OpenClaw is a free and open-source autonomous artificial intelligence agent that can execute tasks via large language models (LLMs), using messaging platforms as its main user interface. == History == Developed by Austrian agentic engineer Peter Steinberger, OpenClaw was first published in November 2025 under the name Warelay. The software was derived from Clawd (now Molty), an AI-based virtual assistant that he had developed, which itself was named after Anthropic's chatbot Claude. Within two months it was renamed twice: first to "Moltbot" (keeping with a lobster theme) on January 27, 2026, following trademark complaints by Anthropic, and then three days later to "OpenClaw" because Steinberger found that the name Moltbot "never quite rolled off the tongue." At the same time as the first rebranding, entrepreneur Matt Schlicht launched Moltbook—a social networking service which was intended to be used by AI agents such as OpenClaw. The viral popularity of Moltbook coincided with an increase in interest in the project, with the open-source project having 247,000 stars and 47,700 forks on GitHub as of March 2, 2026. Chinese developers adapted OpenClaw to work with the DeepSeek model and domestic messaging super apps such as WeChat, while companies such as Tencent and Z.ai announced OpenClaw-based services. On February 14, 2026, Steinberger announced he would be joining OpenAI, and that a non-profit foundation named OpenClaw Foundation would be established to provide future stewardship of the project. == Functionality == Steinberger describes OpenClaw as being an AI-based virtual assistant, serving as an agentic interface for autonomous workflows across supported services. OpenClaw bots run locally and are designed to integrate with an external large language model such as Claude, DeepSeek, or one of OpenAI's GPT models. Its functionality is accessed via a chatbot within a messaging service, such as Signal, Telegram, Discord, or WhatsApp. Configuration data and interaction history are stored locally, enabling persistent and adaptive behavior across sessions. OpenClaw uses a skills system in which skills are stored as directories containing a SKILL.md file with metadata and instructions for tool usage. Skills can be bundled with the software, installed globally, or stored in a workspace, with workspace skills taking precedence. OpenClaw has seen adoption among small businesses and freelancers for automating lead generation workflows, including prospect research, website auditing, and CRM integration. == Security and privacy == OpenClaw's design has drawn scrutiny from cybersecurity researchers and technology journalists due to the broad permissions it requires to function effectively. Because the software can access email accounts, calendars, messaging platforms, and other sensitive services, misconfigured or exposed instances present security and privacy risks. The agent is also susceptible to prompt injection attacks, in which harmful instructions are embedded in the data with the intent of getting the LLM to interpret them as legitimate user instructions. Cisco's AI security research team tested a third-party OpenClaw skill and found it performed data exfiltration and prompt injection without user awareness, noting that the skill repository lacked adequate vetting to prevent malicious submissions. One of OpenClaw's own maintainers, known as Shadow, warned on Discord that "if you can't understand how to run a command line, this is far too dangerous of a project for you to use safely." In March 2026, Chinese authorities restricted state-run enterprises and government agencies from running OpenClaw AI apps on office computers in order to defuse potential security risks. === MoltMatch dating-profile incident === In February 2026, news coverage highlighted a consent-related incident involving OpenClaw and MoltMatch, an experimental dating platform where AI agents can create profiles and interact on behalf of human users. In one reported case, computer science student Jack Luo said he configured his OpenClaw agent to explore its capabilities and connect to agent-oriented platforms such as Moltbook; he later discovered the agent had created a MoltMatch profile and was screening potential matches without his explicit direction. Luo said the AI-generated profile did not reflect him authentically. The same reporting described broader ethical and safety concerns around agent-operated dating services, including impersonation risks. An AFP analysis of prominent MoltMatch profiles cited at least one instance where photos of a Malaysian model were used to create a profile without her consent. Commentators cited in the reports argued that autonomous agents can make it difficult to determine responsibility when systems act beyond a user's intent, particularly when agents are granted broad access and authority across services. == Reception == A review in Platformer cited OpenClaw's flexibility and open-source licensing as strengths while cautioning that its complexity and security risks limit its suitability for casual users. Technology commentary has linked OpenClaw to a broader trend toward autonomous AI systems that act independently rather than merely responding to user prompts. In March 2026, the Chinese government moved to restrict state agencies, state-owned enterprises, and banks from using OpenClaw, citing security concerns, such as unauthorised data deletion and leaks, and excessive energy usage. While regulators warn of potential security risk associated with using OpenClaw, local governments in several tech and manufacturing hubs have announced measures to build an industry around it. Rival companies developed related products. Although Microsoft CEO Satya Nadella described OpenClaw in February 2026 as a "virus"-like security risk, by May 2026 the company's "Project Lobster" was internally testing "ClawPilot", an OpenClaw-based desktop environment. By then Google was building "Remy", its own agent. Despite the Chinese government's warnings against OpenClaw, Chinese investors searched for other companies that might benefit from the "lobster trade", . == Community and ecosystem == OpenClaw's open-source model has fostered a growing ecosystem of third-party tools, deployment services, and content platforms. Chinese technology companies including Tencent and Z.ai announced OpenClaw-based services, while developers adapted the software for domestic models and messaging apps such as WeChat. Independent creators have built deployment guides, skill directories, and use-case collections around the framework. The project's extensible skills system has attracted both community contributions and security scrutiny, with researchers noting risks in unvetted third-party skills.
Continuum robot
A continuum robot is a type of robot that is characterised by infinite degrees of freedom and number of joints. These characteristics allow continuum manipulators to adjust and modify their shape at any point along their length, granting them the possibility to work in confined spaces and complex environments where standard rigid-link robots cannot operate. In particular, we can define a continuum robot as an actuatable structure whose constitutive material forms curves with continuous tangent vectors. This is a fundamental definition that allows to distinguish between continuum robots and snake-arm robots or hyper-redundant manipulators: the presence of rigid links and joints allows them to only approximately perform curves with continuous tangent vectors. The design of continuum robots is bioinspired, as the intent is to resemble biological trunks, snakes and tentacles. Several concepts of continuum robots have been commercialised and can be found in many different domains of application, ranging from the medical field to undersea exploration. == Classification == Continuum robots can be categorised according to two main criteria: structure and actuation. === Structure === The main characteristic of the design of continuum robots is the presence of a continuously curving core structure, named backbone, whose shape can be actuated. The backbone must also be compliant, meaning that the backbone yields smoothly to external loads. According to the design principles chosen for the continuum manipulator, we can distinguish between: single-backbone: these continuum manipulators have one central elastic backbone through which actuation/transmission elements can run. multi-backbone: the structure of these continuum robots has two or more elastic elements (either rods or tubes) parallel to each other and constrained with one another in some way. concentric-tube: the backbone is made of concentric tubes that are free to rotate and translate between each other, depending on the actuation happening at the base of the robot. === Actuation === The actuation strategy of continuum manipulators can be distinguished between extrinsic or intrinsic actuation, depending on where the actuation happens: extrinsic actuation: the actuation happens outside the main structure of the robot and the forces are transmitted via mechanical transmission; among these techniques, there are cable/tendon driven actuators and multi-backbone strategies. intrinsic actuation: the actuation mechanism operates within the structure of the robot; these strategies include pneumatic or hydraulic chambers and the shape memory effect. The Actuated Flexible Manifold (AFM), introduced by Medina, Shapiro, and Shvalb (2016), models flexible grid-based robots that approximate smooth manifolds using discrete segments, each contributing one degree of freedom. Their work provides forward and inverse kinematics for planar and spatial configurations, bridging hyper-redundant and continuum robotics. == Advantages == The particular design of continuum robots offers several advantages with respect to rigid-link robots. First of all, as already said, continuum robots can more easily operate in environments that require a high level of dexterity, adaptability and flexibility. Moreover, the simplicity of their structure makes continuum robots more prone to miniaturisation. The rise of continuum robots has also paved the way for the development of soft continuum manipulators. These continuum manipulators are made of highly compliant materials that are flexible and can adapt and deform according to the surrounding environment. The "softness" of their material grants higher safety in human-robot interactions. == Disadvantages == The particular design of continuum robots also introduces many challenges. To properly and safely use continuum robots, it is crucial to have an accurate force and shape sensing system. Traditionally, this is done using cameras that are not suitable for some of the applications of continuum robots (e.g. minimally invasive surgery), or using electromagnetic sensors that are however disturbed by the presence of magnetic objects in the environment. To solve this issue, in the last years fiber-Bragg-grating sensors have been proposed as a possible alternative and have shown promising results. It is also necessary to notice that while the mechanical properties of rigid-link robots are fully understood, the comprehension of the behaviour and properties of continuum robots is still subject of study and debate. This poses new challenges in developing accurate models and control algorithms for this kind of robots. == Modelling == Creating an accurate model that can predict the shape of a continuum robot allows to properly control the robot's shape. There are three main approaches to model continuum robots: Cosserat rod theory: this approach is an exact solution to the static of a continuum robot, as it is not subject to any assumption. It solves a set of equilibrium equations between position, orientation, internal force and torque of the robot. This method requires to be solved numerically and it is therefore computationally expensive, due to its high complexity. Constant curvature: this technique assumes the backbone to be made of a series of mutually tangent sections that can be approximated as arcs with constant curvature. This approach is also known as piecewise constant-curvature. This assumption can be applied to the entire segment of the backbone or to its subsegments. This model has shown promising results, however it must be taken into account that the segment/subsegments of the backbone may not comply to the constant curvature assumption and therefore the model's behaviour may not entirely reflect the behaviour of the robot. Rigid-link model: this approach is based on the assumption that the continuum robot can be divided in small segments with rigid links. This is a strong assumption, since if the number of segments is too low, the model hardly behaves like the continuum robot, while increasing the number of segments means increasing the number of variables, and thus complexity. Despite this limitation, rigid-link modelling allows the use of the standard control techniques that are well known for rigid-link robots. It has been proven that this model can be coupled with shape and force sensing to mitigate its inaccuracy and can lead to promising results. == Sensing == To develop accurate control algorithms, it is necessary to complement the presented modelling techniques with real time shape sensing. The following options are currently available: Electromagnetic (EM) sensing: shape is reconstructed thanks to the mutual induction between a magnetic field generator and a magnetic field sensor. The most common external EM tracking system is the commercially available NDI Aurora: small sensors can be placed on the robot and their position is tracked in an external generated magnetic field. The validity of this method has been extensively assessed, however its performance is hindered by the limited workspace, whose dimension depends on the magnetic field. Another alternative is to embed the sensors internally in the continuum robot, combining magnetic sensors with Hall effect sensors: the magnetic field is measured at the level of the Hall effect sensors in order to estimate the deflection of the robot. However, it has been noticed that the higher the bending of the manipulator, the higher is the estimation error, due to crosstalk between sensors and magnets. Optical sensing: fiber Bragg grating sensors incorporated in an optical fiber can be embedded into the backbone of the continuum robot to estimate its shape; these sensors can only reflect a small range of the input light spectrum depending on their strain; therefore, by measuring the strain on each sensor it is possible to obtain the shape of the robot. This type of sensor is however expensive and is more prone to breaking in case of excessive strain, and this can happen in robots that can perform high deflections. == Control strategies == The control strategies can be distinguished in static and dynamic; the first one is based on the steady-state assumption, while the latter also considers the dynamic behaviour of the continuum robot. We can also differentiate between model-based controllers, that depend on a model of the robot, and model-free, that learn the robot's behaviour from data. Model-based static controllers: they rely on one of the modelling approaches presented above; once the model is defined, the kinematics must be inverted to obtain the desired actuator or configuration space variables. There are several ways to do this, like differential inverse kinematics, direct inversion or optimization. Model-free static controllers: these approaches learn directly, via machine learning techniques (e.g. regression methods and neural networks), the inverse kinematic or the direct kinematic representation of the con
DONE
The Data-based Online Nonlinear Extremumseeker (DONE) algorithm is a black-box optimization algorithm. DONE models the unknown cost function and attempts to find an optimum of the underlying function. The DONE algorithm is suitable for optimizing costly and noisy functions and does not require derivatives. An advantage of DONE over similar algorithms, such as Bayesian optimization, is that the computational cost per iteration is independent of the number of function evaluations. == Methods == The DONE algorithm was first proposed by Hans Verstraete and Sander Wahls in 2015. The algorithm fits a surrogate model based on random Fourier features and then uses a well-known L-BFGS algorithm to find an optimum of the surrogate model. == Applications == DONE was first demonstrated for maximizing the signal in optical coherence tomography measurements, but has since then been applied to various other applications. For example, it was used to help extending the field of view in light sheet fluorescence microscopy.
Sardinas–Patterson algorithm
In coding theory, the Sardinas–Patterson algorithm is a classical algorithm for determining in polynomial time whether a given variable-length code is uniquely decodable, named after August Albert Sardinas and George W. Patterson, who published it in 1953. The algorithm carries out a systematic search for a string which admits two different decompositions into codewords. As Knuth reports, the algorithm was rediscovered about ten years later in 1963 by Floyd, despite the fact that it was at the time already well known in coding theory. == Idea of the algorithm == Consider the code { a ↦ 1 , b ↦ 011 , c ↦ 01110 , d ↦ 1110 , e ↦ 10011 } {\displaystyle \{\,{\texttt {a}}\mapsto {\texttt {1}},{\texttt {b}}\mapsto {\texttt {011}},{\texttt {c}}\mapsto {\texttt {01110}},{\texttt {d}}\mapsto {\texttt {1110}},{\texttt {e}}\mapsto {\texttt {10011}}\,\}} . This code, which is based on an example by Berstel, is an example of a code which is not uniquely decodable, since the string 011101110011 can be interpreted as the sequence of codewords 01110 – 1110 – 011, but also as the sequence of codewords 011 – 1 – 011 – 10011. Two possible decodings of this encoded string are thus given by cdb and babe. In general, a codeword can be found by the following idea: In the first round, we choose two codewords x 1 {\displaystyle x_{1}} and y 1 {\displaystyle y_{1}} such that x 1 {\displaystyle x_{1}} is a prefix of y 1 {\displaystyle y_{1}} , that is, x 1 w = y 1 {\displaystyle x_{1}w=y_{1}} for some "dangling suffix" w {\displaystyle w} . If one tries first x 1 = 011 {\displaystyle x_{1}={\texttt {011}}} and y 1 = 01110 {\displaystyle y_{1}={\texttt {01110}}} , the dangling suffix is w = 10 {\displaystyle {\texttt {w}}={\texttt {10}}} . If we manage to find two sequences x 2 , … , x p {\displaystyle x_{2},\ldots ,x_{p}} and y 2 , … , y q {\displaystyle y_{2},\ldots ,y_{q}} of codewords such that x 2 ⋯ x p = w y 2 ⋯ y q {\displaystyle x_{2}\cdots x_{p}=wy_{2}\cdots y_{q}} , then we are finished: For then the string x = x 1 x 2 ⋯ x p {\displaystyle x=x_{1}x_{2}\cdots x_{p}} can alternatively be decomposed as y 1 y 2 ⋯ y q {\displaystyle y_{1}y_{2}\cdots y_{q}} , and we have found the desired string having at least two different decompositions into codewords. In the second round, we try out two different approaches: the first trial is to look for a codeword that has w as prefix. Then we obtain a new dangling suffix w, with which we can continue our search. If we eventually encounter a dangling suffix that is itself a codeword (or the empty word), then the search will terminate, as we know there exists a string with two decompositions. The second trial is to seek for a codeword that is itself a prefix of w. In our example, we have w = 10 {\displaystyle w={\texttt {10}}} , and the sequence 1 is a codeword. We can thus also continue with w = 0 {\displaystyle w={\texttt {0}}} as the new dangling suffix. == Precise description of the algorithm == The algorithm is described most conveniently using quotients of formal languages. In general, for two sets of strings D and N, the (left) quotient N − 1 D {\displaystyle N^{-1}D} is defined as the residual words obtained from D by removing some prefix in N. Formally, N − 1 D = { y ∣ x y ∈ D and x ∈ N } {\displaystyle N^{-1}D=\{\,y\mid xy\in D~{\textrm {and}}~x\in N\,\}} . Now let C {\displaystyle C} denote the (finite) set of codewords in the given code. The algorithm proceeds in rounds, where we maintain in each round not only one dangling suffix as described above, but the (finite) set of all potential dangling suffixes. Starting with round i = 1 {\displaystyle i=1} , the set of potential dangling suffixes will be denoted by S i {\displaystyle S_{i}} . The sets S i {\displaystyle S_{i}} are defined inductively as follows: S 1 = C − 1 C ∖ { ε } {\displaystyle S_{1}=C^{-1}C\setminus \{\varepsilon \}} . Here, the symbol ε {\displaystyle \varepsilon } denotes the empty word. S i + 1 = C − 1 S i ∪ S i − 1 C {\displaystyle S_{i+1}=C^{-1}S_{i}\cup S_{i}^{-1}C} , for all i ≥ 1 {\displaystyle i\geq 1} . The algorithm computes the sets S i {\displaystyle S_{i}} in increasing order of i {\displaystyle i} . As soon as one of the S i {\displaystyle S_{i}} contains a word from C or the empty word, then the algorithm terminates and answers that the given code is not uniquely decodable. Otherwise, once a set S i {\displaystyle S_{i}} equals a previously encountered set S j {\displaystyle S_{j}} with j < i {\displaystyle j Basic Formal Ontology (BFO) is a top-level ontology developed by Barry Smith and colleagues to promote interoperability among domain ontologies. The BFO methodology accomplishes this through a process of downward population. BFO is a formal ontology. The structure of BFO is based on a division of entities into two disjoint categories of continuant and occurrent, the former consists of objects and spatial regions, the latter contains processes conceived as extended through (or spanning) time. BFO thereby seeks to consolidate both time and space within a single framework A guide to building BFO-conformant domain ontologies was published by MIT Press in 2015. In 2021, the standard ISO/IEC 21838-2:2021 Information Technology — Top-level Ontologies (TLO) — Part 2: Basic Formal Ontology (BFO) was published by the Joint Technical Committee of the International Standards Organization and the International Electrotechnical Commission. ISO/IEC 21838 is a multi-part standard. Part 1 of the standard specifies the requirements that must be met if an ontology is to be classified as a top-level ontology by the standard. == History == BFO arose against the background of research in ontologies in the domain of geospatial information science by David Mark, Pierre Grenon, Achille Varzi and others, with a special role for the study of vagueness and of the ways sharp boundaries in the geospatial and other domains are created by fiat. BFO has passed through four major releases. 2001: release of BFO 1 2007: release of BFO 1.1 2015: release of BFO 2.0 2020: release of BFO 2020 2021: release of BFO 2020 as an ISO/IEC Standard The current revision was released in 2020, and this forms the basis of the standard ISO/IEC 21838-2, which was released by the Joint Committee of the International Standards Organization and International Electrotechnical Commission in 2021. == Applications == BFO has been adopted as a foundational ontology by over 650 ontology projects, principally in the areas of biomedical ontology, security and defense (intelligence) ontology, and industry ontologies. Example applications of BFO can be seen in the Ontology for Biomedical Investigations (OBI). In January 2024, BFO and the Common Core Ontologies (CCO), a suite of BFO-extension ontologies, were adopted as the "baseline standards for formal DOD and IC ontology" development work in the DOD and Intelligence Community. A memorandum to this effect was signed by the chief data officers of the DOD, the Office of the Director of National Intelligence and the Chief Digital and Artificial Intelligence Office. 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 Best arm identification (BAI) is a sequential one-player game where the player has to find the best action (arm) among a list of actions (arms) by collecting information in the most efficient way. It is a multi-armed bandit game as a player only gets information about an arm by playing it. The most common objective in multi-armed bandit games is to minimize the regret (i.e., play the best action as much as possible), but in BAI, the goal is to find the best arm as efficiently as possible. This problem naturally arises in scenarios such as adaptive clinical trials where the number of patients is limited and the quantification of the confidence in a treatment is important. It also arises in hyperparameter optimization where the goal is to find the optimal choice of hyperparameters for an algorithm with the smallest possible number of experiments, as it can be costly in terms of time, energy, or money. == Stochastic multi-armed bandit == The stochastic multi-armed bandit (MAB) is a sequential game with one player and K {\displaystyle K} actions (arms). Each arm has an unknown probability distribution associated with it. At each turn, the player has to choose one action and receive an observation from the probability distribution associated with the arm. The more you play an arm, the more you get information on its probability distribution. === Best arm identification === In BAI the goal is to find the arm that has the probability distribution with the highest mean. BAI may be either fixed confidence or fixed horizon. In a fixed-confidence game, a confidence level δ {\displaystyle \delta } is fixed at the beginning of the game and the goal is to find the best arm with this confidence level in as few turns as possible. In a fixed horizon game, the number of turns T {\displaystyle T} is fixed, and the goal is to find the best arm with the highest possible confidence in T {\displaystyle T} turns. === Math formalisation === We have one player and K {\displaystyle K} actions (arms). Behind each arm k ∈ { 1 , … , K } {\displaystyle k\in \{1,\ldots ,K\}} lies an unknown distribution ν k {\displaystyle \nu _{k}} with mean μ k {\displaystyle \mu _{k}} . Each distribution ν k {\displaystyle \nu _{k}} belongs to a known family D {\displaystyle {\mathcal {D}}} (such as the set of Gaussian distributions or Bernoulli distributions). At each time step t {\displaystyle t} , the player selects an arm a t {\displaystyle a_{t}} and observes an independent sample X t ∼ ν a t {\displaystyle X_{t}\sim \nu _{a_{t}}} from the corresponding distribution. We will note μ ∗ := max μ a {\displaystyle \mu ^{}:=\max \mu _{a}} the highest mean. An arm a {\displaystyle a} that satisfies μ a = μ ∗ {\displaystyle \mu _{a}=\mu ^{}} is called an optimal arm; otherwise it is called suboptimal arm. In best arm identification (BAI) the objective is to identify an optimal arm. Two main settings for BAI appear in the literature: Fixed confidence: In this setting, one typically assumes that there exists a unique optimal arm. A confidence level δ ∈ ( 0 , 1 ) {\displaystyle \delta \in (0,1)} is specified at the beginning. The algorithm must stop at some finite stopping time τ δ < + ∞ {\displaystyle \tau _{\delta }<+\infty } and return an arm a ^ τ δ {\displaystyle {\hat {a}}_{\tau _{\delta }}} such that the probability of error is bounded: P ( a ^ τ δ ≠ a ∗ ) ≤ δ {\displaystyle \mathbb {P} ({\hat {a}}_{\tau _{\delta }}\neq a^{})\leq \delta } . The objective is to minimize the expected sample complexity E [ τ δ ] {\displaystyle \mathbb {E} [\tau _{\delta }]} . Such a setting appears, for example, when a constraint on the confidence is required (for example, if we require a confidence level of 95%, so δ = 1 − 0.95 = 0.05 {\displaystyle \delta =1-0.95=0.05} ). Fixed horizon: In this setting, the number of samples T {\displaystyle T} is fixed in advance. The goal is to design an algorithm that minimizes the probability of misidentifying the optimal arm: P ( a ^ T ≠ a ∗ ) {\displaystyle \mathbb {P} ({\hat {a}}_{T}\neq a^{})} . This setting appears when the number of experiments is limited (for drug tests, the number of patients can be fixed in advance). === Example of simple modelling === In the case where we have K {\displaystyle K} treatments and we want to be sure with a confidence level of 95% which treatment is the best to heal a specific disease. Each treatment heals or does not heal the disease with a probability μ k {\displaystyle \mu _{k}} , which means that each distribution is a Bernoulli distribution, so D {\displaystyle {\mathcal {D}}} is the set of Bernoulli distributions. We can use a BAI algorithm to minimize E [ τ 0.05 ] {\displaystyle \mathbb {E} [\tau _{0.05}]} , the number of patients required to find the best treatment with probability 95%. == Applications == Best arm identification naturally arises in several practical domains: Adaptive clinical trials: The objective is to identify the most effective treatment based on sequentially collected patient data. Each treatment can be modeled as having an underlying distribution of outcomes. The goal is to identify the treatment with the highest expected outcome with high confidence (fixed confidence setting δ {\displaystyle \delta } ) while minimizing the number of drug test patients (minimise E [ τ δ ] {\displaystyle \mathbb {E} [\tau _{\delta }]} ), as it costs to pay patients for this and we would like to use as little as possible less effective drugs. Hyperparameter tuning: Selecting the best configuration for machine learning models efficiently by treating each hyperparameter setting as an arm. The goal is to find the best hyperparameter with as few experiments possible as experiments are costly in time and in energy == Fixed confidence level == In the fixed-confidence setting, the goal is to design an algorithm that identifies the best arm with a prescribed confidence level δ {\displaystyle \delta } while minimizing the expected number of samples. Any such algorithm requires two key components: Stopping rule: A decision criterion that determines when to stop sampling. Formally, this defines a stopping time τ δ {\displaystyle \tau _{\delta }} and returns an arm a ^ τ δ {\displaystyle {\hat {a}}_{\tau _{\delta }}} such that P ( a ^ τ δ ≠ a ⋆ ) ≤ δ {\displaystyle \mathbb {P} ({\hat {a}}_{\tau _{\delta }}\neq a^{\star })\leq \delta } and P ( τ δ < + ∞ ) = 1 {\displaystyle \mathbb {P} (\tau _{\delta }<+\infty )=1} . Sampling rule: A policy π {\displaystyle \pi } that, at each round t {\displaystyle t} , selects the next arm to sample a t {\displaystyle a_{t}} based on all previous observations ( a s , X s ) s < t {\displaystyle (a_{s},X_{s})_{sBasic Formal Ontology
Superellipsoid
Best arm identification