AI For Students App

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Lazy learning

(Not to be confused with the lazy learning regime, see Neural tangent kernel). In machine learning, lazy learning is a learning method in which generalization of the training data is, in theory, delayed until a query is made to the system, as opposed to eager learning, where the system tries to generalize the training data before receiving queries. The primary motivation for employing lazy learning, as in the K-nearest neighbors algorithm, used by online recommendation systems ("people who viewed/purchased/listened to this movie/item/tune also ...") is that the data set is continuously updated with new entries (e.g., new items for sale at Amazon, new movies to view at Netflix, new clips at YouTube, new music at Spotify or Pandora). Because of the continuous update, the "training data" would be rendered obsolete in a relatively short time especially in areas like books and movies, where new best-sellers or hit movies/music are published/released continuously. Therefore, one cannot really talk of a "training phase". Lazy classifiers are most useful for large, continuously changing datasets with few attributes that are commonly queried. Specifically, even if a large set of attributes exist - for example, books have a year of publication, author/s, publisher, title, edition, ISBN, selling price, etc. - recommendation queries rely on far fewer attributes - e.g., purchase or viewing co-occurrence data, and user ratings of items purchased/viewed. == Advantages == The main advantage gained in employing a lazy learning method is that the target function will be approximated locally, such as in the k-nearest neighbor algorithm. Because the target function is approximated locally for each query to the system, lazy learning systems can simultaneously solve multiple problems and deal successfully with changes in the problem domain. At the same time they can reuse a lot of theoretical and applied results from linear regression modelling (notably PRESS statistic) and control. It is said that the advantage of this system is achieved if the predictions using a single training set are only developed for few objects. This can be demonstrated in the case of the k-NN technique, which is instance-based and function is only estimated locally. == Disadvantages == Theoretical disadvantages with lazy learning include: The large space requirement to store the entire training dataset. In practice, this is not an issue because of advances in hardware and the relatively small number of attributes (e.g., as co-occurrence frequency) that need to be stored. Particularly noisy training data increases the case base unnecessarily, because no abstraction is made during the training phase. In practice, as stated earlier, lazy learning is applied to situations where any learning performed in advance soon becomes obsolete because of changes in the data. Also, for the problems for which lazy learning is optimal, "noisy" data does not really occur - the purchaser of a book has either bought another book or hasn't. Lazy learning methods are usually slower to evaluate. In practice, for very large databases with high concurrency loads, the queries are not postponed until actual query time, but recomputed in advance on a periodic basis - e.g., nightly, in anticipation of future queries, and the answers stored. This way, the next time new queries are asked about existing entries in the database, the answers are merely looked up rapidly instead of having to be computed on the fly, which would almost certainly bring a high-concurrency multi-user system to its knees. Larger training data also entail increased cost. Particularly, there is the fixed amount of computational cost, where a processor can only process a limited amount of training data points. There are standard techniques to improve re-computation efficiency so that a particular answer is not recomputed unless the data that impact this answer has changed (e.g., new items, new purchases, new views). In other words, the stored answers are updated incrementally. This approach, used by large e-commerce or media sites, has long been used in the Entrez portal of the National Center for Biotechnology Information (NCBI) to precompute similarities between the different items in its large datasets: biological sequences, 3-D protein structures, published-article abstracts, etc. Because "find similar" queries are asked so frequently, the NCBI uses highly parallel hardware to perform nightly recomputation. The recomputation is performed only for new entries in the datasets against each other and against existing entries: the similarity between two existing entries need not be recomputed. == Examples of Lazy Learning Methods == K-nearest neighbors, which is a special case of instance-based learning. Local regression. Lazy naive Bayes rules, which are extensively used in commercial spam detection software. Here, the spammers keep getting smarter and revising their spamming strategies, and therefore the learning rules must also be continually updated.
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Sum of absolute transformed differences

The sum of absolute transformed differences (SATD) is a block matching criterion widely used in fractional motion estimation for video compression. It works by taking a frequency transform, usually a Hadamard transform, of the differences between the pixels in the original block and the corresponding pixels in the block being used for comparison. The transform itself is often of a small block rather than the entire macroblock. For example, in x264, a series of 4×4 blocks are transformed rather than doing the more processor-intensive 16×16 transform. == Comparison to other metrics == SATD is slower than the sum of absolute differences (SAD), both due to its increased complexity and the fact that SAD-specific MMX and SSE2 instructions exist, while there are no such instructions for SATD. However, SATD can still be optimized considerably with SIMD instructions on most modern CPUs. The benefit of SATD is that it more accurately models the number of bits required to transmit the residual error signal. As such, it is often used in video compressors, either as a way to drive and estimate rate explicitly, such as in the Theora encoder (since 1.1 alpha2), as an optional metric used in wide motion searches, such as in the Microsoft VC-1 encoder, or as a metric used in sub-pixel refinement, such as in x264.
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Jubatus

Jubatus is an open-source online machine learning and distributed computing framework developed at Nippon Telegraph and Telephone and Preferred Infrastructure. Its features include classification, recommendation, regression, anomaly detection and graph mining. It supports many client languages, including C++, Java, Ruby and Python. It uses Iterative Parameter Mixture for distributed machine learning. == Notable Features == Jubatus supports: Multi-classification algorithms: Perceptron Passive Aggressive Confidence Weighted Adaptive Regularization of Weight Vectors Normal Herd Recommendation algorithms using: Inverted index Minhash Locality-sensitive hashing Regression algorithms: Passive Aggressive feature extraction method for natural language: n-gram Text segmentation
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Mean squared error

In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between the estimated values and the true value. MSE is a risk function, corresponding to the expected value of the squared error loss. The fact that MSE is almost always strictly positive (and not zero) is because of randomness or because the estimator does not account for information that could produce a more accurate estimate. In machine learning, specifically empirical risk minimization, MSE may refer to the empirical risk (the average loss on an observed data set), as an estimate of the true MSE (the true risk: the average loss on the actual population distribution). The MSE is a measure of the quality of an estimator. As it is derived from the square of Euclidean distance, it is always a positive value that decreases as the error approaches zero. The MSE is the second moment (about the origin) of the error, and thus incorporates both the variance of the estimator (how widely spread the estimates are from one data sample to another) and its bias (how far off the average estimated value is from the true value). For an unbiased estimator, the MSE is the variance of the estimator. Like the variance, MSE has the same units of measurement as the square of the quantity being estimated. In an analogy to standard deviation, taking the square root of MSE yields the root-mean-square error or root-mean-square deviation (RMSE or RMSD), which has the same units as the quantity being estimated; for an unbiased estimator, the RMSE is the square root of the variance, known as the standard error. == Definition and basic properties == The MSE either assesses the quality of a predictor (i.e., a function mapping arbitrary inputs to a sample of values of some random variable), or of an estimator (i.e., a mathematical function mapping a sample of data to an estimate of a parameter of the population from which the data is sampled). In the context of prediction, understanding the prediction interval can also be useful as it provides a range within which a future observation will fall, with a certain probability. The definition of an MSE differs according to whether one is describing a predictor or an estimator. === Predictor === If a vector of n {\displaystyle n} predictions is generated from a sample of n {\displaystyle n} data points on all variables, and Y {\displaystyle Y} is the vector of observed values of the variable being predicted, with Y ^ {\displaystyle {\hat {Y}}} being the predicted values (e.g. as from a least-squares fit), then the within-sample MSE of the predictor is computed as MSE = 1 n ∑ i = 1 n ( Y i − Y i ^ ) 2 {\displaystyle \operatorname {MSE} ={\frac {1}{n}}\sum _{i=1}^{n}\left(Y_{i}-{\hat {Y_{i}}}\right)^{2}} In other words, the MSE is the mean ( 1 n ∑ i = 1 n ) {\textstyle \left({\frac {1}{n}}\sum _{i=1}^{n}\right)} of the squares of the errors ( Y i − Y i ^ ) 2 {\textstyle \left(Y_{i}-{\hat {Y_{i}}}\right)^{2}} . This is an easily computable quantity for a particular sample (and hence is sample-dependent). In matrix notation, MSE = 1 n ∑ i = 1 n ( e i ) 2 = 1 n e T e {\displaystyle \operatorname {MSE} ={\frac {1}{n}}\sum _{i=1}^{n}(e_{i})^{2}={\frac {1}{n}}\mathbf {e} ^{\mathsf {T}}\mathbf {e} } where e i {\displaystyle e_{i}} is Y i − Y i ^ {\displaystyle Y_{i}-{\hat {Y_{i}}}} and e {\displaystyle \mathbf {e} } is a n × 1 {\displaystyle n\times 1} column vector. The MSE can also be computed on q data points that were not used in estimating the model, either because they were held back for this purpose, or because these data have been newly obtained. Within this process, known as cross-validation, the MSE is often called the test MSE, and is computed as MSE = 1 q ∑ i = n + 1 n + q ( Y i − Y i ^ ) 2 {\displaystyle \operatorname {MSE} ={\frac {1}{q}}\sum _{i=n+1}^{n+q}\left(Y_{i}-{\hat {Y_{i}}}\right)^{2}} === Estimator === The MSE of an estimator θ ^ {\displaystyle {\hat {\theta }}} with respect to an unknown parameter θ {\displaystyle \theta } is defined as MSE ⁡ ( θ ^ ) = E θ ⁡ [ ( θ ^ − θ ) 2 ] . {\displaystyle \operatorname {MSE} ({\hat {\theta }})=\operatorname {E} _{\theta }\left[({\hat {\theta }}-\theta )^{2}\right].} This definition depends on the unknown parameter, therefore the MSE is a priori property of an estimator. The MSE could be a function of unknown parameters, in which case any estimator of the MSE based on estimates of these parameters would be a function of the data (and thus a random variable). If the estimator θ ^ {\displaystyle {\hat {\theta }}} is derived as a sample statistic and is used to estimate some population parameter, then the expectation is with respect to the sampling distribution of the sample statistic. The MSE can be written as the sum of the variance of the estimator and the squared bias of the estimator, providing a useful way to calculate the MSE and implying that in the case of unbiased estimators, the MSE and variance are equivalent. MSE ⁡ ( θ ^ ) = Var θ ⁡ ( θ ^ ) + Bias ⁡ ( θ ^ , θ ) 2 . {\displaystyle \operatorname {MSE} ({\hat {\theta }})=\operatorname {Var} _{\theta }({\hat {\theta }})+\operatorname {Bias} ({\hat {\theta }},\theta )^{2}.} ==== Proof of variance and bias relationship ==== MSE ⁡ ( θ ^ ) = E θ ⁡ [ ( θ ^ − θ ) 2 ] = E θ ⁡ [ ( θ ^ − E θ ⁡ [ θ ^ ] + E θ ⁡ [ θ ^ ] − θ ) 2 ] = E θ ⁡ [ ( θ ^ − E θ ⁡ [ θ ^ ] ) 2 + 2 ( θ ^ − E θ ⁡ [ θ ^ ] ) ( E θ ⁡ [ θ ^ ] − θ ) + ( E θ ⁡ [ θ ^ ] − θ ) 2 ] = E θ ⁡ [ ( θ ^ − E θ ⁡ [ θ ^ ] ) 2 ] + E θ ⁡ [ 2 ( θ ^ − E θ ⁡ [ θ ^ ] ) ( E θ ⁡ [ θ ^ ] − θ ) ] + E θ ⁡ [ ( E θ ⁡ [ θ ^ ] − θ ) 2 ] = E θ ⁡ [ ( θ ^ − E θ ⁡ [ θ ^ ] ) 2 ] + 2 ( E θ ⁡ [ θ ^ ] − θ ) E θ ⁡ [ θ ^ − E θ ⁡ [ θ ^ ] ] + ( E θ ⁡ [ θ ^ ] − θ ) 2 E θ ⁡ [ θ ^ ] − θ = constant = E θ ⁡ [ ( θ ^ − E θ ⁡ [ θ ^ ] ) 2 ] + 2 ( E θ ⁡ [ θ ^ ] − θ ) ( E θ ⁡ [ θ ^ ] − E θ ⁡ [ θ ^ ] ) + ( E θ ⁡ [ θ ^ ] − θ ) 2 E θ ⁡ [ θ ^ ] = constant = E θ ⁡ [ ( θ ^ − E θ ⁡ [ θ ^ ] ) 2 ] + ( E θ ⁡ [ θ ^ ] − θ ) 2 = Var θ ⁡ ( θ ^ ) + Bias θ ⁡ ( θ ^ , θ ) 2 {\displaystyle {\begin{aligned}\operatorname {MSE} ({\hat {\theta }})&=\operatorname {E} _{\theta }\left[({\hat {\theta }}-\theta )^{2}\right]\\&=\operatorname {E} _{\theta }\left[\left({\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]+\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)^{2}\right]\\&=\operatorname {E} _{\theta }\left[\left({\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]\right)^{2}+2\left({\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]\right)\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)+\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)^{2}\right]\\&=\operatorname {E} _{\theta }\left[\left({\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]\right)^{2}\right]+\operatorname {E} _{\theta }\left[2\left({\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]\right)\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)\right]+\operatorname {E} _{\theta }\left[\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)^{2}\right]\\&=\operatorname {E} _{\theta }\left[\left({\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]\right)^{2}\right]+2\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)\operatorname {E} _{\theta }\left[{\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]\right]+\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)^{2}&&\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta ={\text{constant}}\\&=\operatorname {E} _{\theta }\left[\left({\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]\right)^{2}\right]+2\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\operatorname {E} _{\theta }[{\hat {\theta }}]\right)+\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)^{2}&&\operatorname {E} _{\theta }[{\hat {\theta }}]={\text{constant}}\\&=\operatorname {E} _{\theta }\left[\left({\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]\right)^{2}\right]+\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)^{2}\\&=\operatorname {Var} _{\theta }({\hat {\theta }})+\operatorname {Bias} _{\theta }({\hat {\theta }},\theta )^{2}\end{aligned}}} An even shorter proof can be achieved using the well-known formula that for a random variable X {\textstyle X} , E ( X 2 ) = Var ⁡ ( X ) + ( E ( X ) ) 2 {\textstyle \mathbb {E} (X^{2})=\operatorname {Var} (X)+(\mathbb {E} (X))^{2}} . By substituting X {\textstyle X} with, θ ^ − θ {\textstyle {\hat {\theta }}-\theta } , we have MSE ⁡ ( θ ^ ) = E [ ( θ ^ − θ ) 2 ] = Var ⁡ ( θ ^ − θ ) + ( E [ θ ^ − θ ] ) 2 = Var ⁡ ( θ ^ ) + Bias 2 ⁡ ( θ ^ , θ ) {\displaystyle {\begin{aligned}\operatorname {MSE} ({\hat {\theta }})&=\mathbb {E} [({\hat {\theta }}-\theta )^{2}]\\&=\operator
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Human–robot collaboration

Human-Robot Collaboration is the study of collaborative processes in human and robot agents work together to achieve shared goals. Many new applications for robots require them to work alongside people as capable members of human-robot teams. These include robots for homes, hospitals, and offices, space exploration and manufacturing. Human-Robot Collaboration (HRC) is an interdisciplinary research area comprising classical robotics, human-computer interaction, artificial intelligence, process design, layout planning, ergonomics, cognitive sciences, and psychology. Industrial applications of human-robot collaboration involve Collaborative Robots, or cobots, that physically interact with humans in a shared workspace to complete tasks such as collaborative manipulation or object handovers. == Collaborative Activity == Collaboration is defined as a special type of coordinated activity, one in which two or more agents work jointly with each other, together performing a task or carrying out the activities needed to satisfy a shared goal. The process typically involves shared plans, shared norms and mutually beneficial interactions. Although collaboration and cooperation are often used interchangeably, collaboration differs from cooperation as it involves a shared goal and joint action where the success of both parties depend on each other. For effective human-robot collaboration, it is imperative that the robot is capable of understanding and interpreting several communication mechanisms similar to the mechanisms involved in human-human interaction. The robot must also communicate its own set of intents and goals to establish and maintain a set of shared beliefs and to coordinate its actions to execute the shared plan. In addition, all team members demonstrate commitment to doing their own part, to the others doing theirs, and to the success of the overall task. == Theories Informing Human-Robot Collaboration == Human-human collaborative activities are studied in depth in order to identify the characteristics that enable humans to successfully work together. These activity models usually aim to understand how people work together in teams, how they form intentions and achieve a joint goal. Theories on collaboration inform human-robot collaboration research to develop efficient and fluent collaborative agents. === Belief Desire Intention Model === The belief-desire-intention (BDI) model is a model of human practical reasoning that was originally developed by Michael Bratman. The approach is used in intelligent agents research to describe and model intelligent agents. The BDI model is characterized by the implementation of an agent's beliefs (the knowledge of the world, state of the world), desires (the objective to accomplish, desired end state) and intentions (the course of actions currently under execution to achieve the desire of the agent) in order to deliberate their decision-making processes. BDI agents are able to deliberate about plans, select plans and execute plans. === Shared Cooperative Activity === Shared Cooperative Activity defines certain prerequisites for an activity to be considered shared and cooperative: mutual responsiveness, commitment to the joint activity and commitment to mutual support. An example case to illustrate these concepts would be a collaborative activity where agents are moving a table out the door, mutual responsiveness ensures that movements of the agents are synchronized; a commitment to the joint activity reassures each team member that the other will not at some point drop his side; and a commitment to mutual support deals with possible breakdowns due to one team member's inability to perform part of the plan. === Joint Intention Theory === Joint Intention Theory proposes that for joint action to emerge, team members must communicate to maintain a set of shared beliefs and to coordinate their actions towards the shared plan. In collaborative work, agents should be able to count on the commitment of other members, therefore each agent should inform the others when they reach the conclusion that a goal is achievable, impossible, or irrelevant. == Approaches to Human-Robot Collaboration == The approaches to human-robot collaboration include human emulation (HE) and human complementary (HC) approaches. Although these approaches have differences, there are research efforts to develop a unified approach stemming from potential convergences such as Collaborative Control. === Human Emulation === The human emulation approach aims to enable computers to act like humans or have human-like abilities in order to collaborate with humans. It focuses on developing formal models of human-human collaboration and applying these models to human-computer collaboration. In this approach, humans are viewed as rational agents who form and execute plans for achieving their goals and infer other people's plans. Agents are required to infer the goals and plans of other agents, and collaborative behavior consists of helping other agents to achieve their goals. === Human Complementary === The human complementary approach seeks to improve human-computer interaction by making the computer a more intelligent partner that complements and collaborates with humans. The premise is that the computer and humans have fundamentally asymmetric abilities. Therefore, researchers invent interaction paradigms that divide responsibility between human users and computer systems by assigning distinct roles that exploit the strengths and overcome the weaknesses of both partners. == Key Aspects == Specialization of Roles: Based on the level of autonomy and intervention, there are several human-robot relationships including master-slave, supervisor–subordinate, partner–partner, teacher–learner and fully autonomous robot. In addition to these roles, homotopy (a weighting function that allows a continuous change between leader and follower behaviors) was introduced as a flexible role distribution. Establishing shared goal(s): Through direct discussion about goals or inference from statements and actions, agents must determine the shared goals they are trying to achieve. Allocation of Responsibility and Coordination: Agents must decide how to achieve their goals, determine what actions will be done by each agent, and how to coordinate the actions of individual agents and integrate their results. Shared context: Agents must be able to track progress toward their goals. They must keep track of what has been achieved and what remains to be done. They must evaluate the effects of actions and determine whether an acceptable solution has been achieved. Communication: Any collaboration requires communication to define goals, negotiate over how to proceed and who will do what, and evaluate progress and results. Adaptation and learning: Collaboration over time require partners to adapt themselves to each other and learn from one's partner both directly or indirectly. Time and space: The time-space taxonomy divides human-robot interaction into four categories based on whether the humans and robots are using computing systems at the same time (synchronous) or different times (asynchronous) and while in the same place (collocated) or in different places (non-collocated). Ergonomics: Human factors and ergonomics are one of the key aspects for a sustainable human-robot collaboration. The robot control system can use biomechanical models and sensors to optimize various ergonomic metrics, such as muscle fatigue.
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Transkribus

Transkribus is a platform for the text recognition, image analysis and structure recognition of historical documents. The platform was created in the context of the two EU projects "tranScriptorium" (2013–2015) and "READ" (Recognition and Enrichment of Archival Documents – 2016–2019). It was developed by the University of Innsbruck. Since July 1, 2019 the platform has been directed and further developed by the READ-COOP, a non-profit cooperative. The platform integrates tools developed by research groups throughout Europe, including the Pattern Recognition and Human Language Technology (PRHLT) group of the Technical University of Valencia and the Computational Intelligence Technology Lab (CITlab) group of University of Rostock. Comparable programs that offer similar functions are eScriptorium and OCR4All.
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Error tolerance (PAC learning)

In PAC learning, error tolerance refers to the ability of an algorithm to learn when the examples received have been corrupted in some way. In fact, this is a very common and important issue since in many applications it is not possible to access noise-free data. Noise can interfere with the learning process at different levels: the algorithm may receive data that have been occasionally mislabeled, or the inputs may have some false information, or the classification of the examples may have been maliciously adulterated. == Notation and the Valiant learning model == In the following, let X {\displaystyle X} be our n {\displaystyle n} -dimensional input space. Let H {\displaystyle {\mathcal {H}}} be a class of functions that we wish to use in order to learn a { 0 , 1 } {\displaystyle \{0,1\}} -valued target function f {\displaystyle f} defined over X {\displaystyle X} . Let D {\displaystyle {\mathcal {D}}} be the distribution of the inputs over X {\displaystyle X} . The goal of a learning algorithm A {\displaystyle {\mathcal {A}}} is to choose the best function h ∈ H {\displaystyle h\in {\mathcal {H}}} such that it minimizes e r r o r ( h ) = P x ∼ D ( h ( x ) ≠ f ( x ) ) {\displaystyle error(h)=P_{x\sim {\mathcal {D}}}(h(x)\neq f(x))} . Let us suppose we have a function s i z e ( f ) {\displaystyle size(f)} that can measure the complexity of f {\displaystyle f} . Let Oracle ( x ) {\displaystyle {\text{Oracle}}(x)} be an oracle that, whenever called, returns an example x {\displaystyle x} and its correct label f ( x ) {\displaystyle f(x)} . When no noise corrupts the data, we can define learning in the Valiant setting: Definition: We say that f {\displaystyle f} is efficiently learnable using H {\displaystyle {\mathcal {H}}} in the Valiant setting if there exists a learning algorithm A {\displaystyle {\mathcal {A}}} that has access to Oracle ( x ) {\displaystyle {\text{Oracle}}(x)} and a polynomial p ( ⋅ , ⋅ , ⋅ , ⋅ ) {\displaystyle p(\cdot ,\cdot ,\cdot ,\cdot )} such that for any 0 < ε ≤ 1 {\displaystyle 0<\varepsilon \leq 1} and 0 < δ ≤ 1 {\displaystyle 0<\delta \leq 1} it outputs, in a number of calls to the oracle bounded by p ( 1 ε , 1 δ , n , size ( f ) ) {\displaystyle p\left({\frac {1}{\varepsilon }},{\frac {1}{\delta }},n,{\text{size}}(f)\right)} , a function h ∈ H {\displaystyle h\in {\mathcal {H}}} that satisfies with probability at least 1 − δ {\displaystyle 1-\delta } the condition error ( h ) ≤ ε {\displaystyle {\text{error}}(h)\leq \varepsilon } . In the following we will define learnability of f {\displaystyle f} when data have suffered some modification. == Classification noise == In the classification noise model a noise rate 0 ≤ η < 1 2 {\displaystyle 0\leq \eta <{\frac {1}{2}}} is introduced. Then, instead of Oracle ( x ) {\displaystyle {\text{Oracle}}(x)} that returns always the correct label of example x {\displaystyle x} , algorithm A {\displaystyle {\mathcal {A}}} can only call a faulty oracle Oracle ( x , η ) {\displaystyle {\text{Oracle}}(x,\eta )} that will flip the label of x {\displaystyle x} with probability η {\displaystyle \eta } . As in the Valiant case, the goal of a learning algorithm A {\displaystyle {\mathcal {A}}} is to choose the best function h ∈ H {\displaystyle h\in {\mathcal {H}}} such that it minimizes e r r o r ( h ) = P x ∼ D ( h ( x ) ≠ f ( x ) ) {\displaystyle error(h)=P_{x\sim {\mathcal {D}}}(h(x)\neq f(x))} . In applications it is difficult to have access to the real value of η {\displaystyle \eta } , but we assume we have access to its upperbound η B {\displaystyle \eta _{B}} . Note that if we allow the noise rate to be 1 / 2 {\displaystyle 1/2} , then learning becomes impossible in any amount of computation time, because every label conveys no information about the target function. Definition: We say that f {\displaystyle f} is efficiently learnable using H {\displaystyle {\mathcal {H}}} in the classification noise model if there exists a learning algorithm A {\displaystyle {\mathcal {A}}} that has access to Oracle ( x , η ) {\displaystyle {\text{Oracle}}(x,\eta )} and a polynomial p ( ⋅ , ⋅ , ⋅ , ⋅ ) {\displaystyle p(\cdot ,\cdot ,\cdot ,\cdot )} such that for any 0 ≤ η ≤ 1 2 {\displaystyle 0\leq \eta \leq {\frac {1}{2}}} , 0 ≤ ε ≤ 1 {\displaystyle 0\leq \varepsilon \leq 1} and 0 ≤ δ ≤ 1 {\displaystyle 0\leq \delta \leq 1} it outputs, in a number of calls to the oracle bounded by p ( 1 1 − 2 η B , 1 ε , 1 δ , n , s i z e ( f ) ) {\displaystyle p\left({\frac {1}{1-2\eta _{B}}},{\frac {1}{\varepsilon }},{\frac {1}{\delta }},n,size(f)\right)} , a function h ∈ H {\displaystyle h\in {\mathcal {H}}} that satisfies with probability at least 1 − δ {\displaystyle 1-\delta } the condition e r r o r ( h ) ≤ ε {\displaystyle error(h)\leq \varepsilon } . == Statistical query learning == Statistical Query Learning is a kind of active learning problem in which the learning algorithm A {\displaystyle {\mathcal {A}}} can decide if to request information about the likelihood P f ( x ) {\displaystyle P_{f(x)}} that a function f {\displaystyle f} correctly labels example x {\displaystyle x} , and receives an answer accurate within a tolerance α {\displaystyle \alpha } . Formally, whenever the learning algorithm A {\displaystyle {\mathcal {A}}} calls the oracle Oracle ( x , α ) {\displaystyle {\text{Oracle}}(x,\alpha )} , it receives as feedback probability Q f ( x ) {\displaystyle Q_{f(x)}} , such that Q f ( x ) − α ≤ P f ( x ) ≤ Q f ( x ) + α {\displaystyle Q_{f(x)}-\alpha \leq P_{f(x)}\leq Q_{f(x)}+\alpha } . Definition: We say that f {\displaystyle f} is efficiently learnable using H {\displaystyle {\mathcal {H}}} in the statistical query learning model if there exists a learning algorithm A {\displaystyle {\mathcal {A}}} that has access to Oracle ( x , α ) {\displaystyle {\text{Oracle}}(x,\alpha )} and polynomials p ( ⋅ , ⋅ , ⋅ ) {\displaystyle p(\cdot ,\cdot ,\cdot )} , q ( ⋅ , ⋅ , ⋅ ) {\displaystyle q(\cdot ,\cdot ,\cdot )} , and r ( ⋅ , ⋅ , ⋅ ) {\displaystyle r(\cdot ,\cdot ,\cdot )} such that for any 0 < ε ≤ 1 {\displaystyle 0<\varepsilon \leq 1} the following hold: Oracle ( x , α ) {\displaystyle {\text{Oracle}}(x,\alpha )} can evaluate P f ( x ) {\displaystyle P_{f(x)}} in time q ( 1 ε , n , s i z e ( f ) ) {\displaystyle q\left({\frac {1}{\varepsilon }},n,size(f)\right)} ; 1 α {\displaystyle {\frac {1}{\alpha }}} is bounded by r ( 1 ε , n , s i z e ( f ) ) {\displaystyle r\left({\frac {1}{\varepsilon }},n,size(f)\right)} A {\displaystyle {\mathcal {A}}} outputs a model h {\displaystyle h} such that e r r ( h ) < ε {\displaystyle err(h)<\varepsilon } , in a number of calls to the oracle bounded by p ( 1 ε , n , s i z e ( f ) ) {\displaystyle p\left({\frac {1}{\varepsilon }},n,size(f)\right)} . Note that the confidence parameter δ {\displaystyle \delta } does not appear in the definition of learning. This is because the main purpose of δ {\displaystyle \delta } is to allow the learning algorithm a small probability of failure due to an unrepresentative sample. Since now Oracle ( x , α ) {\displaystyle {\text{Oracle}}(x,\alpha )} always guarantees to meet the approximation criterion Q f ( x ) − α ≤ P f ( x ) ≤ Q f ( x ) + α {\displaystyle Q_{f(x)}-\alpha \leq P_{f(x)}\leq Q_{f(x)}+\alpha } , the failure probability is no longer needed. The statistical query model is strictly weaker than the PAC model: any efficiently SQ-learnable class is efficiently PAC learnable in the presence of classification noise, but there exist efficient PAC-learnable problems such as parity that are not efficiently SQ-learnable. == Malicious classification == In the malicious classification model an adversary generates errors to foil the learning algorithm. This setting describes situations of error burst, which may occur when for a limited time transmission equipment malfunctions repeatedly. Formally, algorithm A {\displaystyle {\mathcal {A}}} calls an oracle Oracle ( x , β ) {\displaystyle {\text{Oracle}}(x,\beta )} that returns a correctly labeled example x {\displaystyle x} drawn, as usual, from distribution D {\displaystyle {\mathcal {D}}} over the input space with probability 1 − β {\displaystyle 1-\beta } , but it returns with probability β {\displaystyle \beta } an example drawn from a distribution that is not related to D {\displaystyle {\mathcal {D}}} . Moreover, this maliciously chosen example may strategically selected by an adversary who has knowledge of f {\displaystyle f} , β {\displaystyle \beta } , D {\displaystyle {\mathcal {D}}} , or the current progress of the learning algorithm. Definition: Given a bound β B < 1 2 {\displaystyle \beta _{B}<{\frac {1}{2}}} for 0 ≤ β < 1 2 {\displaystyle 0\leq \beta <{\frac {1}{2}}} , we say that f {\displaystyle f} is efficiently learnable using H {\displaystyle {\mathcal {H}}} in the malicious classification model, if there exist a learning algorithm A {\displaystyle {\mathcal {A}}} that has access to Oracle ( x , β ) {\displaystyle {\text{Oracle}}(x,\beta )}
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Canonical correspondence analysis

In multivariate analysis, canonical correspondence analysis (CCA) is an ordination technique that determines axes from the response data as a unimodal combination of measured predictors. CCA is commonly used in ecology in order to extract gradients that drive the composition of ecological communities. CCA extends correspondence analysis (CA) with regression, in order to incorporate predictor variables. == History == CCA was developed in 1986 by Cajo ter Braak and implemented in the program CANOCO, an extension of DECORANA. To date, CCA is one of the most popular multivariate methods in ecology, despite the availability of contemporary alternatives. CCA was originally derived and implemented using an algorithm of weighted averaging, though Legendre & Legendre (1998) derived an alternative algorithm. == Assumptions == The requirements of a CCA are that the samples are random and independent. Also, the data are categorical and that the independent variables are consistent within the sample site and error-free. The original publication states the need for equal species tolerances, equal species maxima, and equispaced or uniformly distributed species optima and site scores.
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Universal psychometrics

Universal psychometrics encompasses psychometrics instruments that could measure the psychological properties of any intelligent agent. Up until the early 21st century, psychometrics relied heavily on psychological tests that require the subject to cooperate and answer questions, the most famous example being an intelligence test. Such methods are only applicable to the measurement of human psychological properties. As a result, some researchers have proposed the idea of universal psychometrics - they suggest developing testing methods that allow for the measurement of non-human entities' psychological properties. For example, it has been suggested that the Turing test is a form of universal psychometrics. This test involves having testers (without any foreknowledge) attempt to distinguish a human from a machine by interacting with both (while not being to see either individuals). It is supposed that if the machine is equally intelligent to a human, the testers will not be able to distinguish between the two, i.e., their guesses will not be better than chance. Thus, Turing test could measure the intelligence (a psychological variable) of an AI. Other instruments proposed for universal psychometrics include reinforcement learning and measuring the ability to predict complexity.
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BrownBoost

BrownBoost is a boosting algorithm that may be robust to noisy datasets. BrownBoost is an adaptive version of the boost by majority algorithm. As is the case for all boosting algorithms, BrownBoost is used in conjunction with other machine learning methods. BrownBoost was introduced by Yoav Freund in 2001. == Motivation == AdaBoost performs well on a variety of datasets; however, it can be shown that AdaBoost does not perform well on noisy data sets. This is a result of AdaBoost's focus on examples that are repeatedly misclassified. In contrast, BrownBoost effectively "gives up" on examples that are repeatedly misclassified. The core assumption of BrownBoost is that noisy examples will be repeatedly mislabeled by the weak hypotheses and non-noisy examples will be correctly labeled frequently enough to not be "given up on." Thus only noisy examples will be "given up on," whereas non-noisy examples will contribute to the final classifier. In turn, if the final classifier is learned from the non-noisy examples, the generalization error of the final classifier may be much better than if learned from noisy and non-noisy examples. The user of the algorithm can set the amount of error to be tolerated in the training set. Thus, if the training set is noisy (say 10% of all examples are assumed to be mislabeled), the booster can be told to accept a 10% error rate. Since the noisy examples may be ignored, only the true examples will contribute to the learning process. == Algorithm description == BrownBoost uses a non-convex potential loss function, thus it does not fit into the AdaBoost framework. The non-convex optimization provides a method to avoid overfitting noisy data sets. However, in contrast to boosting algorithms that analytically minimize a convex loss function (e.g. AdaBoost and LogitBoost), BrownBoost solves a system of two equations and two unknowns using standard numerical methods. The only parameter of BrownBoost ( c {\displaystyle c} in the algorithm) is the "time" the algorithm runs. The theory of BrownBoost states that each hypothesis takes a variable amount of time ( t {\displaystyle t} in the algorithm) which is directly related to the weight given to the hypothesis α {\displaystyle \alpha } . The time parameter in BrownBoost is analogous to the number of iterations T {\displaystyle T} in AdaBoost. A larger value of c {\displaystyle c} means that BrownBoost will treat the data as if it were less noisy and therefore will give up on fewer examples. Conversely, a smaller value of c {\displaystyle c} means that BrownBoost will treat the data as more noisy and give up on more examples. During each iteration of the algorithm, a hypothesis is selected with some advantage over random guessing. The weight of this hypothesis α {\displaystyle \alpha } and the "amount of time passed" t {\displaystyle t} during the iteration are simultaneously solved in a system of two non-linear equations ( 1. uncorrelated hypothesis w.r.t example weights and 2. hold the potential constant) with two unknowns (weight of hypothesis α {\displaystyle \alpha } and time passed t {\displaystyle t} ). This can be solved by bisection (as implemented in the JBoost software package) or Newton's method (as described in the original paper by Freund). Once these equations are solved, the margins of each example ( r i ( x j ) {\displaystyle r_{i}(x_{j})} in the algorithm) and the amount of time remaining s {\displaystyle s} are updated appropriately. This process is repeated until there is no time remaining. The initial potential is defined to be 1 m ∑ j = 1 m 1 − erf ( c ) = 1 − erf ( c ) {\displaystyle {\frac {1}{m}}\sum _{j=1}^{m}1-{\mbox{erf}}({\sqrt {c}})=1-{\mbox{erf}}({\sqrt {c}})} . Since a constraint of each iteration is that the potential be held constant, the final potential is 1 m ∑ j = 1 m 1 − erf ( r i ( x j ) / c ) = 1 − erf ( c ) {\displaystyle {\frac {1}{m}}\sum _{j=1}^{m}1-{\mbox{erf}}(r_{i}(x_{j})/{\sqrt {c}})=1-{\mbox{erf}}({\sqrt {c}})} . Thus the final error is likely to be near 1 − erf ( c ) {\displaystyle 1-{\mbox{erf}}({\sqrt {c}})} . However, the final potential function is not the 0–1 loss error function. For the final error to be exactly 1 − erf ( c ) {\displaystyle 1-{\mbox{erf}}({\sqrt {c}})} , the variance of the loss function must decrease linearly w.r.t. time to form the 0–1 loss function at the end of boosting iterations. This is not yet discussed in the literature and is not in the definition of the algorithm below. The final classifier is a linear combination of weak hypotheses and is evaluated in the same manner as most other boosting algorithms. == BrownBoost learning algorithm definition == Input: m {\displaystyle m} training examples ( x 1 , y 1 ) , … , ( x m , y m ) {\displaystyle (x_{1},y_{1}),\ldots ,(x_{m},y_{m})} where x j ∈ X , y j ∈ Y = { − 1 , + 1 } {\displaystyle x_{j}\in X,\,y_{j}\in Y=\{-1,+1\}} The parameter c {\displaystyle c} Initialise: s = c {\displaystyle s=c} . (The value of s {\displaystyle s} is the amount of time remaining in the game) r i ( x j ) = 0 {\displaystyle r_{i}(x_{j})=0} ∀ j {\displaystyle \forall j} . The value of r i ( x j ) {\displaystyle r_{i}(x_{j})} is the margin at iteration i {\displaystyle i} for example x j {\displaystyle x_{j}} . While s > 0 {\displaystyle s>0} : Set the weights of each example: W i ( x j ) = e − ( r i ( x j ) + s ) 2 c {\displaystyle W_{i}(x_{j})=e^{-{\frac {(r_{i}(x_{j})+s)^{2}}{c}}}} , where r i ( x j ) {\displaystyle r_{i}(x_{j})} is the margin of example x j {\displaystyle x_{j}} Find a classifier h i : X → { − 1 , + 1 } {\displaystyle h_{i}:X\to \{-1,+1\}} such that ∑ j W i ( x j ) h i ( x j ) y j > 0 {\displaystyle \sum _{j}W_{i}(x_{j})h_{i}(x_{j})y_{j}>0} Find values α , t {\displaystyle \alpha ,t} that satisfy the equation: ∑ j h i ( x j ) y j e − ( r i ( x j ) + α h i ( x j ) y j + s − t ) 2 c = 0 {\displaystyle \sum _{j}h_{i}(x_{j})y_{j}e^{-{\frac {(r_{i}(x_{j})+\alpha h_{i}(x_{j})y_{j}+s-t)^{2}}{c}}}=0} . (Note this is similar to the condition E W i + 1 [ h i ( x j ) y j ] = 0 {\displaystyle E_{W_{i+1}}[h_{i}(x_{j})y_{j}]=0} set forth by Schapire and Singer. In this setting, we are numerically finding the W i + 1 = exp ⁡ ( ⋯ ⋯ ) {\displaystyle W_{i+1}=\exp \left({\frac {\cdots }{\cdots }}\right)} such that E W i + 1 [ h i ( x j ) y j ] = 0 {\displaystyle E_{W_{i+1}}[h_{i}(x_{j})y_{j}]=0} .) This update is subject to the constraint ∑ ( Φ ( r i ( x j ) + α h ( x j ) y j + s − t ) − Φ ( r i ( x j ) + s ) ) = 0 {\displaystyle \sum \left(\Phi \left(r_{i}(x_{j})+\alpha h(x_{j})y_{j}+s-t\right)-\Phi \left(r_{i}(x_{j})+s\right)\right)=0} , where Φ ( z ) = 1 − erf ( z / c ) {\displaystyle \Phi (z)=1-{\mbox{erf}}(z/{\sqrt {c}})} is the potential loss for a point with margin r i ( x j ) {\displaystyle r_{i}(x_{j})} Update the margins for each example: r i + 1 ( x j ) = r i ( x j ) + α h ( x j ) y j {\displaystyle r_{i+1}(x_{j})=r_{i}(x_{j})+\alpha h(x_{j})y_{j}} Update the time remaining: s = s − t {\displaystyle s=s-t} Output: H ( x ) = sign ( ∑ i α i h i ( x ) ) {\displaystyle H(x)={\textrm {sign}}\left(\sum _{i}\alpha _{i}h_{i}(x)\right)} == Empirical results == In preliminary experimental results with noisy datasets, BrownBoost outperformed AdaBoost's generalization error; however, LogitBoost performed as well as BrownBoost. An implementation of BrownBoost can be found in the open source software JBoost.
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Evolutionary programming

Evolutionary programming is an evolutionary algorithm, where a share of new population is created by mutation of previous population without crossover. Evolutionary programming differs from evolution strategy ES( μ + λ {\displaystyle \mu +\lambda } ) in one detail. All individuals are selected for the new population, while in ES( μ + λ {\displaystyle \mu +\lambda } ), every individual has the same probability to be selected. It is one of the four major evolutionary algorithm paradigms. == History == It was first used by Lawrence J. Fogel in the US in 1960 in order to use simulated evolution as a learning process aiming to generate artificial intelligence. It was used to evolve finite-state machines as predictors.
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Cultural algorithm

Cultural algorithms (CA) are a branch of evolutionary computation where there is a knowledge component that is called the belief space in addition to the population component. In this sense, cultural algorithms can be seen as an extension to a conventional genetic algorithm. Cultural algorithms were introduced by Reynolds (see references). == Belief space == The belief space of a cultural algorithm is divided into distinct categories. These categories represent different domains of knowledge that the population has of the search space. The belief space is updated after each iteration by the best individuals of the population. The best individuals can be selected using a fitness function that assesses the performance of each individual in population much like in genetic algorithms. === List of belief space categories === Normative knowledge A collection of desirable value ranges for the individuals in the population component e.g. acceptable behavior for the agents in population. Domain specific knowledge Information about the domain of the cultural algorithm problem is applied to. Situational knowledge Specific examples of important events - e.g. successful/unsuccessful solutions Temporal knowledge History of the search space - e.g. the temporal patterns of the search process Spatial knowledge Information about the topography of the search space == Population == The population component of the cultural algorithm is approximately the same as that of the genetic algorithm. == Communication protocol == Cultural algorithms require an interface between the population and belief space. The best individuals of the population can update the belief space via the update function. Also, the knowledge categories of the belief space can affect the population component via the influence function. The influence function can affect population by altering the genome or the actions of the individuals. == Pseudocode for cultural algorithms == Initialize population space (choose initial population) Initialize belief space (e.g. set domain specific knowledge and normative value-ranges) Repeat until termination condition is met Perform actions of the individuals in population space Evaluate each individual by using the fitness function Select the parents to reproduce a new generation of offspring Let the belief space alter the genome of the offspring by using the influence function Update the belief space by using the accept function (this is done by letting the best individuals to affect the belief space) == Applications == Various optimization problems Social simulation Real-parameter optimization
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NCover

NCover is a .NET code coverage tool. There are two non-related NCover products that do .NET code coverage. There is an open source NCover that can be found on SourceForge and there is a company called NCover, LLC. There has been additional development on both products since this 2004 reference. The company NCover, LLC began when the founder, Peter Waldschmidt, decided to commercialize the open source tool he created. The commercial versions were launched in 2007, but the last supported free version 1.5.8 is still available on the company site.
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Diffusion model

In machine learning, diffusion models, also known as diffusion-based generative models or score-based generative models, are a class of latent variable generative models. A diffusion model consists of two major components: the forward diffusion process, and the reverse sampling process. The goal of diffusion models is to learn a diffusion process for a given dataset, such that the process can generate new elements that are distributed similarly as the original dataset. A diffusion model models data as generated by a diffusion process, whereby a new datum performs a random walk with drift through the space of all possible data. A trained diffusion model can be sampled in many ways, with different efficiency and quality. There are various equivalent formalisms, including Markov chains, denoising diffusion probabilistic models, noise conditioned score networks, and stochastic differential equations. They are typically trained using variational inference. The model responsible for denoising is typically called its "backbone". The backbone may be of any kind, but they are typically U-nets or transformers. As of 2024, diffusion models are mainly used for computer vision tasks, including image denoising, inpainting, super-resolution, image generation, and video generation. These typically involve training a neural network to sequentially denoise images blurred with Gaussian noise. The model is trained to reverse the process of adding noise to an image. After training to convergence, it can be used for image generation by starting with an image composed of random noise, and applying the network iteratively to denoise the image. Diffusion-based image generators have seen widespread commercial interest, such as Stable Diffusion and DALL-E. These models typically combine diffusion models with other models, such as text-encoders and cross-attention modules to allow text-conditioned generation. Other than computer vision, diffusion models have also found applications in natural language processing such as text generation and summarization, sound generation, and reinforcement learning. == Denoising diffusion model == === Non-equilibrium thermodynamics === Diffusion models were introduced in 2015 as a method to train a model that can sample from a highly complex probability distribution. They used techniques from non-equilibrium thermodynamics, especially diffusion. Consider, for example, how one might model the distribution of all naturally occurring photos. Each image is a point in the space of all images, and the distribution of naturally occurring photos is a "cloud" in space, which, by repeatedly adding noise to the images, diffuses out to the rest of the image space, until the cloud becomes all but indistinguishable from a Gaussian distribution N ( 0 , I ) {\displaystyle {\mathcal {N}}(0,I)} . A model that can approximately undo the diffusion can then be used to sample from the original distribution. This is studied in "non-equilibrium" thermodynamics, as the starting distribution is not in equilibrium, unlike the final distribution. The equilibrium distribution is the Gaussian distribution N ( 0 , I ) {\displaystyle {\mathcal {N}}(0,I)} , with pdf ρ ( x ) ∝ e − 1 2 ‖ x ‖ 2 {\displaystyle \rho (x)\propto e^{-{\frac {1}{2}}\|x\|^{2}}} . This is just the Maxwell–Boltzmann distribution of particles in a potential well V ( x ) = 1 2 ‖ x ‖ 2 {\displaystyle V(x)={\frac {1}{2}}\|x\|^{2}} at temperature 1. The initial distribution, being very much out of equilibrium, would diffuse towards the equilibrium distribution, making biased random steps that are a sum of pure randomness (like a Brownian walker) and gradient descent down the potential well. The randomness is necessary: if the particles were to undergo only gradient descent, then they will all fall to the origin, collapsing the distribution. === Denoising Diffusion Probabilistic Model (DDPM) === The 2020 paper proposed the Denoising Diffusion Probabilistic Model (DDPM), which improves upon the previous method by variational inference. ==== Forward diffusion ==== To present the model, some notation is required. β 1 , . . . , β T ∈ ( 0 , 1 ) {\displaystyle \beta _{1},...,\beta _{T}\in (0,1)} are fixed constants. α t := 1 − β t {\displaystyle \alpha _{t}:=1-\beta _{t}} α ¯ t := α 1 ⋯ α t {\displaystyle {\bar {\alpha }}_{t}:=\alpha _{1}\cdots \alpha _{t}} σ t := 1 − α ¯ t {\displaystyle \sigma _{t}:={\sqrt {1-{\bar {\alpha }}_{t}}}} σ ~ t := σ t − 1 σ t β t {\displaystyle {\tilde {\sigma }}_{t}:={\frac {\sigma _{t-1}}{\sigma _{t}}}{\sqrt {\beta _{t}}}} μ ~ t ( x t , x 0 ) := α t ( 1 − α ¯ t − 1 ) x t + α ¯ t − 1 ( 1 − α t ) x 0 σ t 2 {\displaystyle {\tilde {\mu }}_{t}(x_{t},x_{0}):={\frac {{\sqrt {\alpha _{t}}}(1-{\bar {\alpha }}_{t-1})x_{t}+{\sqrt {{\bar {\alpha }}_{t-1}}}(1-\alpha _{t})x_{0}}{\sigma _{t}^{2}}}} N ( μ , Σ ) {\displaystyle {\mathcal {N}}(\mu ,\Sigma )} is the normal distribution with mean μ {\displaystyle \mu } and variance Σ {\displaystyle \Sigma } , and N ( x | μ , Σ ) {\displaystyle {\mathcal {N}}(x|\mu ,\Sigma )} is the probability density at x {\displaystyle x} . A vertical bar denotes conditioning. A forward diffusion process starts at some starting point x 0 ∼ q {\displaystyle x_{0}\sim q} , where q {\displaystyle q} is the probability distribution to be learned, then repeatedly adds noise to it by x t = 1 − β t x t − 1 + β t z t {\displaystyle x_{t}={\sqrt {1-\beta _{t}}}x_{t-1}+{\sqrt {\beta _{t}}}z_{t}} where z 1 , . . . , z T {\displaystyle z_{1},...,z_{T}} are IID (Independent and identically distributed random variables) samples from N ( 0 , I ) {\displaystyle {\mathcal {N}}(0,I)} . The coefficients 1 − β t {\displaystyle {\sqrt {1-\beta _{t}}}} and β t {\displaystyle {\sqrt {\beta _{t}}}} ensure that Var ( X t ) = I {\displaystyle {\mbox{Var}}(X_{t})=I} assuming that Var ( X 0 ) = I {\displaystyle {\mbox{Var}}(X_{0})=I} . The values of β t {\displaystyle \beta _{t}} are chosen such that for any starting distribution of x 0 {\displaystyle x_{0}} , if it has finite second moment, then lim t → ∞ x t | x 0 {\displaystyle \lim _{t\to \infty }x_{t}|x_{0}} converges to N ( 0 , I ) {\displaystyle {\mathcal {N}}(0,I)} . The entire diffusion process then satisfies q ( x 0 : T ) = q ( x 0 ) q ( x 1 | x 0 ) ⋯ q ( x T | x T − 1 ) = q ( x 0 ) N ( x 1 | α 1 x 0 , β 1 I ) ⋯ N ( x T | α T x T − 1 , β T I ) {\displaystyle q(x_{0:T})=q(x_{0})q(x_{1}|x_{0})\cdots q(x_{T}|x_{T-1})=q(x_{0}){\mathcal {N}}(x_{1}|{\sqrt {\alpha _{1}}}x_{0},\beta _{1}I)\cdots {\mathcal {N}}(x_{T}|{\sqrt {\alpha _{T}}}x_{T-1},\beta _{T}I)} or ln ⁡ q ( x 0 : T ) = ln ⁡ q ( x 0 ) − ∑ t = 1 T 1 2 β t ‖ x t − 1 − β t x t − 1 ‖ 2 + C {\displaystyle \ln q(x_{0:T})=\ln q(x_{0})-\sum _{t=1}^{T}{\frac {1}{2\beta _{t}}}\|x_{t}-{\sqrt {1-\beta _{t}}}x_{t-1}\|^{2}+C} where C {\displaystyle C} is a normalization constant and often omitted. In particular, we note that x 1 : T | x 0 {\displaystyle x_{1:T}|x_{0}} is a Gaussian process, which affords us considerable freedom in reparameterization. For example, by standard manipulation with Gaussian process, x t | x 0 ∼ N ( α ¯ t x 0 , σ t 2 I ) {\displaystyle x_{t}|x_{0}\sim N\left({\sqrt {{\bar {\alpha }}_{t}}}x_{0},\sigma _{t}^{2}I\right)} x t − 1 | x t , x 0 ∼ N ( μ ~ t ( x t , x 0 ) , σ ~ t 2 I ) {\displaystyle x_{t-1}|x_{t},x_{0}\sim {\mathcal {N}}({\tilde {\mu }}_{t}(x_{t},x_{0}),{\tilde {\sigma }}_{t}^{2}I)} In particular, notice that for large t {\displaystyle t} , the variable x t | x 0 ∼ N ( α ¯ t x 0 , σ t 2 I ) {\displaystyle x_{t}|x_{0}\sim N\left({\sqrt {{\bar {\alpha }}_{t}}}x_{0},\sigma _{t}^{2}I\right)} converges to N ( 0 , I ) {\displaystyle {\mathcal {N}}(0,I)} . That is, after a long enough diffusion process, we end up with some x T {\displaystyle x_{T}} that is very close to N ( 0 , I ) {\displaystyle {\mathcal {N}}(0,I)} , with all traces of the original x 0 ∼ q {\displaystyle x_{0}\sim q} gone. For example, since x t | x 0 ∼ N ( α ¯ t x 0 , σ t 2 I ) {\displaystyle x_{t}|x_{0}\sim N\left({\sqrt {{\bar {\alpha }}_{t}}}x_{0},\sigma _{t}^{2}I\right)} we can sample x t | x 0 {\displaystyle x_{t}|x_{0}} directly "in one step", instead of going through all the intermediate steps x 1 , x 2 , . . . , x t − 1 {\displaystyle x_{1},x_{2},...,x_{t-1}} . ==== Backward diffusion ==== The key idea of DDPM is to use a neural network parametrized by θ {\displaystyle \theta } . The network takes in two arguments x t , t {\displaystyle x_{t},t} , and outputs a vector μ θ ( x t , t ) {\displaystyle \mu _{\theta }(x_{t},t)} and a matrix Σ θ ( x t , t ) {\displaystyle \Sigma _{\theta }(x_{t},t)} , such that each step in the forward diffusion process can be approximately undone by x t − 1 ∼ N ( μ θ ( x t , t ) , Σ θ ( x t , t ) ) {\displaystyle x_{t-1}\sim {\mathcal {N}}(\mu _{\theta }(x_{t},t),\Sigma _{\theta }(x_{t},t))} . This then gives us a backward diffusion process p θ {\displaystyle p_{\theta }} defined by p θ ( x T ) = N ( x T | 0 , I ) {\displaystyle p_{\theta }(x
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Hinge loss

In machine learning, the hinge loss is a loss function used for training classifiers. The hinge loss is used for "maximum-margin" classification, most notably for support vector machines (SVMs). For an intended output t = ±1 and a classifier score y, the hinge loss of the prediction y is defined as ℓ ( y ) = max ( 0 , 1 − t ⋅ y ) {\displaystyle \ell (y)=\max(0,1-t\cdot y)} Note that y {\displaystyle y} should be the "raw" output of the classifier's decision function, not the predicted class label. For instance, in linear SVMs, y = w ⋅ x + b {\displaystyle y=\mathbf {w} \cdot \mathbf {x} +b} , where ( w , b ) {\displaystyle (\mathbf {w} ,b)} are the parameters of the hyperplane and x {\displaystyle \mathbf {x} } is the input variable(s). When t and y have the same sign (meaning y predicts the right class) and | y | ≥ 1 {\displaystyle |y|\geq 1} , the hinge loss ℓ ( y ) = 0 {\displaystyle \ell (y)=0} . When they have opposite signs, ℓ ( y ) {\displaystyle \ell (y)} increases linearly with y, and similarly if | y | < 1 {\displaystyle |y|<1} , even if it has the same sign (correct prediction, but not by enough margin). The Hinge loss is not a proper scoring rule. == Extensions == While binary SVMs are commonly extended to multiclass classification in a one-vs.-all or one-vs.-one fashion, it is also possible to extend the hinge loss itself for such an end. Several different variations of multiclass hinge loss have been proposed. For example, Crammer and Singer defined it for a linear classifier as ℓ ( y ) = max ( 0 , 1 + max y ≠ t w y x − w t x ) {\displaystyle \ell (y)=\max(0,1+\max _{y\neq t}\mathbf {w} _{y}\mathbf {x} -\mathbf {w} _{t}\mathbf {x} )} , where t {\displaystyle t} is the target label, w t {\displaystyle \mathbf {w} _{t}} and w y {\displaystyle \mathbf {w} _{y}} are the model parameters. Weston and Watkins provided a similar definition, but with a sum rather than a max: ℓ ( y ) = ∑ y ≠ t max ( 0 , 1 + w y x − w t x ) {\displaystyle \ell (y)=\sum _{y\neq t}\max(0,1+\mathbf {w} _{y}\mathbf {x} -\mathbf {w} _{t}\mathbf {x} )} . In structured prediction, the hinge loss can be further extended to structured output spaces. Structured SVMs with margin rescaling use the following variant, where w denotes the SVM's parameters, y the SVM's predictions, φ the joint feature function, and Δ the Hamming loss: ℓ ( y ) = max ( 0 , Δ ( y , t ) + ⟨ w , ϕ ( x , y ) ⟩ − ⟨ w , ϕ ( x , t ) ⟩ ) = max ( 0 , max y ∈ Y ( Δ ( y , t ) + ⟨ w , ϕ ( x , y ) ⟩ ) − ⟨ w , ϕ ( x , t ) ⟩ ) {\displaystyle {\begin{aligned}\ell (\mathbf {y} )&=\max(0,\Delta (\mathbf {y} ,\mathbf {t} )+\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {y} )\rangle -\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {t} )\rangle )\\&=\max(0,\max _{y\in {\mathcal {Y}}}\left(\Delta (\mathbf {y} ,\mathbf {t} )+\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {y} )\rangle \right)-\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {t} )\rangle )\end{aligned}}} . == Optimization == The hinge loss is a convex function, so many of the usual convex optimizers used in machine learning can work with it. It is not differentiable, but has a subgradient with respect to model parameters w of a linear SVM with score function y = w ⋅ x {\displaystyle y=\mathbf {w} \cdot \mathbf {x} } that is given by ∂ ℓ ∂ w i = { − t ⋅ x i if t ⋅ y < 1 , 0 otherwise . {\displaystyle {\frac {\partial \ell }{\partial w_{i}}}={\begin{cases}-t\cdot x_{i}&{\text{if }}t\cdot y<1,\\0&{\text{otherwise}}.\end{cases}}} However, since the derivative of the hinge loss at t y = 1 {\displaystyle ty=1} is undefined, smoothed versions may be preferred for optimization, such as Rennie and Srebro's ℓ ( y ) = { 1 2 − t y if t y ≤ 0 , 1 2 ( 1 − t y ) 2 if 0 < t y < 1 , 0 if 1 ≤ t y {\displaystyle \ell (y)={\begin{cases}{\frac {1}{2}}-ty&{\text{if}}~~ty\leq 0,\\{\frac {1}{2}}(1-ty)^{2}&{\text{if}}~~0 Read more →