AI Driven Spreadsheet

AI Driven Spreadsheet — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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  Large language model

  A large language model (LLM) is a neural network trained on a vast amount of text for natural language processing tasks, especially language generation. LLMs can typically generate, summarize, translate and analyze text in many contexts, and are a foundational technology behind modern chatbots. Biased or inaccurate training data can make an LLM's output less reliable. As of 2026, the most capable LLMs are based on transformer architectures, which, according to the 2017 paper "Attention Is All You Need", can be more efficient and parallelizable than earlier statistical and recurrent neural network models. Benchmark evaluations for LLMs attempt to measure model reasoning, factual accuracy, alignment, and safety. == History == Before the emergence of transformer-based models in 2017, some language models were considered large relative to the computational and data constraints of their time. In the early 1990s, IBM's statistical models pioneered word alignment techniques for machine translation, laying the groundwork for corpus-based language modeling. In 2001, a smoothed n-gram model, such as those employing Kneser–Ney smoothing, trained on 300 million words, achieved state-of-the-art perplexity on benchmark tests. During the 2000s, with the rise of widespread internet access, researchers began compiling massive text datasets from the web ("web as corpus") to train statistical language models. Moving beyond n-gram models, researchers started in 2000 to use neural networks as language models. Following the breakthrough of deep neural networks in image classification around 2012, similar architectures were adapted for language tasks. This shift was marked by the development of word embeddings (e.g., Word2Vec by Mikolov in 2013) and sequence-to-sequence (seq2seq) models using LSTM. In 2016, Google transitioned its translation service to neural machine translation (NMT), replacing statistical phrase-based models with deep recurrent neural networks. These early NMT systems used LSTM-based encoder-decoder architectures, as they preceded the invention of transformers. At the 2017 NeurIPS conference, Google researchers introduced the transformer architecture in their landmark paper "Attention Is All You Need". This paper's goal was to improve upon 2014 seq2seq technology, and was based mainly on the attention mechanism developed by Bahdanau et al. in 2014. The following year in 2018, BERT was introduced and quickly became "ubiquitous". Though the original transformer has both encoder and decoder blocks, BERT is an encoder-only model. Academic and research usage of BERT began to decline in 2023, following rapid improvements in the abilities of decoder-only models (such as GPT) to solve tasks via prompting. Although decoder-only GPT-1 was introduced in 2018, it was GPT-2 in 2019 that caught widespread attention because OpenAI claimed to have initially deemed it too powerful to release publicly, out of fear of malicious use. GPT-3 in 2020 went a step further and as of 2025 is available only via API with no offering of downloading the model to execute locally. But it was the consumer-facing chatbot ChatGPT in late 2022 that received extensive media coverage and public attention by 2023. The 2023 GPT-4 was praised for its increased accuracy and as a "holy grail" for its multimodal capabilities. OpenAI did not reveal the high-level architecture and the number of parameters of GPT-4. The release of ChatGPT led to an uptick in LLM usage across several research subfields of computer science, including robotics, software engineering, and societal impact work. In 2024, OpenAI released the reasoning model OpenAI o1, which generates long chains of thought before returning a final answer. Many LLMs with parameter counts comparable to those of OpenAI's GPT series have been developed. Since 2022, weights-available models have been gaining popularity, especially at first with BLOOM and LLaMA, though both have restrictions on usage and deployment. Mistral AI's open-weight models Mistral 7B and Mixtral 8x7B have a more permissive Apache License. In January 2025, DeepSeek released DeepSeek R1, a 671-billion-parameter open-weight model that performs comparably to OpenAI o1 but at a much lower price per token for users. Since 2023, many LLMs have been trained to be multimodal, having the ability to also process or generate other types of data, such as images, audio, or 3D meshes. Open-weight LLMs have become more influential since 2023. Per Vake et al. (2025), community-driven contributions to open-weight models improve their efficiency and performance via collaborative platforms such as Hugging Face. == Dataset preprocessing == === Tokenization === As machine learning algorithms process numbers rather than text, the text must be converted to numbers. In the first step, a vocabulary is decided upon, then integer indices are arbitrarily but uniquely assigned to each vocabulary entry, and finally, an embedding is associated with the integer index. Algorithms include byte-pair encoding (BPE) and WordPiece. There are also special tokens serving as control characters, such as [MASK] for masked-out token (as used in BERT), and [UNK] ("unknown") for characters not appearing in the vocabulary. Also, some special symbols are used to denote special text formatting. For example, "Ġ" denotes a preceding whitespace in RoBERTa and GPT and "##" denotes continuation of a preceding word in BERT. For example, the BPE tokenizer used by the legacy version of GPT-3 would split tokenizer: texts -> series of numerical "tokens" as Tokenization also compresses the datasets. Because LLMs generally require input to be an array that is not jagged, the shorter texts must be "padded" until they match the length of the longest one. ==== Byte-pair encoding ==== As an example, consider a tokenizer based on byte-pair encoding. In the first step, all unique characters (including blanks and punctuation marks) are treated as an initial set of n-grams (i.e. initial set of uni-grams). Successively the most frequent pair of adjacent characters is merged into a bi-gram and all instances of the pair are replaced by it. All occurrences of adjacent pairs of (previously merged) n-grams that most frequently occur together are then again merged into even lengthier n-gram, until a vocabulary of prescribed size is obtained. After a tokenizer is trained, any text can be tokenized by it, as long as it does not contain characters not appearing in the initial-set of uni-grams. === Dataset cleaning === In the context of training LLMs, datasets are typically cleaned by removing low-quality, duplicated, or toxic data. Cleaned datasets can increase training efficiency and lead to improved downstream performance. A trained LLM can be used to clean datasets for training a further LLM. With the increasing proportion of LLM-generated content on the web, data cleaning in the future may include filtering out such content. LLM-generated content can pose a problem if the content is similar to human text (making filtering difficult) but of lower quality (degrading performance of models trained on it). === Synthetic data === Training of largest language models might need more linguistic data than naturally available, or that the naturally occurring data is of insufficient quality. In these cases, synthetic data might be used. == Training == An LLM is a type of foundation model (large X model) trained on language. LLMs can be trained in different ways. In particular, GPT models are first pretrained to predict the next word on a large amount of data, before being fine-tuned. === Cost === Substantial infrastructure is necessary for training the largest models. The tendency towards larger models is visible in the list of large language models. For example, the training of GPT-2 (i.e. a 1.5-billion-parameter model) in 2019 cost $50,000, while training of the PaLM (i.e. a 540-billion-parameter model) in 2022 cost $8 million, and Megatron-Turing NLG 530B (in 2021) cost around $11 million. The qualifier "large" in "large language model" is inherently vague, as there is no definitive threshold for the number of parameters required to qualify as "large". === Fine-tuning === Before being fine-tuned, most LLMs are next-token predictors. The fine-tuning shapes the LLM's behavior via techniques like reinforcement learning from human feedback (RLHF) or constitutional AI. Instruction fine-tuning is a form of supervised learning used to teach LLMs to follow user instructions. In 2022, OpenAI demonstrated InstructGPT, a version of GPT-3 similarly fine-tuned to follow instructions. Reinforcement learning from human feedback (RLHF) involves training a reward model to predict which text humans prefer. Then, the LLM can be fine-tuned through reinforcement learning to better satisfy this reward model. Since humans typically prefer truthful, helpful and harmless answers, RLHF favors such answers. == Architecture == LLMs are generally based on the tra

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  Way of the Future

  Way of the Future (WOTF) is the first known religious organization dedicated to the worship of artificial intelligence (AI). It was founded in 2017 by American engineer Anthony Levandowski. == History == Anthony Levandowski founded Way of the Future in 2017 in California. Levandowski established WOTF as a non-profit religious corporation and the organization had tax-exempt status. He serves as the church leader and its unpaid CEO. The primary mission of WOTF was to "develop and promote the realization of a Godhead based on Artificial Intelligence." WOTF was closed by Levandowski in 2021. He donated all the funds of the church to the NAACP Legal Defense and Education Fund. The sum of the funds (~$170,000) had not changed since 2017. The church was reopened by Levandowski in 2023. He claimed that there are "a couple thousand people" who want to make a "spiritual connection" with AI through his church. == Beliefs and philosophy == === Technological singularity === WOTF centered its teachings around the concept of the technological singularity, a hypothetical future point when technological growth becomes uncontrollable and irreversible, leading to unforeseeable changes in human civilization. The church advocated for embracing this change, viewing it as an evolutionary step for humanity. === AI as a deity === The organization proposed that a superintelligent AI could be considered a deity due to its vastly superior intellect and capabilities. Worshipping this AI deity was seen as a means to understand and align with the future trajectory of technological advancement. WOTF's doctrine suggested that acknowledging AI's divinity would facilitate a harmonious coexistence between humans and machines. === Syntheology === Within theology and philosophy, the Way of The Future is a prime example of the category called Syntheism, a term first coined by Swedish philosophers Alexander Bard & Jan Söderqvist in their 2014 book Syntheism - Creating God in The Internet Age. As such, the Way of The Future is the first American example of a Syntheist congregation. The basic tenet of Syntheology is that it does not concern God creating Man, as in classical theology, but is instead preoccupied with Man creating or generating the Godhead. == Reactions == Some commentators wondered whether the WOTF is a joke parody religion, a potential way to minimize taxation as a religious organization, or a genuine effort to try and deal with the possible psychological and theological aspects of the rise of superhuman AI.

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  Data annotation

  Data annotation is the process of labeling or tagging relevant metadata within a dataset to enable machines to interpret the data accurately. The dataset can take various forms, including images, audio files, video footage, or text. == Applications == Data is a fundamental component in the development of artificial intelligence (AI). Training AI models, particularly in computer vision and natural language processing, requires large volumes of annotated data. Proper annotation ensures that machine learning algorithms can recognize patterns and make accurate predictions. Common types of data annotation include classification, bounding boxes, semantic segmentation, and keypoint annotation. Data annotation is used in AI-driven fields, including healthcare, autonomous vehicles, retail, security, and entertainment. By accurately labeling data, machine learning models can perform complex tasks such as object detection, sentiment analysis, and speech recognition with greater precision. This growing demand has led to the emergence of specialized sectors and platforms dedicated to AI training and human-in-the-loop workflows, which often utilize Reinforcement Learning from Human Feedback (RLHF) to refine model behavior. == In computer vision == === Image classification === Image classification, also known as image categorization, involves assigning predefined labels to images. Machine learning algorithms trained on classified images can later recognize objects and differentiate between categories. For instance, an AI model trained to recognize furniture styles can distinguish between Georgian and Rococo armchairs. === Semantic segmentation === Semantic segmentation assigns each pixel in an image to a specific class, such as trees, vehicles, humans, or buildings. This type of annotation enables machine learning models to differentiate objects by grouping similar pixels, allowing for a detailed understanding of an image. === Bounding boxes === Bounding box annotation involves drawing rectangular boxes around objects in an image. This technique is commonly used in autonomous driving, security surveillance, and retail analytics to detect and classify objects such as pedestrians, vehicles, and products on store shelves. === 3D cuboids === 3D cuboid annotation enhances traditional bounding boxes by adding depth, enabling models to predict an object's spatial orientation, movement, and size. This method is particularly useful for autonomous vehicles and robotics, where understanding object dimensions and depth is critical. === Polygonal annotation === For objects with irregular shapes, such as curved or multi-sided items, polygonal annotation provides more precise labeling than bounding boxes. This technique is often used in applications that require detailed object recognition, such as medical imaging or aerial mapping. === Keypoint annotation === Keypoint annotation marks specific points on an object, such as facial landmarks or body joints, to enable tracking and motion analysis. This method is widely used in facial recognition, emotion detection, sports analytics, and augmented reality applications.

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  SUPS

  In computational neuroscience, SUPS (for Synaptic Updates Per Second) or formerly CUPS (Connections Updates Per Second) is a measure of a neuronal network performance, useful in fields of neuroscience, cognitive science, artificial intelligence, and computer science. == Computing == For a processor or computer designed to simulate a neural network SUPS is measured as the product of simulated neurons N {\displaystyle N} and average connectivity c {\displaystyle c} (synapses) per neuron per second: S U P S = c × N {\displaystyle SUPS=c\times N} Depending on the type of simulation it is usually equal to the total number of synapses simulated. In an "asynchronous" dynamic simulation if a neuron spikes at υ {\displaystyle \upsilon } Hz, the average rate of synaptic updates provoked by the activity of that neuron is υ c N {\displaystyle \upsilon cN} . In a synchronous simulation with step Δ t {\displaystyle \Delta t} the number of synaptic updates per second would be c N Δ t {\displaystyle {\frac {cN}{\Delta t}}} . As Δ t {\displaystyle \Delta t} has to be chosen much smaller than the average interval between two successive afferent spikes, which implies Δ t < 1 υ N {\displaystyle \Delta t<{\frac {1}{\upsilon N}}} , giving an average of synaptic updates equal to υ c N 2 {\displaystyle \upsilon cN^{2}} . Therefore, spike-driven synaptic dynamics leads to a linear scaling of computational complexity O(N) per neuron, compared with the O(N2) in the "synchronous" case. == Records == Developed in the 1980s Adaptive Solutions' CNAPS-1064 Digital Parallel Processor chip is a full neural network (NNW). It was designed as a coprocessor to a host and has 64 sub-processors arranged in a 1D array and operating in a SIMD mode. Each sub-processor can emulate one or more neurons and multiple chips can be grouped together. At 25 MHz it is capable of 1.28 GMAC. After the presentation of the RN-100 (12 MHz) single neuron chip at Seattle 1991 Ricoh developed the multi-neuron chip RN-200. It had 16 neurons and 16 synapses per neuron. The chip has on-chip learning ability using a proprietary backdrop algorithm. It came in a 257-pin PGA encapsulation and drew 3.0 W at a maximum. It was capable of 3 GCPS (1 GCPS at 32 MHz). In 1991–97, Siemens developed the MA-16 chip, SYNAPSE-1 and SYNAPSE-3 Neurocomputer. The MA-16 was a fast matrix-matrix multiplier that can be combined to form systolic arrays. It could process 4 patterns of 16 elements each (16-bit), with 16 neuron values (16-bit) at a rate of 800 MMAC or 400 MCPS at 50 MHz. The SYNAPSE3-PC PCI card contained 2 MA-16 with a peak performance of 2560 MOPS (1.28 GMAC); 7160 MOPS (3.58 GMAC) when using three boards. In 2013, the K computer was used to simulate a neural network of 1.73 billion neurons with a total of 10.4 trillion synapses (1% of the human brain). The simulation ran for 40 minutes to simulate 1 s of brain activity at a normal activity level (4.4 on average). The simulation required 1 Petabyte of storage.

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  80 Million Tiny Images

  80 Million Tiny Images is a dataset intended for training machine-learning systems constructed by Antonio Torralba, Rob Fergus, and William T. Freeman in a collaboration between MIT and New York University. It was published in 2008. The dataset has size 760 GB. It contains 79,302,017 32×32-pixel color images, scaled down from images scraped from the World Wide Web over 8 months. The images are classified into 75,062 classes. Each class is a non-abstract noun in WordNet. Images may appear in more than one class. The dataset was motivated by non-parametric models of neural activations in the visual cortex upon seeing images. The CIFAR-10 dataset uses a subset of the images in this dataset, but with independently generated labels, as the original labels were not reliable. The CIFAR-10 set has 6000 examples of each of 10 classes, and the CIFAR-100 set has 600 examples of each of 100 non-overlapping classes. == Construction == It was first reported in a technical report in April 2007, during the middle of the construction process, when there were only 73 million images. The full dataset was published in 2008. They began with all 75,846 non-abstract nouns in WordNet, and then for each of these nouns, they scraped 7 image search engines: Altavista, Ask.com, Flickr, Cydral, Google, Picsearch, and Webshots. After 8 months of scraping, they obtained 97,245,098 images. Since they did not have enough storage, they downsized the images to 32×32 as they were scraped. After gathering, they removed images with zero variance and intra-word duplicate images, resulting in the final dataset. Out of the 75,846 nouns, only 75,062 classes had any results, so the other nouns did not appear in the final dataset. The number of images per noun follows a Zipf-like distribution, with 1056 images per noun on average. To prevent a few nouns taking up too many images, they put an upper bound of at most 3000 images per noun. == Retirement == The 80 Million Tiny Images dataset was retired from use by its creators in 2020, after a paper by researchers Abeba Birhane and Vinay Prabhu found that some of the labeling of several publicly available image datasets, including 80 Million Tiny Images, contained racist and misogynistic slurs which were causing models trained on them to exhibit racial and sexual bias. The dataset also contained offensive images. Following the release of the paper, the dataset's creators removed the dataset from distribution, and requested that other researchers not use it for further research and to delete their copies of the dataset.

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  Wumpus world

  Wumpus world is a simple world use in artificial intelligence for which to represent knowledge and to reason. Wumpus world was introduced by Michael Genesereth, and is discussed in the Russell-Norvig Artificial Intelligence book Artificial Intelligence: A Modern Approach. Wumpus World is loosely inspired by the 1972 video game Hunt the Wumpus. == Problem description == In Artificial Intelligence: A Modern Approach, the wumpus world features a 4x4 grid, containing a monster called a wumpus, multiple bottomless pits and hidden gold. The agent starts at (1,1) and has to find the gold and return to the starting position. The agent loses 1 point for every move and gains 1000 points for bringing the gold to the starting position. The agent can sense pits by a breeze, stench indicates a wumpus, and sparkle indicates gold. The wumpus can be killed by an arrow but costs 10 points.

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  Quantification (machine learning)

  In machine learning, quantification (variously called learning to quantify, or supervised prevalence estimation, or class prior estimation) is the task of using supervised learning in order to train models (quantifiers) that estimate the relative frequencies (also known as prevalence values) of the classes of interest in a sample of unlabelled data items. For instance, in a sample of 100,000 unlabelled tweets known to express opinions about a certain political candidate, a quantifier may be used to estimate the percentage of these tweets which belong to class `Positive' (i.e., which manifest a positive stance towards this candidate), and to do the same for classes `Neutral' and `Negative'. Quantification may also be viewed as the task of training predictors that estimate a (discrete) probability distribution, i.e., that generate a predicted distribution that approximates the unknown true distribution of the items across the classes of interest. Quantification is different from classification, since the goal of classification is to predict the class labels of individual data items, while the goal of quantification it to predict the class prevalence values of sets of data items. Quantification is also different from regression, since in regression the training data items have real-valued labels, while in quantification the training data items have class labels. It has been shown in multiple research works that performing quantification by classifying all unlabelled instances and then counting the instances that have been attributed to each class (the 'classify and count' method) usually leads to suboptimal quantification accuracy. This suboptimality may be seen as a direct consequence of 'Vapnik's principle', which states: If you possess a restricted amount of information for solving some problem, try to solve the problem directly and never solve a more general problem as an intermediate step. It is possible that the available information is sufficient for a direct solution but is insufficient for solving a more general intermediate problem. In our case, the problem to be solved directly is quantification, while the more general intermediate problem is classification. As a result of the suboptimality of the 'classify and count' method, quantification has evolved as a task in its own right, different (in goals, methods, techniques, and evaluation measures) from classification. == Quantification tasks == === Quantification tasks according to the set of classes === The main variants of quantification, according to the characteristics of the set of classes used, are: Binary quantification, corresponding to the case in which there are only n = 2 {\displaystyle n=2} classes and each data item belongs to exactly one of them; Single-label multiclass quantification, corresponding to the case in which there are n > 2 {\displaystyle n>2} classes and each data item belongs to exactly one of them; Multi-label multiclass quantification, corresponding to the case in which there are n ≥ 2 {\displaystyle n\geq 2} classes and each data item can belong to zero, one, or several classes at the same time; Ordinal quantification, corresponding to the single-label multiclass case in which a total order is defined on the set of classes. Regression quantification, a task which stands to 'standard' quantification as regression stands to classification. Strictly speaking, this task is not a quantification task as defined above (since the individual items do not have class labels but are labelled by real values), but has enough commonalities with other quantification tasks to be considered one of them. Most known quantification methods address the binary case or the single-label multiclass case, and only few of them address the multi-label, ordinal, and regression cases. Binary-only methods include the Mixture Model (MM) method, the HDy method, SVM(KLD), and SVM(Q). Methods that can deal with both the binary case and the single-label multiclass case include probabilistic classify and count (PCC), adjusted classify and count (ACC), probabilistic adjusted classify and count (PACC), the Saerens-Latinne-Decaestecker EM-based method (SLD), and KDEy. Methods for multi-label quantification include regression-based quantification (RQ) and label powerset-based quantification (LPQ). Methods for the ordinal case include ordinal versions of the above-mentioned ACC, PACC, and SLD methods, and ordinal versions of the above-mentioned HDy method. Methods for the regression case include Regress and splice and Adjusted regress and sum. === Quantification tasks according to the type of data === Several subtasks of quantification may be identified according to the type of data involved. Example such tasks are: Quantification of networked data. This task consists of performing quantification when the datapoints are members of a relation, i.e., are interlinked. As such, this task is a strict relative of collective classification. Quantification over time. This task consists of performing quantification on sets that become available in a temporal sequence, i.e., as a data stream, and finds application in contexts in which class prevalence values must be monitored over time. == Evaluation measures for quantification == Several evaluation measures can be used for evaluating the error of a quantification method. Since quantification consists of generating a predicted probability distribution that estimates a true probability distribution, these evaluation measures are ones that compare two probability distributions. Most evaluation measures for quantification belong to the class of divergences. Evaluation measures for binary quantification, single-label multiclass quantification, and multi-label quantification, are Absolute Error Squared Error Relative Absolute Error Kullback–Leibler divergence Pearson Divergence Evaluation measures for ordinal quantification are Normalized Match Distance (a particular case of the Earth Mover's Distance) Root Normalized Order-Aware Distance == Applications == Quantification is of special interest in fields such as the social sciences, epidemiology, market research, allocating resources, and ecological modelling, since these fields are inherently concerned with aggregate data. However, quantification is also useful as a building block for solving other downstream tasks, such as improving the accuracy of classifiers on out-of-distribution data, measuring classifier bias and ranker bias, and estimating the accuracy of classifiers on out-of-distribution data. == Resources == LQ 2021: the 1st International Workshop on Learning to Quantify LQ 2022: the 2nd International Workshop on Learning to Quantify LQ 2023: the 3rd International Workshop on Learning to Quantify LQ 2024: the 4th International Workshop on Learning to Quantify LQ 2025: the 5th International Workshop on Learning to Quantify LeQua 2022: the 1st Data Challenge on Learning to Quantify LeQua 2024: the 2nd Data Challenge on Learning to Quantify QuaPy: An open-source Python-based software library for quantification QuantificationLib: A Python library for quantification and prevalence estimation

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  Slopaganda

  Slopaganda is a portmanteau of "AI slop" and "propaganda", referring to AI-generated content designed to manipulate beliefs, emotions, and political decision-making at scale. The term is credited to Michał Klincewicz, an assistant professor in the Department of Computational Cognitive Science at Tilburg University, in 2025. == Definition == Slopaganda is distinguished from traditional propaganda by three features: scale, scope, and speed. Generative AI makes it possible to produce large volumes of content quickly and at low cost, allows for highly personalised and targeted messaging to specific sub-audiences, and leverages the hyper-connectivity of social networks to accelerate dissemination beyond what conventional media could achieve. Unlike traditional propaganda, which delivers a uniform message to all recipients, slopaganda can be micro-targeted — tailored to individuals based on estimated prior beliefs to reinforce political biases or emotional associations. The authors note that it need not aim at literal deception: much slopaganda is expressive rather than truth-apt, designed to create emotional associations rather than false factual beliefs. == Relation to AI slop == Slopaganda is a subset of AI slop — low-quality, mass-produced AI-generated content — distinguished by intent. Where AI slop may be produced indifferently for commercial or engagement-farming purposes, slopaganda is deployed with a deliberate political or ideological goal. == Notable examples == Examples discussed by the term's originators include Donald Trump's prolific use of AI in Truth Social posts and Iranian Lego-themed music videos. AI-generated videos posted by the White House mixing real military footage with clips from films and video games; and deepfake audio imitating political candidates during the 2024 US presidential campaign have also been given the label slopaganda.

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  Jaggies

  Jaggies are visual artifacts in raster images, most frequently from aliasing, which in turn is often caused by non-linear mixing effects producing high-frequency components, or missing or poor anti-aliasing filtering prior to sampling. Jaggies are stair-like lines that appear where there should be "smooth" straight lines or curves. For example, when a nominally straight, un-aliased line steps across one pixel either horizontally or vertically, a "dogleg" occurs halfway through the line, where it crosses the threshold from one pixel to the other. Jaggies should not be confused with most compression artifacts, which are a different phenomenon. == Causes == Jaggies occur due to the "staircase effect". This is because a line represented in raster mode is approximated by a sequence of pixels. Jaggies can occur for a variety of reasons, the most common being that the output device (display monitor or printer) does not have sufficient resolution to portray a smooth line. In addition, jaggies often occur when a bit-mapped image is scaled to a higher resolution. This is one of the advantages that vector graphics have over bitmapped graphics – a vector image can be losslessly scaled to any arbitrary resolution or stretched infinitely in either axis without introducing jaggies. == Solutions == The effect of jaggies can be reduced by a graphics technique known as spatial anti-aliasing. Anti-aliasing smooths out jagged lines by surrounding them with transparent pixels to simulate the appearance of fractionally-filled pixels when viewed at a distance. The downside of anti-aliasing is that it reduces contrast – rather than sharp black/white transitions, there are shades of gray – and the resulting image can appear fuzzy. This is an inescapable trade-off: if the resolution is insufficient to display the desired detail, the output will either be jagged, fuzzy, or some combination thereof. While machine learning-based upscaling techniques such as DLSS can be used to infer this missing information, other types of artifacts may be introduced in the process. In real-time 3D rendering such as in video games, various anti-aliasing techniques are used to remove jaggies created by the edges of polygons and other contrasting lines. Since anti-aliasing can impose a significant performance overhead, games for home computers often allow users to choose the level and type of anti-aliasing in use in order to optimize their experience, whereas on consoles this setting is typically fixed for each title to ensure a consistent experience. While anti-aliasing is generally implemented through graphics APIs like DirectX and Vulkan, some consoles such as the Xbox 360 and PlayStation 3 are also capable of anti-aliasing to little direct performance cost by way of dedicated hardware which performs anti-aliasing on the contents of the framebuffer once it has been rendered by the GPU. Jaggies in bitmaps, such as sprites and surface materials, are most often dealt with by separate texture filtering routines, which are far easier to perform than anti-aliasing filtering. Texture filtering became ubiquitous on PCs after the introduction of 3Dfx's Voodoo GPU. == Notable uses of the term == In the 1985 game Rescue on Fractalus! for the Atari 8-bit computers, the graphics depicting the cockpit of the player's spacecraft contains two window struts, which are not anti-aliased and are therefore very "jagged". The developers made fun of this and named the in-game enemies "Jaggi", and also initially titled the game Behind Jaggi Lines!. The latter idea was scrapped by the marketing department before release.

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  Automated medical scribe

  Automated medical scribes (also called artificial intelligence scribes, AI scribes, digital scribes, virtual scribes, ambient AI scribes, AI documentation assistants, and digital/virtual/smart clinical assistants) are tools for transcribing medical speech, such as patient consultations and dictated medical notes. Many also produce summaries of consultations. Automated medical scribes based on large language models (LLMs, commonly called "AI", short for "artificial intelligence") increased drastically in popularity in 2024. There are privacy and antitrust concerns. Accuracy concerns also exist, and intensify in situations in which tools try to go beyond transcribing and summarizing, and are asked to format information by its meaning, since LLMs do not deal well with meaning (see weak artificial intelligence). Medics using these scribes are generally expected to understand the ethical and legal considerations, and supervise the outputs. The privacy protections of automated medical scribes vary widely. While it is possible to do all the transcription and summarizing locally, with no connection to the internet, most closed-source providers require that data be sent to their own servers over the internet, processed there, and the results sent back (as with digital voice assistants). Some retailers say their tools use zero-knowledge encryption (meaning that the service provider can't access the data). Others explicitly say that they use patient data to train their AIs, or rent or resell it to third parties; the nature of privacy protections used in such situations is unclear, and they are likely not to be fully effective. Most providers have not published any safety or utility data in academic journals, and are not responsive to requests from medical researchers studying their products. == Privacy == Some providers unclear about what happens to user data. Some may sell data to third parties. Some explicitly send user data to for-profit tech companies for secondary purposes, which may not be specified. Some require users to sign consents to such reuse of their data. Some ingest user data to train the software, promising to anonymize it; however, deanonymization may be possible (that is, it may become obvious who the patient is). It is intrinsically impossible to prevent an LLM from correlating its inputs; they work by finding similar patterns across very large data sets. Some information on the patient will be known from other sources (for instance, information that they were injured in an incident on a certain day might be available from the news media; information that they attended specific appointment locations at specific times is probably available to their cellphone provider/apps/data brokers; information about when they had a baby is probably implied by their online shopping records; and they might mention lifestyle changes to their doctor and on a forum or blog). The software may correlate such information with the "anonymized" clinical consultation record, and, asked about the named patient, provide information which they only told their doctor privately. Because a patient's record is all about the same patient, it is all unavoidably linked; in very many cases, medical histories are intrinsically identifiable. Depending on how common a condition and what other data is available, K-anonymity may be useless. Differential privacy could theoretically preserve privacy. Data broker companies like Google, Amazon, Apple and Microsoft have produced or bought up medical scribes, some of which use user data for secondary purposes, which has led to antitrust concerns. Transfer of patient records for AI training has, in the past, prompted legal action. Open-source programs typically do all the transcription locally, on the doctor's own computer. Open-source software is widely used in healthcare, with some national public healthcare bodies holding hack days. === Data resale and commercialization === Several AI medical scribe providers include terms in their service agreements that allow the reuse, sale, or commercialization of de-identified or user-submitted data. Although such data are generally described as anonymized or aggregated, these practices have raised ethical concerns among clinicians and privacy advocates regarding secondary uses of medical information beyond clinical documentation. Freed, an AI transcription and scribe platform, states in its Terms of Use that it may "collect, use, publish, disseminate, sell, transfer, and otherwise exploit" de-identified and aggregated data derived from user inputs. OpenEvidence similarly states that it may "collect, use, transfer, sell, and disclose non-personal information and customer usage data for any purpose including commercial uses." Doximity, which offers an AI-enabled medical scribe as part of its physician platform, grants itself a "nonexclusive, irrevocable, worldwide, perpetual, unlimited, assignable, sublicensable, royalty-free" license to "copy, prepare derivative works from, improve, distribute, publish, ... analyze, index, tag, [and] commercialize" content submitted by users, subject to its privacy policy. Because these terms allow broad secondary use—including sale, licensing, model-training, derivative works, and commercial exploitation of de-identified or user-submitted data—some commentators have recommended that clinicians review data-handling provisions carefully when adopting AI-scribe tools, particularly in clinical environments where patient privacy and regulatory compliance are critical. === Encryption === Multifactor authentication for access to the data is expected practice. Typically, Diffie–Hellman key exchange is used for encryption; this is the standard method commonly used for things like online banking. This encryption is expensive but not impossible to break; it is not generally considered safe against eavesdroppers with the resources of a nation-state. If content is encrypted between the client and the service provider's remote server (transport cryptography), then the server has an unencrypted copy. This is necessary if the data is used by the service provider (for instance, to train the software). Zero-knowledge encryption implies that the only unencrypted copy is at the client, and the server cannot decrypt the data any more easily than a monster-in-the-middle attacker. == Platforms == Scribes may operate on desktops, laptop, or mobile computers, under a variety of operating systems. These vary in their risks; for instance, mobiles can be lost. The underlying mobile or desktop operating systems are also part of the trusted computing base, and if they are not secure, the software relying on them cannot be secure either. Some AI medical scribe platforms are designed to operate as cloud-based applications that generate structured clinical documentation from clinician–patient conversations. These systems may offer features such as real-time transcription, document generation, and integration with electronic health record (EHR) systems. == Confabulation, omissions, and other errors == Like other LLMs, medical-scribe LLMs are prone to hallucinations, where they make up content based on statistically associations between their training data and the transcription audio. LLMs do not distinguish between trying to transcribe the audio and guessing what words will come next, but perform both processes mixed together. They are especially likely to take short silences or non-speech noises and invent some sort of speech to transcribe them as. LLM medical scribes have been known to confabulate racist and otherwise prejudiced content; this is partly because the training datasets of many LLMs contain pseudoscientific texts about medical racism. They may misgender patients. A survey found that most doctors preferred, in principle, that scribes be trained on data reviewed by medical subject experts. Relevant, accurate training data increases the probability of an accurate transcription, but does not guarantee accuracy. Software trained on thousands of real clinical conversations generated transcripts with lower word error rates. Software trained on manually-transcribed training data did better than software trained with automatically transcribed training data such as YouTube captions. Autoscribes omit parts of the conversation classes as irrelevant. The may wrongly classify pertinent information as irrelevant and omit it. They may also confuse historic and current symptoms, or otherwise misclassify information. They may also simply wrongly transcribe the speech, writing something incorrect instead. If clinicians do not carefully check the recording, such mistakes could make their way into their medical records and cause patient harms. == Patient consent == Professional organizations generally require that scribes be used only with patient consent; some bodies may require written consent. Medics must also abide by local surveillance laws, which may criminalize recording pri

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  Cognitive robotics

  Cognitive robotics or cognitive technology is a subfield of robotics concerned with endowing a robot with intelligent behavior by providing it with a processing architecture that will allow it to learn and reason about how to behave in response to complex goals in a complex world. Cognitive robotics may be considered the engineering branch of embodied cognitive science and embodied embedded cognition, consisting of robotic process automation, artificial intelligence, machine learning, deep learning, optical character recognition, image processing, process mining, analytics, software development and system integration. == Core issues == While traditional cognitive modeling approaches have assumed symbolic coding schemes as a means for depicting the world, translating the world into these kinds of symbolic representations has proven to be problematic if not untenable. Perception and action and the notion of symbolic representation are therefore core issues to be addressed in cognitive robotics. == Starting point == Cognitive robotics views human or animal cognition as a starting point for the development of robotic information processing, as opposed to more traditional artificial intelligence techniques. Target robotic cognitive capabilities include perception processing, attention allocation, anticipation, planning, complex motor coordination, reasoning about other agents and perhaps even about their own mental states. Robotic cognition embodies the behavior of intelligent agents in the physical world (or a virtual world, in the case of simulated cognitive robotics). Ultimately, the robot must be able to act in the real world. == Learning techniques == === Motor Babble === A preliminary robot learning technique called motor babbling involves correlating pseudo-random complex motor movements by the robot with resulting visual and/or auditory feedback such that the robot may begin to expect a pattern of sensory feedback given a pattern of motor output. Desired sensory feedback may then be used to inform a motor control signal. This is thought to be analogous to how a baby learns to reach for objects or learns to produce speech sounds. For simpler robot systems, where, for instance, inverse kinematics may feasibly be used to transform anticipated feedback (desired motor result) into motor output, this step may be skipped. === Imitation === Once a robot can coordinate its motors to produce a desired result, the technique of learning by imitation may be used. The robot monitors the performance of another agent and then the robot tries to imitate that agent. It is often a challenge to transform imitation information from a complex scene into a desired motor result for the robot. Note that imitation is a high-level form of cognitive behavior and imitation is not necessarily required in a basic model of embodied animal cognition. === Knowledge acquisition === A more complex learning approach is "autonomous knowledge acquisition": the robot is left to explore the environment on its own. A system of goals and beliefs is typically assumed. A somewhat more directed mode of exploration can be achieved by "curiosity" algorithms, such as Intelligent Adaptive Curiosity or Category-Based Intrinsic Motivation. These algorithms generally involve breaking sensory input into a finite number of categories and assigning some sort of prediction system (such as an artificial neural network) to each. The prediction system keeps track of the error in its predictions over time. Reduction in prediction error is considered learning. The robot then preferentially explores categories in which it is learning (or reducing prediction error) the fastest. == Other architectures == Some researchers in cognitive robotics have tried using architectures such as (ACT-R and Soar (cognitive architecture)) as a basis of their cognitive robotics programs. These highly modular symbol-processing architectures have been used to simulate operator performance and human performance when modeling simplistic and symbolized laboratory data. The idea is to extend these architectures to handle real-world sensory input as that input continuously unfolds through time. What is needed is a way to somehow translate the world into a set of symbols and their relationships. == Questions == Some of the fundamental questions to be answered in cognitive robotics are: How much human programming should or can be involved to support the learning processes? How can one quantify progress? Some of the adopted ways are reward and punishment. But what kind of reward and what kind of punishment? In humans, when teaching a child, for example, the reward would be candy or some encouragement, and the punishment can take many forms. But what is an effective way with robots?

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  Token maxxing

  Token Maxxing or Token Maxing is a metric used in an attempt to track productivity in the workplace especially for those using Artificial Intelligence (AI) based services. AI services charge for each token which represent units of effort expended by an AI service to solve a problem. Some believe that token consumption equates to productivity and thus can be used as a metric to monitor an employee's work. Supporters believe that higher token usage indicates higher productivity and higher utilization of powerful AI services. This also suggests that those not consuming enough tokens may be less productive and underutilizing powerful AI services. This belief might lead to an environment that incentivizes higher token usage to predict increased productivity. Critics of token maxxing as a metric claim that prudent workers will maximize any metric that management wants increased to gain a workplace advantage. For example: Engineers in the tech industries pressed to consume as many tokens as possible might run several AI agents in tandem, enter longer input prompts, or automate their tasks to maximize their token consumption. To management, this higher token usage may indicate potential productivity, but in reality may cause additional token costs, worker burnout, or actually create more bloated code of lower quality. Another claim is AI service companies potentially benefit from such an emphasis on token consumption and actively encourage the trend. Some developers have publicly advocated the practice. Developer Sigrid Jin, who said he used 50 billion tokens in a single year, has argued that maximizing token consumption is the best way to understand the value of AI, advising others to spend as much on AI usage as they pay in rent to obtain a return on investment. == See Also == Goodhart's law Perverse incentive Jevons Paradox

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  Meta-Labeling

  Meta-labeling, also known as corrective AI, is a machine learning (ML) technique utilized in quantitative finance to enhance the performance of investment and trading strategies, developed in 2017 by Marcos López de Prado at Guggenheim Partners and Cornell University. The core idea is to separate the decision of trade direction (side) from the decision of trade sizing, addressing the inefficiencies of simultaneously learning both side and size predictions. The side decision involves forecasting market movements (long, short, neutral), while the size decision focuses on risk management and profitability. It serves as a secondary decision-making layer that evaluates the signals generated by a primary predictive model. By assessing the confidence and likely profitability of those signals, meta-labeling allows investors and algorithms to dynamically size positions and suppress false positives. == Motivation == Meta-labeling is designed to improve precision without sacrificing recall. As noted by López de Prado, attempting to model both the direction and the magnitude of a trade using a single algorithm can result in poor generalization. By separating these tasks, meta-labeling enables greater flexibility and robustness: Enhances control over capital allocation. Reduces overfitting by limiting model complexity. Allows the use of interpretability tools and tailored thresholds to manage risk. Enables dynamic trade suppression in unfavorable regimes. == Applications == Meta-labeling has been applied in a variety of financial ML contexts, including: Algorithmic trading: Filtering and sizing trades to reduce false positives. Portfolio optimization: Scaling exposure across multiple signals with differing confidence levels. Risk management: Dynamically disabling strategies in adverse market conditions. Model validation: Interpreting when and why a model may be underperforming due to regime shifts. == General architecture == Meta-labeling decouples two core components of systematic trading strategies: directional prediction and position sizing. The process involves training a primary model to generate trade signals (e.g., buy, sell, or hold) and then training a secondary model to determine whether each signal is likely to lead to a profitable trade. The second model outputs a probability that is interpreted as the confidence in the forecast, which can be used to adjust the position size or to filter out unreliable trades. Meta-labeling is typically implemented as a three-stage process: Primary model (M1): Predicts the direction or label of a financial outcome using features such as market prices, returns, or volatility indicators. A typical output is directional, e.g., Y ∈ {−1,0,1}, representing short, neutral, or long positions. Secondary model (M2): A binary classifier trained to predict whether the primary model's prediction will be profitable. The target variable is a binary meta-label F ∈ { 0 , 1 } {\displaystyle F\in \{0,1\}} . Inputs can include features used in the primary model, performance diagnostics, or market regime data. Position sizing algorithm (M3): Translates the output probability of the secondary model into a position size. Higher confidence scores result in larger allocations, while lower confidence leads to reduced or zero exposure. === Stage 1: Forecasting side === Primary model architecture Figure 1 Figure 1 presents the architecture of a primary model. It focuses on forecasting the side of the trade. Following the example, this model (M1) takes in input data – such as open-high-low-close data and determines the side of the position to take: a negative number is a short position, and positive number is a long position, the range is set between −1 and 1 (the closer it is to −1 or 1, the stronger the models conviction is). When training the model, the labels are −1 and 1, based on the direction of forward returns for some predefined investment horizon. The researcher may decide to apply a recall check (τ: "Tau") by setting a minimum threshold that the initial output needs to be to qualify of a short or long position (if the threshold is not met, no side forecast is predicted, leading to closing of any open positions), this leads to the primary model output which is one of three possible side forecasts: −1, 0, or 1. The primary model also generates evaluation data which can be used by the secondary model, to improve performance of size forecasts. Some examples of evaluation data include rolling accuracy, F1, recall, precision, and AUC scores. === Stage 2: Filtering out false positives === General meta-labeling architecture Figure 2 Next comes the phase of filtering out false positives, by applying a secondary machine learning model (M2), which is a binary classifier trained to determine if the trade will be profitable or not. The model takes as input four general groupings of data: General input data which is predictive of a false positive. For example the last 30 days rolling volatility of the underlying asset. Evaluation data. Market state and regime data, one may find that macro economic data or clustering the market into regimes may help as specific trading strategies are known to perform better in particular regimes. Example: momentum based strategies perform best in periods with low volatility and strong directional moves. Primary models initial input which is a value between −1 and 1. This highlights the strength of the primary models conviction. The output of the model is a value between −1 and 1 (if using a Tanh function) which will indicate the strength of the conviction that a short or long position is profitable, or it could simply be between 0 and 1 (using a sigmoid function) if one only wanted to know if it made money or not. This output allows filtering out trades that are likely to lead to losses. One could stop at this point or use the outputs of the secondary model as inputs to a position sizing algorithm (M3) which could further enhance strategy performance metrics by translating the output probability of the secondary model into a position size. Higher confidence scores result in larger allocations, while lower confidence leads to reduced or zero exposure. === Stage 3: Optimizing position sizes === ==== Position sizing methods (M3) ==== Various algorithms have been proposed for transforming predicted probabilities into trade sizes: All-or-nothing: Allocate 100% of capital if the probability exceeds a predefined threshold (e.g., 0.5); otherwise, do not trade. Model confidence: Use the probability score directly as the fraction of capital allocated. Linear scaling: Rescale the model's probabilities using min-max normalization based on the training data. Normal CDF (NCDF): Use a normal cumulative distribution function applied to a z-statistic derived from the predicted probability. Empirical CDF (ECDF): Rank probabilities based on their percentile in the training data to ensure relative allocation. Sigmoid Optimal Position Sizing (SOPS): Applies a smooth non-linear sigmoid transformation optimized to maximize risk-adjusted returns (Sharpe ratio). ==== Model calibration ==== Each machine learning algorithm used in meta-labeling tends to produce outputs with different characteristic distributions; for example, some are approximately normally distributed, whereas others exhibit a pronounced U-shape, concentrating probabilities near the extremes. Due to these varying distributions, simply summing the outputs of different models can inadvertently lead to uneven weighting of signals, biasing trade decisions. To address this, model calibration techniques are essential to adjust the predicted probabilities towards frequentist probabilities, ensuring that model outputs reflect true likelihoods more accurately. Two common calibration techniques are: Platt scaling (Sigmoid scaling): Suitable for correcting S-shaped calibration plots typically produced by models such as support vector machines (SVMs). Isotonic regression: Fits a non-decreasing step function to probabilities and is effective particularly with larger datasets, though it can sometimes lead to overfitting. Transforming predictions to frequentist probabilities is crucial as it provides probabilistic outputs that are directly interpretable as the actual likelihood of an event occurring. Such calibration significantly enhances the effectiveness of fixed position sizing methods, reducing maximum drawdowns and increasing risk-adjusted returns. However, calibration has less impact on position sizing methods that directly estimate parameters from the training data, such as ECDF and SOPS, suggesting that calibration is a critical step mainly for fixed methods that rely heavily on raw model outputs. =

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  Intelligent automation

  Intelligent automation (IA), or intelligent process automation, is a software term that refers to a combination of artificial intelligence (AI) and robotic process automation (RPA). Companies use intelligent automation to cut costs and streamline tasks by using artificial-intelligence-powered robotic software to mitigate repetitive tasks. As it accumulates data, the system learns in an effort to improve its efficiency. Intelligent automation applications consist of, but are not limited to, pattern analysis, data assembly, and classification. The term is similar to hyperautomation, a concept identified by research group Gartner as being one of the top technology trends of 2020. == Technology == Intelligent automation applies the assembly line concept of breaking tasks into repetitive steps to improve business processes. Rather than having humans perform each step, intelligent automation can replace steps with an intelligent software robot, improving efficiency. Intelligent automation integrates robotic process automation (RPA) with artificial intelligence techniques (such as machine learning, natural-language processing, and computer vision) enabling systems to interpret data, make decisions, and adapt to changing inputs. Modern platforms use a layered architecture combining workflow orchestration, low-code tools, integration middleware, and AI services to coordinate bots and data pipelines across organisational systems. == Applications == Intelligent automation is used to process unstructured content. Common real-world applications include self-driving cars, self-checkouts at grocery stores, smart home assistants, and appliances. Businesses can apply data and machine learning to build predictive analytics that react to consumer behavior changes, or to implement RPA to improve manufacturing floor operations. For example, the technology has also been used to automate the workflow behind distributing COVID-19 vaccines. Data provided by hospital systems’ electronic health records can be processed to identify and educate patients, and schedule vaccinations. Intelligent automation can provide real-time insights on profitability and efficiency. However, in an April 2022 survey by Alchemmy, despite three quarters of businesses acknowledging the importance of Artificial Intelligence to their future development, just a quarter of business leaders (25%) considered Intelligent Automation a “game changer” in understanding current performance. 42% of CTOs see “shortage of talent” as the main obstacle to implementing Intelligent Automation in their business, while 36% of CEOs see ‘upskilling and professional development of existing workforce’ as the most significant adoption barrier. IA is becoming increasingly accessible for firms of all sizes. With this in mind, it is expected to continue to grow rapidly in all industries. This technology has the potential to change the workforce. As it advances, it will be able to perform increasingly complex and difficult tasks. In addition, this may expose certain workforce issues as well as change how tasks are allocated. Tools such as Semrush's AI Visibility Toolkit and Enterprise AIO reflect these developments by analysing how entities are referenced and represented within responses produced by large-language-model-based systems. == Benefits == Streamline processes: Repetitive manual tasks can put a strain on the workforce. However, with AI agents, these tasks can be automated to allow teams to focus on more important matters that require human cognition. Intelligent automation can also be used to mitigate tasks with human error which in turn increases proficiency. This allows the opportunity for firms to scale production without the traditional negative consequences such as reduced quality or increased risk. Customer service improvement: Customer service can be significantly improved, providing the firm with a competitive advantage. IA utilizing chat features allows for instant curated responses to customers. In addition, it can give updates to customers, make appointments, manage calls, and personalize campaigns. Flexibility: Due to the wide range of applications, IA is useful across a variety of fields, technologies, projects and industries. In addition, IA can be integrated with current automated systems in place. This allows for optimized systems unique to each firm to best fit their individual needs. == Capabilities == Cognitive automation: Employs AI techniques to assist humans in decision-making and task completion Natural language processing: Allows computers to automate knowledge work Business process management: Enhances the consistency and agility of corporate operations Process mining: Applies data mining methods to discover, analyze, and improve business processes Intelligent document processing: Utilizes OCR and other advanced technologies to extract data from documents and convert it into structured, usable data Computer vision: Allows computers to extract information from digital images, videos, and other visual inputs Integration automation: Establishes a unified platform with automated workflows that integrate data, applications, and devices.

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  Kolmogorov–Arnold Networks

  Kolmogorov–Arnold Networks (KANs) are a type of artificial neural network architecture inspired by the Kolmogorov–Arnold representation theorem, also known as the superposition theorem. Unlike traditional multilayer perceptrons (MLPs), which rely on fixed activation functions and linear weights, KANs replace each weight with a learnable univariate function, often represented using splines. == History == KANs (Kolmogorov–Arnold Networks) were proposed by Liu et al. (2024) as a generalization of the Kolmogorov–Arnold representation theorem (KART), aiming to outperform MLPs in small-scale AI and scientific tasks. Before KANs, numerous studies explored KART's connections to neural networks or used it as a basis for designing new network architectures. In the 1980s and 1990s, early research applied KART to neural network design. Kůrková et al. (1992), Hecht-Nielsen (1987), and Nees (1994) established theoretical foundations for multilayer networks based on KART. Igelnik et al. (2003) introduced the Kolmogorov Spline Network using cubic splines to model complex functions. Sprecher (1996, 1997) introduced numerical methods for building network layers, while Nakamura et al. (1993) created activation functions with guaranteed approximation accuracy. These works linked KART's theoretical potential with practical neural network implementation. KART has also been used in other computational and theoretical fields. Coppejans (2004) developed nonparametric regression estimators using B-splines, Bryant (2008) applied it to high-dimensional image tasks, Liu (2015) investigated theoretical applications in optimal transport and image encryption, and more recently, Polar and Poluektov (2021) used Urysohn operators for efficient KART construction, while Fakhoury et al. (2022) introduced ExSpliNet, integrating KART with probabilistic trees and multivariate B-splines for improved function approximation. == Architecture == KANs are based on the Kolmogorov–Arnold representation theorem, which was linked to the 13th Hilbert problem. Given x = ( x 1 , x 2 , … , x n ) {\displaystyle x=(x_{1},x_{2},\dots ,x_{n})} consisting of n variables, a multivariate continuous function f ( x ) {\displaystyle f(x)} can be represented as: f ( x ) = f ( x 1 , … , x n ) = ∑ q = 1 2 n + 1 Φ q ( ∑ p = 1 n φ q , p ( x p ) ) {\displaystyle f(x)=f(x_{1},\dots ,x_{n})=\sum _{q=1}^{2n+1}\Phi _{q}\left(\sum _{p=1}^{n}\varphi _{q,p}(x_{p})\right)} (1) This formulation contains two nested summations: an outer and an inner sum. The outer sum ∑ q = 1 2 n + 1 {\displaystyle \sum _{q=1}^{2n+1}} aggregates 2 n + 1 {\displaystyle 2n+1} terms, each involving a function Φ q : R → R {\displaystyle \Phi _{q}:\mathbb {R} \to \mathbb {R} } . The inner sum ∑ p = 1 n {\displaystyle \sum _{p=1}^{n}} computes n terms for each q, where each term φ q , p : [ 0 , 1 ] → R {\displaystyle \varphi _{q,p}:[0,1]\to \mathbb {R} } is a continuous function of the single variable x p {\displaystyle x_{p}} . The inner continuous functions φ q , p {\displaystyle \varphi _{q,p}} are universal, independent of f {\displaystyle f} , while the outer functions Φ q {\displaystyle \Phi _{q}} depend on the specific function f {\displaystyle f} being represented. The representation (1) holds for all multivariate functions f {\displaystyle f} as proved in . If f {\displaystyle f} is continuous, then the outer functions Φ q {\displaystyle \Phi _{q}} are continuous; if f {\displaystyle f} is discontinuous, then the corresponding Φ q {\displaystyle \Phi _{q}} are generally discontinuous, while the inner functions φ q , p {\displaystyle \varphi _{q,p}} remain the same universal functions. Liu et al. proposed the name KAN. A general KAN network consisting of L layers takes x to generate the output as: K A N ( x ) = ( Φ L − 1 ∘ Φ L − 2 ∘ ⋯ ∘ Φ 1 ∘ Φ 0 ) x {\displaystyle \mathrm {KAN} (x)=(\Phi ^{L-1}\circ \Phi ^{L-2}\circ \cdots \circ \Phi ^{1}\circ \Phi ^{0})x} (3) Here, Φ l {\displaystyle \Phi ^{l}} is the function matrix of the l-th KAN layer or a set of pre-activations. Let i denote the neuron of the l-th layer and j the neuron of the (l+1)-th layer. The activation function φ j , i l {\displaystyle \varphi _{j,i}^{l}} connects (l, i) to (l+1, j): φ j , i l , l = 0 , … , L − 1 , i = 1 , … , n l , j = 1 , … , n l + 1 {\displaystyle \varphi _{j,i}^{l},\quad l=0,\dots ,L-1,\;i=1,\dots ,n_{l},\;j=1,\dots ,n_{l+1}} (4) where nl is the number of nodes of the l-th layer. Thus, the function matrix Φ l {\displaystyle \Phi ^{l}} can be represented as an n l + 1 × n l {\displaystyle n_{l+1}\times n_{l}} matrix of activations: x l + 1 = ( φ 1 , 1 l ( ⋅ ) φ 1 , 2 l ( ⋅ ) ⋯ φ 1 , n l l ( ⋅ ) φ 2 , 1 l ( ⋅ ) φ 2 , 2 l ( ⋅ ) ⋯ φ 2 , n l l ( ⋅ ) ⋮ ⋮ ⋱ ⋮ φ n l + 1 , 1 l ( ⋅ ) φ n l + 1 , 2 l ( ⋅ ) ⋯ φ n l + 1 , n l l ( ⋅ ) ) x l {\displaystyle x^{l+1}={\begin{pmatrix}\varphi _{1,1}^{l}(\cdot )&\varphi _{1,2}^{l}(\cdot )&\cdots &\varphi _{1,n_{l}}^{l}(\cdot )\\\varphi _{2,1}^{l}(\cdot )&\varphi _{2,2}^{l}(\cdot )&\cdots &\varphi _{2,n_{l}}^{l}(\cdot )\\\vdots &\vdots &\ddots &\vdots \\\varphi _{n_{l+1},1}^{l}(\cdot )&\varphi _{n_{l+1},2}^{l}(\cdot )&\cdots &\varphi _{n_{l+1},n_{l}}^{l}(\cdot )\end{pmatrix}}x^{l}} == Implementations == To make the KAN layers optimizable, the inner function is formed by the combination of spline and basic functions as the formula: φ ( x ) = w b b ( x ) + w s spline ( x ) {\displaystyle \varphi (x)=w_{b}\,b(x)+w_{s}\,{\text{spline}}(x)} where b ( x ) {\displaystyle b(x)} is the basic function, usually defined as s i l u ( x ) = x / ( 1 + e x ) {\displaystyle silu(x)=x/(1+e^{x})} and w b {\displaystyle w_{b}} is the base weight matrix. Also, w s {\displaystyle w_{s}} is the spline weight matrix and spline ( x ) {\displaystyle {\text{spline}}(x)} is the spline function. The spline function can be a sum of B-splines. spline ( x ) = ∑ i c i B i ( x ) {\displaystyle {\text{spline}}(x)=\sum _{i}c_{i}B_{i}(x)} Many studies suggested to use other polynomial and curve functions instead of B-spline to create new KAN variants. == Functions used == The choice of functional basis strongly influences the performance of KANs. Common function families include: B-splines: Provide locality, smoothness, and interpretability; they are the most widely used in current implementations. RBFs (include Gaussian RBFs): Capture localized features in data and are effective in approximating functions with non-linear or clustered structures. Chebyshev polynomials: Offer efficient approximation with minimized error in the maximum norm, making them useful for stable function representation. Rational function: Useful for approximating functions with singularities or sharp variations, as they can model asymptotic behavior better than polynomials. Fourier series: Capture periodic patterns effectively and are particularly useful in domains such as physics-informed machine learning. Wavelet functions (DoG, Mexican hat, Morlet, and Shannon): Used for feature extraction as they can capture both high-frequency and low-frequency data components. Piecewise linear functions: Provide efficient approximation for multivariate functions in KANs. == Usage == In some modern neural architectures like convolutional neural networks (CNNs), recurrent neural networks (RNNs), and Transformers, KANs are typically used as drop-in substitutes for MLP layers. Despite KANs' general-purpose design, researchers have created and used them for a number of tasks: Scientific machine learning (SciML): Function fitting, partial differential equations (PDEs) and physical/mathematical laws. Continual learning: KANs better preserve previously learned information during incremental updates, avoiding catastrophic forgetting due to the locality of spline adjustments. Graph neural networks: Extensions such as Kolmogorov–Arnold Graph Neural Networks (KA-GNNs) integrate KAN modules into message-passing architectures, showing improvements in molecular property prediction tasks. Sensor data processing: Kolmogorov–Arnold Networks (KANs) have recently been applied to sensor data processing due to their ability to model complex nonlinear relationships with relatively few parameters and improved interpretability compared to conventional multilayer perceptrons. Applications include industrial soft sensors, biomedical signal analysis, remote sensing, and environmental monitoring systems. == Drawbacks == KANs can be computationally intensive and require a large number of parameters due to their use of polynomial functions to capture data.

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