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  Ampere Computing

  Ampere Computing LLC is an American fabless semiconductor company that designs ARM-based central processing units (CPUs) with high core counts for use in cloud computing and data center environments. Founded in 2017 by former Intel president Renée James, the company is headquartered in Santa Clara, California, and operates as an independent subsidiary of SoftBank Group since November 2025. == History == Ampere Computing was founded in fall 2017 by Renée James, ex-President of Intel, with funding from The Carlyle Group. James acquired a team from MACOM Technology Solutions (formerly AppliedMicro) in addition to several industry hires to start the company. Ampere Computing is an ARM architecture licensee and develops its own server microprocessors. Ampere fabricates its products at TSMC. In April 2019, Ampere announced its second major investment round, including investment from Arm Holdings and Oracle Corporation. In June 2019, Nvidia announced a partnership with Ampere to bring support for Compute Unified Device Architecture (CUDA). In November 2019, Nvidia announced a reference design platform for graphics processing unit (GPU)-accelerated ARM-based servers including Ampere. In the first half of 2020, Ampere announced Ampere Altra, an 80-core processor, and Ampere Altra Max, a 128-core processor, without the use of simultaneous multithreading. In March 2020, the company announced a partnership with Oracle. In September 2020, Oracle said it would launch bare-metal and virtual machine instances in early 2021 based on Ampere Altra. In November 2020, Ampere was named one of the top 10 hottest semiconductor startups by CRN. In May 2021, the company announced a partnership with Microsoft. In April 2022, Ampere said that it had filed a confidential prospectus with the U.S. Securities and Exchange Commission, signaling its intent to go public. In June 2022, HPE announced their Gen11 ProLiant system would use Ampere Altra and Ampere Altra Max Cloud Native Processors. In July 2022, Google announced T2A instances using Ampere Altra in the Google cloud and in August 2022 Microsoft announced their instances of Ampere running in Azure. On March 19, 2025, investment holding company SoftBank Group announced it will acquire Ampere Computing for $6.5 billion. The deal finalized in November 2025, with Ampere remaining as an independent subsidiary with its headquarters in Santa Clara, California. == Products == Ampere develops ARM-based computer processors and CPU cores under their Altra brands. These are used in databases, media encoding, web services, network acceleration, mobile gaming, AI inference processing, and other applications and programs that need to scale. On February 5, 2018, Ampere announced the eMAG 8180 featuring 32x Skylark cores fabricated on TSMC's 16FF+ process. It supports a turbo of up to 3.3 GHz with a TDP of 125 W, 8ch 64-bit DDR4, up to 1 TB DDR4 per socket, and 42x PCIe 3.0 Lanes. The Skylark cores were based on AppliedMicro's X-Gene 3. Packet offers servers with the eMAG 8180 and 128 GB DRAM, 480 GB SSD, and 2x 10 Gbit/s networking. On September 19, 2018, Ampere announced the availability of a version featuring 16x Skylark cores. === 2020 === On March 3, 2020, Ampere announced the Ampere Altra featuring 80 cores fabricated on TSMC's N7 process for hyperscale computing. It was the first server-grade processor to include 80 cores and the Q80-30 conserves power by running at 161 W in use. The cores are semi-custom Arm Neoverse N1 cores with Ampere modifications. It supports a frequency of up to 3.3 GHz with TDP of 250 W, 8ch 72-bit DDR4, up to 4 TB DDR4-3200 per socket, 128x PCIe 4.0 Lanes, 1 MB L2 per core and 32 MB SLC. Ampere also announced their roadmap with Ampere Altra Max (2021) in development and AmpereOne (2022) defined. === 2021 === The 128-core Altra Max was released in 2021 and targeted hyperscale cloud providers. It uses the same server socket and platforms as Ampere Altra, and both products have one thread per core. The Altra Max CPUs provide 128 Arm v8.2+ cores per chip and run up to 3.0 GHz. They also support eight channels of DDR4-3200 memory and 128 lanes of PCIe Gen4. Also in 2021, Oracle launched its Oracle Cloud Infrastructure (OCI) using Ampere Altra processors. === 2022 === In February 2022, Ampere and Rigetti Computing announced a strategic partnership to create hybrid quantum-classical computers. The companies will combine Ampere's Altra Max CPUs with Rigetti's Quantum Processing Units (QPU) in cloud-based High-Performance Computing (HPC) environments. In April, Microsoft previewed its Azure Virtual Machines running on the Ampere Altra. The VMs run scale-out workloads, web servers, application servers, open source databases, cloud native .NET applications, Java applications, gaming servers, media servers, and other processes. In May, Ampere announced the sampling of AmpereOne CPUs, 5 nanometer chips based on its in-house Ampere-developed core. AmpereOne will add support for DDR5 main memory and PCIe Gen5 peripherals. On June 28, 2022, HPE became first tier-one server provider to offer compute with optimized cloud-native silicon for service providers and enterprises embracing cloud-native development with new line of HPE ProLiant RL Gen11 servers, using Ampere® Altra® and Ampere® Altra® Max processors, delivering high performance and power efficiency. === 2023 === During April 2023, Ampere released the Altra developer's kit, an IoT Prototype Kit based on Ampere Altra, aimed at cloud developers, available in 32-core, 64-core, and 80-core formats. === 2024 === In May 2024, Ampere updated its AmpereOne roadmap to 256 cores and announced a joint effort with Qualcomm on CPUs and accelerators. == Customers == Ampere's customers include Microsoft Azure, Tencent Cloud, Oracle, ByteDance, Hewlett Packard Enterprise (HPE), Cloudflare, Equinix, Kingsoft Cloud, Meituan, Scaleway, UCloud, Foxconn Industrial Internet, Gigabyte, Inspur, Cruise, Hetzner, Project Ronin, Wiwynn and Google Cloud Platform Cruise uses an Ampere Altra variant for its autonomous driving unit. The CPU was selected because of its throughput and low power consumption. In 2021, Oracle, Microsoft, Tencent, and ByteDance committed to using Ampere's customized chips, first announced in May. In April 2022, Microsoft previewed Ampere Altra processors in its new Azure D-and E- series virtual machines. The Dpsv5 series is built for Linux enterprise application types, and the Epsv5 series is for memory-intensive Linux workloads. They provide up to 64 vCPUs, include VM sizes with 2GiB, 4GiB, and 8GiB per vCPU memory configurations, up to 40 Gbit/s networking, and high-performance local SSD storage. In 2022, Microsoft's Ampere Altra-based Azure servers became the first cloud solution provider server to be Arm SystemReady SR certified. The Azure VMs, powered by Altra processors, were also the first to be SystemReady Virtual Environment standard certified. SystemReady defines a set of firmware and hardware standards as a baseline for system development for software developers, original equipment vendors, and chipmakers.

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  Information Harvesting

  Information Harvesting (IH) was an early data mining product from the 1990s. It was invented by Ralphe Wiggins and produced by the Ryan Corp, later Information Harvesting Inc., of Cambridge, Massachusetts. Wiggins had a background in genetic algorithms and fuzzy logic. IH sought to infer rules from sets of data. It did this first by classifying various input variables into one of a number of bins, thereby putting some structure on the continuous variables in the input. IH then proceeds to generate rules, trading off generalization against memorization, that will infer the value of the prediction variable, possibly creating many levels of rules in the process. It included strategies for checking if overfitting took place and, if so, correcting for it. Because of its strategies for correcting for overfitting by considering more data, and refining the rules based on that data, IH might also be considered to be a form of machine learning. The advantage of IH, as compared with other data mining products of its time and even later, was that it provided a mechanism for finding multiple rules that would classify the data and determining, according to set criteria, the best rules to use.

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  FERET (facial recognition technology)

  The Facial Recognition Technology (FERET) program was a government-sponsored project that aimed to create a large, automatic face-recognition system for intelligence, security, and law enforcement purposes. The program began in 1993 under the combined leadership of Dr. Harry Wechsler at George Mason University (GMU) and Dr. Jonathon Phillips at the Army Research Laboratory (ARL) in Adelphi, Maryland and resulted in the development of the Facial Recognition Technology (FERET) database. The goal of the FERET program was to advance the field of face recognition technology by establishing a common database of facial imagery for researchers to use and setting a performance baseline for face-recognition algorithms. Potential areas where this face-recognition technology could be used include: Automated searching of mug books using surveillance photos Controlling access to restricted facilities or equipment Checking the credentials of personnel for background and security clearances Monitoring airports, border crossings, and secure manufacturing facilities for particular individuals Finding and logging multiple appearances of individuals over time in surveillance videos Verifying identities at ATM machines Searching photo ID records for fraud detection The FERET database has been used by more than 460 research groups and is currently managed by the National Institute of Standards and Technology (NIST). By 2017, the FERET database has been used to train artificial intelligence programs and computer vision algorithms to identify and sort faces. == History == The origin of facial recognition technology is largely attributed to Woodrow Wilson Bledsoe and his work in the 1960s, when he developed a system to identify faces from a database of thousands of photographs. The FERET program first began as a way to unify a large body of face-recognition technology research under a standard database. Before the program's inception, most researchers created their own facial imagery database that was attuned to their own specific area of study. These personal databases were small and usually consisted of images from less than 50 individuals. The only notable exceptions were the following: Alex Pentland’s database of around 7500 facial images at the Massachusetts Institute of Technology (MIT) Joseph Wilder's database of around 250 individuals at Rutgers University Christoph von der Malsburg’s database of around 100 facial images at the University of Southern California (USC) The lack of a common database made it difficult to compare the results of face recognition studies in the scientific literature because each report involved different assumptions, scoring methods, and images. Most of the papers that were published did not use images from a common database nor follow a standard testing protocol. As a result, researchers were unable to make informed comparisons between the performances of different face-recognition algorithms. In September 1993, the FERET program was spearheaded by Dr. Harry Wechsler and Dr. Jonathon Phillips under the sponsorship of the U.S. Department of Defense Counterdrug Technology Development Program through DARPA with ARL serving as technical agent. === Phase I === The first facial images for the FERET database were collected from August 1993 to December 1994, a time period known as Phase I. The pictures were initially taken with a 35-mm camera at both GMU and ARL facilities, and the same physical setup was used in each photography session to keep the images consistent. For each individual, the pictures were taken in sets, including two frontal views, a right and left profile, a right and left quarter profile, a right and left half profile, and sometimes at five extra locations. Therefore, a set of images consisted of 5 to 11 images per person. At the end of Phase I, the FERET database had collected 673 sets of images, resulting in over 5000 total images. At the end of Phase I, five organizations were given the opportunity to test their face-recognition algorithm on the newly created FERET database in order to compare how they performed against each other. There five principal investigators were: MIT, led by Alex Pentland Rutgers University, led by Joseph Wilder The Analytic Science Company (TASC), led by Gale Gordon The University of Illinois at Chicago (UIC) and the University of Illinois at Urbana-Champaign, led by Lewis Sadler and Thomas Huang USC, led by Christoph von der Malsburg During this evaluation, three different automatic tests were given to the principal investigators without human intervention: The large gallery test, which served to baseline how algorithms performed against a database when it has not been properly tuned. The false-alarm test, which tested how well the algorithm monitored an airport for suspected terrorists. The rotation test, which measured how well the algorithm performed when the images of an individual in the gallery had different poses compared to those in the probe set. For most of the test trials, the algorithms developed by USC and MIT managed to outperform the other three algorithms for the Phase I evaluation. === Phase II === Phase II began after Phase I, and during this time, the FERET database acquired more sets of facial images. By the start of the Phase II evaluation in March 1995, the database contained 1109 sets of images for a total of 8525 images of 884 individuals. During the second evaluation, the same algorithms from the Phase I evaluation were given a single test. However, the database now contained significantly more duplicate images (463, compared to the previous 60), making the test more challenging. === Phase III === Afterwards, the FERET program entered Phase III where another 456 sets of facial images were added to the database. The Phase III evaluation, which took place in September 1996, aimed to not only gauge the progress of the algorithms since the Phase I assessment but also identify the strengths and weaknesses of each algorithm and determine future objectives for research. By the end of 1996, the FERET database had accumulated a total of 14,126 facial images pertaining to 1199 different individuals as well as 365 duplicate sets of images. As a result of the FERET program, researchers were able to establish a common baseline for comparing different face-recognition algorithms and create a large standard database of facial images that is open for research. In 2003, DARPA released a high-resolution, 24-bit color version of the images in the FERET database (existing reference).

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  Polynomial kernel

  In machine learning, the polynomial kernel is a kernel function commonly used with support vector machines (SVMs) and other kernelized models, that represents the similarity of vectors (training samples) in a feature space over polynomials of the original variables, allowing learning of non-linear models. Intuitively, the polynomial kernel looks not only at the given features of input samples to determine their similarity, but also combinations of these. In the context of regression analysis, such combinations are known as interaction features. The (implicit) feature space of a polynomial kernel is equivalent to that of polynomial regression, but without the combinatorial blowup in the number of parameters to be learned. When the input features are binary-valued (booleans), then the features correspond to logical conjunctions of input features. == Definition == For degree-d polynomials, the polynomial kernel is defined as K ( x , y ) = ( x T y + c ) d {\displaystyle K(\mathbf {x} ,\mathbf {y} )=(\mathbf {x} ^{\mathsf {T}}\mathbf {y} +c)^{d}} where x and y are vectors of size n in the input space, i.e. vectors of features computed from training or test samples and c ≥ 0 is a free parameter trading off the influence of higher-order versus lower-order terms in the polynomial. When c = 0, the kernel is called homogeneous. (A further generalized polykernel divides xTy by a user-specified scalar parameter a.) As a kernel, K corresponds to an inner product in a feature space based on some mapping φ: K ( x , y ) = ⟨ φ ( x ) , φ ( y ) ⟩ {\displaystyle K(\mathbf {x} ,\mathbf {y} )=\langle \varphi (\mathbf {x} ),\varphi (\mathbf {y} )\rangle } The nature of φ can be seen from an example. Let d = 2, so we get the special case of the quadratic kernel. After using the multinomial theorem (twice—the outermost application is the binomial theorem) and regrouping, K ( x , y ) = ( ∑ i = 1 n x i y i + c ) 2 = ∑ i = 1 n ( x i 2 ) ( y i 2 ) + ∑ i = 2 n ∑ j = 1 i − 1 ( 2 x i x j ) ( 2 y i y j ) + ∑ i = 1 n ( 2 c x i ) ( 2 c y i ) + c 2 {\displaystyle K(\mathbf {x} ,\mathbf {y} )=\left(\sum _{i=1}^{n}x_{i}y_{i}+c\right)^{2}=\sum _{i=1}^{n}\left(x_{i}^{2}\right)\left(y_{i}^{2}\right)+\sum _{i=2}^{n}\sum _{j=1}^{i-1}\left({\sqrt {2}}x_{i}x_{j}\right)\left({\sqrt {2}}y_{i}y_{j}\right)+\sum _{i=1}^{n}\left({\sqrt {2c}}x_{i}\right)\left({\sqrt {2c}}y_{i}\right)+c^{2}} From this it follows that the feature map is given by: φ ( x ) = ( x n 2 , … , x 1 2 , 2 x n x n − 1 , … , 2 x n x 1 , 2 x n − 1 x n − 2 , … , 2 x n − 1 x 1 , … , 2 x 2 x 1 , 2 c x n , … , 2 c x 1 , c ) {\displaystyle \varphi (x)=\left(x_{n}^{2},\ldots ,x_{1}^{2},{\sqrt {2}}x_{n}x_{n-1},\ldots ,{\sqrt {2}}x_{n}x_{1},{\sqrt {2}}x_{n-1}x_{n-2},\ldots ,{\sqrt {2}}x_{n-1}x_{1},\ldots ,{\sqrt {2}}x_{2}x_{1},{\sqrt {2c}}x_{n},\ldots ,{\sqrt {2c}}x_{1},c\right)} generalizing for ( x T y + c ) d {\displaystyle \left(\mathbf {x} ^{T}\mathbf {y} +c\right)^{d}} , where x ∈ R n {\displaystyle \mathbf {x} \in \mathbb {R} ^{n}} , y ∈ R n {\displaystyle \mathbf {y} \in \mathbb {R} ^{n}} and applying the multinomial theorem: ( x T y + c ) d = ∑ j 1 + j 2 + ⋯ + j n + 1 = d d ! j 1 ! ⋯ j n ! j n + 1 ! x 1 j 1 ⋯ x n j n c j n + 1 d ! j 1 ! ⋯ j n ! j n + 1 ! y 1 j 1 ⋯ y n j n c j n + 1 = φ ( x ) T φ ( y ) {\displaystyle {\begin{alignedat}{2}\left(\mathbf {x} ^{T}\mathbf {y} +c\right)^{d}&=\sum _{j_{1}+j_{2}+\dots +j_{n+1}=d}{\frac {\sqrt {d!}}{\sqrt {j_{1}!\cdots j_{n}!j_{n+1}!}}}x_{1}^{j_{1}}\cdots x_{n}^{j_{n}}{\sqrt {c}}^{j_{n+1}}{\frac {\sqrt {d!}}{\sqrt {j_{1}!\cdots j_{n}!j_{n+1}!}}}y_{1}^{j_{1}}\cdots y_{n}^{j_{n}}{\sqrt {c}}^{j_{n+1}}\\&=\varphi (\mathbf {x} )^{T}\varphi (\mathbf {y} )\end{alignedat}}} The last summation has l d = ( n + d d ) {\displaystyle l_{d}={\tbinom {n+d}{d}}} elements, so that: φ ( x ) = ( a 1 , … , a l , … , a l d ) {\displaystyle \varphi (\mathbf {x} )=\left(a_{1},\dots ,a_{l},\dots ,a_{l_{d}}\right)} where l = ( j 1 , j 2 , . . . , j n , j n + 1 ) {\displaystyle l=(j_{1},j_{2},...,j_{n},j_{n+1})} and a l = d ! j 1 ! ⋯ j n ! j n + 1 ! x 1 j 1 ⋯ x n j n c j n + 1 | j 1 + j 2 + ⋯ + j n + j n + 1 = d {\displaystyle a_{l}={\frac {\sqrt {d!}}{\sqrt {j_{1}!\cdots j_{n}!j_{n+1}!}}}x_{1}^{j_{1}}\cdots x_{n}^{j_{n}}{\sqrt {c}}^{j_{n+1}}\quad |\quad j_{1}+j_{2}+\dots +j_{n}+j_{n+1}=d} == Practical use == Although the RBF kernel is more popular in SVM classification than the polynomial kernel, the latter is quite popular in natural language processing (NLP). The most common degree is d = 2 (quadratic), since larger degrees tend to overfit on NLP problems. Various ways of computing the polynomial kernel (both exact and approximate) have been devised as alternatives to the usual non-linear SVM training algorithms, including: full expansion of the kernel prior to training/testing with a linear SVM, i.e. full computation of the mapping φ as in polynomial regression; basket mining (using a variant of the apriori algorithm) for the most commonly occurring feature conjunctions in a training set to produce an approximate expansion; inverted indexing of support vectors. One problem with the polynomial kernel is that it may suffer from numerical instability: when xTy + c < 1, K(x, y) = (xTy + c)d tends to zero with increasing d, whereas when xTy + c > 1, K(x, y) tends to infinity.

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  Resisting AI

  Resisting AI: An Anti-fascist Approach to Artificial Intelligence is a book on artificial intelligence (AI) by Dan McQuillan, published in 2022 by Bristol University Press. == Content == Resisting AI takes the form of an extended essay, which contrasts optimistic visions about AI's potential by arguing that AI may best be seen as a continuation and reinforcement of bureaucratic forms of discrimination and violence, ultimately fostering authoritarian outcomes. For McQuillan, AI's promise of objective calculability is antithetical to an egalitarian and just society. McQuillan uses the expression "AI violence" to describe how – based on opaque algorithms – various actors can discriminate against categories of people in accessing jobs, loans, medical care, and other benefits. The book suggests that AI has a political resonance with soft eugenic approaches to the valuation of life by modern welfare states, and that AI exhibits eugenic features in its underlying logic, as well as in its technical operations. The parallel is with historical eugenicists achieving saving to the state by sterilizing defectives so the state would not have to care for their offspring. The analysis of McQuillan goes beyond the known critique of AI systems fostering precarious labour markets, addressing "necropolitics", the politics of who is entitled to live, and who to die. Although McQuillan offers a brief history of machine learning at the beginning of the book – with its need for "hidden and undercompensated labour", he is concerned more with the social impacts of AI rather than with its technical aspects. McQuillan sees AI as the continuation of existing bureaucratic systems that already marginalize vulnerable groups – aggravated by the fact that AI systems trained on existing data are likely to reinforce existing discriminations, e.g. in attempting to optimize welfare distribution based on existing data patterns, ultimately creating a system of "self-reinforcing social profiling". In elaborating on the continuation between existing bureaucratic violence and AI, McQuillan connects to Hannah Arendt's concept of the thoughtless bureaucrat in Eichmann in Jerusalem: A Report on the Banality of Evil, which now becomes the algorithm that, lacking intent, cannot be accountable, and is thus endowed with an "algorithmic thoughtlessness". McQuillan defends the "fascist" in the title of the work by arguing that while not all AI is fascist, this emerging technology of control may end up being deployed by fascist or authoritarian regimes. For McQuillan, AI can support the diffusion of states of exception, as a technology impossible to properly regulate and a mechanism for multiplying exceptions more widely. An example of a scenario where AI systems of surveillance could bring discrimination to a new high is the initiative to create LGBT-free zones in Poland. Skeptical of ethical regulations to control the technology, McQuillan suggests people's councils and workers' councils, and other forms of citizens' agency to resist AI. A chapter titled "Post-Machine Learning" makes an appeal for resistance via currents of thought from feminist science (standpoint theory), post-normal science (extended peer communities), and new materialism; McQuillan encourages the reader to question the meaning of "objectivity" and calls for the necessity of alternative ways of knowing. Among the virtuous examples of resistance – possibly to be adopted by the AI workers themselves – McQuillan notes the Lucas Plan of the workers of Lucas Aerospace Corporation, in which a workforce declared redundant took control, reorienting the enterprise toward useful products. McQuillan advocates for what he calls decomputing, an opposition to the sweeping application and expansion of artificial intelligence. Similar to degrowth, the approach criticizes AI as an outgrowth of the systemic issues within capitalist systems. McQuillan argues that a different future is possible, in which distance between people is reduced rather than increased through AI intermediaries. The work of McQuillan warns against "watered-down forms of engagement" with AI, such as citizen juries, which superficially look like democratic deliberation but may actually obscure important decisions about AI that are outside the purview of the engagement situation (McQuillan 2022, 128). In an interview about the book, McQuillan describes himself as an "AI abolitionist". == Reception == The book has been praised for how it "masterfully disassembles AI as an epistemological, social, and political paradigm". On the critical side, a review in the academic journal Justice, Power and Resistance took exception to the "nightmarish visions of Big Brother" offered by McQuillan, and argued that while many elements of AI may pose concern, a critique should not be based on a caricature of what AI is, concluding that McQuillan's work is "less of a theory and more of a Manifesto". Another review notes "a disconnect between the technical aspects of AI and the socio-political analysis McQuillan provides." Although the book was published before the ChatGPT and large language model debate heated up, the book has not lost relevance to the AI discussion. It is noted for suggesting a link between beliefs in artificial intelligence and beliefs in a racialised and gendered visions of intelligence overall, whereby a certain type of rational, measurable intelligence is privileged, leading to "historical notions of hierarchies of being". The blog Reboot praised McQuillan for offering a theory of harm of AI (why AI could end up hurting people and society) that does not just encourage tackling in isolation specific predicted problems with AI-centric systems: bias, non-inclusiveness, exploitativeness, environmental destructiveness, opacity, and non-contestability. For educational policies could also look at AI following the reading of McQuillan: In his book Resisting AI, Dan McQuillan argues that "When we're thinking about the actuality of AI, we can't separate the calculations in the code from the social context of its application" .... McQuillan's particular concern is how many contemporary applications of AI are amplifying existing inequalities and injustices as well as deepening social divisions and instabilities. His book makes a powerful case for anticipating these effects and actively resisting them for the good of societies. Videos and podcasts with an interest in AI and emerging technology have discussed the book.

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  Analogical modeling

  Analogical modeling (AM) is a formal theory of exemplar based analogical reasoning, proposed by Royal Skousen, professor of Linguistics and English language at Brigham Young University in Provo, Utah. It is applicable to language modeling and other categorization tasks. Analogical modeling is related to connectionism and nearest neighbor approaches, in that it is data-based rather than abstraction-based; but it is distinguished by its ability to cope with imperfect datasets (such as caused by simulated short term memory limits) and to base predictions on all relevant segments of the dataset, whether near or far. In language modeling, AM has successfully predicted empirically valid forms for which no theoretical explanation was known (see the discussion of Finnish morphology in Skousen et al. 2002). == Implementation == === Overview === An exemplar-based model consists of a general-purpose modeling engine and a problem-specific dataset. Within the dataset, each exemplar (a case to be reasoned from, or an informative past experience) appears as a feature vector: a row of values for the set of parameters that define the problem. For example, in a spelling-to-sound task, the feature vector might consist of the letters of a word. Each exemplar in the dataset is stored with an outcome, such as a phoneme or phone to be generated. When the model is presented with a novel situation (in the form of an outcome-less feature vector), the engine algorithmically sorts the dataset to find exemplars that helpfully resemble it, and selects one, whose outcome is the model's prediction. The particulars of the algorithm distinguish one exemplar-based modeling system from another. In AM, we think of the feature values as characterizing a context, and the outcome as a behavior that occurs within that context. Accordingly, the novel situation is known as the given context. Given the known features of the context, the AM engine systematically generates all contexts that include it (all of its supracontexts), and extracts from the dataset the exemplars that belong to each. The engine then discards those supracontexts whose outcomes are inconsistent (this measure of consistency will be discussed further below), leaving an analogical set of supracontexts, and probabilistically selects an exemplar from the analogical set with a bias toward those in large supracontexts. This multilevel search exponentially magnifies the likelihood of a behavior's being predicted as it occurs reliably in settings that specifically resemble the given context. === Analogical modeling in detail === AM performs the same process for each case it is asked to evaluate. The given context, consisting of n variables, is used as a template to generate 2 n {\displaystyle 2^{n}} supracontexts. Each supracontext is a set of exemplars in which one or more variables have the same values that they do in the given context, and the other variables are ignored. In effect, each is a view of the data, created by filtering for some criteria of similarity to the given context, and the total set of supracontexts exhausts all such views. Alternatively, each supracontext is a theory of the task or a proposed rule whose predictive power needs to be evaluated. It is important to note that the supracontexts are not equal peers one with another; they are arranged by their distance from the given context, forming a hierarchy. If a supracontext specifies all of the variables that another one does and more, it is a subcontext of that other one, and it lies closer to the given context. (The hierarchy is not strictly branching; each supracontext can itself be a subcontext of several others, and can have several subcontexts.) This hierarchy becomes significant in the next step of the algorithm. The engine now chooses the analogical set from among the supracontexts. A supracontext may contain exemplars that only exhibit one behavior; it is deterministically homogeneous and is included. It is a view of the data that displays regularity, or a relevant theory that has never yet been disproven. A supracontext may exhibit several behaviors, but contain no exemplars that occur in any more specific supracontext (that is, in any of its subcontexts); in this case it is non-deterministically homogeneous and is included. Here there is no great evidence that a systematic behavior occurs, but also no counterargument. Finally, a supracontext may be heterogeneous, meaning that it exhibits behaviors that are found in a subcontext (closer to the given context), and also behaviors that are not. Where the ambiguous behavior of the nondeterministically homogeneous supracontext was accepted, this is rejected because the intervening subcontext demonstrates that there is a better theory to be found. The heterogeneous supracontext is therefore excluded. This guarantees that we see an increase in meaningfully consistent behavior in the analogical set as we approach the given context. With the analogical set chosen, each appearance of an exemplar (for a given exemplar may appear in several of the analogical supracontexts) is given a pointer to every other appearance of an exemplar within its supracontexts. One of these pointers is then selected at random and followed, and the exemplar to which it points provides the outcome. This gives each supracontext an importance proportional to the square of its size, and makes each exemplar likely to be selected in direct proportion to the sum of the sizes of all analogically consistent supracontexts in which it appears. Then, of course, the probability of predicting a particular outcome is proportional to the summed probabilities of all the exemplars that support it. (Skousen 2002, in Skousen et al. 2002, pp. 11–25, and Skousen 2003, both passim) === Formulas === Given a context with n {\displaystyle n} elements: total number of pairings: n 2 {\displaystyle n^{2}} number of agreements for outcome i: n i 2 {\displaystyle n_{i}^{2}} number of disagreements for outcome i: n i ( n − n i ) {\displaystyle n_{i}(n-n_{i})} total number of agreements: ∑ n i 2 {\displaystyle \sum {n_{i}^{2}}} total number of disagreements: ∑ n i ( n − n i ) = n 2 − ∑ n i 2 {\displaystyle \sum {n_{i}(n-n_{i})}=n^{2}-\sum {n_{i}^{2}}} === Example === This terminology is best understood through an example. In the example used in the second chapter of Skousen (1989), each context consists of three variables with potential values 0-3 Variable 1: 0,1,2,3 Variable 2: 0,1,2,3 Variable 3: 0,1,2,3 The two outcomes for the dataset are e and r, and the exemplars are: 3 1 0 e 0 3 2 r 2 1 0 r 2 1 2 r 3 1 1 r We define a network of pointers like so: The solid lines represent pointers between exemplars with matching outcomes; the dotted lines represent pointers between exemplars with non-matching outcomes. The statistics for this example are as follows: n = 5 {\displaystyle n=5} n r = 4 {\displaystyle n_{r}=4} n e = 1 {\displaystyle n_{e}=1} total number of pairings: n 2 = 25 {\displaystyle n^{2}=25} number of agreements for outcome r: n r 2 = 16 {\displaystyle n_{r}^{2}=16} number of agreements for outcome e: n e 2 = 1 {\displaystyle n_{e}^{2}=1} number of disagreements for outcome r: n r ( n − n r ) = 4 {\displaystyle n_{r}(n-n_{r})=4} number of disagreements for outcome e: n e ( n − n e ) = 4 {\displaystyle n_{e}(n-n_{e})=4} total number of agreements: n r 2 + n e 2 = 17 {\displaystyle n_{r}^{2}+n_{e}^{2}=17} total number of disagreements: n r ( n − n r ) + n e ( n − n e ) = n 2 − ( n r 2 + n e 2 ) = 8 {\displaystyle n_{r}(n-n_{r})+n_{e}(n-n_{e})=n^{2}-(n_{r}^{2}+n_{e}^{2})=8} uncertainty or fraction of disagreement: 8 / 25 = .32 {\displaystyle 8/25=.32} Behavior can only be predicted for a given context; in this example, let us predict the outcome for the context "3 1 2". To do this, we first find all of the contexts containing the given context; these contexts are called supracontexts. We find the supracontexts by systematically eliminating the variables in the given context; with m variables, there will generally be 2 m {\displaystyle 2^{m}} supracontexts. The following table lists each of the sub- and supracontexts; x means "not x", and - means "anything". These contexts are shown in the venn diagram below: The next step is to determine which exemplars belong to which contexts in order to determine which of the contexts are homogeneous. The table below shows each of the subcontexts, their behavior in terms of the given exemplars, and the number of disagreements within the behavior: Analyzing the subcontexts in the table above, we see that there is only 1 subcontext with any disagreements: "3 1 2", which in the dataset consists of "3 1 0 e" and "3 1 1 r". There are 2 disagreements in this subcontext; 1 pointing from each of the exemplars to the other (see the pointer network pictured above). Therefore, only supracontexts containing this subcontext will contain any disagreements. We use a simple rule to identify the homogeneous supraco

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  Lattice Miner

  Lattice Miner is a formal concept analysis software tool for the construction, visualization and manipulation of concept lattices. It allows the generation of formal concepts and association rules as well as the transformation of formal contexts via apposition, subposition, reduction and object/attribute generalization, and the manipulation of concept lattices via approximation, projection and selection. Lattice Miner allows also the drawing of nested line diagrams. == Introduction == Formal concept analysis (FCA) is a branch of applied mathematics based on the formalization of concept and concept hierarchy and mainly used as a framework for conceptual clustering and rule mining. Over the last two decades, a collection of tools have emerged to help FCA users visualize and analyze concept lattices. They range from the earliest DOS-based implementations (e.g., ConImp and GLAD) to more recent implementations in Java like ToscanaJ, Galicia, ConExp and Coron. A main issue in the development of FCA tools is to visualize large concept lattices and provide efficient mechanisms to highlight patterns (e.g., concepts, associations) that could be relevant to the user. The initial objective of the FCA tool called Lattice Miner was to focus on visualization mechanisms for the representation of concept lattices, including nested line diagrams. Later on, many other interesting features were integrated into the tool. == Functional architecture of Lattice Miner == Lattice Miner is a Java-based platform whose functions are articulated around a core. The Lattice Miner core provides all low-level operations and structures for the representation and manipulation of contexts, lattices and association rules. Mainly, the core of Lattice Miner consists of three modules: context, concept and association rule modules. The user interface offers a context editor and concept lattice manipulator to assist the user in a set of tasks. The architecture of Lattice Miner is open and modular enough to allow the integration of new features and facilities in each one of its components. === Context module === The context module offers all the basic operations and structures to manipulate binary and valued contexts as well as context decomposition to produce nested line diagrams. Basic context operations include apposition, subposition, generalization, clarification, reduction as well as the complementary context computation. The module provides also the arrow relations (for context reduction and decomposition) [2]. The tool has an input LMB format and recognizes the binary format SLF found in Galicia and the format CEX produced by ConExp. === Concept module === The main function of the concept module is to generate the concepts of the current binary context and construct the corresponding lattice and nested structure (see Figures 2 and 3). It provides the user with basic operators such as projection, selection, and exact search as well as advanced features like pair approximation. Some known algorithms are included in this module such as Bordat’s procedure, Godin’s algorithm and NextClosure algorithm. The approximation feature implemented in Lattice Miner is based on the following idea: given a pair (X,Y) where X ⊆ G, and Y ⊆ M, is there a set of formal concepts (Ai,Bi) which are “close to” (X,Y)? To answer this question, The tool starts to identify the type of couple that the pair (X,Y) represents. It can be a formal concept, a protoconcept, a semiconcept or a preconcept. In the last case, the approximation is given by the interval [(X",X′),(Y′,Y")] and highlighted in the line diagram. === Association rule module === This module includes procedures for computing the (stem) Guigues–Duquenne base using NextClosure algorithm [3], as well as the generic and informative bases. Implications with negation can be obtained using the apposition of a context and its complementary. This module embeds also procedures for the computation of a non-redundant family C of implications and the closure of a set Y of attributes for the given implication set C. === User interface === The initial objective of Lattice Miner was to focus on lattice drawing and visualization either as a flat or nested structure by taking into account the cognitive process of human beings and known principles for lattice drawing (e.g., reducing the number of edge intersections, ensuring diagram symmetry). Some well-known visualization techniques were implemented such as focus & context and fisheye view. The basic idea behind focus & context visualization paradigm is to allow a viewer to see key (important) objects in full detail in the foreground (focus) while at the same time an overview of all the surrounding information (context) remains available in the background. Lattice Miner translates the focus & context paradigm into clear and blurred elements while the size of nodes and the intensity of their color were used to indicate their importance. Various forms of highlighting, labelling and animation are also provided. In order to better handle the display of large lattices, nested line diagrams are offered in the tool. Figure 3 shows the third level of the nested line diagram corresponding to the binary context of Figure 1 where three levels of nesting are defined. Each one of the inner nodes of this diagram represents a combination of attributes from the previous two (outer) levels. Real inner concepts (see the node on the left hand-side of the diagram) are identified by colored nodes while void elements are in grey color. Each node of levels 1 and 2 can be expanded to exhibit its internal line diagram. Both flat and nested diagrams can be saved as an image. Simple (flat) lattices can also be saved as an XML format file.

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  Teacher forcing

  Teacher forcing is an algorithm for training the weights of recurrent neural networks (RNNs). It involves feeding observed sequence values (i.e. ground-truth samples) back into the RNN after each step, thus forcing the RNN to stay close to the ground-truth sequence. The term "teacher forcing" can be motivated by comparing the RNN to a human student taking a multi-part exam where the answer to each part (for example a mathematical calculation) depends on the answer to the preceding part. In this analogy, rather than grading every answer in the end, with the risk that the student fails every single part even though they only made a mistake in the first one, a teacher records the score for each individual part and then tells the student the correct answer, to be used in the next part. The use of an external teacher signal is in contrast to real-time recurrent learning (RTRL). Teacher signals are known from oscillator networks. The promise is, that teacher forcing helps to reduce the training time. The term "teacher forcing" was introduced in 1989 by Ronald J. Williams and David Zipser, who reported that the technique was already being "frequently used in dynamical supervised learning tasks" around that time. A NeurIPS 2016 paper introduced the related method of "professor forcing".

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  OrCam device

  OrCam devices such as OrCam MyEye are portable, artificial vision devices that allow visually impaired people to understand text and identify objects through audio feedback, describing what they are unable to see. Reuters described an important part of how it works as "a wireless smartcamera" which, when attached outside eyeglass frames, can read and verbalize text, and also supermarket barcodes. This information is converted to spoken words and entered "into the user’s ear." Face-recognition is also part of OrCam's feature set. == Devices == OrCam Technologies Ltd has created three devices; OrCam MyEye 2.0, OrCam MyEye 1, and OrCam MyReader. OrCam My Eye 2.0: OrCam debuted the second-generation model, the OrCam MyEye 2.0 in December 2017. About the size of a finger, the MyEye 2.0 is battery-powered, and has been compressed into a self-contained device. The device snaps onto any eyeglass frame magnetically. Orcam 2.0 is small and light (22.5 grams/0.8 ounces) with functionality to restore independence to the visually impaired. It comes in two versions. The basic model can read text, and a more advanced one adds features such as face recognition and barcode reading. As of July 2023, the retail cost is between $4000 and $6000 (USD). == Clinical Studies == JAMA Ophthalmology: In 2016 JAMA Ophthalmology conducted a study involving 12 legally blind participants to evaluate the usefulness of a portable artificial vision device (OrCam) for patients with low vision. The results showed that the OrCam device improved the patient's ability to perform tasks simulating those of daily living, such as reading a message on an electronic device, a newspaper article or a menu. Wills Eye: Wills Eye was a clinical study designed to measure the impact of the OrCam device on the quality of life of patients with End-stage Glaucoma. The conclusion was that OrCam, a novel artificial vision device using a mini-camera mounted on eyeglasses, allowed legally blind patients with end-stage glaucoma to read independently, subsequently improving their quality of life. == Employee testing == The New York Times described how a pre-release OrCam device was used by a Coloboma-impaired employee of the device's developer in 2013 for grocery shopping. It was the small size of the prototype rather than the functionality that gave her added mobility in an Israeli store's aisles. Added life-enhancement was described: "to both recognize and speak .. bus numbers .. traffic lights." == Social aspects == In contrast to an early version of Google Glass, which "failed ... because .. Glass wearers were ..mocked", early OrCam devices used designs that "clip unobtrusively on your shirt or perhaps your belt." In addition, it does not record sounds or images, what was called "the privacy puzzle that stumped Google. One 2018 technology reviewer wrote that he wished it had a headphone jack "so it would be less disruptive in places where others are working." An attempt was made to use bone conduction. == USA introduction == In 2018 a team headed by New York Assemblyman Dov Hikind introduced use of OrCam devices to ten individuals screened for what he termed "new Israeli technology that really makes a difference to the blind." Although not the first USA success, it was more focused than a publicly funded project that was authorized in 2016 by a California government agency. Also in 2016 the Chicago Lighthouse for the Blind demonstrated its use. == Technology == In the area of hardware, miniaturization has been quite important, but one major area, software, was mentioned by Assemblyman Hikind, and reported by The Times of Israel is the "AI-driven algorithms" that "reports .. how many people are in a room. In addition to reading printed text, it can also aid in "seeing" what is on a television or computer screen. Although OrCam can't help with handwritten information, it can reuse information, the basis of recognizing "US currency, and even faces." === Features === While early language support was for English, French, German, Hebrew and Spanish, others now available include Danish, Dutch, Finnish, Italian, Norwegian, Portuguese and Swedish. == History == OrCam Technologies Ltd was founded in 2010 by Professor Amnon Shashua and Ziv Aviram. Before co-founding OrCam, the two in 1999 co-founded Mobileye, an Israeli company that develops vision-based advanced driver-assistance systems (ADAS) providing warnings for collision prevention and mitigation, which was acquired by Intel for $15.3 billion in 2017. OrCam launched OrCam MyEye in 2013 after years of development and testing, and began selling it commercially in 2015. In its early years, the company raised $22 million, $6 million of which came from Intel Capital. By 2014, Intel, which was also investing in Google Glass, had invested $15 million in Orcam. In March 2017, OrCam had raised $41 million in capital, making it worth $600 million. === Marketing === One outcome of initial marketing in the USA was that they "reached a deal with the California Department of Rehabilitation, ...qualifying blind and visually impaired state residents." == OrCam Technologies Ltd == OrCam Technologies Ltd. is the Israeli-based company producing these OrCam devices, which are wearable artificial intelligence space. The company develops and manufactures assistive technology devices for individuals who are visually impaired, partially sighted, blind, print disabilities, or have other disabilities. OrCam headquarters is located in Jerusalem, operating under the company name OrCam Technologies Ltd. OrCam has over 150 employees, is headquartered in Jerusalem, and has offices in New York, Toronto, and London. == Awards == 2018 Last Gadget Standing Winner 2018 CES Innovation Awards Honoree in Accessible Tech 2017 NAIDEX Innovation Award 2016 Louise Braille Corporate Recognition Award 2016 Silmo-d-Or Award

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  Latent class model

  In statistics, a latent class model (LCM) is a model for clustering multivariate discrete data. It assumes that the data arise from a mixture of discrete distributions, within each of which the variables are independent. It is called a latent class model because the class to which each data point belongs is unobserved (or latent). Latent class analysis (LCA) is a subset of structural equation modeling used to find groups or subtypes of cases in multivariate categorical data. These groups or subtypes of cases are called "latent classes". When faced with the following situation, a researcher might opt to use LCA to better understand the data: Symptoms a, b, c, and d have been recorded in a variety of patients diagnosed with diseases X, Y, and Z. Disease X is associated with symptoms a, b, and c; disease Y is linked to symptoms b, c, and d; and disease Z is connected to symptoms a, c, and d. In this context, the LCA would attempt to detect the presence of latent classes (i.e., the disease entities), thus creating patterns of association in the symptoms. As in factor analysis, LCA can also be used to classify cases according to their maximum likelihood class membership probability. The key criterion for resolving the LCA is identifying latent classes in which the observed symptom associations are effectively rendered null. This is because within each class, the diseases responsible for the symptoms create a structure of dependencies. As a result, the symptoms become conditionally independent, meaning that, given the class a case belongs to, the symptoms are no longer related to one another. == Model == Within each latent class, the observed variables are statistically independent—an essential aspect of latent class modeling. Usually, the observed variables are statistically dependent. By introducing the latent variable, independence is restored in the sense that within classes, variables are independent (local independence). Therefore, the association between the observed variables is explained by the classes of the latent variable (McCutcheon, 1987). In one form, the LCM is written as p i 1 , i 2 , … , i N ≈ ∑ t T p t ∏ n N p i n , t n , {\displaystyle p_{i_{1},i_{2},\ldots ,i_{N}}\approx \sum _{t}^{T}p_{t}\,\prod _{n}^{N}p_{i_{n},t}^{n},} where T {\displaystyle T} is the number of latent classes and p t {\displaystyle p_{t}} are the so-called recruitment or unconditional probabilities that should sum to one. p i n , t n {\displaystyle p_{i_{n},t}^{n}} are the marginal or conditional probabilities. For a two-way latent class model, the form is p i j ≈ ∑ t T p t p i t p j t . {\displaystyle p_{ij}\approx \sum _{t}^{T}p_{t}\,p_{it}\,p_{jt}.} This two-way model is related to probabilistic latent semantic analysis and non-negative matrix factorization. The probability model used in LCA is closely related to the Naive Bayes classifier. The main difference is that in LCA, the class membership of an individual is a latent variable, whereas in Naive Bayes classifiers, the class membership is an observed label. == Related methods == There are a number of methods with distinct names and uses that share a common relationship. Cluster analysis is, like LCA, used to discover taxon-like groups of cases in data. Multivariate mixture estimation (MME) is applicable to continuous data and assumes that such data arise from a mixture of distributions, such as a set of heights arising from a mixture of men and women. If a multivariate mixture estimation is constrained so that measures must be uncorrelated within each distribution, it is termed latent profile analysis. Modified to handle discrete data, this constrained analysis is known as LCA. Discrete latent trait models further constrain the classes to form from segments of a single dimension, allocating members to classes based on that dimension. An example would be assigning cases to social classes based on ability or merit. In a practical instance, the variables could be multiple choice items of a political questionnaire. In this case, the data consists of an N-way contingency table with answers to the items for a number of respondents. In this example, the latent variable refers to political opinion, and the latent classes to political groups. Given group membership, the conditional probabilities specify the chance that certain answers are chosen. == Application == LCA may be used in many fields, such as: collaborative filtering, Behavior Genetics and Evaluation of diagnostic tests.

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  Bootstrap aggregating

  Bootstrap aggregating, also called bagging (from bootstrap aggregating) or bootstrapping, is a machine learning (ML) ensemble meta-algorithm designed to improve the stability and accuracy of ML classification and regression algorithms. It also reduces variance and overfitting. Although it is usually applied to decision tree methods, it can be used with any type of method. Bagging is a special case of the ensemble averaging approach. == Description of the technique == Given a standard training set D {\displaystyle D} of size n {\displaystyle n} , bagging generates m {\displaystyle m} new training sets D i {\displaystyle D_{i}} , each of size n ′ {\displaystyle n'} , by sampling from D {\displaystyle D} uniformly and with replacement. By sampling with replacement, some observations may be repeated in each D i {\displaystyle D_{i}} . If n ′ = n {\displaystyle n'=n} , then for large n {\displaystyle n} the set D i {\displaystyle D_{i}} is expected to have the fraction (1 - 1/e) (~63.2%) of the unique samples of D {\displaystyle D} , the rest being duplicates. This kind of sample is known as a bootstrap sample. Sampling with replacement ensures each bootstrap is independent from its peers, as it does not depend on previous chosen samples when sampling. Then, m {\displaystyle m} models are fitted using the above bootstrap samples and combined by averaging the output (for regression) or voting (for classification). Bagging leads to "improvements for unstable procedures", which include, for example, artificial neural networks, classification and regression trees, and subset selection in linear regression. Bagging was shown to improve preimage learning. On the other hand, it can mildly degrade the performance of stable methods such as k-nearest neighbors. == Process of the algorithm == === Key Terms === There are three types of datasets in bootstrap aggregating. These are the original, bootstrap, and out-of-bag datasets. Each section below will explain how each dataset is made except for the original dataset. The original dataset is whatever information is given. === Creating the bootstrap dataset === The bootstrap dataset is made by randomly picking objects from the original dataset. Also, it must be the same size as the original dataset. However, the difference is that the bootstrap dataset can have duplicate objects. Here is a simple example to demonstrate how it works along with the illustration below: Suppose the original dataset is a group of 12 people. Their names are Emily, Jessie, George, Constantine, Lexi, Theodore, John, James, Rachel, Anthony, Ellie, and Jamal. By randomly picking a group of names, let us say our bootstrap dataset had James, Ellie, Constantine, Lexi, John, Constantine, Theodore, Constantine, Anthony, Lexi, Constantine, and Theodore. In this case, the bootstrap sample contained four duplicates for Constantine, and two duplicates for Lexi, and Theodore. === Creating the out-of-bag dataset === The out-of-bag dataset represents the remaining people who were not in the bootstrap dataset. It can be calculated by taking the difference between the original and the bootstrap datasets. In this case, the remaining samples who were not selected are Emily, Jessie, George, Rachel, and Jamal. Keep in mind that since both datasets are sets, when taking the difference the duplicate names are ignored in the bootstrap dataset. The illustration below shows how the math is done: === Application === Creating the bootstrap and out-of-bag datasets is crucial since it is used to test the accuracy of ensemble learning algorithms like random forest. For example, a model that produces 50 trees using the bootstrap/out-of-bag datasets will have a better accuracy than if it produced 10 trees. Since the algorithm generates multiple trees and therefore multiple datasets the chance that an object is left out of the bootstrap dataset is low. The next few sections talk about how the random forest algorithm works in more detail. === Creation of Decision Trees === The next step of the algorithm involves the generation of decision trees from the bootstrapped dataset. To achieve this, the process examines each gene/feature and determines for how many samples the feature's presence or absence yields a positive or negative result. This information is then used to compute a confusion matrix, which lists the true positives, false positives, true negatives, and false negatives of the feature when used as a classifier. These features are then ranked according to various classification metrics based on their confusion matrices. Some common metrics include estimate of positive correctness (calculated by subtracting false positives from true positives), measure of "goodness", and information gain. These features are then used to partition the samples into two sets: those that possess the top feature, and those that do not. The diagram below shows a decision tree of depth two being used to classify data. For example, a data point that exhibits Feature 1, but not Feature 2, will be given a "No". Another point that does not exhibit Feature 1, but does exhibit Feature 3, will be given a "Yes". This process is repeated recursively for successive levels of the tree until the desired depth is reached. At the very bottom of the tree, samples that test positive for the final feature are generally classified as positive, while those that lack the feature are classified as negative. These trees are then used as predictors to classify new data. === Random Forests === The next part of the algorithm involves introducing yet another element of variability amongst the bootstrapped trees. In addition to each tree only examining a bootstrapped set of samples, only a small but consistent number of unique features are considered when ranking them as classifiers. This means that each tree only knows about the data pertaining to a small constant number of features, and a variable number of samples that is less than or equal to that of the original dataset. Consequently, the trees are more likely to return a wider array of answers, derived from more diverse knowledge. This results in a random forest, which possesses numerous benefits over a single decision tree generated without randomness. In a random forest, each tree "votes" on whether or not to classify a sample as positive based on its features. The sample is then classified based on majority vote. An example of this is given in the diagram below, where the four trees in a random forest vote on whether or not a patient with mutations A, B, F, and G has cancer. Since three out of four trees vote yes, the patient is then classified as cancer positive. Because of their properties, random forests are considered one of the most accurate data mining algorithms, are less likely to overfit their data, and run quickly and efficiently even for large datasets. They are primarily useful for classification as opposed to regression, which attempts to draw observed connections between statistical variables in a dataset. This makes random forests particularly useful in such fields as banking, healthcare, the stock market, and e-commerce where it is important to be able to predict future results based on past data. One of their applications would be as a useful tool for predicting cancer based on genetic factors, as seen in the above example. There are several important factors to consider when designing a random forest. If the trees in the random forests are too deep, overfitting can still occur due to over-specificity. If the forest is too large, the algorithm may become less efficient due to an increased runtime. Random forests also do not generally perform well when given sparse data with little variability. However, they still have numerous advantages over similar data classification algorithms such as neural networks, as they are much easier to interpret and generally require less data for training. As an integral component of random forests, bootstrap aggregating is very important to classification algorithms, and provides a critical element of variability that allows for increased accuracy when analyzing new data, as discussed below. == Improving Random Forests and Bagging == While the techniques described above utilize random forests and bagging (otherwise known as bootstrapping), there are certain techniques that can be used in order to improve their execution and voting time, their prediction accuracy, and their overall performance. The following are key steps in creating an efficient random forest: Specify the maximum depth of trees: Instead of allowing the random forest to continue until all nodes are pure, it is better to cut it off at a certain point in order to further decrease chances of overfitting. Prune the dataset: Using an extremely large dataset may create results that are less indicative of the data provided than a smaller set that more accurately represents what is being focused on. Continue pruning the data at each

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  LPBoost

  Linear Programming Boosting (LPBoost) is a supervised classifier from the boosting family of classifiers. LPBoost maximizes a margin between training samples of different classes, and thus also belongs to the class of margin classifier algorithms. Consider a classification function f : X → { − 1 , 1 } , {\displaystyle f:{\mathcal {X}}\to \{-1,1\},} which classifies samples from a space X {\displaystyle {\mathcal {X}}} into one of two classes, labelled 1 and -1, respectively. LPBoost is an algorithm for learning such a classification function, given a set of training examples with known class labels. LPBoost is a machine learning technique especially suited for joint classification and feature selection in structured domains. == LPBoost overview == As in all boosting classifiers, the final classification function is of the form f ( x ) = ∑ j = 1 J α j h j ( x ) , {\displaystyle f({\boldsymbol {x}})=\sum _{j=1}^{J}\alpha _{j}h_{j}({\boldsymbol {x}}),} where α j {\displaystyle \alpha _{j}} are non-negative weightings for weak classifiers h j : X → { − 1 , 1 } {\displaystyle h_{j}:{\mathcal {X}}\to \{-1,1\}} . Each individual weak classifier h j {\displaystyle h_{j}} may be just a little bit better than random, but the resulting linear combination of many weak classifiers can perform very well. LPBoost constructs f {\displaystyle f} by starting with an empty set of weak classifiers. Iteratively, a single weak classifier to add to the set of considered weak classifiers is selected, added and all the weights α {\displaystyle {\boldsymbol {\alpha }}} for the current set of weak classifiers are adjusted. This is repeated until no weak classifiers to add remain. The property that all classifier weights are adjusted in each iteration is known as totally-corrective property. Early boosting methods, such as AdaBoost do not have this property and converge slower. == Linear program == More generally, let H = { h ( ⋅ ; ω ) | ω ∈ Ω } {\displaystyle {\mathcal {H}}=\{h(\cdot ;\omega )|\omega \in \Omega \}} be the possibly infinite set of weak classifiers, also termed hypotheses. One way to write down the problem LPBoost solves is as a linear program with infinitely many variables. The primal linear program of LPBoost, optimizing over the non-negative weight vector α {\displaystyle {\boldsymbol {\alpha }}} , the non-negative vector ξ {\displaystyle {\boldsymbol {\xi }}} of slack variables and the margin ρ {\displaystyle \rho } is the following. min α , ξ , ρ − ρ + D ∑ n = 1 ℓ ξ n sb.t. ∑ ω ∈ Ω y n α ω h ( x n ; ω ) + ξ n ≥ ρ , n = 1 , … , ℓ , ∑ ω ∈ Ω α ω = 1 , ξ n ≥ 0 , n = 1 , … , ℓ , α ω ≥ 0 , ω ∈ Ω , ρ ∈ R . {\displaystyle {\begin{array}{cl}{\underset {{\boldsymbol {\alpha }},{\boldsymbol {\xi }},\rho }{\min }}&-\rho +D\sum _{n=1}^{\ell }\xi _{n}\\{\textrm {sb.t.}}&\sum _{\omega \in \Omega }y_{n}\alpha _{\omega }h({\boldsymbol {x}}_{n};\omega )+\xi _{n}\geq \rho ,\qquad n=1,\dots ,\ell ,\\&\sum _{\omega \in \Omega }\alpha _{\omega }=1,\\&\xi _{n}\geq 0,\qquad n=1,\dots ,\ell ,\\&\alpha _{\omega }\geq 0,\qquad \omega \in \Omega ,\\&\rho \in {\mathbb {R} }.\end{array}}} Note the effects of slack variables ξ ≥ 0 {\displaystyle {\boldsymbol {\xi }}\geq 0} : their one-norm is penalized in the objective function by a constant factor D {\displaystyle D} , which—if small enough—always leads to a primal feasible linear program. Here we adopted the notation of a parameter space Ω {\displaystyle \Omega } , such that for a choice ω ∈ Ω {\displaystyle \omega \in \Omega } the weak classifier h ( ⋅ ; ω ) : X → { − 1 , 1 } {\displaystyle h(\cdot ;\omega ):{\mathcal {X}}\to \{-1,1\}} is uniquely defined. When the above linear program was first written down in early publications about boosting methods it was disregarded as intractable due to the large number of variables α {\displaystyle {\boldsymbol {\alpha }}} . Only later it was discovered that such linear programs can indeed be solved efficiently using the classic technique of column generation. === Column generation for LPBoost === In a linear program a column corresponds to a primal variable. Column generation is a technique to solve large linear programs. It typically works in a restricted problem, dealing only with a subset of variables. By generating primal variables iteratively and on-demand, eventually the original unrestricted problem with all variables is recovered. By cleverly choosing the columns to generate the problem can be solved such that while still guaranteeing the obtained solution to be optimal for the original full problem, only a small fraction of columns has to be created. ==== LPBoost dual problem ==== Columns in the primal linear program corresponds to rows in the dual linear program. The equivalent dual linear program of LPBoost is the following linear program. max λ , γ γ sb.t. ∑ n = 1 ℓ y n h ( x n ; ω ) λ n + γ ≤ 0 , ω ∈ Ω , 0 ≤ λ n ≤ D , n = 1 , … , ℓ , ∑ n = 1 ℓ λ n = 1 , γ ∈ R . {\displaystyle {\begin{array}{cl}{\underset {{\boldsymbol {\lambda }},\gamma }{\max }}&\gamma \\{\textrm {sb.t.}}&\sum _{n=1}^{\ell }y_{n}h({\boldsymbol {x}}_{n};\omega )\lambda _{n}+\gamma \leq 0,\qquad \omega \in \Omega ,\\&0\leq \lambda _{n}\leq D,\qquad n=1,\dots ,\ell ,\\&\sum _{n=1}^{\ell }\lambda _{n}=1,\\&\gamma \in \mathbb {R} .\end{array}}} For linear programs the optimal value of the primal and dual problem are equal. For the above primal and dual problems, the optimal value is equal to the negative 'soft margin'. The soft margin is the size of the margin separating positive from negative training instances minus positive slack variables that carry penalties for margin-violating samples. Thus, the soft margin may be positive although not all samples are linearly separated by the classification function. The latter is called the 'hard margin' or 'realized margin'. ==== Convergence criterion ==== Consider a subset of the satisfied constraints in the dual problem. For any finite subset we can solve the linear program and thus satisfy all constraints. If we could prove that of all the constraints which we did not add to the dual problem no single constraint is violated, we would have proven that solving our restricted problem is equivalent to solving the original problem. More formally, let γ ∗ {\displaystyle \gamma ^{}} be the optimal objective function value for any restricted instance. Then, we can formulate a search problem for the 'most violated constraint' in the original problem space, namely finding ω ∗ ∈ Ω {\displaystyle \omega ^{}\in \Omega } as ω ∗ = argmax ω ∈ Ω ∑ n = 1 ℓ y n h ( x n ; ω ) λ n . {\displaystyle \omega ^{}={\underset {\omega \in \Omega }{\textrm {argmax}}}\sum _{n=1}^{\ell }y_{n}h({\boldsymbol {x}}_{n};\omega )\lambda _{n}.} That is, we search the space H {\displaystyle {\mathcal {H}}} for a single decision stump h ( ⋅ ; ω ∗ ) {\displaystyle h(\cdot ;\omega ^{})} maximizing the left hand side of the dual constraint. If the constraint cannot be violated by any choice of decision stump, none of the corresponding constraint can be active in the original problem and the restricted problem is equivalent. ==== Penalization constant ==== D {\displaystyle D} The positive value of penalization constant D {\displaystyle D} has to be found using model selection techniques. However, if we choose D = 1 ℓ ν {\displaystyle D={\frac {1}{\ell \nu }}} , where ℓ {\displaystyle \ell } is the number of training samples and 0 < ν < 1 {\displaystyle 0<\nu <1} , then the new parameter ν {\displaystyle \nu } has the following properties. ν {\displaystyle \nu } is an upper bound on the fraction of training errors; that is, if k {\displaystyle k} denotes the number of misclassified training samples, then k ℓ ≤ ν {\displaystyle {\frac {k}{\ell }}\leq \nu } . ν {\displaystyle \nu } is a lower bound on the fraction of training samples outside or on the margin. == Algorithm == Input: Training set X = { x 1 , … , x ℓ } {\displaystyle X=\{{\boldsymbol {x}}_{1},\dots ,{\boldsymbol {x}}_{\ell }\}} , x i ∈ X {\displaystyle {\boldsymbol {x}}_{i}\in {\mathcal {X}}} Training labels Y = { y 1 , … , y ℓ } {\displaystyle Y=\{y_{1},\dots ,y_{\ell }\}} , y i ∈ { − 1 , 1 } {\displaystyle y_{i}\in \{-1,1\}} Convergence threshold θ ≥ 0 {\displaystyle \theta \geq 0} Output: Classification function f : X → { − 1 , 1 } {\displaystyle f:{\mathcal {X}}\to \{-1,1\}} Initialization Weights, uniform λ n ← 1 ℓ , n = 1 , … , ℓ {\displaystyle \lambda _{n}\leftarrow {\frac {1}{\ell }},\quad n=1,\dots ,\ell } Edge γ ← 0 {\displaystyle \gamma \leftarrow 0} Hypothesis count J ← 1 {\displaystyle J\leftarrow 1} Iterate h ^ ← argmax ω ∈ Ω ∑ n = 1 ℓ y n h ( x n ; ω ) λ n {\displaystyle {\hat {h}}\leftarrow {\underset {\omega \in \Omega }{\textrm {argmax}}}\sum _{n=1}^{\ell }y_{n}h({\boldsymbol {x}}_{n};\omega )\lambda _{n}} if ∑ n = 1 ℓ y n h ^ ( x n ) λ n + γ ≤ θ {\displaystyle \sum _{n=1}^{\ell }y_{n}{\hat {h}}({\boldsymbol {x}}_{n})\lambda _{n}+\gamma \leq \theta } then break h J ← h ^ {\displaystyle h_{J}\leftarrow {\hat {h}}} J

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  Ameca (robot)

  Ameca is a robotic humanoid created in 2021 by Engineered Arts, headquarters in Falmouth, Cornwall, United Kingdom. The project commenced in February 2021, and the first public demonstration was at the CES 2022 show in Las Vegas. Ameca's appearance features grey rubber skin on the face and hands, and is specifically designed to appear genderless. In 2024, an Ameca unit was installed in Edinburgh in the UK to reside at the National Robotarium. Ameca generation 3 has been released and showcased at ICRA 2025 along with Ami. == History == The first generation of Ameca was developed at Engineered Arts headquarters in Falmouth, Cornwall, United Kingdom. The project started in February 2021, with the first video revealed publicly on 1 December 2021. Ameca gained widespread attention on Twitter and TikTok ahead of its first public demonstration at the Consumer Electronics Show 2022, where it was covered by CNET and other news outlets. In 2022, Ameca presented an Alternative Christmas message by British TV Channel 4 for Christmas Day. Ameca was associated with the Museum of the Future's robotic family, where it could interact with visitors. In 2024, an Ameca unit was installed in Edinburgh in the UK to reside at the National Robotarium. In January 2026, Ameca served as an ambassador for the European Space Agency (ESA) at the 18th European Space Conference. == Features == It is designed as a platform for further developing robotics technologies involving human-robot interaction. utilizes embedded microphones, binocular eye mounted cameras, a chest camera and facial recognition software to interact with the public. Interactions can be governed by either OpenAI's GPT-3 or human telepresence. It also features articulated motorized arms, fingers, neck and facial features. Ameca's appearance features grey rubber skin on the face and hands, and is specifically designed to appear genderless. == Public appearances == Computer History Museum, California Heinz Nixdorf MuseumsForum, Paderborn, Germany Copernicus Science Center, Warsaw, Poland Museum of the Future, Dubai Consumer Electronics Show 2022 Deutsches Museum Nuremberg OMR Festival 2022 Hosted by Vodafone GITEX 2022 International Conference on Robotics and Automation 2023 International Telecommunication Union AI for Good Global Summit 2023 Sphere (Not Ameca, Custom humanoid named Aura built on Ameca technology)

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  Dominance-based rough set approach

  The dominance-based rough set approach (DRSA) is an extension of rough set theory for multi-criteria decision analysis (MCDA), introduced by Greco, Matarazzo and Słowiński. The main change compared to the classical rough sets is the substitution for the indiscernibility relation by a dominance relation, which permits one to deal with inconsistencies typical to consideration of criteria and preference-ordered decision classes. == Multicriteria classification (sorting) == Multicriteria classification (sorting) is one of the problems considered within MCDA and can be stated as follows: given a set of objects evaluated by a set of criteria (attributes with preference-order domains), assign these objects to some pre-defined and preference-ordered decision classes, such that each object is assigned to exactly one class. Due to the preference ordering, improvement of evaluations of an object on the criteria should not worsen its class assignment. The sorting problem is very similar to the problem of classification, however, in the latter, the objects are evaluated by regular attributes and the decision classes are not necessarily preference ordered. The problem of multicriteria classification is also referred to as ordinal classification problem with monotonicity constraints and often appears in real-life application when ordinal and monotone properties follow from the domain knowledge about the problem. As an illustrative example, consider the problem of evaluation in a high school. The director of the school wants to assign students (objects) to three classes: bad, medium and good (notice that class good is preferred to medium and medium is preferred to bad). Each student is described by three criteria: level in Physics, Mathematics and Literature, each taking one of three possible values bad, medium and good. Criteria are preference-ordered and improving the level from one of the subjects should not result in worse global evaluation (class). As a more serious example, consider classification of bank clients, from the viewpoint of bankruptcy risk, into classes safe and risky. This may involve such characteristics as "return on equity (ROE)", "return on investment (ROI)" and "return on sales (ROS)". The domains of these attributes are not simply ordered but involve a preference order since, from the viewpoint of bank managers, greater values of ROE, ROI or ROS are better for clients being analysed for bankruptcy risk . Thus, these attributes are criteria. Neglecting this information in knowledge discovery may lead to wrong conclusions. == Data representation == === Decision table === In DRSA, data are often presented using a particular form of decision table. Formally, a DRSA decision table is a 4-tuple S = ⟨ U , Q , V , f ⟩ {\displaystyle S=\langle U,Q,V,f\rangle } , where U {\displaystyle U\,\!} is a finite set of objects, Q {\displaystyle Q\,\!} is a finite set of criteria, V = ⋃ q ∈ Q V q {\displaystyle V=\bigcup {}_{q\in Q}V_{q}} where V q {\displaystyle V_{q}\,\!} is the domain of the criterion q {\displaystyle q\,\!} and f : U × Q → V {\displaystyle f\colon U\times Q\to V} is an information function such that f ( x , q ) ∈ V q {\displaystyle f(x,q)\in V_{q}} for every ( x , q ) ∈ U × Q {\displaystyle (x,q)\in U\times Q} . The set Q {\displaystyle Q\,\!} is divided into condition criteria (set C ≠ ∅ {\displaystyle C\neq \emptyset } ) and the decision criterion (class) d {\displaystyle d\,\!} . Notice, that f ( x , q ) {\displaystyle f(x,q)\,\!} is an evaluation of object x {\displaystyle x\,\!} on criterion q ∈ C {\displaystyle q\in C} , while f ( x , d ) {\displaystyle f(x,d)\,\!} is the class assignment (decision value) of the object. An example of decision table is shown in Table 1 below. === Outranking relation === It is assumed that the domain of a criterion q ∈ Q {\displaystyle q\in Q} is completely preordered by an outranking relation ⪰ q {\displaystyle \succeq _{q}} ; x ⪰ q y {\displaystyle x\succeq _{q}y} means that x {\displaystyle x\,\!} is at least as good as (outranks) y {\displaystyle y\,\!} with respect to the criterion q {\displaystyle q\,\!} . Without loss of generality, we assume that the domain of q {\displaystyle q\,\!} is a subset of reals, V q ⊆ R {\displaystyle V_{q}\subseteq \mathbb {R} } , and that the outranking relation is a simple order between real numbers ≥ {\displaystyle \geq \,\!} such that the following relation holds: x ⪰ q y ⟺ f ( x , q ) ≥ f ( y , q ) {\displaystyle x\succeq _{q}y\iff f(x,q)\geq f(y,q)} . This relation is straightforward for gain-type ("the more, the better") criterion, e.g. company profit. For cost-type ("the less, the better") criterion, e.g. product price, this relation can be satisfied by negating the values from V q {\displaystyle V_{q}\,\!} . === Decision classes and class unions === Let T = { 1 , … , n } {\displaystyle T=\{1,\ldots ,n\}\,\!} . The domain of decision criterion, V d {\displaystyle V_{d}\,\!} consist of n {\displaystyle n\,\!} elements (without loss of generality we assume V d = T {\displaystyle V_{d}=T\,\!} ) and induces a partition of U {\displaystyle U\,\!} into n {\displaystyle n\,\!} classes Cl = { C l t , t ∈ T } {\displaystyle {\textbf {Cl}}=\{Cl_{t},t\in T\}} , where C l t = { x ∈ U : f ( x , d ) = t } {\displaystyle Cl_{t}=\{x\in U\colon f(x,d)=t\}} . Each object x ∈ U {\displaystyle x\in U} is assigned to one and only one class C l t , t ∈ T {\displaystyle Cl_{t},t\in T} . The classes are preference-ordered according to an increasing order of class indices, i.e. for all r , s ∈ T {\displaystyle r,s\in T} such that r ≥ s {\displaystyle r\geq s\,\!} , the objects from C l r {\displaystyle Cl_{r}\,\!} are strictly preferred to the objects from C l s {\displaystyle Cl_{s}\,\!} . For this reason, we can consider the upward and downward unions of classes, defined respectively, as: C l t ≥ = ⋃ s ≥ t C l s C l t ≤ = ⋃ s ≤ t C l s t ∈ T {\displaystyle Cl_{t}^{\geq }=\bigcup _{s\geq t}Cl_{s}\qquad Cl_{t}^{\leq }=\bigcup _{s\leq t}Cl_{s}\qquad t\in T} == Main concepts == === Dominance === We say that x {\displaystyle x\,\!} dominates y {\displaystyle y\,\!} with respect to P ⊆ C {\displaystyle P\subseteq C} , denoted by x D p y {\displaystyle xD_{p}y\,\!} , if x {\displaystyle x\,\!} is better than y {\displaystyle y\,\!} on every criterion from P {\displaystyle P\,\!} , x ⪰ q y , ∀ q ∈ P {\displaystyle x\succeq _{q}y,\,\forall q\in P} . For each P ⊆ C {\displaystyle P\subseteq C} , the dominance relation D P {\displaystyle D_{P}\,\!} is reflexive and transitive, i.e. it is a partial pre-order. Given P ⊆ C {\displaystyle P\subseteq C} and x ∈ U {\displaystyle x\in U} , let D P + ( x ) = { y ∈ U : y D p x } {\displaystyle D_{P}^{+}(x)=\{y\in U\colon yD_{p}x\}} D P − ( x ) = { y ∈ U : x D p y } {\displaystyle D_{P}^{-}(x)=\{y\in U\colon xD_{p}y\}} represent P-dominating set and P-dominated set with respect to x ∈ U {\displaystyle x\in U} , respectively. === Rough approximations === The key idea of the rough set philosophy is approximation of one knowledge by another knowledge. In DRSA, the knowledge being approximated is a collection of upward and downward unions of decision classes and the "granules of knowledge" used for approximation are P-dominating and P-dominated sets. The P-lower and the P-upper approximation of C l t ≥ , t ∈ T {\displaystyle Cl_{t}^{\geq },t\in T} with respect to P ⊆ C {\displaystyle P\subseteq C} , denoted as P _ ( C l t ≥ ) {\displaystyle {\underline {P}}(Cl_{t}^{\geq })} and P ¯ ( C l t ≥ ) {\displaystyle {\overline {P}}(Cl_{t}^{\geq })} , respectively, are defined as: P _ ( C l t ≥ ) = { x ∈ U : D P + ( x ) ⊆ C l t ≥ } {\displaystyle {\underline {P}}(Cl_{t}^{\geq })=\{x\in U\colon D_{P}^{+}(x)\subseteq Cl_{t}^{\geq }\}} P ¯ ( C l t ≥ ) = { x ∈ U : D P − ( x ) ∩ C l t ≥ ≠ ∅ } {\displaystyle {\overline {P}}(Cl_{t}^{\geq })=\{x\in U\colon D_{P}^{-}(x)\cap Cl_{t}^{\geq }\neq \emptyset \}} Analogously, the P-lower and the P-upper approximation of C l t ≤ , t ∈ T {\displaystyle Cl_{t}^{\leq },t\in T} with respect to P ⊆ C {\displaystyle P\subseteq C} , denoted as P _ ( C l t ≤ ) {\displaystyle {\underline {P}}(Cl_{t}^{\leq })} and P ¯ ( C l t ≤ ) {\displaystyle {\overline {P}}(Cl_{t}^{\leq })} , respectively, are defined as: P _ ( C l t ≤ ) = { x ∈ U : D P − ( x ) ⊆ C l t ≤ } {\displaystyle {\underline {P}}(Cl_{t}^{\leq })=\{x\in U\colon D_{P}^{-}(x)\subseteq Cl_{t}^{\leq }\}} P ¯ ( C l t ≤ ) = { x ∈ U : D P + ( x ) ∩ C l t ≤ ≠ ∅ } {\displaystyle {\overline {P}}(Cl_{t}^{\leq })=\{x\in U\colon D_{P}^{+}(x)\cap Cl_{t}^{\leq }\neq \emptyset \}} Lower approximations group the objects which certainly belong to class union C l t ≥ {\displaystyle Cl_{t}^{\geq }} (respectively C l t ≤ {\displaystyle Cl_{t}^{\leq }} ). This certainty comes from the fact, that object x ∈ U {\displaystyle x\in U} belongs to the lower approximation P _ ( C l t ≥ ) {\displaystyle {\underline {P}}(Cl_{t}^{\geq })} (respectively P _ ( C l t ≤ ) {\displaystyle {\underl

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  Receptron

  The receptron (short for "reservoir perceptron") is a neuromorphic data processing model — specifically neuromorphic computing — that generalizes the traditional perceptron, by incorporating non-linear interactions between inputs. Unlike classical perceptron, which rely on linearly independent weights, the receptron leverages complexity in physical substrates, such as the electric conduction properties of nanostructured materials or optical speckle fields, to perform classification tasks. The receptron bridges unconventional computing and neural network principles, enabling solutions that do not require the training approaches typical of artificial neural networks based on the perceptron model. == Algorithm == The receptron is an algorithm for supervised learning of binary classifiers, so a classification algorithm that makes its predictions based on a predictor function, combining a set of weights with the feature vector. The mathematical model is based on the sum of inputs with non-linear interactions: S = ∑ k = 1 n x j w ~ j ( x → ) | S ∈ R {\displaystyle S=\sum _{k=1}^{n}x_{j}{\widetilde {w}}_{j}({\vec {x}})|S\in R} (1) where j ∈ [ 1 , n ] {\displaystyle j\in [1,n]} and w ~ j {\displaystyle {\widetilde {w}}_{j}} are non-linear weight functions depending on the inputs, x → {\displaystyle {\vec {x}}} . Nonlinearity will typically make the system extremely complex, and allowing for the solution of problems not solvable through the simpler rules of a linear system, such as the perceptron or McCulloch Pitts neurons, which is based on the sum of linearly independent weights: S = ∑ k = 1 n x j w j p {\displaystyle S=\sum _{k=1}^{n}x_{j}w_{j}^{p}} (2) where w j {\displaystyle w_{j}} are constant real values. A consequence of this simplicity is the limitation to linearly separable functions, which necessitates multi-layer architectures and training algorithms like backpropagation As in the perceptron case, the summation in Eq. 1 origins the activation of the receptron output through the thresholding process, Y ( x 1 , . . . , x n ) = { 1 if S > th 0 if S ≤ th {\displaystyle Y(x_{1},...,x_{n})={\begin{cases}1&{\text{if }}S>{\text{th}}\\0&{\text{if }}S\leq {\text{th}}\end{cases}}} (3) where th is a constant threshold parameter. Equation 3 can be written by using the Heaviside step function. The weight functions w ~ ( x → ) {\displaystyle {\widetilde {w}}({\vec {x}})} can be written with a finite number of parameters w j 1 . . . j n {\displaystyle w_{j_{1}...j_{n}}} , simplifying the model representation. One can Taylor-expand w ~ ( x → ) {\displaystyle {\widetilde {w}}({\vec {x}})} and use the idempotency of Boolean variables ( x j ) q = x j ∀ q ≥ 1 {\displaystyle (x_{j})^{q}=x_{j}\forall q\geq 1} such that S ′ = b + ∑ k = 1 n x j w ~ j ( x → ) {\displaystyle S'=b+\sum _{k=1}^{n}x_{j}{\widetilde {w}}_{j}({\vec {x}})} can be written as S ′ ( x → ) = b + ∑ j w j x j + ∑ j < k w j k x j x k + ∑ j < k < l w j k l x j x k x l + . . . {\displaystyle S'({\vec {x}})=b+\sum _{j}w_{j}x_{j}+\sum _{j Read more →