AI Generator Character

AI Generator Character — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

• 

  TikTok

  TikTok is a social media and short-form online video platform. It hosts user-submitted videos, which range in duration from three seconds to 60 minutes. It can be accessed through a mobile app or through its website. Since its launch, TikTok has become one of the world's most popular social media platforms, using recommendation algorithms to connect content creators and influencers with new audiences. In April 2020, TikTok surpassed two billion mobile downloads worldwide. The popularity of TikTok has allowed viral trends in food, fashion, and music to take off and increase the platform's cultural impact worldwide. TikTok has come under scrutiny due to data privacy violations, mental health concerns, misinformation, offensive content, addictive algorithm, its role during the Gaza war, and, following its 2026 divestiture in the U.S., alleged censorship of criticism of Donald Trump and discussions of Jeffrey Epstein. While TikTok remains accessible to users in most countries, a minority of countries (including India and Afghanistan) have implemented full or partial bans. Many other countries limit TikTok's use on government-issued devices for security or privacy reasons. == Corporate structure == TikTok Ltd was incorporated in the Cayman Islands in the Caribbean and is based in both Singapore and Los Angeles. It owns entities which are based respectively in Australia (which also runs the New Zealand business), United Kingdom (also owns subsidiaries in the European Union), and Singapore (owns operations in Southeast Asia and India). A spin-off company, TikTok USDS Joint Venture LLC was formed on 22 January 2026 to handle TikTok and other ByteDance properties in the United States, Oracle Corporation, MGX Fund Management Limited, Silver Lake each holding a 15% stake, ByteDance holds a 19.9% stake and the remaining 35.1% is shared between Dell Technologies founder Michael Dell and Vastmere Strategic Investments. Its parent company, Beijing-based ByteDance, is owned by founders and Chinese investors, other global investors, and employees. One of ByteDance's main domestic subsidiaries is owned by Chinese state funds and entities through a 1% golden share. Employees have reported that multiple overlaps exist between TikTok and ByteDance in terms of personnel management and product development. TikTok says that since 2020, its US-based CEO is responsible for making important decisions, and has downplayed its China connection. == History == === Douyin === Douyin (Chinese: 抖音; pinyin: Dǒuyīn; lit. 'Shaking Sound') was launched on 20 September 2016, by ByteDance, originally under the name A.me, before changing its name to Douyin in December 2016. Douyin was developed in nearly 7 months and within a year had 100 million users, with more than one billion videos viewed every day. While TikTok and Douyin share a similar user interface, the platforms operate separately. Douyin includes an in-video search feature that can search by people's faces for more videos of them, along with other features such as buying, booking hotels, and making geo-tagged reviews. === TikTok === ByteDance planned on Douyin expanding overseas. The founder of ByteDance, Zhang Yiming, stated that "China is home to only one-fifth of Internet users globally. If we don't expand on a global scale, we are bound to lose to peers eyeing the four-fifths. So, going global is a must." ByteDance created TikTok as an overseas version of Douyin. TikTok was launched in the international market in September 2017. On 9 November 2017, ByteDance spent nearly $1 billion to purchase Musical.ly, a startup headquartered in Shanghai with an overseas office in Santa Monica, California. Musical.ly was a social media video platform that allowed users to create short lip-sync and comedy videos, initially released in August 2014. TikTok merged with Musical.ly on 2 August 2018 with existing accounts and data consolidated into one app, keeping the title TikTok. On 23 January 2018, the TikTok app ranked first among free application downloads on app stores in Thailand and other countries. TikTok has been downloaded more than 130 million times in the United States and has reached 2 billion downloads worldwide, according to data from mobile research firm Sensor Tower (those numbers exclude Android users in China). In the United States, Jimmy Fallon, Tony Hawk, and other celebrities began using the app in 2018. Other celebrities like Jennifer Lopez, Jessica Alba, Will Smith, and Justin Bieber joined TikTok. In January 2019, TikTok allowed creators to embed merchandise sale links into their videos. On 3 September 2019, TikTok and the US National Football League (NFL) announced a multi-year partnership. The agreement came just two days before the NFL's 100th season kick-off at Soldier Field in Chicago where TikTok hosted activities for fans in honor of the deal. The partnership entails the launch of an official NFL TikTok account, which is to bring about new marketing opportunities such as sponsored videos and hashtag challenges. In July 2020, TikTok, excluding Douyin, reported close to 800 million monthly active users worldwide after less than four years of existence. In May 2021, TikTok appointed Shou Zi Chew as their new CEO who assumed the position from interim CEO Vanessa Pappas, following the resignation of Kevin A. Mayer on 27 August 2020. In September 2021, TikTok reported that it had reached 1 billion users. In 2021, TikTok earned $4 billion in advertising revenue. In October 2022, TikTok was reported to be planning an expansion into the e-commerce market in the US, following the launch of TikTok Shop in the United Kingdom. The company posted job listings for staff for a series of order fulfillment centers in the US and was reportedly planning to start the new live shopping business before the end of the year. The Financial Times reported that TikTok will launch a video gaming channel, but the report was denied in a statement to Digiday, with TikTok instead aiming to be a social hub for the gaming community. According to data from app analytics group Sensor Tower, advertising on TikTok in the US grew by 11% in March 2023, with companies including Pepsi, DoorDash, Amazon, and Apple among the top spenders. According to estimates from research group Insider Intelligence, TikTok is projected to generate $14.15 billion in revenue in 2023, up from $9.89 billion in 2022. In March 2024, The Wall Street Journal reported that TikTok's growth in the US had stagnated. ==== Plans to sell TikTok's US operations ==== Since at least 2020, following calls to ban TikTok in the country, the Committee on Foreign Investment in the United States (CFIUS) has been investigating the company's 2017 merger with Musical.ly but has not finalized any of its negotiations with TikTok, such as the Project Texas proposal, waiting instead for Congress to act. In January 2025, Chinese officials began preliminary talks about potentially selling TikTok's US operations to Elon Musk if the app faced an impending ban due to national security concerns. While Beijing preferred TikTok remain under ByteDance's control, the sale could happen through a competitive process or with US government involvement. One possibility involved Musk's platform, X, taking over TikTok's US business. The move came ahead of a Supreme Court case that upheld the constitutionality of a law that would force a sale or ban of TikTok in the US by 19 January 2025, due to national security concerns regarding its ties to China. Other potential buyers included Project Liberty's "The People's Bid For TikTok" consortium of Frank McCourt with Kevin O'Leary, Steven Mnuchin, MrBeast and Bobby Kotick, the seriousness of these potential buyers was unclear. The day before the impending ban, California-based conversational search engine company Perplexity AI submitted a bid for a merger with TikTok US. On 14 September 2025, the Wall Street Journal reported the US and China have reached the "framework of a deal" for the US operations of TikTok to be sold to a consortium of investors in the US including close Trump ally Larry Ellison of Oracle. The deal was completed by 22 January 2026, with a consortium of investors—including Oracle, Silver Lake, MGX, and others including the personal investment entity for Michael Dell—owning more than 80% of the new venture. ByteDance retained 19.9% ownership. Under the deal, the app would remain the same, and the algorithm would be adjusted over time to favor American topics for those users. === Expansion in other markets === TikTok was downloaded over 104 million times on Apple's App Store during the first half of 2018, according to data provided to CNBC by Sensor Tower. After merging with musical.ly in August, downloads increased and TikTok subsequently became the most downloaded app in the US in October 2018, which musical.ly had done once before. In February 2019, TikTok, together with Douyin, hit one billion downloads globally, excluding Android

  Read more →
• 

  Triplet loss

  Triplet loss is a machine learning loss function widely used in one-shot learning, a setting where models are trained to generalize effectively from limited examples. It was conceived by Google researchers for their prominent FaceNet algorithm for face detection. Triplet loss is designed to support metric learning. Namely, to assist training models to learn an embedding (mapping to a feature space) where similar data points are closer together and dissimilar ones are farther apart, enabling robust discrimination across varied conditions. In the context of face detection, data points correspond to images. == Definition == The loss function is defined using triplets of training points of the form ( A , P , N ) {\displaystyle (A,P,N)} . In each triplet, A {\displaystyle A} (called an "anchor point") denotes a reference point of a particular identity, P {\displaystyle P} (called a "positive point") denotes another point of the same identity in point A {\displaystyle A} , and N {\displaystyle N} (called a "negative point") denotes a point of an identity different from the identity in point A {\displaystyle A} and P {\displaystyle P} . Let x {\displaystyle x} be some point and let f ( x ) {\displaystyle f(x)} be the embedding of x {\displaystyle x} in the finite-dimensional Euclidean space. It shall be assumed that the L2-norm of f ( x ) {\displaystyle f(x)} is unity (the L2 norm of a vector X {\displaystyle X} in a finite dimensional Euclidean space is denoted by ‖ X ‖ {\displaystyle \Vert X\Vert } .) We assemble m {\displaystyle m} triplets of points from the training dataset. The goal of training here is to ensure that, after learning, the following condition (called the "triplet constraint") is satisfied by all triplets ( A ( i ) , P ( i ) , N ( i ) ) {\displaystyle (A^{(i)},P^{(i)},N^{(i)})} in the training data set: ‖ f ( A ( i ) ) − f ( P ( i ) ) ‖ 2 2 + α < ‖ f ( A ( i ) ) − f ( N ( i ) ) ‖ 2 2 {\displaystyle \Vert f(A^{(i)})-f(P^{(i)})\Vert _{2}^{2}+\alpha <\Vert f(A^{(i)})-f(N^{(i)})\Vert _{2}^{2}} The variable α {\displaystyle \alpha } is a hyperparameter called the margin, and its value must be set manually. In the FaceNet system, its value was set as 0.2. Thus, the full form of the function to be minimized is the following: L = ∑ i = 1 m max ( ‖ f ( A ( i ) ) − f ( P ( i ) ) ‖ 2 2 − ‖ f ( A ( i ) ) − f ( N ( i ) ) ‖ 2 2 + α , 0 ) {\displaystyle L=\sum _{i=1}^{m}\max {\Big (}\Vert f(A^{(i)})-f(P^{(i)})\Vert _{2}^{2}-\Vert f(A^{(i)})-f(N^{(i)})\Vert _{2}^{2}+\alpha ,0{\Big )}} == Intuition == A baseline for understanding the effectiveness of triplet loss is the contrastive loss, which operates on pairs of samples (rather than triplets). Training with the contrastive loss pulls embeddings of similar pairs closer together, and pushes dissimilar pairs apart. Its pairwise approach is greedy, as it considers each pair in isolation. Triplet loss innovates by considering relative distances. Its goal is that the embedding of an anchor (query) point be closer to positive points than to negative points (also accounting for the margin). It does not try to further optimize the distances once this requirement is met. This is approximated by simultaneously considering two pairs (anchor-positive and anchor-negative), rather than each pair in isolation. == Triplet "mining" == One crucial implementation detail when training with triplet loss is triplet "mining", which focuses on the smart selection of triplets for optimization. This process adds an additional layer of complexity compared to contrastive loss. A naive approach to preparing training data for the triplet loss involves randomly selecting triplets from the dataset. In general, the set of valid triplets of the form ( A ( i ) , P ( i ) , N ( i ) ) {\displaystyle (A^{(i)},P^{(i)},N^{(i)})} is very large. To speed-up training convergence, it is essential to focus on challenging triplets. In the FaceNet paper, several options were explored, eventually arriving at the following. For each anchor-positive pair, the algorithm considers only semi-hard negatives. These are negatives that violate the triplet requirement (i.e, are "hard"), but lie farther from the anchor than the positive (not too hard). Restated, for each A ( i ) {\displaystyle A^{(i)}} and P ( i ) {\displaystyle P^{(i)}} , they seek N ( i ) {\displaystyle N^{(i)}} such that: The rationale for this design choice is heuristic. It may appear puzzling that the mining process neglects "very hard" negatives (i.e., closer to the anchor than the positive). Experiments conducted by the FaceNet designers found that this often leads to a convergence to degenerate local minima. Triplet mining is performed at each training step, from within the sample points contained in the training batch (this is known as online mining), after embeddings were computed for all points in the batch. While ideally the entire dataset could be used, this is impractical in general. To support a large search space for triplets, the FaceNet authors used very large batches (1800 samples). Batches are constructed by selecting a large number of same-category sample points (40), and randomly selected negatives for them. == Extensions == Triplet loss has been extended to simultaneously maintain a series of distance orders by optimizing a continuous relevance degree with a chain (i.e., ladder) of distance inequalities. This leads to the Ladder Loss, which has been demonstrated to offer performance enhancements of visual-semantic embedding in learning to rank tasks. In Natural Language Processing, triplet loss is one of the loss functions considered for BERT fine-tuning in the SBERT architecture. Other extensions involve specifying multiple negatives (multiple negatives ranking loss).

  Read more →
• 

  Latent class model

  In statistics, a latent class model (LCM) is a model for clustering multivariate discrete data. It assumes that the data arise from a mixture of discrete distributions, within each of which the variables are independent. It is called a latent class model because the class to which each data point belongs is unobserved (or latent). Latent class analysis (LCA) is a subset of structural equation modeling used to find groups or subtypes of cases in multivariate categorical data. These groups or subtypes of cases are called "latent classes". When faced with the following situation, a researcher might opt to use LCA to better understand the data: Symptoms a, b, c, and d have been recorded in a variety of patients diagnosed with diseases X, Y, and Z. Disease X is associated with symptoms a, b, and c; disease Y is linked to symptoms b, c, and d; and disease Z is connected to symptoms a, c, and d. In this context, the LCA would attempt to detect the presence of latent classes (i.e., the disease entities), thus creating patterns of association in the symptoms. As in factor analysis, LCA can also be used to classify cases according to their maximum likelihood class membership probability. The key criterion for resolving the LCA is identifying latent classes in which the observed symptom associations are effectively rendered null. This is because within each class, the diseases responsible for the symptoms create a structure of dependencies. As a result, the symptoms become conditionally independent, meaning that, given the class a case belongs to, the symptoms are no longer related to one another. == Model == Within each latent class, the observed variables are statistically independent—an essential aspect of latent class modeling. Usually, the observed variables are statistically dependent. By introducing the latent variable, independence is restored in the sense that within classes, variables are independent (local independence). Therefore, the association between the observed variables is explained by the classes of the latent variable (McCutcheon, 1987). In one form, the LCM is written as p i 1 , i 2 , … , i N ≈ ∑ t T p t ∏ n N p i n , t n , {\displaystyle p_{i_{1},i_{2},\ldots ,i_{N}}\approx \sum _{t}^{T}p_{t}\,\prod _{n}^{N}p_{i_{n},t}^{n},} where T {\displaystyle T} is the number of latent classes and p t {\displaystyle p_{t}} are the so-called recruitment or unconditional probabilities that should sum to one. p i n , t n {\displaystyle p_{i_{n},t}^{n}} are the marginal or conditional probabilities. For a two-way latent class model, the form is p i j ≈ ∑ t T p t p i t p j t . {\displaystyle p_{ij}\approx \sum _{t}^{T}p_{t}\,p_{it}\,p_{jt}.} This two-way model is related to probabilistic latent semantic analysis and non-negative matrix factorization. The probability model used in LCA is closely related to the Naive Bayes classifier. The main difference is that in LCA, the class membership of an individual is a latent variable, whereas in Naive Bayes classifiers, the class membership is an observed label. == Related methods == There are a number of methods with distinct names and uses that share a common relationship. Cluster analysis is, like LCA, used to discover taxon-like groups of cases in data. Multivariate mixture estimation (MME) is applicable to continuous data and assumes that such data arise from a mixture of distributions, such as a set of heights arising from a mixture of men and women. If a multivariate mixture estimation is constrained so that measures must be uncorrelated within each distribution, it is termed latent profile analysis. Modified to handle discrete data, this constrained analysis is known as LCA. Discrete latent trait models further constrain the classes to form from segments of a single dimension, allocating members to classes based on that dimension. An example would be assigning cases to social classes based on ability or merit. In a practical instance, the variables could be multiple choice items of a political questionnaire. In this case, the data consists of an N-way contingency table with answers to the items for a number of respondents. In this example, the latent variable refers to political opinion, and the latent classes to political groups. Given group membership, the conditional probabilities specify the chance that certain answers are chosen. == Application == LCA may be used in many fields, such as: collaborative filtering, Behavior Genetics and Evaluation of diagnostic tests.

  Read more →
• 

  Semantic mapping (statistics)

  Semantic mapping (SM) is a statistical method for dimensionality reduction (the transformation of data from a high-dimensional space into a low-dimensional space). SM can be used in a set of multidimensional vectors of features to extract a few new features that preserves the main data characteristics. SM performs dimensionality reduction by clustering the original features in semantic clusters and combining features mapped in the same cluster to generate an extracted feature. Given a data set, this method constructs a projection matrix that can be used to map a data element from a high-dimensional space into a reduced dimensional space. SM can be applied in construction of text mining and information retrieval systems, as well as systems managing vectors of high dimensionality. SM is an alternative to random mapping, principal components analysis and latent semantic indexing methods.

  Read more →
• 

  DeepRoute.ai

  DeepRoute.ai (Chinese: 元戎启行) is a Chinese autonomous driving company founded in 2019 and headquartered in Shenzhen, China. The company develops full-stack self-driving solutions including perception, decision-making, and control systems. == History == DeepRoute.ai was founded in February 2019 in Shenzhen, China, by Zhou Guang (周光), who serves as the company's CEO. In September 2019, the company collaborated with Dongfeng for a live-streamed autonomous driving demonstration. In October 2019, during the 7th Military World Games, DeepRoute.ai conducted Robotaxi demonstration operations. In November 2019, it obtained an intelligent connected vehicle road test permit for public roads in Shenzhen. In October 2020, DeepRoute.ai signed an "Autonomous Driving Leadership Project" with Dongfeng to build one of China's largest autonomous fleets. In August 2020, DeepRoute.ai announced its partnership with Cao Cao Mobility, a Geely-backed ride-hailing company, to test Robotaxis in Hangzhou for daily operations, planning to provide Robotaxis during the 2022 Asian Games. In September 2021, DeepRoute.ai secured US$300 million in a Series B funding round led by Alibaba. In December 2021, the company unveiled its DeepRoute-Driver 2.0, an L4-level autonomous driving solution comprising five solid-state lidar sensors, eight cameras, a proprietary computing system and an optional millimeter-wave radar. with a production cost of under US$10,000. In June 2022, it partnered with Deppon Express to provide autonomous light truck freight transfer services. In March 2023, the company launched its high-precision map-free intelligent driving solution, DeepRoute-Driver 3.0. In November 2024, Great Wall Motor announced a $100 million Series C funding round for Deeproute. With this, Deeproute has completed five rounds of financing, raising a cumulative total of over $500 million. Its shareholders include Fosun RZ Capital, Yunqi Partners, Alibaba, Vision Plus Capital, and Dongfeng, among others. In the same month, Deeproute.ai emphasised that they were in "deep cooperation" with Nvidia and spoke on being part of the first batch of companies in China to get a hold of Nvidia's newer Thor chip for cars which will be used in a new system released next year. This new system will help manage more complex driving scenarios through visual cues. == Products == === VLA Model === VLA Model is a Vision–language–action model designed for autonomous driving systems. It integrates visual perception, semantic understanding, and action decision-making into a unified framework, aiming to enhance the safety and adaptability of advanced driver-assistance systems (ADAS) in complex road environments. The model was officially launched on August 26, 2025, as the core of DeepRoute.ai's DeepRoute IO 2.0 platform. The VLA model is characterized by its "visual-language-action" architecture, which incorporates a chain-of-thought (CoT) reasoning capability inspired by large language models. This design is intended to address the "black box" limitations of traditional end-to-end autonomous driving systems by enabling the model to analyze information, infer causality, and make decisions in a more transparent and interpretable manner. === Appliance === The company has partnered with several automakers including Dongfeng Motor Corporation and Geely to develop and test autonomous vehicles.

  Read more →
• 

  Quantum neural network

  Quantum neural networks are computational neural network models which are based on the principles of quantum mechanics. The first ideas on quantum neural computation were published independently in 1995 by Subhash Kak and Ron Chrisley, engaging with the theory of quantum mind, which posits that quantum effects play a role in cognitive function. However, typical research in quantum neural networks involves combining classical artificial neural network models (which are widely used in machine learning for the important task of pattern recognition) with the advantages of quantum information in order to develop more efficient algorithms. One important motivation for these investigations is the difficulty to train classical neural networks, especially in big data applications. The hope is that features of quantum computing such as quantum parallelism or the effects of interference and entanglement can be used as resources. Since the technological implementation of a quantum computer is still in a premature stage, such quantum neural network models are mostly theoretical proposals that await their full implementation in physical experiments. Most Quantum neural networks are developed as feed-forward networks. Similar to their classical counterparts, this structure intakes input from one layer of qubits, and passes that input onto another layer of qubits. This layer of qubits evaluates this information and passes on the output to the next layer. Eventually the path leads to the final layer of qubits. The layers do not have to be of the same width, meaning they don't have to have the same number of qubits as the layer before or after it. This structure is trained on which path to take similar to classical artificial neural networks. This is discussed in a lower section. Quantum neural networks refer to three different categories: Quantum computer with classical data, classical computer with quantum data, and quantum computer with quantum data. == Examples == Quantum neural network research is still in its infancy, and a conglomeration of proposals and ideas of varying scope and mathematical rigor have been put forward. Most of them are based on the idea of replacing classical binary or McCulloch-Pitts neurons with a qubit (which can be called a "quron"), resulting in neural units that can be in a superposition of the state 'firing' and 'resting'. === Quantum perceptrons === A lot of proposals attempt to find a quantum equivalent for the perceptron unit from which neural nets are constructed. A problem is that nonlinear activation functions do not immediately correspond to the mathematical structure of quantum theory, since a quantum evolution is described by linear operations and leads to probabilistic observation. Ideas to imitate the perceptron activation function with a quantum mechanical formalism reach from special measurements to postulating non-linear quantum operators (a mathematical framework that is disputed). A direct implementation of the activation function using the circuit-based model of quantum computation has recently been proposed by Schuld, Sinayskiy and Petruccione based on the quantum phase estimation algorithm. === Quantum networks === At a larger scale, researchers have attempted to generalize neural networks to the quantum setting. One way of constructing a quantum neuron is to first generalise classical neurons and then generalising them further to make unitary gates. Interactions between neurons can be controlled quantumly, with unitary gates, or classically, via measurement of the network states. This high-level theoretical technique can be applied broadly, by taking different types of networks and different implementations of quantum neurons, such as photonically implemented neurons and quantum reservoir processor (quantum version of reservoir computing). Most learning algorithms follow the classical model of training an artificial neural network to learn the input-output function of a given training set and use classical feedback loops to update parameters of the quantum system until they converge to an optimal configuration. Learning as a parameter optimisation problem has also been approached by adiabatic models of quantum computing. Quantum neural networks can be applied to algorithmic design: given qubits with tunable mutual interactions, one can attempt to learn interactions following the classical backpropagation rule from a training set of desired input-output relations, taken to be the desired output algorithm's behavior. The quantum network thus 'learns' an algorithm. === Quantum associative memory === The first quantum associative memory algorithm was introduced by Dan Ventura and Tony Martinez in 1999. The authors do not attempt to translate the structure of artificial neural network models into quantum theory, but propose an algorithm for a circuit-based quantum computer that simulates associative memory. The memory states (in Hopfield neural networks saved in the weights of the neural connections) are written into a superposition, and a Grover-like quantum search algorithm retrieves the memory state closest to a given input. As such, this is not a fully content-addressable memory, since only incomplete patterns can be retrieved. The first truly content-addressable quantum memory, which can retrieve patterns also from corrupted inputs, was proposed by Carlo A. Trugenberger. Both memories can store an exponential (in terms of n qubits) number of patterns but can be used only once due to the no-cloning theorem and their destruction upon measurement. Trugenberger, however, has shown that his probabilistic model of quantum associative memory can be efficiently implemented and re-used multiples times for any polynomial number of stored patterns, a large advantage with respect to classical associative memories. === Classical neural networks inspired by quantum theory === A substantial amount of interest has been given to a "quantum-inspired" model that uses ideas from quantum theory to implement a neural network based on fuzzy logic. == Training == Quantum Neural Networks can be theoretically trained similarly to training classical/artificial neural networks. A key difference lies in communication between the layers of a neural networks. For classical neural networks, at the end of a given operation, the current perceptron copies its output to the next layer of perceptron(s) in the network. However, in a quantum neural network, where each perceptron is a qubit, this would violate the no-cloning theorem. A proposed generalized solution to this is to replace the classical fan-out method with an arbitrary unitary that spreads out, but does not copy, the output of one qubit to the next layer of qubits. Using this fan-out Unitary ( U f {\displaystyle U_{f}} ) with a dummy state qubit in a known state (Ex. | 0 ⟩ {\displaystyle |0\rangle } in the computational basis), also known as an Ancilla bit, the information from the qubit can be transferred to the next layer of qubits. This process adheres to the quantum operation requirement of reversibility. Using this quantum feed-forward network, deep neural networks can be executed and trained efficiently. A deep neural network is essentially a network with many hidden-layers, as seen in the sample model neural network above. Since the Quantum neural network being discussed uses fan-out Unitary operators, and each operator only acts on its respective input, only two layers are used at any given time. In other words, no Unitary operator is acting on the entire network at any given time, meaning the number of qubits required for a given step depends on the number of inputs in a given layer. Since Quantum Computers are notorious for their ability to run multiple iterations in a short period of time, the efficiency of a quantum neural network is solely dependent on the number of qubits in any given layer, and not on the depth of the network. === Cost functions === To determine the effectiveness of a neural network, a cost function is used, which essentially measures the proximity of the network's output to the expected or desired output. In a Classical Neural Network, the weights ( w {\displaystyle w} ) and biases ( b {\displaystyle b} ) at each step determine the outcome of the cost function C ( w , b ) {\displaystyle C(w,b)} . When training a Classical Neural network, the weights and biases are adjusted after each iteration, and given equation 1 below, where y ( x ) {\displaystyle y(x)} is the desired output and a out ( x ) {\displaystyle a^{\text{out}}(x)} is the actual output, the cost function is optimized when C ( w , b ) {\displaystyle C(w,b)} = 0. For a quantum neural network, the cost function is determined by measuring the fidelity of the outcome state ( ρ out {\displaystyle \rho ^{\text{out}}} ) with the desired outcome state ( ϕ out {\displaystyle \phi ^{\text{out}}} ), seen in Equation 2 below. In this case, the Unitary operators are adjusted after each it

  Read more →
• 

  Random forest

  Random forests or random decision forests is an ensemble learning method for classification, regression and other tasks that works by creating a multitude of decision trees during training. For classification tasks, the output of the random forest is the class selected by most trees. For regression tasks, the output is the average of the predictions of the trees. Random forests correct for decision trees' habit of overfitting to their training set. The first algorithm for random decision forests was created in 1995 by Tin Kam Ho using the random subspace method, which, in Ho's formulation, is a way to implement the "stochastic discrimination" approach to classification proposed by Eugene Kleinberg. An extension of the algorithm was developed by Leo Breiman and Adele Cutler, who registered "Random Forests" as a trademark in 2006 (as of 2019, owned by Minitab, Inc.). The extension combines Breiman's "bagging" idea and random selection of features, introduced first by Ho and later independently by Amit and Geman in order to construct a collection of decision trees with controlled variance. == History == The general method of random decision forests was first proposed by Salzberg and Heath in 1993, with a method that used a randomized decision tree algorithm to create multiple trees and then combine them using majority voting. This idea was developed further by Ho in 1995. Ho established that forests of trees splitting with oblique hyperplanes can gain accuracy as they grow without suffering from overtraining, as long as the forests are randomly restricted to be sensitive to only selected feature dimensions. A subsequent work along the same lines concluded that other splitting methods behave similarly, as long as they are randomly forced to be insensitive to some feature dimensions. This observation that a more complex classifier (a larger forest) gets more accurate nearly monotonically is in sharp contrast to the common belief that the complexity of a classifier can only grow to a certain level of accuracy before being hurt by overfitting. The explanation of the forest method's resistance to overtraining can be found in Kleinberg's theory of stochastic discrimination. The early development of Breiman's notion of random forests was influenced by the work of Amit and Geman who introduced the idea of searching over a random subset of the available decisions when splitting a node, in the context of growing a single tree. The idea of random subspace selection from Ho was also influential in the design of random forests. This method grows a forest of trees, and introduces variation among the trees by projecting the training data into a randomly chosen subspace before fitting each tree or each node. Finally, the idea of randomized node optimization, where the decision at each node is selected by a randomized procedure, rather than a deterministic optimization was first introduced by Thomas G. Dietterich. The proper introduction of random forests was made in a paper by Leo Breiman, that has become one of the world's most cited papers. This paper describes a method of building a forest of uncorrelated trees using a CART like procedure, combined with randomized node optimization and bagging. In addition, this paper combines several ingredients, some previously known and some novel, which form the basis of the modern practice of random forests, in particular: Using out-of-bag error as an estimate of the generalization error. Measuring variable importance through permutation. The report also offers the first theoretical result for random forests in the form of a bound on the generalization error which depends on the strength of the trees in the forest and their correlation. == Algorithm == === Preliminaries: decision tree learning === Decision trees are a popular method for various machine learning tasks. Tree learning is almost "an off-the-shelf procedure for data mining", say Hastie et al., "because it is invariant under scaling and various other transformations of feature values, is robust to inclusion of irrelevant features, and produces inspectable models. However, they are seldom accurate". In particular, trees that are grown very deep tend to learn highly irregular patterns: they overfit their training sets, i.e. have low bias, but very high variance. Random forests are a way of averaging multiple deep decision trees, trained on different parts of the same training set, with the goal of reducing the variance. This comes at the expense of a small increase in the bias and some loss of interpretability, but generally greatly boosts the performance in the final model. === Bagging === The training algorithm for random forests applies the general technique of bootstrap aggregating, or bagging, to tree learners. Given a training set X = x1, ..., xn with responses Y = y1, ..., yn, bagging repeatedly (B times) selects a random sample with replacement of the training set and fits trees to these samples: After training, predictions for unseen samples x' can be made by averaging the predictions from all the individual regression trees on x': f ^ = 1 B ∑ b = 1 B f b ( x ′ ) {\displaystyle {\hat {f}}={\frac {1}{B}}\sum _{b=1}^{B}f_{b}(x')} or by taking the plurality vote in the case of classification trees. This bootstrapping procedure leads to better model performance because it decreases the variance of the model, without increasing the bias. This means that while the predictions of a single tree are highly sensitive to noise in its training set, the average of many trees is not, as long as the trees are not correlated. Simply training many trees on a single training set would give strongly correlated trees (or even the same tree many times, if the training algorithm is deterministic); bootstrap sampling is a way of de-correlating the trees by showing them different training sets. Additionally, an estimate of the uncertainty of the prediction can be made as the standard deviation of the predictions from all the individual regression trees on x′: σ = ∑ b = 1 B ( f b ( x ′ ) − f ^ ) 2 B − 1 . {\displaystyle \sigma ={\sqrt {\frac {\sum _{b=1}^{B}(f_{b}(x')-{\hat {f}})^{2}}{B-1}}}.} The number B of samples (equivalently, of trees) is a free parameter. Typically, a few hundred to several thousand trees are used, depending on the size and nature of the training set. B can be optimized using cross-validation, or by observing the out-of-bag error: the mean prediction error on each training sample xi, using only the trees that did not have xi in their bootstrap sample. The training and test error tend to level off after some number of trees have been fit. === From bagging to random forests === The above procedure describes the original bagging algorithm for trees. Random forests also include another type of bagging scheme: they use a modified tree learning algorithm that selects, at each candidate split in the learning process, a random subset of the features. This process is sometimes called "feature bagging". The reason for doing this is the correlation of the trees in an ordinary bootstrap sample: if one or a few features are very strong predictors for the response variable (target output), these features will be selected in many of the B trees, causing them to become correlated. An analysis of how bagging and random subspace projection contribute to accuracy gains under different conditions is given by Ho. Typically, for a classification problem with p {\displaystyle p} features, p {\displaystyle {\sqrt {p}}} (rounded down) features are used in each split. For regression problems the inventors recommend p / 3 {\displaystyle p/3} (rounded down) with a minimum node size of 5 as the default. In practice, the best values for these parameters should be tuned on a case-to-case basis for every problem. === ExtraTrees === Adding one further step of randomization yields extremely randomized trees, or ExtraTrees. As with ordinary random forests, they are an ensemble of individual trees, but there are two main differences: (1) each tree is trained using the whole learning sample (rather than a bootstrap sample), and (2) the top-down splitting is randomized: for each feature under consideration, a number of random cut-points are selected, instead of computing the locally optimal cut-point (based on, e.g., information gain or the Gini impurity). The values are chosen from a uniform distribution within the feature's empirical range (in the tree's training set). Then, of all the randomly chosen splits, the split that yields the highest score is chosen to split the node. Similar to ordinary random forests, the number of randomly selected features to be considered at each node can be specified. Default values for this parameter are p {\displaystyle {\sqrt {p}}} for classification and p {\displaystyle p} for regression, where p {\displaystyle p} is the number of features in the model. === Random forests for high-dimensional data === The basic random forest procedure may

  Read more →
• 

  Relationship square

  In statistics, the relationship square is a graphical representation for use in the factorial analysis of a table individuals x variables. This representation completes classical representations provided by principal component analysis (PCA) or multiple correspondence analysis (MCA), namely those of individuals, of quantitative variables (correlation circle) and of the categories of qualitative variables (at the centroid of the individuals who possess them). It is especially important in factor analysis of mixed data (FAMD) and in multiple factor analysis (MFA). == Definition of relationship square in the MCA frame == The first interest of the relationship square is to represent the variables themselves, not their categories, which is all the more valuable as there are many variables. For this, we calculate for each qualitative variable j {\displaystyle j} and each factor F s {\displaystyle F_{s}} ( F s {\displaystyle F_{s}} , rank s {\displaystyle s} factor, is the vector of coordinates of the individuals along the axis of rank s {\displaystyle s} ; in PCA, F s {\displaystyle F_{s}} is called principal component of rank s {\displaystyle s} ), the square of the correlation ratio between the F s {\displaystyle F_{s}} and the variable j {\displaystyle j} , usually denoted : η 2 ( j , F s ) {\displaystyle \eta ^{2}(j,F_{s})} Thus, to each factorial plane, we can associate a representation of qualitative variables themselves. Their coordinates being between 0 and 1, the variables appear in the square having as vertices the points (0,0), ( 0,1), (1,0) and (1,1). == Example in MCA == Six individuals ( i 1 , … , i 6 ) {\displaystyle i_{1},\ldots ,i_{6})} are described by three variables ( q 1 , q 2 , q 3 ) {\displaystyle (q_{1},q_{2},q_{3})} having respectively 3, 2 and 3 categories. Example : the individual i 1 {\displaystyle i_{1}} possesses the category a {\displaystyle a} of q 1 {\displaystyle q_{1}} , d {\displaystyle d} of q 2 {\displaystyle q_{2}} and f {\displaystyle f} of q 3 {\displaystyle q_{3}} . Applied to these data, the MCA function included in the R Package FactoMineR provides to the classical graph in Figure 1. The relationship square (Figure 2) makes easier the reading of the classic factorial plane. It indicates that: The first factor is related to the three variables but especially q 3 {\displaystyle q_{3}} (which have a very high coordinate along the first axis) and then q 2 {\displaystyle q_{2}} . The second factor is related only to q 1 {\displaystyle q_{1}} and q 3 {\displaystyle q_{3}} (and not to q 2 {\displaystyle q_{2}} which has a coordinate along axis 2 equal to 0) and that in a strong and equal manner. All this is visible on the classic graphic but not so clearly. The role of the relationship square is first to assist in reading a conventional graphic. This is precious when the variables are numerous and possess numerous coordinates. == Extensions == This representation may be supplemented with those of quantitative variables, the coordinates of the latter being the square of correlation coefficients (and not of correlation ratios). Thus, the second advantage of the relationship square lies in the ability to represent simultaneously quantitative and qualitative variables. The relationship square can be constructed from any factorial analysis of a table individuals x variables. In particular, it is (or should be) used systematically: in multiple correspondences analysis (MCA); in principal components analysis (PCA) when there are many supplementary variables; in factor analysis of mixed data (FAMD). An extension of this graphic to groups of variables (how to represent a group of variables by a single point ?) is used in Multiple Factor Analysis (MFA) == History == The idea of representing the qualitative variables themselves by a point (and not the categories) is due to Brigitte Escofier. The graphic as it is used now has been introduced by Brigitte Escofier and Jérôme Pagès in the framework of multiple factor analysis == Conclusion == In MCA, the relationship square provides a synthetic view of the connections between mixed variables, all the more valuable as there are many variables having many categories. This representation iscan be useful in any factorial analysis when there are numerous mixed variables, active and/or supplementary.

  Read more →
• 

  Kai's Power Tools

  Kai's Power Tools (KPT) are a set of API plugins created by the German computer scientist Kai Krause in 1992 that were designed for use with Adobe Photoshop and Corel Photo-Paint. Kai's Power Tools were sold to Corel in 2000 when MetaCreations was closed. There are various versions of Kai's Power Tools. KPT 3, 5, 6, and X sets are compilations of different filters. The program interface features a reward-based function in which a bonus function is revealed as the user moves towards more complex aspects of the tool. == Filters == The KPT Convolver is a mathematics based filter; the level of precision and varying effects can be achieved by using numerical values of colour, tint, hue, saturation, contrast, brightness, luminosity, and posterize. The KPT Projector takes the current image or selection and offers a number of interactive perspective warp effects. To a large extent, with its draggable distortion handles and its moving, scaling and rotating options, this simply duplicates Adobe Photoshop's Free Transform capabilities. What is completely different is the ability to rotate the bitmap image in 3D space and to tile the results if desired. It can also animate the distortions by dragging keyframes from the preview window into an animation palette. KPT 6 will then preview the animation and output it to various sizes in avi or mov format. This animation capability is even more useful with the KPT Turbulence filter. This is another distortion filter, but one that treats the image as if it was completely liquid. The preview panel shows the animation in real time. The KPT Goo filter is used to produce a single frame freeform liquid distortion. This filter is available both with KPT 6 and the standalone version. It works by effectively turning a bitmap image into a liquid that can be interactively smeared, smudged, twirled, and pinched with the range of tools on offer. The obvious use is to distort photographic portraits into caricatures. KPT Materializer can create advanced surface textures based on bump maps that define troughs and peaks. It can use any external image for the basis of the bump map or alternatively the user can pick out the hue, saturation, luminance or red, green, or blue channel of the current image. It can then offset, scale and rotate the texture map, control its lighting, and even blend in a reflection map. The filter can be used for anything from providing an oil-painting feel to an entire image, to giving the illusion of depth to a selection. Also producing the impression of depth is the KPT Gel filter which uses various paint tools to synthesize photo-realistic 3D materials such as metals, liquids, or plastics. Gel painting is very different from traditional 2D painting as the brush strokes pool together when they touch and refract the underlying image. It can also manipulate 3D paint—once it has been added—by twirling, pinching, and carving it. The opposite is true of the Equalizer filter, which is used for applying variations on sharpening effects. The filter has three modes. The first mode, Equalizer, looks and works rather like the graphic equalizer on a stereo system, enabling adjustment of the level of pixel contrast within nine bands of different visual frequencies. The second mode, Contrast Sharpen, allows for increasing the contrast between light and dark areas in an image. The third mode, Bounded Sharpen, can sharpen an image without causing oversharpening, which can lead to halo effects. This feature is particularly useful when pulling out the detail in an image softened by resizing. KPT SceneBuilder is used for producing photorealistic 3D scenes by importing and rendering 3DS files. The main image window offers three tabs for editing in 2D and 3D mode and for setting up the object's final texture. Many users regard this filter as being the most impressive because it acts as a standalone 3D rendering tool and provides control over everything from transparency, reflection, refraction, bump mapping through to multiple light sources, and so on but without the ability to create or edit objects. The final filter, KPT SkyEffects, also has its roots in Metacreations' experience with 3D programs such as Bryce and RayDream. This filter is designed to simulate the interaction between the light from the sun or moon with no less than six atmospheric layers of haze, fog and cloud. The filter is typical of the KPT 6 collection as a whole: at times the interface is inspired and offers the ability to create beautiful reddening sunsets simply by interactively dragging the sun toward the horizon, producing realistic sunsets and moonscapes. == Other effects == Kai's Power Tools 6 features a lens flare effect for precisely managing the type of glow, halo, streaks, and reflection. The addition of a library of preset effects helps to overcome this by allowing the user to choose a standard effect and then interactively position the flare in the image preview. KPT 6 provides a new engine in the form of the KPT Reaction, which takes a reaction seed and turns it into a seamlessly tiling pattern based on a reaction diffusion process. It offers random noise, regular dots or reticulated voronoi patterns or a bitmap image itself as the seed. Corel has no plans for any updates.

  Read more →
• 

  Neural Networks (journal)

  Neural Networks is a monthly peer-reviewed scientific journal and an official journal of the International Neural Network Society, European Neural Network Society, and Japanese Neural Network Society. == History == The journal was established in 1988 and is published by Elsevier. It covers all aspects of research on artificial neural networks. The founding editor-in-chief was Stephen Grossberg (Boston University). The current editors-in-chief are DeLiang Wang (Ohio State University) and Taro Toyoizumi (RIKEN Center for Brain Science). == Abstracting and indexing == The journal is abstracted and indexed in Scopus and the Science Citation Index Expanded. According to the Journal Citation Reports, the journal has a 2022 impact factor of 7.8.

  Read more →
• 

  Ni1000

  The Ni1000 is an artificial neural network chip developed by Nestor Corporation and Intel, developed in the 1990s. It is Intel's second-generation neural network chip, but the first all-digital chip. The chip is aimed at image analysis applications– containing more than 3 million transistors – and can analyze 40,000 patterns per second. Prototypes running Nestor's OCR software in 1994 were capable of recognizing around 100 handwritten characters per second. The development was funded with money from DARPA and Office of Naval Research.

  Read more →
• 

  Accumulated local effects

  Accumulated local effects (ALE) is a machine learning interpretability method. == Concepts == ALE uses a conditional feature distribution as an input and generates augmented data, creating more realistic data than a marginal distribution. It ignores far out-of-distribution (outlier) values. Unlike partial dependence plots and marginal plots, ALE is not defeated in the presence of correlated predictors. It analyzes differences in predictions instead of averaging them by calculating the average of the differences in model predictions over the augmented data, instead of the average of the predictions themselves. == Example == Given a model that predicts house prices based on its distance from city center and size of the building area, ALE compares the differences of predictions of houses of different sizes. The result separates the impact of the size from otherwise correlated features. == Limitations == Defining evaluation windows is subjective. High correlations between features can defeat the technique. ALE requires more and more uniformly distributed observations than PDP so that the conditional distribution can be reliably determined. The technique may produce inadequate results if the data is highly sparse, which is more common with high-dimensional data (curse of dimensionality).

  Read more →
• 

  Object Data Management Group

  The Object Data Management Group (ODMG) was conceived in the summer of 1991 at a breakfast with object database vendors that was organized by Rick Cattell of Sun Microsystems. In 1998, the ODMG changed its name from the Object Database Management Group to reflect the expansion of its efforts to include specifications for both object database and object–relational mapping products. The primary goal of the ODMG was to put forward a set of specifications that allowed a developer to write portable applications for object database and object–relational mapping products. In order to do that, the data schema, programming language bindings, and data manipulation and query languages needed to be portable. Between 1993 and 2001, the ODMG published five revisions to its specification. The last revision was ODMG version 3.0, after which the group disbanded. == Major components of the ODMG 3.0 specification == Object Model. This was based on the Object Management Group's Object Model. The OMG core model was designed to be a common denominator for object request brokers, object database systems, object programming languages, etc. The ODMG designed a profile by adding components to the OMG core object model. Object Specification Languages. The ODMG Object Definition Language (ODL) was used to define the object types that conform to the ODMG Object Model. The ODMG Object Interchange Format (OIF) was used to dump and load the current state to or from a file or set of files. Object Query Language (OQL). The ODMG OQL was a declarative (nonprocedural) language for query and updating. It used SQL as a basis, where possible, though OQL supports more powerful object-oriented capabilities. C++ Language Binding. This defined a C++ binding of the ODMG ODL and a C++ Object Manipulation Language (OML). The C++ ODL was expressed as a library that provides classes and functions to implement the concepts defined in the ODMG Object Model. The C++ OML syntax and semantics are those of standard C++ in the context of the standard class library. The C++ binding also provided a mechanism to invoke OQL. Smalltalk Language Binding. This defined the mapping between the ODMG ODL and Smalltalk, which was based on the OMG Smalltalk binding for the OMG Interface Definition Language (IDL). The Smalltalk binding also provided a mechanism to invoke OQL. Java Language Binding. This defined the binding between the ODMG ODL and the Java programming language as defined by the Java 2 Platform. The Java binding also provided a mechanism to invoke OQL. == Status == ODMG 3.0 was published in book form in 2000.[1] By 2001, most of the major object database and object-relational mapping vendors claimed conformance to the ODMG Java Language Binding. Compliance to the other components of the specification was mixed.[2] In 2001, the ODMG Java Language Binding was submitted to the Java Community Process as a basis for the Java Data Objects specification. The ODMG member companies then decided to concentrate their efforts on the Java Data Objects specification. As a result, the ODMG disbanded in 2001. In 2004, the Object Management Group (OMG) was granted the right to revise the ODMG 3.0 specification as an OMG specification by the copyright holder, Morgan Kaufmann Publishers. In February 2006, the OMG announced the formation of the Object Database Technology Working Group (ODBT WG) and plans to work on the 4th generation of an object database standard. == ODMG Compliant DBMS == Orient ODBMS: http://www.OrienTechnologies.com Objectivity/DB C++, Java and Smalltalk interfaces.

  Read more →
• 

  Cross-entropy

  In information theory, the cross-entropy between two probability distributions p {\displaystyle p} and q {\displaystyle q} , over the same underlying set of events, measures the average number of bits needed to identify an event drawn from the set when the coding scheme used for the set is optimized for an estimated probability distribution q {\displaystyle q} , rather than the true distribution p {\displaystyle p} . == Definition == The cross-entropy of the distribution q {\displaystyle q} relative to a distribution p {\displaystyle p} over a given set is defined as follows: H ( p , q ) = − E p ⁡ [ log ⁡ q ] , {\displaystyle H(p,q)=-\operatorname {E} _{p}[\log q],} where E p ⁡ [ ⋅ ] {\displaystyle \operatorname {E} _{p}[\cdot ]} is the expected value operator with respect to the distribution p {\displaystyle p} . The definition may be formulated using the Kullback–Leibler divergence D K L ( p ∥ q ) {\displaystyle D_{\mathrm {KL} }(p\parallel q)} , divergence of p {\displaystyle p} from q {\displaystyle q} (also known as the relative entropy of p {\displaystyle p} with respect to q {\displaystyle q} ). H ( p , q ) = H ( p ) + D K L ( p ∥ q ) , {\displaystyle H(p,q)=H(p)+D_{\mathrm {KL} }(p\parallel q),} where H ( p ) {\displaystyle H(p)} is the entropy of p {\displaystyle p} . For discrete probability distributions p {\displaystyle p} and q {\displaystyle q} with the same support X {\displaystyle {\mathcal {X}}} , this means The situation for continuous distributions is analogous. We have to assume that p {\displaystyle p} and q {\displaystyle q} are absolutely continuous with respect to some reference measure r {\displaystyle r} (usually r {\displaystyle r} is a Lebesgue measure on a Borel σ-algebra). Let P {\displaystyle P} and Q {\displaystyle Q} be probability density functions of p {\displaystyle p} and q {\displaystyle q} with respect to r {\displaystyle r} . Then − ∫ X P ( x ) log ⁡ Q ( x ) d x = E p ⁡ [ − log ⁡ Q ] , {\displaystyle -\int _{\mathcal {X}}P(x)\,\log Q(x)\,\mathrm {d} x=\operatorname {E} _{p}[-\log Q],} and therefore NB: The notation H ( p , q ) {\displaystyle H(p,q)} is also used for a different concept, the joint entropy of p {\displaystyle p} and q {\displaystyle q} . == Motivation == In information theory, the Kraft–McMillan theorem establishes that any directly decodable coding scheme for coding a message to identify one value x i {\displaystyle x_{i}} out of a set of possibilities { x 1 , … , x n } {\displaystyle \{x_{1},\ldots ,x_{n}\}} can be seen as representing an implicit probability distribution q ( x i ) = ( 1 2 ) ℓ i {\displaystyle q(x_{i})=\left({\frac {1}{2}}\right)^{\ell _{i}}} over { x 1 , … , x n } {\displaystyle \{x_{1},\ldots ,x_{n}\}} , where ℓ i {\displaystyle \ell _{i}} is the length of the code for x i {\displaystyle x_{i}} in bits. Therefore, cross-entropy can be interpreted as the expected message-length per datum when a wrong distribution q {\displaystyle q} is assumed while the data actually follows a distribution p {\displaystyle p} . That is why the expectation is taken over the true probability distribution p {\displaystyle p} and not q . {\displaystyle q.} Indeed the expected message-length under the true distribution p {\displaystyle p} is E p ⁡ [ ℓ ] = − E p ⁡ [ ln ⁡ q ( x ) ln ⁡ ( 2 ) ] = − E p ⁡ [ log 2 ⁡ q ( x ) ] = − ∑ x i p ( x i ) log 2 ⁡ q ( x i ) = − ∑ x p ( x ) log 2 ⁡ q ( x ) = H ( p , q ) . {\displaystyle {\begin{aligned}\operatorname {E} _{p}[\ell ]&=-\operatorname {E} _{p}\left[{\frac {\ln {q(x)}}{\ln(2)}}\right]\\[1ex]&=-\operatorname {E} _{p}\left[\log _{2}{q(x)}\right]\\[1ex]&=-\sum _{x_{i}}p(x_{i})\,\log _{2}q(x_{i})\\[1ex]&=-\sum _{x}p(x)\,\log _{2}q(x)=H(p,q).\end{aligned}}} == Estimation == There are many situations where cross-entropy needs to be measured but the distribution of p {\displaystyle p} is unknown. An example is language modeling, where a model is created based on a training set T {\displaystyle T} , and then its cross-entropy is measured on a test set to assess how accurate the model is in predicting the test data. In this example, p {\displaystyle p} is the true distribution of words in any corpus, and q {\displaystyle q} is the distribution of words as predicted by the model. Since the true distribution is unknown, cross-entropy cannot be directly calculated. In these cases, an estimate of cross-entropy is calculated using the following formula: H ( T , q ) = − ∑ i = 1 N 1 N log 2 ⁡ q ( x i ) {\displaystyle H(T,q)=-\sum _{i=1}^{N}{\frac {1}{N}}\log _{2}q(x_{i})} where N {\displaystyle N} is the size of the test set, and q ( x ) {\displaystyle q(x)} is the probability of event x {\displaystyle x} estimated from the training set. In other words, q ( x i ) {\displaystyle q(x_{i})} is the probability estimate of the model that the i-th word of the text is x i {\displaystyle x_{i}} . The sum is averaged over the N {\displaystyle N} words of the test. This is a Monte Carlo estimate of the true cross-entropy, where the test set is treated as samples from p ( x ) {\displaystyle p(x)} . == Relation to maximum likelihood == The cross entropy arises in classification problems when introducing a logarithm in the guise of the log-likelihood function. This section concerns the estimation of the probabilities of different discrete outcomes. To this end, denote a parametrized family of distributions by q θ {\displaystyle q_{\theta }} , with θ {\displaystyle \theta } subject to the optimization effort. Consider a given finite sequence of N {\displaystyle N} values x i {\displaystyle x_{i}} from a training set, obtained from conditionally independent sampling. The likelihood assigned to any considered parameter θ {\displaystyle \theta } of the model is then given by the product over all probabilities q θ ( X = x i ) {\displaystyle q_{\theta }(X=x_{i})} . Repeated occurrences are possible, leading to equal factors in the product. If the count of occurrences of the value equal to x {\displaystyle x} is denoted by # x {\displaystyle \#x} , then the frequency of that value equals # x / N {\displaystyle \#x/N} . If p ( X = x ) {\displaystyle p(X=x)} is the underlying probability distribution, for large N {\displaystyle N} we expect p ( X = x ) ≈ # x / N {\displaystyle p(X=x)\approx \#x/N} , by the law of large numbers. Writing our likelihood function as the product of observations from the distribution q θ {\displaystyle q_{\theta }} : L ( θ ; x ) = ∏ i q θ ( X = x i ) = ∏ x q θ ( X = x ) # x ≈ ∏ x q θ ( X = x ) N ⋅ p ( X = x ) = exp ⁡ log ⁡ [ ∏ x q θ ( X = x ) N ⋅ p ( X = x ) ] = exp ⁡ ( ∑ x N ⋅ p ( X = x ) log ⁡ q θ ( X = x ) ) , {\displaystyle {\begin{aligned}{\mathcal {L}}(\theta ;{\mathbf {x} })&=\prod _{i}q_{\theta }(X=x_{i})=\prod _{x}q_{\theta }(X=x)^{\#x}\\&\approx \prod _{x}q_{\theta }(X=x)^{N\cdot p(X=x)}=\exp \log \left[\prod _{x}q_{\theta }(X=x)^{N\cdot p(X=x)}\right]\\&=\exp \left(\sum _{x}N\cdot p(X=x)\log q_{\theta }(X=x)^{}\right),\end{aligned}}} where we have used the calculation rules for the logarithm in the final line. Notice how the exponent contains a − H ( p , q θ ) {\displaystyle -H(p,q_{\theta })} term. Taking the logarithm of both sides gives: log ⁡ L ( θ ; x ) = − N ⋅ H ( p , q θ ) . {\displaystyle \log {\mathcal {L}}(\theta ;{\mathbf {x} })=-N\cdot H(p,q_{\theta }).} Since the logarithm is a monotonically increasing function, the maximizing value of θ {\displaystyle \theta } is unaffected by this final step. Similarly, the maximizing value of θ {\displaystyle \theta } is unaffected by the factor of N {\displaystyle N} . So we observe that the likelihood maximization amounts to minimization of the cross-entropy. == Cross-entropy minimization == Cross-entropy minimization is frequently used in optimization and rare-event probability estimation. When comparing a distribution q {\displaystyle q} against a fixed reference distribution p {\displaystyle p} , cross-entropy and KL divergence are identical up to an additive constant (since p {\displaystyle p} is fixed): According to the Gibbs' inequality, both take on their minimal values when p = q {\displaystyle p=q} , which is 0 {\displaystyle 0} for KL divergence, and H ( p ) {\displaystyle \mathrm {H} (p)} for cross-entropy. In the engineering literature, the principle of minimizing KL divergence (Kullback's "Principle of Minimum Discrimination Information") is often called the Principle of Minimum Cross-Entropy (MCE), or Minxent. However, as discussed in the article Kullback–Leibler divergence, sometimes the distribution q {\displaystyle q} is the fixed prior reference distribution, and the distribution p {\displaystyle p} is optimized to be as close to q {\displaystyle q} as possible, subject to some constraint. In this case the two minimizations are not equivalent. This has led to some ambiguity in the literature, with some authors attempting to resolve the inconsistency by restating cross-entropy to be D K L ( p ∥ q ) {\displaystyle D_{\mathrm {KL} }(p\parallel q)} , rather than H (

  Read more →
• 

  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 →