# Modern computational statistics is turning more and more to high-dimensional optimization

Modern computational statistics is turning more and more to high-dimensional optimization to handle the deluge of big data. must be distinguished from the in a sequence of vectors. To maintain consistency denotes the = (= (≥ 0 for all = and = ?for some design matrix and response vector to 0 gives + 1 to construct the coordinate descent update of is a × matrix whose columns represent data vectors. In many applications it is reasonable to postulate a reduced number of prototypes and write Dipyridamole = (× is small compared to ≈ compresses the data for easier storage and retrieval. Depending on the circumstances one may want to add further constraints . For instance if the entries of are nonnegative then it is often reasonable to demand that the entries of be nonnegative as well [55 68 If we want each to equal a convex combination of the prototypes then constraining the column sums of to equal 1 is indicated. One way of estimating and is to minimize the squared Frobenius norm is fixed then we can update the by minimizing the sum of squares is fixed then we can update the by minimizing the sum of squares motivates the method of steepest descent. In view of the Cauchy-Schwarz inequality the choice of the expansion over the sphere of unit vectors. Of course if ?is a stationary point. The steepest Dipyridamole descent algorithm iterates according to > 0. If is sufficiently small then the descent property by searching for the minimum of the objective function along the direction of steepest descent. Among the many methods of line search the methods of false position cubic interpolation and golden section stand out . These are all local search methods and unless some guarantee of convexity exists confusion of local and global minima can occur. The method of steepest descent often exhibits zigzagging and a painfully slow rate of convergence. For these reasons it was largely replaced in practice by Newton’s method and its variants. However the sheer scale Dipyridamole of modern optimization problems has led to a re-evaluation. The avoidance of second derivatives and Hessian approximations is now viewed as an virtue. Furthermore the method has been generalized to nondifferentiable problems by substituting the forward directional derivative to minimize is both necessary and sufficient for to be a minimum point. If the domain of = with ∈ come into Dipyridamole play. Steepest descent also has a role to play in constrained optimization. Suppose we want to minimize ∈ for some closed convex set. The projected gradient method capitalizes on the steepest descent update (1) by projecting it onto the set [35 56 79 It is well known that for a point external to in is a box Euclidean ball hyperplane or halfspace. Fast algorithms Mouse Monoclonal to RFP tag. for computing in the update (1) is crucial. Current theory suggests taking to equal is a Lipschitz constant for the gradient ?belongs to the interval (0 2 In particular the Lipschitz inequality = sup∥must be estimated. Any induced matrix norm ∥ · ∥? can be substituted for the spectral norm ∥ · ∥ in the defining supremum and will give an upper bound on with i.i.d. standard normal entries a random 50 × 1 parameter vector with i.i.d. uniform [0 1 entries and a random 100 × 1 error vector with i.i.d. standard normal entries. In this setting the response = + equal to the spectral radius of and equal to 1.0 1.75 and 2.0) and the MM algorithm explained later in Example 0.6. All computer runs start from the common point ? Ron the line segment [as with default value 1. Any stationary point of > 0 is small enough and the Hessian matrix in the update (3). Backtracking is crucial to avoid overshooting. In the step-halving version of backtracking one starts with = 1. If the descent property holds then one takes the Newton step. Otherwise is substituted for is generated to guarantee independent responses represents a count between 0 and with success probability connecting every pair of nodes {are independent Poisson random variables with means = and are nonnegative propensities . The loglikelihood of the observed edge counts = amounts to nodes the matrix ?× is large. Fortunately the Sherman-Morrison formula comes to the rescue. If we write ?+ 11with diagonal then the explicit inverse ≥ 0. More generally it is always cheap to invert a low-rank perturbation of an explicitly invertible matrix. In maximum likelihood estimation the method of steepest ascent replaces the observed information matrix ?and immediately supply the asymptotic variances.

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