Download Regularization, Optimization, Kernels, and Support Vector by Johan A.K. Suykens, Marco Signoretto, Andreas Argyriou PDF

By Johan A.K. Suykens, Marco Signoretto, Andreas Argyriou

Regularization, Optimization, Kernels, and help Vector Machines bargains a photograph of the present cutting-edge of large-scale desktop studying, offering a unmarried multidisciplinary resource for the most recent examine and advances in regularization, sparsity, compressed sensing, convex and large-scale optimization, kernel tools, and help vector machines. together with 21 chapters authored through best researchers in computer studying, this finished reference:

  • Covers the connection among aid vector machines (SVMs) and the Lasso
  • Discusses multi-layer SVMs
  • Explores nonparametric function choice, foundation pursuit tools, and strong compressive sensing
  • Describes graph-based regularization tools for unmarried- and multi-task learning
  • Considers regularized tools for dictionary studying and portfolio selection
  • Addresses non-negative matrix factorization
  • Examines low-rank matrix and tensor-based models
  • Presents complex kernel equipment for batch and on-line computer studying, approach id, area variation, and photo processing
  • Tackles large-scale algorithms together with conditional gradient tools, (non-convex) proximal strategies, and stochastic gradient descent

Regularization, Optimization, Kernels, and aid Vector Machines is perfect for researchers in desktop studying, trend popularity, information mining, sign processing, statistical studying, and comparable areas.

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Example text

However, weaker conditions λ λ can be employed, as 0 ∈ sri(R(A) − dom ω), where sri stands for strong relative interior [4]. 1 also gives a stopping criterion for the algorithm. 17). 14) by whatever algorithm just provides a minimizing sequence. We underline that, for the dual problem, no convergence on the minimizers is required, but convergence in value is sufficient. 14). Set u0 = v0 = 0, t0 = 1 and for every k ∈ N define M xtmp = y − λ m=1 A∗m uk,m 2 0 < γk ≤ (λ A )−1 for m = 1, . . 21) 1 + 4t2k /2 for m = 1, .

4) s=1 where h : RJ → R is a penalization function promoting sparsity in the coefficients and B ⊆ RL×J is a constraint set for the matrix of atoms. Regularized Dictionary Learning 31 In the literature, different instances of h and B have been considered. We list some important examples: Sparse coding [25, 18]. h(θ) = τ θ 1 , and B = B | (∀ j) [25] and B = B | (∀ j) bj 2 ≤ c in [18]. p bj sparsity [16]. h(θ) = θ p , with 0 < p ≤ 1, and B = B | B 2 B = B | (∀ j) bj 2 = 1/J . 2 = 1 in F = 1 or = Hierarchical Sparse Coding [13].

Geometric Intuition. 2) is to compute the smallest Euclidean distance of the set A to the point b ∈ Rd . On the other hand the SVM problem — after translating by b — is to minimize the distance of the smaller set A ⊂ A to the point b. Here we have used the notation AS := {Ax | x ∈ S} for subsets S ⊆ Rd and linear maps A (it is easy to check that linear maps do preserve convexity of sets, so that conv(AS) = A conv(S)). Intuitively, the main idea of our reduction is to mirror our SVM points Ai at the origin, so that both the points and their mirrored copies — and therefore the entire larger polytope A — do end up lying “behind” the separating SVM margin.

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