W3C Device Description Working Group

The W3C Device Description Working Group (DDWG), operating as part of the World Wide Web Consortium (W3C) Mobile Web Initiative (MWI), was chartered to "foster the provision and access to device descriptions that can be used in support of Web-enabled applications that provide an appropriate user experience on mobile devices." Mobile devices exhibit the greatest diversity of capabilities, and therefore present the greatest challenge to content adaptation technologies. The group published several documents, including a list of requirements for an interface to a Device Description Repository (DDR) and a standard interface meeting those requirements. The group was rechartered in 2006 to work in public towards the development of the Application Programming Interface (API) for a DDR. Early in 2007, the group launched a wiki and a blog to add to the public mailing list. The group subsequently published a formal vocabulary of core device properties, and an API called the DDR Simple API, which became a W3C Recommendation in December 2008. The group closed at the end of 2008, but with the intention of maintaining the Web pages, blog and wiki through W3C volunteer effort. == Publications == The DDWG published several W3C Working Group Notes and one W3C Recommendation. A W3C WG Note that articulates what the W3C and other organizations are doing or have already done with regard to device information. This document suggests an environment in which these technologies work together to meet the goals of content adaptation. The completed document was published on 31 October 2007. A W3C WG Note describing the ecosystem surrounding creation, maintenance and use of device descriptions. The completed document was published on 31 October 2007. A W3C WG Note describing a set of requirements for a reference repository of device descriptions. The completed document was published on 17 December 2007. A W3C WG Note describing a process to manage contributions to an initial core vocabulary, identification of key device properties, a formal initial core vocabulary and the identification of a maintainer for the core vocabulary. The details were contained in the Working Group Note describing the DDWG Core Vocabulary published on 14 April 2008. A W3C WG Note defining useful grouping and structure patterns in device descriptions. The Device Description Structures document was published as a Working Draft on 5 December 2008. The intention is that this document will be future input to other W3C groups. A W3C Recommendation defining a language-neutral programming interface to a Device Description Repository. The DDR Simple API was published on 5 December 2008. There is the possibility of future publications on the DDWG wiki describing implementations of the API in various languages, including Java, IDL, WSDL, C# etc. Much of the DDWG's material was developed in public via the DDWG Wiki and through their public mailing lists.

BLOOM (language model)

The BigScience Large Open-science Open-access Multilingual Language Model (BLOOM) is an open-access large language model (LLM) released in 2022. It was created by a volunteer-driven research effort to provide a transparently-created alternative to proprietary AI models. With 176 billion parameters, BLOOM is a transformer-based autoregressive model designed to generate text in 46 natural languages and 13 programming languages. The model is distributed under the project's "Responsible AI License". == Development == BLOOM is the main outcome of the BigScience initiative, a one-year-long research workshop. The project was coordinated by Hugging Face using funding from the French government and involved several hundred volunteer researchers and engineers from academia and the private sector. The model was trained between March and July 2022 on the Jean Zay public supercomputer in France, managed by GENCI and IDRIS (CNRS). Unlike GPT-3, BLOOM was trained to be multilingual. The source code is released under the Apache 2.0 license. The model's parameters are released under BigScience's "Responsible AI License" (RAIL), which grants open access and reuse rights but with some usage restrictions. BLOOM was used in the chatbots BLOOMChat and HuggingChat due to its multilingual abilities. BLOOM's training corpus, named ROOTS, combines data extracted from the then-latest version of the web-based OSCAR corpus (38% of ROOTS) and newly collected data extracted from a manually selected and documented list of language data sources. In total, the model was trained on approximately 366 billion (1.6TB) tokens. It was developed using the open-source libraries DeepSpeed Megatron. BigScience then released xP3, a multilingual dataset for LLM supervised learning. It also released BLOOMZ, a variant of BLOOM fine-tuned on xP3 to follow instructions.

Population model (evolutionary algorithm)

The population model of an evolutionary algorithm (EA) describes the structural properties of its population to which its members are subject. A population is the set of all proposed solutions of an EA considered in one iteration, which are also called individuals according to the biological role model. The individuals of a population can generate further individuals as offspring with the help of the genetic operators of the procedure. The simplest and widely used population model in EAs is the global or panmictic model, which corresponds to an unstructured population. It allows each individual to choose any other individual of the population as a partner for the production of offspring by crossover, whereby the details of the selection are irrelevant as long as the fitness of the individuals plays a significant role. Due to global mate selection, the genetic information of even slightly better individuals can prevail in a population after a few generations (iteration of an EA), provided that no better other offspring have emerged in this phase. If the solution found in this way is not the optimum sought, that is called premature convergence. This effect can be observed more often in panmictic populations. In nature global mating pools are rarely found. What prevails is a certain and limited isolation due to spatial distance. The resulting local neighbourhoods initially evolve independently and mutants have a higher chance of persisting over several generations. As a result, genotypic diversity in the gene pool is preserved longer than in a panmictic population. It is therefore obvious to divide the previously global population by substructures. Two basic models were introduced for this purpose, the island models, which are based on a division of the population into fixed subpopulations that exchange individuals from time to time, and the neighbourhood models, which assign individuals to overlapping neighbourhoods, also known as cellular genetic or evolutionary algorithms (cGA or cEA). The associated division of the population also suggests a corresponding parallelization of the procedure. For this reason, the topic of population models is also frequently discussed in the literature in connection with the parallelization of EAs. == Island models == In the island model, also called the migration model or coarse grained model, evolution takes place in strictly divided subpopulations. These can be organised panmictically, but do not have to be. From time to time an exchange of individuals takes place, which is called migration. The time between an exchange is called an epoch and its end can be triggered by various criteria: E.g. after a given time or given number of completed generations, or after the occurrence of stagnation. Stagnation can be detected, for example, by the fact that no fitness improvement has occurred in the island for a given number of generations. Island models introduce a variety of new strategy parameters: Number of subpopulations Size of the subpopulations Neighbourhood relations between islands: they determine which islands are considered neighbouring and can thus exchange individuals, see picture of a simple unidirectional ring (black arrows) and its extension by additional bidirectional neighbourhood relations (additional green arrows) Criteria for the termination of an epoch, synchronous or asynchronous migration Migration rate: number or proportion of individuals involved in migration. Migrant selection: There are many alternatives for this. E.g. the best individuals can replace the worst or randomly selected ones. Depending on the migration rate, this can affect one or more individuals at a time. With these parameters, the selection pressure can be influenced to a considerable extent. For example, it increases with the interconnectedness of the islands and decreases with the number of subpopulations or the epoch length. == Neighbourhood models or cellular evolutionary algorithms == The neighbourhood model, also called diffusion model or fine grained model, defines a topological neighbouhood relation between the individuals of a population that is independent of their phenotypic properties. The fundamental idea of this model is to provide the EA population with a special structure defined as a connected graph, in which each vertex is an individual that communicates with its nearest neighbours. Particularly, individuals are conceptually set in a toroidal mesh, and are only allowed to recombine with close individuals. This leads to a kind of locality known as isolation by distance. The set of potential mates of an individual is called its neighbourhood or deme. The adjacent figure illustrates that by showing two slightly overlapping neighbourhoods of two individuals marked yellow, through which genetic information can spread between the two demes. It is known that in this kind of algorithm, similar individuals tend to cluster and create niches that are independent of the deme boundaries and, in particular, can be larger than a deme. There is no clear borderline between adjacent groups, and close niches could be easily colonized by competitive ones and maybe merge solution contents during this process. Simultaneously, farther niches can be affected more slowly. EAs with this type of population are also well known as cellular EAs (cEA) or cellular genetic algorithms (cGA). A commonly used structure for arranging the individuals of a population is a 2D toroidal grid, although the number of dimensions can be easily extended (to 3D) or reduced (to 1D, e.g. a ring, see the figure on the right). The neighbourhood of a particular individual in the grid is defined in terms of the Manhattan distance from it to others in the population. In the basic algorithm, all the neighbourhoods have the same size and identical shapes. The two most commonly used neighbourhoods for two-dimensional cEAs are L5 and C9, see the figure on the left. Here, L stands for Linear while C stands for Compact. Each deme represents a panmictic subpopulation within which mate selection and the acceptance of offspring takes place by replacing the parent. The rules for the acceptance of offspring are local in nature and based on the neighbourhood: for example, it can be specified that the best offspring must be better than the parent being replaced or, less strictly, only better than the worst individual in the deme. The first rule is elitist and creates a higher selective pressure than the second non-elitist rule. In elitist EAs, the best individual of a population always survives. In this respect, they deviate from the biological model. The overlap of the neighbourhoods causes a mostly slow spread of genetic information across the neighbourhood boundaries, hence the name diffusion model. A better offspring now needs more generations than in panmixy to spread in the population. This promotes the emergence of local niches and their local evolution, thus preserving genotypic diversity over a longer period of time. The result is a better and dynamic balance between breadth and depth search adapted to the search space during a run. Depth search takes place in the niches and breadth search in the niche boundaries and through the evolution of the different niches of the whole population. For the same neighbourhood size, the spread of genetic information is larger for elongated figures like L9 than for a block like C9, and again significantly larger than for a ring. This means that ring neighbourhoods are well suited for achieving high quality results, even if this requires comparatively long run times. On the other hand, if one is primarily interested in fast and good, but possibly suboptimal results, 2D topologies are more suitable. == Comparison == When applying both population models to genetic algorithms, evolutionary strategy and other EAs, the splitting of a total population into subpopulations usually reduces the risk of premature convergence and leads to better results overall more reliably and faster than would be expected with panmictic EAs. Island models have the disadvantage compared to neighbourhood models that they introduce a large number of new strategy parameters. Despite the existing studies on this topic in the literature, a certain risk of unfavourable settings remains for the user. With neighbourhood models, on the other hand, only the size of the neighbourhood has to be specified and, in the case of the two-dimensional model, the choice of the neighbourhood figure is added. == Parallelism == Since both population models imply population partitioning, they are well suited as a basis for parallelizing an EA. This applies even more to cellular EAs, since they rely only on locally available information about the members of their respective demes. Thus, in the extreme case, an independent execution thread can be assigned to each individual, so that the entire cEA can run on a parallel hardware platform. The island model also supports p

Waffles (machine learning)

Waffles is a collection of command-line tools for performing machine learning operations developed at Brigham Young University. These tools are written in C++, and are available under the GNU Lesser General Public License. == Description == The Waffles machine learning toolkit contains command-line tools for performing various operations related to machine learning, data mining, and predictive modeling. The primary focus of Waffles is to provide tools that are simple to use in scripted experiments or processes. For example, the supervised learning algorithms included in Waffles are all designed to support multi-dimensional labels, classification and regression, automatically impute missing values, and automatically apply necessary filters to transform the data to a type that the algorithm can support, such that arbitrary learning algorithms can be used with arbitrary data sets. Many other machine learning toolkits provide similar functionality, but require the user to explicitly configure data filters and transformations to make it compatible with a particular learning algorithm. The algorithms provided in Waffles also have the ability to automatically tune their own parameters (with the cost of additional computational overhead). Because Waffles is designed for script-ability, it deliberately avoids presenting its tools in a graphical environment. It does, however, include a graphical "wizard" tool that guides the user to generate a command that will perform a desired task. This wizard does not actually perform the operation, but requires the user to paste the command that it generates into a command terminal or a script. The idea motivating this design is to prevent the user from becoming "locked in" to a graphical interface. All of the Waffles tools are implemented as thin wrappers around functionality in a C++ class library. This makes it possible to convert scripted processes into native applications with minimal effort. Waffles was first released as an open source project in 2005. Since that time, it has been developed at Brigham Young University, with a new version having been released approximately every 6–9 months. Waffles is not an acronym—the toolkit was named after the food for historical reasons. == Advantages == Some of the advantages of Waffles in contrast with other popular open source machine learning toolkits include: Waffles automatically takes care of many issues related to data format in order to simplify its tools. Because it is implemented in C++, many of its algorithms are particularly fast. Also, the lack of dependency on any virtual machine makes it easier to deploy in conjunction with other applications. The functionality included in Waffles is very broad, including algorithms for dimensionality reduction, collaborative filtering, visualization, clustering, supervised learning, optimization, linear algebra, data transformation, image and signal processing, policy learning, and sparse matrix operations. == Disadvantages == Although Waffles provides significant breadth, it lacks the depth of many toolkits that focus on a particular area of machine learning. The Weka (machine learning) toolkit, for example, provides many more classification algorithms than Waffles provides. Waffles only has a limited graphical interface.

Quadratic classifier

In statistics, a quadratic classifier is a statistical classifier that uses a quadratic decision surface to separate measurements of two or more classes of objects or events. It is a more general version of the linear classifier. == The classification problem == Statistical classification considers a set of vectors of observations x of an object or event, each of which has a known type y. This set is referred to as the training set. The problem is then to determine, for a given new observation vector, what the best class should be. For a quadratic classifier, the correct solution is assumed to be quadratic in the measurements, so y will be decided based on x T A x + b T x + c {\displaystyle \mathbf {x^{T}Ax} +\mathbf {b^{T}x} +c} In the special case where each observation consists of two measurements, this means that the surfaces separating the classes will be conic sections (i.e., either a line, a circle or ellipse, a parabola or a hyperbola). In this sense, we can state that a quadratic model is a generalization of the linear model, and its use is justified by the desire to extend the classifier's ability to represent more complex separating surfaces. == Quadratic discriminant analysis == Quadratic discriminant analysis (QDA) is closely related to linear discriminant analysis (LDA), where it is assumed that the measurements from each class are normally distributed. Unlike LDA however, in QDA there is no assumption that the covariance of each of the classes is identical. When the normality assumption is true, the best possible test for the hypothesis that a given measurement is from a given class is the likelihood ratio test. Suppose there are only two groups, with means μ 0 , μ 1 {\displaystyle \mu _{0},\mu _{1}} and covariance matrices Σ 0 , Σ 1 {\displaystyle \Sigma _{0},\Sigma _{1}} corresponding to y = 0 {\displaystyle y=0} and y = 1 {\displaystyle y=1} respectively. Then the likelihood ratio is given by Likelihood ratio = | 2 π Σ 1 | − 1 exp ⁡ ( − 1 2 ( x − μ 1 ) T Σ 1 − 1 ( x − μ 1 ) ) | 2 π Σ 0 | − 1 exp ⁡ ( − 1 2 ( x − μ 0 ) T Σ 0 − 1 ( x − μ 0 ) ) < t {\displaystyle {\text{Likelihood ratio}}={\frac {{\sqrt {|2\pi \Sigma _{1}|}}^{-1}\exp \left(-{\frac {1}{2}}(\mathbf {x} -{\boldsymbol {\mu }}_{1})^{T}\Sigma _{1}^{-1}(\mathbf {x} -{\boldsymbol {\mu }}_{1})\right)}{{\sqrt {|2\pi \Sigma _{0}|}}^{-1}\exp \left(-{\frac {1}{2}}(\mathbf {x} -{\boldsymbol {\mu }}_{0})^{T}\Sigma _{0}^{-1}(\mathbf {x} -{\boldsymbol {\mu }}_{0})\right)}}

Keka HR

Keka HR is a software company that provides cloud-based human resource management and payroll automation software. Keka HR specializes in providing business services in the field of HR technology, payroll automation, recruiting, leave, attendance and performance management. The company was founded by Vijay Yalamanchili on July 21, 2014. The company is headquartered in Hyderabad, with operations in Singapore and the United States. == History == Keka HR was established in 2014 in Hyderabad, Telangana, India. In 2015, the company entered the Indian HR market and received the HYSEA Startup Award. By 2019, Keka HR had surpassed $1 million in annual recurring revenue (ARR). During the COVID-19 pandemic in 2020, the company reported a sevenfold increase in sales. By 2021, the company had raised $1.6 million through Recur Club. In 2022, Keka HR secured $57 million in Series A funding from West Bridge Capital. The company's headquarters are located in Gachibowli, Hyderabad, with offices in Singapore and Seattle, Washington.

Quadratic unconstrained binary optimization

Quadratic unconstrained binary optimization (QUBO), also known as unconstrained binary quadratic programming (UBQP), is a combinatorial optimization problem with a wide range of applications from finance and economics to machine learning. QUBO is an NP hard problem, and for many classical problems from theoretical computer science, like maximum cut, graph coloring and the partition problem, embeddings into QUBO have been formulated. Embeddings for machine learning models include support-vector machines, clustering and probabilistic graphical models. Moreover, due to its close connection to Ising models, QUBO constitutes a central problem class for adiabatic quantum computation, where it is solved through a physical process called quantum annealing. == Definition == Let B = { 0 , 1 } {\displaystyle \mathbb {B} =\lbrace 0,1\rbrace } the set of binary digits (or bits), then B n {\displaystyle \mathbb {B} ^{n}} is the set of binary vectors of fixed length n ∈ N {\displaystyle n\in \mathbb {N} } . Given a symmetric or upper triangular matrix Q ∈ R n × n {\displaystyle {\boldsymbol {Q}}\in \mathbb {R} ^{n\times n}} , whose entries Q i j {\displaystyle Q_{ij}} define a weight for each pair of indices i , j ∈ { 1 , … , n } {\displaystyle i,j\in \lbrace 1,\dots ,n\rbrace } , we can define the function f Q : B n → R {\displaystyle f_{\boldsymbol {Q}}:\mathbb {B} ^{n}\rightarrow \mathbb {R} } that assigns a value to each binary vector x {\displaystyle {\boldsymbol {x}}} through f Q ( x ) = x ⊺ Q x = ∑ i = 1 n ∑ j = 1 n Q i j x i x j . {\displaystyle f_{\boldsymbol {Q}}({\boldsymbol {x}})={\boldsymbol {x}}^{\intercal }{\boldsymbol {Qx}}=\sum _{i=1}^{n}\sum _{j=1}^{n}Q_{ij}x_{i}x_{j}.} Alternatively, the linear and quadratic parts can be separated as f Q ′ , q ( x ) = x ⊺ Q ′ x + q ⊺ x , {\displaystyle f_{{\boldsymbol {Q}}',{\boldsymbol {q}}}({\boldsymbol {x}})={\boldsymbol {x}}^{\intercal }{\boldsymbol {Q}}'{\boldsymbol {x}}+{\boldsymbol {q}}^{\intercal }{\boldsymbol {x}},} where Q ′ ∈ R n × n {\displaystyle {\boldsymbol {Q}}'\in \mathbb {R} ^{n\times n}} and q ∈ R n {\displaystyle {\boldsymbol {q}}\in \mathbb {R} ^{n}} . This is equivalent to the previous definition through Q = Q ′ + diag ⁡ [ q ] {\displaystyle {\boldsymbol {Q}}={\boldsymbol {Q}}'+\operatorname {diag} [{\boldsymbol {q}}]} using the diag operator, exploiting that x = x ⋅ x {\displaystyle x=x\cdot x} for all binary values x {\displaystyle x} . Intuitively, the weight Q i j {\displaystyle Q_{ij}} is added if both x i = 1 {\displaystyle x_{i}=1} and x j = 1 {\displaystyle x_{j}=1} . The QUBO problem consists of finding a binary vector x ∗ {\displaystyle {\boldsymbol {x}}^{}} that minimizes f Q {\displaystyle f_{\boldsymbol {Q}}} , i.e., ∀ x ∈ B n : f Q ( x ∗ ) ≤ f Q ( x ) {\displaystyle \forall {\boldsymbol {x}}\in \mathbb {B} ^{n}:~f_{\boldsymbol {Q}}({\boldsymbol {x}}^{})\leq f_{\boldsymbol {Q}}({\boldsymbol {x}})} . In general, x ∗ {\displaystyle {\boldsymbol {x}}^{}} is not unique, meaning there may be a set of minimizing vectors with equal value w.r.t. f Q {\displaystyle f_{\boldsymbol {Q}}} . The complexity of QUBO arises from the number of candidate binary vectors to be evaluated, as | B n | = 2 n {\displaystyle \left|\mathbb {B} ^{n}\right|=2^{n}} grows exponentially in n {\displaystyle n} . Sometimes, QUBO is defined as the problem of maximizing f Q {\displaystyle f_{\boldsymbol {Q}}} , which is equivalent to minimizing f − Q = − f Q {\displaystyle f_{-{\boldsymbol {Q}}}=-f_{\boldsymbol {Q}}} . == Properties == QUBO is scale invariant for positive factors α > 0 {\displaystyle \alpha >0} , which leave the optimum x ∗ {\displaystyle {\boldsymbol {x}}^{}} unchanged: f α Q ( x ) = x ⊺ ( α Q ) x = α ( x ⊺ Q x ) = α f Q ( x ) {\displaystyle f_{\alpha {\boldsymbol {Q}}}({\boldsymbol {x}})={\boldsymbol {x}}^{\intercal }(\alpha {\boldsymbol {Q}}){\boldsymbol {x}}=\alpha ({\boldsymbol {x}}^{\intercal }{\boldsymbol {Qx}})=\alpha f_{\boldsymbol {Q}}({\boldsymbol {x}})} . In its general form, QUBO is NP-hard and cannot be solved efficiently by any known polynomial-time algorithm. However, there are polynomially-solvable special cases, where Q {\displaystyle {\boldsymbol {Q}}} has certain properties, for example: If all coefficients are positive, the optimum is trivially x ∗ = ( 0 , … , 0 ) ⊺ {\displaystyle {\boldsymbol {x}}^{}=(0,\dots ,0)^{\intercal }} . Similarly, if all coefficients are negative, the optimum is x ∗ = ( 1 , … , 1 ) ⊺ {\displaystyle {\boldsymbol {x}}^{}=(1,\dots ,1)^{\intercal }} . If Q {\displaystyle {\boldsymbol {Q}}} is diagonal, the bits can be optimized independently, and the problem is solvable in O ( n ) {\displaystyle {\mathcal {O}}(n)} . The optimal variable assignments are simply x i ∗ = 1 {\displaystyle x_{i}^{}=1} if Q i i < 0 {\displaystyle Q_{ii}<0} , and x i ∗ = 0 {\displaystyle x_{i}^{}=0} otherwise. If all off-diagonal elements of Q {\displaystyle {\boldsymbol {Q}}} are non-positive, the corresponding QUBO problem is solvable in polynomial time. QUBO can be solved using integer linear programming solvers like CPLEX or Gurobi Optimizer. This is possible since QUBO can be reformulated as a linear constrained binary optimization problem. To achieve this, substitute the product x i x j {\displaystyle x_{i}x_{j}} by an additional binary variable z i j ∈ B {\displaystyle z_{ij}\in \mathbb {B} } and add the constraints x i ≥ z i j {\displaystyle x_{i}\geq z_{ij}} , x j ≥ z i j {\displaystyle x_{j}\geq z_{ij}} and x i + x j − 1 ≤ z i j {\displaystyle x_{i}+x_{j}-1\leq z_{ij}} . Note that z i j {\displaystyle z_{ij}} can also be relaxed to continuous variables within the bounds zero and one. == Applications == QUBO is a structurally simple, yet computationally hard optimization problem. It can be used to encode a wide range of optimization problems from various scientific areas. === Maximum Cut === Given a graph G = ( V , E ) {\displaystyle G=(V,E)} with vertex set V = { 1 , … , n } {\displaystyle V=\lbrace 1,\dots ,n\rbrace } and edges E ⊆ V × V {\displaystyle E\subseteq V\times V} , the maximum cut (max-cut) problem consists of finding two subsets S , T ⊆ V {\displaystyle S,T\subseteq V} with T = V ∖ S {\displaystyle T=V\setminus S} , such that the number of edges between S {\displaystyle S} and T {\displaystyle T} is maximized. The more general weighted max-cut problem assumes edge weights w i j ≥ 0 ∀ i , j ∈ V {\displaystyle w_{ij}\geq 0~\forall i,j\in V} , with ( i , j ) ∉ E ⇒ w i j = 0 {\displaystyle (i,j)\notin E\Rightarrow w_{ij}=0} , and asks for a partition S , T ⊆ V {\displaystyle S,T\subseteq V} that maximizes the sum of edge weights between S {\displaystyle S} and T {\displaystyle T} , i.e., max S ⊆ V ∑ i ∈ S , j ∉ S w i j . {\displaystyle \max _{S\subseteq V}\sum _{i\in S,j\notin S}w_{ij}.} By setting w i j = 1 {\displaystyle w_{ij}=1} for all ( i , j ) ∈ E {\displaystyle (i,j)\in E} this becomes equivalent to the original max-cut problem above, which is why we focus on this more general form in the following. For every vertex in i ∈ V {\displaystyle i\in V} we introduce a binary variable x i {\displaystyle x_{i}} with the interpretation x i = 0 {\displaystyle x_{i}=0} if i ∈ S {\displaystyle i\in S} and x i = 1 {\displaystyle x_{i}=1} if i ∈ T {\displaystyle i\in T} . As T = V ∖ S {\displaystyle T=V\setminus S} , every i {\displaystyle i} is in exactly one set, meaning there is a 1:1 correspondence between binary vectors x ∈ B n {\displaystyle {\boldsymbol {x}}\in \mathbb {B} ^{n}} and partitions of V {\displaystyle V} into two subsets. We observe that, for any i , j ∈ V {\displaystyle i,j\in V} , the expression x i ( 1 − x j ) + ( 1 − x i ) x j {\displaystyle x_{i}(1-x_{j})+(1-x_{i})x_{j}} evaluates to 1 if and only if i {\displaystyle i} and j {\displaystyle j} are in different subsets, equivalent to logical XOR. Let W ∈ R + n × n {\displaystyle {\boldsymbol {W}}\in \mathbb {R} _{+}^{n\times n}} with W i j = w i j ∀ i , j ∈ V {\displaystyle W_{ij}=w_{ij}~\forall i,j\in V} . By extending above expression to matrix-vector form we find that x ⊺ W ( 1 − x ) + ( 1 − x ) ⊺ W x = − 2 x ⊺ W x + ( W 1 + W ⊺ 1 ) ⊺ x {\displaystyle {\boldsymbol {x}}^{\intercal }{\boldsymbol {W}}({\boldsymbol {1}}-{\boldsymbol {x}})+({\boldsymbol {1}}-{\boldsymbol {x}})^{\intercal }{\boldsymbol {Wx}}=-2{\boldsymbol {x}}^{\intercal }{\boldsymbol {Wx}}+({\boldsymbol {W1}}+{\boldsymbol {W}}^{\intercal }{\boldsymbol {1}})^{\intercal }{\boldsymbol {x}}} is the sum of weights of all edges between S {\displaystyle S} and T {\displaystyle T} , where 1 = ( 1 , 1 , … , 1 ) ⊺ ∈ R n {\displaystyle {\boldsymbol {1}}=(1,1,\dots ,1)^{\intercal }\in \mathbb {R} ^{n}} . As this is a quadratic function over x {\displaystyle {\boldsymbol {x}}} , it is a QUBO problem whose parameter matrix we can read from above expression as Q = 2 W − diag ⁡ [ W 1 + W ⊺ 1 ] , {\displaystyle {\boldsymbol {Q}}=2{\boldsymbol {W}}-\operatorname {diag} [{\boldsymbol {W1}}+{\boldsymbol {W}}^{\intercal }{\bol