POSE: Pseudo Object Space Error for Initialization-Free Bundle Adjustment

Bundle adjustment is a nonlinear refinement method for camera poses and 3D structure requiring sufficiently good initialization. In recent years, it was experimentally observed that useful minima can be reached even from arbitrary initialization for affine bundle adjustment problems (and fixed-rank matrix factorization instances in general). The key success factor lies in the use of the variable projection (VarPro) method, which is known to have a wide basin of convergence for such problems. In this paper, we propose the Pseudo Object Space Error (pOSE), which is an objective with cameras represented as a hybrid between the affine and projective models. This formulation allows us to obtain 3D reconstructions that are close to the true projective reconstructions while retaining a bilinear problem structure suitable for the VarPro method. Experimental results show that using pOSE has a high success rate to yield faithful 3D reconstructions from random initializations, taking one step towards initialization-free structure from motion

Tensor Numerical Methods in Quantum Chemistry: from Hartree-Fock Energy to Excited States

We resume the recent successes of the grid-based tensor numerical methods and discuss their prospects in real-space electronic structure calculations. These methods, based on the low-rank representation of the multidimensional functions and integral operators, led to entirely grid-based tensor-structured 3D Hartree-Fock eigenvalue solver. It benefits from tensor calculation of the core Hamiltonian and two-electron integrals (TEI) in $O(n\log n)$ complexity using the rank-structured approximation of basis functions, electron densities and convolution integral operators all represented on 3D $n\times n\times n$ Cartesian grids. The algorithm for calculating TEI tensor in a form of the Cholesky decomposition is based on multiple factorizations using algebraic 1D ``density fitting`` scheme. The basis functions are not restricted to separable Gaussians, since the analytical integration is substituted by high-precision tensor-structured numerical quadratures. The tensor approaches to post-Hartree-Fock calculations for the MP2 energy correction and for the Bethe-Salpeter excited states, based on using low-rank factorizations and the reduced basis method, were recently introduced. Another direction is related to the recent attempts to develop a tensor-based Hartree-Fock numerical scheme for finite lattice-structured systems, where one of the numerical challenges is the summation of electrostatic potentials of a large number of nuclei. The 3D grid-based tensor method for calculation of a potential sum on a $L\times L\times L$ lattice manifests the linear in $L$ computational work, $O(L)$, instead of the usual $O(L^3 \log L)$ scaling by the Ewald-type approaches

Widening the basin of convergence for the bundle adjustment type of problems in computer vision

Bundle adjustment is the process of simultaneously optimizing camera poses and 3D structure given image point tracks. In structure-from-motion, it is typically used as the final refinement step due to the nonlinearity of the problem, meaning that it requires sufficiently good initialization. Contrary to this belief, recent literature showed that useful solutions can be obtained even from arbitrary initialization for fixed-rank matrix factorization problems, including bundle adjustment with affine cameras. This property of wide convergence basin of high quality optima is desirable for any nonlinear optimization algorithm since obtaining good initial values can often be non-trivial. The aim of this thesis is to find the key factor behind the success of these recent matrix factorization algorithms and explore the potential applicability of the findings to bundle adjustment, which is closely related to matrix factorization. The thesis begins by unifying a handful of matrix factorization algorithms and comparing similarities and differences between them. The theoretical analysis shows that the set of successful algorithms actually stems from the same root of the optimization method called variable projection (VarPro). The investigation then extends to address why VarPro outperforms the joint optimization technique, which is widely used in computer vision. This algorithmic comparison of these methods yields a larger unification, leading to a conclusion that VarPro benefits from an unequal trust region assumption between two matrix factors. The thesis then explores ways to incorporate VarPro to bundle adjustment problems using projective and perspective cameras. Unfortunately, the added nonlinearity causes a substantial decrease in the convergence basin of VarPro, and therefore a bootstrapping strategy is proposed to bypass this issue. Experimental results show that it is possible to yield feasible metric reconstructions and pose estimations from arbitrary initialization given relatively clean point tracks, taking one step towards initialization-free structure-from-motion.Microsoft Toshiba Research Europ

Greedy Algorithms for Cone Constrained Optimization with Convergence Guarantees

Greedy optimization methods such as Matching Pursuit (MP) and Frank-Wolfe (FW) algorithms regained popularity in recent years due to their simplicity, effectiveness and theoretical guarantees. MP and FW address optimization over the linear span and the convex hull of a set of atoms, respectively. In this paper, we consider the intermediate case of optimization over the convex cone, parametrized as the conic hull of a generic atom set, leading to the first principled definitions of non-negative MP algorithms for which we give explicit convergence rates and demonstrate excellent empirical performance. In particular, we derive sublinear ($\mathcal{O}(1/t)$) convergence on general smooth and convex objectives, and linear convergence ($\mathcal{O}(e^{-t})$) on strongly convex objectives, in both cases for general sets of atoms. Furthermore, we establish a clear correspondence of our algorithms to known algorithms from the MP and FW literature. Our novel algorithms and analyses target general atom sets and general objective functions, and hence are directly applicable to a large variety of learning settings.Comment: NIPS 201