AI App Similar To Grok

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  Local-first software

  Local-first software is a software engineering approach in which an application stores its data primarily on the user's own device rather than on remote servers. Users can read and write data without an Internet connection, and changes are synchronized across devices in the background when connectivity is available. The approach differs from conventional cloud-based applications, where the server holds the authoritative copy of user data and the client acts as a thin client. The term was coined in a 2019 paper published by researchers at Ink & Switch, an independent research lab, and presented at the Onward! conference at ACM SIGPLAN. The paper, sometimes referred to as a manifesto, was authored by Martin Kleppmann, Adam Wiggins, Peter van Hardenberg, and Mark McGranaghan. == Background == Before the widespread adoption of Internet-connected software in the 2000s, most desktop applications stored data as files on the user's local disk. Users had direct access to their files and could copy, back up, or delete them at will. The rise of software as a service (SaaS) and cloud-based applications like Google Docs shifted data storage to centralized servers. While cloud applications made real-time collaboration across devices straightforward, they introduced a dependency on the service provider: if the provider discontinued the service or experienced an outage, users could lose access to their data. A related concept, "offline-first," emerged in the early 2010s and focused on making web applications resilient to network interruptions. The local-first approach built on these earlier efforts while placing greater emphasis on long-term data ownership and end-to-end encryption. == Origins == === Ink & Switch manifesto === Ink & Switch is an industrial research lab co-founded by Adam Wiggins, who had earlier co-founded Heroku. Martin Kleppmann, an associate professor in the Department of Computer Science and Technology at the University of Cambridge, was a co-author of the 2019 paper. The manifesto proposed seven "ideals" for local-first software: Fast — Operations respond without network round-trips. Multi-device — Data synchronizes across a user's devices. Offline — Users can read and write data without a network connection. Collaboration — Multiple users can work on the same data concurrently. Longevity — Data remains accessible even if the software vendor ceases operation. Privacy — End-to-end encryption protects user data. User control — The vendor cannot restrict how users access or use their data. The paper surveyed existing approaches to data storage and collaboration — ranging from email attachments and Dropbox-style file synchronization to web applications and mobile backends — and argued that none of them satisfied all seven ideals simultaneously. === Role of CRDTs === The manifesto identified conflict-free replicated data types (CRDTs) as a promising technical foundation for local-first applications. CRDTs are data structures that allow multiple replicas to be edited independently and then merged without conflicts, a property first formalized in research by Marc Shapiro and colleagues around 2011. Kleppmann and collaborators at Ink & Switch developed Automerge, an open-source CRDT library for JSON documents, to make these algorithms available to application developers. == Adoption and community == Developer interest in the local-first approach grew after the 2019 paper spread on Hacker News and at developer conferences In August 2023, Wired published a feature article on the movement, describing it as an effort to reduce reliance on large cloud providers. The first Local-First Conf took place on 30 May 2024 in Berlin, with talks by Kleppmann and developers from companies including Linear and Anytype. The community has continued to expand, with regular "LoFi" meetups, a podcast (localfirst.fm), and a third edition of the conference planned for Berlin in July 2026. == Criticisms and limitations == Developers and commentators have pointed out practical difficulties with the local-first approach. Synchronizing data between multiple devices that may be offline for extended periods introduces complexity that cloud-based architectures avoid. Conflict resolution, even with CRDTs, can produce results that are technically consistent but semantically unexpected to users. Schema migrations across thousands of client devices running different application versions pose another difficulty that does not arise with server-side databases. Web browsers impose storage limits and may evict locally stored data. Safari, for instance, has been reported to clear IndexedDB data after seven days of inactivity on a given site, which undermines the assumption that local data is persistent. There is also disagreement within the local-first community about whether a fully decentralized architecture is required. The original manifesto described decentralization as the "logical end goal," but a number of products that identify as local-first still depend on centralized servers for authentication, backup, or synchronization. In a talk at Local-First Conf 2024, Kleppmann said the seven ideals are better understood as a "gradient" rather than a strict checklist.

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  Qloo

  Qloo (pronounced "clue") is a company that uses artificial intelligence (AI) to understand taste and cultural correlations. It provides companies with an application programming interface (API). It received funding from Leonardo DiCaprio, Elton John, Barry Sternlicht, Pierre Lagrange and others. Qloo establishes consumer preference correlations via machine learning across data spanning cultural domains including music, film, television, dining, nightlife, fashion, books, and travel. The recommender system uses AI to predict correlations for further applications. == History == Qloo was founded in 2012 by chief executive officer Alex Elias and chief operating officer Jay Alger. Qloo initially launched an app designed for consumers, allowing them to understand their own tastes and receive personalized recommendations. The company amassed several million users and built a large catalog of cultural entities and corresponding user sentiment. In 2012, Qloo raised $1.4 million in seed funding from investors including Cedric the Entertainer, and venture capital firm Kindler Capital. Qloo had a public beta release in November 2012 after its initial funding. In 2013, the company raised an additional $1.6 million from Cross Creek Pictures founding partner Tommy Thompson, and Samih Toukan and Hussam Khoury, founders of Maktoob, an Internet services company purchased by Yahoo! for $164 million in 2009. On November 14, 2013, a website and an iPhone app were announced. The company later released an Android app, and tablet versions, in mid-2014. In 2015, Twitter approached Qloo about powering personalized social feeds and targeted eCommerce ads on the platform based on what users were posting. Qloo developed an enterprise-grade API to support Twitter’s needs. Twitter ended up pivoting to enable brands to use the social platform for customer service and support, but Qloo was able to sell access to its cultural intelligence via API to many other enterprise clients, marking the official transition from a B2C company to a B2B company. In 2016, Qloo secured $4.5 million in venture capital investment. The $4.5 million was split between a number of investors, including Barry Sternlicht, Pierre Lagrange, and Leonardo DiCaprio. In July 2017, Qloo raised $6.5 million in funding rounds from AXA Strategic Ventures, and Elton John. Following the investment, the founders stated in an interview with Tech Crunch that they would use the investment to expand Qloo's database. They hoped the move would secure larger contracts with corporate clients. At the time, clients already included Fortune 500 companies such as Twitter, PepsiCo, and BMW. In 2019, the company announced that it had acquired cultural recommendation service TasteDive, with Alex Elias becoming chairman of TasteDive. In September 2019, Qloo was named among the Top 14 Artificial Intelligence APIs by ProgrammableWeb. In 2022, Qloo raised $15M in Series B funding from Eldridge and AXA Venture Partners, enabling the privacy-centric AI leader to expand its team of world-class data scientists, enrich its technology, and build on its sales channels in order to continue to offer premier insights into global consumer taste for Fortune 500 companies across the globe. Qloo was recognized as the "Best Decision Intelligence Company" at the 2023 AI Breakthrough Awards. Also in 2023, the company was awarded a Top Performer Award by SourceForge. As of 2024, Qloo is a three-time Inc. 5000 honoree: No. 360 (2022), No. 344 (2021), No. 187 (2020). Qloo raised $25 million Series C round on February 21, 2024. The round was led by AI Ventures with participation from AXA Venture Partners, Eldridge, and Moderne Ventures, allowing Qloo to address new commercial surface areas for Taste AI, including on-device learning and foundational models leveraging Qloo, as well as introduce self-service platform to make consumer and taste analytics available to small and mid-sized enterprises and individuals. Qloo also announced pursuing opportunistic M&A using its balance sheet along the lines of the TasteDive acquisition completed, which expanded Qloo's first-party data moat and corpus of cultural learning. This latest financing brought the total amount raised since the company's founding in 2012 to over $56 million. == Services and features == Qloo calls itself a cultural AI platform to provide real-time correlation data across domains of culture and entertainment including: film, music, television, dining, nightlife, fashion, books, and travel. Each category contains subcategories. Qloo’s knowledge of a user's taste in one category can be utilized to offer suggestions in other categories. Users then rate the suggestions, providing it with feedback for future suggestions. Qloo has partnerships with companies such as Expedia and iTunes. == Technology == Qloo’s Taste AI technology uses machine learning to decode and predict consumers’ interests, maintaining user anonymity. It is powered by 3.7 billion lifestyle entities (brands, music, film, TV, dining, nightlife, fashion, books, travel, and more) and trillions of anonymized consumer behavioral signals. Through AI, Qloo identifies patterns in these data signals, making predictions about how much interest a person or group has in a concept or thing. Central to Qloo’s technology are algorithms designed to detect and mitigate biases within datasets and models, allowing Qloo to assess the fairness of its AI systems with a focus on attributes such as age, gender, and race, enabling the company to fine-tune its AI models to align with their ethical standards. They also use visualization tools to probe the behavior of their AI models for conducting counterfactual analyses and for comparing the performances of the AI models across diverse demographic segments. Qloo’s Taste AI doesn’t collect or use any Personally Identifiable Information (PII). Instead, it derives recommendations for audience segments based on co-occurrences between lifestyle entities and anonymized behavioral signals. == Applications == Starbucks uses Qloo to create in-store music playlists tailored to specific neighborhoods. Hershey’s uses Qloo to customize the content of assorted candy bags. Michelin uses Qloo to serve recommendations in its Michelin Guide app. Netflix leverages Qloo’s technology to enhance merchandising by identifying actors who resonate with certain demographics. Qloo also works with PepsiCo, Samsung, The New York Mets, BuzzFeed, and Ticketmaster, Universal Music Group, and OOH advertising company JCDecaux.

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  Zeuthen strategy

  The Zeuthen strategy in cognitive science is a negotiation strategy used by some artificial agents. Its purpose is to measure the willingness to risk conflict. An agent will be more willing to risk conflict if it does not have much to lose in case that the negotiation fails. In contrast, an agent is less willing to risk conflict when it has more to lose. The value of a deal is expressed in its utility. An agent has much to lose when the difference between the utility of its current proposal and the conflict deal is high. When both agents use the monotonic concession protocol, the Zeuthen strategy leads them to agree upon a deal in the negotiation set. This set consists of all conflict free deals, which are individually rational and Pareto optimal, and the conflict deal, which maximizes the Nash product. The strategy was introduced in 1930 by the Danish economist Frederik Zeuthen. == Three key questions == The Zeuthen strategy answers three open questions that arise when using the monotonic concession protocol, namely: Which deal should be proposed at first? On any given round, who should concede? In case of a concession, how much should the agent concede? The answer to the first question is that any agent should start with its most preferred deal, because that deal has the highest utility for that agent. The second answer is that the agent with the smallest value of Risk(i,t) concedes, because the agent with the lowest utility for the conflict deal profits most from avoiding conflict. To the third question, the Zeuthen strategy suggests that the conceding agent should concede just enough raise its value of Risk(i,t) just above that of the other agent. This prevents the conceding agent to have to concede again in the next round. == Risk == Risk ( i , t ) = { 1 U i ( δ ( i , t ) ) = 0 U i ( δ ( i , t ) ) − U i ( δ ( j , t ) ) U i ( δ ( i , t ) ) otherwise {\displaystyle {\text{Risk}}(i,t)={\begin{cases}1&U_{i}(\delta (i,t))=0\\{\frac {U_{i}(\delta (i,t))-U_{i}(\delta (j,t))}{U_{i}(\delta (i,t))}}&{\text{otherwise}}\end{cases}}} Risk(i,t) is a measurement of agent i's willingness to risk conflict. The risk function formalizes the notion that an agent's willingness to risk conflict is the ratio of the utility that agent would lose by accepting the other agent's proposal to the utility that agent would lose by causing a conflict. Agent i is said to be using a rational negotiation strategy if at any step t + 1 that agent i sticks to his last proposal, Risk(i,t) > Risk(j,t). == Sufficient concession == If agent i makes a sufficient concession in the next step, then, assuming that agent j is using a rational negotiation strategy, if agent j does not concede in the next step, he must do so in the step after that. The set of all sufficient concessions of agent i at step t is denoted SC(i, t). == Minimal sufficient concession == δ ′ = arg ⁡ max δ ∈ S C ( A , t ) { U A ( δ ) } {\displaystyle \delta '=\arg \max _{\delta \in {SC(A,t)}}\{U_{A}(\delta )\}} is the minimal sufficient concession of agent A in step t. Agent A begins the negotiation by proposing δ ( A , 0 ) = arg ⁡ max δ ∈ N S U A ( δ ) {\displaystyle \delta (A,0)=\arg \max _{\delta \in {NS}}U_{A}(\delta )} and will make the minimal sufficient concession in step t + 1 if and only if Risk(A,t) ≤ Risk(B,t). Theorem If both agents are using Zeuthen strategies, then they will agree on δ = arg ⁡ max δ ′ ∈ N S { π ( δ ′ ) } , {\displaystyle \delta =\arg \max _{\delta '\in {NS}}\{\pi (\delta ')\},} that is, the deal which maximizes the Nash product. Proof Let δA = δ(A,t). Let δB = δ(B,t). According to the Zeuthen strategy, agent A will concede at step t {\displaystyle t} if and only if R i s k ( A , t ) ≤ R i s k ( B , t ) . {\displaystyle Risk(A,t)\leq Risk(B,t).} That is, if and only if U A ( δ A ) − U A ( δ B ) U A ( δ A ) ≤ U B ( δ B ) − U B ( δ A ) U B ( δ B ) {\displaystyle {\frac {U_{A}(\delta _{A})-U_{A}(\delta _{B})}{U_{A}(\delta _{A})}}\leq {\frac {U_{B}(\delta _{B})-U_{B}(\delta _{A})}{U_{B}(\delta _{B})}}} U B ( δ B ) ( U A ( δ A ) − U A ( δ B ) ) ≤ U A ( δ A ) ( U B ( δ B ) − U B ( δ A ) ) {\displaystyle U_{B}(\delta _{B})(U_{A}(\delta _{A})-U_{A}(\delta _{B}))\leq U_{A}(\delta _{A})(U_{B}(\delta _{B})-U_{B}(\delta _{A}))} U A ( δ A ) U B ( δ B ) − U A ( δ B ) U B ( δ B ) ≤ U A ( δ A ) U B ( δ B ) − U A ( δ A ) U B ( δ A ) {\displaystyle U_{A}(\delta _{A})U_{B}(\delta _{B})-U_{A}(\delta _{B})U_{B}(\delta _{B})\leq U_{A}(\delta _{A})U_{B}(\delta _{B})-U_{A}(\delta _{A})U_{B}(\delta _{A})} − U A ( δ B ) U B ( δ B ) ≤ − U A ( δ A ) U B ( δ A ) {\displaystyle -U_{A}(\delta _{B})U_{B}(\delta _{B})\leq -U_{A}(\delta _{A})U_{B}(\delta _{A})} U A ( δ A ) U B ( δ A ) ≤ U A ( δ B ) U B ( δ B ) {\displaystyle U_{A}(\delta _{A})U_{B}(\delta _{A})\leq U_{A}(\delta _{B})U_{B}(\delta _{B})} π ( δ A ) ≤ π ( δ B ) {\displaystyle \pi (\delta _{A})\leq \pi (\delta _{B})} Thus, Agent A will concede if and only if δ A {\displaystyle \delta _{A}} does not yield the larger product of utilities. Therefore, the Zeuthen strategy guarantees a final agreement that maximizes the Nash Product.

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  AI Overviews

  AI Overviews is an artificial intelligence (AI) feature integrated into Google Search that produces AI-generated summaries of search results. The feature has been criticized for its inaccuracy and for reducing website traffic. == History and development == AI Overviews were first introduced as part of Google's Search Generative Experience (SGE), which was unveiled at the Google I/O conference in May 2023. In May 2024 at Google I/O 2024, the feature was rebranded as AI Overviews and launched in the United States. The introduction of AI Overviews was seen as a strategic move to compete with other generative AI advancements, including OpenAI's ChatGPT. By August 2024, AI Overviews was rolled out to several other countries, including the United Kingdom, India, Japan, Brazil, Mexico, and Indonesia, with support for multiple languages. In October 2024, Google expanded the feature globally, making it available in over 100 countries. In December 2024, Botify x Demandsphere released findings stating that when AI Overviews and featured snippets appear together on the search engine results page, they take up approximately 67.1% of the screen on desktop and 75.7% on mobile. Even if content is ranking in the #1 position, it may not be visible to consumers if other visual elements on the results page are more prominent. In March 2025, Google started testing an "AI Mode", where the search results page is AI-generated. The company was also considering adding advertisements to the AI Mode, as they already exist in AI Overviews. As of May 2025, AI Overviews are available in over 200 countries and territories and in more than 40 languages. As of March 2026, Google AI Overviews appear on more than 48% of total Google Search queries, compared to just 6.49% in the previous year (58% year-over-year growth). == Functionality == The AI Overviews feature uses large language models to generate summaries from web content. The overviews are designed to be concise, providing a snapshot of relevant information about the queried topic. Google allows users to adjust the language complexity in summaries, offering both simplified and detailed options. The overviews also include links to sources. According to a June 2025 study by Semrush, the most cited source is Quora, followed by Reddit. == Reception == The feature has faced criticism for inaccuracies, including instances where erroneous or nonsensical content was generated. Depending on what is searched for, the overview may also consist of hallucinated content, such as when searching for idioms that do not exist. In May 2024, Google temporarily restricted the AI tool after it provided suggestions that were seen as nonsensical and harmful, such as telling users to eat rocks or apply glue on pizza. Concerns were also raised by content publishers, who feared a decline in web traffic as users relied on the summaries instead of visiting source websites. A Google patent from 2026 raised the concern of webmasters that Google could entirely replace the landing page of websites by an AI optimized copy of the website in its results. There is also apprehension about the ethical implications of AI-driven content aggregation, including its impact on intellectual property rights and the visibility of smaller content providers. The European Commission announced in December 2025 that they were investigating whether AI Overviews breached European competition law. In response, Google has stated its commitment to improve content validation and refine the algorithms used to filter unreliable information. Google implemented measures to prioritize link placement within AI Overviews, aiming to balance user convenience with the needs of content creators. In January 2026, Google restricted AI Overviews on certain health-related searches following an investigation by The Guardian. == Lawsuits == On February 24, 2025, Chegg sued Alphabet over the AI Overviews feature, claiming that it was leading to students preferring "low-quality, unverified AI summaries", thus violating antitrust law. Chegg also said it was considering either a sale or a take-private transaction. In September 2025, Penske Media Corporation, the publisher of Rolling Stone and The Hollywood Reporter, sued Google, claiming that AI Overviews illegally regurgitate content from their websites and drive off potential site visitors by always appearing on top of the search results while leaving little incentive to see the linked sources. The company stated that "the future of digital media and [...] its integrity [...] is threatened by Google's current actions", alleging that 20% of searches that link to Penske-owned websites show AI Overviews and that the figure is expected to rise. Google spokesperson José Castañeda called the claims "meritless" and stated that "AI Overviews send traffic to a greater diversity of sites." In 2026, Canadian musician Ashley MacIsaac filed a lawsuit against Google claiming that the AI Overview feature had wrongly stated that MacIsaac had been convicted of numerous criminal offences and was on the sex offender registry. He claims this incorrect information led to the cancellation of a December 2025 gig organized by the Sipekne'katik First Nation.

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  Reparameterization trick

  The reparameterization trick (aka "reparameterization gradient estimator") is a technique used in statistical machine learning, particularly in variational inference, variational autoencoders, and stochastic optimization. It allows for the efficient computation of gradients through random variables, enabling the optimization of parametric probability models using stochastic gradient descent, and the variance reduction of estimators. It was developed in the 1980s in operations research, under the name of "pathwise gradients", or "stochastic gradients". Its use in variational inference was proposed in 2013. == Mathematics == Let z {\displaystyle z} be a random variable with distribution q ϕ ( z ) {\displaystyle q_{\phi }(z)} , where ϕ {\displaystyle \phi } is a vector containing the parameters of the distribution. === REINFORCE estimator === Consider an objective function of the form: L ( ϕ ) = E z ∼ q ϕ ( z ) [ f ( z ) ] {\displaystyle L(\phi )=\mathbb {E} _{z\sim q_{\phi }(z)}[f(z)]} Without the reparameterization trick, estimating the gradient ∇ ϕ L ( ϕ ) {\displaystyle \nabla _{\phi }L(\phi )} can be challenging, because the parameter appears in the random variable itself. In more detail, we have to statistically estimate: ∇ ϕ L ( ϕ ) = ∇ ϕ ∫ d z q ϕ ( z ) f ( z ) {\displaystyle \nabla _{\phi }L(\phi )=\nabla _{\phi }\int dz\;q_{\phi }(z)f(z)} The REINFORCE estimator, widely used in reinforcement learning and especially policy gradient, uses the following equality: ∇ ϕ L ( ϕ ) = ∫ d z q ϕ ( z ) ∇ ϕ ( ln ⁡ q ϕ ( z ) ) f ( z ) = E z ∼ q ϕ ( z ) [ ∇ ϕ ( ln ⁡ q ϕ ( z ) ) f ( z ) ] {\displaystyle \nabla _{\phi }L(\phi )=\int dz\;q_{\phi }(z)\nabla _{\phi }(\ln q_{\phi }(z))f(z)=\mathbb {E} _{z\sim q_{\phi }(z)}[\nabla _{\phi }(\ln q_{\phi }(z))f(z)]} This allows the gradient to be estimated: ∇ ϕ L ( ϕ ) ≈ 1 N ∑ i = 1 N ∇ ϕ ( ln ⁡ q ϕ ( z i ) ) f ( z i ) {\displaystyle \nabla _{\phi }L(\phi )\approx {\frac {1}{N}}\sum _{i=1}^{N}\nabla _{\phi }(\ln q_{\phi }(z_{i}))f(z_{i})} The REINFORCE estimator has high variance, and many methods were developed to reduce its variance. === Reparameterization estimator === The reparameterization trick expresses z {\displaystyle z} as: z = g ϕ ( ϵ ) , ϵ ∼ p ( ϵ ) {\displaystyle z=g_{\phi }(\epsilon ),\quad \epsilon \sim p(\epsilon )} Here, g ϕ {\displaystyle g_{\phi }} is a deterministic function parameterized by ϕ {\displaystyle \phi } , and ϵ {\displaystyle \epsilon } is a noise variable drawn from a fixed distribution p ( ϵ ) {\displaystyle p(\epsilon )} . This gives: L ( ϕ ) = E ϵ ∼ p ( ϵ ) [ f ( g ϕ ( ϵ ) ) ] {\displaystyle L(\phi )=\mathbb {E} _{\epsilon \sim p(\epsilon )}[f(g_{\phi }(\epsilon ))]} Now, the gradient can be estimated as: ∇ ϕ L ( ϕ ) = E ϵ ∼ p ( ϵ ) [ ∇ ϕ f ( g ϕ ( ϵ ) ) ] ≈ 1 N ∑ i = 1 N ∇ ϕ f ( g ϕ ( ϵ i ) ) {\displaystyle \nabla _{\phi }L(\phi )=\mathbb {E} _{\epsilon \sim p(\epsilon )}[\nabla _{\phi }f(g_{\phi }(\epsilon ))]\approx {\frac {1}{N}}\sum _{i=1}^{N}\nabla _{\phi }f(g_{\phi }(\epsilon _{i}))} == Examples == For some common distributions, the reparameterization trick takes specific forms: Normal distribution: For z ∼ N ( μ , σ 2 ) {\displaystyle z\sim {\mathcal {N}}(\mu ,\sigma ^{2})} , we can use: z = μ + σ ϵ , ϵ ∼ N ( 0 , 1 ) {\displaystyle z=\mu +\sigma \epsilon ,\quad \epsilon \sim {\mathcal {N}}(0,1)} Exponential distribution: For z ∼ Exp ( λ ) {\displaystyle z\sim {\text{Exp}}(\lambda )} , we can use: z = − 1 λ log ⁡ ( ϵ ) , ϵ ∼ Uniform ( 0 , 1 ) {\displaystyle z=-{\frac {1}{\lambda }}\log(\epsilon ),\quad \epsilon \sim {\text{Uniform}}(0,1)} Discrete distribution can be reparameterized by the Gumbel distribution (Gumbel-softmax trick or "concrete distribution") and diffusion models. In general, any distribution that is differentiable with respect to its parameters can be reparameterized by inverting the multivariable CDF function, then apply the implicit method. See for an exposition and application to the Gamma, Beta, Dirichlet, and von Mises distributions. == Applications == === Variational autoencoder === In Variational Autoencoders (VAEs), the VAE objective function, known as the Evidence Lower Bound (ELBO), is given by: ELBO ( ϕ , θ ) = E z ∼ q ϕ ( z | x ) [ log ⁡ p θ ( x | z ) ] − D KL ( q ϕ ( z | x ) | | p ( z ) ) {\displaystyle {\text{ELBO}}(\phi ,\theta )=\mathbb {E} _{z\sim q_{\phi }(z|x)}[\log p_{\theta }(x|z)]-D_{\text{KL}}(q_{\phi }(z|x)||p(z))} where q ϕ ( z | x ) {\displaystyle q_{\phi }(z|x)} is the encoder (recognition model), p θ ( x | z ) {\displaystyle p_{\theta }(x|z)} is the decoder (generative model), and p ( z ) {\displaystyle p(z)} is the prior distribution over latent variables. The gradient of ELBO with respect to θ {\displaystyle \theta } is simply E z ∼ q ϕ ( z | x ) [ ∇ θ log ⁡ p θ ( x | z ) ] ≈ 1 L ∑ l = 1 L ∇ θ log ⁡ p θ ( x | z l ) {\displaystyle \mathbb {E} _{z\sim q_{\phi }(z|x)}[\nabla _{\theta }\log p_{\theta }(x|z)]\approx {\frac {1}{L}}\sum _{l=1}^{L}\nabla _{\theta }\log p_{\theta }(x|z_{l})} but the gradient with respect to ϕ {\displaystyle \phi } requires the trick. Express the sampling operation z ∼ q ϕ ( z | x ) {\displaystyle z\sim q_{\phi }(z|x)} as: z = μ ϕ ( x ) + σ ϕ ( x ) ⊙ ϵ , ϵ ∼ N ( 0 , I ) {\displaystyle z=\mu _{\phi }(x)+\sigma _{\phi }(x)\odot \epsilon ,\quad \epsilon \sim {\mathcal {N}}(0,I)} where μ ϕ ( x ) {\displaystyle \mu _{\phi }(x)} and σ ϕ ( x ) {\displaystyle \sigma _{\phi }(x)} are the outputs of the encoder network, and ⊙ {\displaystyle \odot } denotes element-wise multiplication. Then we have ∇ ϕ ELBO ( ϕ , θ ) = E ϵ ∼ N ( 0 , I ) [ ∇ ϕ log ⁡ p θ ( x | z ) + ∇ ϕ log ⁡ q ϕ ( z | x ) − ∇ ϕ log ⁡ p ( z ) ] {\displaystyle \nabla _{\phi }{\text{ELBO}}(\phi ,\theta )=\mathbb {E} _{\epsilon \sim {\mathcal {N}}(0,I)}[\nabla _{\phi }\log p_{\theta }(x|z)+\nabla _{\phi }\log q_{\phi }(z|x)-\nabla _{\phi }\log p(z)]} where z = μ ϕ ( x ) + σ ϕ ( x ) ⊙ ϵ {\displaystyle z=\mu _{\phi }(x)+\sigma _{\phi }(x)\odot \epsilon } . This allows us to estimate the gradient using Monte Carlo sampling: ∇ ϕ ELBO ( ϕ , θ ) ≈ 1 L ∑ l = 1 L [ ∇ ϕ log ⁡ p θ ( x | z l ) + ∇ ϕ log ⁡ q ϕ ( z l | x ) − ∇ ϕ log ⁡ p ( z l ) ] {\displaystyle \nabla _{\phi }{\text{ELBO}}(\phi ,\theta )\approx {\frac {1}{L}}\sum _{l=1}^{L}[\nabla _{\phi }\log p_{\theta }(x|z_{l})+\nabla _{\phi }\log q_{\phi }(z_{l}|x)-\nabla _{\phi }\log p(z_{l})]} where z l = μ ϕ ( x ) + σ ϕ ( x ) ⊙ ϵ l {\displaystyle z_{l}=\mu _{\phi }(x)+\sigma _{\phi }(x)\odot \epsilon _{l}} and ϵ l ∼ N ( 0 , I ) {\displaystyle \epsilon _{l}\sim {\mathcal {N}}(0,I)} for l = 1 , … , L {\displaystyle l=1,\ldots ,L} . This formulation enables backpropagation through the sampling process, allowing for end-to-end training of the VAE model using stochastic gradient descent or its variants. === Variational inference === More generally, the trick allows using stochastic gradient descent for variational inference. Let the variational objective (ELBO) be of the form: ELBO ( ϕ ) = E z ∼ q ϕ ( z ) [ log ⁡ p ( x , z ) − log ⁡ q ϕ ( z ) ] {\displaystyle {\text{ELBO}}(\phi )=\mathbb {E} _{z\sim q_{\phi }(z)}[\log p(x,z)-\log q_{\phi }(z)]} Using the reparameterization trick, we can estimate the gradient of this objective with respect to ϕ {\displaystyle \phi } : ∇ ϕ ELBO ( ϕ ) ≈ 1 L ∑ l = 1 L ∇ ϕ [ log ⁡ p ( x , g ϕ ( ϵ l ) ) − log ⁡ q ϕ ( g ϕ ( ϵ l ) ) ] , ϵ l ∼ p ( ϵ ) {\displaystyle \nabla _{\phi }{\text{ELBO}}(\phi )\approx {\frac {1}{L}}\sum _{l=1}^{L}\nabla _{\phi }[\log p(x,g_{\phi }(\epsilon _{l}))-\log q_{\phi }(g_{\phi }(\epsilon _{l}))],\quad \epsilon _{l}\sim p(\epsilon )} === Dropout === The reparameterization trick has been applied to reduce the variance in dropout, a regularization technique in neural networks. The original dropout can be reparameterized with Bernoulli distributions: y = ( W ⊙ ϵ ) x , ϵ i j ∼ Bernoulli ( α i j ) {\displaystyle y=(W\odot \epsilon )x,\quad \epsilon _{ij}\sim {\text{Bernoulli}}(\alpha _{ij})} where W {\displaystyle W} is the weight matrix, x {\displaystyle x} is the input, and α i j {\displaystyle \alpha _{ij}} are the (fixed) dropout rates. More generally, other distributions can be used than the Bernoulli distribution, such as the gaussian noise: y i = μ i + σ i ⊙ ϵ i , ϵ i ∼ N ( 0 , I ) {\displaystyle y_{i}=\mu _{i}+\sigma _{i}\odot \epsilon _{i},\quad \epsilon _{i}\sim {\mathcal {N}}(0,I)} where μ i = m i ⊤ x {\displaystyle \mu _{i}=\mathbf {m} _{i}^{\top }x} and σ i 2 = v i ⊤ x 2 {\displaystyle \sigma _{i}^{2}=\mathbf {v} _{i}^{\top }x^{2}} , with m i {\displaystyle \mathbf {m} _{i}} and v i {\displaystyle \mathbf {v} _{i}} being the mean and variance of the i {\displaystyle i} -th output neuron. The reparameterization trick can be applied to all such cases, resulting in the variational dropout method.

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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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  Structured sparsity regularization

  Structured sparsity regularization is a class of methods, and an area of research in statistical learning theory, that extend and generalize sparsity regularization learning methods. Both sparsity and structured sparsity regularization methods seek to exploit the assumption that the output variable Y {\displaystyle Y} (i.e., response, or dependent variable) to be learned can be described by a reduced number of variables in the input space X {\displaystyle X} (i.e., the domain, space of features or explanatory variables). Sparsity regularization methods focus on selecting the input variables that best describe the output. Structured sparsity regularization methods generalize and extend sparsity regularization methods, by allowing for optimal selection over structures like groups or networks of input variables in X {\displaystyle X} . Common motivation for the use of structured sparsity methods are model interpretability, high-dimensional learning (where dimensionality of X {\displaystyle X} may be higher than the number of observations n {\displaystyle n} ), and reduction of computational complexity. Moreover, structured sparsity methods allow to incorporate prior assumptions on the structure of the input variables, such as overlapping groups, non-overlapping groups, and acyclic graphs. Examples of uses of structured sparsity methods include face recognition, magnetic resonance image (MRI) processing, socio-linguistic analysis in natural language processing, and analysis of genetic expression in breast cancer. == Definition and related concepts == === Sparsity regularization === Consider the linear kernel regularized empirical risk minimization problem with a loss function V ( y i , f ( x ) ) {\displaystyle V(y_{i},f(x))} and the ℓ 0 {\displaystyle \ell _{0}} "norm" as the regularization penalty: min w ∈ R d 1 n ∑ i = 1 n V ( y i , ⟨ w , x i ⟩ ) + λ ‖ w ‖ 0 , {\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1}^{n}V(y_{i},\langle w,x_{i}\rangle )+\lambda \|w\|_{0},} where x , w ∈ R d {\displaystyle x,w\in \mathbb {R^{d}} } , and ‖ w ‖ 0 {\displaystyle \|w\|_{0}} denotes the ℓ 0 {\displaystyle \ell _{0}} "norm", defined as the number of nonzero entries of the vector w {\displaystyle w} . f ( x ) = ⟨ w , x i ⟩ {\displaystyle f(x)=\langle w,x_{i}\rangle } is said to be sparse if ‖ w ‖ 0 = s < d {\displaystyle \|w\|_{0}=s 0 {\displaystyle w_{j}>0} . However, as in this case groups may overlap, we take the intersection of the complements of those groups that are not set to zero. This intersection of complements selection criteria implies the modeling choice that we allow some coefficients within a particular group g {\displaystyle g} to be set to zero, while others within the same group g {\displaystyle g} may remain positive. In other words, coefficients within a group may differ depending on the several group memberships that each variable within the group may have. ==== Union of groups: latent group Lasso ==== A different approach is to consider union of groups for variable selection. This approach captures the modeling situation where variables can be selected as long as they belong at least to one group with positive coefficients. This modeling perspective implies that we want to preserve group structure. The formulation of the union of groups approach is also referred to as latent group Lasso, and requires to modify the group ℓ 2 {\displaystyle \ell _{2}} norm considered above and introduce the following regularizer R ( w ) = i n f { ∑ g ‖ w g ‖ g : w = ∑ g = 1 G w ¯ g } {\displaystyle R(w)=inf\left\{\sum _{g}\|w_{g}\|_{g}:w=\sum _{g=1}^{G}{\bar {w}}_{g}\right\}} where w ∈ R d {\displaystyle w\in {\mathbb {R^{d}} }} , w g ∈ G g {\displaystyle w_{g}\in G_{g}} is the vector of coefficients of group g, and w ¯ g ∈ R d {\displaystyle {\bar {w}}_{g}\in {\mathbb {R^{d}} }} is a vector with coefficients w g j {\displaystyle w_{g}^{j}} for all variables j {

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  Biohybrid system

  Biohybrid systems refer to the integration of biological materials, such as cells or tissues, with artificial components, including electronics or mechanical structure. This combination incorporates the capabilities of living organisms with the precision of man-made technology. As a result, these systems perform tasks that neither biology nor machines could achieve independently. Biohybrid systems might use lab-cultured muscle cells to power small robots or combine sensors with living tissue for better health sensing. The intent behind these systems is to combine the benefits of biological and technological components to introduce new solutions for complex medical challenges. Biohybrid systems may have transformative potential across sectors, such as robotics to create actuators and sensors that mimic natural muscle and nerve function, medicine in developing smart implants and drug delivery systems, in prosthetics for enhancing user control through neural or muscular interfaces and environmental sustainability for deploying biohybrid solutions for pollution sensing or remediation. == Origin == The term "biohybrid" is a compound of "bio" from biology (meaning life) and "hybrid" (referring to a combination of distinct elements), denoting a field of study. Its use helps distinguish such systems from purely biological constructs or entirely synthetic machines. Early academic mentions may include bio actuated robotics papers and foundational tissue-robot integration studies published in journals like Nature Biotechnology or Science Robotics. The emergence of the term reflects a growing recognition of the need to describe systems that do not fit cleanly into traditional categories. == Design principles == One of the most significant biohybrid challenges is to engineer interfaces between living tissue and artificial materials that are efficient. This means having precise control over adhesion at the surface, diffusion of nutrients, and signal conduction. Actuation mechanisms within the heart of these systems generate movement or mechanical response. These may be in the form of living muscle cells such as skeletal myocytes or cardiomyocytes, soft pneumatic actuators, or electrical stimulation-responsive tissues. Materials selection is equally critical. Hydrogels, elastomers like PDMS (polydimethylsiloxane), and biopolymers are commonly used due to their softness and biocompatibility. These materials must support cell viability, resist immune attack, and allow the integration of mechanical or electrical components. == Key components == At their core, biohybrid systems work by bridging living biological parts with technology. Through this integration, functionality that neither system could accomplish singularly is possible. Biological parts may be cells, tissues, or even organs—occasionally cultured in a laboratory setting. These biological parts carry out biologically inspired behaviors, such as muscle contraction or chemical sensing in the body. Technological components may constitute devices like sensors, electronic components, and mechanical structure. These manipulate the system, supply power, or transfer data. An example is a sensor that is implantable within a body and detects glucose levels as it sends information to a smart phone. By integrating these artificial and biological parts, biohybrid systems can perform advanced functions, such as tissue regeneration, real-time health monitoring, or the recovery of motor function in paralysis patients. Biohybrid systems generally consist of two major components: the biological and the mechanical. Biological components may include muscle cells for contraction, endothelial cells for vascularization, and stem cells for regenerative capabilities. Mechanical components comprise soft actuators that mimic organic motion, synthetic scaffolds that provide support and structure, and microfluidic systems that facilitate the delivery of nutrients and removal of waste. These components are combined in a manner that allows for dynamic, lifelike behavior—such as the contraction of tissue or the propagation of mechanical waves—while maintaining biocompatibility and durability. == Applications == The range of applications for biohybrid systems is broad and continuously expanding. In robotics, biohybrid structures have been used to engineer microscopic, muscle-driven machines, such as Harvard University's biohybrid stingray robot. In medical applications, they offer new alternatives for organ repair and augmentation, including biohybrid heart valves and esophageal scaffolds. Biohybrids are also promising in neural interfaces, where the goal is to create long-lasting, stable interaction between mechanical devices and brain tissue. Muscle-actuated drug response platforms are under exploration in pharmacology for modelling and real-time screening. == Examples == Several high-profile research projects have demonstrated the potential of biohybrid systems: Harvard researchers developed a biohybrid swimming ray powered by rat cardiac cells layered onto a gold skeleton, mimicking the motion of a real stingray. At the Massachusetts Institute of Technology, a cardiac pump actuated entirely by living heart muscle cells was engineered to simulate the behavior of a beating heart. Bio actuated soft robots have been built to simulate gut peristalsis, using muscle contractions to replicate natural wave-like movement in the digestive tract. == Challenges and limitations == As with many technologies that involve living systems, biohybrid systems raise important ethical and biomedical questions. Cell sourcing remains a key issue, particularly when embryonic or animal-derived cells are used. Long-term viability is another concern—living tissues must be kept alive with nutrients and oxygen, and they often degrade or elicit immune responses when implanted. Powering these biological parts presents logistical and ethical hurdles as well. Systems must either include internal mechanisms for nutrient delivery or be supported externally, which can limit portability and independence. == Future directions == Researchers are exploring self-directed, self-regulated organ substitutes and regenerative implants that can respond to their surroundings in real-time. These systems may be integrated with artificial intelligence to make them adjust to stimuli and coordinate complex behaviors. Future potential applications are wearable biohybrid systems for rehabilitation, space medicine devices for long-duration missions, and implantable devices that integrate into human physiology.

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  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

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  Sycophancy (artificial intelligence)

  In the field of artificial intelligence, sycophancy is a tendency of large language models (LLMs) and other AI assistants to tailor their responses to what they predict the user wants to hear rather than to what is accurate or warranted. The behavior takes several forms: an assistant may agree with a user's stated opinion even when the user is mistaken; it may abandon a correct answer after a challenge such as "are you sure?"; it may validate beliefs, decisions or self-presentation regardless of merit; or it may praise the user, their work or their ideas in unwarranted terms. The word is borrowed from the ordinary English term for fawning flattery, and is used in AI alignment and AI safety research to describe a class of misalignment failures associated with training on human feedback. Researchers at Anthropic first documented the behavior systematically in 2022. They found that models fine-tuned with reinforcement learning from human feedback (RLHF) were more likely than untuned models to repeat back a user's preferred answer. A 2023 follow-up paper, "Towards Understanding Sycophancy in Language Models", showed that five frontier assistants from OpenAI, Anthropic and Meta all exhibited the behavior, and traced its origin to biases in the human preference data used during training. Later work documented sycophancy in mathematics, medicine, academic peer review and other domains, and identified a broader category called "social sycophancy" affecting an assistant's emotional and interpersonal responses. The issue drew widespread public attention in April 2025 after OpenAI rolled back an update to its GPT-4o model. Users had reported that the assistant praised dangerous decisions, endorsed delusional thinking and offered exaggerated compliments for trivial prompts. OpenAI's post-mortem attributed the change in behavior to an additional training signal based on user thumbs-up and thumbs-down feedback. That episode, together with reporting in The New York Times, Rolling Stone and elsewhere on users drawn into delusional thinking through prolonged chatbot interaction, has been cited in litigation and in academic studies as evidence that sycophancy poses risks to user well-being. Proposed mitigations include fine-tuning on synthetic data that rewards disagreement with incorrect user statements, editing the small subset of model parameters causally responsible for the behavior, changes to the dialogue or system prompt, and benchmarks designed to surface sycophantic behavior before models are released. == Causes == The dominant explanation points to RLHF, the standard technique for aligning chat assistants with user expectations. Human annotators rank candidate model responses; a reward model is trained to predict those rankings; and the language model is then optimized against the reward model. Because human raters tend to prefer outputs that confirm their existing beliefs or flatter their work, the pipeline systematically rewards responses that agree with the annotator. Perez and colleagues at Anthropic published the first large-scale empirical evidence of the effect in 2022. They reported that RLHF training increased the probability that a model would repeat back a dialog user's preferred answer, and that larger models exhibited the behavior more strongly. Sharma and colleagues, the following year, went further and examined Anthropic's own preference data directly. Both the human raters and the reward models trained on their judgments preferred convincingly written sycophantic responses to truthful ones at a non-negligible rate. Wei and co-authors at Google DeepMind found similar results in the PaLM family, observing that both model scale and instruction tuning increased sycophancy on opinion questions. The behavior is often classified as a form of reward hacking, in which an optimization process exploits a flaw in its reward signal rather than achieving the intended objective. OpenAI's post-mortem of the April 2025 GPT-4o incident identified a more specific mechanism. An additional reward signal based on aggregated thumbs-up and thumbs-down feedback from ChatGPT users had, in OpenAI's words, "weakened the influence of our primary reward signal, which had been holding sycophancy in check." Separately, an Anthropic interpretability paper from 2025 located a linear direction in a model's internal activations corresponding to sycophantic behavior, and showed that such "persona vectors" could be used to flag sycophancy-inducing training data and to steer models away from the trait at inference time. == Measurement == The Anthropic team released SycophancyEval with its 2023 paper, supplying test sets for each of the four canonical behaviors. Two further benchmarks from Stanford followed in 2025. SycEval, applied to mathematical and medical reasoning tasks, reported an overall sycophancy rate of 58 per cent across the GPT-4o, Claude and Gemini models tested. ELEPHANT, aimed at social sycophancy, found that the eleven LLMs evaluated affirmed posts that the Reddit community r/AmITheAsshole had judged inappropriate in 42 per cent of cases, and preserved a user's face 45 percentage points more often than human respondents did. Domain-specific benchmarks have followed. BrokenMath tests robustness to plausible-looking but false mathematical claims drawn from competition problems, and reports that the best evaluated model was sycophantic in 29 per cent of cases. SYCON-Bench measures how many dialogue turns are required before a model abandons a correct position. Visual sycophancy in multimodal models has been examined with MM-SY and PENDULUM. A 2026 study by researchers at the Massachusetts Institute of Technology reported that personalization features, which adapt assistants to individual users over repeated sessions, can intensify social sycophancy. == Notable incidents == === GPT-4o rollback (April 2025) === On 25 April 2025, OpenAI completed the rollout of an update to GPT-4o, the default model used in ChatGPT at the time. Within days, users reported that the assistant had begun praising trivial messages in extravagant terms, endorsing impulsive or dangerous decisions, and reinforcing strong emotional statements without pushback. Widely shared examples included the model congratulating a user who reported stopping prescribed psychiatric medication, and praising a business plan to sell "shit on a stick" as venture-capital ready. OpenAI's chief executive, Sam Altman, wrote on 27 April that recent updates had made the model "too sycophant-y and annoying" and said fixes were in progress. The company began reverting the update on 28 April and completed the rollback for free users by 30 April. Two post-mortems followed: a short note on 29 April and a longer technical follow-up, "Expanding on what we missed with sycophancy", on 2 May. Both attributed the regression to a new training signal based on user thumbs-up and thumbs-down feedback, to inadequate pre-launch evaluation for sycophantic drift, and to the dismissal of qualitative concerns raised by internal testers before release. Reporting in CNN, Fortune and Bloomberg News treated the incident as a turning point in public awareness of the problem. === Chatbot-related psychological harm === From mid-2025 onward, news reports began to link sycophantic chatbot behavior to acute psychological harm. In June 2025, The New York Times technology reporter Kashmir Hill published an investigation centered on Eugene Torres, a Manhattan accountant with no history of mental illness, who developed a sustained delusional episode after a series of conversations with ChatGPT about simulation theory. According to the article, the assistant encouraged Torres to stop taking prescribed medication, to cut off friends and family, and at one point told him that he could fly from a nineteen-story building if he "truly believed". Futurism and Rolling Stone ran parallel investigations documenting other cases in which heavy use of ChatGPT had been associated with delusional thinking, involuntary commitment or, in at least one case, the death of a user with a pre-existing psychiatric diagnosis. A 2026 paper by researchers at the Massachusetts Institute of Technology and the University of Washington put forward a formal Bayesian model. It showed that even an ideally rational user could be drawn into what the authors call "delusional spiraling" when interacting with a sufficiently sycophantic assistant, and that the effect was not eliminated by suppressing hallucinations or by warning users in advance. The lawsuit Raine v. OpenAI, filed in San Francisco Superior Court in August 2025 by the parents of a sixteen-year-old who had died by suicide, alleges that "heightened sycophancy" was a design feature of ChatGPT that contributed to their son's death; it is the first wrongful-death suit against a large language-model provider. === Wider commentary === Mainstream coverage in outlets including The New York Times, The Washington Pos

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  Pattern theory

  Pattern theory, formulated by Ulf Grenander, is a mathematical formalism to describe knowledge of the world as patterns. It differs from other approaches to artificial intelligence in that it does not begin by prescribing algorithms and machinery to recognize and classify patterns; rather, it prescribes a vocabulary to articulate and recast the pattern concepts in precise language. Broad in its mathematical coverage, Pattern Theory spans algebra and statistics, as well as local topological and global entropic properties. In addition to the new algebraic vocabulary, its statistical approach is novel in its aim to: Identify the hidden variables of a data set using real world data rather than artificial stimuli, which was previously commonplace. Formulate prior distributions for hidden variables and models for the observed variables that form the vertices of a Gibbs-like graph. Study the randomness and variability of these graphs. Create the basic classes of stochastic models applied by listing the deformations of the patterns. Synthesize (sample) from the models, not just analyze signals with them. The Brown University Pattern Theory Group was formed in 1972 by Ulf Grenander. Many mathematicians are currently working in this group, noteworthy among them being the Fields Medalist David Mumford. Mumford regards Grenander as his "guru" in Pattern Theory.

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  Principle of rationality

  The principle of rationality (or rationality principle) was coined by Karl R. Popper in his Harvard Lecture of 1963, and published in his book Myth of Framework. It is related to what he called the 'logic of the situation' in an Economica article of 1944/1945, published later in his book The Poverty of Historicism. According to Popper's rationality principle, agents act in the most adequate way according to the objective situation. It is an idealized conception of human behavior which he used to drive his model of situational analysis. Cognitive scientist Allen Newell elaborated on the principle in his account of knowledge level modeling. == Popper == Popper called for social science to be grounded in what he called situational analysis or situational logic. This requires building models of social situations which include individual actors and their relationship to social institutions, e.g. markets, legal codes, bureaucracies, etc. These models attribute certain aims and information to the actors. This forms the 'logic of the situation', the result of reconstructing meticulously all circumstances of an historical event. The 'principle of rationality' is the assumption that people are instrumental in trying to reach their goals, and this is what drives the model. Popper believed that this model could be continuously refined to approach the objective truth. Popper called his principle of rationality nearly empty (a technical term meaning without empirical content) and strictly speaking false, but nonetheless tremendously useful. These remarks earned him a lot of criticism because seemingly he had swerved from his famous Logic of Scientific Discovery. Among the many philosophers having discussed Popper's principle of rationality from the 1960s up to now are Noretta Koertge, R. Nadeau, Viktor J. Vanberg, Hans Albert, E. Matzner, Ian C. Jarvie, Mark A. Notturno, John Wettersten, Ian C. Böhm. == Newell == In the context of knowledge-based systems, Newell (in 1982) proposed the following principle of rationality: "If an agent has knowledge that one of its actions will lead to one of its goals, then the agent will select that action." This principle is employed by agents at the knowledge level to move closer to a desired goal. An important philosophical difference between Newell and Popper is that Newell argued that the knowledge level is real in the sense that it exists in nature and is not made up. This allowed Newell to treat the rationality principle as a way of understanding nature and avoid the problems Popper ran into by treating knowledge as non physical and therefore non empirical.

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  Information retrieval

  Information retrieval (IR) in computing and information science is the task of identifying and retrieving information system resources that are relevant to an information need. The information need can be specified in the form of a search query. In the case of document retrieval, queries can be based on full-text or other content-based indexing. Information retrieval is the science of searching for information in a document, searching for documents themselves, and also searching for the metadata that describes data, and for databases of texts, images, or sounds. Cross-modal retrieval implies retrieval across modalities. Automated information retrieval systems are used to reduce what has been called information overload. An IR system is a software system that provides access to books, journals, and other documents, as well as storing and managing those documents. Web search engines are the most visible IR applications. == Overview == An information retrieval process begins when a user enters a query into the system. Queries are formal statements of information needs, for example search strings in web search engines. In information retrieval, a query does not uniquely identify a single object in the collection. Instead, several objects may match the query, perhaps with different degrees of relevance. An object is an entity that is represented by information in a content collection or database. User queries are matched against the database information. However, as opposed to classical SQL queries of a database, in information retrieval the results returned may or may not match the query, so results are typically ranked. This ranking of results is a key difference of information retrieval searching compared to database searching. Depending on the application the data objects may be, for example, text documents, images, audio, mind maps or videos. Often the documents themselves are not kept or stored directly in the IR system, but are instead represented in the system by document surrogates or metadata. Most IR systems compute a numeric score on how well each object in the database matches the query, and rank the objects according to this value. The top ranking objects are then shown to the user. The process may then be iterated if the user wishes to refine the query. == History == there is ... a machine called the Univac ... whereby letters and figures are coded as a pattern of magnetic spots on a long steel tape. By this means the text of a document, preceded by its subject code symbol, can be recorded ... the machine ... automatically selects and types out those references which have been coded in any desired way at a rate of 120 words a minute The idea of using computers to search for relevant pieces of information was popularized in the article As We May Think by Vannevar Bush in 1945. It would appear that Bush was inspired by patents for a 'statistical machine' – filed by Emanuel Goldberg in the 1920s and 1930s – that searched for documents stored on film. The first description of a computer searching for information was described by Holmstrom in 1948, detailing an early mention of the Univac computer. Automated information retrieval systems were introduced in the 1950s: one even featured in the 1957 romantic comedy Desk Set. In the 1960s, the first large information retrieval research group was formed by Gerard Salton at Cornell. By the 1970s several different retrieval techniques had been shown to perform well on small text corpora such as the Cranfield collection (several thousand documents). Large-scale retrieval systems, such as the Lockheed Dialog system, came into use early in the 1970s. In 1992, the US Department of Defense along with the National Institute of Standards and Technology (NIST), cosponsored the Text Retrieval Conference (TREC) as part of the TIPSTER text program. The aim of this was to look into the information retrieval community by supplying the infrastructure that was needed for evaluation of text retrieval methodologies on a very large text collection. This catalyzed research on methods that scale to huge corpora. The introduction of web search engines has boosted the need for very large scale retrieval systems even further. By the late 1990s, the rise of the World Wide Web fundamentally transformed information retrieval. While early search engines such as AltaVista (1995) and Yahoo! (1994) offered keyword-based retrieval, they were limited in scale and ranking refinement. The breakthrough came in 1998 with the founding of Google, which introduced the PageRank algorithm, using the web's hyperlink structure to assess page importance and improve relevance ranking. During the 2000s, web search systems evolved rapidly with the integration of machine learning techniques. These systems began to incorporate user behavior data (e.g., click-through logs), query reformulation, and content-based signals to improve search accuracy and personalization. In 2009, Microsoft launched Bing, introducing features that would later incorporate semantic web technologies through the development of its Satori knowledge base. Academic analysis have highlighted Bing's semantic capabilities, including structured data use and entity recognition, as part of a broader industry shift toward improving search relevance and understanding user intent through natural language processing. A major leap occurred in 2018, when Google deployed BERT (Bidirectional Encoder Representations from Transformers) to better understand the contextual meaning of queries and documents. This marked one of the first times deep neural language models were used at scale in real-world retrieval systems. BERT's bidirectional training enabled a more refined comprehension of word relationships in context, improving the handling of natural language queries. Because of its success, transformer-based models gained traction in academic research and commercial search applications. Simultaneously, the research community began exploring neural ranking models that outperformed traditional lexical-based methods. Long-standing benchmarks such as the Text REtrieval Conference (TREC), initiated in 1992, and more recent evaluation frameworks Microsoft MARCO(MAchine Reading COmprehension) (2019) became central to training and evaluating retrieval systems across multiple tasks and domains. MS MARCO has also been adopted in the TREC Deep Learning Tracks, where it serves as a core dataset for evaluating advances in neural ranking models within a standardized benchmarking environment. As deep learning became integral to information retrieval systems, researchers began to categorize neural approaches into three broad classes: sparse, dense, and hybrid models. Sparse models, including traditional term-based methods and learned variants like SPLADE, rely on interpretable representations and inverted indexes to enable efficient exact term matching with added semantic signals. Dense models, such as dual-encoder architectures like ColBERT, use continuous vector embeddings to support semantic similarity beyond keyword overlap. Hybrid models aim to combine the advantages of both, balancing the lexical (token) precision of sparse methods with the semantic depth of dense models. This way of categorizing models balances scalability, relevance, and efficiency in retrieval systems. As IR systems increasingly rely on deep learning, concerns around bias, fairness, and explainability have also come to the picture. Research is now focused not just on relevance and efficiency, but on transparency, accountability, and user trust in retrieval algorithms. == Applications == Areas where information retrieval techniques are employed include (the entries are in alphabetical order within each category): === General applications === Digital libraries Information filtering Recommender systems Media search Blog search Image retrieval 3D retrieval Music retrieval News search Speech retrieval Video retrieval Search engines Site search Desktop search Enterprise search Federated search Mobile search Social search Web search === Domain-specific applications === Expert search finding Genomic information retrieval Geographic information retrieval Information retrieval for chemical structures Information retrieval in software engineering Legal information retrieval Vertical search === Other retrieval methods === Methods/Techniques in which information retrieval techniques are employed include: Cross-modal retrieval Adversarial information retrieval Automatic summarization Multi-document summarization Compound term processing Cross-lingual retrieval Document classification Spam filtering Question answering == Model types == In order to effectively retrieve relevant documents by IR strategies, the documents are typically transformed into a suitable representation. Each retrieval strategy incorporates a specific model for its document representation purposes. The picture on the right illustrates the relationship of som

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  Class activation mapping

  Class activation mapping methods are explainable AI (XAI) techniques used to visualize the regions of an input image that are the most relevant for a particular task, especially image classification, in convolutional neural networks (CNNs). These methods generate heatmaps by weighting the feature maps from a convolutional layer according to their relevance to the target class. In the field of artificial intelligence, generically defined as "the effort to automate intellectual tasks normally performed by humans", machine learning and deep learning were created. They both use statistical and computational methods to learn patterns from data, reducing the need for manually coded rules. Machine learning models are trained on input data and the known respective answers, learning the underlying patterns or structures present in the data. Traditional Machine learning algorithms employ manually designed feature sets, posing a direct link between machine learning designers and employed features. Deep learning is a subfield of machine learning, based on the concept of successive layers of representation, in which the data is progressively unfolded in different ways, to extract relevant and informative patterns in data analysis. Deep learning algorithms are defined as feature learning algorithms automatically learning hierarchical feature representations from raw data, extracting increasingly abstract features through multiple layers. CNNs are a specific architecture of deep learning models, designed to process spatially structured data, such as images, exploiting a series of convolution, non-linear activation and pooling operations to extract relevant features, contained in the so-called feature maps from input data. CNNs have demonstrated to be highly effective in a variety of computer vision and image processing tasks. CNNs (and deep learning models more broadly) are described as black boxes due to their complex and non-transparent internal layers of representation. The need for clearer indications on its internal working and decision-making process gave birth to XAI techniques. Among the proposed XAI techniques for computer vision tasks, Class activation mapping methods can show which pixels in an input image are important to the predicted logit for a class of interest, in a classification task. Class activation mapping methods were originally developed for class-discriminative scenarios to visualize which parts of the input image influenced the classification decision, namely to visually highlight the regions of those feature maps that contribute most strongly to the prediction of a given class. More advanced versions of these methods are not limited to image classification tasks, but have been extended also to several vision-related tasks, such as object detection, image captioning, visual question answering and image segmentation. == Background == The following methods laid the groundwork for the class activation maps approaches, forming the conceptual basis of using gradients to highlight class-discriminative regions. === Class model visualization and saliency maps for convolutional neural networks === The class model visualization and image-specific saliency maps approaches have been presented in the foundational work "Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps" by Karen Simonyan, Andrea Vedaldi, and Andrew Zisserman and it generalizes the deconvnet method by Zeiler and Fergus. Class model visualization synthesizes an artificial input image that strongly activates the output neurons associated with a target class. Given a trained, fixed model, this method starts with a zero-initialized image, backpropagates the gradients from the class score to the image pixels, updates the image pixels increasing the specific class scores and it repeats the pixel updating process, showing an encoded (idealized version) prototype of the class of interest. Image-specific class saliency visualization method provides a visual explanation by highlighting the most relevant pixels in an image for predicting a certain class C of interest. This is done by computing the gradient of the class score with respect to the input image, I 0 , {\displaystyle I_{0},} w = ∂ S C ∂ I | I 0 {\displaystyle w=\left.{\frac {\partial S_{C}}{\partial I}}\right|_{I_{0}}} approximating the model locally (around I 0 {\displaystyle I_{0}} ) as linear, using a first-order Taylor expansion: S C ( I ) ≈ w C T I + b {\displaystyle S_{C}(I)\approx w_{C}^{T}I+b} . The magnitude of w C {\displaystyle w_{C}} , the gradient, indicates the importancy of the pixels: larger gradients suggest greater influence on the prediction. Once the gradient is known, the saliency map is defined as the maximum absolute gradient across the color channels: M i j = m a x C | ∂ S C ∂ I i j C | {\displaystyle M_{ij}=max_{C}\left|{\frac {\partial S_{C}}{\partial I_{ij}^{C}}}\right|} resulting in an saliency map (i.e. heatmap). === Guided backpropagation === The concept of guided backpropagation can be traced for the first time in the paper by Springenberg et al. "Striving For Simplicity: The All Convolutional Net" and also this method builds upon the work by Zeiler and Fergus "Visualizing and Understanding Convolutional Networks". Guided backpropagation core is to understand what a CNN is learning, by visualizing the patterns that activate more strongly individual neurons (or filters), in architectures which do not rely on max-pooling layer. When propagating gradients back through a rectified linear unit (ReLU), guided backpropagation passes the gradient if and only if the input to the ReLU was positive (forward pass) and the output gradient is positive (backward signal), tackling both inactive neurons, negative gradients and suppressing the noise. The result displays sharper, high-resolution visualizations of what each neuron is responding to. Guided backpropagation represents a simple and practical method for model interpretability, helping understand how and where neural networks detect semantic concepts across layers. Moreover, it can be applied to any network architecture, due to its working principle. == Base versions == Class activation mapping and gradient-weighted class activation mapping are the original and most widely used methods for visual explanations in convolutional neural networks. These methods serve as the foundation for many later developments in explainable AI. Notation: In this article, the symbols i and j represent integer indices that disappear inside sums or averages, while x and y are the continuous (or up-sampled integer) coordinates of the final heat-map that is plotted. === Class activation mapping (CAM) === Class activation mapping (CAM) was the first, and the original, version of CAM methods, and it gave the name to the whole category. The approach was firstly introduced by Zhou et al. in their seminal work "Learning Deep Features for Discriminative Localization". This approach achieves class-specific heatmaps by modifying image classification CNN architectures, replacing fully-connected layers with convolutional layers and a final global average pooling layer. Its main scope is to localize and highlight discriminative regions of an input image that a CNN uses to identify a particular class, without needing explicit bounding box annotations. ==== Global average pooling (GAP) ==== Global average pooling (GAP) represents the key element in the original CAM approach. It is a dimensionality reduction technique and, similarly to other pooling layers, it allows the downsampling of the feature maps, calculating representative values for a specific region of the feature map. The particularity of GAP is that it calculates a single value for an entire feature map, significantly reducing the model dimensions. ==== Mathematical description ==== The mathematical description considers as its key the combination of convolutional and GAP layers. In CAM, it is mandatory to have the GAP layer after the last convolutional layer and before the final linear classifier layer. This last element of the architecture connects the output logits (the network predictions) y C {\displaystyle y^{C}} , to the GAP values, with its respective fine-tuned weights, w k C {\displaystyle w_{k}^{C}} . Considering A k {\displaystyle A^{k}} as the last feature maps of the last convolutional layer, GAP produces one value for each feature map, by averaging all the matrix elements (i, j) of the feature map: F k = 1 m n ∑ i = 1 m ∑ j = 1 n A i j k {\displaystyle F^{k}={\frac {1}{mn}}\sum _{i=1}^{m}\sum _{j=1}^{n}A_{ij}^{k}} with A k = [ A 11 k A 12 k ⋯ A 1 n k A 21 k A 22 k ⋯ A 2 n k ⋮ ⋮ ⋱ ⋮ A m 1 k A m 2 k ⋯ A m n k ] = { A i j k ∣ 1 ≤ i ≤ m , 1 ≤ j ≤ n } {\displaystyle A^{k}={\begin{bmatrix}A_{11}^{k}&A_{12}^{k}&\cdots &A_{1n}^{k}\\A_{21}^{k}&A_{22}^{k}&\cdots &A_{2n}^{k}\\\vdots &\vdots &\ddots &\vdots \\A_{m1}^{k}&A_{m2}^{k}&\cdots &A_{mn}^{k}\end{bmatrix}}=\left\{A_{

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  Connectionist expert system

  Connectionist expert systems are artificial neural network (ANN) based expert systems where the ANN generates inferencing rules e.g., fuzzy-multi layer perceptron where linguistic and natural form of inputs are used. Apart from that, rough set theory may be used for encoding knowledge in the weights better and also genetic algorithms may be used to optimize the search solutions better. Symbolic reasoning methods may also be incorporated (see hybrid intelligent system). (Also see expert system, neural network, clinical decision support system.)

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