The error rate of learning halfspaces using kernel-SVM

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$Lectures on fractal geometry and dynamics$

4. The decomposition of ρ into a sum of irreducible representation is H = ⊕ n∈I Y d n forsome set I ⊂ N 0 . Moreover,∀f, g ∈ H k , 〈f, g〉 Hk = ∑ n∈IWhere {a n } n∈I are positive numbers.5. It holds that ∑ n∈IN d,n|S d−1 | a−2 n = 1.a 2 n〈P d,n f, P d,n g〉 L 2 (S d−1 )Proof Let f ∈ H k , A ∈ O(d). To prove part 1, assume first thatn∑∀x ∈ S d−1 , f(x) = α i k(x, y i ) (9)For some y 1 , . . . , y n ∈ S d−1 and α 1 , . . . , α n ∈ C. We have, since k is symmetric, thatn∑f ◦ A(x) = α i k(Ax, y i )==i=1i=1n∑α i k(A −1 Ax, A −1 y i )i=1n∑α i k(x, A −1 y i )Thus, by Theorem 5.1, f ◦ A ∈ H k . Moreover, it holds that||f ◦ A|| 2 H k= ∑α i ᾱ j k(A −1 y j , A −1 y i )i=11≤i,j≤n= ∑1≤i,j≤nα i ᾱ j k(y j , y i ) = ||f|| 2 H kThus, part 1 holds for function f ∈ H k of the form (9). For general f ∈ H k , by Theorem 5.1,there is a sequence f n ∈ H k of functions of the from (9) that converges to f in H k . From whatwe have shown for functions of the form (9) if follows that ||f n −f m || Hk = ||f n ◦A−f m ◦A|| Hk ,thus f n ◦ A is a Cauchy sequence, hence, has a limit g ∈ H k . By Theorem 5.1, convergencein H k entails point wise convergence, thus, g = f ◦ A. Finally,||f|| Hk = limn→∞||f n || Hk = limn→∞||f n ◦ A|| Hk = ||f ◦ A|| HkWe proceed to part 2. It is not hard to check that ρ is group homomorphism, so itonly remains to validate that for every f ∈ H the mapping A ↦→ ρ(A)f is continuous. Letɛ > 0 and let A ∈ O(d). We must show that there exists a neighbourhood U of A such that∀B ∈ U, ||f ◦ A −1 − f ◦ B −1 || Hk < ɛ. Choose g(·) = ∑ ni=1 α ik(·, y i ) such that ||g − f|| Hk < ɛ . 3By part 1, it holds that||f ◦ A −1 − f ◦ B −1 || Hk ≤ ||f ◦ A −1 − g ◦ A −1 || Hk + ||g ◦ A −1 − g ◦ B −1 || Hk + ||g ◦ B −1 − f ◦ B −1 || Hk= ||f − g|| Hk + ||g ◦ A −1 − g ◦ B −1 || Hk + ||g − f|| Hk< ɛ 3 + ||g ◦ A−1 − g ◦ B −1 || Hk + ɛ 325
Thus, it is enough to find a neighbourhood U of A such that ∀B ∈ U, ||g◦A −1 −g◦B −1 || Hk
Page 1 and 2: The complexity of learning halfspac
Page 3 and 4: exists an equivalent inner product
Page 5 and 6: is enough that we can efficiently c
Page 7 and 8: our terminology, they considered th
Page 9 and 10: It is shown in (Birnbaum and Shalev
Page 11 and 12: We now expand on this brief descrip
Page 13 and 14: and (uniformly and independently) a
Page 15 and 16: The proof of Theorem 2.7To prove Th
Page 17 and 18: attempts to prove a quantitative op
Page 19 and 20: 5.1.3 Harmonic Analysis on the Sphe
Page 21 and 22: Lemma 5.11 (John’s Lemma) (Matous
Page 23 and 24: For 1 ≤ i ≤ t Let v i = x i −
Page 25: ( )that in this case Err µN ,hinge
Page 29 and 30: Legendre polynomials we have|P d,n
Page 31 and 32: Then, for every K ∈ N, 1 8 > γ >
Page 33 and 34: Now, it holds that∫∫∫ ∫∫
Page 35 and 36: We note that ω f◦g ≤ ω f ·
Page 37 and 38: Now, denote δ = ∫ g. It holds th
Page 39 and 40: equivalent formulationminErr D,l (f
Page 41 and 42: Denote ||g|| Hk = C. By Lemma 5.25,
Page 43 and 44: Consequently, every approximated so
Page 45: Kosaku Yosida. Functional Analysis.

The error rate of learning halfspaces using kernel-SVM

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