In time series analysis, dynamic time warping (DTW) is an algorithm for measuring similarity between two temporal sequences, which may vary in speed. For instance, similarities in walking could be detected using DTW, even if one person was walking faster than the other, or if there were accelerations and decelerations during the course of an observation. DTW has been applied to temporal sequences of video, audio, and graphics data — indeed, any data that can be turned into a one-dimensional sequence can be analyzed with DTW. A well-known application has been automatic speech recognition, to cope with different speaking speeds. Other applications include speaker recognition and online signature recognition. It can also be used in partial shape matching applications. In general, DTW is a method that calculates an optimal match between two given sequences (e.g. time series) with certain restriction and rules: Every index from the first sequence must be matched with one or more indices from the other sequence, and vice versa The first index from the first sequence must be matched with the first index from the other sequence (but it does not have to be its only match) The last index from the first sequence must be matched with the last index from the other sequence (but it does not have to be its only match) The mapping of the indices from the first sequence to indices from the other sequence must be monotonically increasing, and vice versa, i.e. if j > i {\displaystyle j>i} are indices from the first sequence, then there must not be two indices l > k {\displaystyle l>k} in the other sequence, such that index i {\displaystyle i} is matched with index l {\displaystyle l} and index j {\displaystyle j} is matched with index k {\displaystyle k} , and vice versa We can plot each match between the sequences 1 : M {\displaystyle 1:M} and 1 : N {\displaystyle 1:N} as a path in a M × N {\displaystyle M\times N} matrix from ( 1 , 1 ) {\displaystyle (1,1)} to ( M , N ) {\displaystyle (M,N)} , such that each step is one of ( 0 , 1 ) , ( 1 , 0 ) , ( 1 , 1 ) {\displaystyle (0,1),(1,0),(1,1)} . In this formulation, we see that the number of possible matches is the Delannoy number. The optimal match is denoted by the match that satisfies all the restrictions and the rules and that has the minimal cost, where the cost is computed as the sum of absolute differences, for each matched pair of indices, between their values. The sequences are "warped" non-linearly in the time dimension to determine a measure of their similarity independent of certain non-linear variations in the time dimension. This sequence alignment method is often used in time series classification. Although DTW measures a distance-like quantity between two given sequences, it doesn't guarantee the triangle inequality to hold. In addition to a similarity measure between the two sequences (a so called "warping path" is produced), by warping according to this path the two signals may be aligned in time. The signal with an original set of points X(original), Y(original) is transformed to X(warped), Y(warped). This finds applications in genetic sequence and audio synchronisation. In a related technique sequences of varying speed may be averaged using this technique see the average sequence section. This is conceptually very similar to the Needleman–Wunsch algorithm. == Implementation == This example illustrates the implementation of the dynamic time warping algorithm when the two sequences s and t are strings of discrete symbols. For two symbols x and y, d ( x , y ) {\displaystyle d(x,y)} is a distance between the symbols, e.g., d ( x , y ) = | x − y | {\displaystyle d(x,y)=|x-y|} . int DTWDistance(s: array [1..n], t: array [1..m]) { DTW := array [0..n, 0..m] for i := 0 to n for j := 0 to m DTW[i, j] := infinity DTW[0, 0] := 0 for i := 1 to n for j := 1 to m cost := d(s[i], t[j]) DTW[i, j] := cost + minimum(DTW[i-1, j ], // insertion DTW[i , j-1], // deletion DTW[i-1, j-1]) // match return DTW[n, m] } where DTW[i, j] is the distance between s[1:i] and t[1:j] with the best alignment. We sometimes want to add a locality constraint. That is, we require that if s[i] is matched with t[j], then | i − j | {\displaystyle |i-j|} is no larger than w, a window parameter. We can easily modify the above algorithm to add a locality constraint (differences marked). However, the above given modification works only if | n − m | {\displaystyle |n-m|} is no larger than w, i.e. the end point is within the window length from diagonal. In order to make the algorithm work, the window parameter w must be adapted so that | n − m | ≤ w {\displaystyle |n-m|\leq w} (see the line marked with () in the code). int DTWDistance(s: array [1..n], t: array [1..m], w: int) { DTW := array [0..n, 0..m] w := max(w, abs(n-m)) // adapt window size () for i := 0 to n for j:= 0 to m DTW[i, j] := infinity DTW[0, 0] := 0 for i := 1 to n for j := max(1, i-w) to min(m, i+w) DTW[i, j] := 0 for i := 1 to n for j := max(1, i-w) to min(m, i+w) cost := d(s[i], t[j]) DTW[i, j] := cost + minimum(DTW[i-1, j ], // insertion DTW[i , j-1], // deletion DTW[i-1, j-1]) // match return DTW[n, m] } == Warping properties == The DTW algorithm produces a discrete matching between existing elements of one series to another. In other words, it does not allow time-scaling of segments within the sequence. Other methods allow continuous warping. For example, Correlation Optimized Warping (COW) divides the sequence into uniform segments that are scaled in time using linear interpolation, to produce the best matching warping. The segment scaling causes potential creation of new elements, by time-scaling segments either down or up, and thus produces a more sensitive warping than DTW's discrete matching of raw elements. == Complexity == The time complexity of the DTW algorithm is O ( N M ) {\displaystyle O(NM)} , where N {\displaystyle N} and M {\displaystyle M} are the lengths of the two input sequences. The 50 years old quadratic time bound was broken in 2016: an algorithm due to Gold and Sharir enables computing DTW in O ( N 2 / log log N ) {\displaystyle O({N^{2}}/\log \log N)} time and space for two input sequences of length N {\displaystyle N} . This algorithm can also be adapted to sequences of different lengths. Despite this improvement, it was shown that a strongly subquadratic running time of the form O ( N 2 − ϵ ) {\displaystyle O(N^{2-\epsilon })} for some ϵ > 0 {\displaystyle \epsilon >0} cannot exist unless the Strong exponential time hypothesis fails. While the dynamic programming algorithm for DTW requires O ( N M ) {\displaystyle O(NM)} space in a naive implementation, the space consumption can be reduced to O ( min ( N , M ) ) {\displaystyle O(\min(N,M))} using Hirschberg's algorithm. == Fast computation == Fast techniques for computing DTW include PrunedDTW, SparseDTW, FastDTW, and the MultiscaleDTW. A common task, retrieval of similar time series, can be accelerated by using lower bounds such as LB_Keogh, LB_Improved, or LB_Petitjean. However, the Early Abandon and Pruned DTW algorithm reduces the degree of acceleration that lower bounding provides and sometimes renders it ineffective. In a survey, Wang et al. reported slightly better results with the LB_Improved lower bound than the LB_Keogh bound, and found that other techniques were inefficient. Subsequent to this survey, the LB_Enhanced bound was developed that is always tighter than LB_Keogh while also being more efficient to compute. LB_Petitjean is the tightest known lower bound that can be computed in linear time. == Average sequence == Averaging for dynamic time warping is the problem of finding an average sequence for a set of sequences. NLAAF is an exact method to average two sequences using DTW. For more than two sequences, the problem is related to that of multiple alignment and requires heuristics. DBA is currently a reference method to average a set of sequences consistently with DTW. COMASA efficiently randomizes the search for the average sequence, using DBA as a local optimization process. == Supervised learning == A nearest-neighbour classifier can achieve state-of-the-art performance when using dynamic time warping as a distance measure. == Amerced Dynamic Time Warping == Amerced Dynamic Time Warping (ADTW) is a variant of DTW designed to better control DTW's permissiveness in the alignments that it allows. The windows that classical DTW uses to constrain alignments introduce a step function. Any warping of the path is allowed within the window and none beyond it. In contrast, ADTW employs an additive penalty that is incurred each time that the path is warped. Any amount of warping is allowed, but each warping action incurs a direct penalty. ADTW significantly outperforms DTW with windowing when applied as a nearest neighbor classifier on a set of benchmark time series classification tasks. == Alternative approaches == In functional data analysis, time series are regarde
Automaton
An automaton ( ; pl.: automata or automatons) is a relatively self-operating machine or control mechanism designed to automatically follow a sequence of operations or respond to predetermined instructions. Some automata, such as bellstrikers in mechanical clocks, are designed to give the illusion to the casual observer that they are operating under their own power or will, like a mechanical robot. The term has long been commonly associated with automated puppets that resemble moving humans or animals, built to impress and/or to entertain people. Animatronics are a modern type of automata with electronics, often used for the portrayal of characters or creatures in films and in theme park attractions. == Etymology == The word automaton is the latinization of the Ancient Greek automaton (αὐτόματον), which means "acting of one's own will". It was first used by Homer to describe an automatic door opening, or automatic movement of wheeled tripods. It is more often used to describe non-electronic moving machines, especially those that have been made to resemble human or animal actions, such as the jacks on old public striking clocks, or the cuckoo and any other animated figures on a cuckoo clock. == History == === Ancient === There are many examples of automata in Greek mythology: Hephaestus created automata for his workshop; Talos was an artificial man of bronze; King Alkinous of the Phaiakians employed gold and silver watchdogs. According to Aristotle, Daedalus used quicksilver to make his wooden statue of Aphrodite move. In other Greek legends he used quicksilver to install voice in his moving statues. The automata in the Hellenistic world were intended as tools, toys, religious spectacles, or prototypes for demonstrating basic scientific principles. Numerous water-powered automata were built by Ktesibios, a Greek inventor and the first head of the Great Library of Alexandria; for example, he "used water to sound a whistle and make a model owl move. He had invented the world's first 'cuckoo clock'". This tradition continued in Alexandria with inventors such as the Greek mathematician Hero of Alexandria (sometimes known as Heron), whose writings on hydraulics, pneumatics, and mechanics described siphons, a fire engine, a water organ, the aeolipile, and a programmable cart. Philo of Byzantium was famous for his inventions. Complex mechanical devices are known to have existed in Hellenistic Greece, though the only surviving example is the Antikythera mechanism, the earliest known analog computer. The clockwork is thought to have come originally from Rhodes, where there was apparently a tradition of mechanical engineering; the island was renowned for its automata; to quote Pindar's seventh Olympic Ode: The animated figures stand Adorning every public street And seem to breathe in stone, or move their marble feet. However, the information gleaned from recent scans of the fragments indicate that it may have come from the colonies of Corinth in Sicily and implies a connection with Archimedes. According to Jewish legend, King Solomon used his wisdom to design a throne with mechanical animals which hailed him as king when he ascended it; upon sitting down an eagle would place a crown upon his head, and a dove would bring him a Torah scroll. It is also said that when King Solomon stepped upon the throne, a mechanism was set in motion. As soon as he stepped upon the first step, a golden ox and a golden lion each stretched out one foot to support him and help him rise to the next step. On each side, the animals helped the King up until he was comfortably seated upon the throne. In ancient China, a curious account of automata is found in the Lie Zi text, believed to have originated around 400 BCE and compiled around the fourth century CE. Within it there is a description of a much earlier encounter between King Mu of Zhou (1023–957 BCE) and a mechanical engineer known as Yan Shi, an 'artificer'. The latter proudly presented the king with a very realistic and detailed life-size, human-shaped figure of his mechanical handiwork: The king stared at the figure in astonishment. It walked with rapid strides, moving its head up and down, so that anyone would have taken it for a live human being. The artificer touched its chin, and it began singing, perfectly in tune. He touched its hand, and it began posturing, keeping perfect time...As the performance was drawing to an end, the robot winked its eye and made advances to the ladies in attendance, whereupon the king became incensed and would have had Yen Shih [Yan Shi] executed on the spot had not the latter, in mortal fear, instantly taken the robot to pieces to let him see what it really was. And, indeed, it turned out to be only a construction of leather, wood, glue and lacquer, variously coloured white, black, red and blue. Examining it closely, the king found all the internal organs complete—liver, gall, heart, lungs, spleen, kidneys, stomach and intestines; and over these again, muscles, bones and limbs with their joints, skin, teeth and hair, all of them artificial...The king tried the effect of taking away the heart, and found that the mouth could no longer speak; he took away the liver and the eyes could no longer see; he took away the kidneys and the legs lost their power of locomotion. The king was delighted. Other notable examples of automata include Archytas' dove, mentioned by Aulus Gellius. Similar Chinese accounts of flying automata are written of the 5th century BC Mohist philosopher Mozi and his contemporary Lu Ban, who made artificial wooden birds (ma yuan) that could successfully fly according to the Han Fei Zi and other texts. === Medieval === The manufacturing tradition of automata continued in the Greek world well into the Middle Ages. On his visit to Constantinople in 949 ambassador Liutprand of Cremona described automata in the emperor Theophilos' palace, including "lions, made either of bronze or wood covered with gold, which struck the ground with their tails and roared with open mouth and quivering tongue," "a tree of gilded bronze, its branches filled with birds, likewise made of bronze gilded over, and these emitted cries appropriate to their species" and "the emperor's throne" itself, which "was made in such a cunning manner that at one moment it was down on the ground, while at another it rose higher and was to be seen up in the air." Similar automata in the throne room (singing birds, roaring and moving lions) were described by Luitprand's contemporary the Byzantine emperor Constantine Porphyrogenitus, in his book De Ceremoniis (Perì tês Basileíou Tákseōs). In the mid-8th century, the first wind powered automata were built: "statues that turned with the wind over the domes of the four gates and the palace complex of the Round City of Baghdad". The "public spectacle of wind-powered statues had its private counterpart in the 'Abbasid palaces where automata of various types were predominantly displayed." Also in the 8th century, the Muslim alchemist, Jābir ibn Hayyān (Geber), included recipes for constructing artificial snakes, scorpions, and humans that would be subject to their creator's control in his coded Book of Stones. In 827, Abbasid caliph al-Ma'mun had a silver and golden tree in his palace in Baghdad, which had the features of an automatic machine. There were metal birds that sang automatically on the swinging branches of this tree built by Muslim inventors and engineers. The Abbasid caliph al-Muqtadir also had a silver and golden tree in his palace in Baghdad in 917, with birds on it flapping their wings and singing. In the 9th century, the Banū Mūsā brothers invented a programmable automatic flute player and which they described in their Book of Ingenious Devices. Al-Jazari described complex programmable humanoid automata amongst other machines he designed and constructed in the Book of Knowledge of Ingenious Mechanical Devices in 1206. His automaton was a boat with four automatic musicians that floated on a lake to entertain guests at royal drinking parties. His mechanism had a programmable drum machine with pegs (cams) that bump into little levers that operate the percussion. The drummer could be made to play different rhythms and drum patterns if the pegs were moved around. Al-Jazari constructed a hand washing automaton first employing the flush mechanism now used in modern toilets. It features a female automaton standing by a basin filled with water. When the user pulls the lever, the water drains and the automaton refills the basin. His "peacock fountain" was another more sophisticated hand washing device featuring humanoid automata as servants who offer soap and towels. Mark E. Rosheim describes it as follows: "Pulling a plug on the peacock's tail releases water out of the beak; as the dirty water from the basin fills the hollow base a float rises and actuates a linkage which makes a servant figure appear from behind a door under the peacock and offer soap.
Regina Barzilay
Regina Barzilay (Hebrew: רגינה ברזילי; born 1970) is an Israeli-American computer scientist. She is a professor at the Massachusetts Institute of Technology and a faculty lead for artificial intelligence at the MIT Jameel Clinic. Her research interests are in natural language processing and applications of deep learning to chemistry and oncology. == Early life and education == Barzilay was born in Chișinău, Moldova and emigrated to Israel with her parents at the age of 20. She received bachelor's and master's degrees from Ben-Gurion University of the Negev in 1993 and 1998, respectively. She obtained a PhD in computer science from Columbia University in 2003 for research supervised by Kathleen McKeown. == Career and research == After her PhD, she spent a year as a postdoctoral researcher at Cornell University. She was appointed as Delta Electronics Professor of Electrical Engineering and Computer Science at MIT in 2016. She was diagnosed with breast cancer in 2014, which prompted her to conduct research in oncology. Barzilay won the MacArthur Fellowship in 2017. For her doctoral dissertation at Columbia University, she led the development of Newsblaster, which recognized stories from different news sources as being about the same basic subject, and then paraphrased elements from the stories to create a summary. In computational linguistics, Barzilay created algorithms that learned annotations from common languages (i.e. English) to analyze less understood languages. Prompted by her experience with breast cancer, Barzilay is applying machine learning to oncology. She is collaborating with physicians and students to devise deep learning models that utilize images, text, and structured data to identify trends that affect early diagnosis, treatment, and disease prevention. Frontline Documentary Following her battle with breast cancer in 2014, and her researching into applying artificial intelligence to improve early detection methods, she collaborated with Dr. Connie Lehman at Massachusetts General Hospital. While there Barzilay developed an AI-based system capable of predicting the likelihood of breast cancer up to five years in advance. The system leverages deep learning techniques to analyze mammograms and diagnostic notes, surpassing traditional pattern recognition by human radiologists. This breakthrough, while still in development, has the potential to significantly enhance early diagnosis and treatment outcomes. [1] Barzilay's work in this area was featured in the FRONTLINE documentary In the Age of AI, which explores the broader impact of artificial intelligence on society. === MIT Jameel Clinic === In 2018, Barzilay was appointed faculty lead for AI at the new MIT Jameel Clinic, a research center in the field of AI health sciences, including disease detection, drug discovery, and the development of medical devices. In 2020, she was part of the team—with fellow MIT Jameel Clinic faculty lead Professor James J. Collins—that announced the discovery through deep learning of halicin, the first new antibiotic compound for 30 years, which kills over 35 powerful bacteria, including antimicrobial-resistant tuberculosis, the superbug C. difficile, and two of the World Health Organization's top-three most deadly bacteria. In 2020, Collins, Barzilay and the MIT Jameel Clinic were also awarded funding through The Audacious Project to expand on the discovery of halicin in using AI to respond to the antibiotic resistance crisis through the development of new classes of antibiotics. == Awards and recognition == In 2017, Barzilay won the MacArthur Fellowship, known as the "Genius Grant", for "developing machine learning methods that enable computers to process and analyze vast amounts of human language data." She is also a recipient of various awards including the NSF Career Award, the MIT Technology Review TR-35 Award, Microsoft Faculty Fellowship and several Best Paper Awards at NAACL and ACL. Her teaching has also been recognized by MIT as she won the Jamieson Teaching Award in 2016. She was nominated an AAAI Fellow in 2018 by the Association for the Advancement of Artificial Intelligence. In 2020, she became the first recipient of the $1 million AAAI Squirrel AI Award for Artificial Intelligence for the Benefit of Humanity. In 2023, she was elected to the National Academy of Medicine and the National Academy of Engineering.
Linguistic Systems
Linguistic Systems, Inc., also known as LSI, provides language translation services (conversion) for all media in over 115 languages. LSI focuses on the translation of legal, medical, business, institutional, academic, government and personal documents. LSI is headquartered in Cambridge, Massachusetts. == About LSI == Linguistic Systems, Inc. (LSI) was founded in 1967 by Martin Roberts. LSI's translates to/from 115 languages, DTP, audio-visual conversions, software localization, consecutive and simultaneous interpreting services, foreign brand name analysis, and machine translation with post-editing. LSI has provided translation services to over half of the Fortune 500 companies and most of the Fortune 100. Among its clients are AT&T, Boeing, Citigroup, Coca-Cola, DuPont, Exxon-Mobil, General Electric, General Motors, Hewlett-Packard, IBM, Johnson & Johnson, Pfizer, Procter & Gamble, Simon & Schuster, Time Warner, Verizon, and Walmart. As of 2013, LSI had a network of more than 7,000 translators who translate into their native languages; These include lawyers, scientists, engineers, and other bilingual professionals.
AI Avatar Generators Reviews: What Actually Works in 2026
Shopping for the best AI avatar generator? An AI avatar generator is software that uses machine learning to help you get more done — it keeps getting smarter as the underlying models improve. Pricing, accuracy, and the size of the model behind the tool are the three factors that most affect daily usefulness. Whether you are a beginner or a pro, the right AI avatar generator slots into your workflow and pays for itself fast. Below we compare features, pricing, and real output so you can choose with confidence.
Fling (social network)
Fling was a social media app available for IOS and Android. It was founded in 2014 by Marco Nardone and was taken offline in August 2016. == Overview == In 2012, Marco Nardone founded the startup Unii and launched Unii.com, a social network intended for students in the UK. While working on this service, Nardone had the idea for a messaging service where pictures could be sent to strangers in January 2014. The app Fling was then developed and released between March and July 2014. After a month, it already had 375,000 downloads and 180,000 active users on iOS. Users were able to take pictures inside the app and send them to 50 random people all over the world. The recipient could then choose to answer via chat or reply by sending a picture themselves. The app was used by many users as a medium to exchange sexually explicit pictures and for sexting with strangers. This led to the app being removed from the App Store in June 2015. In the 19 days that followed, flings developers rewrote the App almost completely from scratch, working around the clock. The feature to message random strangers was removed, and the app was readmitted into the App Store as a messenger App resembling Snapchat. But the redesigned Application did not have the success of its predecessor. The funding ran out and the parent company Unii went bankrupt. The company was not able to pay their content moderation team anymore, leading to a new surge of pornographic content on the App. Shortly after that, the Social Network was taken offline in August 2016. It has been inactive since. During the 2 years Fling was online, $21 million was raised from investors while generating no revenue at all. Of this $21 million (£16.5m), £5 million came from Nardone's father. == Allegations against CEO == Former employees made multiple allegations against Marco Nardone, the Founder and CEO of Unii and Fling. According to these claims, he behaved erratic and abusive, throwing "things across the office". He hired his girlfriend as the head of human resources to handle issues between him and his staff. Employees who left the company often had "some part of their pay held back". According to the reports, he also spent the money raised from investors irresponsibly, having no clear concept of a budget. Some of that money was used on expensive restaurants in London, a luxurious office for CEO Nardone and advertisements for Fling on Twitter and Facebook. Nardone also spent time partying in Ibiza with two employees, while the developer team in London frantically tried to get Fling back online after it being removed from the App Store. In December 2017 he pleaded guilty to assaulting his girlfriend at a domestic violence court.
Structured support vector machine
The structured supportvector machine is a machine learning algorithm that generalizes the support vector machine (SVM) classifier. Whereas the SVM classifier supports binary classification, multiclass classification and regression, the structured SVM allows training of a classifier for general structured output labels. As an example, a sample instance might be a natural language sentence, and the output label is an annotated parse tree. Training a classifier consists of showing pairs of correct sample and output label pairs. After training, the structured SVM model allows one to predict for new sample instances the corresponding output label; that is, given a natural language sentence, the classifier can produce the most likely parse tree. == Training == For a set of n {\displaystyle n} training instances ( x i , y i ) ∈ X × Y {\displaystyle ({\boldsymbol {x}}_{i},y_{i})\in {\mathcal {X}}\times {\mathcal {Y}}} , i = 1 , … , n {\displaystyle i=1,\dots ,n} from a sample space X {\displaystyle {\mathcal {X}}} and label space Y {\displaystyle {\mathcal {Y}}} , the structured SVM minimizes the following regularized risk function. min w ‖ w ‖ 2 + C ∑ i = 1 n max y ∈ Y ( 0 , Δ ( y i , y ) + ⟨ w , Ψ ( x i , y ) ⟩ − ⟨ w , Ψ ( x i , y i ) ⟩ ) {\displaystyle {\underset {\boldsymbol {w}}{\min }}\quad \|{\boldsymbol {w}}\|^{2}+C\sum _{i=1}^{n}{\underset {y\in {\mathcal {Y}}}{\max }}\left(0,\Delta (y_{i},y)+\langle {\boldsymbol {w}},\Psi ({\boldsymbol {x}}_{i},y)\rangle -\langle {\boldsymbol {w}},\Psi ({\boldsymbol {x}}_{i},y_{i})\rangle \right)} The function is convex in w {\displaystyle {\boldsymbol {w}}} because the maximum of a set of affine functions is convex. The function Δ : Y × Y → R + {\displaystyle \Delta :{\mathcal {Y}}\times {\mathcal {Y}}\to \mathbb {R} _{+}} measures a distance in label space and is an arbitrary function (not necessarily a metric) satisfying Δ ( y , z ) ≥ 0 {\displaystyle \Delta (y,z)\geq 0} and Δ ( y , y ) = 0 ∀ y , z ∈ Y {\displaystyle \Delta (y,y)=0\;\;\forall y,z\in {\mathcal {Y}}} . The function Ψ : X × Y → R d {\displaystyle \Psi :{\mathcal {X}}\times {\mathcal {Y}}\to \mathbb {R} ^{d}} is a feature function, extracting some feature vector from a given sample and label. The design of this function depends very much on the application. Because the regularized risk function above is non-differentiable, it is often reformulated in terms of a quadratic program by introducing one slack variable ξ i {\displaystyle \xi _{i}} for each sample, each representing the value of the maximum. The standard structured SVM primal formulation is given as follows. min w , ξ ‖ w ‖ 2 + C ∑ i = 1 n ξ i s.t. ⟨ w , Ψ ( x i , y i ) ⟩ − ⟨ w , Ψ ( x i , y ) ⟩ + ξ i ≥ Δ ( y i , y ) , i = 1 , … , n , ∀ y ∈ Y {\displaystyle {\begin{array}{cl}{\underset {{\boldsymbol {w}},{\boldsymbol {\xi }}}{\min }}&\|{\boldsymbol {w}}\|^{2}+C\sum _{i=1}^{n}\xi _{i}\\{\textrm {s.t.}}&\langle {\boldsymbol {w}},\Psi ({\boldsymbol {x}}_{i},y_{i})\rangle -\langle {\boldsymbol {w}},\Psi ({\boldsymbol {x}}_{i},y)\rangle +\xi _{i}\geq \Delta (y_{i},y),\qquad i=1,\dots ,n,\quad \forall y\in {\mathcal {Y}}\end{array}}} == Inference == At test time, only a sample x ∈ X {\displaystyle {\boldsymbol {x}}\in {\mathcal {X}}} is known, and a prediction function f : X → Y {\displaystyle f:{\mathcal {X}}\to {\mathcal {Y}}} maps it to a predicted label from the label space Y {\displaystyle {\mathcal {Y}}} . For structured SVMs, given the vector w {\displaystyle {\boldsymbol {w}}} obtained from training, the prediction function is the following. f ( x ) = argmax y ∈ Y ⟨ w , Ψ ( x , y ) ⟩ {\displaystyle f({\boldsymbol {x}})={\underset {y\in {\mathcal {Y}}}{\textrm {argmax}}}\quad \langle {\boldsymbol {w}},\Psi ({\boldsymbol {x}},y)\rangle } Therefore, the maximizer over the label space is the predicted label. Solving for this maximizer is the so-called inference problem and similar to making a maximum a-posteriori (MAP) prediction in probabilistic models. Depending on the structure of the function Ψ {\displaystyle \Psi } , solving for the maximizer can be a hard problem. == Separation == The above quadratic program involves a very large, possibly infinite number of linear inequality constraints. In general, the number of inequalities is too large to be optimized over explicitly. Instead the problem is solved by using delayed constraint generation where only a finite and small subset of the constraints is used. Optimizing over a subset of the constraints enlarges the feasible set and will yield a solution that provides a lower bound on the objective. To test whether the solution w {\displaystyle {\boldsymbol {w}}} violates constraints of the complete set inequalities, a separation problem needs to be solved. As the inequalities decompose over the samples, for each sample ( x i , y i ) {\displaystyle ({\boldsymbol {x}}_{i},y_{i})} the following problem needs to be solved. y n ∗ = argmax y ∈ Y ( Δ ( y i , y ) + ⟨ w , Ψ ( x i , y ) ⟩ − ⟨ w , Ψ ( x i , y i ) ⟩ − ξ i ) {\displaystyle y_{n}^{}={\underset {y\in {\mathcal {Y}}}{\textrm {argmax}}}\left(\Delta (y_{i},y)+\langle {\boldsymbol {w}},\Psi ({\boldsymbol {x}}_{i},y)\rangle -\langle {\boldsymbol {w}},\Psi ({\boldsymbol {x}}_{i},y_{i})\rangle -\xi _{i}\right)} The right hand side objective to be maximized is composed of the constant − ⟨ w , Ψ ( x i , y i ) ⟩ − ξ i {\displaystyle -\langle {\boldsymbol {w}},\Psi ({\boldsymbol {x}}_{i},y_{i})\rangle -\xi _{i}} and a term dependent on the variables optimized over, namely Δ ( y i , y ) + ⟨ w , Ψ ( x i , y ) ⟩ {\displaystyle \Delta (y_{i},y)+\langle {\boldsymbol {w}},\Psi ({\boldsymbol {x}}_{i},y)\rangle } . If the achieved right hand side objective is smaller or equal to zero, no violated constraints for this sample exist. If it is strictly larger than zero, the most violated constraint with respect to this sample has been identified. The problem is enlarged by this constraint and resolved. The process continues until no violated inequalities can be identified. If the constants are dropped from the above problem, we obtain the following problem to be solved. y i ∗ = argmax y ∈ Y ( Δ ( y i , y ) + ⟨ w , Ψ ( x i , y ) ⟩ ) {\displaystyle y_{i}^{}={\underset {y\in {\mathcal {Y}}}{\textrm {argmax}}}\left(\Delta (y_{i},y)+\langle {\boldsymbol {w}},\Psi ({\boldsymbol {x}}_{i},y)\rangle \right)} This problem looks very similar to the inference problem. The only difference is the addition of the term Δ ( y i , y ) {\displaystyle \Delta (y_{i},y)} . Most often, it is chosen such that it has a natural decomposition in label space. In that case, the influence of Δ {\displaystyle \Delta } can be encoded into the inference problem and solving for the most violating constraint is equivalent to solving the inference problem.