Zero-Convex Functions, Perturbation Resilience, and Subgradient Projections for Feasibility-Seeking Methods

The convex feasibility problem (CFP) is at the core of the modeling of many problems in various areas of science. Subgradient projection methods are important tools for solving the CFP because they enable the use of subgradient calculations instead of orthogonal projections onto the individual sets of the problem. Working in a real Hilbert space, we show that the sequential subgradient projection method is perturbation resilient. By this we mean that under appropriate conditions the sequence generated by the method converges weakly, and sometimes also strongly, to a point in the intersection of the given subsets of the feasibility problem, despite certain perturbations which are allowed in each iterative step. Unlike previous works on solving the convex feasibility problem, the involved functions, which induce the feasibility problem's subsets, need not be convex. Instead, we allow them to belong to a wider and richer class of functions satisfying a weaker condition, that we call "zero-convexity". This class, which is introduced and discussed here, holds a promise to solve optimization problems in various areas, especially in non-smooth and non-convex optimization. The relevance of this study to approximate minimization and to the recent superiorization methodology for constrained optimization is explained.Comment: Mathematical Programming Series A, accepted for publicatio

Robust and large-scale quasiconvex programming in structure-from-motion

Structure-from-Motion (SfM) is a cornerstone of computer vision. Briefly speaking, SfM is the task of simultaneously estimating the poses of the cameras behind a set of images of a scene, and the 3D coordinates of the points in the scene. Often, the optimisation problems that underpin SfM do not have closed-form solutions, and finding solutions via numerical schemes is necessary. An objective function, which measures the discrepancy of a geometric object (e.g., camera poses, rotations, 3D coordi- nates) with a set of image measurements, is to be minimised. Each image measurement gives rise to an error function. For example, the reprojection error, which measures the distance between an observed image point and the projection of a 3D point onto the image, is a commonly used error function. An influential optimisation paradigm in SfM is the ℓ₀₀ paradigm, where the objective function takes the form of the maximum of all individual error functions (e.g. individual reprojection errors of scene points). The benefit of the ℓ₀₀ paradigm is that the objective function of many SfM optimisation problems become quasiconvex, hence there is a unique minimum in the objective function. The task of formulating and minimising quasiconvex objective functions is called quasiconvex programming. Although tremendous progress in SfM techniques under the ℓ₀₀ paradigm has been made, there are still unsatisfactorily solved problems, specifically, problems associated with large-scale input data and outliers in the data. This thesis describes novel techniques to tackle these problems. A major weakness of the ℓ₀₀ paradigm is its susceptibility to outliers. This thesis improves the robustness of ℓ₀₀ solutions against outliers by employing the least median of squares (LMS) criterion, which amounts to minimising the median error. In the context of triangulation, this thesis proposes a locally convergent robust algorithm underpinned by a novel quasiconvex plane sweep technique. Imposing the LMS criterion achieves significant outlier tolerance, and, at the same time, some properties of quasiconvexity greatly simplify the process of solving the LMS problem. Approximation is a commonly used technique to tackle large-scale input data. This thesis introduces the coreset technique to quasiconvex programming problems. The coreset technique aims find a representative subset of the input data, such that solving the same problem on the subset yields a solution that is within known bound of the optimal solution on the complete input set. In particular, this thesis develops a coreset approximate algorithm to handle large-scale triangulation tasks. Another technique to handle large-scale input data is to break the optimisation into multiple smaller sub-problems. Such a decomposition usually speeds up the overall optimisation process, and alleviates the limitation on memory. This thesis develops a large-scale optimisation algorithm for the known rotation problem (KRot). The proposed method decomposes the original quasiconvex programming problem with potentially hundreds of thousands of parameters into multiple sub-problems with only three parameters each. An efficient solver based on a novel minimum enclosing ball technique is proposed to solve the sub-problems.Thesis (Ph.D.) (Research by Publication) -- University of Adelaide, School of Computer Science, 201

Beyond Convexity: Stochastic Quasi-Convex Optimization

Stochastic convex optimization is a basic and well studied primitive in machine learning. It is well known that convex and Lipschitz functions can be minimized efficiently using Stochastic Gradient Descent (SGD). The Normalized Gradient Descent (NGD) algorithm, is an adaptation of Gradient Descent, which updates according to the direction of the gradients, rather than the gradients themselves. In this paper we analyze a stochastic version of NGD and prove its convergence to a global minimum for a wider class of functions: we require the functions to be quasi-convex and locally-Lipschitz. Quasi-convexity broadens the con- cept of unimodality to multidimensions and allows for certain types of saddle points, which are a known hurdle for first-order optimization methods such as gradient descent. Locally-Lipschitz functions are only required to be Lipschitz in a small region around the optimum. This assumption circumvents gradient explosion, which is another known hurdle for gradient descent variants. Interestingly, unlike the vanilla SGD algorithm, the stochastic normalized gradient descent algorithm provably requires a minimal minibatch size

New algorithmic developments in maximum consensus robust fitting

In many computer vision applications, the task of robustly estimating the set of parameters of a geometric model is a fundamental problem. Despite the longstanding research efforts on robust model fitting, there remains significant scope for investigation. For a large number of geometric estimation tasks in computer vision, maximum consensus is the most popular robust fitting criterion. This thesis makes several contributions in the algorithms for consensus maximization. Randomized hypothesize-and-verify algorithms are arguably the most widely used class of techniques for robust estimation thanks to their simplicity. Though efficient, these randomized heuristic methods do not guarantee finding good maximum consensus estimates. To improve the randomize algorithms, guided sampling approaches have been developed. These methods take advantage of additional domain information, such as descriptor matching scores, to guide the sampling process. Subsets of the data that are more likely to result in good estimates are prioritized for consideration. However, these guided sampling approaches are ineffective when good domain information is not available. This thesis tackles this shortcoming by proposing a new guided sampling algorithm, which is based on the class of LP-type problems and Monte Carlo Tree Search (MCTS). The proposed algorithm relies on a fundamental geometric arrangement of the data to guide the sampling process. Specifically, we take advantage of the underlying tree structure of the maximum consensus problem and apply MCTS to efficiently search the tree. Empirical results show that the new guided sampling strategy outperforms traditional randomized methods. Consensus maximization also plays a key role in robust point set registration. A special case is the registration of deformable shapes. If the surfaces have the same intrinsic shapes, their deformations can be described accurately by a conformal model. The uniformization theorem allows the shapes to be conformally mapped onto a canonical domain, wherein the shapes can be aligned using a M¨obius transformation. The problem of correspondence-free M¨obius alignment of two sets of noisy and partially overlapping point sets can be tackled as a maximum consensus problem. Solving for the M¨obius transformation can be approached by randomized voting-type methods which offers no guarantee of optimality. Local methods such as Iterative Closest Point can be applied, but with the assumption that a good initialization is given or these techniques may converge to a bad local minima. When a globally optimal solution is required, the literature has so far considered only brute-force search. This thesis contributes a new branch-and-bound algorithm that solves for the globally optimal M¨obius transformation much more efficiently. So far, the consensus maximization problems are approached mainly by randomized algorithms, which are efficient but offer no analytical convergence guarantee. On the other hand, there exist exact algorithms that can solve the problem up to global optimality. The global methods, however, are intractable in general due to the NP-hardness of the consensus maximization. To fill the gap between the two extremes, this thesis contributes two novel deterministic algorithms to approximately optimize the maximum consensus criterion. The first method is based on non-smooth penalization supported by a Frank-Wolfe-style optimization scheme, and another algorithm is based on Alternating Direction Method of Multipliers (ADMM). Both of the proposed methods are capable of handling the non-linear geometric residuals commonly used in computer vision. As will be demonstrated, our proposed methods consistently outperform other heuristics and approximate methods.Thesis (Ph.D.) (Research by Publication) -- University of Adelaide, School of Computer Science, 201