AI Video Generator Tools

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  PCPaint

  PCPaint was one of the first IBM PC-based mouse-driven GUI paint programs, released in 1984. It followed after Microsoft Doodle, released in 1983 with the Microsoft Mouse version 1 drivers for DOS, and around the same time as Digital Research’s Draw program. It was developed and created by John Bridges and Doug Wolfgram. It was later developed into Pictor Paint. The hardware manufacturer Mouse Systems bundled PCPaint with millions of computer mice that they sold, making PCPaint one of the best-selling DOS-based paint programs of the mid 1980s. == History == In 1983, Doug Wolfgram bought a Microsoft Mouse and decided to write a drawing program for it. They named it “Mouse Draw”. The interface was primitive but the program functioned well. Wolfgram traveled to SoftCon in New Orleans where he demonstrated the program to Mouse Systems. Mouse Systems was developing an optical mouse and they wanted to bundle a painting program so they agreed to publish Mouse Draw. The original program was written entirely in assembly language with primitive graphics routines developed by Wolfgram. John Bridges worked for an educational software company, Classroom Consortia Media, Inc., developing and writing Apple and IBM graphics libraries for CCM's software. Bridges and Wolfgram were friends who had been connected through a bulletin board system developed and run by Wolfgram. The two collaborated cross country via the BBS, Wolfram in California and Bridges in New York. Mouse Systems wanted the paint program to capture the look and feel of MacPaint. John Bridges and Doug Wolfgram started reworking Mouse Draw into what became PCPaint. The program was completely re-written using Bridge's graphics library and the top-level elements were written in C rather than assembly language. Bridges developed the core graphics code for the first version of PCPaint while Wolfgram worked on the user interface and top-level code. Mouse Systems signed an exclusive agreement with Wolfgram's company, Microtex Industries, Inc., to bundle PCPaint with every mouse they sold. They began publishing PCPaint with their mice in 1984. Microsoft responded in 1985 by bundling a competing product, PC Paintbrush, with version 4 of its DOS drivers for the Microsoft Mouse, replacing its in-house Microsoft Doodle program which it published with version 1 of the DOS drivers in mid-1983. Microsoft’s mouse began to outsell Mouse Systems mouse. In November 1985 Microsoft bundled a cut-down version of PC Paintbrush with Windows 1.0 (called Microsoft Paint), later bundling an updated version of PC Paintbrush with Windows 3.0 (as Paintbrush), impacting PCPaint’s marketshare. In early 1987, Mouse Systems decided that PCPaint wasn't helping to sell mice any longer so they discontinued the bundle deal and returned rights to the code to MicroTex Industries, but retained rights to the name, PCPaint. Wolfgram then combined the paint program with a new animation system he was developing (called GRASP) and Paul Mace Software bought publishing rights to the animation system and PCPaint, which was to be renamed Pictor. Bridges again got involved and took over programming responsibilities for GRASP as well as PCPaint while Wolfgram focused on more of the business details. In creating the first version of PCPaint, Doug had a dual-floppy machine with a Computer Innovations compiler on one disk and source code on the other. John had the "luxury" of a 10MB hard disk in his XT. Data was exchanged daily via 1200, then 2400 baud modems. === Authorship and Ownership === John Bridges and Wolfgram continued to work on PCPaint and GRASP on behalf of Paul Mace Software until 1990. Also in that year, Doug Wolfgram sold his remaining rights to PCPaint (and its animation system, GRASP) to John Bridges. In 1994, GRASP development stopped and so did development of Pictor Paint. John Bridges terminated his GRASP publishing contract with Paul Mace Software, and went off to create GLPro (the next generation of GRASP) with GMEDIA. Along with GLPro, came GLPaint, the successor to PCPaint and Pictor Paint. == Versions == In June 1984, Mouse Systems shipped PCPaint 1.0, the first GUI based Paint program for the IBM PC family of computers. John Bridges and Doug Wolfgram, were the co-authors of PCPaint 1.0. PCPaint 1.0 saved its graphics in a modified BSaved image format with the extension of ".PIC". The release of PCPaint Version 1.5 followed in late 1984, with the additions of graphics image compression for the .PIC format and support for "larger-than-screen" images. PCjr support was also added in this version after overcoming severe memory shortage problems getting PCPaint to run on the 128k PCjr. October 1985 saw the release of PCPaint 2.0. EGA support and publishing features were added to this version. The .PIC format was further refined, offering support for the rapidly expanding graphics capabilities of the PC and efficient image compression. PCPaint 3.1 was released in 1989. Unlike previous versions, it was not bundled with mice but was sold as a stand-alone software product. PCPaint 3.1 offered improved text and image handling, provided 36 types of flood and fill, worked with VGA adapters in hi-res 16-color and 256-color modes, allowed the user to save and retrieve files in a variety of intercompatible formats (.PIC, .GIF, .PCX, .IMG), and printed selected portions of images on color or black-and-white dot matrix, ink jet, and laser printers such as PostScript and HP Laser Jet. PCPaint 3.1 is still in use today by some users of DOS emulation programs like DOSBox and available for free download. Pictor Paint was an improved version, written by John Bridges, and bundled with GRASP GRaphical System for Presentation also written by John Bridges. It was also called "The Painter's Easel". GLPaint, released in 1995, was the last in this series of paint programs written by John Bridges. By 1998 version 7.0 provided support for TrueColor images and the Pictor PIC format was expanded to handle these. == Pictor PIC Image Format == PCPaint 1.0 saved its graphics in a modified BSAVE image format (which was popular at the time) with the file type (extension) of ".PIC". By PCPaint 1.5 this format was extended further to accommodate image compression. With the release of version 2.0 the PICtor PIC image format was developed almost to its present state, with no similarity to the BSAVE format used by earlier versions. Pictor Paint saved its files in a compressed format with the file extension PIC, which was the same format used by PCPaint.

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  Word2vec

  Word2vec is a technique in natural language processing for obtaining vector representations of words. These vectors capture information about the meaning of the word based on the surrounding words. The word2vec algorithm estimates these representations by modeling text in a large corpus. Once trained, such a model can detect synonymous words or suggest additional words for a partial sentence. Word2vec was developed by Tomáš Mikolov, Kai Chen, Greg Corrado, Ilya Sutskever and Jeff Dean at Google, and published in 2013. Word2vec represents a word as a high-dimension vector of numbers which capture relationships between words. In particular, words which appear in similar contexts are mapped to vectors which are nearby as measured by cosine similarity. This indicates the level of semantic similarity between the words, so for example the vectors for walk and ran are nearby, as are those for "but" and "however", and "Berlin" and "Germany". == Approach == Word2vec is a group of related models that are used to produce word embeddings. These models are shallow, two-layer neural networks that are trained to reconstruct linguistic contexts of words. Word2vec takes as its input a large corpus of text and produces a mapping of the set of words to a vector space, typically of several hundred dimensions, with each unique word in the corpus being assigned a vector in the space. Word2vec can use either of two model architectures to produce these distributed representations of words: continuous bag of words (CBOW) or continuously sliding skip-gram. In both architectures, word2vec considers both individual words and a sliding context window as it iterates over the corpus. The CBOW can be viewed as a 'fill in the blank' task, where the word embedding represents the way the word influences the relative probabilities of other words in the context window. Words which are semantically similar should influence these probabilities in similar ways, because semantically similar words should be used in similar contexts. The order of context words does not influence prediction (bag of words assumption). In the continuous skip-gram architecture, the model uses the current word to predict the surrounding window of context words. The skip-gram architecture weighs nearby context words more heavily than more distant context words. According to the authors' note, CBOW is faster while skip-gram does a better job for infrequent words. After the model is trained, the learned word embeddings are positioned in the vector space such that words that share common contexts in the corpus — that is, words that are semantically and syntactically similar — are located close to one another in the space. More dissimilar words are located farther from one another in the space. == Mathematical details == This section is based on expositions. A corpus is a sequence of words. Both CBOW and skip-gram are methods to learn one vector per word appearing in the corpus. Let V {\displaystyle V} ("vocabulary") be the set of all words appearing in the corpus C {\displaystyle C} . Our goal is to learn one vector v w ∈ R d {\displaystyle v_{w}\in \mathbb {R} ^{d}} for each word w ∈ V {\displaystyle w\in V} . The idea of skip-gram is that the vector of a word should be close to the vector of each of its neighbors. The idea of CBOW is that the vector-sum of a word's neighbors should be close to the vector of the word. === Continuous bag-of-words (CBOW) === The idea of CBOW is to represent each word with a vector, such that it is possible to predict a word using the sum of the vectors of its neighbors. Specifically, for each word w i {\displaystyle w_{i}} in the corpus, the one-hot encoding of the word is used as the input to the neural network. The output of the neural network is a probability distribution over the dictionary, representing a prediction of individual words in the neighborhood of w i {\displaystyle w_{i}} . The objective of training is to maximize ∑ i ln ⁡ Pr ( w i ∣ w i + j : j ∈ N ) {\displaystyle \sum _{i}\ln \Pr(w_{i}\mid w_{i+j}\colon j\in N)} where N {\displaystyle N} is a set of (non-zero) indices representing the relative locations of nearby words considered to be in w i {\displaystyle w_{i}} 's neighborhood. For example, if we want each word in the corpus to be predicted by every other word in a small span of 4 words. The set of relative indexes of neighbor words will be: N = { − 2 , − 1 , + 1 , + 2 } {\displaystyle N=\{-2,-1,+1,+2\}} , and the objective is to maximize ∑ i ln ⁡ Pr ( w i ∣ w i − 2 , w i − 1 , w i + 1 , w i + 2 ) {\displaystyle \sum _{i}\ln \Pr(w_{i}\mid w_{i-2},w_{i-1},w_{i+1},w_{i+2})} . In standard bag-of-words, a word's context is represented by a word-count (aka a word histogram) of its neighboring words. For example, the "sat" in "the cat sat on the mat" is represented as {"the": 2, "cat": 1, "on": 1}. Note that the last word "mat" is not used to represent "sat", because it is outside the neighborhood N = { − 2 , − 1 , + 1 , + 2 } {\displaystyle N=\{-2,-1,+1,+2\}} . In continuous bag-of-words, the histogram is multiplied by a matrix V {\displaystyle V} to obtain a continuous representation of the word's context. The matrix V {\displaystyle V} is also called a dictionary. Its columns are the word vectors. It has D {\displaystyle D} columns, where D {\displaystyle D} is the size of the dictionary. Let d {\displaystyle d} be the length of each word vector. We have V ∈ R d × D {\displaystyle V\in \mathbb {R} ^{d\times D}} . For example, multiplying the word histogram {"the": 2, "cat": 1, "on": 1} with V {\displaystyle V} , we obtain 2 v the + v cat + v on {\displaystyle 2v_{\text{the}}+v_{\text{cat}}+v_{\text{on}}} . This is then multiplied with another matrix V ′ {\displaystyle V'} of shape R D × d {\displaystyle \mathbb {R} ^{D\times d}} . Each row of it is a word vector v ′ {\displaystyle v'} . This results in a vector of length D {\displaystyle D} , one entry per dictionary entry. Then, apply the softmax to obtain a probability distribution over the dictionary. This system can be visualized as a neural network, similar in spirit to an autoencoder, of architecture linear-linear-softmax, as depicted in the diagram. The system is trained by gradient descent to minimize the cross-entropy loss. In full formula, the cross-entropy loss is: − ∑ i ln ⁡ e v w i ′ ⋅ ( ∑ j ∈ N v w j + i ) ∑ w ′ e v w ′ ′ ⋅ ( ∑ j ∈ N v w j + i ) {\displaystyle -\sum _{i}\ln {\frac {e^{v_{w_{i}}'\cdot (\sum _{j\in N}v_{w_{j+i}})}}{\sum _{w'}e^{v_{w'}'\cdot (\sum _{j\in N}v_{w_{j+i}})}}}} where the outer summation ∑ i {\displaystyle \sum _{i}} is over the words in a corpus, the quantity ∑ j ∈ N v w j + i {\displaystyle \sum _{j\in N}v_{w_{j+i}}} is the sum of a word's neighbors' vectors, etc. Once such a system is trained, we have two trained matrices V , V ′ {\displaystyle V,V'} . Either the column vectors of V {\displaystyle V} or the row vectors of V ′ {\displaystyle V'} can serve as the dictionary. For example, the word "sat" can be represented as either the "sat"-th column of V {\displaystyle V} or the "sat"-th row of V ′ {\displaystyle V'} . It is also possible to simply define V ′ = V ⊤ {\displaystyle V'=V^{\top }} , in which case there would no longer be a choice. === Skip-gram === The idea of skip-gram is to represent each word with a vector, such that it is possible to predict the vectors of its neighbors using the vector of a word. The architecture is still linear-linear-softmax, the same as CBOW, but the input and the output are switched. Specifically, for each word w i {\displaystyle w_{i}} in the corpus, the one-hot encoding of the word is used as the input to the neural network. The output of the neural network is a probability distribution over the dictionary, representing a prediction of individual words in the neighborhood of w i {\displaystyle w_{i}} . The objective of training is to maximize ∑ i ∑ j ∈ N ln ⁡ Pr ( w j + i ∣ w i ) {\displaystyle \sum _{i}\sum _{j\in N}\ln \Pr(w_{j+i}\mid w_{i})} . In full formula, the loss function is − ∑ i ∑ j ∈ N ln ⁡ e v w j + i ′ ⋅ v w i ∑ w ′ e v w ′ ′ ⋅ v w i {\displaystyle -\sum _{i}\sum _{j\in N}\ln {\frac {e^{v_{w_{j+i}}'\cdot v_{w_{i}}}}{\sum _{w'}e^{v_{w'}'\cdot v_{w_{i}}}}}} Same as CBOW, once such a system is trained, we have two trained matrices V , V ′ {\displaystyle V,V'} . Either the column vectors of V {\displaystyle V} or the row vectors of V ′ {\displaystyle V'} can serve as the dictionary. It is also possible to simply define V ′ = V ⊤ {\displaystyle V'=V^{\top }} , in which case there would no longer be a choice. Essentially, skip-gram and CBOW are exactly the same in architecture. They only differ in the objective function during training. == History == During the 1980s, there were some early attempts at using neural networks to represent words and concepts as vectors. In 2010, Tomáš Mikolov (then at Brno University of Technology) with co-authors applied a simple recurrent neural network with a single hidden

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  Constructing skill trees

  Constructing skill trees (CST) is a hierarchical reinforcement learning algorithm which can build skill trees from a set of sample solution trajectories obtained from demonstration. CST uses an incremental MAP (maximum a posteriori) change point detection algorithm to segment each demonstration trajectory into skills and integrate the results into a skill tree. CST was introduced by George Konidaris, Scott Kuindersma, Andrew Barto and Roderic Grupen in 2010. == Algorithm == CST consists of mainly three parts;change point detection, alignment and merging. The main focus of CST is online change-point detection. The change-point detection algorithm is used to segment data into skills and uses the sum of discounted reward R t {\displaystyle R_{t}} as the target regression variable. Each skill is assigned an appropriate abstraction. A particle filter is used to control the computational complexity of CST. The change point detection algorithm is implemented as follows. The data for times t ∈ T {\displaystyle t\in T} and models Q with prior p ( q ∈ Q ) {\displaystyle p(q\in Q)} are given. The algorithm is assumed to be able to fit a segment from time j + 1 {\displaystyle j+1} to t using model q with the fit probability P ( j , t , q ) {\displaystyle P(j,t,q)_{}^{}} . A linear regression model with Gaussian noise is used to compute P ( j , t , q ) {\displaystyle P(j,t,q)} . The Gaussian noise prior has mean zero, and variance which follows I n v e r s e G a m m a ( v 2 , u 2 ) {\displaystyle \mathrm {InverseGamma} \left({\frac {v}{2}},{\frac {u}{2}}\right)} . The prior for each weight follows N o r m a l ( 0 , σ 2 δ ) {\displaystyle \mathrm {Normal} (0,\sigma ^{2}\delta )} . The fit probability P ( j , t , q ) {\displaystyle P(j,t,q)} is computed by the following equation. P ( j , t , q ) = π − n 2 δ m | ( A + D ) − 1 | 1 2 u v 2 ( y + u ) u + v 2 Γ ( n + v 2 ) Γ ( v 2 ) {\displaystyle P(j,t,q)={\frac {\pi ^{-{\frac {n}{2}}}}{\delta ^{m}}}\left|(A+D)^{-1}\right|^{\frac {1}{2}}{\frac {u^{\frac {v}{2}}}{(y+u)^{\frac {u+v}{2}}}}{\frac {\Gamma ({\frac {n+v}{2}})}{\Gamma ({\frac {v}{2}})}}} Then, CST compute the probability of the changepoint at time j with model q, P t ( j , q ) {\displaystyle P_{t}(j,q)} and P j MAP {\displaystyle P_{j}^{\text{MAP}}} using a Viterbi algorithm. P t ( j , q ) = ( 1 − G ( t − j − 1 ) ) P ( j , t , q ) p ( q ) P j MAP {\displaystyle P_{t}(j,q)=(1-G(t-j-1))P(j,t,q)p(q)P_{j}^{\text{MAP}}} P j MAP = max i , q P j ( i , q ) g ( j − i ) 1 − G ( j − i − 1 ) , ∀ j < t {\displaystyle P_{j}^{\text{MAP}}=\max _{i,q}{\frac {P_{j}(i,q)g(j-i)}{1-G(j-i-1)}},\forall j Read more →

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  Generalized blockmodeling

  In generalized blockmodeling, the blockmodeling is done by "the translation of an equivalence type into a set of permitted block types", which differs from the conventional blockmodeling, which is using the indirect approach. It's a special instance of the direct blockmodeling approach. Generalized blockmodeling was introduced in 1994 by Patrick Doreian, Vladimir Batagelj and Anuška Ferligoj. == Definition == Generalized blockmodeling approach is a direct one, "where the optimal partition(s) is (are) identified based on minimal values of a compatible criterion function defined by the difference between empirical blocks and corresponding ideal blocks". At the same time, the much broader set of block types is introduced (while in conventional blockmodeling only certain types are used). The conventional blockmodeling is inductive due to nonspecification of neither the clusters or the location of block types, while in generalized blockmodeling the blockmodel is specified with more detail than just the permition of certain block types (e.g., prespecification). Further, it's possible to define departures from the permitted (ideal) blocktype, using criterion function. Using local optimization procedure, firstly the initial clustering (with specified number of clusters is done, based on random creation. How the clusters are neighboring to each other, is based on two transformations: 1) a vertex is moved from one to another cluster or 2) a pair of vertices is interchanged between two different clusters. This process of transformation steps is repeated many times, until only the best fitting partitions (with the minimized value of the criterion function) are kept as blockmodels for the future exploration of the network. Different types of generalized blockmodeling are: generalized binary blockmodeling, generalized valued blockmodeling and generalized homogeneity blockmodeling. == Benefits == According to Patrick Doreian, the benefits of generalized blockmodeling, are as follows: usage of explicit criterion function, compatible with a given type of equivalence, results to in-built measure of fit, which is integral to the establishment of the blockmodels (in conventional blockmodeling, there is no compelling and coherent measures of fit); partitions, based on generalized blockmodeling, regularly outperform and never perform less well than the partitions, based on conventional approach; with generalized blockmodeling it's possible to specify new types of blockmodels; this potentially unlimited set of new block types also results in permittion of inclusion of substantively driven blockmodels; in generalized blockmodeling, the specification of the block types and the location of some of them in the blockmodel is possible; researcher can speficy which (pair of) vertices must be (not) clustered together; this approach also allows the imposition of penalties, resulting into identification of empirical null blocks without inconsistencies with a corresponding ideal null block. == Problems == According to Doreian, the problems of generalized blockmodeling, are as follows: unknown sensitivity to particular data features, examination of boundary problems, computationally burdensome, which results in a constraint regarding practical network size (generalized blockmodeling is thus primarily used to analyse smaller networks (below 100 units)), identifying structure from incomplete network information, most of generalized blockmodeling is based on binary networks, but there is also development in the field of valued networks, criterion function is minimized for a specified blockmodel, with results in issues of evaluating statistically, based on the structural data alone, problems regarding three dimensional network data, problems regarding the evolution of fundamental network structure. == Book == The book with the same title, Generalized blockmodeling, written by Patrick Doreian, Vladimir Batagelj and Anuška Ferligoj, was in 2007 awarded the Harrison White Outstanding Book Award by the Mathematical Sociology Section of American Sociological Association.

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  Index locking

  In databases an index is a data structure, part of the database, used by a database system to efficiently navigate access to user data. Index data are system data distinct from user data, and consist primarily of pointers. Changes in a database (by insert, delete, or modify operations), may require indexes to be updated to maintain accurate user data accesses. Index locking is a technique used to maintain index integrity. A portion of an index is locked during a database transaction when this portion is being accessed by the transaction as a result of attempt to access related user data. Additionally, special database system transactions (not user-invoked transactions) may be invoked to maintain and modify an index, as part of a system's self-maintenance activities. When a portion of an index is locked by a transaction, other transactions may be blocked from accessing this index portion (blocked from modifying, and even from reading it, depending on lock type and needed operation). Index Locking Protocol guarantees that phantom read phenomenon won't occur. Index locking protocol states: Every relation must have at least one index. A transaction can access tuples only after finding them through one or more indices on the relation A transaction Ti that performs a lookup must lock all the index leaf nodes that it accesses, in S-mode, even if the leaf node does not contain any tuple satisfying the index lookup (e.g. for a range query, no tuple in a leaf is in the range) A transaction Ti that inserts, updates or deletes a tuple ti in a relation r must update all indices to r and it must obtain exclusive locks on all index leaf nodes affected by the insert/update/delete The rules of the two-phase locking protocol must be observed. Specialized concurrency control techniques exist for accessing indexes. These techniques depend on the index type, and take advantage of its structure. They are typically much more effective than applying to indexes common concurrency control methods applied to user data. Notable and widely researched are specialized techniques for B-trees (B-Tree concurrency control) which are regularly used as database indexes. Index locks are used to coordinate threads accessing indexes concurrently, and typically shorter-lived than the common transaction locks on user data. In professional literature, they are often called latches.

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  Constructing skill trees

  Constructing skill trees (CST) is a hierarchical reinforcement learning algorithm which can build skill trees from a set of sample solution trajectories obtained from demonstration. CST uses an incremental MAP (maximum a posteriori) change point detection algorithm to segment each demonstration trajectory into skills and integrate the results into a skill tree. CST was introduced by George Konidaris, Scott Kuindersma, Andrew Barto and Roderic Grupen in 2010. == Algorithm == CST consists of mainly three parts;change point detection, alignment and merging. The main focus of CST is online change-point detection. The change-point detection algorithm is used to segment data into skills and uses the sum of discounted reward R t {\displaystyle R_{t}} as the target regression variable. Each skill is assigned an appropriate abstraction. A particle filter is used to control the computational complexity of CST. The change point detection algorithm is implemented as follows. The data for times t ∈ T {\displaystyle t\in T} and models Q with prior p ( q ∈ Q ) {\displaystyle p(q\in Q)} are given. The algorithm is assumed to be able to fit a segment from time j + 1 {\displaystyle j+1} to t using model q with the fit probability P ( j , t , q ) {\displaystyle P(j,t,q)_{}^{}} . A linear regression model with Gaussian noise is used to compute P ( j , t , q ) {\displaystyle P(j,t,q)} . The Gaussian noise prior has mean zero, and variance which follows I n v e r s e G a m m a ( v 2 , u 2 ) {\displaystyle \mathrm {InverseGamma} \left({\frac {v}{2}},{\frac {u}{2}}\right)} . The prior for each weight follows N o r m a l ( 0 , σ 2 δ ) {\displaystyle \mathrm {Normal} (0,\sigma ^{2}\delta )} . The fit probability P ( j , t , q ) {\displaystyle P(j,t,q)} is computed by the following equation. P ( j , t , q ) = π − n 2 δ m | ( A + D ) − 1 | 1 2 u v 2 ( y + u ) u + v 2 Γ ( n + v 2 ) Γ ( v 2 ) {\displaystyle P(j,t,q)={\frac {\pi ^{-{\frac {n}{2}}}}{\delta ^{m}}}\left|(A+D)^{-1}\right|^{\frac {1}{2}}{\frac {u^{\frac {v}{2}}}{(y+u)^{\frac {u+v}{2}}}}{\frac {\Gamma ({\frac {n+v}{2}})}{\Gamma ({\frac {v}{2}})}}} Then, CST compute the probability of the changepoint at time j with model q, P t ( j , q ) {\displaystyle P_{t}(j,q)} and P j MAP {\displaystyle P_{j}^{\text{MAP}}} using a Viterbi algorithm. P t ( j , q ) = ( 1 − G ( t − j − 1 ) ) P ( j , t , q ) p ( q ) P j MAP {\displaystyle P_{t}(j,q)=(1-G(t-j-1))P(j,t,q)p(q)P_{j}^{\text{MAP}}} P j MAP = max i , q P j ( i , q ) g ( j − i ) 1 − G ( j − i − 1 ) , ∀ j < t {\displaystyle P_{j}^{\text{MAP}}=\max _{i,q}{\frac {P_{j}(i,q)g(j-i)}{1-G(j-i-1)}},\forall j Read more →

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  Spiking neural network

  Spiking neural networks (SNNs) are artificial neural networks (ANN) that mimic natural neural networks. These models leverage timing of discrete spikes as the main information carrier. In addition to neuronal and synaptic state, SNNs incorporate the concept of time into their operating model. The idea is that neurons in the SNN do not transmit information at each propagation cycle (as it happens with typical multi-layer perceptron networks), but rather transmit information only when a membrane potential—an intrinsic quality of the neuron related to its membrane electrical charge—reaches a specific value, called the threshold. When the membrane potential reaches the threshold, the neuron fires, and generates a signal that travels to other neurons which, in turn, increase or decrease their potentials in response to this signal. A neuron model that fires at the moment of threshold crossing is also called a spiking neuron model. While spike rates can be considered the analogue of the variable output of a traditional ANN, neurobiology research indicated that high speed processing cannot be performed solely through a rate-based scheme. For example humans can perform an image recognition task requiring no more than 10ms of processing time per neuron through the successive layers (going from the retina to the temporal lobe). This time window is too short for rate-based encoding. The precise spike timings in a small set of spiking neurons also has a higher information coding capacity compared with a rate-based approach. The most prominent spiking neuron model is the leaky integrate-and-fire model. In that model, the momentary activation level (modeled as a differential equation) is normally considered to be the neuron's state, with incoming spikes pushing this value higher or lower, until the state eventually either decays or—if the firing threshold is reached—the neuron fires. After firing, the state variable is reset to a lower value. Various decoding methods exist for interpreting the outgoing spike train as a real-value number, relying on either the frequency of spikes (rate-code), the time-to-first-spike after stimulation, or the interval between spikes. == History == Many multi-layer artificial neural networks are fully connected, receiving input from every neuron in the previous layer and signalling every neuron in the subsequent layer. Although these networks have achieved breakthroughs, they do not match biological networks and do not mimic neurons. The biology-inspired Hodgkin–Huxley model of a spiking neuron was proposed in 1952. This model described how action potentials are initiated and propagated. Communication between neurons, which requires the exchange of chemical neurotransmitters in the synaptic gap, is described in models such as the integrate-and-fire model, FitzHugh–Nagumo model (1961–1962), and Hindmarsh–Rose model (1984). The leaky integrate-and-fire model (or a derivative) is commonly used as it is easier to compute than Hodgkin–Huxley. While the notion of an artificial spiking neural network became popular only in the twenty-first century, studies between 1980 and 1995 supported the concept. The first models of this type of ANN appeared to simulate non-algorithmic intelligent information processing systems. However, the notion of the spiking neural network as a mathematical model was first worked on in the early 1970s. As of 2019 SNNs lagged behind ANNs in accuracy, but the gap is decreasing, and has vanished on some tasks. == Underpinnings == Information in the brain is represented as action potentials (neuron spikes), which may group into spike trains or coordinated waves. A fundamental question of neuroscience is to determine whether neurons communicate by a rate or temporal code. Temporal coding implies that a single spiking neuron can replace hundreds of hidden units on a conventional neural net. SNNs define a neuron's current state as its potential (possibly modeled as a differential equation). An input pulse causes the potential to rise and then gradually decline. Encoding schemes can interpret these pulse sequences as a number, considering pulse frequency and pulse interval. Using the precise time of pulse occurrence, a neural network can consider more information and offer better computing properties. SNNs compute in the continuous domain. Such neurons test for activation only when their potentials reach a certain value. When a neuron is activated, it produces a signal that is passed to connected neurons, accordingly raising or lowering their potentials. The SNN approach produces a continuous output instead of the binary output of traditional ANNs. Pulse trains are not easily interpretable, hence the need for encoding schemes. However, a pulse train representation may be more suited for processing spatiotemporal data (or real-world sensory data classification). SNNs connect neurons only to nearby neurons so that they process input blocks separately (similar to CNN using filters). They consider time by encoding information as pulse trains so as not to lose information. This avoids the complexity of a recurrent neural network (RNN). Impulse neurons are more powerful computational units than traditional artificial neurons. SNNs are theoretically more powerful than so called "second-generation networks" defined as ANNs "based on computational units that apply activation function with a continuous set of possible output values to a weighted sum (or polynomial) of the inputs"; however, SNN training issues and hardware requirements limit their use. Although unsupervised biologically inspired learning methods are available such as Hebbian learning and STDP, no effective supervised training method is suitable for SNNs that can provide better performance than second-generation networks. Spike-based activation of SNNs is not differentiable, thus gradient descent-based backpropagation (BP) is not available. SNNs have much larger computational costs for simulating realistic neural models than traditional ANNs. Pulse-coupled neural networks (PCNN) are often confused with SNNs. A PCNN can be seen as a kind of SNN. Researchers are actively working on various topics. The first concerns differentiability. The expressions for both the forward- and backward-learning methods contain the derivative of the neural activation function which is not differentiable because a neuron's output is either 1 when it spikes, and 0 otherwise. This all-or-nothing behavior disrupts gradients and makes these neurons unsuitable for gradient-based optimization. Approaches to resolving it include: resorting to entirely biologically inspired local learning rules for the hidden units translating conventionally trained "rate-based" NNs to SNNs smoothing the network model to be continuously differentiable defining an SG (Surrogate Gradient) as a continuous relaxation of the real gradients The second concerns the optimization algorithm. Standard BP can be expensive in terms of computation, memory, and communication and may be poorly suited to the hardware that implements it (e.g., a computer, brain, or neuromorphic device). Incorporating additional neuron dynamics such as Spike Frequency Adaptation (SFA) is a notable advance, enhancing efficiency and computational power. These neurons sit between biological complexity and computational complexity. Originating from biological insights, SFA offers significant computational benefits by reducing power usage, especially in cases of repetitive or intense stimuli. This adaptation improves signal/noise clarity and introduces an elementary short-term memory at the neuron level, which in turn, improves accuracy and efficiency. This was mostly achieved using compartmental neuron models. The simpler versions are of neuron models with adaptive thresholds, are an indirect way of achieving SFA. It equips SNNs with improved learning capabilities, even with constrained synaptic plasticity, and elevates computational efficiency. This feature lessens the demand on network layers by decreasing the need for spike processing, thus lowering computational load and memory access time—essential aspects of neural computation. Moreover, SNNs utilizing neurons capable of SFA achieve levels of accuracy that rival those of conventional ANNs, while also requiring fewer neurons for comparable tasks. This efficiency streamlines the computational workflow and conserves space and energy, while maintaining technical integrity. High-performance deep spiking neural networks can operate with 0.3 spikes per neuron. == Applications == SNNs can in principle be applied to the same applications as traditional ANNs. In addition, SNNs can model the central nervous system of biological organisms, such as an insect seeking food without prior knowledge of the environment. Due to their relative realism, they can be used to study biological neural circuits. Starting with a hypothesis about the topology of a biological neuronal circuit and its functi

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  Operational taxonomic unit

  An operational taxonomic unit (OTU) is an operational definition used to classify groups of closely related individuals. The term was originally introduced in 1963 by Robert R. Sokal and Peter H. A. Sneath in the context of numerical taxonomy, where an "operational taxonomic unit" is simply the group of organisms currently being studied. In this sense, an OTU is a pragmatic definition to group individuals by similarity, equivalent to but not necessarily in line with classical Linnaean taxonomy or modern evolutionary taxonomy. Nowadays, however, the term is commonly used in a different context and refers to clusters of (uncultivated or unknown) organisms, grouped by DNA sequence similarity of a specific taxonomic marker gene (originally coined as mOTU; molecular OTU). In other words, OTUs are pragmatic proxies for "species" at different taxonomic levels, in the absence of traditional systems of biological classification as are available for macroscopic organisms. For several years, OTUs have been the most commonly used units of diversity, especially when analysing small subunit 16S (for prokaryotes) or 18S rRNA (for eukaryotes) marker gene sequence datasets. == Molecular OTU by clustering of marker gene sequences == In the approach represented by DNA barcoding, a particular locus is chosen to be used as the marker gene for classification. This locus should be universally present in the scope selected, variable enough to be different among close-related species, and be flanked by conservative sequences that allow for easy amplification and detection. There are databases containing sequences for such marker genes from many different species, allowing for comparison. (Sometimes only using one locus does not provide sufficient resolution, so multiple marker genes are used. This is the case for plants, where rbcL+matK is common.) Sequences obtained this way can be clustered according to their similarity to one another, and operational taxonomic units are defined based on the similarity threshold set by the researcher. The exact threshold depends on the taxa in question and the mutational rates of the selected locus in the taxon. 97–99% are commonly used, but "it is now recognized to be somewhat arbitrary as sequence variation within and among species varies across taxa". 100% similarity (fully identical) is also common, also known as single variants. It remains debatable how well this commonly used method recapitulates true microbial species phylogeny or ecology. Although OTUs can be calculated differently when using different algorithms or thresholds, research by Schmidt et al. (2014) demonstrated that 16S-derived microbial OTUs were generally ecologically consistent across habitats and several clustering approaches. The number of OTUs defined may be inflated due to errors in DNA sequencing. === OTU clustering approaches === There are three main approaches to clustering OTUs: De novo, for which the clustering is based on similarities between sequencing reads. Closed-reference, for which the clustering is performed against a reference database of sequences. Open-reference, where clustering is first performed against a reference database of sequences, then any remaining sequences that could not be mapped to the reference are clustered de novo. Using a reference provides taxonomic context for the OTUs found. Alternatively, taxonomic context can be found after the construction of clusters by comparing representative sequences from clusters against a reference database. There are also specialized classifiers for this purpose which are much faster than naive comparison using BLAST. === OTU clustering algorithms === Hierarchical clustering algorithms (HCA): uclust & cd-hit & ESPRIT Bayesian clustering: CROP == Molecular OTU by other methods == In addition to similarity-based grouping, marker gene sequences can be sorted into OTUs using molecular phylogeny, k-mer composition, or hybrid methods combining these methods with similarity. There are also Bayesian tree-less methods and machine learning approaches. Using phylogeny often involves manually assigning terminal clades or single nodes to an OTU, so this is usually only done for refinement. Genome skimming can be used to obtain high-copy DNA without the need to choose marker genes or to design PCR primers for the chosen genes. It can provide fairly good coverage of organelle DNA and repetitive elements such as ribosomal DNA, both of which can be used like marker genes in OTU analysis. Whole-genome sequencing is more expensive and involves the production and processing of more data. By considering the entire genome, many (sometimes over 100) marker genes can be used at the same time, producing highly resolved phylogenies that correctly identify problematic taxa. It is also possible to use entire genomes for OTU assignment. For example, genomes from different bacterial species almost always have an average nucleotide identity lower than 95%, a fact that can be used to define new OTUs (and likely new species).

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  Language-Theoretic Security

  Language-theoretic security, or LangSec, is an approach to software security that focuses on input handling, complexity, and program design as strategies to improve the verifiability of computer programs. It was introduced in 2005 by Robert J. Hansen and Meredith L. Patterson at BlackHat and in 2011 by Len Sassaman and Patterson. It aims to create a formal description of which software is likely to have security vulnerabilities of particular classes, and why. It considers programs to have an inherent parser component, whether or not explicit, composed of that part of the program which operates on external input before that input is fully parsed. A central hypothesis of language-theoretic security is that vulnerabilities in software increase according to the computational power of the notional input-accepting automaton equivalent to this parser, using the definitions of automata theory. The lower bound on this computational power is the input language complexity of the program. The extent to which reducing this complexity is possible is a function of the specification of the communication protocol or file format the program takes as input. == Parsing as a security mechanism == The behaviour of a program is defined with reference to its expected input. Unexpected input being used by a program is a factor in numerous security bugs, including the so-called Android master key vulnerability (CVE-2013-4787), because accepting unexpected input renders the program's specification ambiguous. In that instance, the unexpected ambiguity came in the form of a ZIP file with duplicate filenames. If a program fully parses its input and only acts on input that unambiguously meets the specification, it follows that the program will avoid these types of vulnerabilities. This is an intentional inversion of the Postel principle. Accepting only unambiguous and valid input is a more formal requirement than input validation or sanitization, and narrows the number of possible but unanticipated program states that can be induced in an application via user input. Conversely, failure to do this is associated with security vulnerabilities. Input sanitization in particular is held to be an inadequate approach to avoiding malicious input because it inherently ignores context-sensitive properties of the input; it can therefore result in paradoxical effects, such as sanitization code activating otherwise inert cross-site scripting payloads in browsers. === Parser differentials === If the language of accepted program input is sufficiently simple, it is possible to verify that two implementations parse the same input language consistently. This is advantageous because it shows no parser differential exists between the two implementations. The requisite level of simplicity is theoretically that for which there is a solution to the equivalence problem. If the two parsers involved in CVE-2013-4787 were equivalent - that is, if they rendered the same output state given the same input state - the vulnerability could not have existed. One strategy for doing this is to publish machine-readable specifications of a format or protocol, and then use a parser generator to generate the parser code. An example of a parser generator built for this purpose is DaeDaLus. The combination of Lex with any of GNU Bison, ANTLR, or Yacc also accomplishes this. However, many parser generators allow the mixing of general purpose code with the parsing definitions, which weakens the guarantees provided by parsing. === Analysis of injection attacks === Injection attacks are generally the result of differences between the serializer (or "unparser") and the corresponding parser at a layer boundary in a system; therefore, they are a special case of parser differentials. In a SQL injection attack, for example, an attacker is able to cause the application with which they are interacting to serialize a SQL query that has different semantics than intended. In the simplest case where the payload ends a string and adds new code, the payload has crossed the code-data boundary in SQL. In language-theoretic security, this is treated as a bug in the serializer of the SQL query, which should instead be written in a way that constrains its possible outputs to those within the scope of the intended query. === Parser combinators === If a parser generator is not used, it is still possible to avoid implementation bugs by using parser combinator such as Nom to implement the parser code. This has the drawback of relying on a programmer correctly translating the specification into the language of the parser generator library, though this task is still less error-prone than hand-coding a parser. == Input format complexity == Complexity in computer programs is associated with security vulnerabilities. Within the domain of language-theoretic security, complexity is described with reference to the computational power of the abstract machine necessary to implement the program, or more particularly, to implement the parser for its input language. This complexity describes whether it is possible to show that there is no unintended or undesired functionality in the program which might be exploitable by an attacker. To be bounded in complexity, the program's input must be well-defined both in terms of form and of semantics. === Weird machines === A weird machine is a model of computation in a program that exists in parallel with, but is distinct from, the intended abstract model of computation in that program. Some classes of weird machine arise from the multi-layered nature of computer programs, or the context in which the programs run; others result from the unanticipated functionality a program has due to its complexity or to software bugs. The more complex the computation model of a program, the more likely it is to implement a weird machine. Depending on context, the weird machine may or may not be concretely useful for an attacker. Since the space of weird machines in the context of some program is the universe of all possible states that are not within the program's intended states, many exploited states including remote code execution and injection attacks belong to the domain of weird machines. A reduction in weird machines is therefore a likely correlate with reduced program vulnerability. === SafeDocs project === SafeDocs is a DARPA project undertaken in 2018 to take existing file formats, create safer subsets of them, and develop programming tools to work for the safer formats. The initial test case for this was PDF. The purpose of creating safer subsets in this case is to lower the minimum bound on parser complexity so that it becomes possible to create tools that will generate correct, normative parsers for them. == Relation to programming languages == The analytic framework of language-theoretic security assumes programs to be virtual machines that execute their input. A document that is read by an application is in this sense a form of machine code, in a generalization of the data as code idea, following the automata theory description of parsers. === Type-safe programming languages === Parsing input and serializing output are operations that consume one data type and emit another. A programming language can therefore check that data is correctly parsed and contains the expected structure by checking data types, and correct serializing (or unparsing) can be implemented as operations on the data types that are relevant to the program's output. This approach can be used to show that the recognizer and unparser patterns have been implemented. It is also possible to implement type checking across a distributed system to enforce parsing and unparsing of the expected structures and to verify that the assumptions made in designing the compositional properties of a distributed system have been followed. === Memory-safe programming languages === In the general case, spatial memory correctness is undecidable. If any proof of spatial memory correctness is to be made, it is therefore necessary to bound the complexity of the code. Interpreted languages such as Java and Python effectively accomplish this via runtime bounds checking, and frameworks for runtime bounds checking also exist for C. The effect of these strategies for spatial memory correctness are to create a halt state in place of a spatial memory correctness violation; therefore, it can be shown that the program will not violate spatial memory correctness, but in exchange, it cannot be shown in the general case that programs will not have runtime bounds checking exceptions. Some programming languages, such as Rust, accomplish this using borrow checking. The borrow checker acts to assure spatial memory correctness by compile-time reference counting. Code for which spatial memory correctness cannot be shown to not be violated therefore does not compile, inherently limiting the complexity of the spatial memory correctness of the program to what is decidable. Thi

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  Huber loss

  In statistics, the Huber loss is a loss function used in robust regression, that is less sensitive to outliers in data than the squared error loss. A variant for classification is also sometimes used. == Definition == The Huber loss function describes the penalty incurred by an estimation procedure f. Huber (1964) defines the loss function piecewise by L δ ( a ) = { 1 2 a 2 for | a | ≤ δ , δ ⋅ ( | a | − 1 2 δ ) , otherwise. {\displaystyle L_{\delta }(a)={\begin{cases}{\frac {1}{2}}{a^{2}}&{\text{for }}|a|\leq \delta ,\\[4pt]\delta \cdot \left(|a|-{\frac {1}{2}}\delta \right),&{\text{otherwise.}}\end{cases}}} This function is quadratic for small values of a, and linear for large values, with equal values and slopes of the different sections at the two points where | a | = δ {\displaystyle |a|=\delta } . The variable a often refers to the residuals, that is to the difference between the observed and predicted values a = y − f ( x ) {\displaystyle a=y-f(x)} , so the former can be expanded to L δ ( y , f ( x ) ) = { 1 2 ( y − f ( x ) ) 2 for | y − f ( x ) | ≤ δ , δ ⋅ ( | y − f ( x ) | − 1 2 δ ) , otherwise. {\displaystyle L_{\delta }(y,f(x))={\begin{cases}{\frac {1}{2}}{\left(y-f(x)\right)}^{2}&{\text{for }}\left|y-f(x)\right|\leq \delta ,\\[4pt]\delta \ \cdot \left(\left|y-f(x)\right|-{\frac {1}{2}}\delta \right),&{\text{otherwise.}}\end{cases}}} The Huber loss is the convolution of the absolute value function with the rectangular function, scaled and translated. Thus it "smoothens out" the former's corner at the origin. == Motivation == Two very commonly used loss functions are the squared loss, L ( a ) = a 2 {\displaystyle L(a)=a^{2}} , and the absolute loss, L ( a ) = | a | {\displaystyle L(a)=|a|} . The squared loss function results in an arithmetic mean-unbiased estimator, and the absolute-value loss function results in a median-unbiased estimator (in the one-dimensional case, and a geometric median-unbiased estimator for the multi-dimensional case). The squared loss has the disadvantage that it has the tendency to be dominated by outliers—when summing over a set of a {\displaystyle a} 's (as in ∑ i = 1 n L ( a i ) {\textstyle \sum _{i=1}^{n}L(a_{i})} ), the sample mean is influenced too much by a few particularly large a {\displaystyle a} -values when the distribution is heavy tailed: in terms of estimation theory, the asymptotic relative efficiency of the mean is poor for heavy-tailed distributions. As defined above, the Huber loss function is strongly convex in a uniform neighborhood of its minimum a = 0 {\displaystyle a=0} ; at the boundary of this uniform neighborhood, the Huber loss function has a differentiable extension to an affine function at points a = − δ {\displaystyle a=-\delta } and a = δ {\displaystyle a=\delta } . These properties allow it to combine much of the sensitivity of the mean-unbiased, minimum-variance estimator of the mean (using the quadratic loss function) and the robustness of the median-unbiased estimator (using the absolute value function). == Pseudo-Huber loss function == The Pseudo-Huber loss function can be used as a smooth approximation of the Huber loss function. It combines the best properties of L2 squared loss and L1 absolute loss by being strongly convex when close to the target/minimum and less steep for extreme values. The scale at which the Pseudo-Huber loss function transitions from L2 loss for values close to the minimum to L1 loss for extreme values and the steepness at extreme values can be controlled by the δ {\displaystyle \delta } value. The Pseudo-Huber loss function ensures that derivatives are continuous for all degrees. It is defined as L δ ( a ) = δ 2 ( 1 + ( a / δ ) 2 − 1 ) . {\displaystyle L_{\delta }(a)=\delta ^{2}\left({\sqrt {1+(a/\delta )^{2}}}-1\right).} As such, this function approximates a 2 / 2 {\displaystyle a^{2}/2} for small values of a {\displaystyle a} , and approximates a straight line with slope δ {\displaystyle \delta } for large values of a {\displaystyle a} . While the above is the most common form, other smooth approximations of the Huber loss function also exist. == Variant for classification == For classification purposes, a variant of the Huber loss called modified Huber is sometimes used. Given a prediction f ( x ) {\displaystyle f(x)} (a real-valued classifier score) and a true binary class label y ∈ { + 1 , − 1 } {\displaystyle y\in \{+1,-1\}} , the modified Huber loss is defined as L ( y , f ( x ) ) = { max ( 0 , 1 − y f ( x ) ) 2 for y f ( x ) > − 1 , − 4 y f ( x ) otherwise. {\displaystyle L(y,f(x))={\begin{cases}\max(0,1-y\,f(x))^{2}&{\text{for }}\,\,y\,f(x)>-1,\\[4pt]-4y\,f(x)&{\text{otherwise.}}\end{cases}}} The term max ( 0 , 1 − y f ( x ) ) {\displaystyle \max(0,1-y\,f(x))} is the hinge loss used by support vector machines; the quadratically smoothed hinge loss is a generalization of L {\displaystyle L} . == Applications == The Huber loss function is used in robust statistics, M-estimation and additive modelling.

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  Prototype methods

  Prototype methods are machine learning methods that use data prototypes. A data prototype is a data value that reflects other values in its class, e.g., the centroid in a K-means clustering problem. == Methods == The following are some prototype methods K-means clustering Learning vector quantization (LVQ) Gaussian mixtures == Related Methods == While K-nearest neighbor's does not use prototypes, it is similar to prototype methods like K-means clustering.

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  Quickprop

  Quickprop is an iterative method for determining the minimum of the loss function of an artificial neural network, following an algorithm inspired by the Newton's method. Sometimes, the algorithm is classified to the group of the second order learning methods. It follows a quadratic approximation of the previous gradient step and the current gradient, which is expected to be close to the minimum of the loss function, under the assumption that the loss function is locally approximately square, trying to describe it by means of an upwardly open parabola. The minimum is sought in the vertex of the parabola. The procedure requires only local information of the artificial neuron to which it is applied. The k {\displaystyle k} -th approximation step is given by: Δ ( k ) w i j = Δ ( k − 1 ) w i j ( ∇ i j E ( k ) ∇ i j E ( k − 1 ) − ∇ i j E ( k ) ) {\displaystyle \Delta ^{(k)}\,w_{ij}=\Delta ^{(k-1)}\,w_{ij}\left({\frac {\nabla _{ij}\,E^{(k)}}{\nabla _{ij}\,E^{(k-1)}-\nabla _{ij}\,E^{(k)}}}\right)} Where w i j {\displaystyle w_{ij}} is the weight of input i {\displaystyle i} of neuron j {\displaystyle j} , and E {\displaystyle E} is the loss function. The Quickprop algorithm is an implementation of the error backpropagation algorithm, but the network can behave chaotically during the learning phase due to large step sizes.

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  Foveated rendering

  Foveated rendering is a rendering technique which uses an eye tracker integrated with a virtual reality headset to reduce the rendering workload by greatly reducing the image quality in the peripheral vision (outside of the zone gazed by the fovea). A less sophisticated variant called fixed foveated rendering doesn't utilise eye tracking and instead assumes a fixed focal point. == History == Research into foveated rendering dates back at least to 1991. At Tech Crunch Disrupt SF 2014, Fove unveiled a headset featuring foveated rendering. This was followed by a successful kickstarter in May 2015. At CES 2016, SensoMotoric Instruments (SMI) demoed a new 250 Hz eye tracking system and a working foveated rendering solution. It resulted from a partnership with camera sensor manufacturer Omnivision who provided the camera hardware for the new system. In July 2016, Nvidia demonstrated during SIGGRAPH a new method of foveated rendering claimed to be invisible to users. In February 2017, Qualcomm announced their Snapdragon 835 Virtual Reality Development Kit (VRDK) which includes foveated rendering support called Adreno Foveation. == Use == According to chief scientist Michael Abrash at Oculus, utilising foveated rendering in conjunction with sparse rendering and deep learning image reconstruction has the potential to require an order of magnitude fewer pixels to be rendered in comparison to a full image. Later, these results have been demonstrated and published. In December 2019, fixed foveated rendering support was added to the Oculus Quest SDK. A number of VR headsets have included on-board eye tracking to provide support for foveated rendering, including HTC's Vive Pro Eye (2019), Meta Quest Pro (2022), PlayStation VR2 (2023), and Apple Vision Pro (2024). In 2025, Valve announced the upcoming Steam Frame headset, which applies a variation of the technique known as "foveated streaming" for wireless streaming from a PC to the headset; the method similarly uses variance in bit rate, and is performed at the encoder level rather than the software level.

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  Multidimensional analysis

  In statistics, econometrics and related fields, multidimensional analysis (MDA) is a data analysis process that groups data into two categories: data dimensions and measurements. For example, a data set consisting of the number of wins for a single football team at each of several years is a single-dimensional (in this case, longitudinal) data set. A data set consisting of the number of wins for several football teams in a single year is also a single-dimensional (in this case, cross-sectional) data set. A data set consisting of the number of wins for several football teams over several years is a two-dimensional data set. == Higher dimensions == In many disciplines, two-dimensional data sets are also called panel data. While, strictly speaking, two- and higher-dimensional data sets are "multi-dimensional", the term "multidimensional" tends to be applied only to data sets with three or more dimensions. For example, some forecast data sets provide forecasts for multiple target periods, conducted by multiple forecasters, and made at multiple horizons. The three dimensions provide more information than can be gleaned from two-dimensional panel data sets. == Software == Computer software for MDA include Online analytical processing (OLAP) for data in relational databases, pivot tables for data in spreadsheets, and Array DBMSs for general multi-dimensional data (such as raster data) in science, engineering, and business.

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  Constructing skill trees

  Constructing skill trees (CST) is a hierarchical reinforcement learning algorithm which can build skill trees from a set of sample solution trajectories obtained from demonstration. CST uses an incremental MAP (maximum a posteriori) change point detection algorithm to segment each demonstration trajectory into skills and integrate the results into a skill tree. CST was introduced by George Konidaris, Scott Kuindersma, Andrew Barto and Roderic Grupen in 2010. == Algorithm == CST consists of mainly three parts;change point detection, alignment and merging. The main focus of CST is online change-point detection. The change-point detection algorithm is used to segment data into skills and uses the sum of discounted reward R t {\displaystyle R_{t}} as the target regression variable. Each skill is assigned an appropriate abstraction. A particle filter is used to control the computational complexity of CST. The change point detection algorithm is implemented as follows. The data for times t ∈ T {\displaystyle t\in T} and models Q with prior p ( q ∈ Q ) {\displaystyle p(q\in Q)} are given. The algorithm is assumed to be able to fit a segment from time j + 1 {\displaystyle j+1} to t using model q with the fit probability P ( j , t , q ) {\displaystyle P(j,t,q)_{}^{}} . A linear regression model with Gaussian noise is used to compute P ( j , t , q ) {\displaystyle P(j,t,q)} . The Gaussian noise prior has mean zero, and variance which follows I n v e r s e G a m m a ( v 2 , u 2 ) {\displaystyle \mathrm {InverseGamma} \left({\frac {v}{2}},{\frac {u}{2}}\right)} . The prior for each weight follows N o r m a l ( 0 , σ 2 δ ) {\displaystyle \mathrm {Normal} (0,\sigma ^{2}\delta )} . The fit probability P ( j , t , q ) {\displaystyle P(j,t,q)} is computed by the following equation. P ( j , t , q ) = π − n 2 δ m | ( A + D ) − 1 | 1 2 u v 2 ( y + u ) u + v 2 Γ ( n + v 2 ) Γ ( v 2 ) {\displaystyle P(j,t,q)={\frac {\pi ^{-{\frac {n}{2}}}}{\delta ^{m}}}\left|(A+D)^{-1}\right|^{\frac {1}{2}}{\frac {u^{\frac {v}{2}}}{(y+u)^{\frac {u+v}{2}}}}{\frac {\Gamma ({\frac {n+v}{2}})}{\Gamma ({\frac {v}{2}})}}} Then, CST compute the probability of the changepoint at time j with model q, P t ( j , q ) {\displaystyle P_{t}(j,q)} and P j MAP {\displaystyle P_{j}^{\text{MAP}}} using a Viterbi algorithm. P t ( j , q ) = ( 1 − G ( t − j − 1 ) ) P ( j , t , q ) p ( q ) P j MAP {\displaystyle P_{t}(j,q)=(1-G(t-j-1))P(j,t,q)p(q)P_{j}^{\text{MAP}}} P j MAP = max i , q P j ( i , q ) g ( j − i ) 1 − G ( j − i − 1 ) , ∀ j < t {\displaystyle P_{j}^{\text{MAP}}=\max _{i,q}{\frac {P_{j}(i,q)g(j-i)}{1-G(j-i-1)}},\forall j Read more →