We show that for any 1 ≤ t ≤ c̃n log−5/2 n, the set of unconditional convex bodies in R contains a t-separated subset of cardinality at least exp ( exp ( c t2 log(1 + t) n )) . This implies the existence of an unconditional convex body in R which cannot be approximated within the distance d by a projection of a polytope with N faces unless N ≥ exp(c(d)n). We also show that for t ≥ 2, the cardina...

We introduce a new method of proving lower bounds on the depth of algebraic d-degree decision (resp. computation) trees and apply it to prove a lower bound (log N) (resp. (log N= log log N)) for testing membership to an n-dimensional convex polyhedron having N faces of all dimensions, provided that N > (nd) (n) (resp. N > n (n)). This bound apparently does not follow from the methods developed ...

Hyperbolic polynomials have their origins in partial diierential equations. We show in this paper that they have applications in interior point methods for convex programming. Each homogeneous hyperbolic polynomial p has an associated open and convex cone called its hyperbolicity cone. We give an explicit representation of this cone in terms of polynomial inequalities. The function F (x) = ? lo...

In this paper, we develop various differentiation rules for general smooth matrix-valued functions, and for the class of matrix convex (or concave) functions first introduced by Löwner and Kraus in the 1930s. For a matrix monotone function, we present formulas for its derivatives of any order in an integral form. Moreover, for a general smooth primary matrix function, we derive a formula for al...

We study the numerical computation of an expectation of a bounded function f with respect to a measure given by a non-normalized density on a convex body K ⊂ Rd . We assume that the density is log-concave, satisfies a variability condition and is not too narrow. In [19, 25, 26] it is required that K is the Euclidean unit ball. We consider general convex bodies or even the whole Rd and show that...

The Kalman filter computes the maximum a posteriori (MAP) estimate of the states for linear state space models with Gaussian noise. We interpret the Kalman filter as the solution to a convex optimization problem, and show that we can generalize the MAP state estimator to any noise with log-concave density function and any combination of linear equality and convex inequality constraints on the s...

We introduce a new class of lower bounds on the log partition function of a Markov random field which makes use of a reversed Jensen’s inequality. In particular, our method approximates the intractable distribution using a linear combination of spanning trees with negative weights. This technique is a lower-bound counterpart to the tree-reweighted belief propagation algorithm, which uses a conv...

The Kalman filter computes the maximum a posteriori (MAP) estimate of the states for linear state space models with Gaussian noise. We interpret the Kalman filter as the solution to a convex optimization problem, and show that we can generalize the MAP state estimator to any noise with log-concave density function and any combination of linear equality and convex inequality constraints on the s...

We consider the problem of predicting covariance a zero mean Gaussian vector, based on another feature vector. describe predictor that has form generalized linear model, i.e., an affine function features followed by inverse link maps vectors to symmetric positive definite matrices. The log-likelihood is concave parameters, so fitting involves convex optimization. Such predictors can be combined...

We consider the problem of unconstrained minimization of a smooth function in the derivativefree setting. In particular, we study the direct search method (of directional type). Despite relevant research activity spanning several decades, until recently no complexity guarantees— bounds on the number of function evaluations needed to find a satisfying point—for methods of this type were establis...