The Legal Knowledge Interchange Format (LKIF) was developed in the European ESTRELLA project and was designed with the goal of becoming a standard for representing and interchanging policy, legislation and cases, including their justificatory arguments, in the legal domain. LKIF builds on and uses the Web Ontology Language (OWL) for representing concepts and includes a reusable basic ontology of legal concepts. The core of LKIF consists of a combination of OWL-DL and SWRL. LKIF was designed with two main roles in mind: the translation of legal knowledge bases written in different representation formats and formalisms and to be a knowledge representation formalism which could be part of larger architectures for developing legal knowledge systems.
RagTime
RagTime is a frame-oriented business publishing software which combines word processing, spreadsheets, simple drawings, image processing, and charts, in a single document/program, integrated software. It is often used to create forms, reports, documentation, desktop publishing, and in office environments. Typical users are business clients, educational institutions, administrations, architects, and also private users. Ragtime includes the following modules: Page layout (forms, templates etc.) Word processing Image processing Spreadsheets, similar to Microsoft Excel Formulas and functions which can be used throughout, in text, graphics, and spreadsheets Charts in different types of diagrams Drawings in vector graphics including lines, polygons, Bézier curves and more Slide show (presentation of RagTime documents) Audio/video Buttons (pop-up menus, switches, and more) that can be used within RagTime documents Import/export of various file formats Support of the AppleScript scripting language available system-wide under macOS == Principle == RagTime differs from most other comparable programs or software packages in its strict frame-oriented design: all content is contained within frames on each page. The content can have a fixed position within its frame or, if it is text or a spreadsheet, flow into another frame that is connected to the first frame via a so-called “pipeline”. RagTime has no different document types for different types of data; all content is stored in a single compound document type. Thus, a RagTime document not only can contain multiple pages, but also multiple layouts within the same document; e.g. spreadsheets in addition to text and images. The RagTime filename extension is .rtd (RagTime document); for templates the extension is .rtt (RagTime template). The current version is RagTime 6.6.5. It is available for OS X (10.6-10.14) and Windows (XP/Vista/7/8/10). == Extensions == FileTime – allows accessing “FileMaker Pro” databases from RagTime documents under OS X RagTime Connect – ODBC database connection for RagTime 6 (Mac and Windows) Johannes – print extension for the simple creation of stapled or folded brochures, booklets etc. PowerFunctions – additional functions for a more effective creation of intelligent documents for exchanging data and for use in mixed Mac/Windows environments MetaFormula – SYLK-based extension that allows calculating text as formula == History == RagTime has been developed since 1985 for the Macintosh – originally named MacFrame – and was published in 1986. When released, it already had the present name, which was chosen following the then-available software package Lotus Jazz. In the European Macintosh market, RagTime quickly gained a prominent position that continues to this day, even though the market share has decreased. Despite repeated attempts, the program could not gain acceptance in the North American market due to its high cost ($395 in 1990). The North American sales office closed in 1991, shortly after Claris Corporation released ClarisWorks which duplicated much of the functionality of RagTime for a lower price. After the manufacturer – first Brüning & Everth, followed by B&E Software and today RagTime.de Development – had focused on the Macintosh only for a very long time, it also released a Windows version, RagTime 5.0, in 1999. However, the program could not assume great significance against established competitors, especially Microsoft Office. Until mid-2006 RagTime was, in addition to the commercial version, also available as a free version (RagTime Solo) for personal use. RagTime Solo included the same features and performance (except for spelling and Syllabification) dictionaries), but was not allowed for use in commercial environments. In other languages RagTime Solo was distributed as RagTime Privat. In a press release from July 5, 2006, RagTime announced the discontinuation of RagTime Solo: “… the RagTime Solo license conditions were often misinterpreted or deliberately flouted. Therefore we discontinued RagTime Solo, there will be no private version of RagTime 6 anymore.” After a successful start of the RagTime 6.0 software, sales edged significantly lower in the following years. Disagreements arose among the shareholders about the continuation of the company, which filed for bankruptcy in July 2007. As a result, the rights to RagTime were taken over by the newly established company RagTime.de Development GmbH, which was responsible for the development. The sales partner RagTime.de Sales GmbH distributed the RagTime products until October 2015. Today RagTime.de Development GmbH is also responsible for sales. The last level of development is the extensively revamped version RagTime 6.6 of 8 October 2015, which also includes new OS X features (e.g. high-resolution “Retina” displays) and supports Windows 10. == Programming == RagTime 1-3 were developed in Pascal, since version 4 the development is completely coded in C++. External programming and automation can be implemented via AppleScript on a Mac, and via OLE/COM-API (e.g. Visual Basic) under Windows. On a Mac, RagTime provides a comprehensive AppleScript library, for the automation of almost any task, from automatic document creation to the export of PDF documents. RagTime also supports “recordings” by use of the “AppleScript Editor”, which allows recording the interactive RagTime operation as an AppleScript program sequence. AppleScripts can be saved in the RagTime document and called via menu or shortcut keys. On Windows, RagTime (since version 6) disposes over an OLE/COM API, which allows automating many RagTime components via external programming. For that purpose there is a type library that installs the available RagTime OLE/COM object catalogue. Programming can be realized in all programming languages supported by Microsoft.
Progress in artificial intelligence
Progress in artificial intelligence (AI) refers to the advances, milestones, and breakthroughs that have been achieved in the field of artificial intelligence over time. AI is a branch of computer science that aims to create machines and systems capable of performing tasks that typically require human intelligence. AI applications have been used in a wide range of fields including medical diagnosis, finance, robotics, law, video games, agriculture, and scientific discovery. The society as a whole is looking for artificial intelligence to be on a key factor in the upcming years because of its potential. However, many AI applications are not perceived as AI: "A lot of cutting-edge AI has filtered into general applications, often without being called AI because once something becomes useful enough and common enough it's not labeled AI anymore." "Many thousands of AI applications are deeply embedded in the infrastructure of every industry." In the late 1990s and early 2000s, AI technology became widely used as elements of larger systems, but the field was rarely credited for these successes at the time. Kaplan and Haenlein structure artificial intelligence along three evolutionary stages: Artificial narrow intelligence – AI capable only of specific tasks; Artificial general intelligence – AI with ability in several areas, and able to autonomously solve problems they were never even designed for; Artificial superintelligence – AI capable of general tasks, including scientific creativity, social skills, and general wisdom. To allow comparison with human performance, artificial intelligence can be evaluated on constrained and well-defined problems. Such tests have been termed subject-matter expert Turing tests. Also, smaller problems provide more achievable goals and there are an ever-increasing number of positive results. In 2023, humans still substantially outperformed both GPT-4 and other models tested on the ConceptARC benchmark. Those models scored 60% on most, and 77% on one category, while humans scored 91% on all and 97% on one category. However, later research in 2025 showed that human-generated output grids were only accurate 73% of the time, while AI models available that year managed to score above 77%. == History == Increasing, promoting or constraining AI progress has often be done via controlling or increasing the amount of compute. == Current performance in specific areas == There are many useful abilities that can be described as showing some form of intelligence. This gives better insight into the comparative success of artificial intelligence in different areas. AI, like electricity or the steam engine, is a general-purpose technology. There is no consensus on how to characterize which tasks AI tends to excel at. Some versions of Moravec's paradox observe that humans are more likely to outperform machines in areas such as physical dexterity that have been the direct target of natural selection. While projects such as AlphaZero have succeeded in generating their own knowledge from scratch, many other machine learning projects require large training datasets. Researcher Andrew Ng has suggested, as a "highly imperfect rule of thumb", that "almost anything a typical human can do with less than one second of mental thought, we can probably now or in the near future automate using AI." Games provide a high-profile benchmark for assessing rates of progress; many games have a large professional player base and a well-established competitive rating system. AlphaGo brought the era of classical board-game benchmarks to a close when Artificial Intelligence proved their competitive edge over humans in 2016. Deep Mind's AlphaGo AI software program defeated the world's best professional Go Player Lee Sedol. Games of imperfect knowledge provide new challenges to AI in the area of game theory; the most prominent milestone in this area was brought to a close by Libratus' poker victory in 2017. E-sports continue to provide additional benchmarks; Facebook AI, Deepmind, and others have engaged with the popular StarCraft franchise of videogames. Broad classes of outcome for an AI test may be given as: optimal: it is not possible to perform better (note: some of these entries were solved by humans) super-human: performs better than all humans high-human: performs better than most humans par-human: performs similarly to most humans sub-human: performs worse than most humans === Optimal === Tic-tac-toe Connect Four: 1988 Checkers (aka 8x8 draughts): Weakly solved (2007) Rubik's Cube: Mostly solved (2010) Heads-up limit hold'em poker: Statistically optimal in the sense that "a human lifetime of play is not sufficient to establish with statistical significance that the strategy is not an exact solution" (2015) === Super-human === Othello (aka reversi): c. 1997 Scrabble: 2006 Backgammon: c. 1995–2002 Chess: Supercomputer (c. 1997); Personal computer (c. 2006); Mobile phone (c. 2009); Computer defeats human + computer (c. 2017) Jeopardy!: Question answering, although the machine did not use speech recognition (2011) Arimaa: 2015 Shogi: c. 2017 Go: 2017 Heads-up no-limit hold'em poker: 2017 Six-player no-limit hold'em poker: 2019 Gran Turismo Sport: 2022 === High-human === Crosswords: c. 2012 Freeciv: 2016 Dota 2: 2018 Bridge card-playing: According to a 2009 review, "the best programs are attaining expert status as (bridge) card players", excluding bidding. StarCraft II: 2019 Mahjong: 2019 Stratego: 2022 No-Press Diplomacy: 2022 Hanabi: 2022 Natural language processing === Par-human === Optical character recognition for ISO 1073-1:1976 and similar special characters. Classification of images Handwriting recognition Facial recognition Visual question answering SQuAD 2.0 English reading-comprehension benchmark (2019) SuperGLUE English-language understanding benchmark (2020) Some school science exams (2019) Some tasks based on Raven's Progressive Matrices Many Atari 2600 games (2015) === Sub-human === Optical character recognition for printed text (nearing par-human for Latin-script typewritten text) Object recognition Various robotics tasks that may require advances in robot hardware as well as AI, including: Stable bipedal locomotion: Bipedal robots can walk, but are less stable than human walkers (as of 2017) Humanoid soccer Speech recognition: "nearly equal to human performance" (2017) Explainability. Current medical systems can diagnose certain medical conditions well, but cannot explain to users why they made the diagnosis. Many tests of fluid intelligence (2020) Bongard visual cognition problems, such as the Bongard-LOGO benchmark (2020) Visual Commonsense Reasoning (VCR) benchmark (as of 2020) Stock market prediction: Financial data collection and processing using Machine Learning algorithms Angry Birds video game, as of 2020 Various tasks that are difficult to solve without contextual knowledge, including: Translation Word-sense disambiguation == Proposed tests of artificial intelligence == In his famous Turing test, Alan Turing picked language, the defining feature of human beings, for its basis. The Turing test is now considered too exploitable to be a meaningful benchmark. The Feigenbaum test, proposed by the inventor of expert systems, tests a machine's knowledge and expertise about a specific subject. A paper by Jim Gray of Microsoft in 2003 suggested extending the Turing test to speech understanding, speaking and recognizing objects and behavior. Proposed "universal intelligence" tests aim to compare how well machines, humans, and even non-human animals perform on problem sets that are generic as possible. At an extreme, the test suite can contain every possible problem, weighted by Kolmogorov complexity; however, these problem sets tend to be dominated by impoverished pattern-matching exercises where a tuned AI can easily exceed human performance levels. == Exams == According to OpenAI, in 2023 GPT-4 achieved high scores on several standardized and professional examinations, including around the 90th percentile on the Uniform Bar Exam, the 89th percentile on the mathematics section of the SAT, the 93rd percentile on SAT Reading and Writing, the 54th percentile on the analytical writing section of the GRE, the 88th percentile on GRE quantitative reasoning, and the 99th percentile on GRE verbal reasoning. OpenAI also reported that GPT-4 scored in the 99th to 100th percentile on the 2020 USA Biology Olympiad semifinal exam and earned top scores on several AP exams. Independent researchers found in 2023 that ChatGPT based on GPT-3.5 performed "at or near the passing threshold" on all three parts of the United States Medical Licensing Examination (USMLE), suggesting that large language models could reach passing-level performance on some medical knowledge assessments even without domain-specific fine-tuning. GPT-3.5 was also reported to attain a low but passing grade on examinations for four law school courses at the University of Minnes
Prompt engineering
Prompt engineering is the process of structuring natural language inputs (known as prompts) to produce specified outputs from a generative artificial intelligence (GenAI) model. Context engineering is the related area of software engineering that focuses on the management of non-prompt contexts supplied to the GenAI model, such as metadata, API tools, and tokens. It can also be defined as the practice of designing and refining input instructions given to a generative AI model to produce more accurate, relevant, or useful outputs. Effective prompt engineering involves understanding how a model interprets language, and may include techniques such as few-shot prompting, chain-of-thought prompting, and role assignment. It is increasingly considered a skill for working with large language models (LLMs) in both research and professional contexts. During the 2020s AI boom, prompt engineering became regarded as a business capability across corporations and industries. Employees with the title prompt engineer were hired to create prompts that would increase productivity and efficacy, although the individual title has since lost traction amid AI models that produce better prompts than humans and corporate training in prompting for general employees. Common prompting techniques include multi-shot, chain-of-thought, and tree-of-thought prompting, as well as the use of assigning roles to the model. Automated prompt generation methods, such as retrieval-augmented generation (RAG), provide for greater accuracy and a wider scope of functions for prompt engineers. Prompt injection is a type of cybersecurity attack that targets machine learning models through malicious prompts. == Terminology == The Oxford English Dictionary defines prompt engineering as "The action or process of formulating and refining prompts for an artificial intelligence program, algorithm, etc., in order to optimize its output or to achieve a desired outcome; the discipline or profession concerned with this." In 2023, prompt ("an instruction given to an artificial intelligence program, algorithm, etc., which determines or influences the content it generates") was the runner-up to Oxford's word of the year. === Prompt === A prompt is some natural language text that describes and prescribes the task that an artificial intelligence (AI) should perform. A prompt for a text-to-text language model can be a query, a command, or a longer statement referencing context, instructions, and conversation history. The process of prompt engineering may involve designing clear queries, refining wording, providing relevant context, specifying the style of output, and assigning a character for the AI to mimic in order to guide the model toward more accurate, useful, and consistent responses. When communicating with a text-to-image or a text-to-audio model, a typical prompt contains a description of a desired output such as "a high-quality photo of an astronaut riding a horse" or "Lo-fi slow BPM electro chill with organic samples". Prompt engineering may be applied to text-to-image models to achieve a desired subject, style, layout, lighting, and aesthetic. === Techniques === Common terms used to describe various specific prompt engineering techniques include chain-of-thought, tree-of-thought, and retrieval-augmented generation (RAG). A 2024 survey of the field identified over 50 distinct text-based prompting techniques, 40 multimodal variants, and a vocabulary of 33 terms used across prompting research, highlighting a present lack of standardised terminology for prompt engineering. Vibe coding is an AI-assisted software development method where a user prompts an LLM with a description of what they want and lets it generate or edit the code. In 2025, "vibe coding" was the Collins Dictionary word of the year. === Context engineering === Context engineering is a related process that focuses on the context elements that accompany user prompts, which include system instructions, retrieved knowledge, tool definitions, conversation summaries, and task metadata. Context engineering is performed to improve reliability, provenance and token efficiency in production LLM systems. The concept emphasises operational practices such as token budgeting, provenance tags, versioning of context artifacts, observability (logging which context was supplied), and context regression tests to ensure that changes to supplied context do not silently alter system behaviour. == Rationale == Research has found that the performance of large language models (LLMs) is highly sensitive to choices such as the ordering of examples, the quality of demonstration labels, and even small variations in phrasing. In some cases, reordering examples in a prompt produced accuracy shifts of more than 40 percent. === In-context learning === A model's ability to temporarily learn from prompts is known as in-context learning. In-context learning is an emergent ability of large language models. It is an emergent property of model scale, meaning that breaks in scaling laws occur, leading to its efficacy increasing at a different rate in larger models than in smaller models. Unlike training and fine-tuning, which produce lasting changes, in-context learning is temporary. Training models to perform in-context learning can be viewed as a form of meta-learning, or "learning to learn". === Prompting to estimate model sensitivity === Research consistently demonstrates that LLMs are highly sensitive to subtle variations in prompt formatting, structure, and linguistic properties. Some studies have shown up to 76 accuracy points across formatting changes in few-shot settings. Linguistic features significantly influence prompt effectiveness—such as morphology, syntax, and lexico-semantic changes—which meaningfully enhance task performance across a variety of tasks. Clausal syntax, for example, improves consistency and reduces uncertainty in knowledge retrieval. This sensitivity persists even with larger model sizes, additional few-shot examples, or instruction tuning. To address sensitivity of models and make them more robust, several evaluative methods have been proposed. FormatSpread facilitates systematic analysis by evaluating a range of plausible prompt formats, offering a more comprehensive performance interval. Similarly, PromptEval estimates performance distributions across diverse prompts, enabling robust metrics such as performance quantiles and accurate evaluations under constrained budgets. == Prompting techniques == === Multi-shot === A prompt may include a few examples for a model to learn from in context, an approach called few-shot learning. For example, the prompt may ask the model to complete "maison → house, chat → cat, chien →", with the expected response being dog. === Chain-of-thought === Chain-of-thought (CoT) prompting is a technique that allows large language models (LLMs) to solve a problem as a series of intermediate steps before giving a final answer. In 2022, Google Brain reported that chain-of-thought prompting improves reasoning ability by inducing the model to answer a multi-step problem with steps of reasoning that mimic a train of thought. Chain-of-thought techniques were developed to help LLMs handle multi-step reasoning tasks, such as arithmetic or commonsense reasoning questions. When applied to PaLM, a 540 billion parameter language model, according to Google, CoT prompting significantly aided the model, allowing it to perform comparably with task-specific fine-tuned models on several tasks, achieving state-of-the-art results at the time on the GSM8K mathematical reasoning benchmark. It is possible to fine-tune models on CoT reasoning datasets to enhance this capability further and stimulate better interpretability. As originally proposed by Google, each CoT prompt is accompanied by a set of input/output examples—called exemplars—to demonstrate the desired model output, making it a few-shot prompting technique. However, according to a later paper from researchers at Google and the University of Tokyo, simply appending the words "Let's think step-by-step" was also effective, which allowed for CoT to be employed as a zero-shot technique. ==== Self-consistency ==== Self-consistency performs several chain-of-thought rollouts, then selects the most commonly reached conclusion out of all the rollouts. === Tree-of-thought === Tree-of-thought prompting generalizes chain-of-thought by generating multiple lines of reasoning in parallel, with the ability to backtrack or explore other paths. It can use tree search algorithms like breadth-first, depth-first, or beam. === Text-to-image prompting === In 2022, text-to-image models like DALL-E 2, Stable Diffusion, and Midjourney were released to the public. These models take text prompts as input and use them to generate images. Early text-to-image models typically do not understand negation, grammar and sentence structure in the same way as large language models, and may thus requi
Non-human
Non-human (also spelled nonhuman) is any entity displaying some, but not enough, human characteristics to be considered a human. The term has been used in a variety of contexts and may refer to objects that have been developed with human intelligence, such as robots or vehicles. == Organisms == === Animal rights and personhood === In the animal rights movement, it is common to distinguish between "human animals" and "non-human animals". Participants in the animal rights movement generally recognize that non-human animals have some similar characteristics to those of human persons. For example, various non-human animals have been shown to register pain, compassion, memory, and some cognitive function. Some animal rights activists argue that the similarities between human and non-human animals justify giving non-human animals rights that human society has afforded to humans, such as the right to self-preservation, and some even wish for all non-human animals or at least those that bear a fully thinking and conscious mind, such as vertebrates and some invertebrates such as cephalopods, to be given a full right of personhood. === The non-human in philosophy === Contemporary philosophers have drawn on the work of Henri Bergson, Gilles Deleuze, Félix Guattari, and Claude Lévi-Strauss (among others) to suggest that the non-human poses epistemological and ontological problems for humanist and post-humanist ethics, and have linked the study of non-humans to materialist and ethological approaches to the study of society and culture. == Software and robots == The term non-human has been used to describe computer programs and robot-like devices that display some human-like characteristics. In both science fiction and in the real world, computer programs and robots have been built to perform tasks that require human-computer interactions in a manner that suggests sentience and compassion. There is increasing interest in the use of robots in nursing homes and to provide elder care. Computer programs have been used for years in schools to provide one-on-one education with children. The Tamagotchi toy required children to provide care, attention, and nourishment to keep it "alive".
Gibberlink
GibberLink is an acoustic data transmission project, with an open-source client available on GitHub, in which two conversational AI agents switch from speaking to one another in a Human-listenable language (such as English) to their own unique language that consists of a sound-level protocol after confirming they are both AI agents. The project was created by Anton Pidkuiko and Boris Starkov. == Reception == The project won the global top prize at the ElevenLabs Worldwide Hackathon. It has also been cited as raising questions around AI ethics and oversight. On February 23, 2025, a YouTube video of two independent conversational ElevenLabs AI agents being prompted to chat about booking a hotel (one as a caller, one as a receptionist) received coverage for going viral. In this video, both agents are prompted to switch to ggwave data-over-sound protocol when they identify the other side as AI, and keep speaking in English otherwise.
Multimodal representation learning
Multimodal representation learning is a subfield of representation learning focused on integrating and interpreting information from different modalities, such as text, images, audio, or video, by projecting them into a shared latent space. This allows for semantically similar content across modalities to be mapped to nearby points within that space, facilitating a unified understanding of diverse data types. By automatically learning meaningful features from each modality and capturing their inter-modal relationships, multimodal representation learning enables a unified representation that enhances performance in cross-media analysis tasks such as video classification, event detection, and sentiment analysis. It also supports cross-modal retrieval and translation, including image captioning, video description, and text-to-image synthesis. == Motivation == The primary motivations for multimodal representation learning arise from the inherent nature of real-world data and the limitations of unimodal approaches. Since multimodal data offers complementary and supplementary information about an object or event from different perspectives, it is more informative than relying on a single modality. A key motivation is to narrow the heterogeneity gap that exists between different modalities by projecting their features into a shared semantic subspace. This allows semantically similar content across modalities to be represented by similar vectors, facilitating the understanding of relationships and correlations between them. Multimodal representation learning aims to leverage the unique information provided by each modality to achieve a more comprehensive and accurate understanding of concepts. These unified representations are crucial for improving performance in various cross-media analysis tasks such as video classification, event detection, and sentiment analysis. They also enable cross-modal retrieval, allowing users to search and retrieve content across different modalities. Additionally, it facilitates cross-modal translation, where information can be converted from one modality to another, as seen in applications like image captioning and text-to-image synthesis. The abundance of ubiquitous multimodal data in real-world applications, including understudied areas like healthcare, finance, and human-computer interaction (HCI), further motivates the development of effective multimodal representation learning techniques. == Approaches and methods == === Canonical-correlation analysis based methods === Canonical-correlation analysis (CCA) was first introduced in 1936 by Harold Hotelling and is a fundamental approach for multimodal learning. CCA aims to find linear relationships between two sets of variables. Given two data matrices X ∈ R n × p {\displaystyle X\in \mathbb {R} ^{n\times p}} and Y ∈ R n × q {\displaystyle Y\in \mathbb {R} ^{n\times q}} representing different modalities, CCA finds projection vectors w x ∈ R p {\displaystyle w_{x}\in \mathbb {R} ^{p}} and w y ∈ R q {\displaystyle w_{y}\in \mathbb {R} ^{q}} that maximizes the correlation between the projected variables: ρ = max w x , w y w x ⊤ Σ x y w y w x ⊤ Σ x x w x w y ⊤ Σ y y w y {\displaystyle \rho =\max _{w_{x},w_{y}}{\frac {w_{x}^{\top }\Sigma _{xy}w_{y}}{{\sqrt {w_{x}^{\top }\Sigma _{xx}w_{x}}}{\sqrt {w_{y}^{\top }\Sigma _{yy}w_{y}}}}}} such that Σ x x {\displaystyle \Sigma _{xx}} and Σ y y {\displaystyle \Sigma _{yy}} are the within-modality covariance matrices, and Σ x y {\displaystyle \Sigma _{xy}} is the between-modality covariance matrix. However, standard CCA is limited by its linearity, which led to the development of nonlinear extensions, such as kernel CCA and deep CCA. ==== Kernel CCA ==== Kernel canonical correlation analysis (KCCA) extends traditional CCA to capture nonlinear relationships between modalities by implicitly mapping the data into high dimensional feature spaces using kernel functions. Given kernel functions K x {\displaystyle K_{x}} and K y {\displaystyle K_{y}} with corresponding Gram matrices K x ∈ R n × n {\displaystyle K_{x}\in \mathbb {R} ^{n\times n}} and K y ∈ R n × n {\displaystyle K_{y}\in \mathbb {R} ^{n\times n}} , KCCA seeks coefficients α {\displaystyle \alpha } and β {\displaystyle \beta } that maximize: ρ = max α , β α ⊤ K x K y β α ⊤ K x 2 α β ⊤ K y 2 β {\displaystyle \rho =\max _{\alpha ,\beta }{\frac {\alpha ^{\top }K_{x}Ky\beta }{{\sqrt {\alpha ^{\top }K_{x}^{2}\alpha }}{\sqrt {\beta ^{\top }K_{y}^{2}\beta }}}}} To prevent overfitting, regularization terms are typically added, resulting in: ρ = max α , β α T K x K y β α T ( K x 2 + λ x K x ) α β T ( K y 2 + λ y K y ) β {\displaystyle \rho =\max _{\alpha ,\beta }{\frac {\alpha ^{T}K_{x}K_{y}\beta }{{\sqrt {\alpha ^{T}\left(K_{x}^{2}+\lambda _{x}K_{x}\right)\alpha }}{\sqrt {\;\beta ^{T}\left(K_{y}^{2}+\lambda _{y}K_{y}\right)\beta }}}}} where λ x {\displaystyle \lambda _{x}} and λ y {\displaystyle \lambda _{y}} are regularization parameters. KCCA has proven effective for tasks such as cross-modal retrieval and semantic analysis, though it faces computational challenges with large datasets due to its O ( n 2 ) {\displaystyle O(n^{2})} memory requirement for sorting kernel matrices. KCCA was proposed independently by several researchers. ==== Deep CCA ==== Deep canonical correlation analysis (DCCA), introduced in 2013, employs neural networks to learn nonlinear transformations for maximizing the correlation between modalities. DCCA uses separate neural networks f x {\displaystyle f_{x}} and f y {\displaystyle f_{y}} for each modality to transform the original data before applying CCA: max W x , W y , θ x , θ y corr ( f x ( X ; θ x ) , f y ( Y ; θ y ) ) {\displaystyle \max _{W_{x},W_{y},\theta _{x},\theta _{y}}\operatorname {corr} \left(f_{x}(X;\theta _{x}),f_{y}(Y;\theta _{y})\right)} where θ x {\displaystyle \theta _{x}} and θ y {\displaystyle \theta _{y}} represent the parameters of the neural networks, and W x {\displaystyle W_{x}} and W y {\displaystyle W_{y}} are the CCA projection matrices. The correlation objective is computed as: corr ( H x , H y ) = tr ( T − 1 / 2 H x T H y S − 1 / 2 ) {\displaystyle \operatorname {corr} (H_{x},H_{y})=\operatorname {tr} \left(T^{-1/2}H_{x}^{T}H_{y}S^{-1/2}\right)} where H x = f x ( X ) {\displaystyle H_{x}=f_{x}(X)} and H y = f y ( Y ) {\displaystyle H_{y}=f_{y}(Y)} are the network outputs, T = H x T H x + r x I {\displaystyle T=H_{x}^{T}H_{x}+r_{x}I} , S = H y T H y + r y I {\displaystyle S=H_{y}^{T}H_{y}+r_{y}I} and r x , r y {\displaystyle r_{x},r_{y}} are the regularization parameters. DCCA overcomes the limitations of linear CCA and kernel CCA by learning complex nonlinear relationships while maintaining computational efficiency for large datasets through mini-batch optimization. === Graph-based methods === Graph-based approaches for multimodal representation learning leverage graph structure to model relationships between entities across different modalities. These methods typically represent each modality as a graph and then learn embedding that preserve cross-modal similarities, enabling more effective joint representation of heterogeneous data. One such method is cross-modal graph neural networks (CMGNNs) that extend traditional graph neural networks (GNNs) to handle data from multiple modalities by constructing graphs that capture both intra-modal and inter-modal relationships. These networks model interactions across modalities by representing them as nodes and their relationships as edges. Other graph-based methods include Probabilistic Graphical Models (PGMs) such as deep belief networks (DBN) and deep Boltzmann machines (DBM). These models can learn a joint representation across modalities, for instance, a multimodal DBN achieves this by adding a shared restricted Boltzmann Machine (RBM) hidden layer on top of modality-specific DBNs. Additionally, the structure of data in some domains like Human-Computer Interaction (HCI), such as the view hierarchy of app screens, can potentially be modeled using graph-like structures. The field of graph representation learning is also relevant, with ongoing progress in developing evaluation benchmarks. === Diffusion maps === Another set of methods relevant to multimodal representation learning are based on diffusion maps and their extensions to handle multiple modalities. ==== Multi-view diffusion maps ==== Multi-view diffusion maps address the challenge of achieving multi-view dimensionality reduction by effectively utilizing the availability of multiple views to extract a coherent low-dimensional representation of the data. The core idea is to exploit both the intrinsic relations within each view and the mutual relations between the different views, defining a cross-view model where a random walk process implicitly hops between objects in different views. A multi-view kernel matrix is constructed by combining these relations, defining a cross-view diffusion process and associ