Summary. In this paper a new class of quasi-Newton methods, named LQN, is introduced in order to solve unconstrained minimization problems. The novel approach, which generalizes classical BF GS methods, is based on a Hessian updating formula involving an algebra L of matrices simultaneously diagonalized by a fast unitary transform. The complexity per step of LQN methods is O(n log n), thereby improving considerably BF GS computational efficiency. Moreover, since LQN's iterative scheme utilizes single-indexed arrays, only O(n) memory allocations are required. Global convergence properties are investigated. In particular a global convergence result is obtained under suitable assumptions on f . Numerical experiences [7] confirm that LQN methods are particularly recommended for large scale problems.

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In this paper, we present a new class of quasi-Newton methods for an effective learning in large multilayer perceptron (MLP)-networks. The algorithms introduced in this work, named LQN, utilize an iterative scheme of a generalized BFGS-type method, involving a suitable family of matrix algebras L. The main advantages of these innovative methods are based upon the fact that they have an O(nlogn) complexity per step and that they require O(n) memory allocations. Numerical experiences, performed on a set of standard benchmarks of MLP-networks, show the competitivity of the LQN methods, especially for large values of n.

A set of fast real transforms including the well known Hartley transfom is fully investigated. Mixed radix splitting properties of Hartley-type transforms are examined in detail. The matrix algebras diagonalized by the Hartley-type matrices are expressed in terms of circulant and (-1)-circulant matrices

Abstract-Recently, a two-phase scheme for removing saltand-pepper impulse noise has been proposed [14]. In the first phase, an adaptive median filter is used to identify pixels which are likely to be contaminated by noise (noise candidates). In the second phase, the image is restored by minimizing a specialized regularization functional that applies only to those selected noise candidates. As an extension of this work, we propose an efficient method to accomplish the second phase. The speed of our method can be double as that of the method proposed in [14] for images contaminated by 30% salt-and-pepper noise and will be faster for higher noise level.