By Hongmin Cai (auth.), Petra Perner (eds.)
This publication constitutes the refereed court cases of the eleventh business convention on facts Mining, ICDM 2011, held in long island, united states in September 2011.
The 22 revised complete papers awarded have been rigorously reviewed and chosen from a hundred submissions. The papers are geared up in topical sections on information mining in drugs and agriculture, facts mining in advertising, facts mining for commercial strategies and in telecommunication, Multimedia info Mining, theoretical facets of information mining, information Warehousing, WebMining and knowledge Mining.
Read Online or Download Advances in Data Mining. Applications and Theoretical Aspects: 11th Industrial Conference, ICDM 2011, New York, NY, USA, August 30 – September 3, 2011. Proceedings PDF
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For notational convenience, we deﬁne β(R) = 1 − C bi (R) and δ(R) = bi (R)di , i resulting in ζ1 (R) = β(R) R + Cδ(R). Furthermore, β(0) = 1 − nC and increases by jumps of size (multiples of) C to reach 1 when R = dn . Hence the minimum R∗ is located at the distance di for which β jumps above zero. These considerations allow us to mirror the conclusions we obtained for the original cost function: 1. The optimal value of R∗ coincides with one of the distances di which means that the optimal circle passes through at least one of the data points.
More precisely, for each point they introduce a variable ξi ≥ 0 such that xi − a 2 ≤ R2 + ξi . (2) This relaxation of the constraints is then oﬀset by adding a penalty term to the cost function: n ζ(R, a, ξ) := R2 + C ξi . i=1 The constant C is a (pre-deﬁned) unit cost that governs the trade-oﬀ between the size of the sphere and the number of outliers. After these modiﬁcations the authors in [8,4] arrive at the following constrained optimization problem: given n data points x1 , . . t. ∀i = 1, .
J. Pauwels and O. Ambekar The collection of all subgradients at a point x is called the subdiﬀerential of f at x and denoted by ∂f (x). Notice that the subdiﬀerential is a set-valued function! e. ∇f (x∗ ) = 0) can be generalized to non-diﬀerentiable functions as: 0 ∈ ∂f (x∗ ). e. set-valued) (23) ⎪ ⎩ 1 if x > 0 as at x = 0 any straight line with slope between 0 and 1 will be located under the graph of the ramp function. e. set-valued) (24) ⎪ ⎩ 0 if x < 0 To forestall confusion we point out that, unlike when used as a distribution, this deﬁnition of the Heaviside function insists its value at the origin is between zero and one.