Neurocomputing (journal)

Neurocomputing is a peer-reviewed scientific journal covering research on artificial intelligence, machine learning, and neural computation. It was established in 1989 and is published by Elsevier. The editor-in-chief is Zidong Wang (Brunel University London). Independent scientometric studies noted that despite being one of the most productive journals in the field, it has kept its reputation across the years intact and plays an important role in leading the research in the area. The journal is abstracted and indexed in Scopus and Science Citation Index Expanded. According to the Journal Citation Reports, its 2023 impact factor is 5.5.

Superellipsoid

In mathematics, a superellipsoid (or super-ellipsoid) is a solid whose horizontal sections are superellipses (Lamé curves) with the same squareness parameter ϵ 2 {\displaystyle \epsilon _{2}} , and whose vertical sections through the center are superellipses with the squareness parameter ϵ 1 {\displaystyle \epsilon _{1}} . It is a generalization of an ellipsoid, which is a special case when ϵ 1 = ϵ 2 = 1 {\displaystyle \epsilon _{1}=\epsilon _{2}=1} . Superellipsoids as computer graphics primitives were popularized by Alan H. Barr (who used the name "superquadrics" to refer to both superellipsoids and supertoroids). In modern computer vision and robotics literatures, superquadrics and superellipsoids are used interchangeably, since superellipsoids are the most representative and widely utilized shape among all the superquadrics. Superellipsoids have a rich shape vocabulary, including cuboids, cylinders, ellipsoids, octahedra and their intermediates. It becomes an important geometric primitive widely used in computer vision, robotics, and physical simulation. The main advantage of describing objects and environment with superellipsoids is its conciseness and expressiveness in shape. Furthermore, a closed-form expression of the Minkowski sum between two superellipsoids is available. This makes it a desirable geometric primitive for robot grasping, collision detection, and motion planning. == Special cases == A handful of notable mathematical figures can arise as special cases of superellipsoids given the correct set of values, which are depicted in the above graphic: Cylinder Sphere Steinmetz solid Bicone Regular octahedron Cube, as a limiting case where the exponents tend to infinity Piet Hein's supereggs are also special cases of superellipsoids. == Formulas == === Basic (normalized) superellipsoid === The basic superellipsoid is defined by the implicit function f ( x , y , z ) = ( x 2 ϵ 2 + y 2 ϵ 2 ) ϵ 2 / ϵ 1 + z 2 ϵ 1 {\displaystyle f(x,y,z)=\left(x^{\frac {2}{\epsilon _{2}}}+y^{\frac {2}{\epsilon _{2}}}\right)^{\epsilon _{2}/\epsilon _{1}}+z^{\frac {2}{\epsilon _{1}}}} The parameters ϵ 1 {\displaystyle \epsilon _{1}} and ϵ 2 {\displaystyle \epsilon _{2}} are positive real numbers that control the squareness of the shape. The surface of the superellipsoid is defined by the equation: f ( x , y , z ) = 1 {\displaystyle f(x,y,z)=1} For any given point ( x , y , z ) ∈ R 3 {\displaystyle (x,y,z)\in \mathbb {R} ^{3}} , the point lies inside the superellipsoid if f ( x , y , z ) < 1 {\displaystyle f(x,y,z)<1} , and outside if f ( x , y , z ) > 1 {\displaystyle f(x,y,z)>1} . Any "parallel of latitude" of the superellipsoid (a horizontal section at any constant z between -1 and +1) is a Lamé curve with exponent 2 / ϵ 2 {\displaystyle 2/\epsilon _{2}} , scaled by a = ( 1 − z 2 ϵ 1 ) ϵ 1 2 {\displaystyle a=(1-z^{\frac {2}{\epsilon _{1}}})^{\frac {\epsilon _{1}}{2}}} , which is ( x a ) 2 ϵ 2 + ( y a ) 2 ϵ 2 = 1. {\displaystyle \left({\frac {x}{a}}\right)^{\frac {2}{\epsilon _{2}}}+\left({\frac {y}{a}}\right)^{\frac {2}{\epsilon _{2}}}=1.} Any "meridian of longitude" (a section by any vertical plane through the origin) is a Lamé curve with exponent 2 / ϵ 1 {\displaystyle 2/\epsilon _{1}} , stretched horizontally by a factor w that depends on the sectioning plane. Namely, if x = u cos ⁡ θ {\displaystyle x=u\cos \theta } and y = u sin ⁡ θ {\displaystyle y=u\sin \theta } , for a given θ {\displaystyle \theta } , then the section is ( u w ) 2 ϵ 1 + z 2 ϵ 1 = 1 , {\displaystyle \left({\frac {u}{w}}\right)^{\frac {2}{\epsilon _{1}}}+z^{\frac {2}{\epsilon _{1}}}=1,} where w = ( cos 2 ϵ 2 ⁡ θ + sin 2 ϵ 2 ⁡ θ ) − ϵ 2 2 . {\displaystyle w=(\cos ^{\frac {2}{\epsilon _{2}}}\theta +\sin ^{\frac {2}{\epsilon _{2}}}\theta )^{-{\frac {\epsilon _{2}}{2}}}.} In particular, if ϵ 2 {\displaystyle \epsilon _{2}} is 1, the horizontal cross-sections are circles, and the horizontal stretching w {\displaystyle w} of the vertical sections is 1 for all planes. In that case, the superellipsoid is a solid of revolution, obtained by rotating the Lamé curve with exponent 2 / ϵ 1 {\displaystyle 2/\epsilon _{1}} around the vertical axis. === Superellipsoid === The basic shape above extends from −1 to +1 along each coordinate axis. The general superellipsoid is obtained by scaling the basic shape along each axis by factors a x {\displaystyle a_{x}} , a y {\displaystyle a_{y}} , a z {\displaystyle a_{z}} , the semi-diameters of the resulting solid. The implicit function is F ( x , y , z ) = ( ( x a x ) 2 ϵ 2 + ( y a y ) 2 ϵ 2 ) ϵ 2 ϵ 1 + ( z a z ) 2 ϵ 1 {\displaystyle F(x,y,z)=\left(\left({\frac {x}{a_{x}}}\right)^{\frac {2}{\epsilon _{2}}}+\left({\frac {y}{a_{y}}}\right)^{\frac {2}{\epsilon _{2}}}\right)^{\frac {\epsilon _{2}}{\epsilon _{1}}}+\left({\frac {z}{a_{z}}}\right)^{\frac {2}{\epsilon _{1}}}} . Similarly, the surface of the superellipsoid is defined by the equation F ( x , y , z ) = 1 {\displaystyle F(x,y,z)=1} For any given point ( x , y , z ) ∈ R 3 {\displaystyle (x,y,z)\in \mathbb {R} ^{3}} , the point lies inside the superellipsoid if f ( x , y , z ) < 1 {\displaystyle f(x,y,z)<1} , and outside if f ( x , y , z ) > 1 {\displaystyle f(x,y,z)>1} . Therefore, the implicit function is also called the inside-outside function of the superellipsoid. The superellipsoid has a parametric representation in terms of surface parameters η ∈ [ − π / 2 , π / 2 ) {\displaystyle \eta \in [-\pi /2,\pi /2)} , ω ∈ [ − π , π ) {\displaystyle \omega \in [-\pi ,\pi )} . x ( η , ω ) = a x cos ϵ 1 ⁡ η cos ϵ 2 ⁡ ω {\displaystyle x(\eta ,\omega )=a_{x}\cos ^{\epsilon _{1}}\eta \cos ^{\epsilon _{2}}\omega } y ( η , ω ) = a y cos ϵ 1 ⁡ η sin ϵ 2 ⁡ ω {\displaystyle y(\eta ,\omega )=a_{y}\cos ^{\epsilon _{1}}\eta \sin ^{\epsilon _{2}}\omega } z ( η , ω ) = a z sin ϵ 1 ⁡ η {\displaystyle z(\eta ,\omega )=a_{z}\sin ^{\epsilon _{1}}\eta } === General posed superellipsoid === In computer vision and robotic applications, a superellipsoid with a general pose in the 3D Euclidean space is usually of more interest. For a given Euclidean transformation of the superellipsoid frame g = [ R ∈ S O ( 3 ) , t ∈ R 3 ] ∈ S E ( 3 ) {\displaystyle g=[\mathbf {R} \in SO(3),\mathbf {t} \in \mathbb {R} ^{3}]\in SE(3)} relative to the world frame, the implicit function of a general posed superellipsoid surface defined the world frame is F ( g − 1 ∘ ( x , y , z ) ) = 1 {\displaystyle F\left(g^{-1}\circ (x,y,z)\right)=1} where ∘ {\displaystyle \circ } is the transformation operation that maps the point ( x , y , z ) ∈ R 3 {\displaystyle (x,y,z)\in \mathbb {R} ^{3}} in the world frame into the canonical superellipsoid frame. === Volume of superellipsoid === The volume encompassed by the superelllipsoid surface can be expressed in terms of the beta functions β ( ⋅ , ⋅ ) {\displaystyle \beta (\cdot ,\cdot )} , V ( ϵ 1 , ϵ 2 , a x , a y , a z ) = 2 a x a y a z ϵ 1 ϵ 2 β ( ϵ 1 2 , ϵ 1 + 1 ) β ( ϵ 2 2 , ϵ 2 + 2 2 ) {\displaystyle V(\epsilon _{1},\epsilon _{2},a_{x},a_{y},a_{z})=2a_{x}a_{y}a_{z}\epsilon _{1}\epsilon _{2}\beta ({\frac {\epsilon _{1}}{2}},\epsilon _{1}+1)\beta ({\frac {\epsilon _{2}}{2}},{\frac {\epsilon _{2}+2}{2}})} or equivalently with the Gamma function Γ ( ⋅ ) {\displaystyle \Gamma (\cdot )} , since β ( m , n ) = Γ ( m ) Γ ( n ) Γ ( m + n ) {\displaystyle \beta (m,n)={\frac {\Gamma (m)\Gamma (n)}{\Gamma (m+n)}}} == Recovery from data == Recoverying the superellipsoid (or superquadrics) representation from raw data (e.g., point cloud, mesh, images, and voxels) is an important task in computer vision, robotics, and physical simulation. Traditional computational methods model the problem as a least-square problem. The goal is to find out the optimal set of superellipsoid parameters θ ≐ [ ϵ 1 , ϵ 2 , a x , a y , a z , g ] {\displaystyle \theta \doteq [\epsilon _{1},\epsilon _{2},a_{x},a_{y},a_{z},g]} that minimize an objective function. Other than the shape parameters, g ∈ {\displaystyle g\in } SE(3) is the pose of the superellipsoid frame with respect to the world coordinate. There are two commonly used objective functions. The first one is constructed directly based on the implicit function G 1 ( θ ) = a x a y a z ∑ i = 1 N ( F ϵ 1 ( g − 1 ∘ ( x i , y i , z i ) ) − 1 ) 2 {\displaystyle G_{1}(\theta )=a_{x}a_{y}a_{z}\sum _{i=1}^{N}\left(F^{\epsilon _{1}}\left(g^{-1}\circ (x_{i},y_{i},z_{i})\right)-1\right)^{2}} The minimization of the objective function provides a recovered superellipsoid as close as possible to all the input points { ( x i , y i , z i ) ∈ R 3 , i = 1 , 2 , . . . , N } {\displaystyle \{(x_{i},y_{i},z_{i})\in \mathbb {R} ^{3},i=1,2,...,N\}} . At the mean time, the scalar value a x , a y , a z {\displaystyle a_{x},a_{y},a_{z}} is positively proportional to the volume of the superellipsoid, and thus have the effect of minimizing the volume as well. The other objective function tries to minimized the radial distance between the points and the superellipsoid. That is G 2 ( θ ) = ∑ i = 1 N ( | r

Argumentation framework

In artificial intelligence and related fields, an argumentation framework is a way to deal with contentious information and draw conclusions from it using formalized arguments. In an abstract argumentation framework, entry-level information is a set of abstract arguments that, for instance, represent data or a proposition. Conflicts between arguments are represented by a binary relation on the set of arguments. In concrete terms, an argumentation framework is represented with a directed graph such that the nodes are the arguments, and the arrows represent the attack relation. There exist some extensions of the Dung's framework, like the logic-based argumentation frameworks or the value-based argumentation frameworks. == Abstract argumentation frameworks == === Formal framework === Abstract argumentation frameworks, also called argumentation frameworks à la Dung, are defined formally as a pair: A set of abstract elements called arguments, denoted A {\displaystyle A} A binary relation on A {\displaystyle A} , called attack relation, denoted R {\displaystyle R} For instance, the argumentation system S = ⟨ A , R ⟩ {\displaystyle S=\langle A,R\rangle } with A = { a , b , c , d } {\displaystyle A=\{a,b,c,d\}} and R = { ( a , b ) , ( b , c ) , ( d , c ) } {\displaystyle R=\{(a,b),(b,c),(d,c)\}} contains four arguments ( a , b , c {\displaystyle a,b,c} and d {\displaystyle d} ) and three attacks ( a {\displaystyle a} attacks b {\displaystyle b} , b {\displaystyle b} attacks c {\displaystyle c} and d {\displaystyle d} attacks c {\displaystyle c} ). Dung defines some notions : an argument a ∈ A {\displaystyle a\in A} is acceptable with respect to E ⊆ A {\displaystyle E\subseteq A} if and only if E {\displaystyle E} defends a {\displaystyle a} , that is ∀ b ∈ A {\displaystyle \forall b\in A} such that ( b , a ) ∈ R , ∃ c ∈ E {\displaystyle (b,a)\in R,\exists c\in E} such that ( c , b ) ∈ R {\displaystyle (c,b)\in R} , a set of arguments E {\displaystyle E} is conflict-free if there is no attack between its arguments, formally : ∀ a , b ∈ E , ( a , b ) ∉ R {\displaystyle \forall a,b\in E,(a,b)\not \in R} , a set of arguments E {\displaystyle E} is admissible if and only if it is conflict-free and all its arguments are acceptable with respect to E {\displaystyle E} . === Different semantics of acceptance === ==== Extensions ==== To decide if an argument can be accepted or not, or if several arguments can be accepted together, Dung defines several semantics of acceptance that allows, given an argumentation system, sets of arguments (called extensions) to be computed. For instance, given S = ⟨ A , R ⟩ {\displaystyle S=\langle A,R\rangle } , E {\displaystyle E} is a complete extension of S {\displaystyle S} only if it is an admissible set and every acceptable argument with respect to E {\displaystyle E} belongs to E {\displaystyle E} , E {\displaystyle E} is a preferred extension of S {\displaystyle S} only if it is a maximal element (with respect to the set-theoretical inclusion) among the admissible sets with respect to S {\displaystyle S} , E {\displaystyle E} is a stable extension of S {\displaystyle S} only if it is a conflict-free set that attacks every argument that does not belong in E {\displaystyle E} (formally, ∀ a ∈ A ∖ E , ∃ b ∈ E {\displaystyle \forall a\in A\backslash E,\exists b\in E} such that ( b , a ) ∈ R {\displaystyle (b,a)\in R} , E {\displaystyle E} is the (unique) grounded extension of S {\displaystyle S} only if it is the smallest element (with respect to set inclusion) among the complete extensions of S {\displaystyle S} . There exists some inclusions between the sets of extensions built with these semantics : Every stable extension is preferred, Every preferred extension is complete, The grounded extension is complete, If the system is well-founded (there exists no infinite sequence a 0 , a 1 , … , a n , … {\displaystyle a_{0},a_{1},\dots ,a_{n},\dots } such that ∀ i > 0 , ( a i + 1 , a i ) ∈ R {\displaystyle \forall i>0,(a_{i+1},a_{i})\in R} ), all these semantics coincide—only one extension is grounded, stable, preferred, and complete. Some other semantics have been defined. One introduce the notation E x t σ ( S ) {\displaystyle Ext_{\sigma }(S)} to note the set of σ {\displaystyle \sigma } -extensions of the system S {\displaystyle S} . In the case of the system S {\displaystyle S} in the figure above, E x t σ ( S ) = { { a , d } } {\displaystyle Ext_{\sigma }(S)=\{\{a,d\}\}} for every Dung's semantic—the system is well-founded. That explains why the semantics coincide, and the accepted arguments are: a {\displaystyle a} and d {\displaystyle d} . ==== Labellings ==== Labellings are a more expressive way than extensions to express the acceptance of the arguments. Concretely, a labelling is a mapping that associates every argument with a label in (the argument is accepted), out (the argument is rejected), or undec (the argument is undefined—not accepted or refused). One can also note a labelling as a set of pairs ( a r g u m e n t , l a b e l ) {\displaystyle ({\mathit {argument}},{\mathit {label}})} . Such a mapping does not make sense without additional constraint. The notion of reinstatement labelling guarantees the sense of the mapping. L {\displaystyle L} is a reinstatement labelling on the system S = ⟨ A , R ⟩ {\displaystyle S=\langle A,R\rangle } if and only if : ∀ a ∈ A , L ( a ) = i n {\displaystyle \forall a\in A,L(a)={\mathit {in}}} if and only if ∀ b ∈ A {\displaystyle \forall b\in A} such that ( b , a ) ∈ R , L ( b ) = o u t {\displaystyle (b,a)\in R,L(b)={\mathit {out}}} ∀ a ∈ A , L ( a ) = o u t {\displaystyle \forall a\in A,L(a)={\mathit {out}}} if and only if ∃ b ∈ A {\displaystyle \exists b\in A} such that ( b , a ) ∈ R {\displaystyle (b,a)\in R} and L ( b ) = i n {\displaystyle L(b)={\mathit {in}}} ∀ a ∈ A , L ( a ) = u n d e c {\displaystyle \forall a\in A,L(a)={\mathit {undec}}} if and only if L ( a ) ≠ i n {\displaystyle L(a)\neq {\mathit {in}}} and L ( a ) ≠ o u t {\displaystyle L(a)\neq {\mathit {out}}} One can convert every extension into a reinstatement labelling: the arguments of the extension are in, those attacked by an argument of the extension are out, and the others are undec. Conversely, one can build an extension from a reinstatement labelling just by keeping the arguments in. Indeed, Caminada proved that the reinstatement labellings and the complete extensions can be mapped in a bijective way. Moreover, the other Datung's semantics can be associated to some particular sets of reinstatement labellings. Reinstatement labellings distinguish arguments not accepted because they are attacked by accepted arguments from undefined arguments—that is, those that are not defended cannot defend themselves. An argument is undec if it is attacked by at least another undec. If it is attacked only by arguments out, it must be in, and if it is attacked some argument in, then it is out. The unique reinstatement labelling that corresponds to the system S {\displaystyle S} above is L = { ( a , i n ) , ( b , o u t ) , ( c , o u t ) , ( d , i n ) } {\displaystyle L=\{(a,{\mathit {in}}),(b,{\mathit {out}}),(c,{\mathit {out}}),(d,{\mathit {in}})\}} . === Inference from an argumentation system === In the general case when several extensions are computed for a given semantic σ {\displaystyle \sigma } , the agent that reasons from the system can use several mechanisms to infer information: Credulous inference: the agent accepts an argument if it belongs to at least one of the σ {\displaystyle \sigma } -extensions—in which case, the agent risks accepting some arguments that are not acceptable together ( a {\displaystyle a} attacks b {\displaystyle b} , and a {\displaystyle a} and b {\displaystyle b} each belongs to an extension) Skeptical inference: the agent accepts an argument only if it belongs to every σ {\displaystyle \sigma } -extension. In this case, the agent risks deducing too little information (if the intersection of the extensions is empty or has a very small cardinal). For these two methods to infer information, one can identify the set of accepted arguments, respectively C r σ ( S ) {\displaystyle Cr_{\sigma }(S)} the set of the arguments credulously accepted under the semantic σ {\displaystyle \sigma } , and S c σ ( S ) {\displaystyle Sc_{\sigma }(S)} the set of arguments accepted skeptically under the semantic σ {\displaystyle \sigma } (the σ {\displaystyle \sigma } can be missed if there is no possible ambiguity about the semantic). Of course, when there is only one extension (for instance, when the system is well-founded), this problem is very simple: the agent accepts arguments of the unique extension and rejects others. The same reasoning can be done with labellings that correspond to the chosen semantic : an argument can be accepted if it is in for each labelling and refused if it is out for each labelling, the others being in an undecided state (the status of the arguments can remind the

Superintelligence ban

Superintelligence ban refers to proposed legal, ethical, or policy measures intended to restrict or prohibit the development of artificial superintelligence, AI systems that would surpass human cognitive abilities in nearly all domains. The idea arises from concerns that such systems could become uncontrollable, potentially posing existential threats to humanity or causing severe social and economic disruption. == Background == The concept of limiting or banning superintelligence research has roots in early 21st-century debates on artificial general intelligence (AGI) safety. Thinkers such as Nick Bostrom and Eliezer Yudkowsky warned that self-improving AI could rapidly exceed human oversight. As advanced models like large-scale language models and autonomous agents began demonstrating complex reasoning abilities, policymakers and ethicists increasingly discussed the need for legal constraints on the creation of systems capable of recursive self-improvement. In October 2025, the Future of Life Institute published a statement calling for "a prohibition on the development of superintelligence, not lifted before there is broad scientific consensus that it will be done safely and controllably, and strong public buy-in." This statement was signed by various public personalities, such as Richard Branson and Steve Wozniak, and AI experts, such as Yoshua Bengio and Geoffrey Hinton. == Rationale == Supporters of a superintelligence ban argue that once AI systems surpass human intelligence, traditional containment, alignment, and control methods may fail. They contend that even limited experimentation with such systems could lead to irreversible outcomes, including loss of human decision-making power or unintended global harm. Some propose international treaties modeled after the nuclear non-proliferation framework to prevent a competitive AI arms race. Opponents argue that a ban would be difficult to define and enforce, given the lack of a precise threshold distinguishing advanced AGI from superintelligence. They also warn that excessive restriction could slow scientific progress, hinder beneficial automation, and encourage unregulated underground research. == Global discussion == Although no government has enacted an explicit superintelligence ban, the idea has been debated within the European Union, United Nations, and several independent AI safety organizations. The Future of Life Institute, Center for AI Safety, and other organizations have called for international cooperation to manage risks associated with the pursuit of superintelligent systems. In 2024 and 2025, proposals for a temporary moratorium on frontier AI research were circulated among major technology firms and research institutes, reflecting growing public concern over the trajectory of AI capabilities.

Singularity studies

Singularity studies is an interdisciplinary academic field which examines the idea of technological singularity — the hypothesised point at which artificial intelligence may surpass human intelligence, might be attained by artificial intelligence (AI), robotics, and other technologies and sciences, and its social impacts. In this academic field, the study and research are conducted across a broad array of terrains such as information science, robotics, social informatics, economics, philosophy, and ethics. The primary aim of singularity studies is to gain an integrative understanding of the transformation of social systems occurring in tandem with the explosive evolution of AI and also the changes to be effected by such transformation in the view of humans, ethics, and legal systems. == History == An academic work on technological singurality has appeared in computer science, philosophy, sociology, and law since the early 1990s. Early discussions of an intelligence explosion were popularised by science-fiction writer Vernor Vinge in 1993 and later systematised by futurist Ray Kurzweil. Since the 2010s, universities such as Oxford, Stanford, and Keio have established dedicated programmes, while peer-reviewed journals have begun to publish scenario analyses and policy studies. Ongoing debates question the predictive value of singularity scenarios and warn against a deterministic view of technology. == Characteristics of research == Singularity studies extends beyond mere future predictions and offer an intellectual foundation for proactively designing and creating a desirable future. Principal research themes in this realm include: Ethics of AI; Social implications of technologies; Possibility of harmonious coexistence of humans and AI; Communication with AI; and Redesign of social systems. == Technologists and academics == Vernor Vinge: Propounded the concept of singularity in 1993, making a massive impact on the academic and science-fiction spheres. Ray Kurzweil: Predicted the advent around 2045 of the technological singularity in his 2005 book The Singularity Is Near. Nick Bostrom: Offered philosophical reflections on superintelligence and the risks posed by AI. He is the founding director of the now-dissolved Future of Humanity Institute at the University of Oxford. === Japan === Kento Sasano: A social informatician, AI educator, and inventor. He is the president of the Japan Society of Singularity Studies. == Challenges and outlook == Singularity studies is still evolving as an academic field, and quite a few challenges remain unresolved in regard to the systematization of their theories, research methods, and educational curricula. That said, in this day and age of accelerating technological and societal shifts, interdisciplinary approaches have gained in importance and are drawing much attention in the arenas of scholarly research, intercorporate collaboration, and policy planning.

Mean shift

Mean shift is a non-parametric feature-space mathematical analysis technique for locating the maxima of a density function, a so-called mode-seeking algorithm. Application domains include cluster analysis in computer vision and image processing. == History == The mean shift procedure is usually credited to work by Fukunaga and Hostetler in 1975. It is, however, reminiscent of earlier work by Schnell in 1964. == Overview == Mean shift is a procedure for locating the maxima—the modes—of a density function given discrete data sampled from that function. This is an iterative method, and we start with an initial estimate x {\displaystyle x} . Let a kernel function K ( x i − x ) {\displaystyle K(x_{i}-x)} be given. This function determines the weight of nearby points for re-estimation of the mean. Typically a Gaussian kernel on the distance to the current estimate is used, K ( x i − x ) = e − c | | x i − x | | 2 {\displaystyle K(x_{i}-x)=e^{-c||x_{i}-x||^{2}}} . The weighted mean of the density in the window determined by K {\displaystyle K} is m ( x ) = ∑ x i ∈ N ( x ) K ( x i − x ) x i ∑ x i ∈ N ( x ) K ( x i − x ) {\displaystyle m(x)={\frac {\sum _{x_{i}\in N(x)}K(x_{i}-x)x_{i}}{\sum _{x_{i}\in N(x)}K(x_{i}-x)}}} where N ( x ) {\displaystyle N(x)} is the neighborhood of x {\displaystyle x} , a set of points for which K ( x i − x ) ≠ 0 {\displaystyle K(x_{i}-x)\neq 0} . The difference m ( x ) − x {\displaystyle m(x)-x} is called mean shift in Fukunaga and Hostetler. The mean-shift algorithm now sets x ← m ( x ) {\displaystyle x\leftarrow m(x)} , and repeats the estimation until m ( x ) {\displaystyle m(x)} converges. Although the mean shift algorithm has been widely used in many applications, a rigid proof for the convergence of the algorithm using a general kernel in a high dimensional space is still not known. Aliyari Ghassabeh showed the convergence of the mean shift algorithm in one dimension with a differentiable, convex, and strictly decreasing profile function. However, the one-dimensional case has limited real world applications. Also, the convergence of the algorithm in higher dimensions with a finite number of the stationary (or isolated) points has been proved. However, sufficient conditions for a general kernel function to have finite stationary (or isolated) points have not been provided. Gaussian Mean-Shift is an Expectation–maximization algorithm. == Details == Let data be a finite set S {\displaystyle S} embedded in the n {\displaystyle n} -dimensional Euclidean space, X {\displaystyle X} . Let K {\displaystyle K} be a flat kernel that is the characteristic function of the λ {\displaystyle \lambda } -ball in X {\displaystyle X} , In each iteration of the algorithm, s ← m ( s ) {\displaystyle s\leftarrow m(s)} is performed for all s ∈ S {\displaystyle s\in S} simultaneously. The first question, then, is how to estimate the density function given a sparse set of samples. One of the simplest approaches is to just smooth the data, e.g., by convolving it with a fixed kernel of width h {\displaystyle h} , where x i {\displaystyle x_{i}} are the input samples and k ( r ) {\displaystyle k(r)} is the kernel function (or Parzen window). h {\displaystyle h} is the only parameter in the algorithm and is called the bandwidth. This approach is known as kernel density estimation or the Parzen window technique. Once we have computed f ( x ) {\displaystyle f(x)} from the equation above, we can find its local maxima using gradient ascent or some other optimization technique. The problem with this "brute force" approach is that, for higher dimensions, it becomes computationally prohibitive to evaluate f ( x ) {\displaystyle f(x)} over the complete search space. Instead, mean shift uses a variant of what is known in the optimization literature as multiple restart gradient descent. Starting at some guess for a local maximum, y k {\displaystyle y_{k}} , which can be a random input data point x 1 {\displaystyle x_{1}} , mean shift computes the gradient of the density estimate f ( x ) {\displaystyle f(x)} at y k {\displaystyle y_{k}} and takes an uphill step in that direction. == Types of kernels == Kernel definition: Let X {\displaystyle X} be the n {\displaystyle n} -dimensional Euclidean space, R n {\displaystyle \mathbb {R} ^{n}} . The norm of x {\displaystyle x} is a non-negative number, ‖ x ‖ 2 = x ⊤ x ≥ 0 {\displaystyle \|x\|^{2}=x^{\top }x\geq 0} . A function K : X → R {\displaystyle K:X\rightarrow \mathbb {R} } is said to be a kernel if there exists a profile, k : [ 0 , ∞ ] → R {\displaystyle k:[0,\infty ]\rightarrow \mathbb {R} } , such that K ( x ) = k ( ‖ x ‖ 2 ) {\displaystyle K(x)=k(\|x\|^{2})} and k is non-negative. k is non-increasing: k ( a ) ≥ k ( b ) {\displaystyle k(a)\geq k(b)} if a < b {\displaystyle a

.ai

.ai is the Internet country code top-level domain (ccTLD) for Anguilla, a British Overseas Territory in the Caribbean. It is administered by the government of Anguilla. It is a popular domain hack with companies and projects related to the artificial intelligence industry (AI). Google's ad targeting treats .ai as a generic top-level domain (gTLD) because "users and website owners frequently see [the domain] as being more generic than country-targeted." In 2021, Google Search analyst Gary Illyes announced that ".ai" had been added to Google’s list of generic country-code top-level domains, meaning that Google would no longer infer Anguilla-specific targeting from the ccTLD. Identity Digital began managing the domain as of January 2025. == Second and third level registrations == Registrations within off.ai, com.ai, net.ai, and org.ai are available worldwide without restriction. From 15 September 2009, second level registrations within .ai are available to everyone worldwide. == Registration == The minimum registration term allowed for .ai domains is 2 through 10 years for registration and renewal, and a 2-year renewal for domain transfer. Identity Digital is the authority in charge of managing this extension. Registrations began on 16 February 1995. The limits on the number of characters used for the domain name are, at a minimum, from 1 to 3, depending on the registrar, and always at most 63 characters. The character set supported for .ai domain names includes A–Z, a–z, 0–9, and hyphen. As of November 2022, .ai domains cannot accommodate IDN characters. There are no requirements for registering a domain, including local and foreign residents. A .ai domain can be suspended or revoked, if the domain is involved in illegal activity such as violating trademarks or copyrights. Usage must not violate the laws of Anguilla. Anguilla uses the UDRP. Filing a UDRP challenge requires using one of the ICANN Approved Dispute Resolution Service Providers. If the domain is with an ICANN accredited registrar, they should work with the arbitrator. Usually this means either doing nothing or transferring a domain. .ai domains are transferable to any desired registrars as the registration of domain is done maintaining EPP. There used to be a whois.ai-based platform of expired domains in which those could be procured and auctioned every ten days through a standard online process. The last auctions of such kind closed there in December 2024; the platform had been scheduled for shutdown on 30 June 2025, but remained online in the months following that date. == Valuation == Domains cost depends on the registrar, with yearly fees ranging from US$140 (the base fee, as established by Anguilla) to $200. As of July 2025, the highest-valued .ai domain is an undisclosed one sold on 8 November 2023, on Escrow.com, for US$1,500,000—months after an initial $300,000 sale to the same buyer. Among the publicly disclosed ones, the most valued, fin.ai, was sold for $1,000,000 in March 2025. On 16 December 2017, the .ai registry started supporting the Extensible Provisioning Protocol (EPP) and migrated all of its domains onto an EPP system. Consequently, many registrars are allowed to sell .ai domains. Since that date, the .ai ccTLD has also been popular with artificial intelligence companies and organisations. Though such trends are primarily seen among new AI based companies or startups, many established AI and Tech companies preferred not to opt for .ai domains. For example, DeepMind has its domain retained at .com; Meta has redirected its facebook.ai domain to ai.meta.com. == Impact on Anguilla's economy == The registration fees earned from the .ai domains go to the treasury of the Government of Anguilla. As per a 2018 New York Times report, the total revenue generated out of selling .ai domains was $2.9 million. In 2023, Anguilla's government made about US$32 million from fees collected for registering .ai domains; that amounted to over 10% of gross domestic product for the territory. "In the years before the real breakthrough of AI, revenue from .ai domains made up less than 1% of our state income, by 2025 it will be around 47%," explained Jose Vanterpool, Minister of Infrastructure and Communications (MICUHITES), in an interview with BBC. The high 90% renewal rate of .ai domains and the 2025 renewal wave of domains registered in 2023 are driving another surge in state revenues, according to Domaintechnik.