Eric Poe Xing (Chinese: 邢波) is an American computer scientist who has been serving as president of Mohamed bin Zayed University of Artificial Intelligence (MBZUAI) since January 2021. He is also a professor in the Carnegie Mellon University School of Computer Science where he founded the SAILING Lab in 2004, and is the co-founder of the AI companies Petuum and GenBio AI. Xing's research focuses on statistical machine learning, probabilistic graphical models, and systems for distributed machine learning. He was elected a Fellow of the Institute of Electrical and Electronics Engineers in 2019 for "contributions to machine learning algorithms and systems" and a Fellow of the Association for Computing Machinery in 2022 for "contributions to algorithms, architectures, and applications in machine learning." == Education == Xing earned a B.Sc. in physics from Tsinghua University in 1993, and an M.Sc. in computer science from Rutgers University in 1998. He earned a Ph.D. in molecular biology and biochemistry from Rutgers in 1999, supervised by molecular cancer researcher Chung S. Yang. His dissertation examined the inactivation of the Rb and p53 pathways in human esophageal squamous cell carcinoma. He earned a second Ph.D. in computer science from the University of California, Berkeley in 2004, supervised by Richard Karp, Michael I. Jordan, and Stuart J. Russell. His thesis applied probabilistic graphical models to motif identification and haplotype inference in genomic data. == Career == Xing joined Carnegie Mellon University (CMU) as a faculty member in 2004, where he created the Statistical Artificial Intelligence and Integrative Genomics (SAILING) Lab. He held visiting appointments from 2010 to 2011, serving as a visiting research professor at Facebook Inc. and as a visiting associate professor in the Department of Statistics at Stanford University. He served as co-Program Chair of the International Conference on Machine Learning (ICML) in 2014 and General Chair in 2019. Xing served as the founding director of CMU’s Center for Machine Learning and Health, established in 2015 as part of the Pittsburgh Health Data Alliance, a collaboration between CMU, the University of Pittsburgh, and the University of Pittsburgh Medical Center. In 2016, Xing co-founded Petuum Inc., a US-based startup. In 2017, Petuum raised $93 million in a round of venture funding from SoftBank. In 2018 Petuum was named a World Economic Forum Technology Pioneer. In 2019, Xing received the Carnegie Science Award for Startup Entrepreneurs in recognition of his leadership of Petuum. On 29 November 2020, Xing was appointed president of the Mohamed bin Zayed University of Artificial Intelligence (MBZUAI), with the appointment taking effect in January 2021. In 2024, Xing co-founded GenBio AI where he is chief scientist. The US-based startup, which he co-founded with David Baker, Ziv Bar-Joseph, Emma Lundberg, Le Song and Fred Hu, aims to create AI-driven digital organisms (AIDO) for the purposes of modeling medical treatments. Xing has overseen the launch of the MBZUAI Institute of Foundation Models (IFM), which focuses on research and development of large-scale foundation models. In 2025–2026, IFM released the open-source reasoning model K2 Think, which was covered internationally as part of the UAE’s push to develop domestically controlled (“sovereign”) AI capabilities. IFM presented PAN as a “world model” research project and demonstrated related systems publicly. MBZUAI also collaborated with G42 and Cerebras Systems on the Jais language model, an open-source Arabic–English large language model released in 2023, according to Reuters. == Awards and honors == Xing is a recipient of the National Science Foundation (NSF) Career Award and the Alfred P. Sloan Research Fellowship. Xing is an elected Fellow of the following institutes and associations: Association for the Advancement of Artificial Intelligence (AAAI) 2016 Institute of Electrical and Electronics Engineers (IEEE) 2019 for "contributions to machine learning algorithms and systems" American Statistical Association (ASA) 2022 Association for Computing Machinery (ACM) 2022 for "contributions to algorithms, architectures, and applications in machine learning" Institute of Mathematical Statistics (IMS) 2023 International Society for Computational Biology (ISCB) 2026 == Selected publications == Eric P. Xing; Michael I. Jordan; Stuart J. Russell; Andrew Y. Ng (2003). "Distance Metric Learning with Application to Clustering with Side-Information" (PDF). Advances in Neural Information Processing Systems 15. Advances in Neural Information Processing Systems. Wikidata Q77691192. Edoardo M. Airoldi; David M. Blei; Stephen E Fienberg; Eric P Xing (1 September 2008). "Mixed Membership Stochastic Blockmodels". Journal of Machine Learning Research. 9: 1981–2014. ISSN 1533-7928. PMC 3119541. PMID 21701698. Wikidata Q35058357. Eric P. Xing; Michael I. Jordan; Richard M. Karp (28 June 2001), Feature selection for high-dimensional genomic microarray data, vol. 18, pp. 601–608, Wikidata Q138678867 Xing EP; Karp RM (1 January 2001). "CLIFF: clustering of high-dimensional microarray data via iterative feature filtering using normalized cuts". Bioinformatics. 17 Suppl 1: S306-15. doi:10.1093/BIOINFORMATICS/17.SUPPL_1.S306. ISSN 1367-4803. PMID 11473022. Wikidata Q30657299.
Exposure Notification
The (Google/Apple) Exposure Notification System (GAEN) is a framework and protocol specification developed by Apple Inc. and Google to facilitate digital contact tracing during the COVID-19 pandemic. When used by health authorities, it augments more traditional contact tracing techniques by automatically logging close approaches among notification system users using Android or iOS smartphones. Exposure Notification is a decentralized reporting protocol built on a combination of Bluetooth Low Energy technology and privacy-preserving cryptography. It is an opt-in feature within COVID-19 apps developed and published by authorized health authorities. Unveiled on April 10, 2020, it was made available on iOS on May 20, 2020, as part of the iOS 13.5 update and on December 14, 2020, as part of the iOS 12.5 update for older iPhones. On Android, it was added to devices via a Google Play Services update, supporting all versions since Android Marshmallow. The Apple/Google protocol is similar to the Decentralized Privacy-Preserving Proximity Tracing (DP-3T) protocol created by the European DP-3T consortium and the Temporary Contact Number (TCN) protocol by Covid Watch, but is implemented at the operating system level, which allows for more efficient operation as a background process. Since May 2020, a variant of the DP-3T protocol is supported by the Exposure Notification Interface. Other protocols are constrained in operation because they are not privileged over normal apps. This leads to issues, particularly on iOS devices where digital contact tracing apps running in the background experience significantly degraded performance. The joint approach is also designed to maintain interoperability between Android and iOS devices, which constitute nearly all of the market. The ACLU stated the approach "appears to mitigate the worst privacy and centralization risks, but there is still room for improvement". In late April, Google and Apple shifted the emphasis of the naming of the system, describing it as an "exposure notification service", rather than "contact tracing" system. == Technical specification == Digital contact tracing protocols typically have two major responsibilities: encounter logging and infection reporting. Exposure Notification only involves encounter logging which is a decentralized architecture. The majority of infection reporting is centralized in individual app implementations. To handle encounter logging, the system uses Bluetooth Low Energy to send tracking messages to nearby devices running the protocol to discover encounters with other people. The tracking messages contain unique identifiers that are encrypted with a secret daily key held by the sending device. These identifiers change every 15–20 minutes as well as Bluetooth MAC address in order to prevent tracking of clients by malicious third parties through observing static identifiers over time. The sender's daily encryption keys are generated using a random number generator. Devices record received messages, retaining them locally for 14 days. If a user tests positive for infection, the last 14 days of their daily encryption keys can be uploaded to a central server, where it is then broadcast to all devices on the network. The method through which daily encryption keys are transmitted to the central server and broadcast is defined by individual app developers. The Google-developed reference implementation calls for a health official to request a one-time verification code (VC) from a verification server, which the user enters into the encounter logging app. This causes the app to obtain a cryptographically signed certificate, which is used to authorize the submission of keys to the central reporting server. The received keys are then provided to the protocol, where each client individually searches for matches in their local encounter history. If a match meeting certain risk parameters is found, the app notifies the user of potential exposure to the infection. Google and Apple intend to use the received signal strength (RSSI) of the beacon messages as a source to infer proximity. RSSI and other signal metadata will also be encrypted to resist deanonymization attacks. === Version 1.0 === To generate encounter identifiers, first a persistent 32-byte private Tracing Key ( t k {\displaystyle tk} ) is generated by a client. From this a 16 byte Daily Tracing Key is derived using the algorithm d t k i = H K D F ( t k , N U L L , 'CT-DTK' | | D i , 16 ) {\displaystyle dtk_{i}=HKDF(tk,NULL,{\text{'CT-DTK'}}||D_{i},16)} , where H K D F ( Key, Salt, Data, OutputLength ) {\displaystyle HKDF({\text{Key, Salt, Data, OutputLength}})} is a HKDF function using SHA-256, and D i {\displaystyle D_{i}} is the day number for the 24-hour window the broadcast is in starting from Unix Epoch Time. These generated keys are later sent to the central reporting server should a user become infected. From the daily tracing key a 16-byte temporary Rolling Proximity Identifier is generated every 10 minutes with the algorithm R P I i , j = Truncate ( H M A C ( d t k i , 'CT-RPI' | | T I N j ) , 16 ) {\displaystyle RPI_{i,j}={\text{Truncate}}(HMAC(dtk_{i},{\text{'CT-RPI'}}||TIN_{j}),16)} , where H M A C ( Key, Data ) {\displaystyle HMAC({\text{Key, Data}})} is a HMAC function using SHA-256, and T I N j {\displaystyle TIN_{j}} is the time interval number, representing a unique index for every 10 minute period in a 24-hour day. The Truncate function returns the first 16 bytes of the HMAC value. When two clients come within proximity of each other they exchange and locally store the current R P I i , j {\displaystyle RPI_{i,j}} as the encounter identifier. Once a registered health authority has confirmed the infection of a user, the user's Daily Tracing Key for the past 14 days is uploaded to the central reporting server. Clients then download this report and individually recalculate every Rolling Proximity Identifier used in the report period, matching it against the user's local encounter log. If a matching entry is found, then contact has been established and the app presents a notification to the user warning them of potential infection. === Version 1.1 === Unlike version 1.0 of the protocol, version 1.1 does not use a persistent tracing key, rather every day a new random 16-byte Temporary Exposure Key ( t e k i {\displaystyle tek_{i}} ) is generated. This is analogous to the daily tracing key from version 1.0. Here i {\displaystyle i} denotes the time is discretized in 10 minute intervals starting from Unix Epoch Time. From this two 128-bit keys are calculated, the Rolling Proximity Identifier Key ( R P I K i {\displaystyle RPIK_{i}} ) and the Associated Encrypted Metadata Key ( A E M K i {\displaystyle AEMK_{i}} ). R P I K i {\displaystyle RPIK_{i}} is calculated with the algorithm R P I K i = H K D F ( t e k i , N U L L , 'EN-RPIK' , 16 ) {\displaystyle RPIK_{i}=HKDF(tek_{i},NULL,{\text{'EN-RPIK'}},16)} , and A E M K i {\displaystyle AEMK_{i}} using the algorithm A E M K i = H K D F ( t e k i , N U L L , 'EN-AEMK' , 16 ) {\displaystyle AEMK_{i}=HKDF(tek_{i},NULL,{\text{'EN-AEMK'}},16)} . From these values a temporary Rolling Proximity Identifier ( R P I i , j {\displaystyle RPI_{i,j}} ) is generated every time the BLE MAC address changes, roughly every 15–20 minutes. The following algorithm is used: R P I i , j = A E S 128 ( R P I K i , 'EN-RPI' | | 0 x 000000000000 | | E N I N j ) {\displaystyle RPI_{i,j}=AES128(RPIK_{i},{\text{'EN-RPI'}}||{\mathtt {0x000000000000}}||ENIN_{j})} , where A E S 128 ( Key, Data ) {\displaystyle AES128({\text{Key, Data}})} is an AES cryptography function with a 128-bit key, the data is one 16-byte block, j {\displaystyle j} denotes the Unix Epoch Time at the moment the roll occurs, and E N I N j {\displaystyle ENIN_{j}} is the corresponding 10-minute interval number. Next, additional Associated Encrypted Metadata is encrypted. What the metadata represents is not specified, likely to allow the later expansion of the protocol. The following algorithm is used: Associated Encrypted Metadata i , j = A E S 128 _ C T R ( A E M K i , R P I i , j , Metadata ) {\displaystyle {\text{Associated Encrypted Metadata}}_{i,j}=AES128\_CTR(AEMK_{i},RPI_{i,j},{\text{Metadata}})} , where A E S 128 _ C T R ( Key, IV, Data ) {\displaystyle AES128\_CTR({\text{Key, IV, Data}})} denotes AES encryption with a 128-bit key in CTR mode. The Rolling Proximity Identifier and the Associated Encrypted Metadata are then combined and broadcast using BLE. Clients exchange and log these payloads. Once a registered health authority has confirmed the infection of a user, the user's Temporary Exposure Keys t e k i {\displaystyle tek_{i}} and their respective interval numbers i {\displaystyle i} for the past 14 days are uploaded to the central reporting server. Clients then download this report and individually recalculate every Rolling Proximity Identifier starting from interval number i {\displaystyle i} ,
Nearest centroid classifier
In machine learning, a nearest centroid classifier or nearest prototype classifier is a classification model that assigns to observations the label of the class of training samples whose mean (centroid) is closest to the observation. When applied to text classification using word vectors containing tfidf weights to represent documents, the nearest centroid classifier is known as the Rocchio classifier because of its similarity to the Rocchio algorithm for relevance feedback. An extended version of the nearest centroid classifier has found applications in the medical domain, specifically classification of tumors. == Algorithm == === Training === Given labeled training samples { ( x → 1 , y 1 ) , … , ( x → n , y n ) } {\displaystyle \textstyle \{({\vec {x}}_{1},y_{1}),\dots ,({\vec {x}}_{n},y_{n})\}} with class labels y i ∈ Y {\displaystyle y_{i}\in \mathbf {Y} } , compute the per-class centroids μ → ℓ = 1 | C ℓ | ∑ i ∈ C ℓ x → i {\displaystyle \textstyle {\vec {\mu }}_{\ell }={\frac {1}{|C_{\ell }|}}{\underset {i\in C_{\ell }}{\sum }}{\vec {x}}_{i}} where C ℓ {\displaystyle C_{\ell }} is the set of indices of samples belonging to class ℓ ∈ Y {\displaystyle \ell \in \mathbf {Y} } . === Prediction === The class assigned to an observation x → {\displaystyle {\vec {x}}} is y ^ = arg min ℓ ∈ Y ‖ μ → ℓ − x → ‖ {\displaystyle {\hat {y}}={\arg \min }_{\ell \in \mathbf {Y} }\|{\vec {\mu }}_{\ell }-{\vec {x}}\|} .
Evolutionary programming
Evolutionary programming is an evolutionary algorithm, where a share of new population is created by mutation of previous population without crossover. Evolutionary programming differs from evolution strategy ES( μ + λ {\displaystyle \mu +\lambda } ) in one detail. All individuals are selected for the new population, while in ES( μ + λ {\displaystyle \mu +\lambda } ), every individual has the same probability to be selected. It is one of the four major evolutionary algorithm paradigms. == History == It was first used by Lawrence J. Fogel in the US in 1960 in order to use simulated evolution as a learning process aiming to generate artificial intelligence. It was used to evolve finite-state machines as predictors.
Vladimir Batagelj
Vladimir Batagelj (born June 14, 1948 in Idrija, Yugoslavia) is a Slovenian mathematician and an emeritus professor of mathematics at the University of Ljubljana. He is known for his work in discrete mathematics and combinatorial optimization, particularly analysis of social networks and other large networks (blockmodeling). == Education and career == Vladimir Batagelj completed his Ph.D. at the University of Ljubljana in 1986 under the direction of Tomaž Pisanski. He stayed at the University of Ljubljana as a professor until his retirement, where he was a professor of sociology and statistics, while also being a chair of the Department of Sociology of the Faculty of Social Sciences. As visiting professor, he was taught at the University of Pittsburgh (1990-91) and at the University of Konstanz (2002). He was also a member of editorial boards of two journals: Informatica and Journal of Social Structure. His work has been cited over 11000 times. His book Exploratory Social Network Analysis with Pajek on blockmodeling, coauthored with Wouter de Nooy and Andrej Mrvar, is Batagelj's most cited work and has over 3300 citations. The book was translated into Chinese and Japanese. The revised and expanded third edition has been published by Cambridge University Press. In 1975, 11 years before completing his PhD, Batagelj published a solo paper in Communications of the ACM. Batagelj authored more than 20 textbooks in Slovenian, covering topics like TeX, combinatorics and discrete mathematics. He has also written extensively in the Slovenian popular science journal Presek. Batagelj has advised 9 Ph.D. students. == Pajek == Batagelj is particularly known for his work on Pajek, a freely available software for analysis and visualization of large networks. He began work on Pajek in 1996 with Andrej Mrvar, who was then his PhD student. == Awards and honors == First prizes for contributions (with Andrej Mrvar) to Graph Drawing Contests in years: 1995, 1996, 1997, 1998, 1999, 2000 and 2005 / Graph Drawing Hall of Fame. In 2007 the book Generalized blockmodeling was awarded the Harrison White Outstanding Book Award by the Mathematical Sociology Section of American Sociological Association In 2007 he was awarded (together with Anuška Ferligoj) the Simmel Award by INSNA. In 2013, Vladimir Batagelj and Andrej Mrvar received the INSNA's William D. Richards Software award for their work on Pajek. == Selected bibliography == Vladimir Batagelj, Social Network Analysis, Large-Scale [1]. in R.A. Meyers, ed., Encyclopedia of Complexity and Systems Science, Springer 2009: 8245–8265. Vladimir Batagelj, Complex Networks, Visualization of [2]. in R.A. Meyers, ed., Encyclopedia of Complexity and Systems Science, Springer 2009: 1253–1268. Wouter de Nooy, Andrej Mrvar, Vladimir Batagelj, Mark Granovetter (Series Editor), Exploratory Social Network Analysis with Pajek (Structural Analysis in the Social Sciences), Cambridge University Press 2005 (ISBN 0-521-60262-9). ESNA in Japanese, TDU, 2010. Patrick Doreian, Vladimir Batagelj, Anuška Ferligoj, Mark Granovetter (Series Editor), Generalized Blockmodeling (Structural Analysis in the Social Sciences), Cambridge University Press 2004 (ISBN 0-521-84085-6)
Plum Voice
The Plum Group, Inc. (DBA Plum Voice) is a company. Plum is headquartered in New York City with offices in Boston and Denver. == History == Plum Voice, founded in 2000 as The Plum Group, Inc., was incorporated to create technologies for personalized audio communication. By 2001, Plum had commercialized the open-standard Plum VoiceXML IVR platform which facilitated the creation of dynamic telecom applications. 2001 - Commercial launch of Plum VoiceXML IVR platform for customer-premises deployment 2002 - Launch of Plum Voice Hosting Centers for 24x7x365 managed IVR hosting 2004 - Plum Voice application suite receives a "Product of the Year" award from Customer Interactions magazine 2008 - Plum Survey builder launched, a do-it-yourself IVR survey tool. 2010 - Plum launched QuickFuse, a web-based rapid development platform used to create voice applications. 2013 - Plum launched VoiceTrends, an analytics and reporting toolkit designed specifically for voice applications. Plum achieves PCI-DSS Level 1. 2015 - Plum launched Plum Insight, a multi-channel (voice, web, mobile) survey platform. Plum achieves HIPAA compliance. 2016 - Plum launched a new version of QuickFuse called Fuse+. 2020 - Plum sunsets QuickFuse, rebrands Fuse+ as Plum Fuse.
Radial basis function kernel
In machine learning, the radial basis function kernel, or RBF kernel, is a popular kernel function used in various kernelized learning algorithms. In particular, it is commonly used in support vector machine classification. The RBF kernel on two samples x , x ′ ∈ R k {\displaystyle \mathbf {x} ,\mathbf {x'} \in \mathbb {R} ^{k}} , represented as feature vectors in some input space, is defined as K ( x , x ′ ) = exp ( − ‖ x − x ′ ‖ 2 2 σ 2 ) {\displaystyle K(\mathbf {x} ,\mathbf {x'} )=\exp \left(-{\frac {\|\mathbf {x} -\mathbf {x'} \|^{2}}{2\sigma ^{2}}}\right)} ‖ x − x ′ ‖ 2 {\displaystyle \textstyle \|\mathbf {x} -\mathbf {x'} \|^{2}} may be recognized as the squared Euclidean distance between the two feature vectors. σ {\displaystyle \sigma } is a free parameter. An equivalent definition involves a parameter γ = 1 2 σ 2 {\displaystyle \textstyle \gamma ={\tfrac {1}{2\sigma ^{2}}}} : K ( x , x ′ ) = exp ( − γ ‖ x − x ′ ‖ 2 ) {\displaystyle K(\mathbf {x} ,\mathbf {x'} )=\exp(-\gamma \|\mathbf {x} -\mathbf {x'} \|^{2})} Since the value of the RBF kernel decreases with distance and ranges between zero (in the infinite-distance limit) and one (when x = x'), it has a ready interpretation as a similarity measure. The feature space of the kernel has an infinite number of dimensions; for σ = 1 {\displaystyle \sigma =1} , its expansion using the multinomial theorem is: exp ( − 1 2 ‖ x − x ′ ‖ 2 ) = exp ( 2 2 x ⊤ x ′ − 1 2 ‖ x ‖ 2 − 1 2 ‖ x ′ ‖ 2 ) = exp ( x ⊤ x ′ ) exp ( − 1 2 ‖ x ‖ 2 ) exp ( − 1 2 ‖ x ′ ‖ 2 ) = ∑ j = 0 ∞ ( x ⊤ x ′ ) j j ! exp ( − 1 2 ‖ x ‖ 2 ) exp ( − 1 2 ‖ x ′ ‖ 2 ) = ∑ j = 0 ∞ ∑ n 1 + n 2 + ⋯ + n k = j exp ( − 1 2 ‖ x ‖ 2 ) x 1 n 1 ⋯ x k n k n 1 ! ⋯ n k ! exp ( − 1 2 ‖ x ′ ‖ 2 ) x ′ 1 n 1 ⋯ x ′ k n k n 1 ! ⋯ n k ! = ⟨ φ ( x ) , φ ( x ′ ) ⟩ {\displaystyle {\begin{alignedat}{2}\exp \left(-{\frac {1}{2}}\|\mathbf {x} -\mathbf {x'} \|^{2}\right)&=\exp \left({\frac {2}{2}}\mathbf {x} ^{\top }\mathbf {x'} -{\frac {1}{2}}\|\mathbf {x} \|^{2}-{\frac {1}{2}}\|\mathbf {x'} \|^{2}\right)\\[5pt]&=\exp \left(\mathbf {x} ^{\top }\mathbf {x'} \right)\exp \left(-{\frac {1}{2}}\|\mathbf {x} \|^{2}\right)\exp \left(-{\frac {1}{2}}\|\mathbf {x'} \|^{2}\right)\\[5pt]&=\sum _{j=0}^{\infty }{\frac {(\mathbf {x} ^{\top }\mathbf {x'} )^{j}}{j!}}\exp \left(-{\frac {1}{2}}\|\mathbf {x} \|^{2}\right)\exp \left(-{\frac {1}{2}}\|\mathbf {x'} \|^{2}\right)\\[5pt]&=\sum _{j=0}^{\infty }\quad \sum _{n_{1}+n_{2}+\dots +n_{k}=j}\exp \left(-{\frac {1}{2}}\|\mathbf {x} \|^{2}\right){\frac {x_{1}^{n_{1}}\cdots x_{k}^{n_{k}}}{\sqrt {n_{1}!\cdots n_{k}!}}}\exp \left(-{\frac {1}{2}}\|\mathbf {x'} \|^{2}\right){\frac {{x'}_{1}^{n_{1}}\cdots {x'}_{k}^{n_{k}}}{\sqrt {n_{1}!\cdots n_{k}!}}}\\[5pt]&=\langle \varphi (\mathbf {x} ),\varphi (\mathbf {x'} )\rangle \end{alignedat}}} φ ( x ) = exp ( − 1 2 ‖ x ‖ 2 ) ( a ℓ 0 ( 0 ) , a 1 ( 1 ) , … , a ℓ 1 ( 1 ) , … , a 1 ( j ) , … , a ℓ j ( j ) , … ) {\displaystyle \varphi (\mathbf {x} )=\exp \left(-{\frac {1}{2}}\|\mathbf {x} \|^{2}\right)\left(a_{\ell _{0}}^{(0)},a_{1}^{(1)},\dots ,a_{\ell _{1}}^{(1)},\dots ,a_{1}^{(j)},\dots ,a_{\ell _{j}}^{(j)},\dots \right)} where ℓ j = ( k + j − 1 j ) {\displaystyle \ell _{j}={\tbinom {k+j-1}{j}}} , a ℓ ( j ) = x 1 n 1 ⋯ x k n k n 1 ! ⋯ n k ! | n 1 + n 2 + ⋯ + n k = j ∧ 1 ≤ ℓ ≤ ℓ j {\displaystyle a_{\ell }^{(j)}={\frac {x_{1}^{n_{1}}\cdots x_{k}^{n_{k}}}{\sqrt {n_{1}!\cdots n_{k}!}}}\quad |\quad n_{1}+n_{2}+\dots +n_{k}=j\wedge 1\leq \ell \leq \ell _{j}} == Approximations == Because support vector machines and other models employing the kernel trick do not scale well to large numbers of training samples or large numbers of features in the input space, several approximations to the RBF kernel (and similar kernels) have been introduced. Typically, these take the form of a function z that maps a single vector to a vector of higher dimensionality, approximating the kernel: ⟨ z ( x ) , z ( x ′ ) ⟩ ≈ ⟨ φ ( x ) , φ ( x ′ ) ⟩ = K ( x , x ′ ) {\displaystyle \langle z(\mathbf {x} ),z(\mathbf {x'} )\rangle \approx \langle \varphi (\mathbf {x} ),\varphi (\mathbf {x'} )\rangle =K(\mathbf {x} ,\mathbf {x'} )} where φ {\displaystyle \textstyle \varphi } is the implicit mapping embedded in the RBF kernel. === Fourier random features === One way to construct such a z is to randomly sample from the Fourier transformation of the kernel φ ( x ) = 1 D [ cos ⟨ w 1 , x ⟩ , sin ⟨ w 1 , x ⟩ , … , cos ⟨ w D , x ⟩ , sin ⟨ w D , x ⟩ ] T {\displaystyle \varphi (x)={\frac {1}{\sqrt {D}}}[\cos \langle w_{1},x\rangle ,\sin \langle w_{1},x\rangle ,\ldots ,\cos \langle w_{D},x\rangle ,\sin \langle w_{D},x\rangle ]^{T}} where w 1 , . . . , w D {\displaystyle w_{1},...,w_{D}} are independent samples from the normal distribution N ( 0 , σ − 2 I ) {\displaystyle N(0,\sigma ^{-2}I)} . Theorem: E [ ⟨ φ ( x ) , φ ( y ) ⟩ ] = e ‖ x − y ‖ 2 / ( 2 σ 2 ) . {\displaystyle \operatorname {E} [\langle \varphi (x),\varphi (y)\rangle ]=e^{\|x-y\|^{2}/(2\sigma ^{2})}.} Proof: It suffices to prove the case of D = 1 {\displaystyle D=1} . Use the trigonometric identity cos ( a − b ) = cos ( a ) cos ( b ) + sin ( a ) sin ( b ) {\displaystyle \cos(a-b)=\cos(a)\cos(b)+\sin(a)\sin(b)} , the spherical symmetry of Gaussian distribution, then evaluate the integral ∫ − ∞ ∞ cos ( k x ) e − x 2 / 2 2 π d x = e − k 2 / 2 . {\displaystyle \int _{-\infty }^{\infty }{\frac {\cos(kx)e^{-x^{2}/2}}{\sqrt {2\pi }}}dx=e^{-k^{2}/2}.} Theorem: Var [ ⟨ φ ( x ) , φ ( y ) ⟩ ] = O ( D − 1 ) {\displaystyle \operatorname {Var} [\langle \varphi (x),\varphi (y)\rangle ]=O(D^{-1})} . (Appendix A.2). === Nyström method === Another approach uses the Nyström method to approximate the eigendecomposition of the Gram matrix K, using only a random sample of the training set.