Harmonic

In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the fundamental frequency of a periodic signal. The fundamental frequency is also called the 1st harmonic; the other harmonics are known as higher harmonics. As all harmonics are periodic at the fundamental frequency, the sum of harmonics is also periodic at that frequency. The set of harmonics forms a harmonic series. The term is employed in various disciplines, including music, physics, acoustics, electronic power transmission, radio technology, and other fields. For example, if the fundamental frequency is 50 Hz, a common AC power supply frequency, the frequencies of the first three higher harmonics are 100 Hz (2nd harmonic), 150 Hz (3rd harmonic), 200 Hz (4th harmonic) and any addition of waves with these frequencies is periodic at 50 Hz. An n {\displaystyle \ n} th characteristic mode, for n > 1 , {\displaystyle \ n>1\ ,} will have nodes that are not vibrating. For example, the 3rd characteristic mode will have nodes at 1 3 L {\displaystyle \ {\tfrac {1}{3}}\ L\ } and 2 3 L , {\displaystyle \ {\tfrac {2}{3}}\ L\ ,} where L {\displaystyle \ L\ } is the length of the string. In fact, each n {\displaystyle \ n} th characteristic mode, for n {\displaystyle \ n\ } not a multiple of 3, will not have nodes at these points. These other characteristic modes will be vibrating at the positions 1 3 L {\displaystyle \ {\tfrac {1}{3}}\ L\ } and 2 3 L . {\displaystyle \ {\tfrac {2}{3}}\ L~.} If the player gently touches one of these positions, then these other characteristic modes will be suppressed. The tonal harmonics from these other characteristic modes will then also be suppressed. Consequently, the tonal harmonics from the n {\displaystyle \ n} th characteristic characteristic modes, where n {\displaystyle \ n\ } is a multiple of 3, will be made relatively more prominent. In music, harmonics are used on string instruments and wind instruments as a way of producing sound on the instrument, particularly to play higher notes and, with strings, obtain notes that have a unique sound quality or "tone colour". On strings, bowed harmonics have a "glassy", pure tone. On stringed instruments, harmonics are played by touching (but not fully pressing down the string) at an exact point on the string while sounding the string (plucking, bowing, etc.); this allows the harmonic to sound, a pitch which is always higher than the fundamental frequency of the string. == Terminology == Harmonics may be called "overtones", "partials", or "upper partials", and in some music contexts, the terms "harmonic", "overtone" and "partial" are used fairly interchangeably. But more precisely, the term "harmonic" includes all pitches in a harmonic series (including the fundamental frequency) while the term "overtone" only includes pitches above the fundamental. == Characteristics == A whizzing, whistling tonal character, distinguishes all the harmonics both natural and artificial from the firmly stopped intervals; therefore their application in connection with the latter must always be carefully considered. Most acoustic instruments emit complex tones containing many individual partials (component simple tones or sinusoidal waves), but the untrained human ear typically does not perceive those partials as separate phenomena. Rather, a musical note is perceived as one sound, the quality or timbre of that sound being a result of the relative strengths of the individual partials. Many acoustic oscillators, such as the human voice or a bowed violin string, produce complex tones that are more or less periodic, and thus are composed of partials that are nearly matched to the integer multiples of fundamental frequency and therefore resemble the ideal harmonics and are called "harmonic partials" or simply "harmonics" for convenience (although it's not strictly accurate to call a partial a harmonic, the first being actual and the second being theoretical). Oscillators that produce harmonic partials behave somewhat like one-dimensional resonators, and are often long and thin, such as a guitar string or a column of air open at both ends (as with the metallic modern orchestral transverse flute). Wind instruments whose air column is open at only one end, such as trumpets and clarinets, also produce partials resembling harmonics. However they only produce partials matching the odd harmonics—at least in theory. In practical use, no real acoustic instrument behaves as perfectly as the simplified physical models predict; for example, instruments made of non-linearly elastic wood, instead of metal, or strung with gut instead of brass or steel strings, tend to have not-quite-integer partials. Partials whose frequencies are not integer multiples of the fundamental are referred to as inharmonic partials. Some acoustic instruments emit a mix of harmonic and inharmonic partials but still produce an effect on the ear of having a definite fundamental pitch, such as pianos, strings plucked pizzicato, vibraphones, marimbas, and certain pure-sounding bells or chimes. Antique singing bowls are known for producing multiple harmonic partials or multiphonics. Other oscillators, such as cymbals, drum heads, and most percussion instruments, naturally produce an abundance of inharmonic partials and do not imply any particular pitch, and therefore cannot be used melodically or harmonically in the same way other instruments can. Building on of Sethares (2004), dynamic tonality introduces the notion of pseudo-harmonic partials, in which the frequency of each partial is aligned to match the pitch of a corresponding note in a pseudo-just tuning, thereby maximizing the consonance of that pseudo-harmonic timbre with notes of that pseudo-just tuning. == Partials, overtones, and harmonics == An overtone is any partial higher than the lowest partial in a compound tone. The relative strengths and frequency relationships of the component partials determine the timbre of an instrument. The similarity between the terms overtone and partial sometimes leads to their being loosely used interchangeably in a musical context, but they are counted differently, leading to some possible confusion. In the special case of instrumental timbres whose component partials closely match a harmonic series (such as with most strings and winds) rather than being inharmonic partials (such as with most pitched percussion instruments), it is also convenient to call the component partials "harmonics", but not strictly correct, because harmonics are numbered the same even when missing, while partials and overtones are only counted when present. This chart demonstrates how the three types of names (partial, overtone, and harmonic) are counted (assuming that the harmonics are present): In many musical instruments, it is possible to play the upper harmonics without the fundamental note being present. In a simple case (e.g., recorder) this has the effect of making the note go up in pitch by an octave, but in more complex cases many other pitch variations are obtained. In some cases it also changes the timbre of the note. This is part of the normal method of obtaining higher notes in wind instruments, where it is called overblowing. The extended technique of playing multiphonics also produces harmonics. On string instruments it is possible to produce very pure sounding notes, called harmonics or flageolets by string players, which have an eerie quality, as well as being high in pitch. Harmonics may be used to check at a unison the tuning of strings that are not tuned to the unison. For example, lightly fingering the node found halfway down the highest string of a cello produces the same pitch as lightly fingering the node ⁠ 1 / 3 ⁠ of the way down the second highest string. For the human voice see Overtone singing, which uses harmonics. While it is true that electronically produced periodic tones (e.g. square waves or other non-sinusoidal waves) have "harmonics" that are whole number multiples of the fundamental frequency, practical instruments do not all have this characteristic. For example, higher "harmonics" of piano notes are not true harmonics but are "overtones" and can be very sharp, i.e. a higher frequency than given by a pure harmonic series. This is especially true of instruments other than strings, brass, or woodwinds. Examples of these "other" instruments are xylophones, drums, bells, chimes, etc.; not all of their overtone frequencies make a simple whole number ratio with the fundamental frequency. (The fundamental frequency is the reciprocal of the longest time period of the collection of vibrations in some single periodic phenomenon.) == On stringed instruments == Harmonics may be singly produced [on stringed instruments] (1) by varying the point of contact with the bow, or (2) by slightly pressing the string at the nodes, or divisions of its aliquot parts ( 1 2 {\displaystyle {\tfrac {1}{2}}} , 1

Aphelion (software)

The Aphelion Imaging Software Suite is a software suite that includes three base products - Aphelion Lab, Aphelion Dev, and Aphelion SDK for addressing image processing and image analysis applications. The suite also includes a set of extension programs to implement specific vertical applications that benefit from imaging techniques. The Aphelion software products can be used to prototype and deploy applications, or can be integrated, in whole or in part, into a user's system as processing and visualization libraries whose components are available as both DLLs or .Net components. == History and evolution == The development of Aphelion started in 1995 as a joint project of a French company, ADCIS S.A., and an American company, Amerinex Applied Imaging, Inc. (AAI) Aphelion's image processing and analysis functions were made from operators available from the KBVision software developed and sold by Amerinex's predecessor, Amerinex Artificial Intelligence Inc. In the 1990s, the XLim software library was developed at the Center of Mathematical Morphology of Mines ParisTech, and both companies carried out its development tasks. The first version of Aphelion was completed and released in April 1996. Successive versions were released before the first official stable release in December 1996 at the Photonics East conference in Boston and the Solutions Vision show in Paris in January 1997, where at the latter it competed with Stemmer Imaging's CVB imaging toolbox. In 1998, version 2.3 of Aphelion for Windows 98 was released, and its user base was growing in both France and the United States. Version 3.0, totally rewritten to take advantage of Microsoft's then-recent ActiveX technology, was officially released in 2000. It also became available as a « Developer » version, for rapid prototyping of applications using its intuitive GUI and the macro recording capability, and a « Core » version, including the full library as a set of ActiveX components to be used by software developers, integrators and original equipment manufacturers (OEM). As AAI turned its focus to security, in 2001, ADCIS took the lead on developing Aphelion. AAI focused on millimeter wave scanners for concealed weapon detection at airports, and eventually merged with Millimetrics to become Millivision. In 2004, ADCIS specified version 4.0 of Aphelion. The set of image processing/analysis functions was rewritten one more time to be compatible with the .NET technology and the emergence of 64 bit architecture PCs. In addition, the GUI was redesigned to address two usage types: a semi-automatic use where the user is guided through the different steps of functions, and a fully automatic use where the expert user can quickly invoke imaging functions. Its first release was presented at the IPOT exhibition in Birmingham, UK the same year. During the Vision Show in Paris in October 2008, the new Aphelion Lab product was launched for users that are not specialists in image processing. It is easier to use, and only includes fewer image processing functions. It was then included in the Aphelion Image Processing Suite, consisting of Aphelion Dev (replacing Aphelion Developer), Aphelion Lab, Aphelion SDK (replacing Aphelion Core), and a set of extensions. Nowadays, ADCIS is still working on the suite, and updated versions with new extensions and functionalities continually become available from the websites of both companies. In 2015, support was added for very large images and scan microscope images (virtual slides compound into a very large JPEG 2000 image) for high throughput imaging, and new specific extensions were also added. In late 2015, ADCIS announced Aphelion's port for tablets and smartphones, for vertical applications. The name "Aphelion" comes from the astronomical term of the same name, meaning the point on a planet rotating around the Sun where it lies farthest from it, applying the term in a metaphorical sense. Unix was the operating system used on scientific workstations in the 1990s, such as on the workstations manufactured by market leader Sun Microsystems, which Windows suite Aphelion was quite removed from. == Description == Aphelion is a software suite to be used for image processing and image analysis. It supports 2D and 3D, monochrome, color, and multi-band images. It is developed by ADCIS, a French software house located in Saint-Contest, Calvados, Normandy. Aphelion is widely used in the scientific/industry community to solve basic and complex imaging applications. First, the imaging application is quickly developed from the Graphical User Interface, involving a set of functions that can be automatically recorded into a macro command. The macro languages available in Aphelion (i.e. BasicScript, Python, and C#) help to process batch of images, and prompt the user if needed for specific parameters that are applied to the imaging functions. All Aphelion image processing functions are written in C++, and the Aphelion user interface is written in C#. C++ functions can be called from the C# language thanks the use of dedicated wrappers. The main principle of image processing is to automatically process pixels of a digital image, then extract one or more objects of interest (i.e. cells in the field of biology, inclusions in the field of material science) and compute one or more measurements on those objects to quantify the image and generate a verdict (good image, image with defects, cancerous cells). In other words, starting from an image, pixels are processed by a set of successive functions or operators until only measurements are computed and used as the input of a 3rd party system or a classification software that will classify objects of interest that have been extracted during the imaging process. An acquisition system such as a digital camera, a video camera, an optical or electron microscope, a medical scanner, or a smartphone can be used to capture images. The set of values or pixels can be processed as a 1D image (1D signal), a 2D image (array of pixel values corresponding to a monochrome or color image), or a 3D image displayed using volume rendering (array of voxels in the 3D space) or displaying surfaces by using 3D rendering. A 2D color image is made of 3 value pixels (typically Red, Green, and Blue information or another color space), and a 3D image is made of monochrome, color (indexed color are often used), multispectral, or hyperspectral data. When dealing with videos, an additional band is added corresponding to temporal information. The Aphelion Software Suite includes three base products, and a set of optional extensions for specific applications: Aphelion Lab: Entry-level package for non-experts in image processing. It helps to quickly segment an image in a semi-automatic or manual ways, and compute a set of measurements computed on objects of interest that have been extracted during the segmentation process. A set of wizards guides the user from image acquisition to report generation. Aphelion Dev: Full imaging environment including over 450 functions to develop and deploy an application that involves image processing and analysis. It also includes a set of macro-command languages to automate any application to be invoked from the user interface. It also helps to run the imaging algorithm on more than one image that are stored on disk, available on the network, or captured by an acquisition device. Aphelion libraries for image processing and visualization are provided in Aphelion Dev as DLLs and .Net components. Aphelion SDK: A set of libraries to develop a stand-alone application with a custom interface based on the Aphelion libraries. This software development kit including display, processing and analysis functions that can be used by software developers and OEMs. It is provided as DLLs and .Net components. The stand-alone application is typically developed in C# on one computer, and then deployed on multiple PCs and systems. A set of optional extensions can be added to the « Aphelion Dev » product, depending on the application. An evaluation version of Aphelion can be run on a PC for 30 days. A permanent version of Aphelion is available based on a perpetual license. Upgrades are available through a maintenance agreement based on a yearly fee. Technical support is provided by the engineers who are developing the product. The goal of image processing is usually to extract object(s) of interest in an image, and then to classify them based on some characteristics such as shape, density, position, etc. Using Aphelion, this goal is achieved by performing the following tasks: Load an image from disk or acquire an image using an acquisition device. Enhance the image removing noise or modifying its contrast. Segment the image extracting objects of interest to be measured and analyzed. Typically, for simple applications, a threshold is performed to generate a binary image. Then, morphological operators are applied to clean the image and only keep obj

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

Tucker decomposition

In mathematics, Tucker decomposition decomposes a tensor into a set of matrices and one small core tensor. It is named after Ledyard R. Tucker although it goes back to Hitchcock in 1927. Initially described as a three-mode extension of factor analysis and principal component analysis it may actually be generalized to higher mode analysis, which is also called higher-order singular value decomposition (HOSVD) or the M-mode SVD. The algorithm to which the literature typically refers when discussing the Tucker decomposition or the HOSVD is the M-mode SVD algorithm introduced by Vasilescu and Terzopoulos, but misattributed to Tucker or De Lathauwer etal. It may be regarded as a more flexible PARAFAC (parallel factor analysis) model. In PARAFAC the core tensor is restricted to be "diagonal". In practice, Tucker decomposition is used as a modelling tool. For instance, it is used to model three-way (or higher way) data by means of relatively small numbers of components for each of the three or more modes, and the components are linked to each other by a three- (or higher-) way core array. The model parameters are estimated in such a way that, given fixed numbers of components, the modelled data optimally resemble the actual data in the least squares sense. The model gives a summary of the information in the data, in the same way as principal components analysis does for two-way data. For a 3rd-order tensor T ∈ F n 1 × n 2 × n 3 {\displaystyle T\in F^{n_{1}\times n_{2}\times n_{3}}} , where F {\displaystyle F} is either R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } , Tucker Decomposition can be denoted as follows, T = T × 1 U ( 1 ) × 2 U ( 2 ) × 3 U ( 3 ) {\displaystyle T={\mathcal {T}}\times _{1}U^{(1)}\times _{2}U^{(2)}\times _{3}U^{(3)}} where T ∈ F d 1 × d 2 × d 3 {\displaystyle {\mathcal {T}}\in F^{d_{1}\times d_{2}\times d_{3}}} is the core tensor, a 3rd-order tensor that contains the 1-mode, 2-mode and 3-mode singular values of T {\displaystyle T} , which are defined as the Frobenius norm of the 1-mode, 2-mode and 3-mode slices of tensor T {\displaystyle {\mathcal {T}}} respectively. U ( 1 ) , U ( 2 ) , U ( 3 ) {\displaystyle U^{(1)},U^{(2)},U^{(3)}} are unitary matrices in F d 1 × n 1 , F d 2 × n 2 , F d 3 × n 3 {\displaystyle F^{d_{1}\times n_{1}},F^{d_{2}\times n_{2}},F^{d_{3}\times n_{3}}} respectively. The k-mode product (k = 1, 2, 3) of T {\displaystyle {\mathcal {T}}} by U ( k ) {\displaystyle U^{(k)}} is denoted as T × U ( k ) {\displaystyle {\mathcal {T}}\times U^{(k)}} with entries as ( T × 1 U ( 1 ) ) ( i 1 , j 2 , j 3 ) = ∑ j 1 = 1 d 1 T ( j 1 , j 2 , j 3 ) U ( 1 ) ( j 1 , i 1 ) ( T × 2 U ( 2 ) ) ( j 1 , i 2 , j 3 ) = ∑ j 2 = 1 d 2 T ( j 1 , j 2 , j 3 ) U ( 2 ) ( j 2 , i 2 ) ( T × 3 U ( 3 ) ) ( j 1 , j 2 , i 3 ) = ∑ j 3 = 1 d 3 T ( j 1 , j 2 , j 3 ) U ( 3 ) ( j 3 , i 3 ) {\displaystyle {\begin{aligned}({\mathcal {T}}\times _{1}U^{(1)})(i_{1},j_{2},j_{3})&=\sum _{j_{1}=1}^{d_{1}}{\mathcal {T}}(j_{1},j_{2},j_{3})U^{(1)}(j_{1},i_{1})\\({\mathcal {T}}\times _{2}U^{(2)})(j_{1},i_{2},j_{3})&=\sum _{j_{2}=1}^{d_{2}}{\mathcal {T}}(j_{1},j_{2},j_{3})U^{(2)}(j_{2},i_{2})\\({\mathcal {T}}\times _{3}U^{(3)})(j_{1},j_{2},i_{3})&=\sum _{j_{3}=1}^{d_{3}}{\mathcal {T}}(j_{1},j_{2},j_{3})U^{(3)}(j_{3},i_{3})\end{aligned}}} Altogether, the decomposition may also be written more directly as T ( i 1 , i 2 , i 3 ) = ∑ j 1 = 1 d 1 ∑ j 2 = 1 d 2 ∑ j 3 = 1 d 3 T ( j 1 , j 2 , j 3 ) U ( 1 ) ( j 1 , i 1 ) U ( 2 ) ( j 2 , i 2 ) U ( 3 ) ( j 3 , i 3 ) {\displaystyle T(i_{1},i_{2},i_{3})=\sum _{j_{1}=1}^{d_{1}}\sum _{j_{2}=1}^{d_{2}}\sum _{j_{3}=1}^{d_{3}}{\mathcal {T}}(j_{1},j_{2},j_{3})U^{(1)}(j_{1},i_{1})U^{(2)}(j_{2},i_{2})U^{(3)}(j_{3},i_{3})} Taking d i = n i {\displaystyle d_{i}=n_{i}} for all i {\displaystyle i} is always sufficient to represent T {\displaystyle T} exactly, but often T {\displaystyle T} can be compressed or efficiently approximately by choosing d i < n i {\displaystyle d_{i}

Margin classifier

In machine learning (ML), a margin classifier is a type of classification model which is able to give an associated distance from the decision boundary for each data sample. For instance, if a linear classifier is used, the distance (typically Euclidean, though others may be used) of a sample from the separating hyperplane is the margin of that sample. The notion of margins is important in several ML classification algorithms, as it can be used to bound the generalization error of these classifiers. These bounds are frequently shown using the VC dimension. The generalization error bound in boosting algorithms and support vector machines is particularly prominent. == Margin for boosting algorithms == The margin for an iterative boosting algorithm given a dataset with two classes can be defined as follows: the classifier is given a sample pair ( x , y ) {\displaystyle (x,y)} , where x ∈ X {\displaystyle x\in X} is a domain space and y ∈ Y = { − 1 , + 1 } {\displaystyle y\in Y=\{-1,+1\}} is the sample's label. The algorithm then selects a classifier h j ∈ C {\displaystyle h_{j}\in C} at each iteration j {\displaystyle j} where C {\displaystyle C} is a space of possible classifiers that predict real values. This hypothesis is then weighted by α j ∈ R {\displaystyle \alpha _{j}\in R} as selected by the boosting algorithm. At iteration t {\displaystyle t} , the margin of a sample x {\displaystyle x} can thus be defined as y ∑ j t α j h j ( x ) ∑ | α j | . {\displaystyle {\frac {y\sum _{j}^{t}\alpha _{j}h_{j}(x)}{\sum |\alpha _{j}|}}.} By this definition, the margin is positive if the sample is labeled correctly, or negative if the sample is labeled incorrectly. This definition may be modified and is not the only way to define the margin for boosting algorithms. However, there are reasons why this definition may be appealing. == Examples of margin-based algorithms == Many classifiers can give an associated margin for each sample. However, only some classifiers utilize information of the margin while learning from a dataset. Many boosting algorithms rely on the notion of a margin to assign weight to samples. If a convex loss is utilized (as in AdaBoost or LogitBoost, for instance) then a sample with a higher margin will receive less (or equal) weight than a sample with a lower margin. This leads the boosting algorithm to focus weight on low-margin samples. In non-convex algorithms (e.g., BrownBoost), the margin still dictates the weighting of a sample, though the weighting is non-monotone with respect to the margin. == Generalization error bounds == One theoretical motivation behind margin classifiers is that their generalization error may be bound by the algorithm parameters and a margin term. An example of such a bound is for the AdaBoost algorithm. Let S {\displaystyle S} be a set of m {\displaystyle m} data points, sampled independently at random from a distribution D {\displaystyle D} . Assume the VC-dimension of the underlying base classifier is d {\displaystyle d} and m ≥ d ≥ 1 {\displaystyle m\geq d\geq 1} . Then, with probability 1 − δ {\displaystyle 1-\delta } , we have the bound: P D ( y ∑ j t α j h j ( x ) ∑ | α j | ≤ 0 ) ≤ P S ( y ∑ j t α j h j ( x ) ∑ | α j | ≤ θ ) + O ( 1 m d log 2 ⁡ ( m / d ) / θ 2 + log ⁡ ( 1 / δ ) ) {\displaystyle P_{D}\left({\frac {y\sum _{j}^{t}\alpha _{j}h_{j}(x)}{\sum |\alpha _{j}|}}\leq 0\right)\leq P_{S}\left({\frac {y\sum _{j}^{t}\alpha _{j}h_{j}(x)}{\sum |\alpha _{j}|}}\leq \theta \right)+O\left({\frac {1}{\sqrt {m}}}{\sqrt {d\log ^{2}(m/d)/\theta ^{2}+\log(1/\delta )}}\right)} for all θ > 0 {\displaystyle \theta >0} .

EyeOS

eyeOS was a web desktop for cloud computing, whose main purpose is to enable collaboration and communication among users. It is mainly written in PHP, XML, and JavaScript. It is a private-cloud application platform with a web-based desktop interface. eyeOS delivers a whole desktop from the cloud with file management, personal management information tools, and collaborative tools, with the integration of the client's applications. == History == The first publicly available eyeOS version was released on August 1, 2005, as eyeOS 0.6.0 in Olesa de Montserrat, Barcelona (Spain). A worldwide community of developers soon took part in the project and helped improve it by translating, testing, and developing it. After two years of development, the eyeOS Team published eyeOS 1.0 on June 4, 2007. Compared with previous versions, eyeOS 1.0 introduced a complete reorganization of the code and some new web technologies, like eyeSoft, a portage-based web software installation system. Moreover, eyeOS also included the eyeOS Toolkit, a set of libraries allowing easy and fast development of new web applications. With the release of eyeOS 1.1 on July 2, 2007, eyeOS changed its license and migrated from GNU GPL Version 2 to Version 3. Version 1.2 was released just a month after the 1.1 version and integrated full compatibility with Microsoft Word files. eyeOS 1.5 Gala was released on January 15, 2008. This version was the first to support both Microsoft Office and OpenOffice.org file formats for documents, presentations, and spreadsheets. With this version, eyeOS also gained the ability to import and export documents in both formats using server-side scripting. eyeOS 1.6 was released on April 25, 2008, and included many improvements such as synchronization with local computers, drag and drop, a mobile version, and more. eyeOS 1.8 Lars was released on January 7, 2009, and featured a completely rewritten file manager and a new sound API to develop media-rich applications. Later, on April 1, 2009, 1.8.5 was released with a new default theme and some rewritten apps, such as the Word Processor and the Address Book. On July 13, 2009, 1.8.6 was released with an interface for the iPhone and a new version of eyeMail with support for POP3 and IMAP. eyeOS 1.9 was released on December 29, 2009. It was followed up with the 1.9.0.1 release with minor fixes on February 18, 2010. These releases were the last of the "classic desktop" interfaces. A major re-work was completed in March 2010, now called eyeOS 2.x. However, a small group of eyeOS developers still maintain the code within the eyeOS forum, where support is provided, but the eyeOS group itself has stopped active 1.x development. It is now available as the On-eye project on GitHub. Active development was halted on 1.x as of February 3, 2010. eyeOS 2.0 release took place on March 3, 2010. This was a total restructure of the operating system. The 2.x stable is the new series of eyeOS, which is in active development and will replace 1.x as stable in a few months. It includes live collaboration and more social capabilities than eyeOS 1.x. eyeOS then released 2.2.0.0 on July 28, 2010. On December 14, 2010, a working group inside the eyeOS open-source development community began the structure development and further upgrade of eyeOS 1.9.x. The group's main goal is to continue the work eyeOS has stopped on 1.9.x. eyeOS released 2.5 on May 17, 2011. This was the last release under an open source license. It is available on SourceForge for download under another project called eyeOS 2.5 Open Source Version. On April 1, 2014, Telefónica announced their acquisition of eyeOS. eyeOS would maintain its headquarters in the Catalonia, Spain, where their staff would continue to work but now as part of Telefónica. After its integration into Telefónica, eyeOS would continue to function as an independent subsidiary under CEO Michel Kisfaludi. == Structure and API == For developers, EyeOS provides the eyeOS Toolkit, a set of libraries and functions to develop applications for eyeOS. Using the integrated Portage-based eyeSoft system, one can create their own repository for eyeOS and distribute applications through it. Each core part of the desktop is its own application, using JavaScript to send server commands as the user interacts. As actions are performed using AJAX (such as launching an application), it sends event information to the server. The server then sends back tasks for the client to do in XML format, such as drawing a widget. On the server, eyeOS uses XML files to store information. This makes it simple for a user to set up on the server, as it requires zero configuration other than the account information for the first user, making it simple to deploy. To avoid bottlenecks that flat files present, each user's information and settings are stored in different files, preventing resource starvation from occurring, though this in turn may create issues in high volume user environments due to host operating system open file descriptor limits. == Professional edition == A Professional Edition of eyeOS was launched on September 15, 2011, as an operating system for businesses. It uses a new version number and was released under version 1.0 instead of continuing with the next version number in the open source project. The Professional Edition retains the web desktop interface used by the open source version while targeting enterprise users. A host of new features designed for enterprises, like file sharing and synchronization (called eyeSync), Active Directory/LDAP connectivity, system-wide administration controls, and a local file execution tool called eyeRun were introduced. A new suite of Web Apps (a mail client, calendar, instant messaging, and collaboration tools) was also introduced, specific to the enterprise edition for the web desktop. With eyeOS Professional Edition 1.1, a to-do task manager tool, Citrix XenApp integration, and a Facebook like 'wall' for collaboration were introduced. == Awards == 2007 – Received the Softpedia's Pick award. 2007 – Finalist at SourceForge's 2007 Community Choice Awards at the "Best Project" category. The winner for that category was 7-Zip. 2007 – Won the Yahoo! Spain Web Revelation award in the Technology category. 2008 – Finalist for the Webware 100 awards by CNET, under the "Browsing" category. 2008 – Finalist at the SourceForge's 2008 Community Choice Awards at the "Most Likely to Change the World" category. The winner for that category was Linux. 2009 – Selected Project of the Month (August 2009) by SourceForge. 2009 – BMW Innovation Award. 2010 – Winner of Accelera (Ernst & Young). 2010 – Asturias & Girona Spanish Prince award “IMPULSA”. 2011 – Winner of MIT's TR35 award as Innovator of the Year in Spain. == Community == eyeOS community is formed with the eyeOS forums, which reached 10,000 members on April 4, 2008; the eyeOS wiki; and the eyeOS Application Communities, available at the eyeOS-Apps website, hosted and provided by openDesktop.org as well as Softpedia.

Genetic programming

Genetic programming (GP) is an evolutionary algorithm, an artificial intelligence technique mimicking natural evolution, which operates on a population of programs. It applies the genetic operators selection according to a predefined fitness measure, mutation and crossover. The crossover operation involves swapping specified parts of selected pairs (parents) to produce new and different offspring that become part of the new generation of programs. Some programs not selected for reproduction are copied from the current generation to the new generation. Mutation involves substitution of some random part of a program with some other random part of a program. Then the selection and other operations are recursively applied to the new generation of programs. Typically, members of each new generation are on average more fit than the members of the previous generation, and the best-of-generation program is often better than the best-of-generation programs from previous generations. Termination of the evolution usually occurs when some individual program reaches a predefined proficiency or fitness level. It may and often does happen that a particular run of the algorithm results in premature convergence to some local maximum that is not a globally optimal or even good solution. Multiple runs (dozens to hundreds) are usually necessary to produce a very good result. It may also be necessary to have a large starting population size and variability of the individuals to avoid pathologies. == History == The first record of the proposal to evolve programs is probably that of Alan Turing in 1950 in "Computing Machinery and Intelligence". There was a gap of 25 years before the publication of John Holland's 'Adaptation in Natural and Artificial Systems' laid out the theoretical and empirical foundations of the science. In 1981, Richard Forsyth demonstrated the successful evolution of small programs, represented as trees, to perform classification of crime scene evidence for the UK Home Office. Although the idea of evolving programs, initially in the computer language Lisp, was current amongst John Holland's students, it was not until they organised the first Genetic Algorithms (GA) conference in Pittsburgh that Nichael Cramer published evolved programs in two specially designed languages, which included the first statement of modern "tree-based" genetic programming (that is, procedural languages organized in tree-based structures and operated on by suitably defined GA-operators). In 1988, John Koza (also a PhD student of John Holland) patented his invention of a GA for program evolution. This was followed by publication in the International Joint Conference on Artificial Intelligence IJCAI-89. Koza followed this with 205 publications on "genetic programming", a term coined by David Goldberg, also a PhD student of John Holland. However, it is the series of 4 books by Koza, starting in 1992 with accompanying videos, that really established GP. Subsequently, there was an enormous expansion of the number of publications with the Genetic Programming Bibliography, surpassing 10,000 entries. In 2010, Koza listed 77 results where genetic programming was human competitive. The departure of GP from the rigid, fixed-length representations typical of early GA models was not entirely without precedent. Early work on variable-length representations laid the groundwork. One notable example is messy genetic algorithms, which introduced irregular, variable-length chromosomes to address building block disruption and positional bias in standard GAs. Another precursor was robot trajectory programming, where genome representations encoded program instructions for robotic movements—structures inherently variable in length. Even earlier, unfixed-length representations were proposed in a doctoral dissertation by Cavicchio, who explored adaptive search using simulated evolution. His work provided foundational ideas for flexible program structures. In 1996, Koza started the annual Genetic Programming conference, which was followed in 1998 by the annual EuroGP conference, and the first book in a GP series edited by Koza. 1998 also saw the first GP textbook. GP continued to flourish, leading to the first specialist GP journal and three years later (2003) the annual Genetic Programming Theory and Practice (GPTP) workshop was established by Rick Riolo. Genetic programming papers continue to be published at a diversity of conferences and associated journals. Today there are nineteen GP books including several for students. === Foundational work in GP === Early work that set the stage for current genetic programming research topics and applications is diverse, and includes software synthesis and repair, predictive modeling, data mining, financial modeling, soft sensors, design, and image processing. Applications in some areas, such as design, often make use of intermediate representations, such as Fred Gruau's cellular encoding. Industrial uptake has been significant in several areas including finance, the chemical industry, bioinformatics and the steel industry. == Methods == === Program representation === GP evolves computer programs, traditionally represented in memory as tree structures. Trees can be easily evaluated in a recursive manner. Every internal node has an operator function and every terminal node has an operand, making mathematical expressions easy to evolve and evaluate. Thus traditionally GP favors the use of programming languages that naturally embody tree structures (for example, Lisp; other functional programming languages are also suitable). Non-tree representations have been suggested and successfully implemented, such as linear genetic programming, which perhaps suits the more traditional imperative languages. The commercial GP software Discipulus uses automatic induction of binary machine code ("AIM") to achieve better performance. μGP uses directed multigraphs to generate programs that fully exploit the syntax of a given assembly language. Multi expression programming uses three-address code for encoding solutions. Other program representations on which significant research and development have been conducted include programs for stack-based virtual machines, and sequences of integers that are mapped to arbitrary programming languages via grammars. Cartesian genetic programming is another form of GP, which uses a graph representation instead of the usual tree based representation to encode computer programs. Most representations have structurally noneffective code (introns). Such non-coding genes may seem to be useless because they have no effect on the performance of any one individual. However, they alter the probabilities of generating different offspring under the variation operators, and thus alter the individual's variational properties. Experiments seem to show faster convergence when using program representations that allow such non-coding genes, compared to program representations that do not have any non-coding genes. Instantiations may have both trees with introns and those without; the latter are called canonical trees. Special canonical crossover operators are introduced that maintain the canonical structure of parents in their children. === Initialisation === The methods for creation of the initial population include: Grow creates the individuals sequentially. Every GP tree is created starting from the root, creating functional nodes with children as well as terminal nodes up to a certain depth. Full is similar to the Grow. The difference is that all brunches in a tree are of same predetermined depth. Ramped half-and-half creates a population consisting of m d − 1 {\displaystyle md-1} parts and a maximum depth of m d {\displaystyle md} for its trees. The first part has a maximum depth of 2, second of 3 and so on up to the m d − 1 {\displaystyle md-1} -th part with maximum depth m d {\displaystyle md} . Half of every part is created by Grow, while the other part is created by Full. === Selection === Selection is a process whereby certain individuals are selected from the current generation that would serve as parents for the next generation. The individuals are selected probabilistically such that the better performing individuals have a higher chance of getting selected. The most commonly used selection method in GP is tournament selection, although other methods such as fitness proportionate selection, lexicase selection, and others have been demonstrated to perform better for many GP problems. Elitism, which involves seeding the next generation with the best individual (or best n individuals) from the current generation, is a technique sometimes employed to avoid regression. === Crossover === In genetic programming two fit individuals are chosen from the population to be parents for one or two children. In tree genetic programming, these parents are represented as inverted lisp like trees, with their root nodes at the top. In subtree cro