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111-405: Canonical polyadic decomposition (CPD) is a variant of the tensor rank decomposition, in which the tensor is approximated as a sum of K rank-1 tensors for a user-specified K . The CP decomposition has found some applications in linguistics and chemometrics . It was introduced by Frank Lauren Hitchcock in 1927 and later rediscovered several times, notably in psychometrics. The CP decomposition

222-871: A m = λ m b m {\displaystyle \mathbf {a} _{m}=\lambda _{m}\mathbf {b} _{m}} for all m . For this reason, the parameters { a m } m = 1 M {\displaystyle \{\mathbf {a} _{m}\}_{m=1}^{M}} of a rank-1 tensor A {\displaystyle {\mathcal {A}}} are called identifiable or essentially unique. A rank- r {\displaystyle r} tensor A ∈ F I 1 ⊗ F I 2 ⊗ ⋯ ⊗ F I M {\displaystyle {\mathcal {A}}\in F^{I_{1}}\otimes F^{I_{2}}\otimes \cdots \otimes F^{I_{M}}}

333-403: A m , r ∈ F I m {\displaystyle \mathbf {a} _{m,r}\in {\mathbb {F} }^{I_{m}}} where 1 ≤ m ≤ M {\displaystyle 1\leq m\leq M} . When the number of terms R {\displaystyle R} is minimal in the above expression, then R {\displaystyle R}

444-568: A dense set S {\displaystyle S} : every tensor in the aforementioned space is either of rank less than the generic rank, or it is the limit in the Euclidean topology of a sequence of tensors from S {\displaystyle S} . In the case of real tensors, the set of tensors of rank at most r ( I 1 , … , I M ) {\displaystyle r(I_{1},\ldots ,I_{M})} only forms an open set of positive measure in

555-414: A branch of linguistics. Before the 20th century, linguists analysed language on a diachronic plane, which was historical in focus. This meant that they would compare linguistic features and try to analyse language from the point of view of how it had changed between then and later. However, with the rise of Saussurean linguistics in the 20th century, the focus shifted to a more synchronic approach, where

666-560: A comparison of different time periods in the past and present) or in a synchronic manner (by observing developments between different variations that exist within the current linguistic stage of a language). At first, historical linguistics was the cornerstone of comparative linguistics , which involves a study of the relationship between different languages. At that time, scholars of historical linguistics were only concerned with creating different categories of language families , and reconstructing prehistoric proto-languages by using both

777-807: A conjecture about this maximum rank is missing. Presently, the best general upper bound states that the maximum rank r max ( I 1 , … , I M ) {\displaystyle r_{\mbox{max}}(I_{1},\ldots ,I_{M})} of F I 1 ⊗ ⋯ ⊗ F I M {\displaystyle F^{I_{1}}\otimes \cdots \otimes F^{I_{M}}} , where I 1 ≥ I 2 ≥ ⋯ ≥ I M {\displaystyle I_{1}\geq I_{2}\geq \cdots \geq I_{M}} , satisfies where r ( I 1 , … , I M ) {\displaystyle r(I_{1},\ldots ,I_{M})}

888-427: A general theoretical framework for describing it. Applied linguistics seeks to utilize the scientific findings of the study of language for practical purposes, such as developing methods of improving language education and literacy. Linguistic features may be studied through a variety of perspectives: synchronically (by describing the structure of a language at a specific point in time) or diachronically (through

999-434: A linguistic medium of communication in itself. Palaeography is therefore the discipline that studies the evolution of written scripts (as signs and symbols) in language. The formal study of language also led to the growth of fields like psycholinguistics , which explores the representation and function of language in the mind; neurolinguistics , which studies language processing in the brain; biolinguistics , which studies

1110-408: A major breakthrough was established by Catalisano, Geramita, and Gimigliano who proved that the expected dimension of the set of rank s {\displaystyle s} tensors of format 2 × 2 × ⋯ × 2 {\displaystyle 2\times 2\times \cdots \times 2} is the expected one except for rank 3 tensors in the 4 factor case, yet

1221-609: A matrix is denoted by bold upper case letters A {\displaystyle \mathbf {A} } . A higher order tensor is denoted by calligraphic letters, A {\displaystyle {\mathcal {A}}} . An element of an M {\displaystyle M} -order tensor A ∈ C I 1 × I 2 × … I m × … I M {\displaystyle {\mathcal {A}}\in \mathbb {C} ^{I_{1}\times I_{2}\times \dots I_{m}\times \dots I_{M}}}

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1332-416: A particular feature or usage is "good" or "bad". This is analogous to practice in other sciences: a zoologist studies the animal kingdom without making subjective judgments on whether a particular species is "better" or "worse" than another. Prescription , on the other hand, is an attempt to promote particular linguistic usages over others, often favoring a particular dialect or " acrolect ". This may have

1443-419: A second-language speaker who is attempting to acquire the language. Most contemporary linguists work under the assumption that spoken data and signed data are more fundamental than written data . This is because Nonetheless, linguists agree that the study of written language can be worthwhile and valuable. For research that relies on corpus linguistics and computational linguistics , written language

1554-441: A sequence of tensor of rank at most r {\displaystyle r} , even though the limit of the sequence converges to a tensor of rank strictly higher than r {\displaystyle r} . The rank-3 tensor can be approximated arbitrarily well by the following sequence of rank-2 tensors as n → ∞ {\displaystyle n\to \infty } . This example neatly illustrates

1665-629: A sequence of tensors of rank at most r < s {\displaystyle r<s} whose limit is A {\displaystyle {\mathcal {A}}} . If r {\displaystyle r} is the least value for which such a convergent sequence exists, then it is called the border rank of A {\displaystyle {\mathcal {A}}} . For order-2 tensors, i.e., matrices, rank and border rank always coincide, however, for tensors of order ≥ 3 {\displaystyle \geq 3} they may differ. Border tensors were first studied in

1776-414: A tensor using numerical optimization algorithms. It is sometimes called the problem of diverging components . It was, in addition, shown that a random low-rank tensor over the reals may not admit a rank-2 approximation with positive probability, leading to the understanding that the ill-posedness problem is an important consideration when employing the tensor rank decomposition. A common partial solution to

1887-419: A view towards uncovering the biological underpinnings of language. In Generative Grammar , these underpinning are understood as including innate domain-specific grammatical knowledge. Thus, one of the central concerns of the approach is to discover what aspects of linguistic knowledge are innate and which are not. Cognitive linguistics , in contrast, rejects the notion of innate grammar, and studies how

1998-424: A word. Linguistic structures are pairings of meaning and form. Any particular pairing of meaning and form is a Saussurean linguistic sign . For instance, the meaning "cat" is represented worldwide with a wide variety of different sound patterns (in oral languages), movements of the hands and face (in sign languages ), and written symbols (in written languages). Linguistic patterns have proven their importance for

2109-419: Is 2. Practically, this means that a randomly sampled real tensor (from a continuous probability measure on the space of tensors) of size 2 × 2 × 2 {\displaystyle 2\times 2\times 2} will be a rank-1 tensor with probability zero, a rank-2 tensor with positive probability, and rank-3 with positive probability. On the other hand, a randomly sampled complex tensor of

2220-898: Is a closed set in the Zariski topology, the decomposition on the right-hand side is a sum of a different set of rank-1 tensors than the decomposition on the left-hand side, entailing that order-2 tensors of rank r > 1 {\displaystyle r>1} are generically not identifiable. The situation changes completely for higher-order tensors in F I 1 ⊗ F I 2 ⊗ ⋯ ⊗ F I M {\displaystyle F^{I_{1}}\otimes F^{I_{2}}\otimes \cdots \otimes F^{I_{M}}} with M > 2 {\displaystyle M>2} and all I m ≥ 2 {\displaystyle I_{m}\geq 2} . For simplicity in notation, assume without loss of generality that

2331-415: Is a collection of multivariate observations organized into a M -way array where M = C +1. Every tensor may be represented with a suitably large R {\displaystyle R} as a linear combination of r {\displaystyle r} rank-1 tensors: where λ r ∈ R {\displaystyle \lambda _{r}\in {\mathbb {R} }} and

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2442-458: Is a researcher within the field, or to someone who uses the tools of the discipline to describe and analyse specific languages. An early formal study of language was in India with Pāṇini , the 6th century BC grammarian who formulated 3,959 rules of Sanskrit morphology . Pāṇini's systematic classification of the sounds of Sanskrit into consonants and vowels, and word classes, such as nouns and verbs,

2553-430: Is a system of rules which governs the production and use of utterances in a given language. These rules apply to sound as well as meaning, and include componential subsets of rules, such as those pertaining to phonology (the organization of phonetic sound systems), morphology (the formation and composition of words), and syntax (the formation and composition of phrases and sentences). Modern frameworks that deal with

2664-970: Is an invertible r × r {\displaystyle r\times r} matrix, A = [ a i ] i = 1 r {\displaystyle A=[\mathbf {a} _{i}]_{i=1}^{r}} , B = [ b i ] i = 1 r {\displaystyle B=[\mathbf {b} _{i}]_{i=1}^{r}} , A X − 1 = [ c i ] i = 1 r {\displaystyle AX^{-1}=[\mathbf {c} _{i}]_{i=1}^{r}} and B X T = [ d i ] i = 1 r {\displaystyle BX^{T}=[\mathbf {d} _{i}]_{i=1}^{r}} . It can be shown that for every X ∈ G L n ( F ) ∖ Z {\displaystyle X\in \mathrm {GL} _{n}(F)\setminus Z} , where Z {\displaystyle Z}

2775-981: Is called identifiable if every of its tensor rank decompositions is the sum of the same set of r {\displaystyle r} distinct tensors { A 1 , A 2 , … , A r } {\displaystyle \{{\mathcal {A}}_{1},{\mathcal {A}}_{2},\ldots ,{\mathcal {A}}_{r}\}} where the A i {\displaystyle {\mathcal {A}}_{i}} 's are of rank 1. An identifiable rank- r {\displaystyle r} thus has only one essentially unique decomposition A = ∑ i = 1 r A i , {\displaystyle {\mathcal {A}}=\sum _{i=1}^{r}{\mathcal {A}}_{i},} and all r ! {\displaystyle r!} tensor rank decompositions of A {\displaystyle {\mathcal {A}}} can be obtained by permuting

2886-643: Is called unbalanced whenever and it is called balanced otherwise. When the first factor is very large with respect to the other factors in the tensor product, then the tensor space essentially behaves as a matrix space. The generic rank of tensors living in an unbalanced tensor spaces is known to equal almost everywhere . More precisely, the rank of every tensor in an unbalanced tensor space F I 1 × ⋯ × I M ∖ Z {\displaystyle F^{I_{1}\times \cdots \times I_{M}}\setminus Z} , where Z {\displaystyle Z}

2997-687: Is called the rank of the tensor, and the decomposition is often referred to as a (tensor) rank decomposition , minimal CP decomposition , or Canonical Polyadic Decomposition (CPD) . If the number of terms is not minimal, then the above decomposition is often referred to as CANDECOMP/PARAFAC , Polyadic decomposition'. Contrary to the case of matrices, computing the rank of a tensor is NP-hard . The only notable well-understood case consists of tensors in F I m ⊗ F I n ⊗ F 2 {\displaystyle F^{I_{m}}\otimes F^{I_{n}}\otimes F^{2}} , whose rank can be obtained from

3108-440: Is conventional or "coded" in a given language, pragmatics studies how the transmission of meaning depends not only on the structural and linguistic knowledge (grammar, lexicon, etc.) of the speaker and listener, but also on the context of the utterance, any pre-existing knowledge about those involved, the inferred intent of the speaker, and other factors. Phonetics and phonology are branches of linguistics concerned with sounds (or

3219-471: Is defective (44 and not the expected 45), but the generic rank in that space is still the expected 6. The AOP conjecture has been proved completely in a number of special cases. Lickteig showed already in 1985 that r ( n , n , n ) = r E ( n , n , n ) {\displaystyle r(n,n,n)=r_{E}(n,n,n)} , provided that n ≠ 3 {\displaystyle n\neq 3} . In 2011,

3330-739: Is denoted by a i 1 , i 2 , … , i m , … i M {\displaystyle a_{i_{1},i_{2},\dots ,i_{m},\dots i_{M}}} or A i 1 , i 2 , … , i m , … i M {\displaystyle {\mathcal {A}}_{i_{1},i_{2},\dots ,i_{m},\dots i_{M}}} . A data tensor A ∈ F I 0 × I 1 × … × I C {\displaystyle {\mathcal {A}}\in {\mathbb {F} }^{I_{0}\times I_{1}\times \ldots \times I_{C}}}

3441-440: Is denoted by lower case italic letters, a {\displaystyle a} and an upper bound scalar is denoted by an upper case italic letter, A {\displaystyle A} . Indices are denoted by a combination of lowercase and upper case italic letters, 1 ≤ i ≤ I {\displaystyle 1\leq i\leq I} . Multiple indices that one might encounter when referring to

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3552-469: Is generally hard to find for events long ago, due to the occurrence of chance word resemblances and variations between language groups. A limit of around 10,000 years is often assumed for the functional purpose of conducting research. It is also hard to date various proto-languages. Even though several methods are available, these languages can be dated only approximately. In modern historical linguistics, we examine how languages change over time, focusing on

3663-447: Is often much more convenient for processing large amounts of linguistic data. Large corpora of spoken language are difficult to create and hard to find, and are typically transcribed and written. In addition, linguists have turned to text-based discourse occurring in various formats of computer-mediated communication as a viable site for linguistic inquiry. The study of writing systems themselves, graphemics, is, in any case, considered

3774-675: Is often referred to as the expected generic rank of the tensor space F I 1 × ⋯ × I M {\displaystyle F^{I_{1}\times \cdots \times I_{M}}} because it is only conjecturally correct. It is known that the true generic rank always satisfies The Abo–Ottaviani–Peterson conjecture states that equality is expected, i.e., r ( I 1 , … , I M ) = r E ( I 1 , … , I M ) {\displaystyle r(I_{1},\ldots ,I_{M})=r_{E}(I_{1},\ldots ,I_{M})} , with

3885-480: Is one, provided that a m ∈ F I m ∖ { 0 } {\displaystyle \mathbf {a} _{m}\in F^{I_{m}}\setminus \{0\}} . The rank of a tensor depends on the field over which the tensor is decomposed. It is known that some real tensors may admit a complex decomposition whose rank is strictly less than the rank of a real decomposition of

3996-534: Is referred to as CANDECOMP, PARAFAC, or CANDECOMP/PARAFAC (CP). Note that the PARAFAC2 rank decomposition is a variation of the CP decomposition. Another popular generalization of the matrix SVD known as the higher-order singular value decomposition computes orthonormal mode matrices and has found applications in econometrics , signal processing , computer vision , computer graphics , and psychometrics . A scalar variable

4107-452: Is selected based on specific contexts but also, at a micro level, shapes language as text (spoken or written) down to the phonological and lexico-grammatical levels. Grammar and discourse are linked as parts of a system. A particular discourse becomes a language variety when it is used in this way for a particular purpose, and is referred to as a register . There may be certain lexical additions (new words) that are brought into play because of

4218-556: Is some indeterminate closed set in the Zariski topology , is expected to equal the above value. For real tensors, r E ( I 1 , … , I M ) {\displaystyle r_{E}(I_{1},\ldots ,I_{M})} is the least rank that is expected to occur on a set of positive Euclidean measure. The value r E ( I 1 , … , I M ) {\displaystyle r_{E}(I_{1},\ldots ,I_{M})}

4329-560: Is some indeterminate closed set in the Zariski topology, equals the above value. The expected generic rank of tensors living in a balanced tensor space is equal to almost everywhere for complex tensors and on a Euclidean-open set for real tensors, where More precisely, the rank of every tensor in C I 1 × ⋯ × I M ∖ Z {\displaystyle \mathbb {C} ^{I_{1}\times \cdots \times I_{M}}\setminus Z} , where Z {\displaystyle Z}

4440-488: Is the Frobenius norm . It was shown in a 2008 paper by de Silva and Lim that the above standard approximation problem may be ill-posed . A solution to aforementioned problem may sometimes not exist because the set over which one optimizes is not closed. As such, a minimizer may not exist, even though an infimum would exist. In particular, it is known that certain so-called border tensors may be approximated arbitrarily well by

4551-444: Is the (least) generic rank of F I 1 ⊗ ⋯ ⊗ F I M {\displaystyle F^{I_{1}}\otimes \cdots \otimes F^{I_{M}}} . It is well-known that the foregoing inequality may be strict. For instance, the generic rank of tensors in R 2 × 2 × 2 {\displaystyle \mathbb {R} ^{2\times 2\times 2}}

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4662-451: Is the scientific study of language . The areas of linguistic analysis are syntax (rules governing the structure of sentences), semantics (meaning), morphology (structure of words), phonetics (speech sounds and equivalent gestures in sign languages ), phonology (the abstract sound system of a particular language), and pragmatics (how the context of use contributes to meaning). Subdisciplines such as biolinguistics (the study of

4773-428: Is the study of how language changes over history, particularly with regard to a specific language or a group of languages. Western trends in historical linguistics date back to roughly the late 18th century, when the discipline grew out of philology , the study of ancient texts and oral traditions. Historical linguistics emerged as one of the first few sub-disciplines in the field, and was most widely practised during

4884-406: Is two, so that the above bound yields r max ( 2 , 2 , 2 ) ≤ 4 {\displaystyle r_{\mbox{max}}(2,2,2)\leq 4} , while it is known that the maximum rank equals 3. A rank- s {\displaystyle s} tensor A {\displaystyle {\mathcal {A}}} is called a border tensor if there exists

4995-539: The Kronecker – Weierstrass normal form of the linear matrix pencil that the tensor represents. A simple polynomial-time algorithm exists for certifying that a tensor is of rank 1, namely the higher-order singular value decomposition . The rank of the tensor of zeros is zero by convention. The rank of a tensor a 1 ⊗ ⋯ ⊗ a M {\displaystyle \mathbf {a} _{1}\otimes \cdots \otimes \mathbf {a} _{M}}

5106-503: The Sanskrit language in his Aṣṭādhyāyī . Today, modern-day theories on grammar employ many of the principles that were laid down then. Before the 20th century, the term philology , first attested in 1716, was commonly used to refer to the study of language, which was then predominantly historical in focus. Since Ferdinand de Saussure 's insistence on the importance of synchronic analysis , however, this focus has shifted and

5217-432: The agent or patient . Functional linguistics , or functional grammar, is a branch of structural linguistics. In the humanistic reference, the terms structuralism and functionalism are related to their meaning in other human sciences . The difference between formal and functional structuralism lies in the way that the two approaches explain why languages have the properties they have. Functional explanation entails

5328-626: The comparative method and the method of internal reconstruction . Internal reconstruction is the method by which an element that contains a certain meaning is re-used in different contexts or environments where there is a variation in either sound or analogy. The reason for this had been to describe well-known Indo-European languages , many of which had detailed documentation and long written histories. Scholars of historical linguistics also studied Uralic languages , another European language family for which very little written material existed back then. After that, there also followed significant work on

5439-412: The knowledge engineering field especially with the ever-increasing amount of available data. Linguists focusing on structure attempt to understand the rules regarding language use that native speakers know (not always consciously). All linguistic structures can be broken down into component parts that are combined according to (sub)conscious rules, over multiple levels of analysis. For instance, consider

5550-504: The mind of the individual or the speech community. Construction grammar is a framework which applies the meme concept to the study of syntax. The generative versus evolutionary approach are sometimes called formalism and functionalism , respectively. This reference is however different from the use of the terms in human sciences . Modern linguistics is primarily descriptive . Linguists describe and explain features of language without making subjective judgments on whether

5661-455: The "medical discourse", and so on. The lexicon is a catalogue of words and terms that are stored in a speaker's mind. The lexicon consists of words and bound morphemes , which are parts of words that can not stand alone, like affixes . In some analyses, compound words and certain classes of idiomatic expressions and other collocations are also considered to be part of the lexicon. Dictionaries represent attempts at listing, in alphabetical order,

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5772-410: The "n" sound in "tenth" is made differently from the "n" sound in "ten" spoken alone. Although most speakers of English are consciously aware of the rules governing internal structure of the word pieces of "tenth", they are less often aware of the rule governing its sound structure. Linguists focused on structure find and analyze rules such as these, which govern how native speakers use language. Grammar

5883-543: The 18th century, the first use of the comparative method by William Jones sparked the rise of comparative linguistics . Bloomfield attributes "the first great scientific linguistic work of the world" to Jacob Grimm , who wrote Deutsche Grammatik . It was soon followed by other authors writing similar comparative studies on other language groups of Europe. The study of language was broadened from Indo-European to language in general by Wilhelm von Humboldt , of whom Bloomfield asserts: This study received its foundation at

5994-563: The East, but the grammarians of the classical languages did not use the same methods or reach the same conclusions as their contemporaries in the Indic world. Early interest in language in the West was a part of philosophy, not of grammatical description. The first insights into semantic theory were made by Plato in his Cratylus dialogue , where he argues that words denote concepts that are eternal and exist in

6105-480: The Euclidean topology) as n → ∞ {\displaystyle n\to \infty } , then there should exist at least 1 ≤ i ≠ j ≤ r {\displaystyle 1\leq i\neq j\leq r} such that as n → ∞ {\displaystyle n\to \infty } . This phenomenon is often encountered when attempting to approximate

6216-426: The Euclidean topology. There may exist Euclidean-open sets of tensors of rank strictly higher than the generic rank. All ranks appearing on open sets in the Euclidean topology are called typical ranks . The smallest typical rank is called the generic rank; this definition applies to both complex and real tensors. The generic rank of tensor spaces was initially studied in 1983 by Volker Strassen . As an illustration of

6327-484: The Zariski topology such that every tensor A ∈ S r ∖ Z r {\displaystyle {\mathcal {A}}\in S_{r}\setminus Z_{r}} is identifiable ( S r {\displaystyle S_{r}} is called generically identifiable in this case), unless either one of the following exceptional cases holds: In these exceptional cases,

6438-480: The above concepts, it is known that both 2 and 3 are typical ranks of R 2 ⊗ R 2 ⊗ R 2 {\displaystyle \mathbb {R} ^{2}\otimes \mathbb {R} ^{2}\otimes \mathbb {R} ^{2}} while the generic rank of C 2 ⊗ C 2 ⊗ C 2 {\displaystyle \mathbb {C} ^{2}\otimes \mathbb {C} ^{2}\otimes \mathbb {C} ^{2}}

6549-668: The aim of establishing a linguistic standard , which can aid communication over large geographical areas. It may also, however, be an attempt by speakers of one language or dialect to exert influence over speakers of other languages or dialects (see Linguistic imperialism ). An extreme version of prescriptivism can be found among censors , who attempt to eradicate words and structures that they consider to be destructive to society. Prescription, however, may be practised appropriately in language instruction , like in ELT , where certain fundamental grammatical rules and lexical items need to be introduced to

6660-449: The biological variables and evolution of language) and psycholinguistics (the study of psychological factors in human language) bridge many of these divisions. Linguistics encompasses many branches and subfields that span both theoretical and practical applications. Theoretical linguistics (including traditional descriptive linguistics) is concerned with understanding the universal and fundamental nature of language and developing

6771-404: The biology and evolution of language; and language acquisition , which investigates how children and adults acquire the knowledge of one or more languages. The fundamental principle of humanistic linguistics, especially rational and logical grammar , is that language is an invention created by people. A semiotic tradition of linguistic research considers language a sign system which arises from

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6882-521: The closure in the Zariski topology of the set of tensors of rank at most r {\displaystyle r} is the entire space F I 1 ⊗ ⋯ ⊗ F I M {\displaystyle F^{I_{1}}\otimes \cdots \otimes F^{I_{M}}} . In the case of complex tensors, tensors of rank at most r ( I 1 , … , I M ) {\displaystyle r(I_{1},\ldots ,I_{M})} form

6993-415: The context of fast approximate matrix multiplication algorithms by Bini, Lotti, and Romani in 1980. A classic example of a border tensor is the rank-3 tensor It can be approximated arbitrarily well by the following sequence of rank-2 tensors as m → ∞ {\displaystyle m\to \infty } . Therefore, its border rank is 2, which is strictly less than its rank. When

7104-546: The corpora of other languages, such as the Austronesian languages and the Native American language families . In historical work, the uniformitarian principle is generally the underlying working hypothesis, occasionally also clearly expressed. The principle was expressed early by William Dwight Whitney , who considered it imperative, a "must", of historical linguistics to "look to find the same principle operative also in

7215-462: The development of modern standard varieties of languages, and over the development of a language from its standardized form to its varieties. For instance, some scholars also tried to establish super-families , linking, for example, Indo-European, Uralic, and other language families to Nostratic . While these attempts are still not widely accepted as credible methods, they provide necessary information to establish relatedness in language change. This

7326-450: The distinction between balanced and unbalanced tensor spaces. A tensor space F I 1 ⊗ ⋯ ⊗ F I M {\displaystyle F^{I_{1}}\otimes \cdots \otimes F^{I_{M}}} , where I 1 ≥ I 2 ≥ ⋯ ≥ I M {\displaystyle I_{1}\geq I_{2}\geq \cdots \geq I_{M}} ,

7437-426: The equivalent aspects of sign languages). Phonetics is largely concerned with the physical aspects of sounds such as their articulation , acoustics, production, and perception. Phonology is concerned with the linguistic abstractions and categorizations of sounds, and it tells us what sounds are in a language, how they do and can combine into words, and explains why certain phonetic features are important to identifying

7548-406: The expected rank in that case is still 4. As a consequence, r ( 2 , 2 , … , 2 ) = r E ( 2 , 2 , … , 2 ) {\displaystyle r(2,2,\ldots ,2)=r_{E}(2,2,\ldots ,2)} for all binary tensors. The maximum rank that can be admitted by any of the tensors in a tensor space is unknown in general; even

7659-430: The expertise of the community of people within a certain domain of specialization. Thus, registers and discourses distinguish themselves not only through specialized vocabulary but also, in some cases, through distinct stylistic choices. People in the medical fraternity, for example, may use some medical terminology in their communication that is specialized to the field of medicine. This is often referred to as being part of

7770-558: The factors are ordered such that I 1 ≥ I 2 ≥ ⋯ ≥ I M ≥ 2 {\displaystyle I_{1}\geq I_{2}\geq \cdots \geq I_{M}\geq 2} . Let S r ⊂ F I 1 ⊗ ⋯ F I m ⊗ ⋯ ⊗ F I M {\displaystyle S_{r}\subset F^{I_{1}}\otimes \cdots F^{I_{m}}\otimes \cdots \otimes F^{I_{M}}} denote

7881-450: The field of philology , of which some branches are more qualitative and holistic in approach. Today, philology and linguistics are variably described as related fields, subdisciplines, or separate fields of language study but, by and large, linguistics can be seen as an umbrella term. Linguistics is also related to the philosophy of language , stylistics , rhetoric , semiotics , lexicography , and translation . Historical linguistics

7992-441: The following exceptional cases: In each of these exceptional cases, the generic rank is known to be r ( I 1 , … , I m , … , I M ) = r E ( I 1 , … , I M ) + 1 {\displaystyle r(I_{1},\ldots ,I_{m},\ldots ,I_{M})=r_{E}(I_{1},\ldots ,I_{M})+1} . Note that while

8103-425: The general principle that a sequence of rank- r {\displaystyle r} tensors that converges to a tensor of strictly higher rank needs to admit at least two individual rank-1 terms whose norms become unbounded. Stated formally, whenever a sequence has the property that A n → A {\displaystyle {\mathcal {A}}_{n}\to {\mathcal {A}}} (in

8214-472: The generic (and also minimum) number of complex decompositions is In summary, the generic tensor of order M > 2 {\displaystyle M>2} and rank r < Π Σ + 1 {\textstyle r<{\frac {\Pi }{\Sigma +1}}} that is not identifiability-unbalanced is expected to be identifiable (modulo the exceptional cases in small spaces). The rank approximation problem asks for

8325-621: The hands of the Prussian statesman and scholar Wilhelm von Humboldt (1767–1835), especially in the first volume of his work on Kavi, the literary language of Java, entitled Über die Verschiedenheit des menschlichen Sprachbaues und ihren Einfluß auf die geistige Entwickelung des Menschengeschlechts ( On the Variety of the Structure of Human Language and its Influence upon the Mental Development of

8436-402: The historical development of a language over a period of time), in monolinguals or in multilinguals , among children or among adults, in terms of how it is being learnt or how it was acquired, as abstract objects or as cognitive structures, through written texts or through oral elicitation, and finally through mechanical data collection or through practical fieldwork. Linguistics emerged from

8547-433: The history of a language. The discipline that deals specifically with the sound changes occurring within morphemes is morphophonology . Semantics and pragmatics are branches of linguistics concerned with meaning. These subfields have traditionally been divided according to aspects of meaning: "semantics" refers to grammatical and lexical meanings, while "pragmatics" is concerned with meaning in context. Within linguistics,

8658-414: The human mind creates linguistic constructions from event schemas , and the impact of cognitive constraints and biases on human language. In cognitive linguistics, language is approached via the senses . A closely related approach is evolutionary linguistics which includes the study of linguistic units as cultural replicators . It is possible to study how language replicates and adapts to

8769-461: The idea that language is a tool for communication, or that communication is the primary function of language. Linguistic forms are consequently explained by an appeal to their functional value, or usefulness. Other structuralist approaches take the perspective that form follows from the inner mechanisms of the bilateral and multilayered language system. Approaches such as cognitive linguistics and generative grammar study linguistic cognition with

8880-511: The ill-posedness problem consists of imposing an additional inequality constraint that bounds the norm of the individual rank-1 terms by some constant. Other constraints that result in a closed set, and, thus, well-posed optimization problem, include imposing positivity or a bounded inner product strictly less than unity between the rank-1 terms appearing in the sought decomposition. Alternating algorithms: Direct algorithms: General optimization algorithms: Linguistics Linguistics

8991-498: The interaction of meaning and form. The organization of linguistic levels is considered computational. Linguistics is essentially seen as relating to social and cultural studies because different languages are shaped in social interaction by the speech community . Frameworks representing the humanistic view of language include structural linguistics , among others. Structural analysis means dissecting each linguistic level: phonetic, morphological, syntactic, and discourse, to

9102-412: The late 19th century. Despite a shift in focus in the 20th century towards formalism and generative grammar , which studies the universal properties of language, historical research today still remains a significant field of linguistic inquiry. Subfields of the discipline include language change and grammaticalization . Historical linguistics studies language change either diachronically (through

9213-429: The lexicon of a given language; usually, however, bound morphemes are not included. Lexicography , closely linked with the domain of semantics, is the science of mapping the words into an encyclopedia or a dictionary. The creation and addition of new words (into the lexicon) is called coining or neologization , and the new words are called neologisms . It is often believed that a speaker's capacity for language lies in

9324-409: The multiple modes of a tensor are conveniently denoted by 1 ≤ i m ≤ I m {\displaystyle 1\leq i_{m}\leq I_{m}} where 1 ≤ m ≤ M {\displaystyle 1\leq m\leq M} . A vector is denoted by a lower case bold Times Roman, a {\displaystyle \mathbf {a} } and

9435-426: The nature of crosslinguistic variation, and the relationship between form and meaning. There are numerous approaches to syntax that differ in their central assumptions and goals. Morphology is the study of words , including the principles by which they are formed, and how they relate to one another within a language. Most approaches to morphology investigate the structure of words in terms of morphemes , which are

9546-1000: The observation A = ∑ i = 1 r a i ⊗ b i = ∑ i = 1 r a i b i T = A B T = ( A X − 1 ) ( B X T ) T = ∑ i = 1 r c i d i T = ∑ i = 1 r c i ⊗ d i , {\displaystyle {\mathcal {A}}=\sum _{i=1}^{r}\mathbf {a} _{i}\otimes \mathbf {b} _{i}=\sum _{i=1}^{r}\mathbf {a} _{i}\mathbf {b} _{i}^{T}=AB^{T}=(AX^{-1})(BX^{T})^{T}=\sum _{i=1}^{r}\mathbf {c} _{i}\mathbf {d} _{i}^{T}=\sum _{i=1}^{r}\mathbf {c} _{i}\otimes \mathbf {d} _{i},} where X ∈ G L r ( F ) {\displaystyle X\in \mathrm {GL} _{r}(F)}

9657-757: The order of the summands. Observe that in a tensor rank decomposition all the A i {\displaystyle {\mathcal {A}}_{i}} 's are distinct, for otherwise the rank of A {\displaystyle {\mathcal {A}}} would be at most r − 1 {\displaystyle r-1} . Order-2 tensors in F I 1 ⊗ F I 2 ≃ F I 1 × I 2 {\displaystyle F^{I_{1}}\otimes F^{I_{2}}\simeq F^{I_{1}\times I_{2}}} , i.e., matrices, are not identifiable for r > 1 {\displaystyle r>1} . This follows essentially from

9768-421: The other hand, focuses on an analysis that is based on the paradigms or concepts that are embedded in a given text. In this case, words of the same type or class may be replaced in the text with each other to achieve the same conceptual understanding. The earliest activities in the description of language have been attributed to the 6th-century-BC Indian grammarian Pāṇini who wrote a formal description of

9879-478: The principles of grammar include structural and functional linguistics , and generative linguistics . Sub-fields that focus on a grammatical study of language include the following: Discourse is language as social practice (Baynham, 1995) and is a multilayered concept. As a social practice, discourse embodies different ideologies through written and spoken texts. Discourse analysis can examine or expose these ideologies. Discourse not only influences genre, which

9990-416: The quantity of words stored in the lexicon. However, this is often considered a myth by linguists. The capacity for the use of language is considered by many linguists to lie primarily in the domain of grammar, and to be linked with competence , rather than with the growth of vocabulary. Even a very small lexicon is theoretically capable of producing an infinite number of sentences. Stylistics also involves

10101-447: The rank of real matrices will never decrease under a field extension to C {\displaystyle \mathbb {C} } : real matrix rank and complex matrix rank coincide for real matrices. The generic rank r ( I 1 , … , I M ) {\displaystyle r(I_{1},\ldots ,I_{M})} is defined as the least rank r {\displaystyle r} such that

10212-449: The rank- r {\displaystyle r} decomposition closest (in the usual Euclidean topology) to some rank- s {\displaystyle s} tensor A {\displaystyle {\mathcal {A}}} , where r < s {\displaystyle r<s} . That is, one seeks to solve where ‖ ⋅ ‖ F {\displaystyle \|\cdot \|_{F}}

10323-424: The relationships between dialects within a specific period. This includes studying morphological, syntactical, and phonetic shifts. Connections between dialects in the past and present are also explored. Syntax is the study of how words and morphemes combine to form larger units such as phrases and sentences . Central concerns of syntax include word order , grammatical relations , constituency , agreement ,

10434-476: The same size will be a rank-1 tensor with probability zero, a rank-2 tensor with probability one, and a rank-3 tensor with probability zero. It is even known that the generic rank-3 real tensor in R 2 ⊗ R 2 ⊗ R 2 {\displaystyle \mathbb {R} ^{2}\otimes \mathbb {R} ^{2}\otimes \mathbb {R} ^{2}} will be of complex rank equal to 2. The generic rank of tensor spaces depends on

10545-624: The same tensor. As an example, consider the following real tensor where x i , y j ∈ R 2 {\displaystyle \mathbf {x} _{i},\mathbf {y} _{j}\in \mathbb {R} ^{2}} . The rank of this tensor over the reals is known to be 3, while its complex rank is only 2 because it is the sum of a complex rank-1 tensor with its complex conjugate , namely where z k = x k + i y k {\displaystyle \mathbf {z} _{k}=\mathbf {x} _{k}+i\mathbf {y} _{k}} . In contrast,

10656-401: The scientific study of language, though linguistic science is sometimes used. Linguistics is a multi-disciplinary field of research that combines tools from natural sciences, social sciences, formal sciences , and the humanities. Many linguists, such as David Crystal, conceptualize the field as being primarily scientific. The term linguist applies to someone who studies language or

10767-426: The set of tensors of rank 3 in F 2 × 2 × 2 × 2 {\displaystyle F^{2\times 2\times 2\times 2}} is defective (13 and not the expected 14), the generic rank in that space is still the expected one, 4. Similarly, the set of tensors of rank 5 in F 4 × 4 × 3 {\displaystyle F^{4\times 4\times 3}}

10878-425: The set of tensors of rank bounded by r {\displaystyle r} . Then, the following statement was proved to be correct using a computer-assisted proof for all spaces of dimension Π < 15000 {\displaystyle \Pi <15000} , and it is conjectured to be valid in general: There exists a closed set Z r {\displaystyle Z_{r}} in

10989-744: The smallest units in a language with some independent meaning . Morphemes include roots that can exist as words by themselves, but also categories such as affixes that can only appear as part of a larger word. For example, in English the root catch and the suffix -ing are both morphemes; catch may appear as its own word, or it may be combined with -ing to form the new word catching . Morphology also analyzes how words behave as parts of speech , and how they may be inflected to express grammatical categories including number , tense , and aspect . Concepts such as productivity are concerned with how speakers create words in specific contexts, which evolves over

11100-404: The smallest units. These are collected into inventories (e.g. phoneme, morpheme, lexical classes, phrase types) to study their interconnectedness within a hierarchy of structures and layers. Functional analysis adds to structural analysis the assignment of semantic and other functional roles that each unit may have. For example, a noun phrase may function as the subject or object of the sentence; or

11211-445: The structure of the word "tenth" on two different levels of analysis. On the level of internal word structure (known as morphology), the word "tenth" is made up of one linguistic form indicating a number and another form indicating ordinality. The rule governing the combination of these forms ensures that the ordinality marker "th" follows the number "ten." On the level of sound structure (known as phonology), structural analysis shows that

11322-471: The study of language in canonical works of literature, popular fiction, news, advertisements, and other forms of communication in popular culture as well. It is usually seen as a variation in communication that changes from speaker to speaker and community to community. In short, Stylistics is the interpretation of text. In the 1960s, Jacques Derrida , for instance, further distinguished between speech and writing, by proposing that written language be studied as

11433-531: The study of written, signed, or spoken discourse through varying speech communities, genres, and editorial or narrative formats in the mass media. It involves the study and interpretation of texts for aspects of their linguistic and tonal style. Stylistic analysis entails the analysis of description of particular dialects and registers used by speech communities. Stylistic features include rhetoric , diction, stress, satire, irony , dialogue, and other forms of phonetic variations. Stylistic analysis can also include

11544-436: The study was geared towards analysis and comparison between different language variations, which existed at the same given point of time. At another level, the syntagmatic plane of linguistic analysis entails the comparison between the way words are sequenced, within the syntax of a sentence. For example, the article "the" is followed by a noun, because of the syntagmatic relation between the words. The paradigmatic plane, on

11655-586: The subfield of formal semantics studies the denotations of sentences and how they are composed from the meanings of their constituent expressions. Formal semantics draws heavily on philosophy of language and uses formal tools from logic and computer science . On the other hand, cognitive semantics explains linguistic meaning via aspects of general cognition, drawing on ideas from cognitive science such as prototype theory . Pragmatics focuses on phenomena such as speech acts , implicature , and talk in interaction . Unlike semantics, which examines meaning that

11766-475: The term philology is now generally used for the "study of a language's grammar, history, and literary tradition", especially in the United States (where philology has never been very popularly considered as the "science of language"). Although the term linguist in the sense of "a student of language" dates from 1641, the term linguistics is first attested in 1847. It is now the usual term in English for

11877-907: The two vectors are orthogonal, this example is also known as a W state . It follows from the definition of a pure tensor that A = a 1 ⊗ a 2 ⊗ ⋯ ⊗ a M = b 1 ⊗ b 2 ⊗ ⋯ ⊗ b M {\displaystyle {\mathcal {A}}=\mathbf {a} _{1}\otimes \mathbf {a} _{2}\otimes \cdots \otimes \mathbf {a} _{M}=\mathbf {b} _{1}\otimes \mathbf {b} _{2}\otimes \cdots \otimes \mathbf {b} _{M}} if and only if there exist λ k {\displaystyle \lambda _{k}} such that λ 1 λ 2 ⋯ λ M = 1 {\displaystyle \lambda _{1}\lambda _{2}\cdots \lambda _{M}=1} and

11988-420: The very outset of that [language] history." The above approach of comparativism in linguistics is now, however, only a small part of the much broader discipline called historical linguistics. The comparative study of specific Indo-European languages is considered a highly specialized field today, while comparative research is carried out over the subsequent internal developments in a language: in particular, over

12099-551: The word in its original meaning as " téchnē grammatikḗ " ( Τέχνη Γραμματική ), the "art of writing", which is also the title of one of the most important works of the Alexandrine school by Dionysius Thrax . Throughout the Middle Ages , the study of language was subsumed under the topic of philology, the study of ancient languages and texts, practised by such educators as Roger Ascham , Wolfgang Ratke , and John Amos Comenius . In

12210-582: The world of ideas. This work is the first to use the word etymology to describe the history of a word's meaning. Around 280 BC, one of Alexander the Great 's successors founded a university (see Musaeum ) in Alexandria , where a school of philologists studied the ancient texts in Greek, and taught Greek to speakers of other languages. While this school was the first to use the word "grammar" in its modern sense, Plato had used

12321-507: Was the first known instance of its kind. In the Middle East, Sibawayh , a Persian, made a detailed description of Arabic in AD 760 in his monumental work, Al-kitab fii an-naħw ( الكتاب في النحو , The Book on Grammar ), the first known author to distinguish between sounds and phonemes (sounds as units of a linguistic system) . Western interest in the study of languages began somewhat later than in

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