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Section: Definition. Inductive programming incorporates all approaches which are concerned with learning programs or algorithms from incomplete (formal) specifications. Possible inputs in an IP system are a set of training inputs and corresponding outputs or an output evaluation function, describing the desired behavio... | Wikipedia - Inductive programming - Definition | 316 | 1,796 | null |
Section: History. Research on the inductive synthesis of recursive functional programs started in the early 1970s and was brought onto firm theoretical foundations with the seminal THESIS system of Summers and work of Biermann. These approaches were split into two phases: first, input-output examples are transformed in... | Wikipedia - Inductive programming - History | 340 | 1,735 | null |
But again, after initial success, the community got disappointed by limited progress about the induction of recursive programs with ILP less and less focusing on recursive programs and leaning more and more towards a machine learning setting with applications in relational data mining and knowledge discovery. In parall... | Wikipedia - Inductive programming - History | 332 | 1,800 | null |
Therefore, the synthesis problem has been reformulated on the background of constructor-based term rewriting systems taking into account modern techniques of functional programming, as well as moderate use of search-based strategies and usage of background knowledge as well as automatic invention of subprograms. Many n... | Wikipedia - Inductive programming - History | 218 | 1,194 | null |
Section: Application areas. The first workshop on Approaches and Applications of Inductive Programming (AAIP) held in conjunction with ICML 2005 identified all applications where "learning of programs or recursive rules are called for, [...] first in the domain of software engineering where structural learning, softwar... | Wikipedia - Inductive programming - Application areas | 229 | 1,291 | null |
Article: Inferential theory of learning. Inferential Theory of Learning (ITL) is an area of machine learning which describes inferential processes performed by learning agents. ITL has been continuously developed by Ryszard S. Michalski, starting in the 1980s. The first known publication of ITL was in 1983. In the ITL ... | Wikipedia - Inferential theory of learning - Summary | 185 | 927 | null |
Article: Instance selection. Instance selection (or dataset reduction, or dataset condensation) is an important data pre-processing step that can be applied in many machine learning (or data mining) tasks. Approaches for instance selection can be applied for reducing the original dataset to a manageable volume, leading... | Wikipedia - Instance selection - Summary | 212 | 1,200 | null |
Section: Instance selection algorithms. The literature provides several different algorithms for instance selection. They can be distinguished from each other according to several different criteria. Considering this, instance selection algorithms can be grouped in two main classes, according to what instances they sel... | Wikipedia - Instance selection - Instance selection algorithms | 341 | 1,765 | null |
Besides that, since they search by the representative instances in each class separately, they are faster (in terms of time complexity and effective running time) than other algorithms, such as DROP3 and ICF. Besides that, there is a third category of algorithms that, instead of selecting actual instances of the datase... | Wikipedia - Instance selection - Instance selection algorithms | 166 | 827 | null |
Article: Instance-based learning. In machine learning, instance-based learning (sometimes called memory-based learning) is a family of learning algorithms that, instead of performing explicit generalization, compare new problem instances with instances seen in training, which have been stored in memory. Because computa... | Wikipedia - Instance-based learning - Summary | 283 | 1,489 | null |
Section: Applications. The technology is used to process unstructured content. Common real-world applications include self-driving cars, self-checkouts at grocery stores, smart home assistants, and appliances. Businesses can apply data and machine learning to build predictive analytics that react to consumer behavior c... | Wikipedia - Intelligent automation - Applications | 305 | 1,653 | null |
Section: Benefits. Streamline Processes Repetitive manual tasks can put a strain on the workforce, these tasks can be automated to allow the workforce to work on more important matters that require human cognition. Intelligent automation can also be used to mitigate tasks with human error which in turn increases profic... | Wikipedia - Intelligent automation - Benefits | 188 | 1,084 | null |
Section: Formal definitions. Let D {\textstyle D} be a distribution over vectors in the vector space R n {\textstyle \mathbb {R} ^{n}} . Then D {\textstyle D} is in isotropic position if, for vector v {\textstyle v} sampled from the distribution, E v v T = I d . {\displaystyle \mathbb {E} \,vv^{\mathsf {T}}=\mathrm {Id... | Wikipedia - Isotropic position - Formal definitions | 347 | 1,013 | null |
Article: JAX (software). JAX is a Python library for accelerator-oriented array computation and program transformation, designed for high-performance numerical computing and large-scale machine learning. It is developed by Google with contributions from Nvidia and other community contributors. It is described as bringi... | Wikipedia - JAX (software) - Summary | 216 | 1,092 | null |
Section: History. The journal was established as an open-access alternative to the journal Machine Learning. In 2001, forty editorial board members of Machine Learning resigned, saying that in the era of the Internet, it was detrimental for researchers to continue publishing their papers in expensive journals with pay-... | Wikipedia - Journal of Machine Learning Research - History | 204 | 1,205 | null |
Article: Kernel density estimation. In statistics, kernel density estimation (KDE) is the application of kernel smoothing for probability density estimation, i.e., a non-parametric method to estimate the probability density function of a random variable based on kernels as weights. KDE answers a fundamental data smooth... | Wikipedia - Kernel density estimation - Summary | 165 | 849 | null |
Section: Definition. Let (x1, x2, ..., xn) be independent and identically distributed samples drawn from some univariate distribution with an unknown density f at any given point x. We are interested in estimating the shape of this function f. Its kernel density estimator is f ^ h ( x ) = 1 n ∑ i = 1 n K h ( x − x i ) ... | Wikipedia - Kernel density estimation - Definition | 349 | 1,067 | null |
A range of kernel functions are commonly used: uniform, triangular, biweight, triweight, Epanechnikov (parabolic), normal, and others. The Epanechnikov kernel is optimal in a mean square error sense, though the loss of efficiency is small for the kernels listed previously. Due to its convenient mathematical properties,... | Wikipedia - Kernel density estimation - Definition | 339 | 1,112 | null |
Section: Example. Kernel density estimates are closely related to histograms, but can be endowed with properties such as smoothness or continuity by using a suitable kernel. The diagram below based on these 6 data points illustrates this relationship: For the histogram, first, the horizontal axis is divided into sub-in... | Wikipedia - Kernel density estimation - Example | 219 | 1,050 | null |
Section: Bandwidth selection. The bandwidth of the kernel is a free parameter which exhibits a strong influence on the resulting estimate. To illustrate its effect, we take a simulated random sample from the standard normal distribution (plotted at the blue spikes in the rug plot on the horizontal axis). The grey curve... | Wikipedia - Kernel density estimation - Bandwidth selection | 249 | 1,156 | null |
In the other extreme limit h → ∞ {\displaystyle h\to \infty } the estimate retains the shape of the used kernel, centered on the mean of the samples (completely smooth). The most common optimality criterion used to select this parameter is the expected L2 risk function, also termed the mean integrated squared error: MI... | Wikipedia - Kernel density estimation - Bandwidth selection | 300 | 863 | null |
The AMISE is the asymptotic MISE, i. e. the two leading terms, AMISE ( h ) = R ( K ) n h + 1 4 m 2 ( K ) 2 h 4 R ( f ″ ) {\displaystyle \operatorname {AMISE} (h)={\frac {R(K)}{nh}}+{\frac {1}{4}}m_{2}(K)^{2}h^{4}R(f'')} where R ( g ) = ∫ g ( x ) 2 d x {\textstyle R(g)=\int g(x)^{2}\,dx} for a function g, m 2 ( K ) = ... | Wikipedia - Kernel density estimation - Bandwidth selection | 236 | 493 | null |
the two leading terms, AMISE ( h ) = R ( K ) n h + 1 4 m 2 ( K ) 2 h 4 R ( f ″ ) {\displaystyle \operatorname {AMISE} (h)={\frac {R(K)}{nh}}+{\frac {1}{4}}m_{2}(K)^{2}h^{4}R(f'')} where R ( g ) = ∫ g ( x ) 2 d x {\textstyle R(g)=\int g(x)^{2}\,dx} for a function g, m 2 ( K ) = ∫ x 2 K ( x ) d x {\textstyle m_{2}(K)=\... | Wikipedia - Kernel density estimation - Bandwidth selection | 502 | 1,093 | null |
The minimum of this AMISE is the solution to this differential equation ∂ ∂ h AMISE ( h ) = − R ( K ) n h 2 + m 2 ( K ) 2 h 3 R ( f ″ ) = 0 {\displaystyle {\frac {\partial }{\partial h}}\operatorname {AMISE} (h)=-{\frac {R(K)}{nh^{2}}}+m_{2}(K)^{2}h^{3}R(f'')=0} or h AMISE = R ( K ) 1 / 5 m 2 ( K ) 2 / 5 R ( f ″ ) 1 ... | Wikipedia - Kernel density estimation - Bandwidth selection | 346 | 966 | null |
Several review studies have been undertaken to compare their efficacies, with the general consensus that the plug-in selectors and cross validation selectors are the most useful over a wide range of data sets. Substituting any bandwidth h which has the same asymptotic order n−1/5 as hAMISE into the AMISE gives that AMI... | Wikipedia - Kernel density estimation - Bandwidth selection | 216 | 989 | null |
Section: Bandwidth selection > A rule-of-thumb bandwidth estimator. If Gaussian basis functions are used to approximate univariate data, and the underlying density being estimated is Gaussian, the optimal choice for h (that is, the bandwidth that minimises the mean integrated squared error) is: h = ( 4 σ ^ 5 3 n ) 1 / ... | Wikipedia - Kernel density estimation - Bandwidth selection > A rule-of-thumb bandwidth estimator | 323 | 1,009 | null |
Another modification that will improve the model is to reduce the factor from 1.06 to 0.9. Then the final formula would be: h = 0.9 min ( σ ^ , I Q R 1.34 ) n − 1 / 5 {\displaystyle h=0.9\,\min \left({\hat {\sigma }},{\frac {\mathrm {IQR} }{1.34}}\right)\,n^{-1/5}} where n {\displaystyle n} is the sample size. This app... | Wikipedia - Kernel density estimation - Bandwidth selection > A rule-of-thumb bandwidth estimator | 177 | 600 | null |
While this rule of thumb is easy to compute, it should be used with caution as it can yield widely inaccurate estimates when the density is not close to being normal. For example, when estimating the bimodal Gaussian mixture model 1 2 2 π e − 1 2 ( x − 10 ) 2 + 1 2 2 π e − 1 2 ( x + 10 ) 2 {\displaystyle {\frac {1}{2{\... | Wikipedia - Kernel density estimation - Bandwidth selection > A rule-of-thumb bandwidth estimator | 237 | 707 | null |
Section: Relation to the characteristic function density estimator. Given the sample (x1, x2, ..., xn), it is natural to estimate the characteristic function φ(t) = E[eitX] as φ ^ ( t ) = 1 n ∑ j = 1 n e i t x j {\displaystyle {\hat {\varphi }}(t)={\frac {1}{n}}\sum _{j=1}^{n}e^{itx_{j}}} Knowing the characteristic fun... | Wikipedia - Kernel density estimation - Relation to the characteristic function density estimator | 298 | 956 | null |
The "bandwidth parameter" h controls how fast we try to dampen the function φ ^ ( t ) {\displaystyle {\hat {\varphi }}(t)} . In particular when h is small, then ψh(t) will be approximately one for a large range of t's, which means that φ ^ ( t ) {\displaystyle {\hat {\varphi }}(t)} remains practically unaltered in the ... | Wikipedia - Kernel density estimation - Relation to the characteristic function density estimator | 175 | 583 | null |
Once the function ψ has been chosen, the inversion formula may be applied, and the density estimator will be f ^ ( x ) = 1 2 π ∫ − ∞ + ∞ φ ^ ( t ) ψ h ( t ) e − i t x d t = 1 2 π ∫ − ∞ + ∞ 1 n ∑ j = 1 n e i t ( x j − x ) ψ ( h t ) d t = 1 n h ∑ j = 1 n 1 2 π ∫ − ∞ + ∞ e − i ( h t ) x − x j h ψ ( h t ) d ( h t ) = 1 n h... | Wikipedia - Kernel density estimation - Relation to the characteristic function density estimator | 347 | 640 | null |
{1}{2\pi }}\int _{-\infty }^{+\infty }e^{-i(ht){\frac {x-x_{j}}{h}}}\psi (ht)\,d(ht)\\[1ex]&={\frac {1}{nh}}\sum _{j=1}^{n}K{\left({\frac {x-x_{j}}{h}}\right)},\end{aligned}}} where K is the Fourier transform of the damping function ψ. Thus the kernel density estimator coincides with the characteristic function density... | Wikipedia - Kernel density estimation - Relation to the characteristic function density estimator | 163 | 331 | null |
Section: Geometric and topological features. We can extend the definition of the (global) mode to a local sense and define the local modes: M = { x : g ( x ) = 0 ∣ λ 1 ( x ) < 0 } {\displaystyle M=\{x:g(x)=0\mid \lambda _{1}(x)<0\}} Namely, M {\displaystyle M} is the collection of points for which the density function ... | Wikipedia - Kernel density estimation - Geometric and topological features | 260 | 796 | null |
Section: Statistical implementation. A non-exhaustive list of software implementations of kernel density estimators includes: In Analytica release 4.4, the Smoothing option for PDF results uses KDE, and from expressions it is available via the built-in Pdf function. In C/C++, FIGTree is a library that can be used to co... | Wikipedia - Kernel density estimation - Statistical implementation | 349 | 1,617 | null |
In Excel, the Royal Society of Chemistry has created an add-in to run kernel density estimation based on their Analytical Methods Committee Technical Brief 4. In gnuplot, kernel density estimation is implemented by the smooth kdensity option, the datafile can contain a weight and bandwidth for each point, or the bandwi... | Wikipedia - Kernel density estimation - Statistical implementation | 347 | 1,565 | null |
In KNIME, 1D and 2D Kernel Density distributions can be generated and plotted using nodes from the Vernalis community contribution, e.g. 1D Kernel Density Plot, among others. The underlying implementation is written in Java. In MATLAB, kernel density estimation is implemented through the ksdensity function (Statistics ... | Wikipedia - Kernel density estimation - Statistical implementation | 345 | 1,692 | null |
In Octave, kernel density estimation is implemented by the kernel_density option (econometrics package). In Origin, 2D kernel density plot can be made from its user interface, and two functions, Ksdensity for 1D and Ks2density for 2D can be used from its LabTalk, Python, or C code. In Perl, an implementation can be fou... | Wikipedia - Kernel density estimation - Statistical implementation | 349 | 1,369 | null |
In R, it is implemented through density in the base distribution, and bw.nrd0 function is used in stats package, this function uses the optimized formula in Silverman's book. bkde in the KernSmooth library, ParetoDensityEstimation in the DataVisualizations library (for pareto distribution density estimation), kde in th... | Wikipedia - Kernel density estimation - Statistical implementation | 285 | 1,203 | null |
Article: Kernel embedding of distributions. In machine learning, the kernel embedding of distributions (also called the kernel mean or mean map) comprises a class of nonparametric methods in which a probability distribution is represented as an element of a reproducing kernel Hilbert space (RKHS). A generalization of t... | Wikipedia - Kernel embedding of distributions - Summary | 347 | 1,726 | null |
The analysis of distributions is fundamental in machine learning and statistics, and many algorithms in these fields rely on information theoretic approaches such as entropy, mutual information, or Kullback–Leibler divergence. However, to estimate these quantities, one must first either perform density estimation, or e... | Wikipedia - Kernel embedding of distributions - Summary | 318 | 1,824 | null |
Methods based on the kernel embedding of distributions sidestep these problems and also possess the following advantages: Data may be modeled without restrictive assumptions about the form of the distributions and relationships between variables Intermediate density estimation is not needed Practitioners may specify th... | Wikipedia - Kernel embedding of distributions - Summary | 238 | 1,408 | null |
Section: Definitions. Let X {\displaystyle X} denote a random variable with domain Ω {\displaystyle \Omega } and distribution P {\displaystyle P} . Given a symmetric, positive-definite kernel k : Ω × Ω → R {\displaystyle k:\Omega \times \Omega \rightarrow \mathbb {R} } the Moore–Aronszajn theorem asserts the existence ... | Wikipedia - Kernel embedding of distributions - Definitions | 289 | 856 | null |
Given a symmetric, positive-definite kernel k : Ω × Ω → R {\displaystyle k:\Omega \times \Omega \rightarrow \mathbb {R} } the Moore–Aronszajn theorem asserts the existence of a unique RKHS H {\displaystyle {\mathcal {H}}} on Ω {\displaystyle \Omega } (a Hilbert space of functions f : Ω → R {\displaystyle f:\Omega \to \... | Wikipedia - Kernel embedding of distributions - Definitions | 473 | 1,272 | null |
{\displaystyle \langle f,k(x,\cdot )\rangle _{\mathcal {H}}=f(x)\qquad \forall f\in {\mathcal {H}},\quad \forall x\in \Omega .} One may alternatively consider x ↦ k ( x , ⋅ ) {\displaystyle x\mapsto k(x,\cdot )} as an implicit feature mapping φ : Ω → H {\displaystyle \varphi :\Omega \rightarrow {\mathcal {H}}} (which i... | Wikipedia - Kernel embedding of distributions - Definitions | 261 | 741 | null |
Section: Definitions > Kernel embedding. The kernel embedding of the distribution P {\displaystyle P} in H {\displaystyle {\mathcal {H}}} (also called the kernel mean or mean map) is given by: μ X := E [ k ( X , ⋅ ) ] = E [ φ ( X ) ] = ∫ Ω φ ( x ) d P ( x ) {\displaystyle \mu _{X}:=\mathbb {E} [k(X,\cdot )]=\mathbb {E}... | Wikipedia - Kernel embedding of distributions - Definitions > Kernel embedding | 332 | 989 | null |
Section: Definitions > Empirical kernel embedding. Given n {\displaystyle n} training examples { x 1 , … , x n } {\displaystyle \{x_{1},\ldots ,x_{n}\}} drawn independently and identically distributed (i.i.d.) from P , {\displaystyle P,} the kernel embedding of P {\displaystyle P} can be empirically estimated as μ ^ X ... | Wikipedia - Kernel embedding of distributions - Definitions > Empirical kernel embedding | 164 | 428 | null |
If Y {\displaystyle Y} denotes another random variable (for simplicity, assume the co-domain of Y {\displaystyle Y} is also Ω {\displaystyle \Omega } with the same kernel k {\displaystyle k} which satisfies ⟨ φ ( x ) ⊗ φ ( y ) , φ ( x ′ ) ⊗ φ ( y ′ ) ⟩ = k ( x , x ′ ) k ( y , y ′ ) {\displaystyle \langle \varphi (x)\ot... | Wikipedia - Kernel embedding of distributions - Definitions > Joint distribution embedding | 350 | 875 | null |
embedding may be interpreted as an uncentered cross-covariance operator C X Y : H → H {\displaystyle {\mathcal {C}}_{XY}:{\mathcal {H}}\to {\mathcal {H}}} from which the cross-covariance of functions f , g ∈ H {\displaystyle f,g\in {\mathcal {H}}} can be computed as Cov ( f ( X ) , g ( Y ) ) := E [ f ( X ) g ( Y ) ] ... | Wikipedia - Kernel embedding of distributions - Definitions > Joint distribution embedding | 345 | 750 | null |
\{(x_{1},y_{1}),\dots ,(x_{n},y_{n})\}} drawn i.i.d. from P {\displaystyle P} , we can also empirically estimate the joint distribution kernel embedding via C ^ X Y = 1 n ∑ i = 1 n φ ( x i ) ⊗ φ ( y i ) {\displaystyle {\widehat {\mathcal {C}}}_{XY}={\frac {1}{n}}\sum _{i=1}^{n}\varphi (x_{i})\otimes \varphi (y_{i})} | Wikipedia - Kernel embedding of distributions - Definitions > Joint distribution embedding | 159 | 317 | null |
Section: Definitions > Conditional distribution embedding. Given a conditional distribution P ( y ∣ x ) , {\displaystyle P(y\mid x),} one can define the corresponding RKHS embedding as μ Y ∣ x = E [ φ ( Y ) ∣ X ] = ∫ Ω φ ( y ) d P ( y ∣ x ) {\displaystyle \mu _{Y\mid x}=\mathbb {E} [\varphi (Y)\mid X]=\int _{\Omega }\v... | Wikipedia - Kernel embedding of distributions - Definitions > Conditional distribution embedding | 193 | 559 | null |
Given a conditional distribution P ( y ∣ x ) , {\displaystyle P(y\mid x),} one can define the corresponding RKHS embedding as μ Y ∣ x = E [ φ ( Y ) ∣ X ] = ∫ Ω φ ( y ) d P ( y ∣ x ) {\displaystyle \mu _{Y\mid x}=\mathbb {E} [\varphi (Y)\mid X]=\int _{\Omega }\varphi (y)\ \mathrm {d} P(y\mid x)} Note that the embedding ... | Wikipedia - Kernel embedding of distributions - Definitions > Conditional distribution embedding | 380 | 1,004 | null |
By fixing X {\displaystyle X} to a particular value, we obtain a single element in H {\displaystyle {\mathcal {H}}} , and thus it is natural to define the operator { C Y ∣ X : H → H C Y ∣ X = C Y X C X X − 1 {\displaystyle {\begin{cases}{\mathcal {C}}_{Y\mid X}:{\mathcal {H}}\to {\mathcal {H}}\\{\mathcal {C}}_{Y\mid X}... | Wikipedia - Kernel embedding of distributions - Definitions > Conditional distribution embedding | 345 | 904 | null |
{\displaystyle X=x.} Assuming that for all g ∈ H : E [ g ( Y ) ∣ X ] ∈ H , {\displaystyle g\in {\mathcal {H}}:\mathbb {E} [g(Y)\mid X]\in {\mathcal {H}},} it can be shown that μ Y ∣ x = C Y ∣ X φ ( x ) {\displaystyle \mu _{Y\mid x}={\mathcal {C}}_{Y\mid X}\varphi (x)} This assumption is always true for finite domains w... | Wikipedia - Kernel embedding of distributions - Definitions > Conditional distribution embedding | 303 | 873 | null |
Given training examples { ( x 1 , y 1 ) , … , ( x n , y n ) } , {\displaystyle \{(x_{1},y_{1}),\dots ,(x_{n},y_{n})\},} the empirical kernel conditional embedding operator may be estimated as C ^ Y ∣ X = Φ ( K + λ I ) − 1 Υ T {\displaystyle {\widehat {C}}_{Y\mid X}={\boldsymbol {\Phi }}(\mathbf {K} +\lambda \mathbf {I}... | Wikipedia - Kernel embedding of distributions - Definitions > Conditional distribution embedding | 350 | 758 | null |
samples of X {\displaystyle X} , and λ {\displaystyle \lambda } is a regularization parameter needed to avoid overfitting. Thus, the empirical estimate of the kernel conditional embedding is given by a weighted sum of samples of Y {\displaystyle Y} in the feature space: μ ^ Y ∣ x = ∑ i = 1 n β i ( x ) φ ( y i ) = Φ β (... | Wikipedia - Kernel embedding of distributions - Definitions > Conditional distribution embedding | 312 | 723 | null |
Section: Properties. The expectation of any function f {\displaystyle f} in the RKHS can be computed as an inner product with the kernel embedding: E [ f ( X ) ] = ⟨ f , μ X ⟩ H {\displaystyle \mathbb {E} [f(X)]=\langle f,\mu _{X}\rangle _{\mathcal {H}}} In the presence of large sample sizes, manipulations of the n × n... | Wikipedia - Kernel embedding of distributions - Properties | 174 | 665 | null |
If k {\displaystyle k} is defined such that f {\displaystyle f} takes values in [ 0 , 1 ] {\displaystyle [0,1]} for all f ∈ H {\displaystyle f\in {\mathcal {H}}} with ‖ f ‖ H ≤ 1 {\displaystyle \|f\|_{\mathcal {H}}\leq 1} (as is the case for the widely used radial basis function kernels), then with probability at least... | Wikipedia - Kernel embedding of distributions - Properties > Convergence of empirical kernel mean to the true distribution embedding | 348 | 749 | null |
)}{2n}}}} where B ( 0 , 1 ) {\displaystyle {\mathcal {B}}(0,1)} denotes the unit ball in H {\displaystyle {\mathcal {H}}} and K = ( k i j ) {\displaystyle \mathbf {K} =(k_{ij})} is the Gram matrix with k i j = k ( x i , x j ) . {\displaystyle k_{ij}=k(x_{i},x_{j}).} The rate of convergence (in RKHS norm) of the empiric... | Wikipedia - Kernel embedding of distributions - Properties > Convergence of empirical kernel mean to the true distribution embedding | 342 | 1,119 | null |
For the embedding of conditional distributions, the empirical estimate can be seen as a weighted average of feature mappings (where the weights β i ( x ) {\displaystyle \beta _{i}(x)} depend on the value of the conditioning variable and capture the effect of the conditioning on the kernel embedding). In this case, the ... | Wikipedia - Kernel embedding of distributions - Properties > Convergence of empirical kernel mean to the true distribution embedding | 181 | 705 | null |
Section: Properties > Universal kernels. Let X ⊆ R b {\displaystyle {\mathcal {X}}\subseteq \mathbb {R} ^{b}} be a compact metric space and C ( X ) {\displaystyle C({\mathcal {X}})} the set of continuous functions. The reproducing kernel k : X × X → R {\displaystyle k:{\mathcal {X}}\times {\mathcal {X}}\rightarrow \mat... | Wikipedia - Kernel embedding of distributions - Properties > Universal kernels | 307 | 845 | null |
All universal kernels defined on a compact space are characteristic kernels but the converse is not always true. Let k {\displaystyle k} be a continuous translation invariant kernel k ( x , x ′ ) = h ( x − x ′ ) {\displaystyle k(x,x')=h(x-x')} with x ∈ R b {\displaystyle x\in \mathbb {R} ^{b}} . Then Bochner's theorem ... | Wikipedia - Kernel embedding of distributions - Properties > Universal kernels | 308 | 870 | null |
{\displaystyle h(t)=\int _{\mathbb {R} ^{b}}e^{-i\langle t,\omega \rangle }d\mu (\omega ),\quad \forall t\in \mathbb {R} ^{b}.} For k {\displaystyle k} to be universal it suffices that the continuous part of μ {\displaystyle \mu } in its unique Lebesgue decomposition μ = μ c + μ s {\displaystyle \mu =\mu _{c}+\mu _{s}}... | Wikipedia - Kernel embedding of distributions - Properties > Universal kernels | 318 | 924 | null |
Section: Properties > Parameter selection for conditional distribution kernel embeddings. The empirical kernel conditional distribution embedding operator C ^ Y | X {\displaystyle {\widehat {\mathcal {C}}}_{Y|X}} can alternatively be viewed as the solution of the following regularized least squares (function-valued) re... | Wikipedia - Kernel embedding of distributions - Properties > Parameter selection for conditional distribution kernel embeddings | 292 | 856 | null |
The following notation is adopted: P ( X , Y ) = {\displaystyle P(X,Y)=} joint distribution over random variables X , Y {\displaystyle X,Y} P ( X ) = ∫ Ω P ( X , d y ) = {\displaystyle P(X)=\int _{\Omega }P(X,\mathrm {d} y)=} marginal distribution of X {\displaystyle X} ; P ( Y ) = {\displaystyle P(Y)=} marginal distri... | Wikipedia - Kernel embedding of distributions - Rules of probability as operations in the RKHS | 335 | 982 | null |
Section: Rules of probability as operations in the RKHS > Kernel sum rule. In probability theory, the marginal distribution of X {\displaystyle X} can be computed by integrating out Y {\displaystyle Y} from the joint density (including the prior distribution on Y {\displaystyle Y} ) Q ( X ) = ∫ Ω P ( X ∣ Y ) d π ( Y ) ... | Wikipedia - Kernel embedding of distributions - Rules of probability as operations in the RKHS > Kernel sum rule | 349 | 894 | null |
{\displaystyle \pi (Y).} In practical implementations, the kernel sum rule takes the following form μ ^ X π = C ^ X ∣ Y μ ^ Y π = Υ ( G + λ I ) − 1 G ~ α {\displaystyle {\widehat {\mu }}_{X}^{\pi }={\widehat {\mathcal {C}}}_{X\mid Y}{\widehat {\mu }}_{Y}^{\pi }={\boldsymbol {\Upsilon }}(\mathbf {G} +\lambda \mathbf {I}... | Wikipedia - Kernel embedding of distributions - Rules of probability as operations in the RKHS > Kernel sum rule | 345 | 752 | null |
{\boldsymbol {\Upsilon }}=\left(\varphi (x_{1}),\ldots ,\varphi (x_{n})\right)} , and G , G ~ {\displaystyle \mathbf {G} ,{\widetilde {\mathbf {G} }}} are Gram matrices with entries G i j = k ( y i , y j ) , G ~ i j = k ( y i , y ~ j ) {\displaystyle \mathbf {G} _{ij}=k(y_{i},y_{j}),{\widetilde {\mathbf {G} }}_{ij}=k(y... | Wikipedia - Kernel embedding of distributions - Rules of probability as operations in the RKHS > Kernel sum rule | 186 | 361 | null |
Section: Rules of probability as operations in the RKHS > Kernel chain rule. In probability theory, a joint distribution can be factorized into a product between conditional and marginal distributions Q ( X , Y ) = P ( X ∣ Y ) π ( Y ) {\displaystyle Q(X,Y)=P(X\mid Y)\pi (Y)} The analog of this rule in the kernel embedd... | Wikipedia - Kernel embedding of distributions - Rules of probability as operations in the RKHS > Kernel chain rule | 338 | 898 | null |
In probability theory, a joint distribution can be factorized into a product between conditional and marginal distributions Q ( X , Y ) = P ( X ∣ Y ) π ( Y ) {\displaystyle Q(X,Y)=P(X\mid Y)\pi (Y)} The analog of this rule in the kernel embedding framework states that C X Y π , {\displaystyle {\mathcal {C}}_{XY}^{\pi }... | Wikipedia - Kernel embedding of distributions - Rules of probability as operations in the RKHS > Kernel chain rule | 581 | 1,366 | null |
Section: Rules of probability as operations in the RKHS > Kernel Bayes' rule. In probability theory, a posterior distribution can be expressed in terms of a prior distribution and a likelihood function as Q ( Y ∣ x ) = P ( x ∣ Y ) π ( Y ) Q ( x ) {\displaystyle Q(Y\mid x)={\frac {P(x\mid Y)\pi (Y)}{Q(x)}}} where Q ( x ... | Wikipedia - Kernel embedding of distributions - Rules of probability as operations in the RKHS > Kernel Bayes' rule | 346 | 915 | null |
{\displaystyle {\mathcal {C}}_{YX}^{\pi }=\left({\mathcal {C}}_{X\mid Y}{\mathcal {C}}_{YY}^{\pi }\right)^{T}.} In practical implementations, the kernel Bayes' rule takes the following form μ ^ Y ∣ x π = C ^ Y X π ( ( C ^ X X ) 2 + λ ~ I ) − 1 C ^ X X π φ ( x ) = Φ ~ Λ T ( ( D K ) 2 + λ ~ I ) − 1 K D K x {\displaystyle... | Wikipedia - Kernel embedding of distributions - Rules of probability as operations in the RKHS > Kernel Bayes' rule | 343 | 676 | null |
{\displaystyle {\boldsymbol {\Lambda }}=\left(\mathbf {G} +{\widetilde {\lambda }}\mathbf {I} \right)^{-1}{\widetilde {\mathbf {G} }}\operatorname {diag} ({\boldsymbol {\alpha }}),\qquad \mathbf {D} =\operatorname {diag} \left(\left(\mathbf {G} +{\widetilde {\lambda }}\mathbf {I} \right)^{-1}{\widetilde {\mathbf {G} }}... | Wikipedia - Kernel embedding of distributions - Rules of probability as operations in the RKHS > Kernel Bayes' rule | 350 | 811 | null |
) − 1 C ^ X X π . {\displaystyle {\widehat {\mathcal {C}}}_{Y\mid X}^{\pi }={\widehat {\mathcal {C}}}_{YX}^{\pi }\left(\left({\widehat {\mathcal {C}}}_{XX}^{\pi }\right)^{2}+{\widetilde {\lambda }}\mathbf {I} \right)^{-1}{\widehat {\mathcal {C}}}_{XX}^{\pi }.} The latter regularization is done on square of C ^ X X π {\... | Wikipedia - Kernel embedding of distributions - Rules of probability as operations in the RKHS > Kernel Bayes' rule | 201 | 429 | null |
Section: Applications > Measuring distance between distributions. The maximum mean discrepancy (MMD) is a distance-measure between distributions P ( X ) {\displaystyle P(X)} and Q ( Y ) {\displaystyle Q(Y)} which is defined as the distance between their embeddings in the RKHS MMD ( P , Q ) = ‖ μ X − μ Y ‖ H . {\display... | Wikipedia - Kernel embedding of distributions - Applications > Measuring distance between distributions | 204 | 730 | null |
{\displaystyle {\text{MMD}}(P,Q)=\left\|\mu _{X}-\mu _{Y}\right\|_{\mathcal {H}}.} While most distance-measures between distributions such as the widely used Kullback–Leibler divergence either require density estimation (either parametrically or nonparametrically) or space partitioning/bias correction strategies, the M... | Wikipedia - Kernel embedding of distributions - Applications > Measuring distance between distributions | 275 | 889 | null |
Given n training examples from P ( X ) {\displaystyle P(X)} and m samples from Q ( Y ) {\displaystyle Q(Y)} , one can formulate a test statistic based on the empirical estimate of the MMD MMD ^ ( P , Q ) = ‖ 1 n ∑ i = 1 n φ ( x i ) − 1 m ∑ i = 1 m φ ( y i ) ‖ H 2 = 1 n 2 ∑ i = 1 n ∑ j = 1 n k ( x i , x j ) + 1 m 2 ∑ i ... | Wikipedia - Kernel embedding of distributions - Applications > Kernel two-sample test | 341 | 658 | null |
{1}{m^{2}}}\sum _{i=1}^{m}\sum _{j=1}^{m}k(y_{i},y_{j})-{\frac {2}{nm}}\sum _{i=1}^{n}\sum _{j=1}^{m}k(x_{i},y_{j})\end{aligned}}} to obtain a two-sample test of the null hypothesis that both samples stem from the same distribution (i.e. P = Q {\displaystyle P=Q} ) against the broad alternative P ≠ Q {\displaystyle P\n... | Wikipedia - Kernel embedding of distributions - Applications > Kernel two-sample test | 160 | 327 | null |
Section: Applications > Density estimation via kernel embeddings. Although learning algorithms in the kernel embedding framework circumvent the need for intermediate density estimation, one may nonetheless use the empirical embedding to perform density estimation based on n samples drawn from an underlying distribution... | Wikipedia - Kernel embedding of distributions - Applications > Density estimation via kernel embeddings | 345 | 1,254 | null |
Section: Applications > Measuring dependence of random variables. A measure of the statistical dependence between random variables X {\displaystyle X} and Y {\displaystyle Y} (from any domains on which sensible kernels can be defined) can be formulated based on the Hilbert–Schmidt Independence Criterion HSIC ( X , Y ) ... | Wikipedia - Kernel embedding of distributions - Applications > Measuring dependence of random variables | 260 | 926 | null |
images and text captions). Given n i.i.d. samples of each random variable, a simple parameter-free unbiased estimator of HSIC which exhibits concentration about the true value can be computed in O ( n ( d f 2 + d g 2 ) ) {\displaystyle O(n(d_{f}^{2}+d_{g}^{2}))} time, where the Gram matrices of the two datasets are app... | Wikipedia - Kernel embedding of distributions - Applications > Measuring dependence of random variables | 346 | 1,158 | null |
Section: Applications > Kernel belief propagation. Belief propagation is a fundamental algorithm for inference in graphical models in which nodes repeatedly pass and receive messages corresponding to the evaluation of conditional expectations. In the kernel embedding framework, the messages may be represented as RKHS f... | Wikipedia - Kernel embedding of distributions - Applications > Kernel belief propagation | 184 | 732 | null |
The kernel belief propagation update message from t to node s is then given by m ^ t s = ( ⊙ u ∈ N ( t ) ∖ s K t β u t ) T ( K s + λ I ) − 1 Υ s T φ ( x s ) {\displaystyle {\widehat {m}}_{ts}=\left(\odot _{u\in N(t)\backslash s}\mathbf {K} _{t}{\boldsymbol {\beta }}_{ut}\right)^{T}(\mathbf {K} _{s}+\lambda \mathbf {I} ... | Wikipedia - Kernel embedding of distributions - Applications > Kernel belief propagation | 348 | 814 | null |
X_{t},X_{s}} , respectively, and Υ s = ( φ ( x s 1 ) , … , φ ( x s n ) ) {\displaystyle {\boldsymbol {\Upsilon }}_{s}=\left(\varphi (x_{s}^{1}),\dots ,\varphi (x_{s}^{n})\right)} is the feature matrix for the samples from X s {\displaystyle X_{s}} . Thus, if the incoming messages to node t are linear combinations of fe... | Wikipedia - Kernel embedding of distributions - Applications > Kernel belief propagation | 221 | 759 | null |
Section: Applications > Nonparametric filtering in hidden Markov models. In the hidden Markov model (HMM), two key quantities of interest are the transition probabilities between hidden states P ( S t ∣ S t − 1 ) {\displaystyle P(S^{t}\mid S^{t-1})} and the emission probabilities P ( O t ∣ S t ) {\displaystyle P(O^{t}\... | Wikipedia - Kernel embedding of distributions - Applications > Nonparametric filtering in hidden Markov models | 251 | 960 | null |
One common use of HMMs is filtering in which the goal is to estimate posterior distribution over the hidden state s t {\displaystyle s^{t}} at time step t given a history of previous observations h t = ( o 1 , … , o t ) {\displaystyle h^{t}=(o^{1},\dots ,o^{t})} from the system. In filtering, a belief state P ( S t + 1... | Wikipedia - Kernel embedding of distributions - Applications > Nonparametric filtering in hidden Markov models | 364 | 922 | null |
In filtering, a belief state P ( S t + 1 ∣ h t + 1 ) {\displaystyle P(S^{t+1}\mid h^{t+1})} is recursively maintained via a prediction step (where updates P ( S t + 1 ∣ h t ) = E [ P ( S t + 1 ∣ S t ) ∣ h t ] {\displaystyle P(S^{t+1}\mid h^{t})=\mathbb {E} [P(S^{t+1}\mid S^{t})\mid h^{t}]} are computed by marginalizing... | Wikipedia - Kernel embedding of distributions - Applications > Nonparametric filtering in hidden Markov models | 485 | 1,106 | null |
Assuming a training sample ( s ~ 1 , … , s ~ T , o ~ 1 , … , o ~ T ) {\displaystyle ({\widetilde {s}}^{1},\dots ,{\widetilde {s}}^{T},{\widetilde {o}}^{1},\dots ,{\widetilde {o}}^{T})} is given, one can in practice estimate μ ^ S t + 1 ∣ h t + 1 = ∑ i = 1 T α i t φ ( s ~ t ) {\displaystyle {\widehat {\mu }}_{S^{t+1}\mi... | Wikipedia - Kernel embedding of distributions - Applications > Nonparametric filtering in hidden Markov models | 348 | 764 | null |
{\alpha }}^{t}\right)} α t + 1 = D t + 1 K ( ( D t + 1 K ) 2 + λ ~ I ) − 1 D t + 1 K o t + 1 {\displaystyle {\boldsymbol {\alpha }}^{t+1}=\mathbf {D} ^{t+1}\mathbf {K} \left((\mathbf {D} ^{t+1}K)^{2}+{\widetilde {\lambda }}\mathbf {I} \right)^{-1}\mathbf {D} ^{t+1}\mathbf {K} _{o^{t+1}}} where G , K {\displaystyle \mat... | Wikipedia - Kernel embedding of distributions - Applications > Nonparametric filtering in hidden Markov models | 344 | 715 | null |
}}_{ij}=k({\widetilde {s}}_{i},{\widetilde {s}}_{j+1}),} and K o t + 1 = ( k ( o ~ 1 , o t + 1 ) , … , k ( o ~ T , o t + 1 ) ) T . {\displaystyle \mathbf {K} _{o^{t+1}}=(k({\widetilde {o}}^{1},o^{t+1}),\dots ,k({\widetilde {o}}^{T},o^{t+1}))^{T}.} | Wikipedia - Kernel embedding of distributions - Applications > Nonparametric filtering in hidden Markov models | 159 | 247 | null |
Section: Applications > Support measure machines. The support measure machine (SMM) is a generalization of the support vector machine (SVM) in which the training examples are probability distributions paired with labels { P i , y i } i = 1 n , y i ∈ { + 1 , − 1 } {\displaystyle \{P_{i},y_{i}\}_{i=1}^{n},\ y_{i}\in \{+1... | Wikipedia - Kernel embedding of distributions - Applications > Support measure machines | 300 | 890 | null |
the Gaussian kernel or polynomial kernel), or can be accurately empirically estimated from i.i.d. samples { x i } i = 1 n ∼ P ( X ) , { z j } j = 1 m ∼ Q ( Z ) {\displaystyle \{x_{i}\}_{i=1}^{n}\sim P(X),\{z_{j}\}_{j=1}^{m}\sim Q(Z)} via K ^ ( X , Z ) = 1 n m ∑ i = 1 n ∑ j = 1 m k ( x i , z j ) {\displaystyle {\widehat... | Wikipedia - Kernel embedding of distributions - Applications > Support measure machines | 373 | 877 | null |
Given training examples { ( x i tr , y i tr ) } i = 1 n {\displaystyle \{(x_{i}^{\text{tr}},y_{i}^{\text{tr}})\}_{i=1}^{n}} and a test set { ( x j te , y j te ) } j = 1 m {\displaystyle \{(x_{j}^{\text{te}},y_{j}^{\text{te}})\}_{j=1}^{m}} where the y j te {\displaystyle y_{j}^{\text{te}}} are unknown, three types of di... | Wikipedia - Kernel embedding of distributions - Applications > Domain adaptation under covariate, target, and conditional shift | 350 | 876 | null |
shift in which P ( Y ) {\displaystyle P(Y)} remains the same across domains, but the conditional distributions differ: P tr ( X ∣ Y ) ≠ P te ( X ∣ Y ) {\displaystyle P^{\text{tr}}(X\mid Y)\neq P^{\text{te}}(X\mid Y)} . In general, the presence of conditional shift leads to an ill-posed problem, and the additional assum... | Wikipedia - Kernel embedding of distributions - Applications > Domain adaptation under covariate, target, and conditional shift | 277 | 1,043 | null |
Target shift, which cannot be similarly dealt with since no samples from Y {\displaystyle Y} are available in the test domain, is accounted for by weighting training examples using the vector β ∗ ( y tr ) {\displaystyle {\boldsymbol {\beta }}^{*}(\mathbf {y} ^{\text{tr}})} which solves the following optimization proble... | Wikipedia - Kernel embedding of distributions - Applications > Domain adaptation under covariate, target, and conditional shift | 350 | 920 | null |
new transformed training data X new = X tr ⊙ W + B {\displaystyle \mathbf {X} ^{\text{new}}=\mathbf {X} ^{\text{tr}}\odot \mathbf {W} +\mathbf {B} } (where ⊙ {\displaystyle \odot } denotes the element-wise vector product). To ensure similar distributions between the new transformed training samples and the test data, W... | Wikipedia - Kernel embedding of distributions - Applications > Domain adaptation under covariate, target, and conditional shift | 392 | 1,110 | null |
Section: Applications > Domain generalization via invariant feature representation. Given N sets of training examples sampled i.i.d. from distributions P ( 1 ) ( X , Y ) , P ( 2 ) ( X , Y ) , … , P ( N ) ( X , Y ) {\displaystyle P^{(1)}(X,Y),P^{(2)}(X,Y),\ldots ,P^{(N)}(X,Y)} , the goal of domain generalization is to f... | Wikipedia - Kernel embedding of distributions - Applications > Domain generalization via invariant feature representation | 333 | 1,423 | null |
Defining a probability distribution P {\displaystyle {\mathcal {P}}} on the RKHS H {\displaystyle {\mathcal {H}}} with P ( μ X ( i ) Y ( i ) ) = 1 N for i = 1 , … , N , {\displaystyle {\mathcal {P}}\left(\mu _{X^{(i)}Y^{(i)}}\right)={\frac {1}{N}}\qquad {\text{ for }}i=1,\dots ,N,} DICA measures dissimilarity between d... | Wikipedia - Kernel embedding of distributions - Applications > Domain generalization via invariant feature representation | 350 | 733 | null |
{\displaystyle \mathbf {G} } is a N × N {\displaystyle N\times N} Gram matrix over the distributions from which the training data are sampled. Finding an orthogonal transform onto a low-dimensional subspace B (in the feature space) which minimizes the distributional variance, DICA simultaneously ensures that B aligns w... | Wikipedia - Kernel embedding of distributions - Applications > Domain generalization via invariant feature representation | 193 | 822 | null |
Given ( { X i , n } n = 1 N i , y i ) i = 1 ℓ {\displaystyle {\left(\{X_{i,n}\}_{n=1}^{N_{i}},y_{i}\right)}_{i=1}^{\ell }} training data, where the X i ^ := { X i , n } n = 1 N i {\displaystyle {\hat {X_{i}}}:=\{X_{i,n}\}_{n=1}^{N_{i}}} bag contains samples from a probability distribution X i {\displaystyle X_{i}} and ... | Wikipedia - Kernel embedding of distributions - Applications > Distribution regression | 238 | 580 | null |
In other words, one can consider the following kernel ridge regression problem ( λ > 0 ) {\displaystyle (\lambda >0)} J ( f ) = 1 ℓ ∑ i = 1 ℓ [ f ( μ X i ^ ) − y i ] 2 + λ ‖ f ‖ H ( K ) 2 → min f ∈ H ( K ) , {\displaystyle J(f)={\frac {1}{\ell }}\sum _{i=1}^{\ell }\left[f\left(\mu _{\hat {X_{i}}}\right)-y_{i}\right]^{2... | Wikipedia - Kernel embedding of distributions - Applications > Distribution regression | 350 | 664 | null |
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