In computer science, a suffix automaton is an efficient data structure for representing the substring index of a given string which allows the storage, processing, and retrieval of compressed information about all its substrings. The suffix automaton of a string S {\displaystyle S} is the smallest directed acyclic graph with a dedicated initial vertex and a set of "final" vertices, such that paths from the initial vertex to final vertices represent the suffixes of the string. In terms of automata theory, a suffix automaton is the minimal partial deterministic finite automaton that recognizes the set of suffixes of a given string S = s 1 s 2 … s n {\displaystyle S=s_{1}s_{2}\dots s_{n}} . The state graph of a suffix automaton is called a directed acyclic word graph (DAWG), a term that is also sometimes used for any deterministic acyclic finite state automaton. Suffix automata were introduced in 1983 by a group of scientists from the University of Denver and the University of Colorado Boulder. They suggested a linear time online algorithm for its construction and showed that the suffix automaton of a string S {\displaystyle S} having length at least two characters has at most 2 | S | − 1 {\textstyle 2|S|-1} states and at most 3 | S | − 4 {\textstyle 3|S|-4} transitions. Further works have shown a close connection between suffix automata and suffix trees, and have outlined several generalizations of suffix automata, such as compacted suffix automaton obtained by compression of nodes with a single outgoing arc. Suffix automata provide efficient solutions to problems such as substring search and computation of the largest common substring of two and more strings. == History == The concept of suffix automaton was introduced in 1983 by a group of scientists from University of Denver and University of Colorado Boulder consisting of Anselm Blumer, Janet Blumer, Andrzej Ehrenfeucht, David Haussler and Ross McConnell, although similar concepts had earlier been studied alongside suffix trees in the works of Peter Weiner, Vaughan Pratt and Anatol Slissenko. In their initial work, Blumer et al. showed a suffix automaton built for the string S {\displaystyle S} of length greater than 1 {\displaystyle 1} has at most 2 | S | − 1 {\displaystyle 2|S|-1} states and at most 3 | S | − 4 {\displaystyle 3|S|-4} transitions, and suggested a linear algorithm for automaton construction. In 1983, Mu-Tian Chen and Joel Seiferas independently showed that Weiner's 1973 suffix-tree construction algorithm while building a suffix tree of the string S {\displaystyle S} constructs a suffix automaton of the reversed string S R {\textstyle S^{R}} as an auxiliary structure. In 1987, Blumer et al. applied the compressing technique used in suffix trees to a suffix automaton and invented the compacted suffix automaton, which is also called the compacted directed acyclic word graph (CDAWG). In 1997, Maxime Crochemore and Renaud Vérin developed a linear algorithm for direct CDAWG construction. In 2001, Shunsuke Inenaga et al. developed an algorithm for construction of CDAWG for a set of words given by a trie. == Definitions == Usually when speaking about suffix automata and related concepts, some notions from formal language theory and automata theory are used, in particular: "Alphabet" is a finite set Σ {\displaystyle \Sigma } that is used to construct words. Its elements are called "characters"; "Word" is a finite sequence of characters ω = ω 1 ω 2 … ω n {\displaystyle \omega =\omega _{1}\omega _{2}\dots \omega _{n}} . "Length" of the word ω {\displaystyle \omega } is denoted as | ω | = n {\displaystyle |\omega |=n} ; "Formal language" is a set of words over given alphabet; "Language of all words" is denoted as Σ ∗ {\displaystyle \Sigma ^{}} (where the "" character stands for Kleene star), "empty word" (the word of zero length) is denoted by the character ε {\displaystyle \varepsilon } ; "Concatenation of words" α = α 1 α 2 … α n {\displaystyle \alpha =\alpha _{1}\alpha _{2}\dots \alpha _{n}} and β = β 1 β 2 … β m {\displaystyle \beta =\beta _{1}\beta _{2}\dots \beta _{m}} is denoted as α ⋅ β {\displaystyle \alpha \cdot \beta } or α β {\displaystyle \alpha \beta } and corresponds to the word obtained by writing β {\displaystyle \beta } to the right of α {\displaystyle \alpha } , that is, α β = α 1 α 2 … α n β 1 β 2 … β m {\displaystyle \alpha \beta =\alpha _{1}\alpha _{2}\dots \alpha _{n}\beta _{1}\beta _{2}\dots \beta _{m}} ; "Concatenation of languages" A {\displaystyle A} and B {\displaystyle B} is denoted as A ⋅ B {\displaystyle A\cdot B} or A B {\displaystyle AB} and corresponds to the set of pairwise concatenations A B = { α β : α ∈ A , β ∈ B } {\displaystyle AB=\{\alpha \beta :\alpha \in A,\beta \in B\}} ; If the word ω ∈ Σ ∗ {\displaystyle \omega \in \Sigma ^{}} may be represented as ω = α γ β {\displaystyle \omega =\alpha \gamma \beta } , where α , β , γ ∈ Σ ∗ {\displaystyle \alpha ,\beta ,\gamma \in \Sigma ^{}} , then words α {\displaystyle \alpha } , β {\displaystyle \beta } and γ {\displaystyle \gamma } are called "prefix", "suffix" and "subword" (substring) of the word ω {\displaystyle \omega } correspondingly; If T = T 1 … T n {\displaystyle T=T_{1}\dots T_{n}} and T l T l + 1 … T r = S {\displaystyle T_{l}T_{l+1}\dots T_{r}=S} (with 1 ≤ l ≤ r ≤ n {\displaystyle 1\leq l\leq r\leq n} ) then S {\displaystyle S} is said to "occur" in T {\displaystyle T} as a subword. Here l {\displaystyle l} and r {\displaystyle r} are called left and right positions of occurrence of S {\displaystyle S} in T {\displaystyle T} correspondingly. == Automaton structure == Formally, deterministic finite automaton is determined by 5-tuple A = ( Σ , Q , q 0 , F , δ ) {\displaystyle {\mathcal {A}}=(\Sigma ,Q,q_{0},F,\delta )} , where: Σ {\displaystyle \Sigma } is an "alphabet" that is used to construct words, Q {\displaystyle Q} is a set of automaton "states", q 0 ∈ Q {\displaystyle q_{0}\in Q} is an "initial" state of automaton, F ⊂ Q {\displaystyle F\subset Q} is a set of "final" states of automaton, δ : Q × Σ ↦ Q {\displaystyle \delta :Q\times \Sigma \mapsto Q} is a partial "transition" function of automaton, such that δ ( q , σ ) {\displaystyle \delta (q,\sigma )} for q ∈ Q {\displaystyle q\in Q} and σ ∈ Σ {\displaystyle \sigma \in \Sigma } is either undefined or defines a transition from q {\displaystyle q} over character σ {\displaystyle \sigma } . Most commonly, deterministic finite automaton is represented as a directed graph ("diagram") such that: Set of graph vertices corresponds to the state of states Q {\displaystyle Q} , Graph has a specific marked vertex corresponding to initial state q 0 {\displaystyle q_{0}} , Graph has several marked vertices corresponding to the set of final states F {\displaystyle F} , Set of graph arcs corresponds to the set of transitions δ {\displaystyle \delta } , Specifically, every transition δ ( q 1 , σ ) = q 2 {\textstyle \delta (q_{1},\sigma )=q_{2}} is represented by an arc from q 1 {\displaystyle q_{1}} to q 2 {\displaystyle q_{2}} marked with the character σ {\displaystyle \sigma } . This transition also may be denoted as q 1 σ ⟶ q 2 {\textstyle q_{1}{\begin{smallmatrix}{\sigma }\\[-5pt]{\longrightarrow }\end{smallmatrix}}q_{2}} . In terms of its diagram, the automaton recognizes the word ω = ω 1 ω 2 … ω m {\displaystyle \omega =\omega _{1}\omega _{2}\dots \omega _{m}} only if there is a path from the initial vertex q 0 {\displaystyle q_{0}} to some final vertex q ∈ F {\displaystyle q\in F} such that concatenation of characters on this path forms ω {\displaystyle \omega } . The set of words recognized by an automaton forms a language that is set to be recognized by the automaton. In these terms, the language recognized by a suffix automaton of S {\displaystyle S} is the language of its (possibly empty) suffixes. === Automaton states === "Right context" of the word ω {\displaystyle \omega } with respect to language L {\displaystyle L} is a set [ ω ] R = { α : ω α ∈ L } {\displaystyle [\omega ]_{R}=\{\alpha :\omega \alpha \in L\}} that is a set of words α {\displaystyle \alpha } such that their concatenation with ω {\displaystyle \omega } forms a word from L {\displaystyle L} . Right contexts induce a natural equivalence relation [ α ] R = [ β ] R {\displaystyle [\alpha ]_{R}=[\beta ]_{R}} on the set of all words. If language L {\displaystyle L} is recognized by some deterministic finite automaton, there exists unique up to isomorphism automaton that recognizes the same language and has the minimum possible number of states. Such an automaton is called a minimal automaton for the given language L {\displaystyle L} . Myhill–Nerode theorem allows it to define it explicitly in terms of right contexts: In these terms, a "suffix automaton" is the minimal deterministic finite automaton recognizing the language of suffixes of the word S = s 1 s 2 … s n {\displaystyle S=s_{1}s_{2}\dots s_{n}} . The right context of the word ω {\displaystyle \omeg
Niki.ai
Niki was an artificial intelligence company headquartered in Bangalore, Karnataka. It was founded in May 2015 by IIT Kharagpur graduates Sachin Jaiswal, Keshav Prawasi, Shishir Modi, and Nitin Babel. The Niki android app was launched for a limited beta in June 2015, then released for public during YourStory's TechSparks 2015, and is a Tech30 company. The company raised an undisclosed amount in seed funding from Unilazer Ventures, a Mumbai-based VC firm founded by Ronnie Screwvala, in October 2015. This was followed by another seed funding round by Ratan Tata in May 2016. The company then raised US$2 million in Series A round of funding from SAP.iO, existing investors and some US and German-based investors, among others. Niki.ai shut down in October 2021 as per media reports. Website not working. == Product == The product is an artificial intelligence-powered chatbot which works as an intelligent personal assistant, named Niki. Leveraging natural language processing and machine learning, Niki presents a chat-based natural language user interface to the users where they can interact with Niki in their natural language. Niki understands how users chat in India, deciphers the words, in the context of product/services that they would like to purchase, and comes up with apt recommendations. Initially, it was only available on the Android platform as a mobile app. The company has expanded its operations to the Facebook Messenger and Apple iOS platforms. The company aims to soon be present on more messaging platforms like Slack and WhatsApp. The company currently provides 20+ services to over 2 million consumers, covering a wide spectrum ranging from utility services like mobile recharge, bill payments, travel services like cabs, buses, hotels and entertainment services like movies and events. Services such as flights and healthcare are also planned. == Partnerships == In September 2017, Infosys Finacle joined with Niki.ai to provide chat-based service to banking customers. In August 2017, Niki partnered with LazyPay to enable a 'buy now, pay later' feature for its users.
Reparameterization trick
The reparameterization trick (aka "reparameterization gradient estimator") is a technique used in statistical machine learning, particularly in variational inference, variational autoencoders, and stochastic optimization. It allows for the efficient computation of gradients through random variables, enabling the optimization of parametric probability models using stochastic gradient descent, and the variance reduction of estimators. It was developed in the 1980s in operations research, under the name of "pathwise gradients", or "stochastic gradients". Its use in variational inference was proposed in 2013. == Mathematics == Let z {\displaystyle z} be a random variable with distribution q ϕ ( z ) {\displaystyle q_{\phi }(z)} , where ϕ {\displaystyle \phi } is a vector containing the parameters of the distribution. === REINFORCE estimator === Consider an objective function of the form: L ( ϕ ) = E z ∼ q ϕ ( z ) [ f ( z ) ] {\displaystyle L(\phi )=\mathbb {E} _{z\sim q_{\phi }(z)}[f(z)]} Without the reparameterization trick, estimating the gradient ∇ ϕ L ( ϕ ) {\displaystyle \nabla _{\phi }L(\phi )} can be challenging, because the parameter appears in the random variable itself. In more detail, we have to statistically estimate: ∇ ϕ L ( ϕ ) = ∇ ϕ ∫ d z q ϕ ( z ) f ( z ) {\displaystyle \nabla _{\phi }L(\phi )=\nabla _{\phi }\int dz\;q_{\phi }(z)f(z)} The REINFORCE estimator, widely used in reinforcement learning and especially policy gradient, uses the following equality: ∇ ϕ L ( ϕ ) = ∫ d z q ϕ ( z ) ∇ ϕ ( ln q ϕ ( z ) ) f ( z ) = E z ∼ q ϕ ( z ) [ ∇ ϕ ( ln q ϕ ( z ) ) f ( z ) ] {\displaystyle \nabla _{\phi }L(\phi )=\int dz\;q_{\phi }(z)\nabla _{\phi }(\ln q_{\phi }(z))f(z)=\mathbb {E} _{z\sim q_{\phi }(z)}[\nabla _{\phi }(\ln q_{\phi }(z))f(z)]} This allows the gradient to be estimated: ∇ ϕ L ( ϕ ) ≈ 1 N ∑ i = 1 N ∇ ϕ ( ln q ϕ ( z i ) ) f ( z i ) {\displaystyle \nabla _{\phi }L(\phi )\approx {\frac {1}{N}}\sum _{i=1}^{N}\nabla _{\phi }(\ln q_{\phi }(z_{i}))f(z_{i})} The REINFORCE estimator has high variance, and many methods were developed to reduce its variance. === Reparameterization estimator === The reparameterization trick expresses z {\displaystyle z} as: z = g ϕ ( ϵ ) , ϵ ∼ p ( ϵ ) {\displaystyle z=g_{\phi }(\epsilon ),\quad \epsilon \sim p(\epsilon )} Here, g ϕ {\displaystyle g_{\phi }} is a deterministic function parameterized by ϕ {\displaystyle \phi } , and ϵ {\displaystyle \epsilon } is a noise variable drawn from a fixed distribution p ( ϵ ) {\displaystyle p(\epsilon )} . This gives: L ( ϕ ) = E ϵ ∼ p ( ϵ ) [ f ( g ϕ ( ϵ ) ) ] {\displaystyle L(\phi )=\mathbb {E} _{\epsilon \sim p(\epsilon )}[f(g_{\phi }(\epsilon ))]} Now, the gradient can be estimated as: ∇ ϕ L ( ϕ ) = E ϵ ∼ p ( ϵ ) [ ∇ ϕ f ( g ϕ ( ϵ ) ) ] ≈ 1 N ∑ i = 1 N ∇ ϕ f ( g ϕ ( ϵ i ) ) {\displaystyle \nabla _{\phi }L(\phi )=\mathbb {E} _{\epsilon \sim p(\epsilon )}[\nabla _{\phi }f(g_{\phi }(\epsilon ))]\approx {\frac {1}{N}}\sum _{i=1}^{N}\nabla _{\phi }f(g_{\phi }(\epsilon _{i}))} == Examples == For some common distributions, the reparameterization trick takes specific forms: Normal distribution: For z ∼ N ( μ , σ 2 ) {\displaystyle z\sim {\mathcal {N}}(\mu ,\sigma ^{2})} , we can use: z = μ + σ ϵ , ϵ ∼ N ( 0 , 1 ) {\displaystyle z=\mu +\sigma \epsilon ,\quad \epsilon \sim {\mathcal {N}}(0,1)} Exponential distribution: For z ∼ Exp ( λ ) {\displaystyle z\sim {\text{Exp}}(\lambda )} , we can use: z = − 1 λ log ( ϵ ) , ϵ ∼ Uniform ( 0 , 1 ) {\displaystyle z=-{\frac {1}{\lambda }}\log(\epsilon ),\quad \epsilon \sim {\text{Uniform}}(0,1)} Discrete distribution can be reparameterized by the Gumbel distribution (Gumbel-softmax trick or "concrete distribution") and diffusion models. In general, any distribution that is differentiable with respect to its parameters can be reparameterized by inverting the multivariable CDF function, then apply the implicit method. See for an exposition and application to the Gamma, Beta, Dirichlet, and von Mises distributions. == Applications == === Variational autoencoder === In Variational Autoencoders (VAEs), the VAE objective function, known as the Evidence Lower Bound (ELBO), is given by: ELBO ( ϕ , θ ) = E z ∼ q ϕ ( z | x ) [ log p θ ( x | z ) ] − D KL ( q ϕ ( z | x ) | | p ( z ) ) {\displaystyle {\text{ELBO}}(\phi ,\theta )=\mathbb {E} _{z\sim q_{\phi }(z|x)}[\log p_{\theta }(x|z)]-D_{\text{KL}}(q_{\phi }(z|x)||p(z))} where q ϕ ( z | x ) {\displaystyle q_{\phi }(z|x)} is the encoder (recognition model), p θ ( x | z ) {\displaystyle p_{\theta }(x|z)} is the decoder (generative model), and p ( z ) {\displaystyle p(z)} is the prior distribution over latent variables. The gradient of ELBO with respect to θ {\displaystyle \theta } is simply E z ∼ q ϕ ( z | x ) [ ∇ θ log p θ ( x | z ) ] ≈ 1 L ∑ l = 1 L ∇ θ log p θ ( x | z l ) {\displaystyle \mathbb {E} _{z\sim q_{\phi }(z|x)}[\nabla _{\theta }\log p_{\theta }(x|z)]\approx {\frac {1}{L}}\sum _{l=1}^{L}\nabla _{\theta }\log p_{\theta }(x|z_{l})} but the gradient with respect to ϕ {\displaystyle \phi } requires the trick. Express the sampling operation z ∼ q ϕ ( z | x ) {\displaystyle z\sim q_{\phi }(z|x)} as: z = μ ϕ ( x ) + σ ϕ ( x ) ⊙ ϵ , ϵ ∼ N ( 0 , I ) {\displaystyle z=\mu _{\phi }(x)+\sigma _{\phi }(x)\odot \epsilon ,\quad \epsilon \sim {\mathcal {N}}(0,I)} where μ ϕ ( x ) {\displaystyle \mu _{\phi }(x)} and σ ϕ ( x ) {\displaystyle \sigma _{\phi }(x)} are the outputs of the encoder network, and ⊙ {\displaystyle \odot } denotes element-wise multiplication. Then we have ∇ ϕ ELBO ( ϕ , θ ) = E ϵ ∼ N ( 0 , I ) [ ∇ ϕ log p θ ( x | z ) + ∇ ϕ log q ϕ ( z | x ) − ∇ ϕ log p ( z ) ] {\displaystyle \nabla _{\phi }{\text{ELBO}}(\phi ,\theta )=\mathbb {E} _{\epsilon \sim {\mathcal {N}}(0,I)}[\nabla _{\phi }\log p_{\theta }(x|z)+\nabla _{\phi }\log q_{\phi }(z|x)-\nabla _{\phi }\log p(z)]} where z = μ ϕ ( x ) + σ ϕ ( x ) ⊙ ϵ {\displaystyle z=\mu _{\phi }(x)+\sigma _{\phi }(x)\odot \epsilon } . This allows us to estimate the gradient using Monte Carlo sampling: ∇ ϕ ELBO ( ϕ , θ ) ≈ 1 L ∑ l = 1 L [ ∇ ϕ log p θ ( x | z l ) + ∇ ϕ log q ϕ ( z l | x ) − ∇ ϕ log p ( z l ) ] {\displaystyle \nabla _{\phi }{\text{ELBO}}(\phi ,\theta )\approx {\frac {1}{L}}\sum _{l=1}^{L}[\nabla _{\phi }\log p_{\theta }(x|z_{l})+\nabla _{\phi }\log q_{\phi }(z_{l}|x)-\nabla _{\phi }\log p(z_{l})]} where z l = μ ϕ ( x ) + σ ϕ ( x ) ⊙ ϵ l {\displaystyle z_{l}=\mu _{\phi }(x)+\sigma _{\phi }(x)\odot \epsilon _{l}} and ϵ l ∼ N ( 0 , I ) {\displaystyle \epsilon _{l}\sim {\mathcal {N}}(0,I)} for l = 1 , … , L {\displaystyle l=1,\ldots ,L} . This formulation enables backpropagation through the sampling process, allowing for end-to-end training of the VAE model using stochastic gradient descent or its variants. === Variational inference === More generally, the trick allows using stochastic gradient descent for variational inference. Let the variational objective (ELBO) be of the form: ELBO ( ϕ ) = E z ∼ q ϕ ( z ) [ log p ( x , z ) − log q ϕ ( z ) ] {\displaystyle {\text{ELBO}}(\phi )=\mathbb {E} _{z\sim q_{\phi }(z)}[\log p(x,z)-\log q_{\phi }(z)]} Using the reparameterization trick, we can estimate the gradient of this objective with respect to ϕ {\displaystyle \phi } : ∇ ϕ ELBO ( ϕ ) ≈ 1 L ∑ l = 1 L ∇ ϕ [ log p ( x , g ϕ ( ϵ l ) ) − log q ϕ ( g ϕ ( ϵ l ) ) ] , ϵ l ∼ p ( ϵ ) {\displaystyle \nabla _{\phi }{\text{ELBO}}(\phi )\approx {\frac {1}{L}}\sum _{l=1}^{L}\nabla _{\phi }[\log p(x,g_{\phi }(\epsilon _{l}))-\log q_{\phi }(g_{\phi }(\epsilon _{l}))],\quad \epsilon _{l}\sim p(\epsilon )} === Dropout === The reparameterization trick has been applied to reduce the variance in dropout, a regularization technique in neural networks. The original dropout can be reparameterized with Bernoulli distributions: y = ( W ⊙ ϵ ) x , ϵ i j ∼ Bernoulli ( α i j ) {\displaystyle y=(W\odot \epsilon )x,\quad \epsilon _{ij}\sim {\text{Bernoulli}}(\alpha _{ij})} where W {\displaystyle W} is the weight matrix, x {\displaystyle x} is the input, and α i j {\displaystyle \alpha _{ij}} are the (fixed) dropout rates. More generally, other distributions can be used than the Bernoulli distribution, such as the gaussian noise: y i = μ i + σ i ⊙ ϵ i , ϵ i ∼ N ( 0 , I ) {\displaystyle y_{i}=\mu _{i}+\sigma _{i}\odot \epsilon _{i},\quad \epsilon _{i}\sim {\mathcal {N}}(0,I)} where μ i = m i ⊤ x {\displaystyle \mu _{i}=\mathbf {m} _{i}^{\top }x} and σ i 2 = v i ⊤ x 2 {\displaystyle \sigma _{i}^{2}=\mathbf {v} _{i}^{\top }x^{2}} , with m i {\displaystyle \mathbf {m} _{i}} and v i {\displaystyle \mathbf {v} _{i}} being the mean and variance of the i {\displaystyle i} -th output neuron. The reparameterization trick can be applied to all such cases, resulting in the variational dropout method.
Hybrid intelligent system
Hybrid intelligent system denotes a software system which employs, in parallel, a combination of methods and techniques from artificial intelligence subfields, such as: Neuro-symbolic systems Neuro-fuzzy systems Hybrid connectionist-symbolic models Fuzzy expert systems Connectionist expert systems Evolutionary neural networks Genetic fuzzy systems Rough fuzzy hybridization Reinforcement learning with fuzzy, neural, or evolutionary methods as well as symbolic reasoning methods. From the cognitive science perspective, every natural intelligent system is hybrid because it performs mental operations on both the symbolic and subsymbolic levels. For the past few years, there has been an increasing discussion of the importance of A.I. Systems Integration. Based on notions that there have already been created simple and specific AI systems (such as systems for computer vision, speech synthesis, etc., or software that employs some of the models mentioned above) and now is the time for integration to create broad AI systems. Proponents of this approach are researchers such as Marvin Minsky, Ron Sun, Aaron Sloman, Angelo Dalli and Michael A. Arbib. An example hybrid is a hierarchical control system in which the lowest, reactive layers are sub-symbolic. The higher layers, having relaxed time constraints, are capable of reasoning from an abstract world model and performing planning (even by hybrid wisdom). Intelligent systems usually rely on hybrid reasoning processes, which include induction, deduction, abduction and reasoning by analogy.
Learning curve (machine learning)
In machine learning (ML), a learning curve (or training curve) is a graphical representation that shows how a model's performance on a training set (and usually a validation set) changes with the number of training iterations (epochs) or the amount of training data. Typically, the number of training epochs or training set size is plotted on the x-axis, and the value of the loss function (and possibly some other metric such as the cross-validation score) on the y-axis. Synonyms include error curve, experience curve, improvement curve and generalization curve. More abstractly, learning curves plot the difference between learning effort and predictive performance, where "learning effort" usually means the number of training samples, and "predictive performance" means accuracy on testing samples. Learning curves have many useful purposes in ML, including: choosing model parameters during design, adjusting optimization to improve convergence, and diagnosing problems such as overfitting (or underfitting). Learning curves can also be tools for determining how much a model benefits from adding more training data, and whether the model suffers more from a variance error or a bias error. If both the validation score and the training score converge to a certain value, then the model will no longer significantly benefit from more training data. == Formal definition == When creating a function to approximate the distribution of some data, it is necessary to define a loss function L ( f θ ( X ) , Y ) {\displaystyle L(f_{\theta }(X),Y)} to measure how good the model output is (e.g., accuracy for classification tasks or mean squared error for regression). We then define an optimization process which finds model parameters θ {\displaystyle \theta } such that L ( f θ ( X ) , Y ) {\displaystyle L(f_{\theta }(X),Y)} is minimized, referred to as θ ∗ {\displaystyle \theta ^{}} . === Training curve for amount of data === If the training data is { x 1 , x 2 , … , x n } , { y 1 , y 2 , … y n } {\displaystyle \{x_{1},x_{2},\dots ,x_{n}\},\{y_{1},y_{2},\dots y_{n}\}} and the validation data is { x 1 ′ , x 2 ′ , … x m ′ } , { y 1 ′ , y 2 ′ , … y m ′ } {\displaystyle \{x_{1}',x_{2}',\dots x_{m}'\},\{y_{1}',y_{2}',\dots y_{m}'\}} , a learning curve is the plot of the two curves i ↦ L ( f θ ∗ ( X i , Y i ) ( X i ) , Y i ) {\displaystyle i\mapsto L(f_{\theta ^{}(X_{i},Y_{i})}(X_{i}),Y_{i})} i ↦ L ( f θ ∗ ( X i , Y i ) ( X i ′ ) , Y i ′ ) {\displaystyle i\mapsto L(f_{\theta ^{}(X_{i},Y_{i})}(X_{i}'),Y_{i}')} where X i = { x 1 , x 2 , … x i } {\displaystyle X_{i}=\{x_{1},x_{2},\dots x_{i}\}} === Training curve for number of iterations === Many optimization algorithms are iterative, repeating the same step (such as backpropagation) until the process converges to an optimal value. Gradient descent is one such algorithm. If θ i ∗ {\displaystyle \theta _{i}^{}} is the approximation of the optimal θ {\displaystyle \theta } after i {\displaystyle i} steps, a learning curve is the plot of i ↦ L ( f θ i ∗ ( X , Y ) ( X ) , Y ) {\displaystyle i\mapsto L(f_{\theta _{i}^{}(X,Y)}(X),Y)} i ↦ L ( f θ i ∗ ( X , Y ) ( X ′ ) , Y ′ ) {\displaystyle i\mapsto L(f_{\theta _{i}^{}(X,Y)}(X'),Y')}
Cone tracing
Cone tracing and beam tracing are a derivative of the ray tracing algorithm that replaces rays, which have no thickness, with thick rays. == Principles == In ray tracing, rays are often modeled as geometric ray with no thickness to perform efficient geometric queries such as a ray-triangle intersection. From a physics of light transport point of view, however, this is an inaccurate model provided the pixel on the sensor plane has non-zero area. In the simplified pinhole camera optics model, the energy reaching the pixel comes from the integral of radiance from the solid angle by which the sensor pixel sees the scene through the pinhole at the focal plane. This yields the key notion of pixel footprint on surfaces or in the texture space, which is the back projection of the pixel on to the scene. Note that this approach can also represent a lens-based camera and thus depth of field effects, using a cone whose cross-section decreases from the lens size to zero at the focal plane, and then increases. Real optical system do not focus on exact points because of diffraction and imperfections. This can be modeled with a point spread function (PSF) weighted within a solid angle larger than the pixel. From a signal processing point of view, ignoring the point spread function and approximating the integral of radiance with a single, central sample (through a ray with no thickness) can lead to strong aliasing because the "projected geometric signal" has very high frequencies exceeding the Nyquist-Shannon maximal frequency that can be represented using the uniform pixel sampling rate. The physically based image formation model can be approximated by the convolution with the point spread function assuming the function is shift-invariant and linear. In practice, techniques such as multisample anti-aliasing estimate this cone-based model by oversampling the signal and then performing a convolution (the reconstruction filter). The backprojected cone footprint onto the scene can also be used to directly pre-filter the geometry and textures of the scene. Note that contrary to intuition, the reconstruction filter should not be the pixel footprint (as the pinhole camera model would suggest), since a box filter has poor spectral properties. Conversely, the ideal sinc function is not practical, having infinite support with possibly negative values which often creates ringing artifacts due to the Gibbs phenomenon. A Gaussian or a Lanczos filter are considered good compromises. == Computer graphics models == Cone and Beam early papers rely on different simplifications: the first considers a circular section and treats the intersection with various possible shapes. The second treats an accurate pyramidal beam through the pixel and along a complex path, but it only works for polyhedrical shapes. Cone tracing solves certain problems related to sampling and aliasing, which can plague conventional ray tracing. However, cone tracing creates a host of problems of its own. For example, just intersecting a cone with scene geometry leads to an enormous variety of possible results. For this reason, cone tracing has remained mostly unpopular. In recent years, increases in computer speed have made Monte Carlo algorithms like distributed ray tracing - i.e. stochastic explicit integration of the pixel - much more used than cone tracing because the results are exact provided enough samples are used. But the convergence is so slow that even in the context of off-line rendering a huge amount of time can be required to avoid noise. Differential cone-tracing, considering a differential angular neighborhood around a ray, avoids the complexity of exact geometry intersection but requires a LOD representation of the geometry and appearance of the objects. MIPmapping is an approximation of it limited to the integration of the surface texture within a cone footprint. Differential ray-tracing extends it to textured surfaces viewed through complex paths of cones reflected or refracted by curved surfaces. Raymarching methods over signed distance fields (SDFs) naturally allow easy use of cone-like tracing, at zero additional cost to the tracing, and both speeds up tracing and improves quality. Voxel cone tracing is a real-time algorithm that uses a hierarchical voxel representation of scene geometry, such as a sparse voxel octree, to support fast cone tracing for indirect illumination. This approach allows for the approximation of effects like glossy reflections and ambient occlusion at interactive framerates without the need for precomputation.
Computational humor
Computational humor is a branch of computational linguistics and artificial intelligence which uses computers in humor research. It is a relatively new area, with the first dedicated conference organized in 1996. The first "computer model of a sense of humor" was suggested by Suslov as early as 1992. Investigation of the general scheme of the information processing show a possibility of a specific malfunction, conditioned by the necessity of a quick deletion from consciousness of a false version. This specific malfunction can be identified with a humorous effect on the psychological grounds; however, an essentially new ingredient, a role of timing, is added to a well known role of ambiguity. In biological systems, a sense of humour inevitably develops in the course of evolution, because its biological function consists in quickening the transmission of processed information into consciousness and in a more effective use of brain resources. A realization of this algorithm in neural networks explains naturally the mechanism of laughter: deletion of a false version corresponds to zeroing of some part of the neural network and excessive energy of neurons is thrown out to the motor cortex, arousing muscular contractions. Unfortunately, a practical realization of this algorithm needs extensive databases, whose creation in the automatic regime was suggested only recently . As a result, this magistral direction was not developed properly and subsequent investigations (see below) accepted somewhat specialized colouring. == Joke generators == === Pun generation === An approach to analysis of humor is classification of jokes. A further step is an attempt to generate jokes basing on the rules that underlie classification. Simple prototypes for computer pun generation were reported in the early 1990s, based on a natural language generator program, VINCI. Graeme Ritchie and Kim Binsted in their 1994 research paper described a computer program, JAPE, designed to generate question-answer-type puns from a general, i.e., non-humorous, lexicon. (The program name is an acronym for "Joke Analysis and Production Engine".) Some examples produced by JAPE are: Q: What is the difference between leaves and a car? A: One you brush and rake, the other you rush and brake. Q: What do you call a strange market? A: A bizarre bazaar. Since then the approach has been improved, and the latest report, dated 2007, describes the STANDUP joke generator, implemented in the Java programming language. The STANDUP generator was tested on children within the framework of analyzing its usability for language skills development for children with communication disabilities, e.g., because of cerebral palsy. (The project name is an acronym for "System To Augment Non-speakers' Dialog Using Puns" and an allusion to standup comedy.) Children responded to this "language playground" with enthusiasm, and showed marked improvement on certain types of language tests. The two young people, who used the system over a ten-week period, regaled their peers, staff, family and neighbors with jokes such as: "What do you call a spicy missile? A hot shot!" Their joy and enthusiasm at entertaining others was inspirational. === Other === Stock and Strapparava described a program to generate funny acronyms. == Joke recognition == A statistical machine learning algorithm to detect whether a sentence contained a "That's what she said" double entendre was developed by Kiddon and Brun (2011). There is an open-source Python implementation of Kiddon & Brun's TWSS system. A program to recognize knock-knock jokes was reported by Taylor and Mazlack. This kind of research is important in analysis of human–computer interaction. An application of machine learning techniques for the distinguishing of joke texts from non-jokes was described by Mihalcea and Strapparava (2006). Takizawa et al. (1996) reported on a heuristic program for detecting puns in the Japanese language. == Applications == A possible application for assistance in language acquisition is described in the section "Pun generation". Another envisioned use of joke generators is in cases of a steady supply of jokes where quantity is more important than quality. Another obvious, yet remote, direction is automated joke appreciation. It is known that humans interact with computers in ways similar to interacting with other humans that may be described in terms of personality, politeness, flattery, and in-group favoritism. Therefore, the role of humor in human–computer interaction is being investigated. In particular, humor generation in user interface to ease communications with computers was suggested. Craig McDonough implemented the Mnemonic Sentence Generator, which converts passwords into humorous sentences. Based on the incongruity theory of humor, it is suggested that the resulting meaningless but funny sentences are easier to remember. For example, the password AjQA3Jtv is converted into "Arafat joined Quayle's Ant, while TARAR Jeopardized thurmond's vase," an example chosen by combining politicians names with verbs and common nouns. == Related research == John Allen Paulos is known for his interest in mathematical foundations of humor. His book Mathematics and Humor: A Study of the Logic of Humor demonstrates structures common to humor and formal sciences (mathematics, linguistics) and develops a mathematical model of jokes based on catastrophe theory. Conversational systems which have been designed to take part in Turing test competitions generally have the ability to learn humorous anecdotes and jokes. Because many people regard humor as something particular to humans, its appearance in conversation can be quite useful in convincing a human interrogator that a hidden entity, which could be a machine or a human, is in fact a human.