By Dana Ron (auth.), Nader H. Bshouty, Claudio Gentile (eds.)

ISBN-10: 3540729259

ISBN-13: 9783540729259

ISBN-10: 3540729275

ISBN-13: 9783540729273

This booklet constitutes the refereed court cases of the 20 th Annual convention on studying concept, COLT 2007, held in San Diego, CA, united states in June 2007.

The forty-one revised complete papers offered including five articles on open difficulties and a couple of invited lectures have been conscientiously reviewed and chosen from a complete of ninety two submissions. The papers hide a variety of issues and are prepared in topical sections on unsupervised, semisupervised and lively studying, statistical studying concept, inductive inference, regularized studying, kernel equipment, SVM, on-line and reinforcement studying, studying algorithms and obstacles on studying, dimensionality relief, different methods, and open problems.

**Read Online or Download Learning Theory: 20th Annual Conference on Learning Theory, COLT 2007, San Diego, CA, USA; June 13-15, 2007. Proceedings PDF**

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**Extra info for Learning Theory: 20th Annual Conference on Learning Theory, COLT 2007, San Diego, CA, USA; June 13-15, 2007. Proceedings**

**Example text**

Ck of partition C are uniquely determined by matrix A. To this end, we view A as the adjacency matrix of a graph G with nodes x1 , x2 , . . , xn , where nodes xp , xq are connected by an edge if and only if Ap,q = 0. Let K1 , K2 , . . , Kℓ be the connected components of G. Note that there is an edge between xp and xq only if p and q belong to the same cluster Stability of k-Means Clustering 29 in C. Thus, the connected components of G represent a reﬁnement of partition C. Consider a ﬁxed cluster Cj in C with center cj .

U. Simon Here, c1 , c2 , . . , ck are the optimal centers ci = and C1 , C2 , . . , Ck ⊆ F are the clusters of C. xs ∈Ci μs xs / xs ∈Ci μs , Proof. Straightforward but long calculation, starting with formula (2). See the extended version paper [1] available online. ⊓ ⊔ Lemma 7 (Weights of Clusters). Let C and D be two partitions of F . Consider the weights μ assigned to points in F . Then, either for every point in F the weight of its cluster in C is the same as the weight of its cluster in D.

The technical lemmas are stated formally, and some of them are proved in Section 5. Proofs of the rest of the lemmas can be found in the extended version [1] available online. Section 2 is devoted to setting the ground in terms of deﬁnitions notation and basic observations. 2 Definitions In the rest of the paper we use the following standard notation. We consider a data space X endowed with probability measure P . A ﬁnite multi-set S = {x1 , x2 , . . , xm } of X is called a sample. d from (X, P ).

### Learning Theory: 20th Annual Conference on Learning Theory, COLT 2007, San Diego, CA, USA; June 13-15, 2007. Proceedings by Dana Ron (auth.), Nader H. Bshouty, Claudio Gentile (eds.)

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