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Section: Methods > Exploiting unrelated tasks. One can attempt learning a group of principal tasks using a group of auxiliary tasks, unrelated to the principal ones. In many applications, joint learning of unrelated tasks which use the same input data can be beneficial. The reason is that prior knowledge about task rel... | Wikipedia - Multi-task learning - Methods > Exploiting unrelated tasks | 163 | 934 | null |
Section: Methods > Multi-task optimization. Multi-task optimization focuses on solving optimizing the whole process. The paradigm has been inspired by the well-established concepts of transfer learning and multi-task learning in predictive analytics. The key motivation behind multi-task optimization is that if optimiza... | Wikipedia - Multi-task learning - Methods > Multi-task optimization | 326 | 1,841 | null |
Section: Methods > Multi-task optimization > Game-theoretic optimization. Game-theoretic approaches to multi-task optimization propose to view the optimization problem as a game, where each task is a player. All players compete through the reward matrix of the game, and try to reach a solution that satisfies all player... | Wikipedia - Multi-task learning - Methods > Multi-task optimization > Game-theoretic optimization | 189 | 968 | null |
Section: Applications. Algorithms for multi-task optimization span a wide array of real-world applications. Recent studies highlight the potential for speed-ups in the optimization of engineering design parameters by conducting related designs jointly in a multi-task manner. In machine learning, the transfer of optimiz... | Wikipedia - Multi-task learning - Applications | 167 | 992 | null |
Section: Mathematics > Reproducing Hilbert space of vector valued functions (RKHSvv) > RKHSvv concepts. Suppose the training data set is S t = { ( x i t , y i t ) } i = 1 n t {\displaystyle {\mathcal {S}}_{t}=\{(x_{i}^{t},y_{i}^{t})\}_{i=1}^{n_{t}}} , with x i t β X {\displaystyle x_{i}^{t}\in {\mathcal {X}}} , y i t β... | Wikipedia - Multi-task learning - Mathematics > Reproducing Hilbert space of vector valued functions (RKHSvv) > RKHSvv concepts | 311 | 718 | null |
In this setting there is a consistent input and output space and the same loss function L : R Γ R β R + {\displaystyle {\mathcal {L}}:\mathbb {R} \times \mathbb {R} \rightarrow \mathbb {R} _{+}} for each task: . This results in the regularized machine learning problem: where H {\displaystyle {\mathcal {H}}} is a vector... | Wikipedia - Multi-task learning - Mathematics > Reproducing Hilbert space of vector valued functions (RKHSvv) > RKHSvv concepts | 186 | 545 | null |
This results in the regularized machine learning problem: where H {\displaystyle {\mathcal {H}}} is a vector valued reproducing kernel Hilbert space with functions f : X β Y T {\displaystyle f:{\mathcal {X}}\rightarrow {\mathcal {Y}}^{T}} having components f t : X β Y {\displaystyle f_{t}:{\mathcal {X}}\rightarrow {\ma... | Wikipedia - Multi-task learning - Mathematics > Reproducing Hilbert space of vector valued functions (RKHSvv) > RKHSvv concepts | 306 | 885 | null |
Section: Mathematics > Reproducing Hilbert space of vector valued functions (RKHSvv) > Separable kernels. The form of the kernel Ξ induces both the representation of the feature space and structures the output across tasks. A natural simplification is to choose a separable kernel, which factors into separate kernels on... | Wikipedia - Multi-task learning - Mathematics > Reproducing Hilbert space of vector valued functions (RKHSvv) > Separable kernels | 261 | 703 | null |
In this case the kernel relating scalar components f t {\displaystyle f_{t}} and f s {\displaystyle f_{s}} is given by Ξ³ ( ( x i , t ) , ( x j , s ) ) = k ( x i , x j ) k T ( s , t ) = k ( x i , x j ) A s , t {\textstyle \gamma ((x_{i},t),(x_{j},s))=k(x_{i},x_{j})k_{T}(s,t)=k(x_{i},x_{j})A_{s,t}} . For vector valued fu... | Wikipedia - Multi-task learning - Mathematics > Reproducing Hilbert space of vector valued functions (RKHSvv) > Separable kernels | 335 | 736 | null |
Henceforth denote S + T = { PSD matrices } β R T Γ T {\displaystyle S_{+}^{T}=\{{\text{PSD matrices}}\}\subset \mathbb {R} ^{T\times T}} . This factorization property, separability, implies the input feature space representation does not vary by task. That is, there is no interaction between the input kernel and the ta... | Wikipedia - Multi-task learning - Mathematics > Reproducing Hilbert space of vector valued functions (RKHSvv) > Separable kernels | 333 | 1,022 | null |
Note the second term in P can be derived as follows: β f β H 2 = β¨ β i = 1 n k ( β
, x i ) A c i , β j = 1 n k ( β
, x j ) A c j β© H = β i , j = 1 n β¨ k ( β
, x i ) A c i , k ( β
, x j ) A c j β© H (bilinearity) = β i , j = 1 n β¨ k ( x i , x j ) A c i , c j β© R T (reproducing property) = β i , j = 1 n k ( x i , x j ) c ... | Wikipedia - Multi-task learning - Mathematics > Reproducing Hilbert space of vector valued functions (RKHSvv) > Separable kernels | 350 | 648 | null |
Section: Mathematics > Reproducing Hilbert space of vector valued functions (RKHSvv) > Known task structure > Task structure examples. Via the regularizer formulation, one can represent a variety of task structures easily. Letting A β = Ξ³ I T + ( Ξ³ β Ξ» ) 1 T 1 1 β€ {\textstyle A^{\dagger }=\gamma I_{T}+(\gamma -\lambda ... | Wikipedia - Multi-task learning - Mathematics > Reproducing Hilbert space of vector valued functions (RKHSvv) > Known task structure > Task structure examples | 329 | 956 | null |
For example, blood levels of some biomarker may be taken on T patients at n t {\displaystyle n_{t}} time points during the course of a day and interest may lie in regularizing the variance of the predictions across patients. Letting A β = Ξ± I T + ( Ξ± β Ξ» ) M {\displaystyle A^{\dagger }=\alpha I_{T}+(\alpha -\lambda )M}... | Wikipedia - Multi-task learning - Mathematics > Reproducing Hilbert space of vector valued functions (RKHSvv) > Known task structure > Task structure examples | 328 | 831 | null |
(Here | G r | {\displaystyle |G_{r}|} the cardinality of group r, and I {\displaystyle \mathbb {I} } is the indicator function). For example, people in different political parties (groups) might be regularized together with respect to predicting the favorability rating of a politician. Note that this penalty reduces to... | Wikipedia - Multi-task learning - Mathematics > Reproducing Hilbert space of vector valued functions (RKHSvv) > Known task structure > Task structure examples | 303 | 932 | null |
Ξ΄ {\displaystyle \delta } regularizes β t , s | | f t β f s | | H k 2 M t , s {\displaystyle \sum _{t,s}||f_{t}-f_{s}||_{{\mathcal {H}}_{k}}^{2}M_{t,s}} . All of the above choices of A also induce the additional regularization term Ξ» β t | | f | | H k 2 {\textstyle \lambda \sum _{t}||f||_{{\mathcal {H}}_{k}}^{2}} which... | Wikipedia - Multi-task learning - Mathematics > Reproducing Hilbert space of vector valued functions (RKHSvv) > Known task structure > Task structure examples | 156 | 360 | null |
Section: Mathematics > Reproducing Hilbert space of vector valued functions (RKHSvv) > Learning tasks together with their structure > Optimization of Q. Restricting to the case of convex losses and coercive penalties Ciliberto et al. have shown that although Q is not convex jointly in C and A, a related problem is join... | Wikipedia - Multi-task learning - Mathematics > Reproducing Hilbert space of vector valued functions (RKHSvv) > Learning tasks together with their structure > Optimization of Q | 276 | 791 | null |
And if ( C R , A R ) {\displaystyle (C_{R},A_{R})} is a minimizer for R then ( C R A R β , A R ) {\displaystyle (C_{R}A_{R}^{\dagger },A_{R})} is a minimizer for Q. R may be solved by a barrier method on a closed set by introducing the following perturbation: The perturbation via the barrier Ξ΄ 2 t r ( A β ) {\displayst... | Wikipedia - Multi-task learning - Mathematics > Reproducing Hilbert space of vector valued functions (RKHSvv) > Learning tasks together with their structure > Optimization of Q | 281 | 805 | null |
Section: Mathematics > Reproducing Hilbert space of vector valued functions (RKHSvv) > Learning tasks together with their structure > Special cases. Spectral penalties - Dinnuzo et al suggested setting F as the Frobenius norm t r ( A β€ A ) {\displaystyle {\sqrt {tr(A^{\top }A)}}} . They optimized Q directly using block... | Wikipedia - Multi-task learning - Mathematics > Reproducing Hilbert space of vector valued functions (RKHSvv) > Learning tasks together with their structure > Special cases | 256 | 799 | null |
In this case let E β { 0 , 1 } T Γ R {\displaystyle E\in \{0,1\}^{T\times R}} be the matrix with E t , r = I ( task t β group r ) {\displaystyle E_{t,r}=\mathbb {I} ({\text{task }}t\in {\text{group }}r)} . Setting M = I β E β E T {\displaystyle M=I-E^{\dagger }E^{T}} , and U = 1 T 11 β€ {\displaystyle U={\frac {1}{T}}\m... | Wikipedia - Multi-task learning - Mathematics > Reproducing Hilbert space of vector valued functions (RKHSvv) > Learning tasks together with their structure > Special cases | 280 | 713 | null |
Setting M = I β E β E T {\displaystyle M=I-E^{\dagger }E^{T}} , and U = 1 T 11 β€ {\displaystyle U={\frac {1}{T}}\mathbf {11} ^{\top }} , the task matrix A β {\displaystyle A^{\dagger }} can be parameterized as a function of M {\displaystyle M} : A β ( M ) = Ο΅ M U + Ο΅ B ( M β U ) + Ο΅ ( I β M ) {\displaystyle A^{\dagger ... | Wikipedia - Multi-task learning - Mathematics > Reproducing Hilbert space of vector valued functions (RKHSvv) > Learning tasks together with their structure > Special cases | 282 | 695 | null |
M is not convex, but there is a convex relaxation S c = { M β S + T : I β M β S + T β§ t r ( M ) = r } {\displaystyle {\mathcal {S}}_{c}=\{M\in S_{+}^{T}:I-M\in S_{+}^{T}\land tr(M)=r\}} . In this formulation, F ( A ) = I ( A ( M ) β { A : M β S C } ) {\displaystyle F(A)=\mathbb {I} (A(M)\in \{A:M\in {\mathcal {S}}_{C}\... | Wikipedia - Multi-task learning - Mathematics > Reproducing Hilbert space of vector valued functions (RKHSvv) > Learning tasks together with their structure > Special cases | 166 | 325 | null |
Article: Multimodal representation learning. Multimodal representation learning is a subfield of representation learning focused on integrating and interpreting information from different modalities, such as text, images, audio, or video, by projecting them into a shared latent space. This allows for semantically simil... | Wikipedia - Multimodal representation learning - Summary | 160 | 896 | null |
Section: Motivation. The primary motivations for multimodal representation learning arise from the inherent nature of real-world data and the limitations of unimodal approaches. Since multimodal data offers complementary and supplementary information about an object or event from different perspectives, it is more info... | Wikipedia - Multimodal representation learning - Motivation | 289 | 1,611 | null |
Given two data matrices X β R n Γ p {\displaystyle X\in \mathbb {R} ^{n\times p}} and Y β R n Γ q {\displaystyle Y\in \mathbb {R} ^{n\times q}} representing different modalities, CCA finds projection vectors w x β R p {\displaystyle w_{x}\in \mathbb {R} ^{p}} and w y β R q {\displaystyle w_{y}\in \mathbb {R} ^{q}} that... | Wikipedia - Multimodal representation learning - Approaches and methods > Canonical-correlation analysis based methods | 350 | 820 | null |
Given two data matrices X β R n Γ p {\displaystyle X\in \mathbb {R} ^{n\times p}} and Y β R n Γ q {\displaystyle Y\in \mathbb {R} ^{n\times q}} representing different modalities, CCA finds projection vectors w x β R p {\displaystyle w_{x}\in \mathbb {R} ^{p}} and w y β R q {\displaystyle w_{y}\in \mathbb {R} ^{q}} that... | Wikipedia - Multimodal representation learning - Approaches and methods > Canonical-correlation analysis based methods | 381 | 958 | null |
Given kernel functions K x {\displaystyle K_{x}} and K y {\displaystyle K_{y}} with corresponding Gram matrices K x β R n Γ n {\displaystyle K_{x}\in \mathbb {R} ^{n\times n}} and K y β R n Γ n {\displaystyle K_{y}\in \mathbb {R} ^{n\times n}} , KCCA seeks coefficients Ξ± {\displaystyle \alpha } and Ξ² {\displaystyle \be... | Wikipedia - Multimodal representation learning - Approaches and methods > Canonical-correlation analysis based methods > Kernel CCA | 345 | 811 | null |
^{T}\left(K_{x}^{2}+\lambda _{x}K_{x}\right)\alpha }}{\sqrt {\;\beta ^{T}\left(K_{y}^{2}+\lambda _{y}K_{y}\right)\beta }}}}} where Ξ» x {\displaystyle \lambda _{x}} and Ξ» y {\displaystyle \lambda _{y}} are regularization parameters. KCCA has proven effective for tasks such as cross-modal retrieval and semantic analysis,... | Wikipedia - Multimodal representation learning - Approaches and methods > Canonical-correlation analysis based methods > Kernel CCA | 180 | 531 | null |
Section: Approaches and methods > Canonical-correlation analysis based methods > Deep CCA. Deep canonical correlation analysis (DCCA), introduced in 2013, employs neural networks to learn nonlinear transformations for maximizing the correlation between modalities. DCCA uses separate neural networks f x {\displaystyle f... | Wikipedia - Multimodal representation learning - Approaches and methods > Canonical-correlation analysis based methods > Deep CCA | 273 | 847 | null |
DCCA uses separate neural networks f x {\displaystyle f_{x}} and f y {\displaystyle f_{y}} for each modality to transform the original data before applying CCA: max W x , W y , ΞΈ x , ΞΈ y corr β‘ ( f x ( X ; ΞΈ x ) , f y ( Y ; ΞΈ y ) ) {\displaystyle \max _{W_{x},W_{y},\theta _{x},\theta _{y}}\operatorname {corr} \left(f_{... | Wikipedia - Multimodal representation learning - Approaches and methods > Canonical-correlation analysis based methods > Deep CCA | 497 | 1,132 | null |
The correlation objective is computed as: corr β‘ ( H x , H y ) = tr β‘ ( T β 1 / 2 H x T H y S β 1 / 2 ) {\displaystyle \operatorname {corr} (H_{x},H_{y})=\operatorname {tr} \left(T^{-1/2}H_{x}^{T}H_{y}S^{-1/2}\right)} where H x = f x ( X ) {\displaystyle H_{x}=f_{x}(X)} and H y = f y ( Y ) {\displaystyle H_{y}=f_{y}(Y)... | Wikipedia - Multimodal representation learning - Approaches and methods > Canonical-correlation analysis based methods > Deep CCA | 305 | 748 | null |
Section: Approaches and methods > Graph-based methods. Graph-based approaches for multimodal representation learning leverage graph structure to model relationships between entities across different modalities. These methods typically represent each modality as a graph and then learn embedding that preserve cross-modal... | Wikipedia - Multimodal representation learning - Approaches and methods > Graph-based methods | 284 | 1,420 | null |
Section: Approaches and methods > Diffusion maps > Multi-view diffusion maps. Multi-view diffusion maps address the challenge of achieving multi-view dimensionality reduction by effectively utilizing the availability of multiple views to extract a coherent low-dimensional representation of the data. The core idea is to... | Wikipedia - Multimodal representation learning - Approaches and methods > Diffusion maps > Multi-view diffusion maps | 164 | 951 | null |
Article: Multiple instance learning. In machine learning, multiple-instance learning (MIL) is a type of supervised learning. Instead of receiving a set of instances which are individually labeled, the learner receives a set of labeled bags, each containing many instances. In the simple case of multiple-instance binary ... | Wikipedia - Multiple instance learning - Summary | 265 | 1,267 | null |
Section: Machine learning. Depending on the type and variation in training data, machine learning can be roughly categorized into three frameworks: supervised learning, unsupervised learning, and reinforcement learning. Multiple instance learning (MIL) falls under the supervised learning framework, where every training... | Wikipedia - Multiple instance learning - Machine learning | 155 | 799 | null |
Section: History. Keeler et al., in his work in the early 1990s was the first one to explore the area of MIL. The actual term multi-instance learning was introduced in the middle of the 1990s, by Dietterich et al. while they were investigating the problem of drug activity prediction. They tried to create a learning sys... | Wikipedia - Multiple instance learning - History | 332 | 1,639 | null |
It attempts to search for appropriate axis-parallel rectangles constructed by the conjunction of the features. They tested the algorithm on Musk dataset, which is a concrete test data of drug activity prediction and the most popularly used benchmark in multiple-instance learning. APR algorithm achieved the best result,... | Wikipedia - Multiple instance learning - History | 222 | 1,135 | null |
Section: Examples. Take image classification for example Amores (2013). Given an image, we want to know its target class based on its visual content. For instance, the target class might be "beach", where the image contains both "sand" and "water". In MIL terms, the image is described as a bag X = { X 1 , . . , X N } {... | Wikipedia - Multiple instance learning - Examples | 340 | 1,364 | null |
Section: Definitions. If the space of instances is X {\displaystyle {\mathcal {X}}} , then the set of bags is the set of functions N X = { B : X β N } {\displaystyle \mathbb {N} ^{\mathcal {X}}=\{B:{\mathcal {X}}\rightarrow \mathbb {N} \}} , which is isomorphic to the set of multi-subsets of X {\displaystyle {\mathcal ... | Wikipedia - Multiple instance learning - Definitions | 333 | 935 | null |
Section: Assumptions > Standard assumption. The standard assumption takes each instance x β X {\displaystyle x\in {\mathcal {X}}} to have an associated label y β { 0 , 1 } {\displaystyle y\in \{0,1\}} which is hidden to the learner. The pair ( x , y ) {\displaystyle (x,y)} is called an "instance-level concept". A bag i... | Wikipedia - Multiple instance learning - Assumptions > Standard assumption | 321 | 1,015 | null |
Because of that, when we use this assumption, we need to be clear which label should be the positive one. The standard assumption might be viewed as too strict, and therefore in the recent years, researchers tried to relax that position, which gave rise to other more loose assumptions. The reason for this is the belief... | Wikipedia - Multiple instance learning - Assumptions > Standard assumption | 312 | 1,594 | null |
Section: Assumptions > Presence-, threshold-, and count-based assumptions. The presence-based assumption is a generalization of the standard assumption, wherein a bag must contain all instances that belong to a set of required instance-level concepts in order to be labeled positive. Formally, let C R β X Γ Y {\displays... | Wikipedia - Multiple instance learning - Assumptions > Presence-, threshold-, and count-based assumptions | 322 | 1,131 | null |
A further generalization comes with the threshold-based assumption, where each required instance-level concept must occur not only once in a bag, but some minimum (threshold) number of times in order for the bag to be labeled positive. With the notation above, to each required instance-level concept c i β C R {\display... | Wikipedia - Multiple instance learning - Assumptions > Presence-, threshold-, and count-based assumptions | 334 | 1,031 | null |
Each required instance-level concept c i β C R {\displaystyle c_{i}\in C_{R}} has a lower threshold l i β N {\displaystyle l_{i}\in \mathbb {N} } and upper threshold u i β N {\displaystyle u_{i}\in \mathbb {N} } with l i β€ u i {\displaystyle l_{i}\leq u_{i}} . A bag B {\displaystyle B} is labeled according to c ( B ) =... | Wikipedia - Multiple instance learning - Assumptions > Presence-, threshold-, and count-based assumptions | 205 | 471 | null |
Section: Assumptions > GMIL assumption. Scott, Zhang, and Brown (2005) describe another generalization of the standard model, which they call "generalized multiple instance learning" (GMIL). The GMIL assumption specifies a set of required instances Q β X {\displaystyle Q\subseteq {\mathcal {X}}} . A bag X {\displaystyl... | Wikipedia - Multiple instance learning - Assumptions > GMIL assumption | 282 | 1,167 | null |
Section: Assumptions > Collective assumption. In contrast to the previous assumptions where the bags were viewed as fixed, the collective assumption views a bag B {\displaystyle B} as a distribution p ( x | B ) {\displaystyle p(x|B)} over instances X {\displaystyle {\mathcal {X}}} , and similarly view labels as a distr... | Wikipedia - Multiple instance learning - Assumptions > Collective assumption | 322 | 952 | null |
Since p ( x | B ) {\displaystyle p(x|B)} is typically considered fixed but unknown, algorithms instead focus on computing the empirical version: p ^ ( y | B ) = 1 n B β i = 1 n B p ( y | x i ) {\displaystyle {\widehat {p}}(y|B)={\frac {1}{n_{B}}}\sum _{i=1}^{n_{B}}p(y|x_{i})} , where n B {\displaystyle n_{B}} is the nu... | Wikipedia - Multiple instance learning - Assumptions > Collective assumption | 217 | 722 | null |
While the collective assumption weights every instance with equal importance, Foulds extended the collective assumption to incorporate instance weights. The weighted collective assumption is then that p ^ ( y | B ) = 1 w B β i = 1 n B w ( x i ) p ( y | x i ) {\displaystyle {\widehat {p}}(y|B)={\frac {1}{w_{B}}}\sum _{i... | Wikipedia - Multiple instance learning - Assumptions > Collective assumption | 206 | 536 | null |
Section: Algorithms > Instance-based algorithms > Iterated-discrimination. Broadly, all of the iterated-discrimination algorithms consist of two phases. The first phase is to grow an axis parallel rectangle (APR) which contains at least one instance from each positive bag and no instances from any negative bags. This i... | Wikipedia - Multiple instance learning - Algorithms > Instance-based algorithms > Iterated-discrimination | 336 | 1,627 | null |
Let B + = { B i + } 1 m {\displaystyle {\mathcal {B}}^{+}=\{B_{i}^{+}\}_{1}^{m}} be the set of positively labeled bags and let B β = { B i β } 1 n {\displaystyle {\mathcal {B}}^{-}=\{B_{i}^{-}\}_{1}^{n}} be the set of negatively labeled bags, then the best candidate for the representative instance is given by t ^ = arg... | Wikipedia - Multiple instance learning - Algorithms > Instance-based algorithms > Diverse Density | 349 | 776 | null |
Letting B i j {\displaystyle B_{ij}} denote the jth instance of bag i, the noisy-or model gives: P r ( t | B i + ) = 1 β β j ( 1 β P r ( t | B i j + ) ) {\displaystyle Pr(t|B_{i}^{+})=1-\prod _{j}\left(1-Pr\left(t|B_{ij}^{+}\right)\right)} P r ( t | B i β ) = β j ( 1 β P r ( t | B i j β ) ) {\displaystyle Pr(t|B_{i}^{-... | Wikipedia - Multiple instance learning - Algorithms > Instance-based algorithms > Diverse Density | 350 | 671 | null |
is the scaling vector. This way, if every positive bag has an instance close to t {\displaystyle t} , then P r ( t | B i + ) {\displaystyle Pr(t|B_{i}^{+})} will be high for each i {\displaystyle i} , but if any negative bag B i β {\displaystyle B_{i}^{-}} has an instance close to t {\displaystyle t} , P r ( t | B i β ... | Wikipedia - Multiple instance learning - Algorithms > Instance-based algorithms > Diverse Density | 340 | 1,305 | null |
in 1998, more recent MIL algorithms use the DD framework, such as EM-DD in 2001 and DD-SVM in 2004, and MILES in 2006 A number of single-instance algorithms have also been adapted to a multiple-instance context under the standard assumption, including Support vector machines Artificial neural networks Decision trees Bo... | Wikipedia - Multiple instance learning - Algorithms > Instance-based algorithms > Diverse Density | 263 | 1,201 | null |
GMIL-1 enumerates all axis-parallel rectangles { R i } i β I {\displaystyle \{R_{i}\}_{i\in I}} in the original space of instances, and defines a new feature space of Boolean vectors. A bag B {\displaystyle B} is mapped to a vector b = ( b i ) i β I {\displaystyle \mathbf {b} =(b_{i})_{i\in I}} in this new feature spac... | Wikipedia - Multiple instance learning - Algorithms > Instance-based algorithms > Diverse Density | 350 | 1,308 | null |
Section: Algorithms > Metadata-based (or embedding-based) algorithms. By mapping each bag to a feature vector of metadata, metadata-based algorithms allow the flexibility of using an arbitrary single-instance algorithm to perform the actual classification task. Future bags are simply mapped (embedded) into the feature ... | Wikipedia - Multiple instance learning - Algorithms > Metadata-based (or embedding-based) algorithms | 332 | 1,690 | null |
Similar approaches are taken by MILES and MInD. MILES represents a bag by its similarities to instances in the training set, while MInD represents a bag by its distances to other bags. A modification of k-nearest neighbors (kNN) can also be considered a metadata-based algorithm with geometric metadata, though the mappi... | Wikipedia - Multiple instance learning - Algorithms > Metadata-based (or embedding-based) algorithms | 304 | 1,021 | null |
Section: Generalizations. So far this article has considered multiple instance learning exclusively in the context of binary classifiers. However, the generalizations of single-instance binary classifiers can carry over to the multiple-instance case. One such generalization is the multiple-instance multiple-label probl... | Wikipedia - Multiple instance learning - Generalizations | 345 | 1,503 | null |
Article: Multiplicative weight update method. The multiplicative weights update method is an algorithmic technique most commonly used for decision making and prediction, and also widely deployed in game theory and algorithm design. The simplest use case is the problem of prediction from expert advice, in which a decisi... | Wikipedia - Multiplicative weight update method - Summary | 174 | 902 | null |
Section: History and background. The earliest known version of this technique was in an algorithm named "fictitious play" which was proposed in game theory in the early 1950s. Grigoriadis and Khachiyan applied a randomized variant of "fictitious play" to solve two-player zero-sum games efficiently using the multiplicat... | Wikipedia - Multiplicative weight update method - History and background | 287 | 1,530 | null |
Section: Algorithm analysis > Weighted majority algorithm. Source: Unlike halving algorithm which dismisses experts who have made mistakes, weighted majority algorithm discounts their advice. Given the same "expert advice" setup, suppose we have n decisions, and we need to select one decision for each loop. In each loo... | Wikipedia - Multiplicative weight update method - Algorithm analysis > Weighted majority algorithm | 308 | 1,310 | null |
. . , w n t {\displaystyle \mathbb {w_{1}} ^{t},...,\mathbb {w_{n}} ^{t}} . That is, choose 0 or 1 depending on which prediction has a higher total weight of experts advising it (breaking ties arbitrarily). 2. For every expert i that predicted wrongly, decrease his weight for the next round by multiplying it by a facto... | Wikipedia - Multiplicative weight update method - Algorithm analysis > Weighted majority algorithm | 294 | 902 | null |
After T {\displaystyle T} steps, let m i T {\displaystyle m_{i}^{T}} be the number of mistakes of expert i and M T {\displaystyle M^{T}} be the number of mistakes our algorithm has made. Then we have the following bound for every i {\displaystyle i} : M T β€ 2 ( 1 + Ξ· ) m i T + 2 ln β‘ ( n ) Ξ· {\displaystyle M^{T}\leq 2(... | Wikipedia - Multiplicative weight update method - Algorithm analysis > Weighted majority algorithm | 202 | 587 | null |
Section: Algorithm analysis > Randomized weighted majority algorithm. This algorithm can be understood as follows: Given the same setup with N experts. Consider the special situation where the proportions of experts predicting positive and negative, counting the weights, are both close to 50%. Then, there might be a ti... | Wikipedia - Multiplicative weight update method - Algorithm analysis > Randomized weighted majority algorithm | 228 | 850 | null |
The algorithm calculates the probabilities of experts predicting positive or negatives, and then makes a random decision based on the computed fraction: predict f ( x ) = { 1 with probability q 1 W 0 otherwise {\displaystyle f(x)={\begin{cases}1&{\text{with probability}}{\frac {q_{1}}{W}}\\0&{\text{otherwise}}\end{case... | Wikipedia - Multiplicative weight update method - Algorithm analysis > Randomized weighted majority algorithm | 349 | 914 | null |
The number of mistakes made by the randomized weighted majority algorithm is bounded as: E [ # mistakes of the learner ] β€ Ξ± Ξ² ( # mistakes of the best expert ) + c Ξ² ln β‘ ( N ) {\displaystyle E\left[\#{\text{mistakes of the learner}}\right]\leq \alpha _{\beta }\left(\#{\text{ mistakes of the best expert}}\right)+c_{\b... | Wikipedia - Multiplicative weight update method - Algorithm analysis > Randomized weighted majority algorithm | 309 | 974 | null |
Section: Applications > Solving zero-sum games approximately (Oracle algorithm). Source: Suppose we were given the distribution P {\displaystyle P} on experts. Let A {\displaystyle A} = payoff matrix of a finite two-player zero-sum game, with n {\displaystyle n} rows. When the row player p r {\displaystyle p_{r}} uses ... | Wikipedia - Multiplicative weight update method - Applications > Solving zero-sum games approximately (Oracle algorithm) | 337 | 985 | null |
If player p r {\displaystyle p_{r}} chooses action i {\displaystyle i} from a distribution P {\displaystyle P} over the rows, then the expected result for player p c {\displaystyle p_{c}} selecting action j {\displaystyle j} is A ( P , j ) = E i β P [ A ( i , j ) ] {\displaystyle A\left(P,j\right)=E_{i\in P}\left[A\lef... | Wikipedia - Multiplicative weight update method - Applications > Solving zero-sum games approximately (Oracle algorithm) | 269 | 724 | null |
Choosing plan i {\displaystyle i} would minimize this payoff. By John Von Neumann's Min-Max Theorem, we obtain: min P max j A ( P , j ) = max Q min i A ( i , Q ) {\displaystyle \min _{P}\max _{j}A\left(P,j\right)=\max _{Q}\min _{i}A\left(i,Q\right)} where P and i changes over the distributions over rows, Q and j change... | Wikipedia - Multiplicative weight update method - Applications > Solving zero-sum games approximately (Oracle algorithm) | 172 | 525 | null |
Let Ξ΄ > 0 {\displaystyle \delta >0} be an error parameter. To solve the zero-sum game bounded by additive error of Ξ΄ {\displaystyle \delta } , Ξ» β β Ξ΄ β€ min i A ( i , q ) {\displaystyle \lambda ^{*}-\delta \leq \min _{i}A\left(i,q\right)} max j A ( p , j ) β€ Ξ» β + Ξ΄ {\displaystyle \max _{j}A\left(p,j\right)\leq \lambda... | Wikipedia - Multiplicative weight update method - Applications > Solving zero-sum games approximately (Oracle algorithm) | 237 | 741 | null |
Section: Applications > Machine learning > Winnow algorithm. Based on current knowledge in algorithms, the multiplicative weight update method was first used in Littlestone's winnow algorithm. It is used in machine learning to solve a linear program. Given m {\displaystyle m} labeled examples ( a 1 , l 1 ) , β¦ , ( a m ... | Wikipedia - Multiplicative weight update method - Applications > Machine learning > Winnow algorithm | 285 | 906 | null |
Without loss of generality, assume the total weight is 1 so that they form a distribution. Thus, for notational convenience, redefine a j {\displaystyle a_{j}} to be l j a j {\displaystyle l_{j}a_{j}} , the problem reduces to finding a solution to the following LP: β j = 1 , 2 , β¦ , m : a j x β₯ 0 {\displaystyle \forall... | Wikipedia - Multiplicative weight update method - Applications > Machine learning > Winnow algorithm | 176 | 471 | null |
Section: Applications > Machine learning > Hedge algorithm > Analysis. Assume the learning rate Ξ· > 0 {\displaystyle \eta >0} and for t β [ T ] {\displaystyle t\in [T]} , p t {\displaystyle p^{t}} is picked by Hedge. Then for all experts i {\displaystyle i} , β t β€ T p t m t β€ β t β€ T m i t + ln β‘ ( N ) Ξ· + Ξ· T {\displ... | Wikipedia - Multiplicative weight update method - Applications > Machine learning > Hedge algorithm > Analysis | 334 | 789 | null |
Section: Applications > Machine learning > AdaBoost algorithm. This algorithm maintains a set of weights w t {\displaystyle w^{t}} over the training examples. On every iteration t {\displaystyle t} , a distribution p t {\displaystyle p^{t}} is computed by normalizing these weights. This distribution is fed to the weak ... | Wikipedia - Multiplicative weight update method - Applications > Machine learning > AdaBoost algorithm | 212 | 815 | null |
The hypothesis h f {\displaystyle h_{f}} combines the outputs of the T weak hypotheses using a weighted majority vote. Input: Sequence of N {\displaystyle N} labeled examples ( x 1 {\displaystyle x_{1}} , y 1 {\displaystyle y_{1}} ),...,( x N {\displaystyle x_{N}} , y N {\displaystyle y_{N}} ) Distribution D {\displays... | Wikipedia - Multiplicative weight update method - Applications > Machine learning > AdaBoost algorithm | 301 | 755 | null |
2. Call WeakLearn, providing it with the distribution p t {\displaystyle p^{t}} ; get back a hypothesis h t : X β {\displaystyle h_{t}:X\rightarrow } [0,1]. 3. Calculate the error of h t : Ο΅ t = β i = 1 N p i t | h t ( x i ) β y i | {\displaystyle h_{t}:\epsilon _{t}=\sum _{i=1}^{N}p_{i}^{t}|h_{t}(x_{i})-y_{i}|} . 4. S... | Wikipedia - Multiplicative weight update method - Applications > Machine learning > AdaBoost algorithm | 286 | 568 | null |
Set the new weight vector to be w i t + 1 = w i t Ξ² t 1 β | h t ( x i ) β y i | {\displaystyle w_{i}^{t+1}=w_{i}^{t}\beta _{t}^{1-|h_{t}(x_{i})-y_{i}|}} . Output the hypothesis: f ( x ) = h f ( x ) = { 1 if β t = 1 T ( log β‘ ( 1 / Ξ² t ) ) h t ( x ) β₯ 1 2 β t = 1 T log β‘ ( 1 / Ξ² t ) 0 otherwise {\displaystyle f(x)=h_{f}... | Wikipedia - Multiplicative weight update method - Applications > Machine learning > AdaBoost algorithm | 266 | 481 | null |
Section: Applications > Solving linear programs approximately > Solution. Given vector p β Ξ n {\displaystyle p\in \Delta _{n}} , solves the following relaxed problem β ? x : p T A x β₯ p T b {\displaystyle \exists ?x:p^{\textsf {T}}\!\!Ax\geq p^{\textsf {T}}\!b} (2) If there exists a x satisfying (1), then x satisfies ... | Wikipedia - Multiplicative weight update method - Applications > Solving linear programs approximately > Solution | 344 | 1,060 | null |
Section: Applications > Other applications. Evolutionary game theory Multiplicative weights update is the discrete-time variant of the replicator equation (replicator dynamics), which is a commonly used model in evolutionary game theory. It converges to Nash equilibrium when applied to a congestion game. Operations res... | Wikipedia - Multiplicative weight update method - Applications > Other applications | 225 | 1,224 | null |
Section: The basics. This section introduces MARS using a few examples. We start with a set of data: a matrix of input variables x, and a vector of the observed responses y, with a response for each row in x. For example, the data could be: Here there is only one independent variable, so the x matrix is just a single c... | Wikipedia - Multivariate adaptive regression spline - The basics | 280 | 1,098 | null |
We thus turn to MARS to automatically build a model taking into account non-linearities. MARS software constructs a model from the given x and y as follows y ^ = 25 + 6.1 max ( 0 , x β 13 ) β 3.1 max ( 0 , 13 β x ) {\displaystyle {\begin{aligned}{\widehat {y}}=&\ 25\\&{}+6.1\max(0,x-13)\\&{}-3.1\max(0,13-x)\end{aligned... | Wikipedia - Multivariate adaptive regression spline - The basics | 329 | 1,124 | null |
In this simple example, we can easily see from the plot that y has a non-linear relationship with x (and might perhaps guess that y varies with the square of x). However, in general there will be multiple independent variables, and the relationship between y and these variables will be unclear and not easily visible by... | Wikipedia - Multivariate adaptive regression spline - The basics | 328 | 966 | null |
An example MARS expression with multiple variables is o z o n e = 5.2 + 0.93 max ( 0 , t e m p β 58 ) β 0.64 max ( 0 , t e m p β 68 ) β 0.046 max ( 0 , 234 β i b t ) β 0.016 max ( 0 , w i n d β 7 ) max ( 0 , 200 β v i s ) {\displaystyle {\begin{aligned}\mathrm {ozone} =&\ 5.2\\&{}+0.93\max(0,\mathrm {temp} -58)\\&{}-0.... | Wikipedia - Multivariate adaptive regression spline - The basics | 299 | 754 | null |
Note that the last term in the formula (on the last line) incorporates an interaction between w i n d {\displaystyle \mathrm {wind} } and v i s {\displaystyle \mathrm {vis} } . The figure on the right plots the predicted o z o n e {\displaystyle \mathrm {ozone} } as w i n d {\displaystyle \mathrm {wind} } and v i s {\d... | Wikipedia - Multivariate adaptive regression spline - The basics | 196 | 833 | null |
Section: The MARS model. MARS builds models of the form f ^ ( x ) = β i = 1 k c i B i ( x ) . {\displaystyle {\widehat {f}}(x)=\sum _{i=1}^{k}c_{i}B_{i}(x).} The model is a weighted sum of basis functions B i ( x ) {\displaystyle B_{i}(x)} . Each c i {\displaystyle c_{i}} is a constant coefficient. For example, each li... | Wikipedia - Multivariate adaptive regression spline - The MARS model | 341 | 1,175 | null |
Section: Hinge functions. A key part of MARS models are hinge functions taking the form max ( 0 , x β c ) {\displaystyle \max(0,x-c)} or max ( 0 , c β x ) {\displaystyle \max(0,c-x)} where c {\displaystyle c} is a constant, called the knot. The figure on the right shows a mirrored pair of hinge functions with a knot at... | Wikipedia - Multivariate adaptive regression spline - Hinge functions | 350 | 1,220 | null |
Section: The model building process > The forward pass. MARS starts with a model which consists of just the intercept term (which is the mean of the response values). MARS then repeatedly adds basis function in pairs to the model. At each step it finds the pair of basis functions that gives the maximum reduction in sum... | Wikipedia - Multivariate adaptive regression spline - The model building process > The forward pass | 346 | 1,659 | null |
Section: The model building process > The backward pass. The forward pass usually overfits the model. To build a model with better generalization ability, the backward pass prunes the model, deleting the least effective term at each step until it finds the best submodel. Model subsets are compared using the Generalized... | Wikipedia - Multivariate adaptive regression spline - The model building process > The backward pass | 194 | 884 | null |
Section: The model building process > The backward pass > Generalized cross validation. The backward pass compares the performance of different models using Generalized Cross-Validation (GCV), a minor variant on the Akaike information criterion that approximates the leave-one-out cross-validation score in the special c... | Wikipedia - Multivariate adaptive regression spline - The model building process > The backward pass > Generalized cross validation | 325 | 1,449 | null |
Section: The model building process > Constraints. One constraint has already been mentioned: the user can specify the maximum number of terms in the forward pass. A further constraint can be placed on the forward pass by specifying a maximum allowable degree of interaction. Typically only one or two degrees of interac... | Wikipedia - Multivariate adaptive regression spline - The model building process > Constraints | 151 | 790 | null |
Section: Pros and cons. No regression modeling technique is best for all situations. The guidelines below are intended to give an idea of the pros and cons of MARS, but there will be exceptions to the guidelines. It is useful to compare MARS to recursive partitioning and this is done below. (Recursive partitioning is a... | Wikipedia - Multivariate adaptive regression spline - Pros and cons | 324 | 1,594 | null |
MARS models tend to have a good bias-variance trade-off. The models are flexible enough to model non-linearity and variable interactions (thus MARS models have fairly low bias), yet the constrained form of MARS basis functions prevents too much flexibility (thus MARS models have fairly low variance). MARS is suitable f... | Wikipedia - Multivariate adaptive regression spline - Pros and cons | 244 | 1,243 | null |
Section: Extensions and related concepts. Generalized linear models (GLMs) can be incorporated into MARS models by applying a link function after the MARS model is built. Thus, for example, MARS models can incorporate logistic regression to predict probabilities. Non-linear regression is used when the underlying form o... | Wikipedia - Multivariate adaptive regression spline - Extensions and related concepts | 330 | 1,669 | null |
Section: Methodology. Natural language processing methods are used to extract and identify language usage patterns common to speakers of an L1-group. This is done using language learner data, usually from a learner corpus. Next, machine learning is applied to train classifiers, like support vector machines, for predict... | Wikipedia - Native-language identification - Methodology | 172 | 869 | null |
Article: Neural modeling fields. Neural modeling field (NMF) is a mathematical framework for machine learning which combines ideas from neural networks, fuzzy logic, and model based recognition. It has also been referred to as modeling fields, modeling fields theory (MFT), Maximum likelihood artificial neural networks ... | Wikipedia - Neural modeling fields - Summary | 194 | 960 | null |
Section: Concept models and similarity measures. In the general case, NMF system consists of multiple processing levels. At each level, output signals are the concepts recognized in (or formed from) input, bottom-up signals. Input signals are associated with (or recognized, or grouped into) concepts according to the mo... | Wikipedia - Neural modeling fields - Concept models and similarity measures | 339 | 1,333 | null |
M . {\displaystyle {\vec {M}}_{m}({\vec {S}}_{m},n),m=1..M.} , where M is the number of models. Each model is characterized by its parameters, Sm; in the neuron structure of the brain they are encoded by strength of synaptic connections, mathematically, they are given by a set of numbers, S β m = { S m a } , a = 1.. A ... | Wikipedia - Neural modeling fields - Concept models and similarity measures | 339 | 1,228 | null |
This is a simplified description of perception. The most benign everyday visual perception uses many levels from retina to object perception. The NMF premise is that the same laws describe the basic interaction dynamics at each level. Perception of minute features, or everyday objects, or cognition of complex abstract ... | Wikipedia - Neural modeling fields - Concept models and similarity measures | 282 | 1,375 | null |
Important information could be contained in any bottom-up signal; Therefore, the similarity measure is constructed so that it accounts for all bottom-up signals, X(n), L ( { X β ( n ) } , { M β m ( S β m , n ) } ) = β n = 1 N l ( X β ( n ) ) . {\displaystyle L(\{{\vec {X}}(n)\},\{{\vec {M}}_{m}({\vec {S}}_{m},n)\})=\pr... | Wikipedia - Neural modeling fields - Concept models and similarity measures | 323 | 1,086 | null |
Its constituent elements are conditional partial similarities between signal X(n) and model Mm, l(X(n)|m). This measure is "conditional" on object m being present, therefore, when combining these quantities into the overall similarity measure, L, they are multiplied by r(m), which represent a probabilistic measure of o... | Wikipedia - Neural modeling fields - Concept models and similarity measures | 330 | 1,086 | null |
If learning is successful, it approximates probabilistic description and leads to near-optimal Bayesian decisions. The name "conditional partial similarity" for l(X(n)|m) (or simply l(n|m)) follows the probabilistic terminology. If learning is successful, l(n|m) becomes a conditional probability density function, a pro... | Wikipedia - Neural modeling fields - Concept models and similarity measures | 338 | 1,572 | null |
This requires a modification of the similarity measure L; The reason is that more models always result in a better fit between the models and data. This is a well known problem, it is addressed by reducing similarity L using a "skeptic penalty function," (Penalty method) p(N,M) that grows with the number of models M, a... | Wikipedia - Neural modeling fields - Concept models and similarity measures | 161 | 686 | null |
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