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## Label Ranking by Learning Pairwise Preferences

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4327 |
Data mining: Practical machine learning tools and techniques
- Witten, Frank, et al.
- 2011
(Show Context)
Citation Context ...ively, can frequently be found. More precisely, our experiments considered two types of genetic data, namely phylogenetic 8 We employed the implementation offered by the Weka machine learning package =-=[65]-=- in its default setting. To obtain a ranking of labels, classification scores were transformed into (pseudo-)probabilities using a logistic regression technique [54]. 9 This algorithm is based on the ... |

3418 |
UCI Repository of machine learning databases
- Blake, Merz
- 1998
(Show Context)
Citation Context ...tive computational demands of the constraint classification approach. 22nating from several different domains, the following multiclass datasets from the UCI Repository of machine learning databases =-=[7]-=- and the Statlog collection [50] were included in the experimental evaluation: iris, wine, glass, vowel, vehicle (a summary of dataset properties is given in Table 2). These datasets were also used in... |

1250 | Optimizing search engines using clickthrough data
- Joachims
- 2002
(Show Context)
Citation Context ...a set of objects O ⊆ X and returns a permutation (ranking) of this set Fig. 1. Learning from object preferences As an example consider the problem of learning to rank query results of a search engine =-=[39, 56]-=-. The training information is provided implicitly by the user who clicks on some of the links in the query result and not on others. This information can be turned into binary preferences by assuming ... |

1028 | Probabilistic Outputs for Support Vector Machines and Comparisons to Regularized Likelihood Methods
- Platt
- 1999
(Show Context)
Citation Context ... the Weka machine learning package [65] in its default setting. To obtain a ranking of labels, classification scores were transformed into (pseudo-)probabilities using a logistic regression technique =-=[54]-=-. 9 This algorithm is based on the “alpha-trick”. We set the corresponding parameter α to 500. 10 Again, we used the implementation offered by the Weka package. 11 All linear models also incorporate a... |

927 | A comparison of methods for multiclass support vector machines
- Hsu, Lin
- 2002
(Show Context)
Citation Context ...h the highest number of votes is proposed as a final prediction. 3 Pairwise classification has been tried in the areas of statistics [9, 23], neural networks [44, 45, 55, 48], support vector machines =-=[58, 32, 46, 35]-=-, and others. Typically, the technique learns more accurate theories than the more commonly used one-against-all classification method, which learns one theory for each class, using the examples of th... |

557 | Reducing multiclass to binary: A unifying approach for margin classifiers - Allwein, Schapire, et al. |

517 |
The Proof and Measurement of Association between Two Things
- Spearman
- 1904
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Citation Context ... 6 in our case, Y is given by Sm. 5.2 Spearman’s Rank Correlation An important and frequently applied similarity measure for rankings is the Spearman rank correlation, originally proposed by Spearman =-=[61]-=- as a nonparametric rank statistic to measure the strength of the associations between two variables [47]. It is defined as follows 1 − 6D(τ, τ ′ ) m(m 2 − 1) (5.2) as a linear transformation (normali... |

497 |
Probabilistic Metric Spaces
- Schweizer, Sklar
- 1983
(Show Context)
Citation Context ... 1] 2 → [0, 1] which is associative, commutative, monotone, and satisfies ⊤(0, x) = 0, ⊤(1, x) = x for all x. Operators of that kind have been introduced in the context of probabilistic metric spaces =-=[59]-=- and have been studied intensively in fuzzy set theory in recent years [43]. A binary relation R ⊂ A × A is called ⊤-transitive if it satisfies R(a, c) ≥ ⊤(R(a, b), R(b, c)) for all a, b, c ∈ A. There... |

471 | Rank aggregation methods for the web - Dwork, Kumar, et al. - 2001 |

444 |
JM D’Abrera. Nonparametrics: statistical methods based on ranks
- Lehmann, Howard
- 2006
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Citation Context ...ilarity measure for rankings is the Spearman rank correlation, originally proposed by Spearman [61] as a nonparametric rank statistic to measure the strength of the associations between two variables =-=[47]-=-. It is defined as follows 1 − 6D(τ, τ ′ ) m(m 2 − 1) (5.2) as a linear transformation (normalization) of the sum of squared rank distances D(τ ′ , τ) df = m∑ ( τ i=1 ′ (i) − τ(i) ) 2 (5.3) to the int... |

422 | Accurately interpreting clickthrough data as implicit feedback
- Joachims, Granka, et al.
- 2005
(Show Context)
Citation Context ... of the links in the query result and not on others. This information can be turned into binary preferences by assuming that the selected pages are preferred over nearby pages that are not clicked on =-=[40]-=-. 2.2 Learning from Label Preferences In this learning scenario, the problem is to predict, for any instance x (e.g., a person) from an instance space X , a preference relation ≻x ⊆ L × L among a fini... |

404 | Learning to order things
- Cohen, Schapire, et al.
- 1999
(Show Context)
Citation Context ...AI-2006 conferences (the second and fifth organized by two of the authors). 2modeling utility functions modeling pairwise preferences object ranking comparison training [62] learning to order things =-=[14]-=- label ranking constraint classification [30] this work [26] Table 1 Four different approaches to learning from preference information together with representative references 2 Learning from Preferenc... |

390 |
Fuzzy Preference Modelling and Multicriteria Decision Support
- Fodor, Roubens
- 1994
(Show Context)
Citation Context ... problem of inducing a ranking from a (valued) preference relation has received a lot of attention in several research fields, e.g., in fuzzy preference modeling and (multi-attribute) decision making =-=[22]-=-. In the context of pairwise classification and preference learning, several studies have empirically compared different ways of combining the predictions of individual classifiers [66, 2, 38, 25]. A ... |

372 | R.: Classification by pairwise coupling
- Hastie, Tibshirani
- 1998
(Show Context)
Citation Context ...h the highest number of votes is proposed as a final prediction. 3 Pairwise classification has been tried in the areas of statistics [9, 23], neural networks [44, 45, 55, 48], support vector machines =-=[58, 32, 46, 35]-=-, and others. Typically, the technique learns more accurate theories than the more commonly used one-against-all classification method, which learns one theory for each class, using the examples of th... |

358 |
Rank analysis of incomplete block designs. I. The method of paired comparisons
- Bradley, Terry
- 1952
(Show Context)
Citation Context ...Mij is interpreted as a vote for either λi or λj, and the label with the highest number of votes is proposed as a final prediction. 3 Pairwise classification has been tried in the areas of statistics =-=[9, 23]-=-, neural networks [44, 45, 55, 48], support vector machines [58, 32, 46, 35], and others. Typically, the technique learns more accurate theories than the more commonly used one-against-all classificat... |

312 | Ultraconservative online algorithms for multiclass problems
- Crammer, Singer
- 2003
(Show Context)
Citation Context ...ors αi, αj appropriately. In particular, using perceptron training, the algorithm can be implemented in terms of a multi-output perceptron in a way quite similar to the approach of Crammer and Singer =-=[16]-=-. 2.3.2 Log-Linear Models for Label Ranking So-called log-linear models for label ranking have been proposed in Dekel et al. [18]. Here, utility functions are expressed in terms of linear combinations... |

310 | In defense of one-vs-all classification
- Rifkin, Klautau
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Citation Context ...n about the preference between λi and λj are ignored. 3 Ties can be broken in favor or prevalent classes, i.e., according to the class distribution in the classification setting. 4 Rifkin and Klautau =-=[57]-=- have argued that, at least in the case of support vector machines, one-against-all can be as effective provided that the binary base classifiers are carefully tuned. 9dataset with preferences for ea... |

306 | CP-nets: A tool for representing and reasoning with conditional ceteris paribus preference statements
- Boutilier, Brafman, et al.
(Show Context)
Citation Context ...to rank learning algorithms according to their suitability for a new dataset, based on the characteristics of this dataset [11]. Finally, every preference statement in the well-known CP-nets approach =-=[8]-=-, a qualitative graphical representation that reflects conditional dependence and independence of preferences under a ceteris paribus interpretation, formally corresponds to a label ranking. In additi... |

291 | Probability estimates for multiclass classification by pairwise coupling
- Wu, Lin, et al.
- 2004
(Show Context)
Citation Context ...ecision making [22]. In the context of pairwise classification and preference learning, several studies have empirically compared different ways of combining the predictions of individual classifiers =-=[66, 2, 38, 25]-=-. A simple though effective strategy is a generalization of the aforementioned voting strategy: each alternative λi is evaluated by the sum of (weighted) votes S(λi) = ∑ λj=λi Rx(λi, λj), (3.3) and a... |

181 |
Another approach to polychotomous classification
- Friedman
- 1996
(Show Context)
Citation Context ...Mij is interpreted as a vote for either λi or λj, and the label with the highest number of votes is proposed as a final prediction. 3 Pairwise classification has been tried in the areas of statistics =-=[9, 23]-=-, neural networks [44, 45, 55, 48], support vector machines [58, 32, 46, 35], and others. Typically, the technique learns more accurate theories than the more commonly used one-against-all classificat... |

179 |
Voting schemes for which it can be difficult to tell who won the election. Social Choice and Welfare 6:157–165
- Bartholdi, Tovey, et al.
- 1989
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Citation Context ...ck of using Kendall’s tau instead of rank correlation as a distance measure in (5.11) is a loss of computational efficiency. In fact, the computation of Kemeny-optimal rankings is known to be NP-hard =-=[6]-=-. 7 See, e.g., Marden’s book [49], which also contains results closely related to our results from Section 5.2. 206 Empirical Evaluation The experimental evaluation presented in this section compares... |

141 |
Analyzing and Modeling Rank Data
- Marden, editor
- 1995
(Show Context)
Citation Context ...d of rank correlation as a distance measure in (5.11) is a loss of computational efficiency. In fact, the computation of Kemeny-optimal rankings is known to be NP-hard [6]. 7 See, e.g., Marden’s book =-=[49]-=-, which also contains results closely related to our results from Section 5.2. 206 Empirical Evaluation The experimental evaluation presented in this section compares, in terms of accuracy and comput... |

116 |
Single-layer learning revisited: a stepwise procedure for building and training a neural network
- Knerr, Personnaz, et al.
- 1990
(Show Context)
Citation Context ...ote for either λi or λj, and the label with the highest number of votes is proposed as a final prediction. 3 Pairwise classification has been tried in the areas of statistics [9, 23], neural networks =-=[44, 45, 55, 48]-=-, support vector machines [58, 32, 46, 35], and others. Typically, the technique learns more accurate theories than the more commonly used one-against-all classification method, which learns one theor... |

107 | Log-linear models for label ranking
- Dekel, Keshet, et al.
- 2004
(Show Context)
Citation Context ...t reflects conditional dependence and independence of preferences under a ceteris paribus interpretation, formally corresponds to a label ranking. In addition, it has been observed by several authors =-=[30, 26, 18]-=- that many conventional learning problems, such as classification and multi-label classification, may be formulated in terms of label preferences: • Classification: A single class label λi is assigned... |

86 | D.: Constraint classification: A new approach to multiclass classification. Algorithmic Learning Theory
- Har-Peled, Roth, et al.
- 2002
(Show Context)
Citation Context ...anized by two of the authors). 2modeling utility functions modeling pairwise preferences object ranking comparison training [62] learning to order things [14] label ranking constraint classification =-=[30]-=- this work [26] Table 1 Four different approaches to learning from preference information together with representative references 2 Learning from Preferences In this section, we will motivate preferen... |

85 | Ranking tournaments
- Alon
- 2006
(Show Context)
Citation Context ...rs) can be used. Finding the ranking that minimizes (5.10) is formally equivalent to solving the graph-theoretical feedback arc set problem (for weighted tournaments) which is known to be NP complete =-=[3]-=-. Of course, in the context of label ranking, this result should be put into perspective, because the set of class labels is typically of small to moderate size. Nevertheless, from a computational poi... |

83 |
Voting procedures
- Brams, Fishburn
- 2002
(Show Context)
Citation Context ...g Theory It is worth mentioning that the voting strategy in RPC, as discussed in Section 5.2, is closely related to the so-called Borda-count, a voting rule that is well-known in social choice theory =-=[10]-=-: Suppose that the preferences of n voters are expressed in terms of rankings τ1, τ2 . . . τn of m alternatives. From a ranking τi, the following scores are derived for the alternatives: The best alte... |

83 | Task decomposition and module combination based on class relations: A modular neural network for pattern classification
- Lu, Ito
- 1999
(Show Context)
Citation Context ...ote for either λi or λj, and the label with the highest number of votes is proposed as a final prediction. 3 Pairwise classification has been tried in the areas of statistics [9, 23], neural networks =-=[44, 45, 55, 48]-=-, support vector machines [58, 32, 46, 35], and others. Typically, the technique learns more accurate theories than the more commonly used one-against-all classification method, which learns one theor... |

77 |
Pairwise classification and support vector machines
- Kreßel
- 1999
(Show Context)
Citation Context ...h the highest number of votes is proposed as a final prediction. 3 Pairwise classification has been tried in the areas of statistics [9, 23], neural networks [44, 45, 55, 48], support vector machines =-=[58, 32, 46, 35]-=-, and others. Typically, the technique learns more accurate theories than the more commonly used one-against-all classification method, which learns one theory for each class, using the examples of th... |

75 | Ordering by weighted number of wins gives a good ranking for weighted tournaments
- Coppersmith, Fleischer, et al.
(Show Context)
Citation Context ... employed the implementation for naive Bayes classification on numerical datasets (NaiveBayesSimple) contained in the Weka machine learning package [65]. 15 For example, it has recently been shown in =-=[15]-=- that optimizing rank correlation yields a 5-approximation to the ranking which is optimal for the Kendall measure. 23Table 3 Experimental results (mean and standard deviation) in terms of Kendall’s ... |

73 |
DIETTERICH AND GHULUM BAKIRI. Solving multiclass learning problems via error-correcting output codes
- THOMAS
- 1995
(Show Context)
Citation Context ...el problem, but also to select an appropriate threshold between relevant and irrelevant labels. It is well-known that pairwise classification is a special case of Error Correcting Output Codes (ECOC) =-=[19]-=- or, more precisely, their generalization that has been introduced in [2]. Even though ECOC allows for a more flexible decomposition of the original problem into simpler ones, the pairwise approach ha... |

68 | A family of additive online algorithms for category ranking
- Crammer, Singer
- 2003
(Show Context)
Citation Context ...n, and its relation to other learning class binarization techniques. 28Another special scenario is the application of label ranking algorithms to multilabel problems. For example, Crammer and Singer =-=[17]-=- consider a variety of on-line learning algorithms for the problem of ranking possible labels in a multi-label text categorization task. They investigate a set of algorithms that maintain a prototype ... |

67 | Ranking learning algorithms: Using ibl and meta-learning on accuracy and time results
- Brazdil, Soares, et al.
- 2003
(Show Context)
Citation Context ...atures [4]. Another application scenario is meta-learning, where the task is to rank learning algorithms according to their suitability for a new dataset, based on the characteristics of this dataset =-=[11]-=-. Finally, every preference statement in the well-known CP-nets approach [8], a qualitative graphical representation that reflects conditional dependence and independence of preferences under a ceteri... |

66 | Constraint classification for multiclass classification and ranking
- Har-Peled, Roth, et al.
- 2003
(Show Context)
Citation Context ...re instances with partial preference information, provided the (expected) total number of pairwise preferences is the same. 7 Related Work As noted in Section 6, the work on constraint classification =-=[30, 31]-=- appears to be a natural counterpart to our algorithm. In the same section, we have also discussed the log-linear models for label ranking proposed by Dekel et al. [18]. As both CC and LL are directly... |

63 | Prospects for preferences
- Doyle
- 2004
(Show Context)
Citation Context ...ttracted considerable attention in Artificial Intelligence (AI) research, notably in fields such as agents, non-monotonic reasoning, constraint satisfaction, planning, and qualitative decision theory =-=[20]-=-. 1 Preferences provide a means for specifying desires in a declarative way, which is a point of critical importance for AI. In fact, consider AI’s paradigm of a rationally acting (decision-theoretic)... |

43 |
Speaker identification via support vector classifiers
- Schmidt, Gish
- 1996
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Citation Context |

35 | Connectionist learning of expert preferences by comparison training
- Tesauro
- 1988
(Show Context)
Citation Context ...S-04, GfKl-05, IJCAI-05 and ECAI-2006 conferences (the second and fifth organized by two of the authors). 2modeling utility functions modeling pairwise preferences object ranking comparison training =-=[62]-=- learning to order things [14] label ranking constraint classification [30] this work [26] Table 1 Four different approaches to learning from preference information together with representative refere... |

34 | Pairwise neural network classifiers with probabilistic outputs
- Price, Knerr, et al.
- 1995
(Show Context)
Citation Context ...ote for either λi or λj, and the label with the highest number of votes is proposed as a final prediction. 3 Pairwise classification has been tried in the areas of statistics [9, 23], neural networks =-=[44, 45, 55, 48]-=-, support vector machines [58, 32, 46, 35], and others. Typically, the technique learns more accurate theories than the more commonly used one-against-all classification method, which learns one theor... |

34 | Efficient learning of label ranking by soft projections onto polyhedra
- Shalev-Shwartz, Singer
(Show Context)
Citation Context ...o (7.2), the number of these functions is predetermined by the number of labels (m), and each of them has the same relevance (i.e., weighing coefficients αi are not needed). Shalev-Shwartz and Singer =-=[60]-=- learn utility functions fi(·) on the basis of a different type of training information, namely real values g(λi) that reflect the relevance of the labels λi for an input x. Binary preferences between... |

33 | Noise tolerant variants of the perceptron algorithm
- Khardon, Wachman
- 2007
(Show Context)
Citation Context ... as a binary classifier (CC-SVM). 8 Apart from CC in its original version, we also included an online-variant (CC-P) as proposed in [30], using a noise-tolerant perceptron algorithm as a base learner =-=[41]-=-. 9 To guarantee a fair comparison, we use LL with (2.2) as base ranking functions, which means that it is based on the same underlying model class as CC. Moreover, we implement RPC with simple logist... |

28 | Efficient pairwise classification, in
- Park, Fürnkranz
- 2007
(Show Context)
Citation Context ...ch preference constraint.) As all the model parameters have to be used for deriving a label ranking, this may also affect the prediction time. However, for the classification setting, it was shown in =-=[52]-=- that a more efficient algorithm yields the same predictions as voting in almost linear time (≈ O(ℓ · m)). To what extent this algorithm can be generalized to label ranking is currently under investig... |

24 | An Analysis of Alternative Strategies for Implementing Matching Algorithms - Ball, Derigs - 1983 |

21 | Similarity of personal preferences: Theoretical foundations and empirical analysis
- Ha, Haddawy
(Show Context)
Citation Context ...e for decomposing a preference graph into subgraphs, and defines the generalized error as the fraction of subgraphs that are not exactly in agreement with the learned utility function. Ha and Haddawy =-=[28]-=- discuss a variety of different ranking loss functions and introduce a different extension of Kendall’s tau. With respect to predictive performance, Usunier et al. [63] analyze the generalization prop... |

20 | Preference elicitation via theory refinement
- Haddawy, Restificar, et al.
- 2003
(Show Context)
Citation Context ...components of the network with a single network, which can subsequently provide a real-valued evaluation of single states. Later works on learning utility function from object preference data include =-=[64, 34, 39, 29]-=- Subsequently, we outline two approaches, constraint classification (CC) and log-linear models for label ranking (LL), that are direct alternatives to our method of ranking by pairwise comparison, and... |

17 |
robin classification
- Round
- 2002
(Show Context)
Citation Context ...s functions on label rankings without the need for re-training the pairwise classifiers. 83.1 Pairwise Classification The key idea of pairwise learning is well-known in the context of classification =-=[24]-=-, where it allows one to transform a multi-class classification problem, i.e., a problem involving m > 2 classes L = {λ1 . . . λm}, into a number of binary problems. To this end, a separate model (bas... |

16 | Fürnkranz and Eyke Hüllermeier. Preference learning - Johannes - 2005 |

16 | Comparison of ranking procedures in pairwise preference learning
- Hullermeier, Furnkranz
- 2004
(Show Context)
Citation Context ...ecision making [22]. In the context of pairwise classification and preference learning, several studies have empirically compared different ways of combining the predictions of individual classifiers =-=[66, 2, 38, 25]-=-. A simple though effective strategy is a generalization of the aforementioned voting strategy: each alternative λi is evaluated by the sum of (weighted) votes S(λi) = ∑ λj=λi Rx(λi, λj), (3.3) and a... |

14 | Ranking by pairwise comparison: a note on risk minimization
- Hullermeier, Furnkranz
- 2004
(Show Context)
Citation Context ...h alternative approaches applicable to the same learning problem. The paper closes with a discussion of related work in Section 7 and concluding remarks in Section 8. Parts of this paper are based on =-=[26, 27, 36]-=-. 1 The increasing activity in this area is also witnessed by several workshops that have been devoted to preference learning and related topics, such as those at the NIPS-02, KI-03, SIGIR-03, NIPS-04... |

14 | Generalization error bounds for classifiers trained with interdependent data
- Usunier, Amini, et al.
- 2006
(Show Context)
Citation Context ...tility function. Ha and Haddawy [28] discuss a variety of different ranking loss functions and introduce a different extension of Kendall’s tau. With respect to predictive performance, Usunier et al. =-=[63]-=- analyze the generalization properties of binary classifiers trained on interdependent data for certain types of structured learning problems such as bipartite ranking. As mentioned in Section 2, labe... |

13 | Learning label preferences: Ranking error versus position error - Hüllermeier, Fürnkranz - 2005 |

11 |
Fürnkranz and Eyke Hüllermeier. Pairwise preference learning and ranking
- Johannes
- 2003
(Show Context)
Citation Context ...h alternative approaches applicable to the same learning problem. The paper closes with a discussion of related work in Section 7 and concluding remarks in Section 8. Parts of this paper are based on =-=[26, 27, 36]-=-. 1 The increasing activity in this area is also witnessed by several workshops that have been devoted to preference learning and related topics, such as those at the NIPS-02, KI-03, SIGIR-03, NIPS-04... |

9 |
Hüllermeier E.: A unified model for multilabel classification and ranking
- Brinker, Fürnkranz
- 2006
(Show Context)
Citation Context ...n task. They investigate a set of algorithms that maintain a prototype for each possible label, and order the labels of an example according to the response signal returned by each of the prototypes. =-=[12]-=- demonstrates a general technique that not only allows one to rank all possible labels in multi-label problem, but also to select an appropriate threshold between relevant and irrelevant labels. It is... |

9 | Algorithms for the Computation of T-Transitive Closures - Naessens, Meyer, et al. |

8 | Supervised learning of preference relations
- Herbrich, Graepel, et al.
- 1998
(Show Context)
Citation Context ...components of the network with a single network, which can subsequently provide a real-valued evaluation of single states. Later works on learning utility function from object preference data include =-=[64, 34, 39, 29]-=- Subsequently, we outline two approaches, constraint classification (CC) and log-linear models for label ranking (LL), that are direct alternatives to our method of ranking by pairwise comparison, and... |

6 |
robin ensembles
- Round
(Show Context)
Citation Context ...ecision making [22]. In the context of pairwise classification and preference learning, several studies have empirically compared different ways of combining the predictions of individual classifiers =-=[66, 2, 38, 25]-=-. A simple though effective strategy is a generalization of the aforementioned voting strategy: each alternative λi is evaluated by the sum of (weighted) votes S(λi) = ∑ λj=λi Rx(λi, λj), (3.3) and a... |

6 |
Artificial neural networks versus natural neural networks: a connectionist paradigm for preference assessment. Decision Support Systems
- Wang
- 1994
(Show Context)
Citation Context ...components of the network with a single network, which can subsequently provide a real-valued evaluation of single states. Later works on learning utility function from object preference data include =-=[64, 34, 39, 29]-=- Subsequently, we outline two approaches, constraint classification (CC) and log-linear models for label ranking (LL), that are direct alternatives to our method of ranking by pairwise comparison, and... |

4 |
Eyke Hüllermeier, Nils Weskamp, and Jörg Kämper. Clustering of gene expression data using a local shape-based similarity measure
- Balasubramaniyan
(Show Context)
Citation Context ...m λi ≻xk λj Find: • a ranking function that maps any x ∈ X to a ranking ≻x of L (permutation τx ∈ Sm) Fig. 2. Learning from label preferences by microarray analysis from phylogenetic profile features =-=[4]-=-. Another application scenario is meta-learning, where the task is to rank learning algorithms according to their suitability for a new dataset, based on the characteristics of this dataset [11]. Fina... |

3 |
Fürnkranz, Eyke Hüllermeier. Label Ranking by Learning Pairwise Preferences
- Brinker, Johannes
- 2007
(Show Context)
Citation Context ...ill focus on other key works related to label ranking and pairwise decomposition techniques that have recently appeared in the literature; a somewhat more exhaustive literature survey can be found in =-=[13]-=-. We are not aware of any other work that, as our method, approaches the label ranking problem by learning pairwise preference predicates Rx(λi, λj), 1 ≤ i < j ≤ m, and, thereby, reduces the problem t... |

3 |
Léon Personnaz, and Gérard Dreyfus. Handwritten digit recognition by neural networks with single-layer training
- Knerr
- 1992
(Show Context)
Citation Context |

2 |
G.: Rank correlation methods Charles
- Kendall
- 1975
(Show Context)
Citation Context ...functions. As mentioned previously, this can be accomplished by replacing the ranking procedure in the second step of RPC in a suitable way. To illustrate, consider the well-known Kendall tau measure =-=[42]-=- as an alternative loss function. This measure essentially calculates the number of pairwise rank inversions on labels to measure the ordinal correlation of two rankings; more formally, with D(τ ′ , τ... |

2 |
Data available at ftp://ftp.ncc.up.pt/pub/statlog
- Michie, Spiegelhalter, et al.
- 1994
(Show Context)
Citation Context ...he constraint classification approach. 22nating from several different domains, the following multiclass datasets from the UCI Repository of machine learning databases [7] and the Statlog collection =-=[50]-=- were included in the experimental evaluation: iris, wine, glass, vowel, vehicle (a summary of dataset properties is given in Table 2). These datasets were also used in a recent experimental study on ... |

1 |
Aiolli A preference model for structured supervised learning tasks
- Fabio
- 2005
(Show Context)
Citation Context ...ematic illustration of learning by pairwise comparison. The mapping (3.1) can be realized by any binary classifier. Alternatively, one may also employ base classifiers that map into the unit interval =-=[0, 1]-=- instead of {0, 1}, and thereby assign a valued preference relation Rx to every (query) instance x ∈ X : Rx(λi, λj) = { Mij(x) if i < j 1 − Mji(x) if i > j (3.2) for all λi = λj ∈ L. The output of a ... |

1 | Pursuing the best ECOC dimension for multiclass problems
- Pimenta, Gama, et al.
- 2007
(Show Context)
Citation Context ...good overall performance. In several experimental studies, including [2], it performed en par or better with competing decoding matrices. While finding a good encoding matrix still is an open problem =-=[53]-=-, it can be said that pairwise classification is among the most efficient decoding schemes. Even though we have to train a quadratic number of classifiers, both training (and to some extent also testi... |