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SYMAN is an artificial intelligence technology that uses data from social media profiles to identify trends in the job market. SYMAN is designed to organize actionable data for products and services including recruiting, human capital management, CRM, and marketing. SYMAN was developed with a $21 million series B financing round secured by Identified, which was led by VantagePoint Capital Partners and Capricorn Investment Group.
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Averaged one-dependence estimators (AODE) is a probabilistic classification learning technique. It was developed to address the attribute-independence problem of the popular naive Bayes classifier. It frequently develops substantially more accurate classifiers than naive Bayes at the cost of a modest increase in the amount of computation. == The AODE classifier == AODE seeks to estimate the probability of each class y given a specified set of features x1, ... xn, P(y | x1, ... xn). To do so it uses the formula P ^ ( y ∣ x 1 , … x n ) = ∑ i : 1 ≤ i ≤ n ∧ F ( x i ) ≥ m P ^ ( y , x i ) ∏ j = 1 n P ^ ( x j ∣ y , x i ) ∑ y ′ ∈ Y ∑ i : 1 ≤ i ≤ n ∧ F ( x i ) ≥ m P ^ ( y ′ , x i ) ∏ j = 1 n P ^ ( x j ∣ y ′ , x i ) {\displaystyle {\hat {P}}(y\mid x_{1},\ldots x_{n})={\frac {\sum _{i:1\leq i\leq n\wedge F(x_{i})\geq m}{\hat {P}}(y,x_{i})\prod _{j=1}^{n}{\hat {P}}(x_{j}\mid y,x_{i})}{\sum _{y^{\prime }\in Y}\sum _{i:1\leq i\leq n\wedge F(x_{i})\geq m}{\hat {P}}(y^{\prime },x_{i})\prod _{j=1}^{n}{\hat {P}}(x_{j}\mid y^{\prime },x_{i})}}} where P ^ ( ⋅ ) {\displaystyle {\hat {P}}(\cdot )} denotes an estimate of P ( ⋅ ) {\displaystyle P(\cdot )} , F ( ⋅ ) {\displaystyle F(\cdot )} is the frequency with which the argument appears in the sample data and m is a user specified minimum frequency with which a term must appear in order to be used in the outer summation. In recent practice m is usually set at 1. == Derivation of the AODE classifier == We seek to estimate P(y | x1, ... xn). By the definition of conditional probability P ( y ∣ x 1 , … x n ) = P ( y , x 1 , … x n ) P ( x 1 , … x n ) . {\displaystyle P(y\mid x_{1},\ldots x_{n})={\frac {P(y,x_{1},\ldots x_{n})}{P(x_{1},\ldots x_{n})}}.} For any 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} , P ( y , x 1 , … x n ) = P ( y , x i ) P ( x 1 , … x n ∣ y , x i ) . {\displaystyle P(y,x_{1},\ldots x_{n})=P(y,x_{i})P(x_{1},\ldots x_{n}\mid y,x_{i}).} Under an assumption that x1, ... xn are independent given y and xi, it follows that P ( y , x 1 , … x n ) = P ( y , x i ) ∏ j = 1 n P ( x j ∣ y , x i ) . {\displaystyle P(y,x_{1},\ldots x_{n})=P(y,x_{i})\prod _{j=1}^{n}P(x_{j}\mid y,x_{i}).} This formula defines a special form of One Dependence Estimator (ODE), a variant of the naive Bayes classifier that makes the above independence assumption that is weaker (and hence potentially less harmful) than the naive Bayes' independence assumption. In consequence, each ODE should create a less biased estimator than naive Bayes. However, because the base probability estimates are each conditioned by two variables rather than one, they are formed from less data (the training examples that satisfy both variables) and hence are likely to have more variance. AODE reduces this variance by averaging the estimates of all such ODEs. == Features of the AODE classifier == Like naive Bayes, AODE does not perform model selection and does not use tuneable parameters. As a result, it has low variance. It supports incremental learning whereby the classifier can be updated efficiently with information from new examples as they become available. It predicts class probabilities rather than simply predicting a single class, allowing the user to determine the confidence with which each classification can be made. Its probabilistic model can directly handle situations where some data are missing. AODE has computational complexity O ( l n 2 ) {\displaystyle O(ln^{2})} at training time and O ( k n 2 ) {\displaystyle O(kn^{2})} at classification time, where n is the number of features, l is the number of training examples and k is the number of classes. This makes it infeasible for application to high-dimensional data. However, within that limitation, it is linear with respect to the number of training examples and hence can efficiently process large numbers of training examples. == Implementations == The free Weka machine learning suite includes an implementation of AODE.
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In combinatorics and computational learning theory, a witness set is a set of elements that distinguishes a given Boolean function from a given class of other Boolean functions. Let C {\displaystyle C} be a concept class over a domain X {\displaystyle X} (that is, a family of Boolean functions over X {\displaystyle X} ) and c {\displaystyle c} be a concept in X {\displaystyle X} (a single Boolean function). A subset S {\displaystyle S} of X {\displaystyle X} is a witness set for c {\displaystyle c} in X {\displaystyle X} if S {\displaystyle S} distinguishes c {\displaystyle c} from all the other functions in C {\displaystyle C} , in the sense that no other function in C {\displaystyle C} has the same values on S {\displaystyle S} . For a concept class with | C | {\displaystyle |C|} concepts, there exists a concept that has a witness of size at most log 2 | C | {\displaystyle \log _{2}|C|} ; this bound is tight when C {\displaystyle C} consists of all Boolean functions over X {\displaystyle X} . By a result of Bondy (1972) there exists a single witness set of size at most | C | − 1 {\displaystyle |C|-1} that is valid for all concepts in C {\displaystyle C} ; this bound is tight when C {\displaystyle C} consists of the indicator functions of the empty set and some singleton sets. One way to construct this set is to interpret the concepts as bitstrings, and the domain elements as positions in these bitstrings. Then the set of positions at which a trie of the bitstrings branches forms the desired witness set. This construction is central to the operation of the fusion tree data structure. The minimum size of a witness set for c {\displaystyle c} is called the witness size or specification number and is denoted by w C ( c ) {\displaystyle w_{C}(c)} . The value max { w C ( c ) : c ∈ C } {\displaystyle \max\{w_{C}(c):c\in C\}} is called the teaching dimension of C {\displaystyle C} . It represents the number of examples of a concept that need to be presented by a teacher to a learner, in the worst case, to enable the learner to determine which concept is being presented. Witness sets have also been called teaching sets, keys, specifying sets, or discriminants. The "witness set" terminology is from Kushilevitz et al. (1996), who trace the concept of witness sets to work by Cover (1965).
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In computational learning theory, Occam learning is a model of algorithmic learning where the objective of the learner is to output a succinct representation of received training data. This is closely related to probably approximately correct (PAC) learning, where the learner is evaluated on its predictive power of a test set. Occam learnability implies PAC learning, and for a wide variety of concept classes, the converse is also true: PAC learnability implies Occam learnability. == Introduction == Occam Learning is named after Occam's razor, which is a principle stating that, given all other things being equal, a shorter explanation for observed data should be favored over a lengthier explanation. The theory of Occam learning is a formal and mathematical justification for this principle. It was first shown by Blumer, et al. that Occam learning implies PAC learning, which is the standard model of learning in computational learning theory. In other words, parsimony (of the output hypothesis) implies predictive power. == Definition of Occam learning == The succinctness of a concept c {\displaystyle c} in concept class C {\displaystyle {\mathcal {C}}} can be expressed by the length s i z e ( c ) {\displaystyle size(c)} of the shortest bit string that can represent c {\displaystyle c} in C {\displaystyle {\mathcal {C}}} . Occam learning connects the succinctness of a learning algorithm's output to its predictive power on unseen data. Let C {\displaystyle {\mathcal {C}}} and H {\displaystyle {\mathcal {H}}} be concept classes containing target concepts and hypotheses respectively. Then, for constants α ≥ 0 {\displaystyle \alpha \geq 0} and 0 ≤ β < 1 {\displaystyle 0\leq \beta <1} , a learning algorithm L {\displaystyle L} is an ( α , β ) {\displaystyle (\alpha ,\beta )} -Occam algorithm for C {\displaystyle {\mathcal {C}}} using H {\displaystyle {\mathcal {H}}} iff, given a set S = { x 1 , … , x m } {\displaystyle S=\{x_{1},\dots ,x_{m}\}} of m {\displaystyle m} samples labeled according to a concept c ∈ C {\displaystyle c\in {\mathcal {C}}} , L {\displaystyle L} outputs a hypothesis h ∈ H {\displaystyle h\in {\mathcal {H}}} such that h {\displaystyle h} is consistent with c {\displaystyle c} on S {\displaystyle S} (that is, h ( x ) = c ( x ) , ∀ x ∈ S {\displaystyle h(x)=c(x),\forall x\in S} ), and s i z e ( h ) ≤ ( n ⋅ s i z e ( c ) ) α m β {\displaystyle size(h)\leq (n\cdot size(c))^{\alpha }m^{\beta }} where n {\displaystyle n} is the maximum length of any sample x ∈ S {\displaystyle x\in S} . An Occam algorithm is called efficient if it runs in time polynomial in n {\displaystyle n} , m {\displaystyle m} , and s i z e ( c ) . {\displaystyle size(c).} We say a concept class C {\displaystyle {\mathcal {C}}} is Occam learnable with respect to a hypothesis class H {\displaystyle {\mathcal {H}}} if there exists an efficient Occam algorithm for C {\displaystyle {\mathcal {C}}} using H . {\displaystyle {\mathcal {H}}.} == The relation between Occam and PAC learning == Occam learnability implies PAC learnability, as the following theorem of Blumer, et al. shows: === Theorem (Occam learning implies PAC learning) === Let L {\displaystyle L} be an efficient ( α , β ) {\displaystyle (\alpha ,\beta )} -Occam algorithm for C {\displaystyle {\mathcal {C}}} using H {\displaystyle {\mathcal {H}}} . Then there exists a constant a > 0 {\displaystyle a>0} such that for any 0 < ϵ , δ < 1 {\displaystyle 0<\epsilon ,\delta <1} , for any distribution D {\displaystyle {\mathcal {D}}} , given m ≥ a ( 1 ϵ log 1 δ + ( ( n ⋅ s i z e ( c ) ) α ϵ ) 1 1 − β ) {\displaystyle m\geq a\left({\frac {1}{\epsilon }}\log {\frac {1}{\delta }}+\left({\frac {(n\cdot size(c))^{\alpha }}{\epsilon }}\right)^{\frac {1}{1-\beta }}\right)} samples drawn from D {\displaystyle {\mathcal {D}}} and labelled according to a concept c ∈ C {\displaystyle c\in {\mathcal {C}}} of length n {\displaystyle n} bits each, the algorithm L {\displaystyle L} will output a hypothesis h ∈ H {\displaystyle h\in {\mathcal {H}}} such that e r r o r ( h ) ≤ ϵ {\displaystyle error(h)\leq \epsilon } with probability at least 1 − δ {\displaystyle 1-\delta } .Here, e r r o r ( h ) {\displaystyle error(h)} is with respect to the concept c {\displaystyle c} and distribution D {\displaystyle {\mathcal {D}}} . This implies that the algorithm L {\displaystyle L} is also a PAC learner for the concept class C {\displaystyle {\mathcal {C}}} using hypothesis class H {\displaystyle {\mathcal {H}}} . A slightly more general formulation is as follows: === Theorem (Occam learning implies PAC learning, cardinality version) === Let 0 < ϵ , δ < 1 {\displaystyle 0<\epsilon ,\delta <1} . Let L {\displaystyle L} be an algorithm such that, given m {\displaystyle m} samples drawn from a fixed but unknown distribution D {\displaystyle {\mathcal {D}}} and labeled according to a concept c ∈ C {\displaystyle c\in {\mathcal {C}}} of length n {\displaystyle n} bits each, outputs a hypothesis h ∈ H n , m {\displaystyle h\in {\mathcal {H}}_{n,m}} that is consistent with the labeled samples. Then, there exists a constant b {\displaystyle b} such that if log | H n , m | ≤ b ϵ m − log 1 δ {\displaystyle \log |{\mathcal {H}}_{n,m}|\leq b\epsilon m-\log {\frac {1}{\delta }}} , then L {\displaystyle L} is guaranteed to output a hypothesis h ∈ H n , m {\displaystyle h\in {\mathcal {H}}_{n,m}} such that e r r o r ( h ) ≤ ϵ {\displaystyle error(h)\leq \epsilon } with probability at least 1 − δ {\displaystyle 1-\delta } . While the above theorems show that Occam learning is sufficient for PAC learning, it doesn't say anything about necessity. Board and Pitt show that, for a wide variety of concept classes, Occam learning is in fact necessary for PAC learning. They proved that for any concept class that is polynomially closed under exception lists, PAC learnability implies the existence of an Occam algorithm for that concept class. Concept classes that are polynomially closed under exception lists include Boolean formulas, circuits, deterministic finite automata, decision-lists, decision-trees, and other geometrically defined concept classes. A concept class C {\displaystyle {\mathcal {C}}} is polynomially closed under exception lists if there exists a polynomial-time algorithm A {\displaystyle A} such that, when given the representation of a concept c ∈ C {\displaystyle c\in {\mathcal {C}}} and a finite list E {\displaystyle E} of exceptions, outputs a representation of a concept c ′ ∈ C {\displaystyle c'\in {\mathcal {C}}} such that the concepts c {\displaystyle c} and c ′ {\displaystyle c'} agree except on the set E {\displaystyle E} . == Proof that Occam learning implies PAC learning == We first prove the Cardinality version. Call a hypothesis h ∈ H {\displaystyle h\in {\mathcal {H}}} bad if e r r o r ( h ) ≥ ϵ {\displaystyle error(h)\geq \epsilon } , where again e r r o r ( h ) {\displaystyle error(h)} is with respect to the true concept c {\displaystyle c} and the underlying distribution D {\displaystyle {\mathcal {D}}} . The probability that a set of samples S {\displaystyle S} is consistent with h {\displaystyle h} is at most ( 1 − ϵ ) m {\displaystyle (1-\epsilon )^{m}} , by the independence of the samples. By the union bound, the probability that there exists a bad hypothesis in H n , m {\displaystyle {\mathcal {H}}_{n,m}} is at most | H n , m | ( 1 − ϵ ) m {\displaystyle |{\mathcal {H}}_{n,m}|(1-\epsilon )^{m}} , which is less than δ {\displaystyle \delta } if log | H n , m | ≤ O ( ϵ m ) − log 1 δ {\displaystyle \log |{\mathcal {H}}_{n,m}|\leq O(\epsilon m)-\log {\frac {1}{\delta }}} . This concludes the proof of the second theorem above. Using the second theorem, we can prove the first theorem. Since we have a ( α , β ) {\displaystyle (\alpha ,\beta )} -Occam algorithm, this means that any hypothesis output by L {\displaystyle L} can be represented by at most ( n ⋅ s i z e ( c ) ) α m β {\displaystyle (n\cdot size(c))^{\alpha }m^{\beta }} bits, and thus log | H n , m | ≤ ( n ⋅ s i z e ( c ) ) α m β {\displaystyle \log |{\mathcal {H}}_{n,m}|\leq (n\cdot size(c))^{\alpha }m^{\beta }} . This is less than O ( ϵ m ) − log 1 δ {\displaystyle O(\epsilon m)-\log {\frac {1}{\delta }}} if we set m ≥ a ( 1 ϵ log 1 δ + ( ( n ⋅ s i z e ( c ) ) α ) ϵ ) 1 1 − β ) {\displaystyle m\geq a\left({\frac {1}{\epsilon }}\log {\frac {1}{\delta }}+\left({\frac {(n\cdot size(c))^{\alpha })}{\epsilon }}\right)^{\frac {1}{1-\beta }}\right)} for some constant a > 0 {\displaystyle a>0} . Thus, by the Cardinality version Theorem, L {\displaystyle L} will output a consistent hypothesis h {\displaystyle h} with probability at least 1 − δ {\displaystyle 1-\delta } . This concludes the proof of the first theorem above. == Improving sample complexity for common problems == Though Occam and PAC learnability are equivalent, the Occam framework can be used to produce tighter bounds on the sample complexity of classical problems including conjunctions, co
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WebGPU Shading Language (WGSL, internet media type: text/wgsl) is a high-level shading language and the normative shader language for the WebGPU API on the web. WGSL's syntax is influenced by Rust and is designed with strong static validation, explicit resource binding, and portability in mind for secure execution in browsers. In web contexts, WebGPU implementations accept WGSL source and perform compilation to platform-specific intermediate forms (for example, to SPIR‑V, DXIL, or MSL via the user agent), but such backends are not exposed to web content. == History and background == Graphics on the web historically used WebGL, with shaders written in GLSL ES. As applications demanded more modern GPU features and finer control over compute and graphics pipelines, the W3C's GPU for the Web Community Group and Working Group created WebGPU and its companion shading language, WGSL, to provide a secure, portable model suitable for the web platform. WGSL was developed to be human-readable, avoid undefined behavior common in legacy shading languages, and align closely with WebGPU's resource and validation model. == Design goals == WGSL's design emphasizes: Safety and determinism suitable for web security constraints (extensive static validation and well-defined semantics). Portability across diverse GPU backends via an abstract resource model shared with WebGPU. Readability and explicitness (no preprocessor, minimal implicit conversions, explicit address spaces and bindings). Alignment with modern GPU features (compute, storage buffers, textures, atomics) while retaining a familiar C/Rust-like syntax. == Language overview == === Types and values === Core scalar types include bool, i32, u32, and f32. Vectors (e.g., vec2, vec3, vec4) and matrices (up to 4×4) are available for floating-point element types. Optional f16 (half precision) may be enabled via a WebGPU feature; availability is implementation-dependent. Atomic types (atomic
In machine learning, the radial basis function kernel, or RBF kernel, is a popular kernel function used in various kernelized learning algorithms. In particular, it is commonly used in support vector machine classification. The RBF kernel on two samples x , x ′ ∈ R k {\displaystyle \mathbf {x} ,\mathbf {x'} \in \mathbb {R} ^{k}} , represented as feature vectors in some input space, is defined as K ( x , x ′ ) = exp ( − ‖ x − x ′ ‖ 2 2 σ 2 ) {\displaystyle K(\mathbf {x} ,\mathbf {x'} )=\exp \left(-{\frac {\|\mathbf {x} -\mathbf {x'} \|^{2}}{2\sigma ^{2}}}\right)} ‖ x − x ′ ‖ 2 {\displaystyle \textstyle \|\mathbf {x} -\mathbf {x'} \|^{2}} may be recognized as the squared Euclidean distance between the two feature vectors. σ {\displaystyle \sigma } is a free parameter. An equivalent definition involves a parameter γ = 1 2 σ 2 {\displaystyle \textstyle \gamma ={\tfrac {1}{2\sigma ^{2}}}} : K ( x , x ′ ) = exp ( − γ ‖ x − x ′ ‖ 2 ) {\displaystyle K(\mathbf {x} ,\mathbf {x'} )=\exp(-\gamma \|\mathbf {x} -\mathbf {x'} \|^{2})} Since the value of the RBF kernel decreases with distance and ranges between zero (in the infinite-distance limit) and one (when x = x'), it has a ready interpretation as a similarity measure. The feature space of the kernel has an infinite number of dimensions; for σ = 1 {\displaystyle \sigma =1} , its expansion using the multinomial theorem is: exp ( − 1 2 ‖ x − x ′ ‖ 2 ) = exp ( 2 2 x ⊤ x ′ − 1 2 ‖ x ‖ 2 − 1 2 ‖ x ′ ‖ 2 ) = exp ( x ⊤ x ′ ) exp ( − 1 2 ‖ x ‖ 2 ) exp ( − 1 2 ‖ x ′ ‖ 2 ) = ∑ j = 0 ∞ ( x ⊤ x ′ ) j j ! exp ( − 1 2 ‖ x ‖ 2 ) exp ( − 1 2 ‖ x ′ ‖ 2 ) = ∑ j = 0 ∞ ∑ n 1 + n 2 + ⋯ + n k = j exp ( − 1 2 ‖ x ‖ 2 ) x 1 n 1 ⋯ x k n k n 1 ! ⋯ n k ! exp ( − 1 2 ‖ x ′ ‖ 2 ) x ′ 1 n 1 ⋯ x ′ k n k n 1 ! ⋯ n k ! = ⟨ φ ( x ) , φ ( x ′ ) ⟩ {\displaystyle {\begin{alignedat}{2}\exp \left(-{\frac {1}{2}}\|\mathbf {x} -\mathbf {x'} \|^{2}\right)&=\exp \left({\frac {2}{2}}\mathbf {x} ^{\top }\mathbf {x'} -{\frac {1}{2}}\|\mathbf {x} \|^{2}-{\frac {1}{2}}\|\mathbf {x'} \|^{2}\right)\\[5pt]&=\exp \left(\mathbf {x} ^{\top }\mathbf {x'} \right)\exp \left(-{\frac {1}{2}}\|\mathbf {x} \|^{2}\right)\exp \left(-{\frac {1}{2}}\|\mathbf {x'} \|^{2}\right)\\[5pt]&=\sum _{j=0}^{\infty }{\frac {(\mathbf {x} ^{\top }\mathbf {x'} )^{j}}{j!}}\exp \left(-{\frac {1}{2}}\|\mathbf {x} \|^{2}\right)\exp \left(-{\frac {1}{2}}\|\mathbf {x'} \|^{2}\right)\\[5pt]&=\sum _{j=0}^{\infty }\quad \sum _{n_{1}+n_{2}+\dots +n_{k}=j}\exp \left(-{\frac {1}{2}}\|\mathbf {x} \|^{2}\right){\frac {x_{1}^{n_{1}}\cdots x_{k}^{n_{k}}}{\sqrt {n_{1}!\cdots n_{k}!}}}\exp \left(-{\frac {1}{2}}\|\mathbf {x'} \|^{2}\right){\frac {{x'}_{1}^{n_{1}}\cdots {x'}_{k}^{n_{k}}}{\sqrt {n_{1}!\cdots n_{k}!}}}\\[5pt]&=\langle \varphi (\mathbf {x} ),\varphi (\mathbf {x'} )\rangle \end{alignedat}}} φ ( x ) = exp ( − 1 2 ‖ x ‖ 2 ) ( a ℓ 0 ( 0 ) , a 1 ( 1 ) , … , a ℓ 1 ( 1 ) , … , a 1 ( j ) , … , a ℓ j ( j ) , … ) {\displaystyle \varphi (\mathbf {x} )=\exp \left(-{\frac {1}{2}}\|\mathbf {x} \|^{2}\right)\left(a_{\ell _{0}}^{(0)},a_{1}^{(1)},\dots ,a_{\ell _{1}}^{(1)},\dots ,a_{1}^{(j)},\dots ,a_{\ell _{j}}^{(j)},\dots \right)} where ℓ j = ( k + j − 1 j ) {\displaystyle \ell _{j}={\tbinom {k+j-1}{j}}} , a ℓ ( j ) = x 1 n 1 ⋯ x k n k n 1 ! ⋯ n k ! | n 1 + n 2 + ⋯ + n k = j ∧ 1 ≤ ℓ ≤ ℓ j {\displaystyle a_{\ell }^{(j)}={\frac {x_{1}^{n_{1}}\cdots x_{k}^{n_{k}}}{\sqrt {n_{1}!\cdots n_{k}!}}}\quad |\quad n_{1}+n_{2}+\dots +n_{k}=j\wedge 1\leq \ell \leq \ell _{j}} == Approximations == Because support vector machines and other models employing the kernel trick do not scale well to large numbers of training samples or large numbers of features in the input space, several approximations to the RBF kernel (and similar kernels) have been introduced. Typically, these take the form of a function z that maps a single vector to a vector of higher dimensionality, approximating the kernel: ⟨ z ( x ) , z ( x ′ ) ⟩ ≈ ⟨ φ ( x ) , φ ( x ′ ) ⟩ = K ( x , x ′ ) {\displaystyle \langle z(\mathbf {x} ),z(\mathbf {x'} )\rangle \approx \langle \varphi (\mathbf {x} ),\varphi (\mathbf {x'} )\rangle =K(\mathbf {x} ,\mathbf {x'} )} where φ {\displaystyle \textstyle \varphi } is the implicit mapping embedded in the RBF kernel. === Fourier random features === One way to construct such a z is to randomly sample from the Fourier transformation of the kernel φ ( x ) = 1 D [ cos ⟨ w 1 , x ⟩ , sin ⟨ w 1 , x ⟩ , … , cos ⟨ w D , x ⟩ , sin ⟨ w D , x ⟩ ] T {\displaystyle \varphi (x)={\frac {1}{\sqrt {D}}}[\cos \langle w_{1},x\rangle ,\sin \langle w_{1},x\rangle ,\ldots ,\cos \langle w_{D},x\rangle ,\sin \langle w_{D},x\rangle ]^{T}} where w 1 , . . . , w D {\displaystyle w_{1},...,w_{D}} are independent samples from the normal distribution N ( 0 , σ − 2 I ) {\displaystyle N(0,\sigma ^{-2}I)} . Theorem: E [ ⟨ φ ( x ) , φ ( y ) ⟩ ] = e ‖ x − y ‖ 2 / ( 2 σ 2 ) . {\displaystyle \operatorname {E} [\langle \varphi (x),\varphi (y)\rangle ]=e^{\|x-y\|^{2}/(2\sigma ^{2})}.} Proof: It suffices to prove the case of D = 1 {\displaystyle D=1} . Use the trigonometric identity cos ( a − b ) = cos ( a ) cos ( b ) + sin ( a ) sin ( b ) {\displaystyle \cos(a-b)=\cos(a)\cos(b)+\sin(a)\sin(b)} , the spherical symmetry of Gaussian distribution, then evaluate the integral ∫ − ∞ ∞ cos ( k x ) e − x 2 / 2 2 π d x = e − k 2 / 2 . {\displaystyle \int _{-\infty }^{\infty }{\frac {\cos(kx)e^{-x^{2}/2}}{\sqrt {2\pi }}}dx=e^{-k^{2}/2}.} Theorem: Var [ ⟨ φ ( x ) , φ ( y ) ⟩ ] = O ( D − 1 ) {\displaystyle \operatorname {Var} [\langle \varphi (x),\varphi (y)\rangle ]=O(D^{-1})} . (Appendix A.2). === Nyström method === Another approach uses the Nyström method to approximate the eigendecomposition of the Gram matrix K, using only a random sample of the training set.
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XGBoost (eXtreme Gradient Boosting) is an open-source software library which provides a regularizing gradient boosting framework for C++, Java, Python, R, Julia, Perl, and Scala. It works on Linux, Microsoft Windows, and macOS. From the project description, it aims to provide a "Scalable, Portable and Distributed Gradient Boosting (GBM, GBRT, GBDT) Library". It runs on a single machine, as well as the distributed processing frameworks Apache Hadoop, Apache Spark, Apache Flink, and Dask. XGBoost gained much popularity and attention in the mid-2010s as the algorithm of choice for many winning teams of machine learning competitions. == History == XGBoost initially started as a research project by Tianqi Chen as part of the Distributed (Deep) Machine Learning Community (DMLC) group at the University of Washington. Initially, it began as a terminal application which could be configured using a libsvm configuration file. It became well known in the ML competition circles after its use in the winning solution of the Higgs Machine Learning Challenge. Soon after, the Python and R packages were built, and XGBoost now has package implementations for Java, Scala, Julia, Perl, and other languages. This brought the library to more developers and contributed to its popularity among the Kaggle community, where it has been used for a large number of competitions. It was soon integrated with a number of other packages making it easier to use in their respective communities. It has now been integrated with scikit-learn for Python users and with the caret package for R users. It can also be integrated into Data Flow frameworks like Apache Spark, Apache Hadoop, and Apache Flink using the abstracted Rabit and XGBoost4J. XGBoost is also available on OpenCL for FPGAs. An efficient, scalable implementation of XGBoost has been published by Tianqi Chen and Carlos Guestrin. While the XGBoost model often achieves higher accuracy than a single decision tree, it sacrifices the intrinsic interpretability of decision trees. For example, following the path that a decision tree takes to make its decision is trivial and self-explained, but following the paths of hundreds or thousands of trees is much harder. == Features == Salient features of XGBoost which make it different from other gradient boosting algorithms include: Clever penalization of trees A proportional shrinking of leaf nodes Newton Boosting Extra randomization parameter Implementation on single, distributed systems and out-of-core computation Automatic feature selection Theoretically justified weighted quantile sketching for efficient computation Parallel tree structure boosting with sparsity Efficient cacheable block structure for decision tree training == The algorithm == XGBoost works as Newton–Raphson in function space unlike gradient boosting that works as gradient descent in function space, a second order Taylor approximation is used in the loss function to make the connection to Newton–Raphson method. A generic unregularized XGBoost algorithm is: == Parameters == XGBoost has parameters which can be specified to affect how it functions and performs. Some parameters include: Learning rate (also known as the "step size" or “"shrinkage"), it is a number between 0 and 1 (default is 0.3), which determines the rate the algorithm learns from each iteration. n_estimators, sets the number of trees to be built in the ensemble, where more trees generally increases the complexity of the model, but can lead to overfitting with too many trees. Gamma (also known as Lagrange multiplier or the minimum loss reduction parameter), controls the minimum amount of loss reduction required to make a further split on a leaf node of the tree. The default in XGBoost is 0 . max_depth, represents how deeply each tree in the boosting process can grow during training, where the default is 6. == Awards == John Chambers Award (2016) High Energy Physics meets Machine Learning award (HEP meets ML) (2016)
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In statistics, a quadratic classifier is a statistical classifier that uses a quadratic decision surface to separate measurements of two or more classes of objects or events. It is a more general version of the linear classifier. == The classification problem == Statistical classification considers a set of vectors of observations x of an object or event, each of which has a known type y. This set is referred to as the training set. The problem is then to determine, for a given new observation vector, what the best class should be. For a quadratic classifier, the correct solution is assumed to be quadratic in the measurements, so y will be decided based on x T A x + b T x + c {\displaystyle \mathbf {x^{T}Ax} +\mathbf {b^{T}x} +c} In the special case where each observation consists of two measurements, this means that the surfaces separating the classes will be conic sections (i.e., either a line, a circle or ellipse, a parabola or a hyperbola). In this sense, we can state that a quadratic model is a generalization of the linear model, and its use is justified by the desire to extend the classifier's ability to represent more complex separating surfaces. == Quadratic discriminant analysis == Quadratic discriminant analysis (QDA) is closely related to linear discriminant analysis (LDA), where it is assumed that the measurements from each class are normally distributed. Unlike LDA however, in QDA there is no assumption that the covariance of each of the classes is identical. When the normality assumption is true, the best possible test for the hypothesis that a given measurement is from a given class is the likelihood ratio test. Suppose there are only two groups, with means μ 0 , μ 1 {\displaystyle \mu _{0},\mu _{1}} and covariance matrices Σ 0 , Σ 1 {\displaystyle \Sigma _{0},\Sigma _{1}} corresponding to y = 0 {\displaystyle y=0} and y = 1 {\displaystyle y=1} respectively. Then the likelihood ratio is given by Likelihood ratio = | 2 π Σ 1 | − 1 exp ( − 1 2 ( x − μ 1 ) T Σ 1 − 1 ( x − μ 1 ) ) | 2 π Σ 0 | − 1 exp ( − 1 2 ( x − μ 0 ) T Σ 0 − 1 ( x − μ 0 ) ) < t {\displaystyle {\text{Likelihood ratio}}={\frac {{\sqrt {|2\pi \Sigma _{1}|}}^{-1}\exp \left(-{\frac {1}{2}}(\mathbf {x} -{\boldsymbol {\mu }}_{1})^{T}\Sigma _{1}^{-1}(\mathbf {x} -{\boldsymbol {\mu }}_{1})\right)}{{\sqrt {|2\pi \Sigma _{0}|}}^{-1}\exp \left(-{\frac {1}{2}}(\mathbf {x} -{\boldsymbol {\mu }}_{0})^{T}\Sigma _{0}^{-1}(\mathbf {x} -{\boldsymbol {\mu }}_{0})\right)}}
iMazing is mobile device management software that allows users to transfer files and data between iOS devices (iPhone, iPad and iPod Touch) and macOS or Windows computers, in addition to many other features beyond the scope of what Apple's own tools enable. == History == Developed by DigiDNA, iMazing was initially released in 2008 as DiskAid, enabling users to transfer data and files from the iPhone or iPod Touch to Mac or Windows computers. DiskAid was renamed iMazing in 2014. Version 2.0 was released on September 13, 2016. In August 2021, version 2.14 of iMazing added a spyware detection feature. The feature is based on Amnesty International’s Mobile Verification Toolkit to detect Pegasus Spyware following the publication of Pegasus Project. == Description == With iMazing, an iPhone or iPad can be used similarly to an external hard drive. It performs tasks that iTunes doesn’t offer, including incremental backups of iOS devices, browsing and exporting text and voicemail messages, managing apps, encryption, and migrating data from an old phone to a new one. The menu bar app iMazing Mini enables automatic, wireless and encrypted backups of iPhones. The iMazing HEIC Converter is a free desktop app for Mac and PC that lets users convert photos from HEIC format to JPG or PNG.
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Waikato Environment for Knowledge Analysis (Weka) is a collection of machine learning and data analysis free software licensed under the GNU General Public License. It was developed at the University of Waikato, New Zealand, and is the companion software to the book "Data Mining: Practical Machine Learning Tools and Techniques". == Description == Weka contains a collection of visualization tools and algorithms for data analysis and predictive modeling, together with graphical user interfaces for easy access to these functions. The original non-Java version of Weka was a Tcl/Tk front-end to (mostly third-party) modeling algorithms implemented in other programming languages, plus data preprocessing utilities in C, and a makefile-based system for running machine learning experiments. This original version was primarily designed as a tool for analyzing data from agricultural domains, but the more recent fully Java-based version (Weka 3), for which development started in 1997, is now used in many different application areas, in particular for educational purposes and research. Advantages of Weka include: Free availability under the GNU General Public License. Portability, since it is fully implemented in the Java programming language and thus runs on almost any modern computing platform. A comprehensive collection of data preprocessing and modeling techniques. Ease of use due to its graphical user interfaces. Weka supports several standard data mining tasks, more specifically, data preprocessing, clustering, classification, regression, visualization, and feature selection. Input to Weka is expected to be formatted according the Attribute-Relational File Format and with the filename bearing the .arff extension. All of Weka's techniques are predicated on the assumption that the data is available as one flat file or relation, where each data point is described by a fixed number of attributes (normally, numeric or nominal attributes, but some other attribute types are also supported). Weka provides access to SQL databases using Java Database Connectivity and can process the result returned by a database query. Weka provides access to deep learning with Deeplearning4j. It is not capable of multi-relational data mining, but there is separate software for converting a collection of linked database tables into a single table that is suitable for processing using Weka. Another important area that is currently not covered by the algorithms included in the Weka distribution is sequence modeling. == Extension packages == In version 3.7.2, a package manager was added to allow the easier installation of extension packages. Some functionality that used to be included with Weka prior to this version has since been moved into such extension packages, but this change also makes it easier for others to contribute extensions to Weka and to maintain the software, as this modular architecture allows independent updates of the Weka core and individual extensions. == History == In 1993, the University of Waikato in New Zealand began development of the original version of Weka, which became a mix of Tcl/Tk, C, and makefiles. In 1997, the decision was made to redevelop Weka from scratch in Java, including implementations of modeling algorithms. In 2005, Weka received the SIGKDD Data Mining and Knowledge Discovery Service Award. In 2006, Pentaho Corporation acquired an exclusive licence to use Weka for business intelligence. It forms the data mining and predictive analytics component of the Pentaho business intelligence suite. Pentaho has since been acquired by Hitachi Vantara, and Weka now underpins the PMI (Plugin for Machine Intelligence) open source component. == Related tools == Auto-WEKA is an automated machine learning system for Weka. Environment for DeveLoping KDD-Applications Supported by Index-Structures (ELKI) is a similar project to Weka with a focus on cluster analysis, i.e., unsupervised methods. H2O.ai is an open-source data science and machine learning platform KNIME is a machine learning and data mining software implemented in Java. Massive Online Analysis (MOA) is an open-source project for large scale mining of data streams, also developed at the University of Waikato in New Zealand. Neural Designer is a data mining software based on deep learning techniques written in C++. Orange is a similar open-source project for data mining, machine learning and visualization based on scikit-learn. RapidMiner is a commercial machine learning framework implemented in Java which integrates Weka. scikit-learn is a popular machine learning library in Python.
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Implicit blockmodeling is an approach in blockmodeling, similar to a valued and homogeneity blockmodeling, where initially an additional normalization is used and then while specifying the parameter of the relevant link is replaced by the block maximum. This approach was first proposed by Batagelj and Ferligoj in 2000, and developed by Aleš Žiberna in 2007/08. Comparing with homogeneity, the implicit blockmodeling will perform similarly with max-regular equivalence, but slightly worse in other settings. It will perform worse than valued and homogeneity blockmodeling with a pre-specified blockmodel.
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In computational learning theory, the teaching dimension of a concept class C is defined to be max c ∈ C { w C ( c ) } {\displaystyle \max _{c\in C}\{w_{C}(c)\}} , where w C ( c ) {\displaystyle {w_{C}(c)}} is the minimum size of a witness set for c in C. Intuitively, this measures the number of instances that are needed to identify a concept in the class, using supervised learning with examples provided by a helpful teacher who is trying to convey the concept as succinctly as possible. This definition was formulated in 1995 by Sally Goldman and Michael Kearns, based on earlier work by Goldman, Ron Rivest, and Robert Schapire. The teaching dimension of a finite concept class can be used to give a lower and an upper bound on the membership query cost of the concept class. In Stasys Jukna's book "Extremal Combinatorics", a lower bound is given for the teaching dimension in general: Let C be a concept class over a finite domain X. If the size of C is greater than 2 k ( | X | k ) , {\displaystyle 2^{k}{|X| \choose k},} then the teaching dimension of C is greater than k. However, there are more specific teaching models that make assumptions about teacher or learner, and can get lower values for the teaching dimension. For instance, several models are the classical teaching (CT) model, the optimal teacher (OT) model, recursive teaching (RT), preference-based teaching (PBT), and non-clashing teaching (NCT).
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Cozi is a family organization website and mobile app designed to streamline household management. It offers shared calendars, to-do lists, shopping lists, and messaging tools, allowing multiple users to coordinate under one account. Founded in 2005 by former Microsoft employees, Cozi has evolved through acquisitions and now operates under OurFamilyWizard. The app is available in both free and premium versions on iOS, Android, and desktop platforms. == History == Cozi was founded in 2005 by Robbie Cape and Jan Miksovsky, two former Microsoft employees who sought to simplify family logistics with technology. The company's first product, Cozi Central, was released on September 25, 2006, and included a family calendar, shopping lists, family messaging and a photo collage screensaver. The company is based in Seattle, Washington. Cozi has both a freemium version, and a paid version called Cozi Gold. Cozi Gold's additional features include Cozi Contacts, a birthday tracker, more reminders, mobile month view, and change notifications. The software can be used on desktop or mobile applications for iOS and Android. On June 5, 2011, Cozi set a Guinness World Record for the longest line of ducks in a row. The line stretched for one mile and was made up of 17,782 rubber ducks. Cozi was acquired by Time Inc. in 2014. After the Meredith Corporation acquired Time in 2018, Cozi was moved into the Parents Network division. On May 4, 2022, Cozi was acquired by OurFamilyWizard of Minneapolis, Minnesota, reporting more than 20 million registered users.
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An oscillatory neural network (ONN) is an artificial neural network that uses coupled oscillators as neurons. Oscillatory neural networks are closely linked to the Kuramoto model, and are inspired by the phenomenon of neural oscillations in the brain. Oscillatory neural networks have been trained to recognize images. Complex-Valued Oscillatory network has also been shown to store and retrieve multidimensional aperiodic signals. An oscillatory autoencoder has also been demonstrated, which uses a combination of oscillators and rate-coded neurons. A neuron made of two coupled oscillators, one having a fixed and the other having a tunable natural frequency, has been shown able to run logic gates such as XOR that conventional sigmoid neurons cannot.
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Multidimensional scaling (MDS) is a means of visualizing the level of similarity of individual cases of a data set. MDS is used to translate distances between each pair of n {\textstyle n} objects in a set into a configuration of n {\textstyle n} points mapped into an abstract Cartesian space. More technically, MDS refers to a set of related ordination techniques used in information visualization, in particular to display the information contained in a distance matrix. It is a form of non-linear dimensionality reduction. Given a distance matrix with the distances between each pair of objects in a set, and a chosen number of dimensions, N, an MDS algorithm places each object into N-dimensional space (a lower-dimensional representation) such that the between-object distances are preserved as well as possible. For N = 1, 2, and 3, the resulting points can be visualized on a scatter plot. Core theoretical contributions to MDS were made by James O. Ramsay of McGill University, who is also regarded as the founder of functional data analysis. == Types == MDS algorithms fall into a taxonomy, depending on the meaning of the input matrix: === Classical multidimensional scaling === It is also known as Principal Coordinates Analysis (PCoA), Torgerson Scaling or Torgerson–Gower scaling. It takes an input matrix giving dissimilarities between pairs of items and outputs a coordinate matrix whose configuration minimizes a loss function called strain, which is given by Strain D ( x 1 , x 2 , . . . , x n ) = ( ∑ i , j ( b i j − x i T x j ) 2 ∑ i , j b i j 2 ) 1 / 2 , {\displaystyle {\text{Strain}}_{D}(x_{1},x_{2},...,x_{n})={\Biggl (}{\frac {\sum _{i,j}{\bigl (}b_{ij}-x_{i}^{T}x_{j}{\bigr )}^{2}}{\sum _{i,j}b_{ij}^{2}}}{\Biggr )}^{1/2},} where x i {\displaystyle x_{i}} denote vectors in N-dimensional space, x i T x j {\displaystyle x_{i}^{T}x_{j}} denotes the scalar product between x i {\displaystyle x_{i}} and x j {\displaystyle x_{j}} , and b i j {\displaystyle b_{ij}} are the elements of the matrix B {\displaystyle B} defined on step 2 of the following algorithm, which are computed from the distances. Steps of a Classical MDS algorithm: Classical MDS uses the fact that the coordinate matrix X {\displaystyle X} can be derived by eigenvalue decomposition from B = X X ′ {\textstyle B=XX'} . And the matrix B {\textstyle B} can be computed from proximity matrix D {\textstyle D} by using double centering. Set up the squared proximity matrix D ( 2 ) = [ d i j 2 ] {\textstyle D^{(2)}=[d_{ij}^{2}]} Apply double centering: B = − 1 2 C D ( 2 ) C {\textstyle B=-{\frac {1}{2}}CD^{(2)}C} using the centering matrix C = I − 1 n J n {\textstyle C=I-{\frac {1}{n}}J_{n}} , where n {\textstyle n} is the number of objects, I {\textstyle I} is the n × n {\textstyle n\times n} identity matrix, and J n {\textstyle J_{n}} is an n × n {\textstyle n\times n} matrix of all ones. Determine the m {\textstyle m} largest eigenvalues λ 1 , λ 2 , . . . , λ m {\textstyle \lambda _{1},\lambda _{2},...,\lambda _{m}} and corresponding eigenvectors e 1 , e 2 , . . . , e m {\textstyle e_{1},e_{2},...,e_{m}} of B {\textstyle B} (where m {\textstyle m} is the number of dimensions desired for the output). Now, X = E m Λ m 1 / 2 {\textstyle X=E_{m}\Lambda _{m}^{1/2}} , where E m {\textstyle E_{m}} is the matrix of m {\textstyle m} eigenvectors and Λ m {\textstyle \Lambda _{m}} is the diagonal matrix of m {\textstyle m} eigenvalues of B {\textstyle B} . Classical MDS assumes metric distances. So this is not applicable for direct dissimilarity ratings. === Metric multidimensional scaling (mMDS) === It is a superset of classical MDS that generalizes the optimization procedure to a variety of loss functions and input matrices of known distances with weights and so on. A useful loss function in this context is called stress, which is often minimized using a procedure called stress majorization. Metric MDS minimizes the cost function called “stress” which is a residual sum of squares: Stress D ( x 1 , x 2 , . . . , x n ) = ∑ i ≠ j = 1 , . . . , n ( d i j − ‖ x i − x j ‖ ) 2 . {\displaystyle {\text{Stress}}_{D}(x_{1},x_{2},...,x_{n})={\sqrt {\sum _{i\neq j=1,...,n}{\bigl (}d_{ij}-\|x_{i}-x_{j}\|{\bigr )}^{2}}}.} Metric scaling uses a power transformation with a user-controlled exponent p {\textstyle p} : d i j p {\textstyle d_{ij}^{p}} and − d i j 2 p {\textstyle -d_{ij}^{2p}} for distance. In classical scaling p = 1. {\textstyle p=1.} Non-metric scaling is defined by the use of isotonic regression to nonparametrically estimate a transformation of the dissimilarities. === Non-metric multidimensional scaling (NMDS) === In contrast to metric MDS, non-metric MDS finds both a non-parametric monotonic relationship between the dissimilarities in the item-item matrix and the Euclidean distances between items, and the location of each item in the low-dimensional space. Let d i j {\displaystyle d_{ij}} be the dissimilarity between points i , j {\displaystyle i,j} . Let d ^ i j = ‖ x i − x j ‖ {\displaystyle {\hat {d}}_{ij}=\|x_{i}-x_{j}\|} be the Euclidean distance between embedded points x i , x j {\displaystyle x_{i},x_{j}} . Now, for each choice of the embedded points x i {\displaystyle x_{i}} and is a monotonically increasing function f {\displaystyle f} , define the "stress" function: S ( x 1 , . . . , x n ; f ) = ∑ i < j ( f ( d i j ) − d ^ i j ) 2 ∑ i < j d ^ i j 2 . {\displaystyle S(x_{1},...,x_{n};f)={\sqrt {\frac {\sum _{i