before it saturates. Also if there is a group of highly correlated variables, then the LASSO tends to select one variable from a group and ignore the others. To overcome these limitations, the elastic net adds a quadratic part ( ‖ β ‖ 2 {\displaystyle \|\beta \|^{2}} ) to the penalty, which when used alone is ridge regression (known also as Tikhonov regularization). The estimates from the elastic net method are defined by β ^ ≡ argmin β ( ‖ y − X β ‖ 2 + λ 2 ‖ β ‖ 2 + λ 1 ‖ β ‖ 1 ) . {\displaystyle {\hat {\beta }}\equiv {\underset {\beta }{\operatorname {argmin} }}(\|y-X\beta \|^{2}+\lambda _{2}\|\beta \|^{2}+\lambda _{1}\|\beta \|_{1}).} The quadratic penalty term makes the loss function strongly convex, and it therefore has a unique minimum. The elastic net method includes the LASSO and ridge regression: in other words, each of them is a special case where λ 1 = λ , λ 2 = 0 {\displaystyle \lambda _{1}=\lambda ,\lambda _{2}=0} or λ 1 = 0 , λ 2 = λ {\displaystyle \lambda _{1}=0,\lambda _{2}=\lambda } . Meanwhile, the naive version of elastic net method finds an estimator in a two-stage procedure : first for each fixed λ 2 {\displaystyle \lambda _{2}} it finds the ridge regression coefficients, and then does a LASSO type shrinkage. This kind of estimation incurs a double amount of shrinkage, which leads to increased bias and poor predictions. To improve the prediction performance, sometimes the coefficients of the naive version of elastic net is rescaled by multiplying the estimated coefficients by ( 1 + λ 2 ) {\displaystyle (1+\lambda _{2})} . Examples of where the elastic net method has been applied are: Support vector machine Metric learning Portfolio optimization Cancer prognosis == Reduction to support vector machine == It was proven in 2014 that the elastic net can be reduced to the linear support vector machine. A similar reduction was previously proven for the LASSO in 2014. The authors showed that for every instance of the elastic net, an artificial binary classification problem can be constructed such that the hyper-plane solution of a linear support vector machine (SVM) is identical to the solution β {\displaystyle \beta } (after re-scaling). The reduction immediately enables the use of highly optimized SVM solvers for elastic net problems. It also enables the use of GPU acceleration, which is often already used for large-scale SVM solvers. The reduction is a simple transformation of the original data and regularization constants X ∈ R n × p , y ∈ R n , λ 1 ≥ 0 , λ 2 ≥ 0 {\displaystyle X\in {\mathbb {R} }^{n\times p},y\in {\mathbb {R} }^{n},\lambda _{1}\geq 0,\lambda _{2}\geq 0} into new artificial data instances and a regularization constant that specify a binary classification problem and the SVM regularization constant X 2 ∈ R 2 p × n , y 2 ∈ { − 1 , 1 } 2 p , C ≥ 0. {\displaystyle X_{2}\in {\mathbb {R} }^{2p\times n},y_{2}\in \{-1,1\}^{2p},C\geq 0.} Here, y 2 {\displaystyle y_{2}} consists of binary labels − 1 , 1 {\displaystyle {-1,1}} . When 2 p > n {\displaystyle 2p>n} it is typically faster to solve the linear SVM in the primal, whereas otherwise the dual formulation is faster. Some authors have referred to the transformation as Support Vector Elastic Net (SVEN), and provided the following MATLAB pseudo-code: == Software == "Glmnet: Lasso and elastic-net regularized generalized linear models" is a software which is implemented as an R source package and as a MATLAB toolbox. This includes fast algorithms for estimation of generalized linear models with ℓ1 (the lasso), ℓ2 (ridge regression) and mixtures of the two penalties (the elastic net) using cyclical coordinate descent, computed along a regularization path. JMP Pro 11 includes elastic net regularization, using the Generalized Regression personality with Fit Model. "pensim: Simulation of high-dimensional data and parallelized repeated penalized regression" implements an alternate, parallelised "2D" tuning method of the ℓ parameters, a method claimed to result in improved prediction accuracy. scikit-learn includes linear regression and logistic regression with elastic net regularization. SVEN, a Matlab implementation of Support Vector Elastic Net. This solver reduces the Elastic Net problem to an instance of SVM binary classification and uses a Matlab SVM solver to find the solution. Because SVM is easily parallelizable, the code can be faster than Glmnet on modern hardware. SpaSM, a Matlab implementation of sparse regression, classification and principal component analysis, including elastic net regularized regression. Apache Spark provides support for Elastic Net Regression in its MLlib machine learning library. The method is available as a parameter of the more general LinearRegression class. SAS (software) The SAS procedure Glmselect and SAS Viya procedure Regselect support the use of elastic net regularization for model selection.

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  TabPFN

  TabPFN (Tabular Prior-data Fitted Network) is a machine learning model for tabular datasets proposed in 2022. It uses a transformer architecture. It is intended for supervised classification and regression analysis on tabular datasets, particularly focusing on small- to medium-sized datasets. The latest version, TabPFN-3, was released in May 2026 and supports datasets with up to one million rows and 200 features. == History == TabPFN was first introduced in a 2022 pre-print and presented at ICLR 2023. TabPFN v2 was published in 2025 in Nature by Hollmann and co-authors. The source code is published on GitHub under a modified Apache License and on PyPi. Writing for ICLR blogs, McCarter states that the model has attracted attention due to its performance on small dataset benchmarks. TabPFN v2.5 was released on November 6, 2025. TabPFN-3 was released on May 12, 2026. Prior Labs, founded in 2024, aims to commercialize TabPFN. As of April 2026, the open-source TabPFN repository had more than 6,000 stars on GitHub. == Overview and pre-training == TabPFN supports classification, regression and generative tasks. It leverages "Prior-Data Fitted Networks" models to model tabular data. By using a transformer pre-trained on synthetic tabular datasets, TabPFN avoids benchmark contamination and costs of curating real-world data. TabPFN v2 was pre-trained on approximately 130 million such datasets. Synthetic datasets are generated using causal models or Bayesian neural networks; this can include simulating missing values, imbalanced data, and noise. Random inputs are passed through these models to generate outputs, with a bias towards simpler causal structures. During pre-training, TabPFN predicts the masked target values of new data points given training data points and their known targets, effectively learning a generic learning algorithm that is executed by running a neural network forward pass. The new dataset is then processed in a single forward pass without retraining. The model's transformer encoder processes features and labels by alternating attention across rows and columns. TabPFN v2 handles numerical and categorical features, missing values, and supports tasks like regression and synthetic data generation, while TabPFN-2.5 scales this approach to datasets with up to 50,000 rows and 2,000 features. TabPFN-3 introduced a redesigned architecture with row-compression, an attention-based many-class decoder, native missing-value handling, and inference optimizations such as row chunking and a reduced key-value cache, with benchmark-validated regimes of up to 1 million rows with 200 features, 100,000 rows with 2,000 features, or 1,000 rows with 20,000 features. Since TabPFN is pre-trained, in contrast to other deep learning methods, it does not require costly hyperparameter optimization. == Research == TabPFN is the subject of on-going research. Applications for TabPFN have been investigated for domains such as chemoproteomics, insurance risk classification, and metagenomics. In clinical research, TabPFN was used in a study on the early detection of pancreatic cancer from blood samples, where it was combined with metabolomic data and reported high diagnostic performance. == Applications == TabPFN has been used in industrial and biomedical contexts. Hitachi Ltd. has been reported to use the model for predictive maintenance in rail networks, with its use described as helping to identify track issues earlier and reduce manual inspections. In the biomedical domain, Oxford Cancer Analytics has used TabPFN in the analysis of proteomic data in lung disease research. A 2025 ML Contests report noted that the winners of DrivenData's PREPARE challenge used TabPFN to generate features for gradient-boosted decision tree models. == Limitations == TabPFN has been criticized for its "one large neural network is all you need" approach to modeling problems. Further, its performance is limited in high-dimensional and large-scale datasets. == Scaling Mode == In late November 2025, Prior Labs introduced ‘‘Scaling Mode’’, an operating mode for TabPFN designed to remove the fixed upper bound on training set size. Earlier versions of TabPFN had been optimized and validated primarily for datasets of up to 100,000 rows, whereas Scaling Mode was reported to extend support to substantially larger datasets, with benchmarked experiments on datasets containing up to 10 million rows. According to Prior Labs, Scaling Mode preserves the existing TabPFN architecture, including its alternating row-attention and column-attention design, as well as the same feature-count limits as prior releases. Inference remains based on a single forward pass rather than dataset-specific gradient-descent training, while scalability is described as being constrained primarily by available compute and memory resources. Prior Labs reported benchmark results on an internal collection of datasets ranging from 1 million to 10 million rows across industry and scientific applications. In these benchmarks, Scaling Mode was compared with CatBoost, XGBoost, LightGBM, and TabPFN 2.5 using 50,000-row subsampling. The company stated that predictive performance improved monotonically with increasing training set size and that no diminishing returns from scaling were observed within the tested range. Prior Labs also announced the release through company and executive social media channels. TabPFN-3 later incorporated related scaling improvements, including row chunking and a reduced key-value cache, into the model architecture and inference pipeline.

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  Autoencoder

  An autoencoder is a type of artificial neural network used to learn efficient codings of unlabeled data (unsupervised learning). An autoencoder learns two functions: an encoding function that transforms the input data, and a decoding function that recreates the input data from the encoded representation. The autoencoder learns an efficient representation (encoding) for a set of data, typically for dimensionality reduction, to generate lower-dimensional embeddings for subsequent use by other machine learning algorithms. Variants exist which aim to make the learned representations assume useful properties. Examples are regularized autoencoders (sparse, denoising and contractive autoencoders), which are effective in learning representations for subsequent classification tasks, and variational autoencoders, which can be used as generative models. Autoencoders are applied to many problems, including facial recognition, feature detection, anomaly detection, and learning the meaning of words. In terms of data synthesis, autoencoders can also be used to randomly generate new data that is similar to the input (training) data. == Mathematical principles == === Definition === An autoencoder is defined by the following components: Two sets: the space of encoded messages Z {\displaystyle {\mathcal {Z}}} ; the space of decoded messages X {\displaystyle {\mathcal {X}}} . Typically X {\displaystyle {\mathcal {X}}} and Z {\displaystyle {\mathcal {Z}}} are Euclidean spaces, that is, X = R m , Z = R n {\displaystyle {\mathcal {X}}=\mathbb {R} ^{m},{\mathcal {Z}}=\mathbb {R} ^{n}} with m > n . {\displaystyle m>n.} Two parametrized families of functions: the encoder family E ϕ : X → Z {\displaystyle E_{\phi }:{\mathcal {X}}\rightarrow {\mathcal {Z}}} , parametrized by ϕ {\displaystyle \phi } ; the decoder family D θ : Z → X {\displaystyle D_{\theta }:{\mathcal {Z}}\rightarrow {\mathcal {X}}} , parametrized by θ {\displaystyle \theta } .For any x ∈ X {\displaystyle x\in {\mathcal {X}}} , we usually write z = E ϕ ( x ) {\displaystyle z=E_{\phi }(x)} , and refer to it as the code, the latent variable, latent representation, latent vector, etc. Conversely, for any z ∈ Z {\displaystyle z\in {\mathcal {Z}}} , we usually write x ′ = D θ ( z ) {\displaystyle x'=D_{\theta }(z)} , and refer to it as the (decoded) message. Usually, both the encoder and the decoder are defined as multilayer perceptrons (MLPs). For example, a one-layer-MLP encoder E ϕ {\displaystyle E_{\phi }} is: E ϕ ( x ) = σ ( W x + b ) {\displaystyle E_{\phi }(\mathbf {x} )=\sigma (Wx+b)} where σ {\displaystyle \sigma } is an element-wise activation function, W {\displaystyle W} is a "weight" matrix, and b {\displaystyle b} is a "bias" vector. === Training an autoencoder === An autoencoder, by itself, is simply a tuple of two functions. To judge its quality, we need a task. A task is defined by a reference probability distribution μ r e f {\displaystyle \mu _{ref}} over X {\displaystyle {\mathcal {X}}} , and a "reconstruction quality" function d : X × X → [ 0 , ∞ ] {\displaystyle d:{\mathcal {X}}\times {\mathcal {X}}\to [0,\infty ]} , such that d ( x , x ′ ) {\displaystyle d(x,x')} measures how much x ′ {\displaystyle x'} differs from x {\displaystyle x} . With those, we can define the loss function for the autoencoder as L ( θ , ϕ ) := E x ∼ μ r e f [ d ( x , D θ ( E ϕ ( x ) ) ) ] {\displaystyle L(\theta ,\phi ):=\mathbb {\mathbb {E} } _{x\sim \mu _{ref}}[d(x,D_{\theta }(E_{\phi }(x)))]} The optimal autoencoder for the given task ( μ r e f , d ) {\displaystyle (\mu _{ref},d)} is then arg ⁡ min θ , ϕ L ( θ , ϕ ) {\displaystyle \arg \min _{\theta ,\phi }L(\theta ,\phi )} . The search for the optimal autoencoder can be accomplished by any mathematical optimization technique, but usually by gradient descent. This search process is referred to as "training the autoencoder". In most situations, the reference distribution is just the empirical distribution given by a dataset { x 1 , . . . , x N } ⊂ X {\displaystyle \{x_{1},...,x_{N}\}\subset {\mathcal {X}}} , so that μ r e f = 1 N ∑ i = 1 N δ x i {\displaystyle \mu _{ref}={\frac {1}{N}}\sum _{i=1}^{N}\delta _{x_{i}}} where δ x i {\displaystyle \delta _{x_{i}}} is the Dirac measure, the quality function is just L 2 {\displaystyle L^{2}} loss: d ( x , x ′ ) = ‖ x − x ′ ‖ 2 2 {\displaystyle d(x,x')=\|x-x'\|_{2}^{2}} , and ‖ ⋅ ‖ 2 {\displaystyle \|\cdot \|_{2}} is the Euclidean norm. Then the problem of searching for the optimal autoencoder is just a least-squares optimization: min θ , ϕ L ( θ , ϕ ) , where L ( θ , ϕ ) = 1 N ∑ i = 1 N ‖ x i − D θ ( E ϕ ( x i ) ) ‖ 2 2 {\displaystyle \min _{\theta ,\phi }L(\theta ,\phi ),\qquad {\text{where }}L(\theta ,\phi )={\frac {1}{N}}\sum _{i=1}^{N}\|x_{i}-D_{\theta }(E_{\phi }(x_{i}))\|_{2}^{2}} === Interpretation === An autoencoder has two main parts: an encoder that maps the message to a code, and a decoder that reconstructs the message from the code. An optimal autoencoder would perform as close to perfect reconstruction as possible, with "close to perfect" defined by the reconstruction quality function d {\displaystyle d} . The simplest way to perform the copying task perfectly would be to duplicate the signal. To suppress this behavior, the code space Z {\displaystyle {\mathcal {Z}}} usually has fewer dimensions than the message space X {\displaystyle {\mathcal {X}}} . Such an autoencoder is called undercomplete. It can be interpreted as compressing the message, or reducing its dimensionality. At the limit of an ideal undercomplete autoencoder, every possible code z {\displaystyle z} in the code space is used to encode a message x {\displaystyle x} that really appears in the distribution μ r e f {\displaystyle \mu _{ref}} , and the decoder is also perfect: D θ ( E ϕ ( x ) ) = x {\displaystyle D_{\theta }(E_{\phi }(x))=x} . This ideal autoencoder can then be used to generate messages indistinguishable from real messages, by feeding its decoder arbitrary code z {\displaystyle z} and obtaining D θ ( z ) {\displaystyle D_{\theta }(z)} , which is a message that really appears in the distribution μ r e f {\displaystyle \mu _{ref}} . If the code space Z {\displaystyle {\mathcal {Z}}} has dimension larger than (overcomplete), or equal to, the message space X {\displaystyle {\mathcal {X}}} , or the hidden units are given enough capacity, an autoencoder can learn the identity function and become useless. However, experimental results found that overcomplete autoencoders might still learn useful features. In the ideal setting, the code dimension and the model capacity could be set on the basis of the complexity of the data distribution to be modeled. A standard way to do so is to add modifications to the basic autoencoder, to be detailed below. == Variations == === Variational autoencoder (VAE) === Variational autoencoders (VAEs) belong to the families of variational Bayesian methods. Despite the architectural similarities with basic autoencoders, VAEs are architected with different goals and have a different mathematical formulation. The latent space is, in this case, composed of a mixture of distributions instead of fixed vectors. Given an input dataset x {\displaystyle x} characterized by an unknown probability function P ( x ) {\displaystyle P(x)} and a multivariate latent encoding vector z {\displaystyle z} , the objective is to model the data as a distribution p θ ( x ) {\displaystyle p_{\theta }(x)} , with θ {\displaystyle \theta } defined as the set of the network parameters so that p θ ( x ) = ∫ z p θ ( x , z ) d z {\displaystyle p_{\theta }(x)=\int _{z}p_{\theta }(x,z)dz} . === Sparse autoencoder (SAE) === Inspired by the sparse coding hypothesis in neuroscience, sparse autoencoders (SAE) are variants of autoencoders, such that the codes E ϕ ( x ) {\displaystyle E_{\phi }(x)} for messages tend to be sparse codes, that is, E ϕ ( x ) {\displaystyle E_{\phi }(x)} is close to zero in most entries. Sparse autoencoders may include more (rather than fewer) hidden units than inputs, but only a small number of the hidden units are allowed to be active at the same time. Encouraging sparsity improves performance on classification tasks. There are two main ways to enforce sparsity. One way is to simply clamp all but the highest-k activations of the latent code to zero. This is the k-sparse autoencoder. The k-sparse autoencoder inserts the following "k-sparse function" in the latent layer of a standard autoencoder: f k ( x 1 , . . . , x n ) = ( x 1 b 1 , . . . , x n b n ) {\displaystyle f_{k}(x_{1},...,x_{n})=(x_{1}b_{1},...,x_{n}b_{n})} where b i = 1 {\displaystyle b_{i}=1} if | x i | {\displaystyle |x_{i}|} ranks in the top k, and 0 otherwise. Backpropagating through f k {\displaystyle f_{k}} is simple: set gradient to 0 for b i = 0 {\displaystyle b_{i}=0} entries, and keep gradient for b i = 1 {\displaystyle b_{i}=1} entries. This is essentially a generalized ReLU function. The other way is a relaxed version of the k-

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  Digital image processing

  Digital image processing is the use of a digital computer to process digital images through an algorithm. As a subcategory or field of digital signal processing, digital image processing has many advantages over analog image processing. It allows a much wider range of algorithms to be applied to the input data and can avoid problems such as the build-up of noise and distortion during processing. Since images are defined over two dimensions (perhaps more), digital image processing may be modeled in the form of multidimensional systems. The generation and development of digital image processing are mainly affected by three factors: first, the development of computers; second, the development of mathematics (especially the creation and improvement of discrete mathematics theory); and third, the demand for a wide range of applications in environment, agriculture, military, industry and medical science has increased. == History == Many of the techniques of digital image processing, or digital picture processing as it often was called, were developed in the 1960s, at Bell Laboratories, the Jet Propulsion Laboratory, Massachusetts Institute of Technology, University of Maryland, and a few other research facilities, with application to satellite imagery, wire-photo standards conversion, medical imaging, videophone, character recognition, and photograph enhancement. The purpose of early image processing was to improve the quality of the image. In image processing, the input is a low-quality image, and the output is an image with improved quality. Common image processing includes image enhancement, restoration, encoding, and compression. The first successful application was the American Jet Propulsion Laboratory (JPL). They used image processing techniques such as geometric correction, gradation transformation, noise removal, etc. on the thousands of lunar photos sent back by the Space Detector Ranger 7 in 1964, taking into account the position of the Sun and the environment of the Moon. The impact of the successful mapping of the Moon's surface map by the computer has been a success. Later, more complex image processing was performed on the nearly 100,000 photos sent back by the spacecraft, so that the topographic map, color map and panoramic mosaic of the Moon were obtained, which achieved extraordinary results and laid a solid foundation for human landing on the Moon. The cost of processing was fairly high, however, with the computing equipment of that era. That changed in the 1970s, when digital image processing proliferated as cheaper computers and dedicated hardware became available. This led to images being processed in real-time, for some dedicated problems such as television standards conversion. As general-purpose computers became faster, they started to take over the role of dedicated hardware for all but the most specialized and computer-intensive operations. With the fast computers and signal processors available in the 2000s, digital image processing has become the most common form of image processing, and is generally used because it is not only the most versatile method, but also the cheapest. === Image sensors === The basis for modern image sensors is metal–oxide–semiconductor (MOS) technology, invented at Bell Labs between 1955 and 1960, This led to the development of digital semiconductor image sensors, including the charge-coupled device (CCD) and later the CMOS sensor. The charge-coupled device was invented by Willard S. Boyle and George E. Smith at Bell Labs in 1969. While researching MOS technology, they realized that an electric charge was the analogy of the magnetic bubble and that it could be stored on a tiny MOS capacitor. As it was fairly straightforward to fabricate a series of MOS capacitors in a row, they connected a suitable voltage to them so that the charge could be stepped along from one to the next. The CCD is a semiconductor circuit that was later used in the first digital video cameras for television broadcasting. The NMOS active-pixel sensor (APS) was invented by Olympus in Japan during the mid-1980s. This was enabled by advances in MOS semiconductor device fabrication, with MOSFET scaling reaching smaller micron and then sub-micron levels. The NMOS APS was fabricated by Tsutomu Nakamura's team at Olympus in 1985. The CMOS active-pixel sensor (CMOS sensor) was later developed by Eric Fossum's team at the NASA Jet Propulsion Laboratory in 1993. By 2007, sales of CMOS sensors had surpassed CCD sensors. MOS image sensors are widely used in optical mouse technology. The first optical mouse, invented by Richard F. Lyon at Xerox in 1980, used a 5 μm NMOS integrated circuit sensor chip. Since the first commercial optical mouse, the IntelliMouse introduced in 1999, most optical mouse devices use CMOS sensors. === Image compression === An important development in digital image compression technology was the discrete cosine transform (DCT), a lossy compression technique first proposed by Nasir Ahmed in 1972. DCT compression became the basis for JPEG, which was introduced by the Joint Photographic Experts Group in 1992. JPEG compresses images down to much smaller file sizes, and has become the most widely used image file format on the Internet. Its highly efficient DCT compression algorithm was largely responsible for the wide proliferation of digital images and digital photos, with several billion JPEG images produced every day as of 2015. Medical imaging techniques produce very large amounts of data, especially from CT, MRI and PET modalities. As a result, storage and communications of electronic image data are prohibitive without the use of compression. JPEG 2000 image compression is used by the DICOM standard for storage and transmission of medical images. The cost and feasibility of accessing large image data sets over low or various bandwidths are further addressed by use of another DICOM standard, called JPIP, to enable efficient streaming of the JPEG 2000 compressed image data. === Digital signal processor (DSP) === Electronic signal processing was revolutionized by the wide adoption of MOS technology in the 1970s. MOS integrated circuit technology was the basis for the first single-chip microprocessors and microcontrollers in the early 1970s, and then the first single-chip digital signal processor (DSP) chips in the late 1970s. DSP chips have since been widely used in digital image processing. The discrete cosine transform (DCT) image compression algorithm has been widely implemented in DSP chips, with many companies developing DSP chips based on DCT technology. DCTs are widely used for encoding, decoding, video coding, audio coding, multiplexing, control signals, signaling, analog-to-digital conversion, formatting luminance and color differences, and color formats such as YUV444 and YUV411. DCTs are also used for encoding operations such as motion estimation, motion compensation, inter-frame prediction, quantization, perceptual weighting, entropy encoding, variable encoding, and motion vectors, and decoding operations such as the inverse operation between different color formats (YIQ, YUV and RGB) for display purposes. DCTs are also commonly used for high-definition television (HDTV) encoder/decoder chips. == Tasks == Digital image processing allows the use of much more complex algorithms, and hence, can offer both more sophisticated performance at simple tasks, and the implementation of methods which would be impossible by analogue means. In particular, digital image processing is a concrete application of, and a practical technology based on: Classification Feature extraction Multi-scale signal analysis Pattern recognition Projection Some techniques that are used in digital image processing include: Anisotropic diffusion Hidden Markov models Image editing Image restoration Independent component analysis Linear filtering Neural networks Partial differential equations Pixelation Point feature matching Principal components analysis Self-organizing maps Wavelets == Digital image transformations == === Filtering === Digital filters are used to blur and sharpen digital images. Filtering can be performed by: convolution with specifically designed kernels (filter array) in the spatial domain masking specific frequency regions in the frequency (Fourier) domain The following examples show both methods: ==== Image padding in Fourier domain filtering ==== Images are typically padded before being transformed to the Fourier space, the highpass filtered images below illustrate the consequences of different padding techniques: Notice that the highpass filter shows extra edges when zero padded compared to the repeated edge padding. ==== Filtering code examples ==== MATLAB example for spatial domain highpass filtering. === Affine transformations === Affine transformations enable basic image transformations including scale, rotate, translate, mirror and shear as is shown in the following examples: To apply the affine

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  Information gain ratio

  In decision tree learning, information gain ratio is a ratio of information gain to the intrinsic information. It was proposed by Ross Quinlan, to reduce a bias towards multi-valued attributes by taking the number and size of branches into account when choosing an attribute. Information gain is also known as mutual information. == Information gain calculation == Information gain is the reduction in entropy produced from partitioning a set with attributes a {\displaystyle a} and finding the optimal candidate that produces the highest value: IG ( T , a ) = H ( T ) − H ( T | a ) , {\displaystyle {\text{IG}}(T,a)=\mathrm {H} {(T)}-\mathrm {H} {(T|a)},} where T {\displaystyle T} is a random variable and H ( T | a ) {\displaystyle \mathrm {H} {(T|a)}} is the entropy of T {\displaystyle T} given the value of attribute a {\displaystyle a} . The information gain is equal to the total entropy for an attribute if for each of the attribute values a unique classification can be made for the result attribute. In this case the relative entropies subtracted from the total entropy are 0. == Split information calculation == The split information value for a test is defined as follows: SplitInformation ( X ) = − ∑ i = 1 n N ( x i ) N ( x ) ∗ log ⁡ 2 N ( x i ) N ( x ) {\displaystyle {\text{SplitInformation}}(X)=-\sum _{i=1}^{n}{{\frac {\mathrm {N} (x_{i})}{\mathrm {N} (x)}}\log {_{2}}{\frac {\mathrm {N} (x_{i})}{\mathrm {N} (x)}}}} where X {\displaystyle X} is a discrete random variable with possible values x 1 , x 2 , . . . , x i {\displaystyle {x_{1},x_{2},...,x_{i}}} and N ( x i ) {\displaystyle N(x_{i})} being the number of times that x i {\displaystyle x_{i}} occurs divided by the total count of events N ( x ) {\displaystyle N(x)} where x {\displaystyle x} is the set of events. The split information value is a positive number that describes the potential worth of splitting a branch from a node. This in turn is the intrinsic value that the random variable possesses and will be used to remove the bias in the information gain ratio calculation. == Information gain ratio calculation == The information gain ratio is the ratio between the information gain and the split information value: IGR ( T , a ) = IG ( T , a ) / SplitInformation ( T ) {\displaystyle {\text{IGR}}(T,a)={\text{IG}}(T,a)/{\text{SplitInformation}}(T)} IGR ( T , a ) = − ∑ i = 1 n P ( T ) log ⁡ P ( T ) − ( − ∑ i = 1 n P ( T | a ) log ⁡ P ( T | a ) ) − ∑ i = 1 n N ( t i ) N ( t ) ∗ log ⁡ 2 N ( t i ) N ( t ) {\displaystyle {\text{IGR}}(T,a)={\frac {-\sum _{i=1}^{n}{\mathrm {P} (T)\log \mathrm {P} (T)}-(-\sum _{i=1}^{n}{\mathrm {P} (T|a)\log \mathrm {P} (T|a)})}{-\sum _{i=1}^{n}{{\frac {\mathrm {N} (t_{i})}{\mathrm {N} (t)}}\log {_{2}}{\frac {\mathrm {N} (t_{i})}{\mathrm {N} (t)}}}}}} == Example == Using weather data published by Fordham University, the table was created below: Using the table above, one can find the entropy, information gain, split information, and information gain ratio for each variable (outlook, temperature, humidity, and wind). These calculations are shown in the tables below: Using the above tables, one can deduce that Outlook has the highest information gain ratio. Next, one must find the statistics for the sub-groups of the Outlook variable (sunny, overcast, and rainy), for this example one will only build the sunny branch (as shown in the table below): One can find the following statistics for the other variables (temperature, humidity, and wind) to see which have the greatest effect on the sunny element of the outlook variable: Humidity was found to have the highest information gain ratio. One will repeat the same steps as before and find the statistics for the events of the Humidity variable (high and normal): Since the play values are either all "No" or "Yes", the information gain ratio value will be equal to 1. Also, now that one has reached the end of the variable chain with Wind being the last variable left, they can build an entire root to leaf node branch line of a decision tree. Once finished with reaching this leaf node, one would follow the same procedure for the rest of the elements that have yet to be split in the decision tree. This set of data was relatively small, however, if a larger set was used, the advantages of using the information gain ratio as the splitting factor of a decision tree can be seen more. == Advantages == Information gain ratio biases the decision tree against considering attributes with a large number of distinct values. For example, suppose that we are building a decision tree for some data describing a business's customers. Information gain ratio is used to decide which of the attributes are the most relevant. These will be tested near the root of the tree. One of the input attributes might be the customer's telephone number. This attribute has a high information gain, because it uniquely identifies each customer. Due to its high amount of distinct values, this will not be chosen to be tested near the root. == Disadvantages == Although information gain ratio solves the key problem of information gain, it creates another problem. If one is considering an amount of attributes that have a high number of distinct values, these will never be above one that has a lower number of distinct values. == Difference from information gain == Information gain's shortcoming is created by not providing a numerical difference between attributes with high distinct values from those that have less. Example: Suppose that we are building a decision tree for some data describing a business's customers. Information gain is often used to decide which of the attributes are the most relevant, so they can be tested near the root of the tree. One of the input attributes might be the customer's credit card number. This attribute has a high information gain, because it uniquely identifies each customer, but we do not want to include it in the decision tree: deciding how to treat a customer based on their credit card number is unlikely to generalize to customers we haven't seen before. Information gain ratio's strength is that it has a bias towards the attributes with the lower number of distinct values. Below is a table describing the differences of information gain and information gain ratio when put in certain scenarios.

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  Nonlinear dimensionality reduction

  Nonlinear dimensionality reduction (NLDR), also known as manifold learning, is any of various related techniques that aim to project high-dimensional data, potentially existing across non-linear manifolds which cannot be adequately captured by linear decomposition methods, onto lower-dimensional latent manifolds, with the goal of either visualizing the data in the low-dimensional space, or learning the mapping (either from the high-dimensional space to the low-dimensional embedding or vice versa) itself. The techniques described below can be understood as generalizations of linear decomposition methods used for dimensionality reduction, such as singular value decomposition and principal component analysis. == Applications of NLDR == High dimensional data can be hard for machines to work with, requiring significant time and space for analysis. It also presents a challenge for humans, since it's hard to visualize or understand data in more than three dimensions. Reducing the dimensionality of a data set, while keeping its essential features relatively intact, can make algorithms more efficient and allow analysts to visualize trends and patterns. The reduced-dimensional representations of data are often referred to as "intrinsic variables". This description implies that these are the values from which the data was produced. For example, consider a dataset that contains images of a letter 'A', which has been scaled and rotated by varying amounts. Each image has 32×32 pixels. Each image can be represented as a vector of 1024 pixel values. Each row is a sample on a two-dimensional manifold in 1024-dimensional space (a Hamming space). The intrinsic dimensionality is two, because two variables (rotation and scale) were varied in order to produce the data. Information about the shape or look of a letter 'A' is not part of the intrinsic variables because it is the same in every instance. Nonlinear dimensionality reduction will discard the correlated information (the letter 'A') and recover only the varying information (rotation and scale). By comparison, if principal component analysis, which is a linear dimensionality reduction algorithm, is used to reduce this same dataset into two dimensions, the resulting values are not so well organized. This demonstrates that the high-dimensional vectors (each representing a letter 'A') that sample this manifold vary in a non-linear manner. It should be apparent, therefore, that NLDR has several applications in the field of computer-vision. For example, consider a robot that uses a camera to navigate in a closed static environment. The images obtained by that camera can be considered to be samples on a manifold in high-dimensional space, and the intrinsic variables of that manifold will represent the robot's position and orientation. Invariant manifolds are of general interest for model order reduction in dynamical systems. In particular, if there is an attracting invariant manifold in the phase space, nearby trajectories will converge onto it and stay on it indefinitely, rendering it a candidate for dimensionality reduction of the dynamical system. While such manifolds are not guaranteed to exist in general, the theory of spectral submanifolds (SSM) gives conditions for the existence of unique attracting invariant objects in a broad class of dynamical systems. Active research in NLDR seeks to unfold the observation manifolds associated with dynamical systems to develop modeling techniques. Some of the more prominent nonlinear dimensionality reduction techniques are listed below. == Important concepts == === Sammon's mapping === Sammon's mapping is one of the first and most popular NLDR techniques. === Self-organizing map === The self-organizing map (SOM, also called Kohonen map) and its probabilistic variant generative topographic mapping (GTM) use a point representation in the embedded space to form a latent variable model based on a non-linear mapping from the embedded space to the high-dimensional space. These techniques are related to work on density networks, which also are based around the same probabilistic model. === Kernel principal component analysis === Perhaps the most widely used algorithm for dimensional reduction is kernel PCA. PCA begins by computing the covariance matrix of the m × n {\displaystyle m\times n} matrix X {\displaystyle \mathbf {X} } C = 1 m ∑ i = 1 m x i x i T . {\displaystyle C={\frac {1}{m}}\sum _{i=1}^{m}{\mathbf {x} _{i}\mathbf {x} _{i}^{\mathsf {T}}}.} It then projects the data onto the first k eigenvectors of that matrix. By comparison, KPCA begins by computing the covariance matrix of the data after being transformed into a higher-dimensional space, C = 1 m ∑ i = 1 m Φ ( x i ) Φ ( x i ) T . {\displaystyle C={\frac {1}{m}}\sum _{i=1}^{m}{\Phi (\mathbf {x} _{i})\Phi (\mathbf {x} _{i})^{\mathsf {T}}}.} It then projects the transformed data onto the first k eigenvectors of that matrix, just like PCA. It uses the kernel trick to factor away much of the computation, such that the entire process can be performed without actually computing Φ ( x ) {\displaystyle \Phi (\mathbf {x} )} . Of course Φ {\displaystyle \Phi } must be chosen such that it has a known corresponding kernel. Unfortunately, it is not trivial to find a good kernel for a given problem, so KPCA does not yield good results with some problems when using standard kernels. For example, it is known to perform poorly with these kernels on the Swiss roll manifold. However, one can view certain other methods that perform well in such settings (e.g., Laplacian Eigenmaps, LLE) as special cases of kernel PCA by constructing a data-dependent kernel matrix. KPCA has an internal model, so it can be used to map points onto its embedding that were not available at training time. === Principal curves and manifolds === Principal curves and manifolds give the natural geometric framework for nonlinear dimensionality reduction and extend the geometric interpretation of PCA by explicitly constructing an embedded manifold, and by encoding using standard geometric projection onto the manifold. This approach was originally proposed by Trevor Hastie in his 1984 thesis, which he formally introduced in 1989. This idea has been explored further by many authors. How to define the "simplicity" of the manifold is problem-dependent, however, it is commonly measured by the intrinsic dimensionality and/or the smoothness of the manifold. Usually, the principal manifold is defined as a solution to an optimization problem. The objective function includes a quality of data approximation and some penalty terms for the bending of the manifold. The popular initial approximations are generated by linear PCA and Kohonen's SOM. === Laplacian eigenmaps === Laplacian eigenmaps uses spectral techniques to perform dimensionality reduction. This technique relies on the basic assumption that the data lies in a low-dimensional manifold in a high-dimensional space. This algorithm cannot embed out-of-sample points, but techniques based on Reproducing kernel Hilbert space regularization exist for adding this capability. Such techniques can be applied to other nonlinear dimensionality reduction algorithms as well. Traditional techniques like principal component analysis do not consider the intrinsic geometry of the data. Laplacian eigenmaps builds a graph from neighborhood information of the data set. Each data point serves as a node on the graph and connectivity between nodes is governed by the proximity of neighboring points (using e.g. the k-nearest neighbor algorithm). The graph thus generated can be considered as a discrete approximation of the low-dimensional manifold in the high-dimensional space. Minimization of a cost function based on the graph ensures that points close to each other on the manifold are mapped close to each other in the low-dimensional space, preserving local distances. The eigenfunctions of the Laplace–Beltrami operator on the manifold serve as the embedding dimensions, since under mild conditions this operator has a countable spectrum that is a basis for square integrable functions on the manifold (compare to Fourier series on the unit circle manifold). Attempts to place Laplacian eigenmaps on solid theoretical ground have met with some success, as under certain nonrestrictive assumptions, the graph Laplacian matrix has been shown to converge to the Laplace–Beltrami operator as the number of points goes to infinity. === Isomap === Isomap is a combination of the Floyd–Warshall algorithm with classic Multidimensional Scaling (MDS). Classic MDS takes a matrix of pair-wise distances between all points and computes a position for each point. Isomap assumes that the pair-wise distances are only known between neighboring points, and uses the Floyd–Warshall algorithm to compute the pair-wise distances between all other points. This effectively estimates the full matrix of pair-wise geodesic distances between all of the points. Isomap th

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  Weka (software)

  Waikato Environment for Knowledge Analysis (Weka) is a collection of machine learning and data analysis free software licensed under the GNU General Public License. It was developed at the University of Waikato, New Zealand, and is the companion software to the book "Data Mining: Practical Machine Learning Tools and Techniques". == Description == Weka contains a collection of visualization tools and algorithms for data analysis and predictive modeling, together with graphical user interfaces for easy access to these functions. The original non-Java version of Weka was a Tcl/Tk front-end to (mostly third-party) modeling algorithms implemented in other programming languages, plus data preprocessing utilities in C, and a makefile-based system for running machine learning experiments. This original version was primarily designed as a tool for analyzing data from agricultural domains, but the more recent fully Java-based version (Weka 3), for which development started in 1997, is now used in many different application areas, in particular for educational purposes and research. Advantages of Weka include: Free availability under the GNU General Public License. Portability, since it is fully implemented in the Java programming language and thus runs on almost any modern computing platform. A comprehensive collection of data preprocessing and modeling techniques. Ease of use due to its graphical user interfaces. Weka supports several standard data mining tasks, more specifically, data preprocessing, clustering, classification, regression, visualization, and feature selection. Input to Weka is expected to be formatted according the Attribute-Relational File Format and with the filename bearing the .arff extension. All of Weka's techniques are predicated on the assumption that the data is available as one flat file or relation, where each data point is described by a fixed number of attributes (normally, numeric or nominal attributes, but some other attribute types are also supported). Weka provides access to SQL databases using Java Database Connectivity and can process the result returned by a database query. Weka provides access to deep learning with Deeplearning4j. It is not capable of multi-relational data mining, but there is separate software for converting a collection of linked database tables into a single table that is suitable for processing using Weka. Another important area that is currently not covered by the algorithms included in the Weka distribution is sequence modeling. == Extension packages == In version 3.7.2, a package manager was added to allow the easier installation of extension packages. Some functionality that used to be included with Weka prior to this version has since been moved into such extension packages, but this change also makes it easier for others to contribute extensions to Weka and to maintain the software, as this modular architecture allows independent updates of the Weka core and individual extensions. == History == In 1993, the University of Waikato in New Zealand began development of the original version of Weka, which became a mix of Tcl/Tk, C, and makefiles. In 1997, the decision was made to redevelop Weka from scratch in Java, including implementations of modeling algorithms. In 2005, Weka received the SIGKDD Data Mining and Knowledge Discovery Service Award. In 2006, Pentaho Corporation acquired an exclusive licence to use Weka for business intelligence. It forms the data mining and predictive analytics component of the Pentaho business intelligence suite. Pentaho has since been acquired by Hitachi Vantara, and Weka now underpins the PMI (Plugin for Machine Intelligence) open source component. == Related tools == Auto-WEKA is an automated machine learning system for Weka. Environment for DeveLoping KDD-Applications Supported by Index-Structures (ELKI) is a similar project to Weka with a focus on cluster analysis, i.e., unsupervised methods. H2O.ai is an open-source data science and machine learning platform KNIME is a machine learning and data mining software implemented in Java. Massive Online Analysis (MOA) is an open-source project for large scale mining of data streams, also developed at the University of Waikato in New Zealand. Neural Designer is a data mining software based on deep learning techniques written in C++. Orange is a similar open-source project for data mining, machine learning and visualization based on scikit-learn. RapidMiner is a commercial machine learning framework implemented in Java which integrates Weka. scikit-learn is a popular machine learning library in Python.

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  MetaMask

  MetaMask is a software cryptocurrency wallet developed by ConsenSys for interacting with the Ethereum blockchain and other EVM-compatible networks. It enables users to manage Ethereum accounts and connect to decentralized applications (dApps) via a browser extension or mobile app. As of early 2026, MetaMask reports over 100 million users worldwide. == Overview == MetaMask allows users to store and manage private keys, send and receive Ethereum-based cryptocurrencies and tokens (including ERC-20 and ERC-721 standards), broadcast transactions, and interact with dApps. dApps connect to the wallet via JavaScript interfaces, prompting users to approve signatures or transactions. The wallet features MetaMask Swaps, an in-app token swap aggregator sourcing liquidity from multiple decentralized exchanges (DEXs), with a service fee of 0.875%. In 2025, MetaMask introduced the MetaMask Rewards program (initially mobile-only), where users earn points for activities such as swaps, bridging, and referrals. Season 1 (October 2025 – January 2026) distributed over $30 million in Linea tokens and other perks to participants. == History == MetaMask launched in 2016 as open-source software under the MIT license. It initially supported browser extensions for Chrome and Firefox. Mobile versions were in closed beta from 2019 and publicly released for iOS and Android in September 2020. In August 2020, the license changed to a custom proprietary one. MetaMask Swaps launched on desktop in October 2020 and on mobile in March 2021. The Rewards program launched in late 2025 with Linea integration. == Criticism == MetaMask has faced criticism over privacy, including default analytics settings that share some user data (which can be disabled). Its reliance on Infura (acquired by ConsenSys in 2019) has raised concerns about centralization in Ethereum infrastructure. The wallet regularly issues warnings about phishing scams and fake airdrops impersonating MetaMask.

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  Extremal Ensemble Learning

  Extremal Ensemble Learning (EEL) is a machine learning algorithmic paradigm for graph partitioning. EEL creates an ensemble of partitions and then uses information contained in the ensemble to find new and improved partitions. The ensemble evolves and learns how to form improved partitions through extremal updating procedure. The final solution is found by achieving consensus among its member partitions about what the optimal partition is. == Reduced-Network Extremal Ensemble Learning (RenEEL) == A particular implementation of the EEL paradigm is the Reduced-Network Extremal Ensemble Learning (RenEEL) scheme for partitioning a graph. RenEEL uses consensus across many partitions in an ensemble to create a reduced network that can be efficiently analyzed to find more accurate partitions. These better quality partitions are subsequently used to update the ensemble. An algorithm that utilizes the RenEEL scheme is currently the best algorithm for finding the graph partition with maximum modularity, which is an NP-hard problem.

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  Spatial Analysis of Principal Components

  Spatial Principal Component Analysis (sPCA) is a multivariate statistical technique that complements the traditional Principal Component Analysis (PCA) by incorporating spatial information into the analysis of genetic variation. While traditional PCA can be used to find spatial patterns, it focuses on reducing data dimensionality by identifying uncorrelated principal components that capture maximum variance, thus often lacking power to identify non-trivial spatial genetic patterns. By accounting for spatial autocorrelation, sPCA is able to uncover spatial patterns in the data and find the spatial structure of datasets where observations are either geographically or topologically linked. This statistical power improvement allows the investigation of cryptic spatial patterns of genetic variability otherwise overlooked. sPCA has been applied in various fields, including geography, ecology and genetics. == History == sPCA was introduced in 2008 by Thibaut Jombart, Sébastien Devillard, Anne-Béatrice Dufour, and D. Pontier as a spatially explicit method to investigate the spatial pattern of genetic variation among individuals or populations. In 2017, Valeria Montano and Thibaut Jombart published an alternative non-parametric test to evaluate the significance of global and local spatial genetic patterns with improved statistical power. == Details == sPCA modifies the PCA framework by integrating spatial weights, typically in the form of connectivity matrices or spatial adjacency graphs. It identifies principal components (PCs) that maximize both genentic variance and spatial autocorreation, as measured by Moran's I. These weights represent relationships between observations based on geographic distance or other spatial criteria. The method decomposes variance into two components: Global structures, correspond to positive autocorrelation, that is, reflect broad-scale spatial patterns where similar values cluster over large regions. Local structures, correspond to negative autocorrelation, that is, capture fine-scale spatial variations or localized patterns. The core of sPCA relies on the eigenanalysis of a spatially weighted covariance or correlation matrix. The spatial weight matrix can be constructed using techniques such as Delaunay triangulation, nearest-neighbor graphs, or distance-based criteria. Applications of sPCA should be used only as an explorative tool. == Applications == sPCA has been widely used in many fields, including: Ecology: To find spatial patterns in species distributions and environmental gradients. Genetics: Population structure and gene flow analysis while allowing for spatial autocorrelation considerations. Biogeography: To identify historical dispersal routes, and barriers to gene flow, providing insights into species distribution patterns and evolutionary history. == Software/Source Code == sPCA implementations are available in R in adegenet and ntbox . These tools facilitate the application of sPCA by providing functions for constructing spatial weight matrices, performing eigenanalysis, and obtaining spatial principal components in an easy-to-read form.

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  Principal component analysis

  Principal component analysis (PCA) is a linear dimensionality reduction technique with applications in exploratory data analysis, visualization and data preprocessing. The data are linearly transformed onto a new coordinate system such that the directions (principal components) capturing the largest variation in the data can be easily identified. The principal components of a collection of points in a real coordinate space are a sequence of p {\displaystyle p} unit vectors, where the i {\displaystyle i} -th vector is the direction of a line that best fits the data while being orthogonal to the first i − 1 {\displaystyle i-1} vectors. Here, a best-fitting line is defined as one that minimizes the average squared perpendicular distance from the points to the line. These directions (i.e., principal components) constitute an orthonormal basis in which different individual dimensions of the data are linearly uncorrelated. Many studies use the first two principal components in order to plot the data in two dimensions and to visually identify clusters of closely related data points. Principal component analysis has applications in many fields such as population genetics, microbiome studies, and atmospheric science. == Overview == When performing PCA, the first principal component of a set of p {\displaystyle p} variables is the derived variable formed as a linear combination of the original variables that explains the most variance. The second principal component explains the most variance in what is left once the effect of the first component is removed, and we may proceed through p {\displaystyle p} iterations until all the variance is explained. PCA is most commonly used when many of the variables are highly correlated with each other and it is desirable to reduce their number to an independent set. The first principal component can equivalently be defined as a direction that maximizes the variance of the projected data. The i {\displaystyle i} -th principal component can be taken as a direction orthogonal to the first i − 1 {\displaystyle i-1} principal components that maximizes the variance of the projected data. For either objective, it can be shown that the principal components are eigenvectors of the data's covariance matrix. Thus, the principal components are often computed by eigendecomposition of the data covariance matrix or singular value decomposition of the data matrix. PCA is the simplest of the true eigenvector-based multivariate analyses and is closely related to factor analysis. Factor analysis typically incorporates more domain-specific assumptions about the underlying structure and solves eigenvectors of a slightly different matrix. PCA is also related to canonical correlation analysis (CCA). CCA defines coordinate systems that optimally describe the cross-covariance between two datasets while PCA defines a new orthogonal coordinate system that optimally describes variance in a single dataset. Robust and L1-norm-based variants of standard PCA have also been proposed. == History == PCA was invented in 1901 by Karl Pearson, as an analogue of the principal axis theorem in mechanics; it was later independently developed and named by Harold Hotelling in the 1930s. Depending on the field of application, it is also named the discrete Karhunen–Loève transform (KLT) in signal processing, the Hotelling transform in multivariate quality control, proper orthogonal decomposition (POD) in mechanical engineering, singular value decomposition (SVD) of X (invented in the last quarter of the 19th century), eigenvalue decomposition (EVD) of XTX in linear algebra, factor analysis (for a discussion of the differences between PCA and factor analysis see Ch. 7 of Jolliffe's Principal Component Analysis), Eckart–Young theorem (Harman, 1960), or empirical orthogonal functions (EOF) in meteorological science (Lorenz, 1956), empirical eigenfunction decomposition (Sirovich, 1987), quasiharmonic modes (Brooks et al., 1988), spectral decomposition in noise and vibration, and empirical modal analysis in structural dynamics. == Intuition == PCA can be thought of as fitting a p-dimensional ellipsoid to the data, where each axis of the ellipsoid represents a principal component. If some axis of the ellipsoid is small, then the variance along that axis is also small. To find the axes of the ellipsoid, we must first center the values of each variable in the dataset on 0 by subtracting the mean of the variable's observed values from each of those values. These transformed values are used instead of the original observed values for each of the variables. Then, we compute the covariance matrix of the data and calculate the eigenvalues and corresponding eigenvectors of this covariance matrix. Then we must normalize each of the orthogonal eigenvectors to turn them into unit vectors. Once this is done, each of the mutually-orthogonal unit eigenvectors can be interpreted as an axis of the ellipsoid fitted to the data. This choice of basis will transform the covariance matrix into a diagonalized form, in which the diagonal elements represent the variance of each axis. The proportion of the variance that each eigenvector represents can be calculated by dividing the eigenvalue corresponding to that eigenvector by the sum of all eigenvalues. Biplots and scree plots (degree of explained variance) are used to interpret findings of the PCA. == Details == PCA is defined as an orthogonal linear transformation on a real inner product space that transforms the data to a new coordinate system such that the greatest variance by some scalar projection of the data comes to lie on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on. Consider an n × p {\displaystyle n\times p} data matrix, X, with column-wise zero empirical mean (the sample mean of each column has been shifted to zero), where each of the n rows represents a different repetition of the experiment, and each of the p columns gives a particular kind of feature (say, the results from a particular sensor). Mathematically, the transformation is defined by a set of size l {\displaystyle l} (where l {\displaystyle l} is usually selected to be strictly less than p {\displaystyle p} to reduce dimensionality) of p {\displaystyle p} -dimensional vectors of weights or coefficients w ( k ) = ( w 1 , … , w p ) ( k ) {\displaystyle \mathbf {w} _{(k)}=(w_{1},\dots ,w_{p})_{(k)}} that map each row vector x ( i ) = ( x 1 , … , x p ) ( i ) {\displaystyle \mathbf {x} _{(i)}=(x_{1},\dots ,x_{p})_{(i)}} of X to a new vector of principal component scores t ( i ) = ( t 1 , … , t l ) ( i ) {\displaystyle \mathbf {t} _{(i)}=(t_{1},\dots ,t_{l})_{(i)}} , given by t k ( i ) = x ( i ) ⋅ w ( k ) f o r i = 1 , … , n k = 1 , … , l {\displaystyle {t_{k}}_{(i)}=\mathbf {x} _{(i)}\cdot \mathbf {w} _{(k)}\qquad \mathrm {for} \qquad i=1,\dots ,n\qquad k=1,\dots ,l} in such a way that the individual variables t 1 , … , t l {\displaystyle t_{1},\dots ,t_{l}} of t considered over the data set successively inherit the maximum possible variance from X, with each coefficient vector w constrained to be a unit vector. The above may equivalently be written in matrix form as T = X W {\displaystyle \mathbf {T} =\mathbf {X} \mathbf {W} } where T i k = t k ( i ) {\displaystyle {\mathbf {T} }_{ik}={t_{k}}_{(i)}} , X i j = x j ( i ) {\displaystyle {\mathbf {X} }_{ij}={x_{j}}_{(i)}} , and W j k = w j ( k ) {\displaystyle {\mathbf {W} }_{jk}={w_{j}}_{(k)}} . === First component === In order to maximize variance, the first weight vector w(1) thus has to satisfy w ( 1 ) = arg ⁡ max ‖ w ‖ = 1 { ∑ i ( t 1 ) ( i ) 2 } = arg ⁡ max ‖ w ‖ = 1 { ∑ i ( x ( i ) ⋅ w ) 2 } {\displaystyle \mathbf {w} _{(1)}=\arg \max _{\Vert \mathbf {w} \Vert =1}\,\left\{\sum _{i}(t_{1})_{(i)}^{2}\right\}=\arg \max _{\Vert \mathbf {w} \Vert =1}\,\left\{\sum _{i}\left(\mathbf {x} _{(i)}\cdot \mathbf {w} \right)^{2}\right\}} Equivalently, writing this in matrix form gives w ( 1 ) = arg ⁡ max ‖ w ‖ = 1 { ‖ X w ‖ 2 } = arg ⁡ max ‖ w ‖ = 1 { w T X T X w } {\displaystyle \mathbf {w} _{(1)}=\arg \max _{\left\|\mathbf {w} \right\|=1}\left\{\left\|\mathbf {Xw} \right\|^{2}\right\}=\arg \max _{\left\|\mathbf {w} \right\|=1}\left\{\mathbf {w} ^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {Xw} \right\}} Since w(1) has been defined to be a unit vector, it equivalently also satisfies w ( 1 ) = arg ⁡ max { w T X T X w w T w } {\displaystyle \mathbf {w} _{(1)}=\arg \max \left\{{\frac {\mathbf {w} ^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {Xw} }{\mathbf {w} ^{\mathsf {T}}\mathbf {w} }}\right\}} The quantity to be maximised can be recognised as a Rayleigh quotient. A standard result for a positive semidefinite matrix such as XTX is that the quotient's maximum possible value is the largest eigenvalue of the matrix, which occurs when w is the corresponding eigenvector. With w(1) found, the first principal component of a data vector

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  Web development

  Web development is the process of designing, developing and maintaining websites and web apps. Web development encompasses several different fields, most commonly referring to the programming of websites. Front-end development is the act of developing the user interface and client-side code, while back-end development focuses on the infrastructure behind a website, mainly server-side code. Since the World Wide Web was released publicly in 1993, web development has evolved greatly, with websites changing from a collection of static HTML pages to complex projects using frameworks, servers, and databases. == Overview == Web development includes many individual tasks, including web design, web content development, networking, and coding. Among web professionals, "web development" usually refers to the main non-design aspects of building websites: writing markup and coding. Web development is generally split into two fields: front-end development and back-end development. Front-end developers create the user interface of websites, turning web designs into HTML, CSS, and JavaScript code. Front-end developers must also make sure that websites work consistently across different browsers and devices. Back-end development, also known as server-side development, focuses on the infrastructure behind a website, including APIs, database management, and security. Some choose to be full-stack developers, meaning they work on both the front-end and back-end. == History == The World Wide Web is often categorised into three generations: Web 1.0, Web 2.0, and Web 3.0 (or Web3). It was invented in 1989, and released to the public in 1993. In the early years of the web, restrospecitvely referred to as Web 1.0, websites were simply a collection of static HTML files, and had limited interactivity. After the introduction of JavaScript in 1995, websites could contain logic, allowing for interactivity. The following year CSS was released, allowing greater control over the styling of web pages. In 1999, the term Web 2.0 was coined by Darcy DiNucci. The term later resurfaced in the early 2000s, as websites started to increase in complexity, requiring server-side services in addition to JavaScript. This led to the emergence of various new programming languages and frameworks designed for backend services, such as PHP, Active Server Pages, and Jakarta Server Pages. This enabled websites to do additional server-side processing, such as accessing databases. Another shift in web development was the release of the iPhone in 2007. This created a new medium for accessing the web, requiring a new approach to web development, and resulting in responsive web design, which allows a single website to appear different depending on the device running it. Later, progressive web apps were introduced, allowing websites to be installed on a device as an independent application. In the 2010s, JavaScript frameworks began to emerge, creating new ways to manipulate web pages, and increasing compatibility between web browsers. JQuery was popular in the early 2010s, but was later surpassed by other frameworks such as React and Vue.js. In the mid 2020s, use of AI became prevalent among web developers, with the 2025 Stack Overflow survey showing over 80% of developers saying the use AI at least monthly in their development process.

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  ID3 algorithm

  In decision tree learning, ID3 (Iterative Dichotomiser 3) is a greedy algorithm invented by Ross Quinlan used to generate a decision tree from a dataset. ID3 is the precursor to the C4.5 algorithm. The 3 in the name is meant to signify that this was Quinlan's third attempt at a model based on entropy-based splitting, and the term dichotimser is a misnomer as it implies a binary split, but the ID3 algorithm can split on multi-valued attributes. == Algorithm == The ID3 algorithm begins with the original set S {\displaystyle S} as the root node. On each iteration of the algorithm, it iterates through every unused attribute of the set S {\displaystyle S} and calculates the entropy H ( S ) {\displaystyle \mathrm {H} {(S)}} or the information gain I G ( S ) {\displaystyle IG(S)} of that attribute. It then selects the attribute which has the smallest entropy (or largest information gain) value. The set S {\displaystyle S} is then split or partitioned by the selected attribute to produce subsets of the data. (For example, a node can be split into child nodes based upon the subsets of the population whose ages are less than 50, between 50 and 100, and greater than 100.) The algorithm continues to recurse on each subset, considering only attributes never selected before. Recursion on a subset may stop in one of these cases: every element in the subset belongs to the same class; in which case the node is turned into a leaf node and labelled with the class of the examples. there are no more attributes to be selected, but the examples still do not belong to the same class. In this case, the node is made a leaf node and labelled with the most common class of the examples in the subset. there are no examples in the subset, which happens when no example in the parent set was found to match a specific value of the selected attribute. An example could be the absence of a person among the population with age over 100 years. Then a leaf node is created and labelled with the most common class of the examples in the parent node's set. Throughout the algorithm, the decision tree is constructed with each non-terminal node (internal node) representing the selected attribute on which the data was split, and terminal nodes (leaf nodes) representing the class label of the final subset of this branch. === Summary === Calculate the entropy of every attribute a {\displaystyle a} of the data set S {\displaystyle S} . Partition ("split") the set S {\displaystyle S} into subsets using the attribute for which the resulting entropy after splitting is minimized; or, equivalently, information gain is maximum. Make a decision tree node containing that attribute. Recurse on subsets using the remaining attributes. === Properties === ID3 does not guarantee an optimal solution. It can converge upon local optima. It uses a greedy strategy by selecting the locally best attribute to split the dataset on each iteration. The algorithm's optimality can be improved by using backtracking during the search for the optimal decision tree at the cost of possibly taking longer. ID3 can overfit the training data. To avoid overfitting, smaller decision trees should be preferred over larger ones. This algorithm usually produces small trees, but it does not always produce the smallest possible decision tree. ID3 is harder to use on continuous data than on factored data (factored data has a discrete number of possible values, thus reducing the possible branch points). If the values of any given attribute are continuous, then there are many more places to split the data on this attribute, and searching for the best value to split by can be time-consuming. === Usage === The ID3 algorithm is used by training on a data set S {\displaystyle S} to produce a decision tree which is stored in memory. At runtime, this decision tree is used to classify new test cases (feature vectors) by traversing the decision tree using the features of the datum to arrive at a leaf node. == The ID3 metrics == === Entropy === Entropy H ( S ) {\displaystyle \mathrm {H} {(S)}} is a measure of the amount of uncertainty in the (data) set S {\displaystyle S} (i.e. entropy characterizes the (data) set S {\displaystyle S} ). H ( S ) = ∑ x ∈ X − p ( x ) log 2 ⁡ p ( x ) {\displaystyle \mathrm {H} {(S)}=\sum _{x\in X}{-p(x)\log _{2}p(x)}} Where, S {\displaystyle S} – The current dataset for which entropy is being calculated This changes at each step of the ID3 algorithm, either to a subset of the previous set in the case of splitting on an attribute or to a "sibling" partition of the parent in case the recursion terminated previously. X {\displaystyle X} – The set of classes in S {\displaystyle S} p ( x ) {\displaystyle p(x)} – The proportion of the number of elements in class x {\displaystyle x} to the number of elements in set S {\displaystyle S} When H ( S ) = 0 {\displaystyle \mathrm {H} {(S)}=0} , the set S {\displaystyle S} is perfectly classified (i.e. all elements in S {\displaystyle S} are of the same class). In ID3, entropy is calculated for each remaining attribute. The attribute with the smallest entropy is used to split the set S {\displaystyle S} on this iteration. Entropy in information theory measures how much information is expected to be gained upon measuring a random variable; as such, it can also be used to quantify the amount to which the distribution of the quantity's values is unknown. A constant quantity has zero entropy, as its distribution is perfectly known. In contrast, a uniformly distributed random variable (discretely or continuously uniform) maximizes entropy. Therefore, the greater the entropy at a node, the less information is known about the classification of data at this stage of the tree; and therefore, the greater the potential to improve the classification here. As such, ID3 is a greedy heuristic performing a best-first search for locally optimal entropy values. Its accuracy can be improved by preprocessing the data. === Information gain === Information gain I G ( A ) {\displaystyle IG(A)} is the measure of the difference in entropy from before to after the set S {\displaystyle S} is split on an attribute A {\displaystyle A} . In other words, how much uncertainty in S {\displaystyle S} was reduced after splitting set S {\displaystyle S} on attribute A {\displaystyle A} . I G ( S , A ) = H ( S ) − ∑ t ∈ T p ( t ) H ( t ) = H ( S ) − H ( S | A ) . {\displaystyle IG(S,A)=\mathrm {H} {(S)}-\sum _{t\in T}p(t)\mathrm {H} {(t)}=\mathrm {H} {(S)}-\mathrm {H} {(S|A)}.} Where, H ( S ) {\displaystyle \mathrm {H} (S)} – Entropy of set S {\displaystyle S} T {\displaystyle T} – The subsets created from splitting set S {\displaystyle S} by attribute A {\displaystyle A} such that S = ⋃ t ∈ T t {\displaystyle S=\bigcup _{t\in T}t} p ( t ) {\displaystyle p(t)} – The proportion of the number of elements in t {\displaystyle t} to the number of elements in set S {\displaystyle S} H ( t ) {\displaystyle \mathrm {H} (t)} – Entropy of subset t {\displaystyle t} In ID3, information gain can be calculated (instead of entropy) for each remaining attribute. The attribute with the largest information gain is used to split the set S {\displaystyle S} on this iteration.

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  Information Harvesting

  Information Harvesting (IH) was an early data mining product from the 1990s. It was invented by Ralphe Wiggins and produced by the Ryan Corp, later Information Harvesting Inc., of Cambridge, Massachusetts. Wiggins had a background in genetic algorithms and fuzzy logic. IH sought to infer rules from sets of data. It did this first by classifying various input variables into one of a number of bins, thereby putting some structure on the continuous variables in the input. IH then proceeds to generate rules, trading off generalization against memorization, that will infer the value of the prediction variable, possibly creating many levels of rules in the process. It included strategies for checking if overfitting took place and, if so, correcting for it. Because of its strategies for correcting for overfitting by considering more data, and refining the rules based on that data, IH might also be considered to be a form of machine learning. The advantage of IH, as compared with other data mining products of its time and even later, was that it provided a mechanism for finding multiple rules that would classify the data and determining, according to set criteria, the best rules to use.

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