Multilevel optimisation for computer vision

The recent spark in machine learning and computer vision methods requiring increasingly larger datasets has motivated the introduction of optimisation algorithms specifically tailored to solve very large problems within practical time constraints. This demand in algorithms challenges the practicability of state of the art methods requiring new approaches that can take advantage of not only the problem’s mathematical structure, but also its data structure. Fortunately, such structure is present in many computer vision applications, where the problems can be modelled with varying degrees of fidelity. This structure suggests using multiscale models and thus multilevel algorithms. The objective of this thesis is to develop, implement and test provably convergent multilevel optimisation algorithms for convex composite optimisation problems in general and its applications in computer vision in particular. Our first multilevel algorithm solves convex composite optimisation problem and it is most efficient particularly for the robust facial recognition task. The method uses concepts from proximal gradient, mirror descent and multilevel optimisation algorithms, thus we call it multilevel accelerated gradient mirror descent algorithm (MAGMA). We first show that MAGMA has the same theoretical convergence rate as the state of the art first order methods and has much lower per iteration complexity. Then we demonstrate its practical advantage on many facial recognition problems. The second part of the thesis introduces new multilevel procedure most appropriate for the robust PCA problems requiring iterative SVD computations. We propose to exploit the multiscale structure present in these problems by constructing lower dimensional matrices and use its singular values for each iteration of the optimisation procedure. We implement this approach on three different optimisation algorithms - inexact ALM, Frank-Wolfe Thresholding and non-convex alternating projections. In this case as well we show that these multilevel algorithms converge (to an exact or approximate) solution with the same convergence rate as their standard counterparts and test all three methods on numerous synthetic and real life problems demonstrating that the multilevel algorithms are not only much faster, but also solve problems that often cannot be solved by their standard counterparts.Open Acces

MAGMA: Multi-level accelerated gradient mirror descent algorithm for large-scale convex composite minimization

Composite convex optimization models arise in several applications, and are especially prevalent in inverse problems with a sparsity inducing norm and in general convex optimization with simple constraints. The most widely used algorithms for convex composite models are accelerated first order methods, however they can take a large number of iterations to compute an acceptable solution for large-scale problems. In this paper we propose to speed up first order methods by taking advantage of the structure present in many applications and in image processing in particular. Our method is based on multi-level optimization methods and exploits the fact that many applications that give rise to large scale models can be modelled using varying degrees of fidelity. We use Nesterov's acceleration techniques together with the multi-level approach to achieve $\mathcal{O}(1/\sqrt{\epsilon})$ convergence rate, where $\epsilon$ denotes the desired accuracy. The proposed method has a better convergence rate than any other existing multi-level method for convex problems, and in addition has the same rate as accelerated methods, which is known to be optimal for first-order methods. Moreover, as our numerical experiments show, on large-scale face recognition problems our algorithm is several times faster than the state of the art

Multilevel approximate robust principal component analysis

Robust principal component analysis (RPCA) is currently the method of choice for recovering a low-rank matrix from sparse corruptions that are of unknown value and support by decomposing the observation matrix into low-rank and sparse matrices. RPCA has many applications including background subtraction, learning of robust subspaces from visual data, etc. Nevertheless, the application of SVD in each iteration of optimisation methods renders the application of RPCA challenging in cases when data is large. In this paper, we propose the first, to the best of our knowledge, multilevel approach for solving convex and non-convex RPCA models. The basic idea is to construct lower dimensional models and perform SVD on them instead of the original high dimensional problem. We show that the proposed approach gives a good approximate solution to the original problem for both convex and non-convex formulations, while being many times faster than original RPCA methods in several real world datasets

Determining construction method patterns to automate and optimise scheduling – a graph-based approach

Construction projects have been experiencing project delays for decades. As an executive guide to construction activities, construction schedules can mitigate delay risks and are essential to project success. Yet, creating a quality construction schedule is often the outcome of experienced schedulers, and what makes it harder is the fact that historic information including decision reasoning was not documented and disseminated for future use. This study proposes a graph-based method to find the time- and risk-efficient construction method patterns from historic projects to help schedulers improve productivity and accuracy. The method leverages schedule data (including activity names, Work Breakdown Structure, and start and finish date) that were obtained from a Tier-1 contractor for this study. The method was validated for excavation activities. The results indicate that the most time-efficient excavation activities can be done in 0.6% of total project time. The proposed method can help industry professionals standardise scheduling guidelines and automate the generation of construction schedules for critical subtasks

Computation of microcanonical entropy at fixed magnetization without direct counting

We discuss a method to compute the microcanonical entropy at fixed magnetization without direct counting. Our approach is based on the evaluation of a saddle-point leading to an optimization problem. The method is applied to a benchmark Ising model with simultaneous presence of mean-field and nearest-neighbour interactions for which direct counting is indeed possible, thus allowing a comparison. Moreover, we apply the method to an Ising model with mean-field, nearest-neighbour and next-nearest-neighbour interactions, for which direct counting is not straightforward. For this model, we compare the solution obtained by our method with the one obtained from the formula for the entropy in terms of all correlation functions. This example shows that for general couplings our method is much more convenient than direct counting methods to compute the microcanonical entropy at fixed magnetization

Efficacy and safety of curcumin and its combination with boswellic acid in osteoarthritis: a comparative, randomized, double-blind, placebo-controlled study

Abstract Background The aim of this clinical trial was to assess the efficacy and safety of curcuminoid complex extract from turmeric rhizome with turmeric volatile oil (CuraMed®) and its combination with boswellic acid extract from Indian frankincense root (Curamin®) vs placebo for the treatment of 40- to 70-year-old patients with osteoarthritis (OA). Methods The effects of CuraMed® 500-mg capsules (333 mg curcuminoids) and Curamin® 500-mg capsules (350 mg curcuminoids and 150 mg boswellic acid) taken orally three times a day for 12 weeks in 201 patients was investigated in a three-arm, parallel-group, randomized, double-blinded, placebo-controlled trial. Primary outcome efficacy measures included OA physical function performance-based tests, the WOMAC recommended index of joint pain, morning stiffness, limitations of physical function, and the patients’ global assessment of disease severity. Results Favorable effects of both preparations compared to placebo were observed after only 3 months of continuous treatment. A significant effect of Curamin® compared to placebo was observed both in physical performance tests and the WOMAC joint pain index, while superior efficacy of CuraMed vs placebo was observed only in physical performance tests. The effect size compared to placebo was comparable for both treatment groups but was superior in the Curamin® group. The treatments were well tolerated. Conclusions Twelve-week use of curcumin complex or its combination with boswellic acid reduces pain-related symptoms in patients with OA. Curcumin in combination with boswellic acid is more effective. Combining Curcuma longa and Boswellia serrata extracts in Curamin® increases the efficacy of OA treatment presumably due to synergistic effects of curcumin and boswellic acid. Trial registration This trial is registered at the database www.clinicaltrials.gov . https://clinicaltrials.gov/ct2/show/NCT02390349?term=EuroPharma&rank=1 . Study registration number: NCT02390349

Ising chains with competing interactions in the presence of long-range couplings

In this paper we study an Ising spin chain with short-range competing interactions in the presence of long-range ferromagnetic interactions in the canonical ensemble. The simultaneous presence of the frustration induced by the short-range couplings together with their competition with the long-range term gives rise to a rich thermodynamic-phase diagram. We compare our results with the limit in which one of two local interactions is turned off, which was previously studied in the literature. Eight regions of parameters with qualitatively distinct properties are featured, with different first-and second-order phase transition lines and critical points