In computer science, a single address space operating system (or SASOS) is an operating system that provides only one globally shared address space for all processes. In a single address space operating system, numerically identical (virtual memory) logical addresses in different processes all refer to exactly the same byte of data. In a traditional OS with private per-process address space, memory protection is based on address space boundaries ("address space isolation"). Single address-space operating systems make translation and protection orthogonal, which in no way weakens protection. The core advantage is that pointers (i.e. memory references) have global validity, meaning their meaning is independent of the process using it. This allows sharing pointer-connected data structures across processes, and making them persistent, i.e. storing them on backup store. Some processor architectures have direct support for protection independent of translation. On such architectures, a SASOS may be able to perform context switches faster than a traditional OS. Such architectures include Itanium, and Version 5 of the Arm architecture, as well as capability architectures such as CHERI. A SASOS should not be confused with a flat memory model, which provides no address translation and generally no memory protection. In contrast, a SASOS makes protection orthogonal to translation: it may be possible to name a data item (i.e. know its virtual address) while not being able to access it. SASOS projects using hardware-based protection include the following: Angel IBM i (formerly called OS/400) Iguana at NICTA, Australia Mungi at NICTA, Australia Nemesis Opal Scout Sombrero Related are OSes that provide protection through language-level type safety: Br1X Genera JX a research Java OS Phantom OS Singularity Theseus OS Torsion
Avid Free DV
Avid Free DV is a non-linear editing video editing software application developed by Avid Technology. Avid introduced Free DV in January 2003 at the 2003 MacWorld Expo; the company discontinued it in September 2007. Free DV was intended to give editors a sample of the Avid interface to use in deciding whether or not to purchase Avid software, so when compared with other Avid products its features were relatively minimal. When it was available it was not limited by time or watermarking, so it could be used as a non-linear editor for as long as desired. == Comparisons == When compared with other consumer-end non-linear editors such as iMovie and Windows Movie Maker, it sported more powerful video processing tools, but lacked the ease-of-use and shallow learning curve emphasized in similar programs because it had the full interface of the professional Avid system. However, Avid did offer a number of flash-based tutorials to help new users learn how to use the program for capturing, editing, clipping, processing, and outputting audio/video, among other things. == Limitations == The limitations of Avid Free DV included that it allowed only two video and audio tracks, had fewer editing tools than other Avid products, had few import and export formats, and allowed capture and output of standard-definition DV only, via FireWire. Avid Free DV projects and media were not compatible with other Avid systems. As the name implied, Avid Free DV was available as a free download, although users were required to complete a short survey on the Avid website before they were given a download link and key. In addition to using Free DV to evaluate Avid prior to purchase, it could also act as a stepping stone for people wishing to learn to use Avid's other editing products, such as Xpress Pro, Media Composer and Symphony. While additional skills and techniques are necessary to use these professionally geared systems, the basic operation remains the same. == Operating systems == Avid Free DV was available for Windows XP and Mac OS X. The officially supported Mac OS X versions were Panther versions up to 10.3.5, and Tiger versions up to 10.4.3 only. == Supported formats == Avid Free DV supported QuickTime (MOV) and DV AVIs. == Reception == John P. Mello Jr. of The Boston Globe gave Free DV a negative review, finding the user interface obfuscatory and the process of ingesting video error-prone. He summarized: "Professional video editors who use an Avid competitor may jump at the chance to take a free look at how Avid does things. But for the merely curious, this software is a nightmare". Video Systems's Steve Mullen opined that its lack of interoperability with Avid's professional editing software contracted Avid's stated goal to entice budding video editors into buying into the company's software ecosystem.
Three-factor learning
In neuroscience and machine learning, three-factor learning is the combination of Hebbian plasticity with a third modulatory factor to stabilise and enhance synaptic learning. This third factor can represent various signals such as reward, punishment, error, surprise, or novelty, often implemented through neuromodulators. == Description == Three-factor learning introduces the concept of eligibility traces, which flag synapses for potential modification pending the arrival of the third factor, and helps temporal credit assignement by bridging the gap between rapid neuronal firing and slower behavioral timescales, from which learning can be done. Biological basis for Three-factor learning rules have been supported by experimental evidence. This approach addresses the instability of classical Hebbian learning by minimizing autocorrelation and maximizing cross-correlation between inputs.
Learning rate
In machine learning and statistics, the learning rate is a tuning parameter in an optimization algorithm that determines the step size at each iteration while moving toward a minimum of a loss function. Since it influences to what extent newly acquired information overrides old information, it metaphorically represents the speed at which a machine learning model "learns". In the adaptive control literature, the learning rate is commonly referred to as gain. In setting a learning rate, there is a trade-off between the rate of convergence and overshooting. While the descent direction is usually determined from the gradient of the loss function, the learning rate determines how big a step is taken in that direction. Too high a learning rate will make the learning jump over minima, but too low a learning rate will either take too long to converge or get stuck in an undesirable local minimum. In order to achieve faster convergence, prevent oscillations and getting stuck in undesirable local minima the learning rate is often varied during training either in accordance to a learning rate schedule or by using an adaptive learning rate. The learning rate and its adjustments may also differ per parameter, in which case it is a diagonal matrix that can be interpreted as an approximation to the inverse of the Hessian matrix in Newton's method. The learning rate is related to the step length determined by inexact line search in quasi-Newton methods and related optimization algorithms. == Learning rate schedule == Initial rate can be left as system default or can be selected using a range of techniques. A learning rate schedule changes the learning rate during learning and is most often changed between epochs/iterations. This is mainly done with two parameters: decay and momentum. There are many different learning rate schedules but the most common are time-based, step-based and exponential. Decay serves to settle the learning in a nice place and avoid oscillations, a situation that may arise when too high a constant learning rate makes the learning jump back and forth over a minimum, and is controlled by a hyperparameter. Momentum is analogous to a ball rolling down a hill; we want the ball to settle at the lowest point of the hill (corresponding to the lowest error). Momentum both speeds up the learning (increasing the learning rate) when the error cost gradient is heading in the same direction for a long time and also avoids local minima by 'rolling over' small bumps. Momentum is controlled by a hyperparameter analogous to a ball's mass which must be chosen manually—too high and the ball will roll over minima which we wish to find, too low and it will not fulfil its purpose. The formula for factoring in the momentum is more complex than for decay but is most often built in with deep learning libraries such as Keras. Time-based learning schedules alter the learning rate depending on the learning rate of the previous time iteration. Factoring in the decay the mathematical formula for the learning rate is: η n + 1 = η 0 1 + d n {\displaystyle \eta _{n+1}={\frac {\eta _{0}}{1+dn}}} where η {\displaystyle \eta } is the learning rate, η 0 {\displaystyle \eta _{0}} is the original learning rate, d {\displaystyle d} is a decay parameter and n {\displaystyle n} is the iteration step. Step-based learning schedules changes the learning rate according to some predefined steps. The decay application formula is here defined as: η n = η 0 d ⌊ 1 + n r ⌋ {\displaystyle \eta _{n}=\eta _{0}d^{\left\lfloor {\frac {1+n}{r}}\right\rfloor }} where η n {\displaystyle \eta _{n}} is the learning rate at iteration n {\displaystyle n} , η 0 {\displaystyle \eta _{0}} is the initial learning rate, d {\displaystyle d} is how much the learning rate should change at each drop (0.5 corresponds to a halving) and r {\displaystyle r} corresponds to the drop rate, or how often the rate should be dropped (10 corresponds to a drop every 10 iterations). The floor function ( ⌊ … ⌋ {\displaystyle \lfloor \dots \rfloor } ) here drops the value of its input to 0 for all values smaller than 1. Exponential learning schedules are similar to step-based, but instead of steps, a decreasing exponential function is used. The mathematical formula for factoring in the decay is: η n = η 0 e − d n {\displaystyle \eta _{n}=\eta _{0}e^{-dn}} where d {\displaystyle d} is a decay parameter. == Adaptive learning rate == The issue with learning rate schedules is that they all depend on hyperparameters that must be manually chosen for each given learning session and may vary greatly depending on the problem at hand or the model used. To combat this, there are many different types of adaptive gradient descent algorithms such as Adagrad, Adadelta, RMSprop, and Adam which are generally built into deep learning libraries such as Keras.
AI safety
AI safety is an interdisciplinary field focused on preventing accidents, misuse, or other harmful consequences arising from artificial intelligence systems. It encompasses AI alignment (which aims to ensure AI systems behave as intended), monitoring AI systems for risks, and enhancing their robustness. The field is particularly concerned with existential risks posed by advanced AI models. Beyond technical research, AI safety involves developing norms and policies that promote safety, including advocacy for regulations at different levels of government. The field gained significant popularity in 2023, with rapid progress in generative AI and public concerns voiced by researchers and CEOs about potential dangers. During the 2023 AI Safety Summit, the United States and the United Kingdom both established their own AI Safety Institute. However, researchers have expressed concern that AI safety measures are not keeping pace with the rapid development of AI capabilities. == Motivations == Scholars discuss current risks from critical systems failures, bias, and AI-enabled surveillance, as well as emerging risks like technological unemployment, digital manipulation, weaponization, AI-enabled cyberattacks and bioterrorism. They also discuss speculative risks from losing control of future artificial general intelligence (AGI) agents, or from AI enabling perpetually stable dictatorships. === Existential safety === Some have criticized concerns about AGI, such as Andrew Ng who compared them in 2015 to "worrying about overpopulation on Mars when we have not even set foot on the planet yet". Stuart J. Russell on the other side urges caution, arguing that "it is better to anticipate human ingenuity than to underestimate it". AI researchers have widely differing opinions about the severity and primary sources of risk posed by AI technology – though surveys suggest that experts take high consequence risks seriously. In two surveys of AI researchers, the median respondent was optimistic about AI overall, but placed a 5% probability on an "extremely bad (e.g. human extinction)" outcome of advanced AI. In a 2022 survey of the natural language processing community, 37% agreed or weakly agreed that it is plausible that AI decisions could lead to a catastrophe that is "at least as bad as an all-out nuclear war". == History == Risks from AI began to be seriously discussed at the start of the computer age: Moreover, if we move in the direction of making machines which learn and whose behavior is modified by experience, we must face the fact that every degree of independence we give the machine is a degree of possible defiance of our wishes. In 1988 Blay Whitby published a book outlining the need for AI to be developed along ethical and socially responsible lines. From 2008 to 2009, the Association for the Advancement of Artificial Intelligence (AAAI) commissioned a study to explore and address potential long-term societal influences of AI research and development. The panel was generally skeptical of the radical views expressed by science-fiction authors but agreed that "additional research would be valuable on methods for understanding and verifying the range of behaviors of complex computational systems to minimize unexpected outcomes". In 2011, Roman Yampolskiy introduced the term "AI safety engineering" at the Philosophy and Theory of Artificial Intelligence conference, listing prior failures of AI systems and arguing that "the frequency and seriousness of such events will steadily increase as AIs become more capable". In 2014, philosopher Nick Bostrom published the book Superintelligence: Paths, Dangers, Strategies. He has the opinion that the rise of AGI has the potential to create various societal issues, ranging from the displacement of the workforce by AI, manipulation of political and military structures, to even the possibility of human extinction. His argument that future advanced systems may pose a threat to human existence prompted Elon Musk, Bill Gates, and Stephen Hawking to voice similar concerns. In 2015, dozens of artificial intelligence experts signed an open letter on artificial intelligence calling for research on the societal impacts of AI and outlining concrete directions. To date, the letter has been signed by over 8000 people including Yann LeCun, Shane Legg, Yoshua Bengio, and Stuart Russell. In the same year, a group of academics led by professor Stuart J. Russell founded the Center for Human-Compatible AI at the University of California Berkeley and the Future of Life Institute awarded $6.5 million in grants for research aimed at "ensuring artificial intelligence (AI) remains safe, ethical and beneficial". In 2016, the White House Office of Science and Technology Policy and Carnegie Mellon University announced The Public Workshop on Safety and Control for Artificial Intelligence, which was one of a sequence of four White House workshops aimed at investigating "the advantages and drawbacks" of AI. In the same year, Concrete Problems in AI Safety – one of the first and most influential technical AI Safety agendas – was published. In 2017, the Future of Life Institute sponsored the Asilomar Conference on Beneficial AI, where more than 100 thought leaders formulated principles for beneficial AI including "Race Avoidance: Teams developing AI systems should actively cooperate to avoid corner-cutting on safety standards". In 2018, the DeepMind Safety team outlined AI safety problems in specification, robustness, and assurance. The following year, researchers organized a workshop at ICLR that focused on these problem areas. In 2021, Unsolved Problems in ML Safety was published, outlining research directions in robustness, monitoring, alignment, and systemic safety. In 2023, Rishi Sunak said he wants the United Kingdom to be the "geographical home of global AI safety regulation" and to host the first global summit on AI safety. The AI safety summit took place in November 2023, and focused on the risks of misuse and loss of control associated with frontier AI models. During the summit the intention to create the International Scientific Report on the Safety of Advanced AI was announced. In 2024, The US and UK forged a new partnership on the science of AI safety. The MoU was signed on 1 April 2024 by US commerce secretary Gina Raimondo and UK technology secretary Michelle Donelan to jointly develop advanced AI model testing, following commitments announced at an AI Safety Summit in Bletchley Park in November. In 2025, an international team of 96 experts chaired by Yoshua Bengio published the first International AI Safety Report. The report, commissioned by 30 nations and the United Nations, represents the first global scientific review of potential risks associated with advanced artificial intelligence. It details potential threats stemming from misuse, malfunction, and societal disruption, with the objective of informing policy through evidence-based findings, without providing specific recommendations. == Research focus == AI safety research areas include robustness, monitoring, and alignment. === Robustness === ==== Adversarial robustness ==== AI systems are often vulnerable to adversarial examples or "inputs to machine learning (ML) models that an attacker has intentionally designed to cause the model to make a mistake". For example, in 2013, Szegedy et al. discovered that adding specific imperceptible perturbations to an image could cause it to be misclassified with high confidence. This continues to be an issue with neural networks, though in recent work the perturbations are generally large enough to be perceptible. The image on the right is predicted to be an ostrich after the perturbation is applied. (Left) is a correctly predicted sample, (center) perturbation applied magnified by 10x, (right) adversarial example. Adversarial robustness is often associated with security. Researchers demonstrated that an audio signal could be imperceptibly modified so that speech-to-text systems transcribe it to any message the attacker chooses. Network intrusion and malware detection systems also must be adversarially robust since attackers may design their attacks to fool detectors. Models that represent objectives (reward models) must also be adversarially robust. For example, a reward model might estimate how helpful a text response is and a language model might be trained to maximize this score. Researchers have shown that if a language model is trained for long enough, it will leverage the vulnerabilities of the reward model to achieve a better score and perform worse on the intended task. This issue can be addressed by improving the adversarial robustness of the reward model. More generally, any AI system used to evaluate another AI system must be adversarially robust. This could include monitoring tools, since they could also potentially be tampered with to produce a higher reward. Large language models (LLMs) can be vulnerable to prom
Deep Learning Super Sampling
Deep Learning Super Sampling (DLSS) is a suite of real-time deep learning image enhancement and upscaling technologies developed by Nvidia that are available in a number of video games. The goal of these technologies is to allow the majority of the graphics pipeline to run at a lower resolution for increased performance, and then infer a higher resolution image from this that approximates the same level of detail as if the image had been rendered at this higher resolution. This allows for higher graphical settings or frame rates for a given output resolution, depending on user preference. All generations of DLSS are available on all RTX-branded cards from Nvidia in supported titles. However, the Frame Generation feature is only supported on RTX 40 series GPUs or newer and Multi Frame Generation is only available on 50 series GPUs. == History == Nvidia advertised DLSS as a key feature of GeForce RTX 20 series GPUs when they launched in September 2018. At that time, the results were limited to a few video games, namely Battlefield V, or Metro Exodus, because the algorithm had to be trained specifically on each game on which it was applied and the results were usually not as good as simple resolution upscaling. In 2019, Control shipped with ray tracing and an image processing algorithm that approximated DLSS, which did not use the Tensor Cores. In April 2020, Nvidia advertised and shipped an improved version of DLSS named DLSS 2 with driver version 445.75. DLSS 2.0 was available for a few existing games including Control and Wolfenstein: Youngblood, and would later be added to many newly released games and game engines such as Unreal Engine and Unity. This time Nvidia said that it used the Tensor Cores again, and that the AI did not need to be trained specifically on each game. Despite sharing the DLSS branding, the two iterations of DLSS differ significantly and are not backwards-compatible. In January 2025, Nvidia stated that there are over 540 games and apps supporting DLSS, and that over 80% of Nvidia RTX users activate DLSS. In March 2025, there were more than 100 games that support DLSS 4, according to Nvidia. By May 2025, over 125 games supported DLSS 4. The first video game console to use DLSS, the Nintendo Switch 2, was released on June 5, 2025. Nvidia announced DLSS 4.5 at CES 2026. In January 2026, Nvidia stated that over 250 games and applications support Multi Frame Generation. On March 16, 2026, at GTC 2026, Nvidia CEO Jensen Huang presented DLSS 5, a real-time AI model based on neural rendering that realistically enhances lighting and material surfaces at up to 4K resolution while retaining the developer's intended art style. It is planned to release in fall of 2026. In a blog post on its website, Nvidia has announced that DLSS 5 will be available in such games as Assassin's Creed Shadows, Delta Force, Hogwarts Legacy, Naraka: Bladepoint, Phantom Blade Zero, Resident Evil Requiem, Starfield, The Elder Scrolls IV: Oblivion Remastered, and more. On May 31, 2026, Nvidia announced an updated version of Ray Reconstruction for DLSS 4.5 in a blog post, scheduled for release on all RTX GPUs in August of the same year. They said it is designed to better embed spatial awareness into scenes and analyze engine data on movements and lighting conditions, resulting in a sharper, more stable, and less noisy image. === Release timeline === == Technology == === DLSS 1 === The first iteration of DLSS is a predominantly spatial image upscaler with two stages, both relying on convolutional auto-encoder neural networks. The first step is an image enhancement network which uses the current frame and motion vectors to perform edge enhancement, and spatial anti-aliasing. The second stage is an image upscaling step which uses the single raw, low-resolution frame to upscale the image to the desired output resolution. Using just a single frame for upscaling means the neural network itself must generate a large amount of new information to produce the high-resolution output, which can result in slight hallucinations such as leaves that differ in style to the source content. The neural networks are trained on a per-game basis by generating a "perfect frame" using traditional supersampling to 64 samples per pixel, as well as the motion vectors for each frame. The data collected must be as comprehensive as possible, including as many levels, times of day, graphical settings, resolutions, etc. as possible. This data is also augmented using common augmentations such as rotations, colour changes, and random noise to help generalize the test data. Training is performed on Nvidia's Saturn V supercomputer. This first iteration received a mixed response, with many criticizing the often soft appearance and artifacts along with glitches in certain situations; likely a side effect of the limited data from only using a single frame input to the neural networks which could not be trained to perform optimally in all scenarios and edge-cases. Nvidia also demonstrated the ability for the auto-encoder networks to learn the ability to recreate depth-of-field and motion blur, although this functionality has never been included in a publicly released product. === DLSS 2 === DLSS 2 is a temporal anti-aliasing upsampling (TAAU) implementation, using data from previous frames extensively through sub-pixel jittering to resolve fine detail and reduce aliasing. The data DLSS 2 collects includes: the raw low-resolution input, motion vectors, depth buffers, and exposure / brightness information. It can also be used as a simpler TAA implementation where the image is rendered at 100% resolution, rather than being upsampled by DLSS, Nvidia brands this as DLAA (Deep Learning Anti-Aliasing). TAA(U) is used in many modern video games and game engines; however, all previous implementations have used some form of manually written heuristics to prevent temporal artifacts such as ghosting and flickering. One example of this is neighborhood clamping which forcefully prevents samples collected in previous frames from deviating too much compared to nearby pixels in newer frames. This helps to identify and fix many temporal artifacts, but deliberately removing fine details in this way is analogous to applying a blur filter, and thus the final image can appear blurry when using this method. DLSS 2 uses a convolutional auto-encoder neural network trained to identify and fix temporal artifacts, instead of manually programmed heuristics as mentioned above. Because of this, DLSS 2 can generally resolve detail better than other TAA and TAAU implementations, while also removing most temporal artifacts. This is why DLSS 2 can sometimes produce a sharper image than rendering at higher, or even native resolutions using traditional TAA. However, no temporal solution is perfect, and artifacts (ghosting in particular) are still visible in some scenarios when using DLSS 2. Because temporal artifacts occur in most art styles and environments in broadly the same way, the neural network that powers DLSS 2 does not need to be retrained when being used in different games. Despite this, Nvidia does frequently ship new minor revisions of DLSS 2 with new titles, so this could suggest some minor training optimizations may be performed as games are released, although Nvidia does not provide changelogs for these minor revisions to confirm this. The main advancements compared to DLSS 1 include: Significantly improved detail retention, a generalized neural network that does not need to be re-trained per-game, and ~2x less overhead (~1–2 ms vs ~2–4 ms). It should also be noted that forms of TAAU such as DLSS 2 are not upscalers in the same sense as techniques such as ESRGAN or DLSS 1, which attempt to create new information from a low-resolution source; instead, TAAU works to recover data from previous frames, rather than creating new data. In practice, this means low resolution textures in games will still appear low-resolution when using current TAAU techniques. This is why Nvidia recommends game developers use higher resolution textures than they would normally for a given rendering resolution by applying a mip-map bias when DLSS 2 is enabled. === DLSS 3 === Augments DLSS 2 with improved image quality and the introduction of a new motion interpolation feature, called Frame Generation. The DLSS Frame Generation algorithm takes two rendered frames from the rendering pipeline and generates a new frame that smoothly transitions between them. For every frame rendered, one additional frame is generated. DLSS 3.0 makes use of a new generation Optical Flow Accelerator (OFA) included in the Ada Lovelace architecture of GeForce RTX 40 series GPUs and with that is exclusive to them. The new OFA is said to be faster and more accurate than the one already available in previous Turing and Ampere RTX GPUs. === DLSS 3.5 === DLSS 3.5 adds Ray Reconstruction, replacing multiple denoising algorithms with a single AI model trained o
Similarity learning
Similarity learning is an area of supervised machine learning in artificial intelligence. It is closely related to regression and classification, but the goal is to learn a similarity function that measures how similar or related two objects are. It has applications in ranking, in recommendation systems, visual identity tracking, face verification, and speaker verification. == Learning setup == There are four common setups for similarity and metric distance learning. Regression similarity learning In this setup, pairs of objects are given ( x i 1 , x i 2 ) {\displaystyle (x_{i}^{1},x_{i}^{2})} together with a measure of their similarity y i ∈ R {\displaystyle y_{i}\in R} . The goal is to learn a function that approximates f ( x i 1 , x i 2 ) ∼ y i {\displaystyle f(x_{i}^{1},x_{i}^{2})\sim y_{i}} for every new labeled triplet example ( x i 1 , x i 2 , y i ) {\displaystyle (x_{i}^{1},x_{i}^{2},y_{i})} . This is typically achieved by minimizing a regularized loss min W ∑ i l o s s ( w ; x i 1 , x i 2 , y i ) + r e g ( w ) {\displaystyle \min _{W}\sum _{i}loss(w;x_{i}^{1},x_{i}^{2},y_{i})+reg(w)} . Classification similarity learning Given are pairs of similar objects ( x i , x i + ) {\displaystyle (x_{i},x_{i}^{+})} and non similar objects ( x i , x i − ) {\displaystyle (x_{i},x_{i}^{-})} . An equivalent formulation is that every pair ( x i 1 , x i 2 ) {\displaystyle (x_{i}^{1},x_{i}^{2})} is given together with a binary label y i ∈ { 0 , 1 } {\displaystyle y_{i}\in \{0,1\}} that determines if the two objects are similar or not. The goal is again to learn a classifier that can decide if a new pair of objects is similar or not. Ranking similarity learning Given are triplets of objects ( x i , x i + , x i − ) {\displaystyle (x_{i},x_{i}^{+},x_{i}^{-})} whose relative similarity obey a predefined order: x i {\displaystyle x_{i}} is known to be more similar to x i + {\displaystyle x_{i}^{+}} than to x i − {\displaystyle x_{i}^{-}} . The goal is to learn a function f {\displaystyle f} such that for any new triplet of objects ( x , x + , x − ) {\displaystyle (x,x^{+},x^{-})} , it obeys f ( x , x + ) > f ( x , x − ) {\displaystyle f(x,x^{+})>f(x,x^{-})} (contrastive learning). This setup assumes a weaker form of supervision than in regression, because instead of providing an exact measure of similarity, one only has to provide the relative order of similarity. For this reason, ranking-based similarity learning is easier to apply in real large-scale applications. Locality sensitive hashing (LSH) Hashes input items so that similar items map to the same "buckets" in memory with high probability (the number of buckets being much smaller than the universe of possible input items). It is often applied in nearest neighbor search on large-scale high-dimensional data, e.g., image databases, document collections, time-series databases, and genome databases. A common approach for learning similarity is to model the similarity function as a bilinear form. For example, in the case of ranking similarity learning, one aims to learn a matrix W that parametrizes the similarity function f W ( x , z ) = x T W z {\displaystyle f_{W}(x,z)=x^{T}Wz} . When data is abundant, a common approach is to learn a siamese network – a deep network model with parameter sharing. == Metric learning == Similarity learning is closely related to distance metric learning. Metric learning is the task of learning a distance function over objects. A metric or distance function has to obey four axioms: non-negativity, identity of indiscernibles, symmetry and subadditivity (or the triangle inequality). In practice, metric learning algorithms ignore the condition of identity of indiscernibles and learn a pseudo-metric. When the objects x i {\displaystyle x_{i}} are vectors in R d {\displaystyle R^{d}} , then any matrix W {\displaystyle W} in the symmetric positive semi-definite cone S + d {\displaystyle S_{+}^{d}} defines a distance pseudo-metric of the space of x through the form D W ( x 1 , x 2 ) 2 = ( x 1 − x 2 ) ⊤ W ( x 1 − x 2 ) {\displaystyle D_{W}(x_{1},x_{2})^{2}=(x_{1}-x_{2})^{\top }W(x_{1}-x_{2})} . When W {\displaystyle W} is a symmetric positive definite matrix, D W {\displaystyle D_{W}} is a metric. Moreover, as any symmetric positive semi-definite matrix W ∈ S + d {\displaystyle W\in S_{+}^{d}} can be decomposed as W = L ⊤ L {\displaystyle W=L^{\top }L} where L ∈ R e × d {\displaystyle L\in R^{e\times d}} and e ≥ r a n k ( W ) {\displaystyle e\geq rank(W)} , the distance function D W {\displaystyle D_{W}} can be rewritten equivalently D W ( x 1 , x 2 ) 2 = ( x 1 − x 2 ) ⊤ L ⊤ L ( x 1 − x 2 ) = ‖ L ( x 1 − x 2 ) ‖ 2 2 {\displaystyle D_{W}(x_{1},x_{2})^{2}=(x_{1}-x_{2})^{\top }L^{\top }L(x_{1}-x_{2})=\|L(x_{1}-x_{2})\|_{2}^{2}} . The distance D W ( x 1 , x 2 ) 2 = ‖ x 1 ′ − x 2 ′ ‖ 2 2 {\displaystyle D_{W}(x_{1},x_{2})^{2}=\|x_{1}'-x_{2}'\|_{2}^{2}} corresponds to the Euclidean distance between the transformed feature vectors x 1 ′ = L x 1 {\displaystyle x_{1}'=Lx_{1}} and x 2 ′ = L x 2 {\displaystyle x_{2}'=Lx_{2}} . Many formulations for metric learning have been proposed. Some well-known approaches for metric learning include learning from relative comparisons, which is based on the triplet loss, large margin nearest neighbor, and information theoretic metric learning (ITML). In statistics, the covariance matrix of the data is sometimes used to define a distance metric called Mahalanobis distance. == Applications == Similarity learning is used in information retrieval for learning to rank, in face verification or face identification, and in recommendation systems. Also, many machine learning approaches rely on some metric. This includes unsupervised learning such as clustering, which groups together close or similar objects. It also includes supervised approaches like K-nearest neighbor algorithm which rely on labels of nearby objects to decide on the label of a new object. Metric learning has been proposed as a preprocessing step for many of these approaches. == Scalability == Metric and similarity learning scale quadratically with the dimension of the input space, as can easily see when the learned metric has a bilinear form f W ( x , z ) = x T W z {\displaystyle f_{W}(x,z)=x^{T}Wz} . Scaling to higher dimensions can be achieved by enforcing a sparseness structure over the matrix model, as done with HDSL, and with COMET. == Software == metric-learn is a free software Python library which offers efficient implementations of several supervised and weakly-supervised similarity and metric learning algorithms. The API of metric-learn is compatible with scikit-learn. OpenMetricLearning is a Python framework to train and validate the models producing high-quality embeddings. == Further information == For further information on this topic, see the surveys on metric and similarity learning by Bellet et al. and Kulis.