By Nicolò Cesa-Bianchi, Masayuki Numao, Rüdiger Reischuk (eds.)

This quantity includes the papers provided on the thirteenth Annual convention on Algorithmic studying thought (ALT 2002), which used to be held in Lub ¨ eck (Germany) in the course of November 24–26, 2002. the most target of the convention was once to p- vide an interdisciplinary discussion board discussing the theoretical foundations of computing device studying in addition to their relevance to useful functions. The convention was once colocated with the 5th foreign convention on Discovery technology (DS 2002). the amount comprises 26 technical contributions that have been chosen by means of this system committee from forty nine submissions. It additionally includes the ALT 2002 invited talks provided by means of Susumu Hayashi (Kobe collage, Japan) on “Mathematics in accordance with Learning”, by means of John Shawe-Taylor (Royal Holloway collage of L- don, united kingdom) on “On the Eigenspectrum of the Gram Matrix and Its courting to the Operator Eigenspectrum”, and via Ian H. Witten (University of Waikato, New Zealand) on “Learning constitution from Sequences, with functions in a electronic Library” (joint invited speak with DS 2002). in addition, this quantity - cludes abstracts of the invited talks for DS 2002 awarded via Gerhard Widmer (Austrian learn Institute for Arti?cial Intelligence, Vienna) on “In seek of the Horowitz issue: meantime record on a Musical Discovery venture” and by way of Rudolf Kruse (University of Magdeburg, Germany) on “Data Mining with Graphical Models”. the entire types of those papers are released within the DS 2002 court cases (Lecture Notes in Arti?cial Intelligence, Vol. 2534). ALT has been awarding the E.

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**Extra resources for Algorithmic Learning Theory: 13th International Conference, ALT 2002 Lübeck, Germany, November 24–26, 2002 Proceedings**

**Example text**

Furthermore the sum of the last m − k eigenvalues is similarly concentrated as is the residual when the data is projected into a ﬁxed subspace. Furthermore, we have shown that estimating the projection subspace on a random sample can give a good model for future data provided the number of examples is much larger than the dimension of the subspace. The results provide 38 J. Shawe-Taylor et al. 02 0 0 10 20 30 40 50 60 Projection Dimensionality 70 80 90 100 (b) Fig. 2. Plots of the fraction of the average squared norm captured in the subspace spanned by the ﬁrst k eigenvectors for diﬀerent values of k.

The projection norm PVˆk (ψ(x)) 2 as a linear function fˆ in a feature space Fˆ for which the kernel function is given by ˆ z) = k(x, z)2 . k(x, √ Furthermore the 2-norm of the function fˆ is k. 36 J. Shawe-Taylor et al. Proof : Let X = U ΣV be the singular value decomposition of the sample matrix X in the feature space. The projection norm is then given by fˆ(x) = PVˆk (ψ(x)) 2 = ψ(x) Uk Uk ψ(x), where Uk is the matrix containing the ﬁrst k columns of U . Hence we can write PVˆk (ψ(x)) NF NF 2 = ˆ αij ψ(x) ij , αij ψ(x)i ψ(x)j = ij=1 ij=1 ˆ is the projection mapping into the feature space Fˆ consisting of all where ψ pairs of F features and αij = (Uk Uk )ij .

Proof : The result follows from an application of Theorem 2 provided sup |P¯V (S) − P¯ (S \ {xi } ∪ {ˆ xi )| ≤ R2 /m. S,ˆ xi Clearly the largest change will occur if one of the points ψ(xi ) and ψ(ˆ xi ) is lies in the subspace V and the other does not. In this case the change will be at most R2 /m. On the Eigenspectrum of the Gram Matrix 35 The concentration results of this section are very tight. In the notation of the earlier sections they show that with high probability ˆ E PVˆk (ψ(x)) 2 = and 1 m k ˆ i ≈ Em E ˆ λ PVˆk (ψ(x)) i=1 E PVk (ψ(x)) k 2 ˆ λi ≈ E = 2 = Em 1 m PVk (ψ(x)) k ˆi λ i=1 2 , (12) i=1 where we have used Theorem 5 to obtain the ﬁrst approximate equality and Theorem 6 with V = Vk to obtain the second approximate equality.