Related papers: Majorizing Measures for the Optimizer
The majorizing measure theorem of Fernique and Talagrand is a fundamental result in the theory of random processes. It relates the boundedness of random processes indexed by elements of a metric space to complexity measures arising from…
In this paper, we give a short Bayesian proof of Talagrand's celebrated majorizing-measure theorem (MMT). While the upper-bound direction of MMT follows relatively directly from standard arguments, the lower-bound direction is widely…
We present three methods to construct majorizing measures in various settings. These methods are based on direct constructions of increasing sequences of partitions through a simple exhaustion procedure rather than on the construction of…
Let $(T,d)$ be a metric space and $\phi:\mathbb{R}_+\to \mathbb{R}$ an increasing, convex function with $\phi(0)=0$. We prove that if $m$ is a probability measure $m$ on $T$ which is majorizing with respect to $d,\phi$, that is,…
Suprema of random processes appear naturally in a plethora of disciplines, and Talagrand's majorizing theorem yields a geometric interpretation for them: for a centered Gaussian random process $(X_t)_{t \in T},$ $\mathbb{E}[\sup_{t \in…
In this paper we describe the alternative approach to the sample boundedness and continuity of stochastic processes. We show that the regularity of paths can be understood in terms of a distribution of the argument maximum. For a centered…
Purpose: To describe and mathematically validate the superiorization methodology, which is a recently-developed heuristic approach to optimization, and to discuss its applicability to medical physics problem formulations that specify the…
Majorization-minimization (MM) is a family of optimization methods that iteratively reduce a loss by minimizing a locally-tight upper bound, called a majorizer. Traditionally, majorizers were derived by hand, and MM was only applicable to a…
Majorization-minimization algorithms consist of successively minimizing a sequence of upper bounds of the objective function. These upper bounds are tight at the current estimate, and each iteration monotonically drives the objective…
Majorization-minimization algorithms consist of iteratively minimizing a majorizing surrogate of an objective function. Because of its simplicity and its wide applicability, this principle has been very popular in statistics and in signal…
We give a dual, separated-tree formulation of Latala's majorizing measure theorem for canonical processes with log-concave tails. Under the same assumptions as in Latala's characterization, we introduce parameterized separation trees and…
The superiorization methodology is intended to work with input data of constrained minimization problems, i.e., a target function and a constraints set. However, it is based on an antipodal way of thinking to the thinking that leads…
For appropriate Gaussian processes, as a corollary of the majorizing measure theorem, Michel Talagrand (1987) proved that the event that the supremum is significantly larger than its expectation can be covered by a set of half-spaces whose…
The Majorization Principle is a fundamental statement governing the dynamics of information processing in optimal and efficient quantum algorithms. While quantum computation can be modeled to be reversible, due to the unitary evolution…
Gaussian processes can be treated as subsets of a standard Hilbert space, however, the volume size relation between the underlying index space of random processes and its convex hull is not clear. The understanding of such volume size…
Recently a majorization method for optimizing partition functions of log-linear models was proposed alongside a novel quadratic variational upper-bound. In the batch setting, it outperformed state-of-the-art first- and second-order…
Majorization theory is a powerful mathematical tool to compare the disorder in distributions, with wide-ranging applications in many fields including mathematics, physics, information theory, and economics. While majorization theory…
We address the fundamental problem of selection under uncertainty by modeling it from the perspective of Bayesian persuasion. In our model, a decision maker with imperfect information always selects the option with the highest expected…
The minimization of a nonconvex composite function can model a variety of imaging tasks. A popular class of algorithms for solving such problems are majorization-minimization techniques which iteratively approximate the composite nonconvex…
We introduce the technique of generic chaining and majorizing measures for controlling sequential Rademacher complexity. We relate majorizing measures to the notion of fractional covering numbers, which we show to be dominated in terms of…