1,578 research outputs found
On Correcting Inputs: Inverse Optimization for Online Structured Prediction
Algorithm designers typically assume that the input data is correct, and then
proceed to find "optimal" or "sub-optimal" solutions using this input data.
However this assumption of correct data does not always hold in practice,
especially in the context of online learning systems where the objective is to
learn appropriate feature weights given some training samples. Such scenarios
necessitate the study of inverse optimization problems where one is given an
input instance as well as a desired output and the task is to adjust the input
data so that the given output is indeed optimal. Motivated by learning
structured prediction models, in this paper we consider inverse optimization
with a margin, i.e., we require the given output to be better than all other
feasible outputs by a desired margin. We consider such inverse optimization
problems for maximum weight matroid basis, matroid intersection, perfect
matchings, minimum cost maximum flows, and shortest paths and derive the first
known results for such problems with a non-zero margin. The effectiveness of
these algorithmic approaches to online learning for structured prediction is
also discussed.Comment: Conference version to appear in FSTTCS, 201
On the tradeoff between stability and fit
In computing, as in many aspects of life, changes incur cost. Many optimization problems are formulated as a one-time instance starting from scratch. However, a common case that arises is when we already have a set of prior assignments and must decide how to respond to a new set of constraints, given that each change from the current assignment comes at a price. That is, we would like to maximize the fitness or efficiency of our system, but we need to balance it with the changeout cost from the previous state.
We provide a precise formulation for this tradeoff and analyze the resulting stable extensions of some fundamental problems in measurement and analytics. Our main technical contribution is a stable extension of Probability Proportional to Size (PPS) weighted random sampling, with applications to monitoring and anomaly detection problems. We also provide a general framework that applies to top-k, minimum spanning tree, and assignment. In both cases, we are able to provide exact solutions and discuss efficient incremental algorithms that can find new solutions as the input changes
Pairwise MRF Calibration by Perturbation of the Bethe Reference Point
We investigate different ways of generating approximate solutions to the
pairwise Markov random field (MRF) selection problem. We focus mainly on the
inverse Ising problem, but discuss also the somewhat related inverse Gaussian
problem because both types of MRF are suitable for inference tasks with the
belief propagation algorithm (BP) under certain conditions. Our approach
consists in to take a Bethe mean-field solution obtained with a maximum
spanning tree (MST) of pairwise mutual information, referred to as the
\emph{Bethe reference point}, for further perturbation procedures. We consider
three different ways following this idea: in the first one, we select and
calibrate iteratively the optimal links to be added starting from the Bethe
reference point; the second one is based on the observation that the natural
gradient can be computed analytically at the Bethe point; in the third one,
assuming no local field and using low temperature expansion we develop a dual
loop joint model based on a well chosen fundamental cycle basis. We indeed
identify a subclass of planar models, which we refer to as \emph{Bethe-dual
graph models}, having possibly many loops, but characterized by a singly
connected dual factor graph, for which the partition function and the linear
response can be computed exactly in respectively O(N) and operations,
thanks to a dual weight propagation (DWP) message passing procedure that we set
up. When restricted to this subclass of models, the inverse Ising problem being
convex, becomes tractable at any temperature. Experimental tests on various
datasets with refined or regularization procedures indicate that
these approaches may be competitive and useful alternatives to existing ones.Comment: 54 pages, 8 figure. section 5 and refs added in V
Robust Rotation Synchronization via Low-rank and Sparse Matrix Decomposition
This paper deals with the rotation synchronization problem, which arises in
global registration of 3D point-sets and in structure from motion. The problem
is formulated in an unprecedented way as a "low-rank and sparse" matrix
decomposition that handles both outliers and missing data. A minimization
strategy, dubbed R-GoDec, is also proposed and evaluated experimentally against
state-of-the-art algorithms on simulated and real data. The results show that
R-GoDec is the fastest among the robust algorithms.Comment: The material contained in this paper is part of a manuscript
submitted to CVI
Combinatorial algorithms for inverse network flow problems
"(Revised January 25, 1998)"--T.p. -- "February 1998."--Cover.Includes bibliographical references (p. 23-25).Supported by a grant from the United Parcel Service and a contract from the Office of Naval Research. ONR N00014-96-1-0051Ravindra K. Ahuja, James B. Orlin
Synchronization Problems in Computer Vision
The goal of \u201csynchronization\u201d is to infer the unknown states of a network of nodes, where only the ratio (or difference) between pairs of states can be measured. Typically, states are represented by elements of a group, such as the Symmetric Group or the Special Euclidean Group. The former can represent local labels of a set of features, which refer to the multi-view matching application, whereas the latter can represent camera reference frames, in which case we are in the context of structure from motion, or local coordinates where 3D points are represented, in which case we are dealing with multiple point-set registration. A related problem is that of \u201cbearing-based network localization\u201d where each node is located at a fixed (unknown) position in 3-space and pairs of nodes can measure the direction of the line joining their locations. In this thesis we are interested in global techniques where all the measures are considered at once, as opposed to incremental approaches that grow a solution by adding pieces iteratively
Playing with Duality: An Overview of Recent Primal-Dual Approaches for Solving Large-Scale Optimization Problems
Optimization methods are at the core of many problems in signal/image
processing, computer vision, and machine learning. For a long time, it has been
recognized that looking at the dual of an optimization problem may drastically
simplify its solution. Deriving efficient strategies which jointly brings into
play the primal and the dual problems is however a more recent idea which has
generated many important new contributions in the last years. These novel
developments are grounded on recent advances in convex analysis, discrete
optimization, parallel processing, and non-smooth optimization with emphasis on
sparsity issues. In this paper, we aim at presenting the principles of
primal-dual approaches, while giving an overview of numerical methods which
have been proposed in different contexts. We show the benefits which can be
drawn from primal-dual algorithms both for solving large-scale convex
optimization problems and discrete ones, and we provide various application
examples to illustrate their usefulness
Directed shortest paths via approximate cost balancing
We present an O(nm) algorithm for all-pairs shortest paths computations in a directed graph with n nodes, m arcs, and nonnegative integer arc costs. This matches the complexity bound attained by Thorup [31] for the all-pairs problems in undirected graphs. The main insight is that shortest paths problems with approximately balanced directed cost functions can be solved similarly to the undirected case. The algorithm finds an approximately balanced reduced cost function in an O(m â n log n) preprocessing step. Using these reduced costs, every shortest path query can be solved in O(m) time using an adaptation of Thorupâs component hierarchy method. The balancing result can also be applied to the ââ-matrix balancing problem
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