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  Digistar

  Digistar is the first computer graphics-based planetarium projection and content system. It was designed by Evans & Sutherland and released in 1983. The technology originally focused on accurate and high quality display of stars, including for the first time showing stars from points of view other than Earth's surface, travelling through the stars, and accurately showing celestial bodies from different times in the past and future. Beginning with the Digistar 3 the system now projects full-dome video. == Projector == Unlike modern full-dome systems, which use LCD, DLP, SXRD, or laser projection technology, the Digistar projection system was designed for projecting bright pinpoints of light representing stars. This was accomplished using a calligraphic display, a form of vector graphics, rather than raster graphics. The heart of the Digistar projector is a large cathode-ray tube (CRT). A phosphor plate is mounted atop the tube, and light is then dispersed by a large lens with a 160 degree field of view to cover the planetarium dome. The original lens bore the inscription: "August 1979 mfg. by Lincoln Optical Corp., L.A., CA for Evans and Sutherland Computer Corp., SLC, UT, Digital planetarium CRT projection lens, 43mm, f2.8, 160 degree field of view". The coordinates of the stars and wire-frame models to be displayed by the projector were stored in computer RAM in a display list. The display would read each set of coordinates in turn and drive the CRT's electron beam directly to those coordinates. If the electron beam was enabled while being moved a line would be painted on the phosphor plate. Otherwise, the electron beam would be enabled once at its destination and a star would be painted. Once all coordinates in the display list had been processed, the display would repeat from the top of the display list. Thus, the shorter the display list the more frequently the electron beam would refresh the charge on a given point on the phosphor plate, making the projection of the points brighter. In this way, the stars projected by Digistar were substantially brighter than could be achieved using a raster display, which has to touch every point on the phosphor plate before repeating. Likewise, the calligraphic technology allowed Digistar to have a darker black-level than full-dome projectors, since the portions of the phosphor plate representing dark sky were never hit by the electron beam. As it is only one tube, with no pixelated color filter screen, the Digistar projector is monochromatic. The Digistar projects a bright, phosphorescent green, though many (including both visitors and planetarians) report they cannot distinguish between this green and white. Additionally, unlike a raster display, the calligraphic display is not discretized into pixels, so the displayed stars were a more realistic single spot of light, without the blocky or ropy artifacts that are hard to avoid with raster graphics. Due to the use of vector graphics, as opposed to raster imaging, the Digistar does not have the resolution issues that many full-dome systems have. Thanks to this, and the brightness of the CRT, only one projector is needed to project on the entire dome, whereas most full-dome systems require up to six raster projectors, depending on dome size. The projector in the original Digistar was housed in a square pyramid-shaped sheathing. When powered on, the four sides at the tip of the pyramid would recede into the housing, exposing the lens and appearing as a cut-off pyramid. As Digistar II was being developed, many planetaria were sold Digistar LEA projectors. The LEA, called Digistar 1.5 by many users, was effectively a prototype of the D2 projector, compatible with Digistar and upgradable to Digistar II. There are no significant differences in performance between the LEA and the true D2. == History == Digistar was the brainchild of Stephen McAllister and Brent Watson, both of whom were long-time amateur astronomers and computer graphics engineers. In 1977, E&S had been consulting with Johnson Space Center regarding training simulators for astronauts. McAllister had been writing proof-of-concept software for this consultation and in summer 1977 entered the data for 400 bright stars and wrote the software to display them. Steve and Brent both originally saw the system's purpose as celestial navigation training. Brent, who had until recently worked at Hansen planetarium, asked his planetarium coworkers what they thought of a potential digital planetarium system, and then Steve and Brent both targeted the system toward planetaria. The primary goal of the planetarium system was to use computer graphics to overcome the limitation of traditional star ball technology that only allowed display of star fields from the point of view of Earth's surface. By using computer graphics the stars could be displayed from viewpoints in space, including simulating the appearance of space flight. Likewise, planets and moons within the Solar System could be displayed accurately for any time in history, from any point of view. The system used the location of real stars from the Yale Bright Star Catalogue, as well as random stars. A laboratory prototype of Digistar was used to generate the star fields and tactical displays in the 1982 science fiction film Star Trek II: The Wrath of Khan. Filming was done directly from the Digistar display in the lab. ILM projected the effort would take two weeks, but in fact it took from late November 1981 until mid-February 1982. The last shot recorded was what became the first entirely computer generated feature film sequence. It was the opening scene of the film, a rotating forward translation through a star field that lasted 3.5 minutes. It was recorded in one take, at a rate of one frame every 3.5 seconds, taking four hours for the shoot. The Digistar team members are credited in the film. After prototyping in labs at Evans and Sutherland the team repeatedly used Salt Lake City's Hansen planetarium to beta test the system at the planetarium at night. The Digistar team performed one week of shows at the planetarium as a fund raiser to benefit the planetarium. The company also later gave the planetarium an improved prototype Digistar to replace "Jake", the planetarium's aging Spitz planetarium projector. The first customer installation was to the newly constructed Universe Planetarium at the Science Museum of Virginia in 1983, the largest planetarium dome in the world at the time, for $595,000. By September 1986 there were four installed Digistars. Even at this point the long-term success of the product was very much in doubt, but as of 2019 Digistar has an installed base of over 550 planetaria. === Versions === Digistar (1983) Digistar II (1995) Digistar 3 (2002) Digistar 4 (2010?) Digistar 5 (2012) Digistar 6 (2016) Digistar 7 (2021) == Hardware == Digistar was driven by a VAX-11/780 minicomputer, with custom graphics hardware related to the E&S Picture System 2. Later versions of Digistar 1 used a DEC MicroVAX 2, driving a custom version of a PS/300. The original Digistar and Digistar 2 had a physical control panel that was used for running the star shows. This control panel was approximately 3' x 4' and contained a keyboard, a 6 DOF joystick, and a large array of back-lit buttons. One button that was used for moving the viewpoint forward in space was labeled "Boldly Go". Later iterations of Digistar replaced the physical control panel with a common graphical user interface. Digistar 3 was the first Digistar system to offer full-dome video in 2002, using six projectors. Digistar 4 was able to cover the dome using only two projectors. == System limitations == Though technologically advanced in its day, and the closest system to true full-dome video at the time of its release, the original Digistar and Digistar 2 are limited to only projecting dots and lines—meaning only wireframe models can be projected. To compensate for this, the projector is capable of defocusing specific models, blurring lines and dots together. An example of this is in the Digistar 2's built-in Milky Way model. The model is a circle of parallel lines that, when defocused, appear as the continuous band of the Milky Way across the sky. On more complex models, especially three-dimensional ones, brightness and details may be lost in this process, so it is not useful in all situations. The Digistar and Digistar 2 also suffer focus limitations. Because they use a single lens to cover the entire dome, it is difficult to gain perfect focus across the dome. Coupled with this, stars greater than a certain brightness are "multihit" points, meaning the projector draws two dots at the given position to accommodate the brightness of the star. Errors in the projector can lead the second dot to be slightly out-of-place with the first one. These two issues together, along with other issues that can occur within the projector's focus system, give the stars a blobby look. Some p

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  Markov chain Monte Carlo

  In statistics, Markov chain Monte Carlo (MCMC) is a class of algorithms used to draw samples from a probability distribution. Given a probability distribution, one can construct a Markov chain whose elements' distribution approximates it, i.e. the Markov chain's equilibrium distribution matches the target distribution. The more steps that are included, the more closely the distribution of the sample matches the actual desired distribution. Markov chain Monte Carlo methods are used to study probability distributions that are too complex or too high dimensional to study with analytic techniques alone. Various algorithms exist for constructing such Markov chains, including the Metropolis–Hastings algorithm. == General explanation == Markov chain Monte Carlo methods create samples from a continuous random variable, with probability density proportional to a known function. These samples can be used to evaluate an integral over that variable, as its expected value or variance. Practically, an ensemble of chains is generally developed, starting from a set of points arbitrarily chosen and sufficiently distant from each other. These chains are stochastic processes of "walkers" which move around randomly according to an algorithm that looks for places with a reasonably high contribution to the integral to move into next, assigning them higher probabilities. Random walk Monte Carlo methods are a kind of random simulation or Monte Carlo method. However, whereas the random samples of the integrand used in a conventional Monte Carlo integration are statistically independent, those used in MCMC are autocorrelated. Correlations of samples introduces the need to use the Markov chain central limit theorem when estimating the error of mean values. These algorithms create Markov chains such that they have an equilibrium distribution which is proportional to the function given. == History == The development of MCMC methods is deeply rooted in the early exploration of Monte Carlo (MC) techniques in the mid-20th century, particularly in physics. These developments were marked by the Metropolis algorithm proposed by Nicholas Metropolis, Arianna W. Rosenbluth, Marshall Rosenbluth, Augusta H. Teller, and Edward Teller in 1953, which was designed to tackle high-dimensional integration problems using early computers. Then in 1970, W. K. Hastings generalized this algorithm and inadvertently introduced the component-wise updating idea, later known as Gibbs sampling. Simultaneously, the theoretical foundations for Gibbs sampling were being developed, such as the Hammersley–Clifford theorem from Julian Besag's 1974 paper. Although the seeds of MCMC were sown earlier, including the formal naming of Gibbs sampling in image processing by Stuart Geman and Donald Geman (1984) and the data augmentation method by Martin A. Tanner and Wing Hung Wong (1987), its "revolution" in mainstream statistics largely followed demonstrations of the universality and ease of implementation of sampling methods (especially Gibbs sampling) for complex statistical (particularly Bayesian) problems, spurred by increasing computational power and software like BUGS. This transformation was accompanied by significant theoretical advancements, such as Luke Tierney's (1994) rigorous treatment of MCMC convergence, and Jun S. Liu, Wong, and Augustine Kong's (1994, 1995) analysis of Gibbs sampler structure. Subsequent developments further expanded the MCMC toolkit, including particle filters (Sequential Monte Carlo) for sequential problems, Perfect sampling aiming for exact simulation (Jim Propp and David B. Wilson, 1996), RJMCMC (Peter J. Green, 1995) for handling variable-dimension models, and deeper investigations into convergence diagnostics and the central limit theorem. Overall, the evolution of MCMC represents a paradigm shift in statistical computation, enabling the analysis of numerous previously intractable complex models and continually expanding the scope and impact of statistics. == Mathematical setting == Suppose (Xn) is a Markov Chain in the general state space X {\displaystyle {\mathcal {X}}} with specific properties. We are interested in the limiting behavior of the partial sums: S n ( h ) = 1 n ∑ i = 1 n h ( X i ) {\displaystyle S_{n}(h)={\dfrac {1}{n}}\sum _{i=1}^{n}h(X_{i})} as n goes to infinity. Particularly, we hope to establish the Law of Large Numbers and the Central Limit Theorem for MCMC. In the following, we state some definitions and theorems necessary for the important convergence results. In short, we need the existence of invariant measure and Harris recurrent to establish the Law of Large Numbers of MCMC (Ergodic Theorem). And we need aperiodicity, irreducibility and extra conditions such as reversibility to ensure the Central Limit Theorem holds in MCMC. === Irreducibility and aperiodicity === Recall that in the discrete setting, a Markov chain is said to be irreducible if it is possible to reach any state from any other state in a finite number of steps with positive probability. However, in the continuous setting, point-to-point transitions have zero probability. In this case, φ-irreducibility generalizes irreducibility by using a reference measure φ on the measurable space ( X , B ( X ) ) {\displaystyle ({\mathcal {X}},{\mathcal {B}}({\mathcal {X}}))} . Definition (φ-irreducibility) Given a measure φ {\displaystyle \varphi } defined on ( X , B ( X ) ) {\displaystyle ({\mathcal {X}},{\mathcal {B}}({\mathcal {X}}))} , the Markov chain ( X n ) {\displaystyle (X_{n})} with transition kernel K ( x , y ) {\displaystyle K(x,y)} is φ-irreducible if, for every A ∈ B ( X ) {\displaystyle A\in {\mathcal {B}}({\mathcal {X}})} with φ ( A ) > 0 {\displaystyle \varphi (A)>0} , there exists n {\displaystyle n} such that K n ( x , A ) > 0 {\displaystyle K^{n}(x,A)>0} for all x ∈ X {\displaystyle x\in {\mathcal {X}}} (Equivalently, P x ( τ A < ∞ ) > 0 {\displaystyle P_{x}(\tau _{A}<\infty )>0} , here τ A = inf { n ≥ 1 ; X n ∈ A } {\displaystyle \tau _{A}=\inf\{n\geq 1;X_{n}\in A\}} is the first n {\displaystyle n} for which the chain enters the set A {\displaystyle A} ). This is a more general definition for irreducibility of a Markov chain in non-discrete state space. In the discrete case, an irreducible Markov chain is said to be aperiodic if it has period 1. Formally, the period of a state ω ∈ X {\displaystyle \omega \in {\mathcal {X}}} is defined as: d ( ω ) := g c d { m ≥ 1 ; K m ( ω , ω ) > 0 } {\displaystyle d(\omega ):=\mathrm {gcd} \{m\geq 1\,;\,K^{m}(\omega ,\omega )>0\}} For the general (non-discrete) case, we define aperiodicity in terms of small sets: Definition (Cycle length and small sets) A φ-irreducible Markov chain ( X n ) {\displaystyle (X_{n})} has a cycle of length d if there exists a small set C {\displaystyle C} , an associated integer M {\displaystyle M} , and a probability distribution ν M {\displaystyle \nu _{M}} such that d is the greatest common divisor of: { m ≥ 1 ; ∃ δ m > 0 such that C is small for ν m ≥ δ m ν M } . {\displaystyle \{m\geq 1\,;\,\exists \,\delta _{m}>0{\text{ such that }}C{\text{ is small for }}\nu _{m}\geq \delta _{m}\nu _{M}\}.} A set C {\displaystyle C} is called small if there exists m ∈ N ∗ {\displaystyle m\in \mathbb {N} ^{}} and a nonzero measure ν m {\displaystyle \nu _{m}} such that: K m ( x , A ) ≥ ν m ( A ) , ∀ x ∈ C , ∀ A ∈ B ( X ) . {\displaystyle K^{m}(x,A)\geq \nu _{m}(A),\quad \forall x\in C,\,\forall A\in {\mathcal {B}}({\mathcal {X}}).} === Harris recurrent === Definition (Harris recurrence) A set A {\displaystyle A} is Harris recurrent if P x ( η A = ∞ ) = 1 {\displaystyle P_{x}(\eta _{A}=\infty )=1} for all x ∈ A {\displaystyle x\in A} , where η A = ∑ n = 1 ∞ I A ( X n ) {\displaystyle \eta _{A}=\sum _{n=1}^{\infty }\mathbb {I} _{A}(X_{n})} is the number of visits of the chain ( X n ) {\displaystyle (X_{n})} to the set A {\displaystyle A} . The chain ( X n ) {\displaystyle (X_{n})} is said to be Harris recurrent if there exists a measure ψ {\displaystyle \psi } such that the chain is ψ {\displaystyle \psi } -irreducible and every measurable set A {\displaystyle A} with ψ ( A ) > 0 {\displaystyle \psi (A)>0} is Harris recurrent. A useful criterion for verifying Harris recurrence is the following: Proposition If for every A ∈ B ( X ) {\displaystyle A\in {\mathcal {B}}({\mathcal {X}})} , we have P x ( τ A < ∞ ) = 1 {\displaystyle P_{x}(\tau _{A}<\infty )=1} for every x ∈ A {\displaystyle x\in A} , then P x ( η A = ∞ ) = 1 {\displaystyle P_{x}(\eta _{A}=\infty )=1} for all x ∈ X {\displaystyle x\in {\mathcal {X}}} , and the chain ( X n ) {\displaystyle (X_{n})} is Harris recurrent. This definition is only needed when the state space X {\displaystyle {\mathcal {X}}} is uncountable. In the countable case, recurrence corresponds to E x [ η x ] = ∞ {\displaystyle \mathbb {E} _{x}[\eta _{x}]=\infty } , which is equivalent to P x ( τ x < ∞ ) = 1 {\displaystyle P_{x}(\tau _{x}<\infty )=1} for all x ∈ X {\displaystyle x\i

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  Permutation automaton

  In automata theory, a permutation automaton, or pure-group automaton, is a deterministic finite automaton such that each input symbol permutes the set of states. Formally, a deterministic finite automaton A may be defined by the tuple (Q, Σ, δ, q0, F), where Q is the set of states of the automaton, Σ is the set of input symbols, δ is the transition function that takes a state q and an input symbol x to a new state δ(q,x), q0 is the initial state of the automaton, and F is the set of accepting states (also: final states) of the automaton. A is a permutation automaton if and only if, for every two distinct states qi and qj in Q and every input symbol x in Σ, δ(qi,x) ≠ δ(qj,x). A formal language is p-regular (also: a pure-group language) if it is accepted by a permutation automaton. For example, the set of strings of even length forms a p-regular language: it may be accepted by a permutation automaton with two states in which every transition replaces one state by the other. == Applications == The pure-group languages were the first interesting family of regular languages for which the star height problem was proved to be computable. Another mathematical problem on regular languages is the separating words problem, which asks for the size of a smallest deterministic finite automaton that distinguishes between two given words of length at most n – by accepting one word and rejecting the other. The known upper bound in the general case is O ( n 2 / 5 ( log ⁡ n ) 3 / 5 ) {\displaystyle O(n^{2/5}(\log n)^{3/5})} . The problem was later studied for the restriction to permutation automata. In this case, the known upper bound changes to O ( n 1 / 2 ) {\displaystyle O(n^{1/2})} .

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  Is an AI Image Generator Worth It in 2026?

  Comparing the best AI image generator? An AI image generator is software that uses machine learning to help you get more done — it lowers the barrier so anyone can produce professional output. Privacy matters too: check whether your data trains the model and whether a no-log or enterprise tier is available. Whether you are a beginner or a pro, the right AI image generator slots into your workflow and pays for itself fast. We tested the leading options and ranked them by quality, value, and ease of use.

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  Vector database

  A vector database, vector store or vector search engine is a database that stores and retrieves embeddings of data in vector space. Vector databases typically implement approximate nearest neighbor algorithms so users can search for records semantically similar to a given input, unlike traditional databases which primarily look up records by exact match. Use-cases for vector databases include similarity search, semantic search, multi-modal search, recommendations engines, object detection, and retrieval-augmented generation (RAG). Vector embeddings are mathematical representations of data in a high-dimensional space. In this space, each dimension corresponds to a feature of the data, with the number of dimensions ranging from a few hundred to tens of thousands, depending on the complexity of the data being represented. Each data item is represented by one vector in this space. Words, phrases, or entire documents, as well as images, audio, and other types of data, can all be vectorized. These feature vectors may be computed from the raw data using machine learning methods such as feature extraction algorithms, word embeddings or deep learning networks. The goal is that semantically similar data items receive feature vectors close to each other. Vector retrieval can be combined with metadata filtering or lexical search to support filtered and hybrid retrieval workflows. == Techniques == Common techniques for similarity search on high-dimensional vectors include: Hierarchical Navigable Small World (HNSW) graphs Locality-sensitive hashing (LSH) and sketching Product quantization (PQ) Inverted files These techniques may also be combined in vector search systems. In recent benchmarks, HNSW-based implementations have been among the best performers. Conferences such as the International Conference on Similarity Search and Applications (SISAP) and the Conference on Neural Information Processing Systems (NeurIPS) have hosted competitions on vector search in large databases. == Applications == Vector databases are used in a wide range of machine learning applications including similarity search, semantic search, multi-modal search, recommendations engines, object detection, and retrieval-augmented generation. === Retrieval-augmented generation === An especially common use-case for vector databases is in retrieval-augmented generation (RAG), a method to improve domain-specific responses of large language models. The retrieval component of a RAG can be any search system, but is most often implemented as a vector database. Text documents describing the domain of interest are collected, and for each document or document section, a feature vector (known as an "embedding") is computed, typically using a deep learning network, and stored in a vector database along with a link to the document. Given a user prompt, the feature vector of the prompt is computed, and the database is queried to retrieve the most relevant documents. These are then automatically added into the context window of the large language model, and the large language model proceeds to create a response to the prompt given this context. == Implementations ==

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  How to Choose an AI Writing Assistant

  Comparing the best AI writing assistant? An AI writing assistant is software that uses machine learning to help you get more done — it lowers the barrier so anyone can produce professional output. Privacy matters too: check whether your data trains the model and whether a no-log or enterprise tier is available. Whether you are a beginner or a pro, the right AI writing assistant slots into your workflow and pays for itself fast. We tested the leading options and ranked them by quality, value, and ease of use.

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  Heng Ji

  Heng Ji is a computer scientist who works on information extraction and natural language processing. She is well known for her work on joined named entity recognition and relation extraction, as well as for her work on cross-document event extraction. She has been coordinating the popular NIST TAC Knowledge Base Population task since 2010. She has been recognised as one of AI's 10 to watch by IEEE Intelligent Systems in 2013, and has won multiple awards, including a NSF Career Award in 2009, Google Research awards in 2009 and 2014, and an IBM Watson Faculty Award in 2012. == Education == Heng Ji obtained a Bachelor's and master's degree in Computational Linguistics from Tsinghua University. She subsequently obtained a MSc, then PhD in Computer Science from New York University in 2008 under the supervision of Ralph Grishman. Her PhD thesis was on the topic of information extraction, with a particular focus on joint training of multiple components in the information extraction pipeline, as well as cross-lingual learning. == Career == Upon graduating with a PhD from New York University, Ji took up a position as assistant professor at Queens College, City University of New York, where she founded the BLENDER Lab, which focuses on research on cross-lingual, cross-documents, cross-media information extraction and fusion. In 2013, she joined Rensselaer Polytechnic Institute as an Edward P. Hamilton Development Chair and Tenured associate professor in Computer Science. Since 2019, she has been a full professor at the University of Illinois at Urbana–Champaign, as well as an Amazon Scholar. == Research == Heng Ji works in the area of natural language processing, machine learning and information extraction. She has published over 300 peer-reviewed research papers. Her work is published in the proceedings of computer science conferences, including the Annual Meeting of the Association for Computational Linguistics, The Web Conference, and the ACM Conference on Knowledge Discovery and Data Mining (KDD). Ji is a leading researcher in information extraction, having coordinated the popular NIST TAC Knowledge Base Population shared task since 2010. She is most recognised for her work on modelling interactions between subtasks in information extraction, which was also the topic of her PhD thesis, and for her work on event detection using cross-document signals. == Selected honors and distinctions == 2009 NSF Career Award 2009 Google Research Award 2012 IBM Watson Faculty Award 2013 IEEE AI's 10 to Watch 2014 Google Research Award 2016 World Economic Forum, 'Young Scientist' 2017 World Economic Forum, 'Young Scientist' 2020 Annual Meeting of the Association for Computational Linguistics, best demonstration paper

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  Best AI Art Generators in 2026

  Curious about the best AI art generator? An AI art generator is software that uses machine learning to help you get more done — it combines speed, accuracy, and an interface that just works. Hands-on testing shows real-world results vary, so a short free trial is the smartest way to decide. Whether you are a beginner or a pro, the right AI art generator slots into your workflow and pays for itself fast. This guide breaks down the top picks, their pros and cons, and who each one is best for.

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  Cross-entropy method

  The cross-entropy (CE) method is a Monte Carlo method for importance sampling and optimization. It is applicable to both combinatorial and continuous problems, with either a static or noisy objective. The method approximates the optimal importance sampling estimator by repeating two phases: Draw a sample from a probability distribution. Minimize the cross-entropy between this distribution and a target distribution to produce a better sample in the next iteration. Reuven Rubinstein developed the method in the context of rare-event simulation, where tiny probabilities must be estimated, for example in network reliability analysis, queueing models, or performance analysis of telecommunication systems. The method has also been applied to the traveling salesman, quadratic assignment, DNA sequence alignment, max-cut and buffer allocation problems. == Estimation via importance sampling == Consider the general problem of estimating the quantity ℓ = E u [ H ( X ) ] = ∫ H ( x ) f ( x ; u ) d x {\displaystyle \ell =\mathbb {E} _{\mathbf {u} }[H(\mathbf {X} )]=\int H(\mathbf {x} )\,f(\mathbf {x} ;\mathbf {u} )\,{\textrm {d}}\mathbf {x} } , where H {\displaystyle H} is some performance function and f ( x ; u ) {\displaystyle f(\mathbf {x} ;\mathbf {u} )} is a member of some parametric family of distributions. Using importance sampling this quantity can be estimated as ℓ ^ = 1 N ∑ i = 1 N H ( X i ) f ( X i ; u ) g ( X i ) {\displaystyle {\hat {\ell }}={\frac {1}{N}}\sum _{i=1}^{N}H(\mathbf {X} _{i}){\frac {f(\mathbf {X} _{i};\mathbf {u} )}{g(\mathbf {X} _{i})}}} , where X 1 , … , X N {\displaystyle \mathbf {X} _{1},\dots ,\mathbf {X} _{N}} is a random sample from g {\displaystyle g\,} . For positive H {\displaystyle H} , the theoretically optimal importance sampling density (PDF) is given by g ∗ ( x ) = H ( x ) f ( x ; u ) / ℓ {\displaystyle g^{}(\mathbf {x} )=H(\mathbf {x} )f(\mathbf {x} ;\mathbf {u} )/\ell } . This, however, depends on the unknown ℓ {\displaystyle \ell } . The CE method aims to approximate the optimal PDF by adaptively selecting members of the parametric family that are closest (in the Kullback–Leibler sense) to the optimal PDF g ∗ {\displaystyle g^{}} . == Generic CE algorithm == Choose initial parameter vector v ( 0 ) {\displaystyle \mathbf {v} ^{(0)}} ; set t = 1. Generate a random sample X 1 , … , X N {\displaystyle \mathbf {X} _{1},\dots ,\mathbf {X} _{N}} from f ( ⋅ ; v ( t − 1 ) ) {\displaystyle f(\cdot ;\mathbf {v} ^{(t-1)})} Solve for v ( t ) {\displaystyle \mathbf {v} ^{(t)}} , where v ( t ) = argmax v ⁡ 1 N ∑ i = 1 N H ( X i ) f ( X i ; u ) f ( X i ; v ( t − 1 ) ) log ⁡ f ( X i ; v ) {\displaystyle \mathbf {v} ^{(t)}=\mathop {\textrm {argmax}} _{\mathbf {v} }{\frac {1}{N}}\sum _{i=1}^{N}H(\mathbf {X} _{i}){\frac {f(\mathbf {X} _{i};\mathbf {u} )}{f(\mathbf {X} _{i};\mathbf {v} ^{(t-1)})}}\log f(\mathbf {X} _{i};\mathbf {v} )} If convergence is reached then stop; otherwise, increase t by 1 and reiterate from step 2. In several cases, the solution to step 3 can be found analytically. Situations in which this occurs are When f {\displaystyle f\,} belongs to the natural exponential family When f {\displaystyle f\,} is discrete with finite support When H ( X ) = I { x ∈ A } {\displaystyle H(\mathbf {X} )=\mathrm {I} _{\{\mathbf {x} \in A\}}} and f ( X i ; u ) = f ( X i ; v ( t − 1 ) ) {\displaystyle f(\mathbf {X} _{i};\mathbf {u} )=f(\mathbf {X} _{i};\mathbf {v} ^{(t-1)})} , then v ( t ) {\displaystyle \mathbf {v} ^{(t)}} corresponds to the maximum likelihood estimator based on those X k ∈ A {\displaystyle \mathbf {X} _{k}\in A} . == Continuous optimization—example == The same CE algorithm can be used for optimization, rather than estimation. Suppose the problem is to maximize some function S {\displaystyle S} , for example, S ( x ) = e − ( x − 2 ) 2 + 0.8 e − ( x + 2 ) 2 {\displaystyle S(x)={\textrm {e}}^{-(x-2)^{2}}+0.8\,{\textrm {e}}^{-(x+2)^{2}}} . To apply CE, one considers first the associated stochastic problem of estimating P θ ( S ( X ) ≥ γ ) {\displaystyle \mathbb {P} _{\boldsymbol {\theta }}(S(X)\geq \gamma )} for a given level γ {\displaystyle \gamma \,} , and parametric family { f ( ⋅ ; θ ) } {\displaystyle \left\{f(\cdot ;{\boldsymbol {\theta }})\right\}} , for example the 1-dimensional Gaussian distribution, parameterized by its mean μ t {\displaystyle \mu _{t}\,} and variance σ t 2 {\displaystyle \sigma _{t}^{2}} (so θ = ( μ , σ 2 ) {\displaystyle {\boldsymbol {\theta }}=(\mu ,\sigma ^{2})} here). Hence, for a given γ {\displaystyle \gamma \,} , the goal is to find θ {\displaystyle {\boldsymbol {\theta }}} so that D K L ( I { S ( x ) ≥ γ } ‖ f θ ) {\displaystyle D_{\mathrm {KL} }({\textrm {I}}_{\{S(x)\geq \gamma \}}\|f_{\boldsymbol {\theta }})} is minimized. This is done by solving the sample version (stochastic counterpart) of the KL divergence minimization problem, as in step 3 above. It turns out that parameters that minimize the stochastic counterpart for this choice of target distribution and parametric family are the sample mean and sample variance corresponding to the elite samples, which are those samples that have objective function value ≥ γ {\displaystyle \geq \gamma } . The worst of the elite samples is then used as the level parameter for the next iteration. This yields the following randomized algorithm that happens to coincide with the so-called Estimation of Multivariate Normal Algorithm (EMNA), an estimation of distribution algorithm. === Pseudocode === // Initialize parameters μ := −6 σ2 := 100 t := 0 maxits := 100 N := 100 Ne := 10 // While maxits not exceeded and not converged while t < maxits and σ2 > ε do // Obtain N samples from current sampling distribution X := SampleGaussian(μ, σ2, N) // Evaluate objective function at sampled points S := exp(−(X − 2) ^ 2) + 0.8 exp(−(X + 2) ^ 2) // Sort X by objective function values in descending order X := sort(X, S) // Update parameters of sampling distribution via elite samples μ := mean(X(1:Ne)) σ2 := var(X(1:Ne)) t := t + 1 // Return mean of final sampling distribution as solution return μ == Related methods == Simulated annealing Genetic algorithms Harmony search Estimation of distribution algorithm Tabu search Natural Evolution Strategy Ant colony optimization algorithms

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  How to Choose an Conversational AI Platform

  Trying to pick the best conversational AI platform? An conversational AI platform is software that uses machine learning to help you get more done — it scales effortlessly from a single task to thousands. The best picks balance beginner-friendly simplicity with the depth power users need, and they ship updates often. Whether you are a beginner or a pro, the right conversational AI platform slots into your workflow and pays for itself fast. This guide breaks down the top picks, their pros and cons, and who each one is best for.

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  Ocrad

  Ocrad is an optical character recognition program and part of the GNU Project. It is free software licensed under the GNU GPL. Based on a feature extraction method, it reads images in portable pixmap formats known as Portable anymap and produces text in byte (8-bit) or UTF-8 formats. Also included is a layout analyser, able to separate the columns or blocks of text normally found on printed pages. == User interface == Ocrad can be used as a stand-alone command-line application or as a back-end to other programs. Kooka, which was the KDE environment's default scanning application until KDE 4, can use Ocrad as its OCR engine. Since conversion to newer Qt versions, current versions of KDE no longer contain Kooka; development continues in the KDE git repository. Ocrad can be also used as an OCR engine in OCRFeeder. == History == Ocrad has been developed by Antonio Diaz Diaz since 2003. Version 0.7 was released in February 2004, 0.14 in February 2006 and 0.18 in May 2009. It is written in C++. Archives of the bug-ocrad mailing list go back to October 2003.

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  Corinna Cortes

  Corinna Cortes (born 31 March 1961) is a Danish computer scientist known for her contributions to machine learning. She is a Vice President at Google Research in New York City. Cortes is an ACM Fellow and a recipient of the Paris Kanellakis Award for her work on theoretical foundations of support vector machines. == Early life and education == Corinna Cortes was born in 1961 in Denmark. Cortes received her Master of Science degree in physics from University of Copenhagen in 1989. She received her PhD in computer science from the University of Rochester in 1993 for research supervised by Randal C. Nelson. == Career and research == Cortes joined AT&T Bell Labs as a researcher in 1993. Since 2003, she has served as Vice President of Google Research, New York City, and since 2011, as adjunct professor at the UCPH Department of Computer Science. She is serves as an editorial board member of the journal Machine Learning. Cortes' research covers a wide range of topics in machine learning, including support vector machines (SVM) and data mining. SVM is one of the most frequently used algorithms in machine learning, which is used in many practical applications, including medical diagnosis and weather forecasting. At AT&T, Cortes was a contributor to the design of Hancock programming language. === Awards and honours === In 2008, she jointly with Vladimir Vapnik received the Paris Kanellakis Award for the development of a highly effective algorithm for supervised learning known as support vector machines (SVM). She was named an ACM Fellow in 2023 for theoretical and practical contributions to machine learning, industrial leadership and service to the field. == Personal life == Corinna has two children and is also a competitive runner.

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  Docic

  Docic is a Tunisian digital health platform available as a web and mobile application, headquartered in Tunis, Tunisia. Founded in 2022 by Sami Kallel, an orthopedic surgeon, and Sofiane Trabelsi. The service helps patients and healthcare professionals store, organize, and share medical records digitally and to connect with the doctor online. == History == Docic was founded in 2022 as a health-technology company based in Tunisia, after which the mobile application was subsequently developed and made available to users. The platform was designed to provide healthcare professionals with access to patients’ complete medical history, including updates and recent changes, aiming at supporting clinical decision-making and reducing the risk of medical errors. In January 2025, Docic was listed amongst companies that have received the Startup Act label, which is a recognition under the Tunisian legal framework made to support innovative startups.

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  Bob Coecke

  Bob Coecke (born 23 July 1968) is a Belgian theoretical physicist and logician. He was Professor of Quantum foundations, Logics, and Structures at Oxford University until 2020. He was Chief Scientist at quantum computing company Quantinuum, until 2025 and founded a startup called Relational Intelligence in 2026. He is also Distinguished Visiting Research Chair at the Perimeter Institute for Theoretical Physics, and Emeritus Fellow at Wolfson College, Oxford. He pioneered categorical quantum mechanics (entry 18M40 in Mathematics Subject Classification 2020), Quantum Picturalism, ZX-calculus, DisCoCat model for natural language,, quantum natural language processing (QNLP) and quantum education through the book Quantum in Pictures. He is a founder of the Quantum Physics and Logic community and the Applied Category Theory communities and conference series, and of the journal Compositionality. Coecke is also a composer and musician, who has been called a pioneer of industrial music, and is also one of the pioneers of employing quantum computers in music. == Education and career == Coecke obtained his doctorate in sciences at the Vrije Universiteit Brussel in 1996, and performed postdoctoral work in the Theoretical Physics Group of Imperial College, London in the Category Theory Group of the Mathematics and Statistics Department at McGill University in Montreal, in the Department of Pure Mathematics and Mathematical Statistics of Cambridge University, and in the Department of Computer Science, University of Oxford. He was an EPSRC Advanced Research Fellow at the Department of Computer Science, University of Oxford, where he became Lecturer in Quantum Computer Science in 2007, and jointly with Samson Abramsky built and headed the Quantum Group. In July 2011, he was nominated professor of Quantum Foundations, Logics and Structures at Oxford University, with retroactive effect as of October 2010. He was a Governing Body Fellow of Wolfson College, Oxford since 2007, where he now is an Emeritus Fellow. In January 2019, Coecke became Senior Scientific Advisor of Cambridge Quantum Computing, and in January 2021 he resigned from his Professorship at Oxford, to become Chief Scientist of Cambridge Quantum Computing. After the merger of Cambridge Quantum Computing with Honeywell Quantum Systems, he stayed on as Chief Scientist of the joint entity Quantinuum until 2025. In January 2023 he also became Distinguished Visiting Research Chair at the Perimeter Institute for Theoretical Physics. == Work == Coecke's research focuses on the foundations of physics, more particularly category theory, logic, and diagrammatic reasoning, with application to quantum informatics, quantum gravity, and NLP. He has pioneered categorical quantum mechanics together with Samson Abramsky, and spearheaded the development of a diagrammatic quantum formalism based on Penrose graphical notation, on which he wrote a textbook entitled Picturing Quantum Processes with Aleks Kissinger. With Ross Duncan he pioneered ZX-calculus. He pioneered the DisCoCat model for natural language, with Stephen Clark and Mehrnoosh Sadrzadeh. He also pioneered quantum natural language processing (QNLP), with Will Zeng, and colleagues at Cambridge Quantum Computing. == Music == Coecke is also a musician, performing and recording since the eighties. He retrospectively has been named a pioneer of industrial music. His band, Black Tish, "used cutting edge sampling techniques for the time, a host of synth and sound loops and metal-style guitars to create a heavy rock/electronica fusion unlike anything heard before", and "bridge the gap between the pure experimental nature of bands like Throbbing Gristle and Einstürzende Neubauten and the (comparatively) more radio accessible Ministry or Nine Inch Nails". Coecke is also one of the pioneers of employing quantum computers in music. == Selected publications == Textbooks Bob Coecke, Aleks Kissinger:Picturing Quantum Processes. A First Course in Quantum Theory and Diagrammatic Reasoning, Cambridge University Press, 2017, ISBN 978-1316219317 Bob Coecke, Stefano Gogioso:Quantum in Pictures, Quantinuum, 2022, ISBN 978-1-7392147-1-5 Books (as editor) Bob Coecke, David Moore, Alexander Wilce (eds.): Current Research in Operational Quantum Logic: Algebras, Categories, Languages, Fundamental Theories of Physics, Kluwer Academic, 2010, ISBN 978-9048154371 Bob Coecke (ed.): New Structures for Physics, Lecture Notes in Physics 813, Springer, 2011, ISBN 978-3642128202 Articles Bob Coecke: Kindergarten quantum mechanics, arXiv:quant-ph/0510032 Samson Abramsky, Bob Coecke: A categorical semantics of quantum protocols, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004, pp. 415–425 Bob Coecke, Ross Duncan: Interacting quantum observables, Automata, Languages and Programming, pp. 298–310, 2008 Konstantinos Meichanetzidis, Alexis Toumi, Giovanni de Felice, Bob Coecke: Grammar-Aware Question-Answering on Quantum Computers, arXiv:2012.03756 Bob Coecke: The Mathematics of Text Structure, arXiv:1904.03478 Will Zeng, Bob Coecke: Quantum Algorithms for Compositional Natural Language Processing, arXiv:1608.01406 Bob Coecke, Tobias Fritz, Robert Spekkens: A mathematical theory of resources, arXiv:1409.5531 Bob Coecke: An Alternative Gospel of structure: order, composition, processes, arxiv:1307.4038 Bob Coecke, Mehrnoosh Sadrzadeh, Steven Clark: Mathematical Foundations for a Compositional Distributional Model of Meaning, arXiv:1003.4394 Bob Coecke: Quantum Picturalism, arXiv:0908.1787 Software articles Eduardo Reck Miranda, Richie Yeung, Anna Pearson, Konstantinos Meichanetzidis, Bob Coecke: A quantum natural language processing approach to musical intelligence, arXiv:2111.06741 Dimitri Kartsaklis, Ian Fan, Richie Yeung, Anna Pearson, Robin Lorenz, Alexis Toumi, Giovanni de Felice, Konstantinos Meichanetzidis, Stephen Clark, Bob Coecke: lambeq: An efficient high-level python library for quantum NLP, arXiv:2110.04236 Giovanni de Felice, Alexis Toumi, Bob Coecke: Discopy: monoidal categories in Python, arXiv:2111.06741

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  Noisy channel model

  The noisy channel model is a framework used in spell checkers, question answering, speech recognition, and machine translation. In this model, the goal is to find the intended word given a word where the letters have been scrambled in some manner. == In spell-checking == See Chapter B of. Given an alphabet Σ {\displaystyle \Sigma } , let Σ ∗ {\displaystyle \Sigma ^{}} be the set of all finite strings over Σ {\displaystyle \Sigma } . Let the dictionary D {\displaystyle D} of valid words be some subset of Σ ∗ {\displaystyle \Sigma ^{}} , i.e., D ⊆ Σ ∗ {\displaystyle D\subseteq \Sigma ^{}} . The noisy channel is the matrix Γ w s = Pr ( s | w ) {\displaystyle \Gamma _{ws}=\Pr(s|w)} , where w ∈ D {\displaystyle w\in D} is the intended word and s ∈ Σ ∗ {\displaystyle s\in \Sigma ^{}} is the scrambled word that was actually received. The goal of the noisy channel model is to find the intended word given the scrambled word that was received. The decision function σ : Σ ∗ → D {\displaystyle \sigma :\Sigma ^{}\to D} is a function that, given a scrambled word, returns the intended word. Methods of constructing a decision function include the maximum likelihood rule, the maximum a posteriori rule, and the minimum distance rule. In some cases, it may be better to accept the scrambled word as the intended word rather than attempt to find an intended word in the dictionary. For example, the word schönfinkeling may not be in the dictionary, but might in fact be the intended word. === Example === Consider the English alphabet Σ = { a , b , c , . . . , y , z , A , B , . . . , Z , . . . } {\displaystyle \Sigma =\{a,b,c,...,y,z,A,B,...,Z,...\}} . Some subset D ⊆ Σ ∗ {\displaystyle D\subseteq \Sigma ^{}} makes up the dictionary of valid English words. There are several mistakes that may occur while typing, including: Missing letters, e.g., leter instead of letter Accidental letter additions, e.g., misstake instead of mistake Swapping letters, e.g., recieved instead of received Replacing letters, e.g., fimite instead of finite To construct the noisy channel matrix Γ {\displaystyle \Gamma } , we must consider the probability of each mistake, given the intended word ( Pr ( s | w ) {\displaystyle \Pr(s|w)} for all w ∈ D {\displaystyle w\in D} and s ∈ Σ ∗ {\displaystyle s\in \Sigma ^{}} ). These probabilities may be gathered, for example, by considering the Damerau–Levenshtein distance between s {\displaystyle s} and w {\displaystyle w} or by comparing the draft of an essay with one that has been manually edited for spelling. == In machine translation == One naturally wonders if the problem of translation could conceivably be treated as a problem in cryptography. When I look at an article in Russian, I say: 'This is really written in English, but it has been coded in some strange symbols. I will now proceed to decode. See chapter 1, and chapter 25 of. Suppose we want to translate a foreign language to English, we could model P ( E | F ) {\displaystyle P(E|F)} directly: the probability that we have English sentence E given foreign sentence F, then we pick the most likely one E ^ = arg ⁡ max E P ( E | F ) {\displaystyle {\hat {E}}=\arg \max _{E}P(E|F)} . However, by Bayes law, we have the equivalent equation: E ^ = argmax E ∈ English P ( F ∣ E ) ⏞ translation model P ( E ) ⏞ language model {\displaystyle {\hat {E}}={\underset {E\in {\text{ English }}}{\operatorname {argmax} }}\overbrace {P(F\mid E)} ^{\text{translation model }}\overbrace {P(E)} ^{\text{language model}}} The benefit of the noisy-channel model is in terms of data: If collecting a parallel corpus is costly, then we would have only a small parallel corpus, so we can only train a moderately good English-to-foreign translation model, and a moderately good foreign-to-English translation model. However, we can collect a large corpus in the foreign language only, and a large corpus in the English language only, to train two good language models. Combining these four models, we immediately get a good English-to-foreign translator and a good foreign-to-English translator. The cost of noisy-channel model is that using Bayesian inference is more costly than using a translation model directly. Instead of reading out the most likely translation by arg ⁡ max E P ( E | F ) {\displaystyle \arg \max _{E}P(E|F)} , it would have to read out predictions by both the translation model and the language model, multiply them, and search for the highest number. == In speech recognition == Speech recognition can be thought of as translating from a sound-language to a text-language. Consequently, we have T ^ = argmax T ∈ Text P ( S ∣ T ) ⏞ speech model P ( T ) ⏞ language model {\displaystyle {\hat {T}}={\underset {T\in {\text{ Text }}}{\operatorname {argmax} }}\overbrace {P(S\mid T)} ^{\text{speech model }}\overbrace {P(T)} ^{\text{language model}}} where P ( S | T ) {\displaystyle P(S|T)} is the probability that a speech sound S is produced if the speaker is intending to say text T. Intuitively, this equation states that the most likely text is a text that's both a likely text in the language, and produces the speech sound with high probability. The utility of the noisy-channel model is not in capacity. Theoretically, any noisy-channel model can be replicated by a direct P ( T | S ) {\displaystyle P(T|S)} model. However, the noisy-channel model factors the model into two parts which are appropriate for the situation, and consequently it is generally more well-behaved. When a human speaks, it does not produce the sound directly, but first produces the text it wants to speak in the language centers of the brain, then the text is translated into sound by the motor cortex, vocal cords, and other parts of the body. The noisy-channel model matches this model of the human, and so it is appropriate. This is justified in the practical success of noisy-channel model in speech recognition. === Example === Consider the sound-language sentence (written in IPA for English) S = aɪ wʊd laɪk wʌn tuː. There are three possible texts T 1 , T 2 , T 3 {\displaystyle T_{1},T_{2},T_{3}} : T 1 = {\displaystyle T_{1}=} I would like one to. T 2 = {\displaystyle T_{2}=} I would like one too. T 3 = {\displaystyle T_{3}=} I would like one two. that are equally likely, in the sense that P ( S | T 1 ) = P ( S | T 2 ) = P ( S | T 3 ) {\displaystyle P(S|T_{1})=P(S|T_{2})=P(S|T_{3})} . With a good English language model, we would have P ( T 2 ) > P ( T 1 ) > P ( T 3 ) {\displaystyle P(T_{2})>P(T_{1})>P(T_{3})} , since the second sentence is grammatical, the first is not quite, but close to a grammatical one (such as "I would like one to [go]."), while the third one is far from grammatical. Consequently, the noisy-channel model would output T 2 {\displaystyle T_{2}} as the best transcription.

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