AI Generator Reddit

AI Generator Reddit — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

Anti-Grain Geometry

Anti-Grain Geometry (AGG) is a 2D rendering graphics library written in C++. It features anti-aliasing and sub-pixel resolution. It is not a graphics library, per se, but rather a framework to build a graphics library upon. The library is operating system independent and renders to an abstract memory object. It comes with examples interfaced to the X Window System, Microsoft Windows, Mac OS X, AmigaOS, BeOS, SDL. The examples also include an SVG viewer. The design of AGG uses C++ templates only at a very high level, rather than extensively, to achieve the flexibility to plug custom classes into the rendering pipeline, without requiring a rigid class hierarchy, and allows the compiler to inline many of the method calls for high performance. For a library of its complexity, it is remarkably lightweight: it has no dependencies above the standard C++ libraries and it avoids the C++ STL in the implementation of the basic algorithms. The implicit interfaces are not well documented, however, and this can make the learning process quite cumbersome. While AGG version 2.5 is licensed under the GNU General Public License, version 2 or greater, AGG version 2.4 is still available under the 3-clause BSD license and is virtually the same as version 2.5. == History == Active development of the AGG codebase stalled in 2006, around the time of the v2.5 release, due to shifting priorities of its main developer and maintainer Maxim Shemanarev. M. Shemanarev remained active in the community until his sudden death in 2013. Development has continued on a fork of the more liberally licensed v2.4 on SourceForge.net. == Usage == The Haiku operating system uses AGG in its windowing system. It is one of the renderers available for use in GNU's Gnash Flash player. Graphical version of Rebol language interpreter is using AGG for scalable vector graphics DRAW dialect. Hilti uses it in some of their rebar detection tools, like the PS 1000. Matplotlib uses AGG as its canonical renderer for interactive user interfaces. fpGUI Toolkit has an optional AggPas back-end rendering engine. Work is being done to make AggPas the default or sole rendering engine for fpGUI. Mapnik, the toolkit that renders the maps on the OpenStreetMap website, uses AGG for all its bitmap map rendering by default. HTTPhotos uses AGG to scale photos. Pdfium, the PDF rendering engine used by Google Chrome makes use of AGG, although work is progressing to replace this with Skia Graphics Engine. Graphics Mill, the .NET imaging SDK uses AGG as its drawing engine. Image-Line FL Studio, a digital audio workstation, since version 10.8 released on September 30, 2012, uses AGG for drawing. Native Instruments's Supercharger and Supercharger GT compressors use AGG for its user interface. == Author == The main author of the library was Maxim Shemanarev (Russian: Максим Шеманарёв). On November 26, 2013 Shemanarev (born June 15, 1966, Nizhny Novgorod, Russia) was reported dead at the age of 47 at his home in Columbia, Maryland (US). He died suddenly, allegedly from an epileptic seizure that he had suffered for a while. He was a graduate from Nizhny Novgorod State Technical University. Little is known about his personal life. It's known though that he was divorced and his mother was alive at the time of his death. He used to love skiing, snowboarding (in Colorado), and inline skating. He was praised by his friends for his intelligent programming skills.
Read more →
Hostile Waters: Antaeus Rising

Hostile Waters, released as Hostile Waters: Antaeus Rising in America, is a hybrid vehicle and strategy game developed and published by Rage Software for Microsoft Windows. It was inspired by Carrier Command (Realtime Games, 1988). It has won several awards and one unofficial award from Rock Paper Shotgun as a "lost classic" or "The best game you've never played". == Plot == Hostile Waters takes place in a Utopian future where war has been abolished. In the year 2012, a revolutionary war takes place between the corrupt and power-hungry politicians, leaders and businessmen (described as the "Old Guard") and the people. The Old Guard were defeated, with only a few of their leaders escaping. By 2032, the world has been rebuilt as a utopia, with the help of nano-technological assemblers, which are used in "creation engines" to create matter from energy and waste, for free. The newly united world is governed from a capital city known as Central. Missile attacks are suddenly launched against major cities all over the world from an unknown location. This is eventually discovered to be an island chain in the South Pacific Ocean. A response to the missile attacks was a special forces team sent in to investigate the area for preliminary investigations. The Ministry of Intelligence (MinIntel) loses contact with it shortly thereafter. The world government authorises a reactivation of the Antaeus program, a series of warships able to create any weapon of their choosing using their on-board nano-technological creation engine. Two of these were left on the seabed in the case of an emergency, capable of being re-activated and refloating itself. On board are a series of "soulcatcher" chips, a classified 1990s military program researched into for the storage of human brain functions on a silicon chip. The soulcatcher technology was used to store the minds of every crew member ever assigned to an Antaeus vessel. It is soon discovered that one of the cruisers does not respond to the awakening signal. The other cruiser, however, is refloated and re-activated, with heavy damage to vital ship components. A course is plotted for a nearby disused wet-dock. As the Antaeus progresses from the wet-dock, unusual biological life-forms are discovered amongst the enemy bases on the islands. The identity of the aggressor firing the missiles is confirmed as the leftovers of the old, pre-Central forces, known as the Cabal. Outnumbering Central's army a thousand to one, they are fighting with thousands of troops and weapons that they hid away when it was apparent that the war was lost. The Antaeus is deployed into the chicane to stop the Cabal's operations there. It's later discovered that along with their superior numbers, they have also biologically engineered a species of organic machines, designed in the popular likeness of extraterrestrials, which they intend to use to create the fear of an alien invasion, to facilitate their taking over the world and the removal of the public use of creation engines. The Cabal later lose control of the species, which eventually turns on its masters, destroying them. The species starts spreading, modifying the planetary climate and geographical features in an attempt to exterminate humanity and make the planet more hospitable to itself. Having exterminated its creators, the species resolves to cleanse humanity as a whole from the planet using a massive 'disassembler cannon', only to be stopped by the Antaeus. The species subsequently attempts to flee into the cosmos and colonise the surrounding planets and stars, by launching a massive number of 'culture stones' (information devices that also double as creation engines) into space from an enormous, artificially-grown organic "island", the final staging point. Central's only option is to bind the Antaeus' creation engine and the disassembler cannon stolen from the aliens together to create a makeshift bomb, and detonate it at the central "column" containing the culture stones. The plan succeeds, and the Antaeus is sacrificed to save the world. The final cinematic show the organic disassembler cannon and the Antaeus' creation engine moving closer together and fusing, creating something new. A post-credits scene also shows that two of the species' culture stones have managed to get into space. == Gameplay == Each Mission takes place on and or near a fortified enemy island containing various forms of anti-air and ground defence, with scattered unit-production complexes powered by oil-derricks and fuel containers (which are dependent on the oil-derricks) that the player can destroy to keep the enemy from replacing destroyed forces. Vehicles are built on the Antaeus and, if desired, land vehicles can be delivered to a location by the air-lifting "magpie". Units are created by providing Antaeus with a number of resources which are obtained at the beginning of the level and debris which are taken from destroyed enemy units and structures. Transport helicopters such as the "Pegasus" can fly to an object and airlift it to the ship-board recycling system with little resources required. The carrier can analyse objects it disassembles at the rear of the Antaeus cruiser, and several of the game's vehicles and items are unlocked by "sampling" them in this fashion. The game has a number of vehicles that are progressively unlocked as the missions progress. Vehicles contain a number of slots for equipment and a selection of different types of weapons to use in the vehicle. A variety of vehicle equipment combinations can be designed. Vehicles have an individual damage multiplier such that different vehicles with the same weapon will do different damage. In addition to this, each soul-chip personality specializes in one unit along with specific equipment, which, if equipped will gain them a bonus in efficiency. == Development == The game was developed by 12 people. == Reception == The game received "favourable" reviews according to the review aggregation website Metacritic. Carla Harker of NextGen said, "You'll feel like a real battlefield general when you take to the field in Antaeus Rising." Jake The Snake of GamePro said, "If the usual game categories leave you unscathed, get bloodied in these Hostile Waters."
Read more →
TuVox

TuVox is a company that produces VXML-based telephone speech-recognition applications to replace DTMF touch-tone systems for their clients. == History == TuVox was founded in 2001 by Steven S. Pollock and Ashok Khosla, formerly of Apple Computer Corporation and Claris Corporation. Since then, TuVox has grown to over 150 employees and has US offices in Cupertino, California and Boca Raton, Florida as well as international offices in London, Vancouver and Sydney. In 2005, TuVox acquired the customers and hosting facilities of Net-By-Tel. In 2007, the company raised $20m for its speech recognition, and phone menu software. On July 22, 2010, West Interactive — a subsidiary of West Corporation — announced its acquisition of TuVox. == Customers == TuVox clients include: 1-800-Flowers.com, AMC Entertainment, American Airlines, British Airways, M&T Bank, Canon Inc., Gateway, Inc., Motorola, Progress Energy Inc., Telecom New Zealand, Time, Inc., BECU, Virgin America and USAA.
Read more →
Pommerman Challenge

The Pommerman Challenge is a multi-agent game to test autonomous artificial intelligence systems. == Game structure == Two-agent team compete against each other on an 11 x 11 board. Each agent can observe only part of the board, and the agents cannot communicate. The goal is to knock down the opponents. Agents place explosives to destroy walls and collect power-ups that appear from those walls, while avoiding death. Game objects can move unpredictably or be moved by an agent. == Play == The game involves real-time decision making. Agents must choose moves in about .1 seconds. == Algorithms == The real-time requirement limits the use of compute-heavy techniques such as Monte Carlo tree search. The branching factor at each move can be as large as 1,296, because all four agents act in each step, choosing among six possibilities. The agents choose by accounting for explosions, which have lifetimes of 10 steps. Explosions derail tree search techniques, as searches with less than 10 levels ignore explosions while deeper searches consider too many choices (given the branching factor). A hybrid approach uses a limited-depth tree search followed by exploring a deterministic/pessimistic scenario. Limiting the depth keeps the search tree small. The deterministic approach can predict far in the future, by omitting branching. "Good" actions are often those that perform well under pessimistic scenarios, particularly if safety is important. Identifying the worst sequence of positions for an object can suggest where to move it. After generating pessimistic scenarios, the agent quantifies the survivability of each move, notionally the number of positions in which the agent can then remain safely (without encountering other agents). == Competitions == 3 competitions were organized with slightly changing rules during 2018–2019. === Online - FFA === This round was a warm-up online event, where each competitor controlled only one agent. Results: 1st: Agent47Agent by Yichen Gong 2nd: aiKiller by Márton Görög === NeurIPS 2018 - Team === The first Pommerman competition with in-person finals. Results: 1st: hakozakijunctions by Toshihiro Takahashi 2nd: eisenach by Márton Görög 3rd: dypm by Takayuki Osogami The 3 best performing solutions used online tree search. === NeurIPS 2019 - Team Radio === The second competition with in-person finals improved communication between teammate agents. Results: 1st: Márton Görög 2nd: Paul Jasek 3rd: Yifan Zhang
Read more →
Deluxe Paint Animation

DeluxePaint Animation is a 1990 graphics editor and animation creation package for MS-DOS, based on Deluxe Paint for the Amiga. It was adapted by Brent Iverson with additional animation features by Steve Shaw and released by Electronic Arts. The program requires VGA graphics, MS-DOS 2.1 or higher, and a mouse. == Features == Listed from the back of the box. Complete selection of painting tools — Draw any shape you want, any way you want. Turn any image into a brush. You can rotate, flip, shear, resize, smear, and shade it. 7 levels of magnification — Paint in magnified mode if you want. Use variable zoom for detailed editing at the pixel level. 3-D perspective — Move and rotate images in full 3-D, automatically. Use color cycling and gradient fills to create great special effects. Stencils — Protect your designs from the slip of the hand or a bad idea. A stencil masks your image so you can paint "behind" and "in front of" it. Use the handy Move Dialog to animate brushes in full 3-D — automatically! Ideal for creating spinning titles for low-cost videos. 37 multi-sized fonts
Read more →
Resistance Database Initiative

HIV Resistance Response Database Initiative (RDI) was formed in 2002 to use artificial intelligence (AI) to predict how patients will respond to HIV drugs using data from more 250,000 patients from around 50 countries around the world. The RDI used its models to power its HIV Treatment Response Prediction System (HIV-TRePS). Launched in 2010, this free online tool enabled healthcare professionals to upload their patient’s data and obtain highly accurate predictions of how they would respond to different combinations of the 30 or more drugs available. The tool enabled physicians to individualize their patients’ treatment, using these predictions based on more than a million patient-years of treatment experience. HIV-TRePS was possibly the first ever AI-based system for medical decision-making to be developed, successfully tested, and used in clinical practice. It has since been used by thousands of healthcare professionals to optimise the treatment of tens of thousands of patients. Since the RDI’s inception the treatment of HIV infection has progressed enormously, with more effective and better tolerated drugs available in ever more convenient combination formulations. In most countries HIV is now considered a chronic, manageable condition. Moreover, the success of the drugs in reducing the amount of virus is substantially reducing the onward transmission of the virus and cases of new infections are falling in many settings. This improvement in HIV treatment means the need for sophisticated AI to support HIV treatment decisions has significantly reduced. In response, the RDI ceased development of further models and, in March 2024, withdrew its HIV-TRePS system. == Background == Human immunodeficiency virus (HIV) is the virus that causes acquired immunodeficiency syndrome (AIDS), a condition in which the immune system begins to fail, leading to life-threatening opportunistic infections. There are approximately 30 HIV antiretroviral drugs that have been approved for the treatment of HIV infection, from six different classes, based on the point in the HIV life-cycle at which they act. They are used in combination; typically 3 or more drugs from 2 or more different classes, a form of therapy known as highly active antiretroviral therapy (HAART). The aim of therapy is to suppress the virus to very low, ideally undetectable, levels in the blood. This prevents the virus from depleting the immune cells that it preferentially attacks CD4 cells and prevents or delays illness and death. Despite the expanding availability of these drugs and the impact of their use, treatments continue to fail, often involving to the development of resistance. During drug therapy, low-level virus replication may still occur, particularly when a patient misses a dose. HIV makes errors in copying its genetic material and, if a mutation makes the virus resistant to one or more of the drugs in the patient's treatment, it may begin to replicate more successfully in the presence of that drug and undermine the effect of the treatment. If this happens, the treatment needs to be changed to re-establish control over the virus. == RDI's Approach == The RDI’s approach was to use artificial intelligence (including neural network and random forest models), trained with data from hundreds of thousands of patients, treated with different drugs in a variety of clinical settings all over the world, to predict how an individual patient will respond to any new combination of HIV drugs. The models were tested with independent data sets and consistently achieved accuracy of approximately 80%.
Read more →
The Future of Truth (Rosenbaum book)

The Future of Truth: How AI Reshapes Reality is a 2026 book by American filmmaker and author Steven Rosenbaum about how artificial intelligence affects the concept of truth. It was published by Matt Holt Books on May 12, 2026, to positive media attention; on May 19, in response to an inquiry from The New York Times, Rosenbaum acknowledged that the book itself contains multiple misattributed or false quotes that were hallucinated by AIs. == Synopsis == == Development == Rosenbaum has said that he developed the book using AI chatbots as research tools, indicating in his notes what information came from AI and sending those claims to a fact-checker affiliated with the publisher. He has said that he did not use AI tools to write the book itself. He has described AI tools as "a delightful writing companion ... strangely creative and crafty and unusual in all these ways", while acknowledging that sometimes "then it betrays you in ways that are just really quite horrible". Journalist and Nobel laureate Maria Ressa wrote the book's foreword. Taylor Lorenz, Michael Wolff, and Nicholas Thompson wrote blurbs promoting it. == Release and reception == The Future of Truth was published by Matt Holt Books, an imprint of BenBella Books, and distributed by Simon & Schuster. The book's release on May 12, 2026, was described by Futurism as "buzzy" and by The New York Times as "to great fanfare". On May 14, an excerpt was published in Wired under the title "Gen Z Is Pioneering a New Understanding of Truth". On May 17, the Times contacted Rosenbaum regarding a number of quotes that appeared to be falsified or misattributed; the following evening he confirmed that they were the result of AI hallucinations:As I disclosed in the book's acknowledgments, I used AI tools ChatGPT and Claude during the research, writing and editing process. That does not excuse these errors, of which I take full responsibility. I am now working with the editors to thoroughly review and quickly correct any affected passages; any future editions will be corrected. The Times documented several of the errors, including a quote from Kara Swisher that Swisher described as making it "sound like I have a stick up my butt" and a quote from Lisa Feldman Barrett that Barrett described as misrepresenting her views on the nature of emotions, social signals, and truth. The book also misattributed a quote by Meredith Broussard from an interview with Marketplace Tech as having been from her book Artificial Unintelligence and hallucinated several words in a quote from Lee McIntyre, although according to McIntyre it did not misrepresent his views. Wired's editors, in an addendum to the excerpt they published, said that all quotes included in it had been verified as part of their fact-checking process. Rosenbaum told the Times that the series of errors "serves as a warning about the risks of AI-assisted research and verification, that is why I wrote the book. These AI errors do not, in fact, diminish the larger questions that the book raises about truth, trust and AI and its impact on society, democracy and editorial." Maggie Harrison Dupré in Futurism expressed skepticism, writing "The risk of AI hallucinations ... is well-known. If you're going to literally write the book on post-AI truth, you should probably put some more elbow grease into fact-checking your AI-assisted research." Kyle Orland in Ars Technica, responding to Rosenbaum's statement that his error "demonstrates the problem more vividly than any abstract argument could", was similarly skeptical, writing that "if we accept this take, every avoidably obvious mess in the world might be a disguised good because it really helps illuminate the huge mistake. And that can't be right; sometimes 'negligence' is just that." Subsequent comments by Rosenbaum placed more blame on the chatbots, which he told The Atlantic "fucked up the book". Rosenbaum told Ars Technica that fact-checking occurred "incredibly effectively, but not a hundred percent"; Orland observed that "it's worth noting that most writers manage to include zero made-up quotes when they write a book". Rosenbaum said that he had "learned a lesson" and would be "much more suspicious" of AI in the future, but would continue to use AI in his research. Orland responded to Rosenbaum's characterization of AI as "magical" by comparing it to the One Ring from The Lord of the Rings, in that it "convinces many of those who use it that they can control its power properly" when many cannot. Orland highlighted the limits of traditional fact-checking regarding AI, given that fact-checkers are used to assuming that direct quotes are copied word-for-word from the source. Rosenbaum told Orland that the future of fact-checking for AI-researched works "probably includes mandatory source tracing for quotations, better provenance tracking, clearer standards around AI-assisted research, and potentially (more irony here) AI tools that audit citations against primary materials". Patrick Redford in Defector criticized Rosenbaum, alongside other artists tricked by AI, for failing to recognize AI as "the enemy". Will Oremus in The Atlantic described Redford's approach of stigmatizing AI writing as "reasonable", noting the presence of low-quality, seemingly AI-generated verbiage in The Future of Truth—a claim denied by Rosenbaum—before saying that the greater issue is finding the line at which AI assistance in writing becomes a problem. Oremus concluded, "The scandal can't just be that [Rosenbaum] used AI while working on his book, because he acknowledged that up front. He got in trouble because he had used AI badly, failing to check its work on a task at which it is famously unreliable."
Read more →
Estimation of distribution algorithm

Estimation of distribution algorithms (EDAs), sometimes called probabilistic model-building genetic algorithms (PMBGAs), are stochastic optimization methods that guide the search for the optimum by building and sampling explicit probabilistic models of promising candidate solutions. Optimization is viewed as a series of incremental updates of a probabilistic model, starting with the model encoding an uninformative prior over admissible solutions and ending with the model that generates only the global optima. EDAs belong to the class of evolutionary algorithms. The main difference between EDAs and most conventional evolutionary algorithms is that evolutionary algorithms generate new candidate solutions using an implicit distribution defined by one or more variation operators, whereas EDAs use an explicit probability distribution encoded by a Bayesian network, a multivariate normal distribution, or another model class. Similarly as other evolutionary algorithms, EDAs can be used to solve optimization problems defined over a number of representations from vectors to LISP style S expressions, and the quality of candidate solutions is often evaluated using one or more objective functions. The general procedure of an EDA is outlined in the following: t := 0 initialize model M(0) to represent uniform distribution over admissible solutions while (termination criteria not met) do P := generate N>0 candidate solutions by sampling M(t) F := evaluate all candidate solutions in P M(t + 1) := adjust_model(P, F, M(t)) t := t + 1 Using explicit probabilistic models in optimization allowed EDAs to feasibly solve optimization problems that were notoriously difficult for most conventional evolutionary algorithms and traditional optimization techniques, such as problems with high levels of epistasis. Nonetheless, the advantage of EDAs is also that these algorithms provide an optimization practitioner with a series of probabilistic models that reveal a lot of information about the problem being solved. This information can in turn be used to design problem-specific neighborhood operators for local search, to bias future runs of EDAs on a similar problem, or to create an efficient computational model of the problem. For example, if the population is represented by bit strings of length 4, the EDA can represent the population of promising solution using a single vector of four probabilities (p1, p2, p3, p4) where each component of p defines the probability of that position being a 1. Using this probability vector it is possible to create an arbitrary number of candidate solutions. == Estimation of distribution algorithms (EDAs) == This section describes the models built by some well known EDAs of different levels of complexity. It is always assumed a population P ( t ) {\displaystyle P(t)} at the generation t {\displaystyle t} , a selection operator S {\displaystyle S} , a model-building operator α {\displaystyle \alpha } and a sampling operator β {\displaystyle \beta } . == Univariate factorizations == The most simple EDAs assume that decision variables are independent, i.e. p ( X 1 , X 2 ) = p ( X 1 ) ⋅ p ( X 2 ) {\displaystyle p(X_{1},X_{2})=p(X_{1})\cdot p(X_{2})} . Therefore, univariate EDAs rely only on univariate statistics and multivariate distributions must be factorized as the product of N {\displaystyle N} univariate probability distributions, D Univariate := p ( X 1 , … , X N ) = ∏ i = 1 N p ( X i ) . {\displaystyle D_{\text{Univariate}}:=p(X_{1},\dots ,X_{N})=\prod _{i=1}^{N}p(X_{i}).} Such factorizations are used in many different EDAs, next we describe some of them. === Univariate marginal distribution algorithm (UMDA) === The UMDA is a simple EDA that uses an operator α U M D A {\displaystyle \alpha _{UMDA}} to estimate marginal probabilities from a selected population S ( P ( t ) ) {\displaystyle S(P(t))} . By assuming S ( P ( t ) ) {\displaystyle S(P(t))} contain λ {\displaystyle \lambda } elements, α U M D A {\displaystyle \alpha _{UMDA}} produces probabilities: p t + 1 ( X i ) = 1 λ ∑ x ∈ S ( P ( t ) ) x i , ∀ i ∈ 1 , 2 , … , N . {\displaystyle p_{t+1}(X_{i})={\dfrac {1}{\lambda }}\sum _{x\in S(P(t))}x_{i},~\forall i\in 1,2,\dots ,N.} Every UMDA step can be described as follows D ( t + 1 ) = α UMDA ∘ S ∘ β λ ( D ( t ) ) . {\displaystyle D(t+1)=\alpha _{\text{UMDA}}\circ S\circ \beta _{\lambda }(D(t)).} === Population-based incremental learning (PBIL) === The PBIL, represents the population implicitly by its model, from which it samples new solutions and updates the model. At each generation, μ {\displaystyle \mu } individuals are sampled and λ ≤ μ {\displaystyle \lambda \leq \mu } are selected. Such individuals are then used to update the model as follows p t + 1 ( X i ) = ( 1 − γ ) p t ( X i ) + ( γ / λ ) ∑ x ∈ S ( P ( t ) ) x i , ∀ i ∈ 1 , 2 , … , N , {\displaystyle p_{t+1}(X_{i})=(1-\gamma )p_{t}(X_{i})+(\gamma /\lambda )\sum _{x\in S(P(t))}x_{i},~\forall i\in 1,2,\dots ,N,} where γ ∈ ( 0 , 1 ] {\displaystyle \gamma \in (0,1]} is a parameter defining the learning rate, a small value determines that the previous model p t ( X i ) {\displaystyle p_{t}(X_{i})} should be only slightly modified by the new solutions sampled. PBIL can be described as D ( t + 1 ) = α PIBIL ∘ S ∘ β μ ( D ( t ) ) {\displaystyle D(t+1)=\alpha _{\text{PIBIL}}\circ S\circ \beta _{\mu }(D(t))} === Compact genetic algorithm (cGA) === The CGA, also relies on the implicit populations defined by univariate distributions. At each generation t {\displaystyle t} , two individuals x , y {\displaystyle x,y} are sampled, P ( t ) = β 2 ( D ( t ) ) {\displaystyle P(t)=\beta _{2}(D(t))} . The population P ( t ) {\displaystyle P(t)} is then sorted in decreasing order of fitness, S Sort ( f ) ( P ( t ) ) {\displaystyle S_{{\text{Sort}}(f)}(P(t))} , with u {\displaystyle u} being the best and v {\displaystyle v} being the worst solution. The CGA estimates univariate probabilities as follows p t + 1 ( X i ) = p t ( X i ) + γ ( u i − v i ) , ∀ i ∈ 1 , 2 , … , N , {\displaystyle p_{t+1}(X_{i})=p_{t}(X_{i})+\gamma (u_{i}-v_{i}),\quad \forall i\in 1,2,\dots ,N,} where, γ ∈ ( 0 , 1 ] {\displaystyle \gamma \in (0,1]} is a constant defining the learning rate, usually set to γ = 1 / N {\displaystyle \gamma =1/N} . The CGA can be defined as D ( t + 1 ) = α CGA ∘ S Sort ( f ) ∘ β 2 ( D ( t ) ) {\displaystyle D(t+1)=\alpha _{\text{CGA}}\circ S_{{\text{Sort}}(f)}\circ \beta _{2}(D(t))} == Bivariate factorizations == Although univariate models can be computed efficiently, in many cases they are not representative enough to provide better performance than GAs. In order to overcome such a drawback, the use of bivariate factorizations was proposed in the EDA community, in which dependencies between pairs of variables could be modeled. A bivariate factorization can be defined as follows, where π i {\displaystyle \pi _{i}} contains a possible variable dependent to X i {\displaystyle X_{i}} , i.e. | π i | = 1 {\displaystyle |\pi _{i}|=1} . D Bivariate := p ( X 1 , … , X N ) = ∏ i = 1 N p ( X i | π i ) . {\displaystyle D_{\text{Bivariate}}:=p(X_{1},\dots ,X_{N})=\prod _{i=1}^{N}p(X_{i}|\pi _{i}).} Bivariate and multivariate distributions are usually represented as probabilistic graphical models (graphs), in which edges denote statistical dependencies (or conditional probabilities) and vertices denote variables. To learn the structure of a PGM from data linkage-learning is employed. === Mutual information maximizing input clustering (MIMIC) === The MIMIC factorizes the joint probability distribution in a chain-like model representing successive dependencies between variables. It finds a permutation of the decision variables, r : i ↦ j {\displaystyle r:i\mapsto j} , such that x r ( 1 ) x r ( 2 ) , … , x r ( N ) {\displaystyle x_{r(1)}x_{r(2)},\dots ,x_{r(N)}} minimizes the Kullback–Leibler divergence in relation to the true probability distribution, i.e. π r ( i + 1 ) = { X r ( i ) } {\displaystyle \pi _{r(i+1)}=\{X_{r(i)}\}} . MIMIC models a distribution p t + 1 ( X 1 , … , X N ) = p t ( X r ( N ) ) ∏ i = 1 N − 1 p t ( X r ( i ) | X r ( i + 1 ) ) . {\displaystyle p_{t+1}(X_{1},\dots ,X_{N})=p_{t}(X_{r(N)})\prod _{i=1}^{N-1}p_{t}(X_{r(i)}|X_{r(i+1)}).} New solutions are sampled from the leftmost to the rightmost variable, the first is generated independently and the others according to conditional probabilities. Since the estimated distribution must be recomputed each generation, MIMIC uses concrete populations in the following way P ( t + 1 ) = β μ ∘ α MIMIC ∘ S ( P ( t ) ) . {\displaystyle P(t+1)=\beta _{\mu }\circ \alpha _{\text{MIMIC}}\circ S(P(t)).} === Bivariate marginal distribution algorithm (BMDA) === The BMDA factorizes the joint probability distribution in bivariate distributions. First, a randomly chosen variable is added as a node in a graph, the most dependent variable to one of those in the graph is chosen among those not yet in the graph, this procedure is repeated until no remain
Read more →
OpenFog Consortium

The OpenFog Consortium (sometimes stylized as Open Fog Consortium) was a consortium of high tech industry companies and academic institutions across the world aimed at the standardization and promotion of fog computing in various capacities and fields. The consortium was founded by Cisco Systems, Intel, Microsoft, Princeton University, Dell, and ARM Holdings in 2015 and now has 57 members across the North America, Asia, and Europe, including Forbes 500 companies and noteworthy academic institutions. The OpenFog consortium merged with the Industrial Internet Consortium, now the Industry IoT Consortium, on January 31, 2019. == History == OpenFog was created on November 19, 2015, by ARM Holdings, Cisco Systems, Dell, Intel, Microsoft, and Princeton University. The idea for a consortium centered on the advancement and dissemination of fog computing was thought up by Helder Antunes, a Cisco executive with a history in IoT, Mung Chiang, then a Princeton University professor and now President of Purdue University, and Dr. Tao Zhang, a Cisco Distinguished Engineer and CIO for the IEEE Communications Society then and now a manager at the National Institute of Standards and Technologies (NIST). The project was executed from concept to launch by Armando Pereira at PVentures Consulting, a Silicon Valley–based high-tech consulting firm. OpenFog released its reference architecture for fog computing on February 13, 2017. The Fog World Congress 2017, with Dr. Tao Zhang as its General Chair, was hosted in October 2017 by OpenFog, in conjunction with the IEEE Communications Society, as the first congress devoted to fog computing. == Administration == The OpenFog Consortium was governed by its board of directors, which is chaired by Cisco Senior Director Helder Antunes. The board of directors is made up of 11 seats, each representing one of the following companies and institutions: ARM, AT&T, Cisco, Dell, Intel, Microsoft, Princeton University, IEEE, GE, ZTE and Shanghai Tech University. The consortium's general membership comprised 13 academic members: Aalto University, Arizona State University, California Institute of Technology, Georgia State University, National Chiao Tung University, National Taiwan University, Shanghai Research Centre for Wireless Communication, Chinese University of Hong Kong, University of Colorado Boulder, University of Southern California, University of Pisa, Vanderbilt University, Wayne State University, and 20 additional members: Hitachi, Internet Initiative Japan, Itochu, Kii, Nebbiolo, PrismTech, NEC, NGD Systems, NTT Communications, OSIsoft, Real-time Innovations, relayr, Sakura Internet, Stichting imec Nederland, Toshiba, TTT Tech, Fujitsu, FogHorn Systems, TTTech and MARSEC. == Published work == The OpenFog Consortium published the white paper, "OpenFog Reference Architecture". This document outlines the eight pillars of an OpenFog architecture:Security; Scalability; Open; Autonomy; Programmability; RAS (reliability, availability and serviceability); Agility; and Hierarchy. It also incorporates a glossary for fog computing terms. In July 2018, the IEEE Standards Association announced it had adopted the OpenFog Reference Architecture as the first standard for fog computing.
Read more →
Polyworld

Polyworld is a cross-platform (Linux, Mac OS X) program written by Larry Yaeger to evolve Artificial Intelligence through natural selection and evolutionary algorithms. It uses the Qt graphics toolkit and OpenGL to display a graphical environment in which a population of trapezoid agents search for food, mate, have offspring, and prey on each other. The population is typically only in the hundreds, as each individual is rather complex and the environment consumes considerable computer resources. The graphical environment is necessary since the individuals actually move around the 2-D plane and must be able to "see." Since some basic abilities, like eating carcasses or randomly generated food, seeing other individuals, mating or fighting with them, etc., are possible, a number of interesting behaviours have been observed to spontaneously arise after prolonged evolution, such as cannibalism, predators and prey, and mimicry. Each individual makes decisions based on a neural net using Hebbian learning; the neural net is derived from each individual's genome. The genome does not merely specify the wiring of the neural nets, but also determines their size, speed, color, mutation rate and a number of other factors. The genome is randomly mutated at a set probability, which are also changed in descendant organisms.
Read more →
Probabilistic database

Most real databases contain data whose correctness is uncertain. In order to work with such data, there is a need to quantify the integrity of the data. This is achieved by using probabilistic databases. A probabilistic database is an uncertain database in which the possible worlds have associated probabilities. Probabilistic database management systems are currently an active area of research. "While there are currently no commercial probabilistic database systems, several research prototypes exist..." Probabilistic databases distinguish between the logical data model and the physical representation of the data much like relational databases do in the ANSI-SPARC Architecture. In probabilistic databases this is even more crucial since such databases have to represent very large numbers of possible worlds, often exponential in the size of one world (a classical database), succinctly. == Terminology == In a probabilistic database, each tuple is associated with a probability between 0 and 1, with 0 representing that the data is certainly incorrect, and 1 representing that it is certainly correct. === Possible worlds === A probabilistic database could exist in multiple states. For example, if there is uncertainty about the existence of a tuple in the database, then the database could be in two different states with respect to that tuple—the first state contains the tuple, while the second one does not. Similarly, if an attribute can take one of the values x, y or z, then the database can be in three different states with respect to that attribute. Each of these states is called a possible world. Consider the following database: (Here {b3, b3′, b3′′} denotes that the attribute can take any of the values b3, b3′ or b3′′) Assuming that there is uncertainty about the first tuple, certainty about the second tuple, and uncertainty about the value of attribute B in the third tuple. Then the actual state of the database may or may not contain the first tuple (depending on whether it is correct or not). Similarly, the value of the attribute B may be b3, b3′ or b3′′. Consequently, the possible worlds corresponding to the database are as follows: === Types of Uncertainties === There are essentially two kinds of uncertainties that could exist in a probabilistic database, as described in the table below: By assigning values to random variables associated with the data items, different possible worlds can be represented. == History == The first published use of the term "probabilistic database" was probably in the 1987 VLDB conference paper "The theory of probabilistic databases", by Cavallo and Pittarelli. The title (of the 11 page paper) was intended as a bit of a joke, since David Maier's 600 page monograph, The Theory of Relational Databases, would have been familiar at that time to many of the conference participants and readers of the conference proceedings.
Read more →
TuVox

TuVox is a company that produces VXML-based telephone speech-recognition applications to replace DTMF touch-tone systems for their clients. == History == TuVox was founded in 2001 by Steven S. Pollock and Ashok Khosla, formerly of Apple Computer Corporation and Claris Corporation. Since then, TuVox has grown to over 150 employees and has US offices in Cupertino, California and Boca Raton, Florida as well as international offices in London, Vancouver and Sydney. In 2005, TuVox acquired the customers and hosting facilities of Net-By-Tel. In 2007, the company raised $20m for its speech recognition, and phone menu software. On July 22, 2010, West Interactive — a subsidiary of West Corporation — announced its acquisition of TuVox. == Customers == TuVox clients include: 1-800-Flowers.com, AMC Entertainment, American Airlines, British Airways, M&T Bank, Canon Inc., Gateway, Inc., Motorola, Progress Energy Inc., Telecom New Zealand, Time, Inc., BECU, Virgin America and USAA.
Read more →
Rclone

Rclone is an open source, multi threaded, command line computer program to manage or migrate content on cloud and other high latency storage. Its capabilities include sync, transfer, crypt, cache, union, compress and mount. The rclone website lists supported backends including S3 and Google Drive. Descriptions of rclone often carry the strapline "Rclone syncs your files to cloud storage". Those prior to 2020 include the alternative "Rsync for Cloud Storage". Rclone is well known for its rclone sync and rclone mount commands. It provides further management functions analogous to those ordinarily used for files on local disks, but which tolerate some intermittent and unreliable service. Rclone is commonly used with media servers such as Plex, Emby or Jellyfin to stream content direct from consumer file storage services. Official Ubuntu, Debian, Fedora, Gentoo, Arch, Brew, Chocolatey, and other package managers include rclone. == History == Nick Craig-Wood was inspired by rsync. Concerns about the noise and power costs arising from home computer servers prompted him to embrace cloud storage and he began developing rclone as open source software in 2012 under the name swiftsync. Rclone was promoted to stable version 1.00 in July 2014. In May 2017, Amazon Drive barred new users of rclone and other upload utilities, citing security concerns. Amazon Drive had been advertised as offering unlimited storage for £55 per year. Amazon's AWS S3 service continues to support new rclone users. The original rclone logo was updated in September 2018. In March 2020, Nick Craig-Wood resigned from Memset Ltd, a cloud hosting company he founded, to focus on open source software. Amazon's AWS April 2020 public sector blog explained how the Fred Hutch Cancer Research Center were using rclone in their Motuz tool to migrate very large biomedical research datasets in and out of AWS S3 object stores. In November 2020, rclone was updated to correct a weakness in the way it generated passwords. Passwords for encrypted remotes can be generated randomly by rclone or supplied by the user. In all versions of rclone from 1.49.0 to 1.53.2 the seed value for generated passwords was based on the number of seconds elapsed in the day, and therefore not truly random. CVE-2020-28924 recommended users upgrade to the latest version of rclone and check the passwords protecting their encrypted remotes. Release 1.55 of rclone in March 2021 included features sponsored by CERN and their CS3MESH4EOSC project. The work was EU funded to promote vendor-neutral application programming interfaces and protocols for synchronisation and sharing of academic data on cloud storage. == Backends and commands == Rclone supports the following services as backends. There are others, built on standard protocols such as WebDAV or S3, that work. WebDAV backends do not support rclone functionality dependent on server side checksum or modtime. Remotes are usually defined interactively from these backends, local disk, or memory (as S3), with rclone config. Rclone can further wrap those remotes with one or more of alias, chunk, compress, crypt or union, remotes. Once defined, the remotes are referenced by other rclone commands interchangeably with the local drive. Remote names are followed by a colon to distinguish them from local drives. For example, a remote example_remote containing a folder, or pseudofolder, myfolder is referred to within a command as a path example_remote:/myfolder. Rclone commands directly apply to remotes, or mount them for file access or streaming. With appropriate cache options the mount can be addressed as if a conventional, block level disk. Commands are provided to serve remotes over SFTP, HTTP, WebDAV, FTP and DLNA. Commands can have sub-commands and flags. Filters determine which files on a remote that rclone commands are applied to. rclone rc passes commands or new parameters to existing rclone sessions and has an experimental web browser interface. === Crypt remotes === Rclone's crypt implements encryption of files at rest in cloud storage. It layers an encrypted remote over a pre-existing, cloud or other remote. Crypt is commonly used to encrypt / decrypt media, for streaming, on consumer storage services such as Google Drive. Rclone's configuration file contains the crypt password. The password can be lightly obfuscated, or the whole rclone.conf file can be encrypted. Crypt can either encrypt file content and name, or additionally full paths. In the latter case there is a potential clash with encryption for cloud backends, such as Microsoft OneDrive, having limited path lengths. Crypt remotes do not encrypt object modification time or size. The encryption mechanism for content, name and path is available, for scrutiny, on the rclone website. Key derivation is with scrypt. === Example syntax (Linux) === These examples describe paths and file names but object keys behave similarly. To recursively copy files from directory remote_stuff, at the remote xmpl, to directory stuff in the home folder:- -v enables logging and -P, progress information. By default rclone checks the file integrity (hash) after copy; can retry each file up to three times if the operation is interrupted; uses up to four parallel transfer threads, and does not apply bandwidth throttling. Running the above command again copies any new or changed files at the remote to the local folder but, like default rsync behaviour, will not delete from the local directory, files which have been removed from the remote. To additionally delete files from the local folder which have been removed from the remote - more like the behaviour of rsync with a --delete flag:- And to delete files from the source after they have been transferred to the local directory - more like the behaviour of rsync with a --remove-source-file flag:- To mount the remote directory at a mountpoint in the pre-existing, empty stuff directory in the home directory (the ampersand at the end makes the mount command run as a background process):- Default rclone syntax can be modified. Alternative transfer, filter, conflict and backend specific flags are available. Performance choices include number of concurrent transfer threads; chunk size; bandwidth limit profiling, and cache aggression. == Academic evaluation == In 2018, University of Kentucky researchers published a conference paper comparing use of rclone and other command line, cloud data transfer agents for big data. The paper was published as a result of funding by the National Science Foundation. Later that year, University of Utah's Center for High Performance Computing examined the impact of rclone options on data transfer rates. == Rclone use at HPC research sites == Examples are University of Maryland, Iowa State University, Trinity College Dublin, NYU, BYU, Indiana University, CSC Finland, Utrecht University, University of Nebraska, University of Utah, North Carolina State University, Stony Brook, Tulane University, Washington State University, Georgia Tech, National Institutes of Health, Wharton, Yale, Harvard, Minnesota, Michigan State, Case Western Reserve University, University of South Dakota, Northern Arizona University, University of Pennsylvania, Stanford, University of Southern California, UC Santa Barbara, UC Irvine, UC Berkeley, and SURFnet. == Rclone and cybercrime == May 2020 reports stated rclone had been used by hackers to exploit Diebold Nixdorf ATMs with ProLock ransomware. The FBI issued a Flash Alert MI-000125-MW on May 4, 2020, in relation to the compromise. They issued a further, related alert 20200901–001 in September 2020. Attackers had exfiltrated / encrypted data from organisations involved in healthcare, construction, finance, and legal services. Multiple US government agencies, and industrial entities were affected. Researchers established the hackers spent about a month exploring the breached networks, using rclone to archive stolen data to cloud storage, before encrypting the target system. Reported targets included LaSalle County, and the city of Novi Sad. The FBI warned January 2021, in Private Industry Notification 20210106–001, of extortion activity using Egregor ransomware and rclone. Organisations worldwide had been threatened with public release of exfiltrated data. In some cases rclone had been disguised under the name svchost. Bookseller Barnes & Noble, US retailer Kmart, games developer Ubisoft and the Vancouver metro system have been reported as victims. An April 2021, cybersecurity investigation into SonicWall VPN zero-day vulnerability SNWLID-2021-0001 by FireEye's Mandiant team established attackers UNC2447 used rclone for reconnaissance and exfiltration of victims' files. Cybersecurity and Infrastructure Security Agency Analysis Report AR21-126A confirmed this use of rclone in FiveHands ransomware attacks. A June 2021, Microsoft Security Intelligence Twitter post identified use of rclone in BazaCall cyber attacks. The attackers sent emails e
Read more →
Oasis (Minecraft clone)

Oasis is a 2024 video game that attempts to replicate the 2011 sandbox game Minecraft, run entirely using generative artificial intelligence. The project, which began development in 2022 between the AI company Decart and the computer hardware startup Etched, was released by Decart to the public on October 31, 2024. The AI-driven simulation uses "next-frame prediction" to anticipate player actions based on keyboard and mouse inputs, trained on millions of hours of gameplay footage. Without memory or code, the game often outputs unpredictable changes in scenery and inventory, limiting its functionality as a traditional video game. Critics noted its lack of sound, low frame rate, and "dream-like" appearance, though some praised its unpredictability as entertaining. The project is seen as a potential proof of concept for AI-driven video games. == Creation and gameplay == The demo "proof of concept" version of the game was developed by Israeli San Francisco–based AI company Decart and Silicon Valley hardware startup Etched. The idea originated in 2022 when Robert Wachen, a Harvard graduate and co-founder of Etched, met Dean Leitersdorf, an Israel Institute of Technology graduate and co-founder of Decart. Sharing an interest in OpenAI's GPT-3, they collaborated to create the game, naming it after the setting of the novel and film Ready Player One. It was funded by a $21 million grant from Israeli-American billionaire Oren Zeev and New York–based Sequoia Capital. Decart released the game to the public for free on October 31, 2024. The AI replicates Minecraft's gameplay without code using "next-frame prediction", in which the AI tries to predict what the player will see after each keyboard and mouse input, which it was trained to do on millions of hours of Minecraft footage. The game used Nvidia graphics processing units or GPUs for its demo but plans to transition to more energy-efficient Sohu GPUs, under development by Etched, capable of supporting up to 4K graphics. Etched has also suggested the possibility of making the game open source in the future. Alongside Oasis, the company is co-developing AI-generated video and educational content. == Reception == Upon its launch, many players posted videos of their experience with the game online, which often showed Oasis could not maintain coherent logic in its actions or setting. The game also presented low-quality graphics, running between 360p and 720p consistently at 20 FPS, no in-game sound, and could only be played for five minutes at a time before restarting. These issues led some news outlets to refer to the game as a "nightmarish hallucination", and drawing comparisons to dementia and dreams. Despite the negative reviews, Leitersdorf, as well as a number of commentators, have commented that while the game may have fallen short of replicating Minecraft in its demo launch, it was the first step towards something more advanced, which could one day resemble Minecraft or any other game. Online publication The Backdash commented the game could be a "glimpse at the future of game development", while others like Tom's Hardware expressed doubts a game without code could ever look as good as one with, arguing they fail to capture "the point of what makes games fun—or even coherent". In terms of legality, Decart and Etched did not receive permission from Microsoft to create a copy of their game using generative artificial intelligence. No legal actions have been taken by the latter, however, as artificial intelligence and copyright remains largely vague legally.
Read more →
Model collapse

Model collapse, also known by other names such as "AI inbreeding", "AI cannibalism", "Habsburg AI", and "model autophagy disorder" or "MAD" is a phenomenon noted in artificial intelligence studies, where machine learning models gradually degrade due to errors coming from uncurated synthetic data, or due to training on the outputs of another model such as prior versions of itself. It is unclear to what extent the phenomenon threatens the long-term development of such models, and some techniques have been proposed to mitigate the effect. == Characteristics == Shumailov et al. coined the term to describe two specific stages to the degradation of machine learning models: early model collapse and late model collapse: In early model collapse, the model begins losing information about the tails of the distribution – mostly affecting minority data. Later work highlighted that early model collapse is hard to notice, since overall performance may appear to improve, while the model loses performance on minority data. In late model collapse, the model loses a significant proportion of its performance, confusing concepts and losing most of its variance. == Mechanism == Using synthetic data as training data can lead to issues with the quality and reliability of the trained model. Model collapse occurs for three main reasons: functional approximation errors sampling errors learning errors Importantly, it happens in even the simplest of models, where not all of the error sources are present. In more complex models the errors often compound, leading to faster collapse. == Disagreement over real-world impact == Some researchers and commentators on model collapse warn that the phenomenon could fundamentally threaten future generative AI development: As AI-generated data is shared on the Internet, it will inevitably end up in future training datasets, which are often crawled from the Internet. If training on "slop" (large quantities of unlabeled synthetic data) inevitably leads to model collapse, this could therefore pose a difficult problem. However, recently, other researchers have disagreed with this argument, showing that if synthetic data accumulates alongside human-generated data, model collapse is avoided. The researchers argue that data accumulating over time is a more realistic description of reality than deleting all existing data every year, and that the real-world impact of model collapse may not be as catastrophic as feared. An alternative branch of the literature investigates the use of machine learning detectors and watermarking to identify model generated data and filter it out. == Mathematical models of the phenomenon == === 1D Gaussian model === In 2024, a first attempt has been made at illustrating collapse for the simplest possible model — a single dimensional normal distribution fit using unbiased estimators of mean and variance, computed on samples from the previous generation. To make this more precise, we say that original data follows a normal distribution X 0 ∼ N ( μ , σ 2 ) {\displaystyle X^{0}\sim {\mathcal {N}}(\mu ,\sigma ^{2})} , and we possess M 0 {\displaystyle M_{0}} samples X j 0 {\displaystyle X_{j}^{0}} for j ∈ { 1 , … , M 0 } {\displaystyle j\in {\{\,1,\dots ,M_{0}\,{}\}}} . Denoting a general sample X j i {\displaystyle X_{j}^{i}} as sample j ∈ { 1 , … , M i } {\displaystyle j\in {\{\,1,\dots ,M_{i}\,{}\}}} at generation i {\displaystyle i} , then the next generation model is estimated using the sample mean and variance: μ i + 1 = 1 M i ∑ j X j i ; σ i + 1 2 = 1 M i − 1 ∑ j ( X j i − μ i + 1 ) 2 . {\displaystyle \mu _{i+1}={\frac {1}{M_{i}}}\sum _{j}X_{j}^{i};\quad \sigma _{i+1}^{2}={\frac {1}{M_{i}-1}}\sum _{j}(X_{j}^{i}-\mu _{i+1})^{2}.} Leading to a conditionally normal next generation model X j i + 1 | μ i + 1 , σ i + 1 ∼ N ( μ i + 1 , σ i + 1 2 ) {\displaystyle X_{j}^{i+1}|\mu _{i+1},\;\sigma _{i+1}\sim {\mathcal {N}}(\mu _{i+1},\sigma _{i+1}^{2})} . In theory, this is enough to calculate the full distribution of X j i {\displaystyle X_{j}^{i}} . However, even after the first generation, the full distribution is no longer normal: It follows a variance-gamma distribution. To continue the analysis, instead of writing the probability density function at each generation, it is possible to explicitly construct them in terms of independent random variables using Cochran's theorem. To be precise, μ 1 {\displaystyle \mu _{1}} and σ 1 {\displaystyle \sigma _{1}} are independent, with μ 1 ∼ N ( μ , σ 2 M 0 ) {\displaystyle \mu _{1}\sim {\mathcal {N}}\left(\mu ,{\frac {\sigma ^{2}}{M_{0}}}\right)} and ( M 0 − 1 ) σ 1 2 ∼ σ 2 Γ ( M 0 − 1 2 , 1 2 ) {\displaystyle (M_{0}-1)\,\sigma _{1}^{2}\sim \sigma ^{2}\,\Gamma \left({\frac {M_{0}-1}{2}},{\frac {1}{2}}\right)} , following a Gamma distribution. Denoting with Z {\displaystyle Z} Gaussian random variables distributed according to N ( 0 , 1 ) {\displaystyle {\mathcal {N}}(0,1)} and with S i {\displaystyle S^{i}} random variables distributed with 1 M i − 1 − 1 Γ ( M i − 1 − 1 2 , 1 2 ) {\displaystyle {\frac {1}{M_{i-1}-1}}\Gamma \left({\frac {M_{i-1}-1}{2}},{\frac {1}{2}}\right)} , it turns out to be possible to write samples at each generation as X j 0 = μ + σ Z j 0 , {\textstyle X_{j}^{0}=\mu +\sigma Z_{j}^{0},} X j 1 = μ + σ M 0 Z 1 + σ S 1 Z j 1 , {\textstyle X_{j}^{1}=\mu +{\frac {\sigma }{\sqrt {M_{0}}}}Z^{1}+\sigma {\sqrt {S^{1}}}Z_{j}^{1},} and more generally X j n = μ + σ M 0 Z 1 + σ M 1 S 1 Z 2 + ⋯ + σ M n − 1 S 1 × ⋯ × S n − 1 Z n + σ S 1 × ⋯ × S n Z j n . {\displaystyle X_{j}^{n}=\mu +{\frac {\sigma }{\sqrt {M_{0}}}}Z^{1}+{\frac {\sigma }{\sqrt {M_{1}}}}{\sqrt {S^{1}}}Z^{2}+\dots +{\frac {\sigma }{\sqrt {M_{n-1}}}}{\sqrt {S^{1}\times \dots \times S^{n-1}}}Z^{n}+\sigma {\sqrt {S^{1}\times \dots \times S^{n}}}Z_{j}^{n}.} Note, that these are not joint distributions, as Z n {\displaystyle Z^{n}} and S n {\displaystyle S^{n}} depend directly on Z j n − 1 {\displaystyle Z_{j}^{n-1}} , but when considering X j n {\displaystyle X_{j}^{n}} on its own the formula above provides all the information about the full distribution. To analyse the model collapse, we can first calculate variance and mean of samples at generation n {\displaystyle n} . This would tell us what kind of distributions we expect to arrive at after n {\displaystyle n} generations. It is possible to find its exact value in closed form, but the mean and variance of the square root of gamma distribution are expressed in terms of gamma functions, making the result quite clunky. Following, it is possible to expand all results to second order in each of 1 / M i {\displaystyle 1/M_{i}} , assuming each sample size to be large. It is then possible to show that 1 σ 2 Var ⁡ ( X j n ) = 1 M 0 + 1 M 1 + ⋯ + 1 M n − 1 + 1 + O ( M i − 2 ) . {\displaystyle {\frac {1}{\sigma ^{2}}}\operatorname {Var} (X_{j}^{n})={\frac {1}{M_{0}}}+{\frac {1}{M_{1}}}+\dots +{\frac {1}{M_{n-1}}}+1+{\mathcal {O}}\left(M_{i}^{-2}\right).} And if all sample sizes M i = M {\displaystyle M_{i}=M} are constant, this diverges linearly as n → ∞ {\displaystyle n\to \infty } : Var ⁡ ( X j n ) = σ 2 ( 1 + n M ) ; E ( X j n ) = μ . {\displaystyle \operatorname {Var} (X_{j}^{n})=\sigma ^{2}\left(1+{\frac {n}{M}}\right);\quad \mathbb {E} (X_{j}^{n})=\mu .} This is the same scaling as for a single dimensional Gaussian random walk. However, divergence of the variance of X j n {\displaystyle X_{j}^{n}} does not directly provide any information about the corresponding estimates of μ n + 1 {\displaystyle \mu _{n+1}} and σ n + 1 {\displaystyle \sigma _{n+1}} , particularly how different they are from the original μ {\displaystyle \mu } and σ {\displaystyle \sigma } . It turns out to be possible to calculate the distance between the true distribution and the approximated distribution at step n + 1 {\displaystyle n+1} , using the Wasserstein-2 distance (which is also sometimes referred to as risk): E [ W 2 2 ( N ( μ , σ 2 ) , N ( μ n + 1 , σ n + 1 2 ) ) ] = 3 2 σ 2 ( 1 M 0 + 1 M 1 + ⋯ + 1 M n ) + O ( M i − 2 ) , {\displaystyle \mathbb {E} \left[\mathbb {W} _{2}^{2}\left({\mathcal {N}}(\mu ,\sigma ^{2}),{\mathcal {N}}(\mu _{n+1},\sigma _{n+1}^{2})\right)\right]={\frac {3}{2}}\sigma ^{2}\left({\frac {1}{M_{0}}}+{\frac {1}{M_{1}}}+\dots +{\frac {1}{M_{n}}}\right)+{\mathcal {O}}\left(M_{i}^{-2}\right),} Var ⁡ [ W 2 2 ( N ( μ , σ 2 ) , N ( μ n + 1 , σ n + 1 2 ) ) ] = 1 2 σ 4 ( 3 M 0 2 + 3 M 1 2 + ⋯ + 3 M n 2 + ∑ i ≠ j 4 M i M j ) + O ( M i − 3 ) . {\displaystyle \operatorname {Var} \left[\mathbb {W} _{2}^{2}\left({\mathcal {N}}(\mu ,\sigma ^{2}),{\mathcal {N}}(\mu _{n+1},\sigma _{n+1}^{2})\right)\right]={\frac {1}{2}}\sigma ^{4}\left({\frac {3}{M_{0}^{2}}}+{\frac {3}{M_{1}^{2}}}+\dots +{\frac {3}{M_{n}^{2}}}+\sum _{i\neq j}{\frac {4}{M_{i}M_{j}}}\right)+{\mathcal {O}}\left(M_{i}^{-3}\right).} This directly shows why model collapse occurs in this simple model. Due to errors from re-sampling the approximated distribution, each generation ends up corresponding to a
Read more →