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## Learning distributions by their density levels: A paradigm for learning without a teacher (1997)

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### Other Repositories/Bibliography

Venue: | Journal of Computer and System Sciences |

Citations: | 33 - 3 self |

### Citations

4769 |
Pattern Classification and scene analysis
- Duda, Hart
- 1973
(Show Context)
Citation Context ...y area in the feature vector space. A common demand is to identify these clusters and to report their number, position, size and shape, thereby getting insight to the nature or structure of the data, =-=[DH73]-=-. The identification of high-probability-density areas plays a central role in such task of classification via clustering. Yet another relevant scenario arises in computer vision, when one wishes to i... |

1204 | Probability, random variables, and stochastic processes (4th ed - Papoulis, Pillai - 2002 |

1133 |
On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications
- Vapnik, Chervonenkis
- 1971
(Show Context)
Citation Context ...he probability of the event c.) It turns out that if a class, C, has a small VC-dimension, then there are many small ffl-approximations for it. The following theorem is due to Vapnik and Chervonenkis =-=[VC71]-=-. Theorem 1: [[VC71]] There is a positive constant s such that if C is any concept class of VC-dimension at most d, and ffl; ffi ? 0, then with probability at least 1 \Gamma ffi, a randomly selected s... |

713 | Learnability and the Vapnik-Chervonenkis dimension
- Blumer, Ehrenfeucht, et al.
- 1989
(Show Context)
Citation Context ...nt density levels and ignoring the behaviour of the target distributions on all irrelevant levels. 3 Characterizing density level learnability. The fundamental theorem of PAC learnability, namely the =-=[BEHW89]-=- characterization of learnability, states that the finiteness of the VC-dimension of a concept class is both necessary and sufficient for its PAC-learnability. Furthermore, Blumer et al show that for ... |

477 |
A survey of the hough transform
- Illingworth, Kittler
- 1988
(Show Context)
Citation Context ..., (minimum spanning tree,) for example, are used for clustering in a feature space [DH73]. Methods that look for maximal consistency of a concept with the data are used for finding the straight edges =-=[IK88]-=-. In the context of Computational Learning Theory such tasks fall into the realm of unsupervisedslearning. It seems that un-supervised learning has, so far, attracted only limited attention in the Com... |

254 |
Learning from noisy examples
- Angluin, Laird
- 1988
(Show Context)
Citation Context ...l was 1 and finally outputting only the positively labeled sample points. There are some differences between this 'noisy-PAC` scenario and the common PAC classification-noise model (see, for example, =-=[AL88]-=-). First we assume that the student receives, as input, only positively labeled examples (rather than revealing to the student all the drawn examples and providing him their labels). The second differ... |

211 | Efficient distribution-free learning of probabilistic concepts
- Kearns, Schapire
- 1994
(Show Context)
Citation Context ... of p-concepts learning. For the relevant definitions of p-concept learning and the related notions of fl-shattering and the dimension dC (fl) we refer the reader to the papers of Kearns and Schapire =-=[KS90]-=- and of Simon [S93]. Proof: Given a class C of sets over a domain (X ; B; ) and a parameter 0 ! fl ! 0:5, we define a class of distributions, D C = fD t : t 2 Cg. Each distribution D t is defined by s... |

90 |
A course on empirical processes
- Dudley
- 1984
(Show Context)
Citation Context ...rem implies part 1. That is because, using standard VC calculation arguments 2 if VC-dim(C lev D ) = l then VC-dim(C r;ae )s2l log l. We therefore proceed to prove part 2. 2 The argument that Dudley, =-=[D84]-=-, uses for the calculation of the dimension of classes of intersections are applicable here as well. 8 The proof is in two parts. First, we show that if a sample is an ffl-approximation for C r;ae , t... |

37 | Polynomial bounds for VC dimension of sigmoidal neural networks
- Karpinski, Macintyre
- 1995
(Show Context)
Citation Context ...wnomials, which generalizes the topological properties of polynomials to other simple functions [K91], together with the technique developed in [BL93], imply that the VC-dimension is finite (see also =-=[KM95]-=-). 9 4 Un-Supervised Concept Learning So far we have been following the approach that views the example-generating distribution as the primal target of learning process. A different approach, prevalen... |

29 |
General bounds on the number of examples needed for learning probabilistic concepts
- Simon
- 1993
(Show Context)
Citation Context ...vel 1 \Gamma ffi, when the x i 's are generated by the Bernouli-p distribution (i.e., the distribution defined by prob(x i = 1) = p), the statisticssp = 1 n P n i=1 x i , is higher than than q. Simon =-=[S93]-=- proves that, if m ! LB(p; q; ffi) then no function of the variables (x 1 ; : : : ; xm ) can distiguish, with probability greater than (1 \Gamma ffi), between the case where the x i 's are generated b... |

27 | A survey of the Hough transforms,” Computer vision, Graphics and Image processing 44 - Illingworth, Kittler - 1988 |

24 | Lower bounds for sampling algorithms for estimating the average - Canetti, Even, et al. - 1995 |

17 | E cient distribution-free learning of probabilistic concepts - Kearns, Schapire - 1990 |

13 | Localization vs. identification of semi-algebraic sets
- Ben-David, Lindenbaum
- 1998
(Show Context)
Citation Context ...e not polynomial anymore. Nevertheless, the theory of fewnomials, which generalizes the topological properties of polynomials to other simple functions [K91], together with the technique developed in =-=[BL93]-=-, imply that the VC-dimension is finite (see also [KM95]). 9 4 Un-Supervised Concept Learning So far we have been following the approach that views the example-generating distribution as the primal ta... |

12 | Fewnomials, Translations of Mathematical Monographs - Khovanskĭı - 1991 |

8 |
Fewnomials
- Khovanskii
- 1991
(Show Context)
Citation Context ...n IR n is more complicated as the r + -levels are not polynomial anymore. Nevertheless, the theory of fewnomials, which generalizes the topological properties of polynomials to other simple functions =-=[K91]-=-, together with the technique developed in [BL93], imply that the VC-dimension is finite (see also [KM95]). 9 4 Un-Supervised Concept Learning So far we have been following the approach that views the... |

7 |
Probably approximate learning of sets and functions
- Natarajan
- 1991
(Show Context)
Citation Context ...ing. It seems that un-supervised learning has, so far, attracted only limited attention in the Computational Learnability research, mainly under the title of learning from positive examples.Natarajan =-=[Nat91]-=- provides a necessary and sufficient condition for distribution-free learnability from positive examples. This condition is very restrictive and rules out most of the interesting examples one may wish... |

3 | Learnability andTheVapnik-Chervonenkis Dimension - Blumer, Ehrenfeucht, et al. - 1989 |

1 |
Private communication
- Dichterman
(Show Context)
Citation Context ...\Gamma q) 2 log 1 ffi ! : To obtain our next lower bound, we shall apply the following non-asymptotic version of this lower bound on tail probabilities. The lemma was communicated to us by Dichterman =-=[D95]-=-. Lemma 4: [Dichterman] LB(p; q; ffi = 0=05)s/ p(1 \Gamma p) p \Gamma q ! 2 . Now, we may prove the following lower bound on the sample complexity required for (ff; fi)- learnability. Lemma 5: For any... |

1 |
94, "On the Learnability of Discrete Distributions
- Kearns, Mansour, et al.
(Show Context)
Citation Context ...tudied problem in statistics and pattern recognition literature. Some variants of this fundamental task have been recently investigated in the context of computational learning theory by Kearns et al =-=[KMRRSS94]-=-. The starting point of this work is the observation that, for many un-supervised learning tasks (including those mentioned above), a much weaker type of information suffices. Rather than attempting t... |

1 |
Learning by Smoothing: a Morphological approach
- Kim
- 1991
(Show Context)
Citation Context ...ee learnability from positive examples. This condition is very restrictive and rules out most of the interesting examples one may wish to consider in Computer Vision or Pattern Recognition tasks. Kim =-=[Kim91]-=-, restricts his attention to limited classes of `nicely 2 behaving' distributions over R n and offers an algorithm for learning geometrical objects with respect to such classes. From a wider mathemati... |

1 | Localization vs. Identi cation of SemiAlgebraic Sets - Ben-David, Lindenbaum - 1993 |

1 | Private communication. 16 [D84] [IK88] [KM95] [K91 - Dichterman - 1973 |