Large-scale Binary Quadratic Optimization Using Semidefinite Relaxation and Applications

In computer vision, many problems such as image segmentation, pixel labelling, and scene parsing can be formulated as binary quadratic programs (BQPs). For submodular problems, cuts based methods can be employed to efficiently solve large-scale problems. However, general nonsubmodular problems are significantly more challenging to solve. Finding a solution when the problem is of large size to be of practical interest, however, typically requires relaxation. Two standard relaxation methods are widely used for solving general BQPs--spectral methods and semidefinite programming (SDP), each with their own advantages and disadvantages. Spectral relaxation is simple and easy to implement, but its bound is loose. Semidefinite relaxation has a tighter bound, but its computational complexity is high, especially for large scale problems. In this work, we present a new SDP formulation for BQPs, with two desirable properties. First, it has a similar relaxation bound to conventional SDP formulations. Second, compared with conventional SDP methods, the new SDP formulation leads to a significantly more efficient and scalable dual optimization approach, which has the same degree of complexity as spectral methods. We then propose two solvers, namely, quasi-Newton and smoothing Newton methods, for the dual problem. Both of them are significantly more efficiently than standard interior-point methods. In practice, the smoothing Newton solver is faster than the quasi-Newton solver for dense or medium-sized problems, while the quasi-Newton solver is preferable for large sparse/structured problems. Our experiments on a few computer vision applications including clustering, image segmentation, co-segmentation and registration show the potential of our SDP formulation for solving large-scale BQPs.Comment: Fixed some typos. 18 pages. Accepted to IEEE Transactions on Pattern Analysis and Machine Intelligenc

Adaptive Normalized Risk-Averting Training For Deep Neural Networks

This paper proposes a set of new error criteria and learning approaches, Adaptive Normalized Risk-Averting Training (ANRAT), to attack the non-convex optimization problem in training deep neural networks (DNNs). Theoretically, we demonstrate its effectiveness on global and local convexity lower-bounded by the standard $L_p$-norm error. By analyzing the gradient on the convexity index $\lambda$, we explain the reason why to learn $\lambda$ adaptively using gradient descent works. In practice, we show how this method improves training of deep neural networks to solve visual recognition tasks on the MNIST and CIFAR-10 datasets. Without using pretraining or other tricks, we obtain results comparable or superior to those reported in recent literature on the same tasks using standard ConvNets + MSE/cross entropy. Performance on deep/shallow multilayer perceptrons and Denoised Auto-encoders is also explored. ANRAT can be combined with other quasi-Newton training methods, innovative network variants, regularization techniques and other specific tricks in DNNs. Other than unsupervised pretraining, it provides a new perspective to address the non-convex optimization problem in DNNs.Comment: AAAI 2016, 0.39%~0.4% ER on MNIST with single 32-32-256-10 ConvNets, code available at https://github.com/cauchyturing/ANRA

Regularized Optimal Transport and the Rot Mover's Distance

This paper presents a unified framework for smooth convex regularization of discrete optimal transport problems. In this context, the regularized optimal transport turns out to be equivalent to a matrix nearness problem with respect to Bregman divergences. Our framework thus naturally generalizes a previously proposed regularization based on the Boltzmann-Shannon entropy related to the Kullback-Leibler divergence, and solved with the Sinkhorn-Knopp algorithm. We call the regularized optimal transport distance the rot mover's distance in reference to the classical earth mover's distance. We develop two generic schemes that we respectively call the alternate scaling algorithm and the non-negative alternate scaling algorithm, to compute efficiently the regularized optimal plans depending on whether the domain of the regularizer lies within the non-negative orthant or not. These schemes are based on Dykstra's algorithm with alternate Bregman projections, and further exploit the Newton-Raphson method when applied to separable divergences. We enhance the separable case with a sparse extension to deal with high data dimensions. We also instantiate our proposed framework and discuss the inherent specificities for well-known regularizers and statistical divergences in the machine learning and information geometry communities. Finally, we demonstrate the merits of our methods with experiments using synthetic data to illustrate the effect of different regularizers and penalties on the solutions, as well as real-world data for a pattern recognition application to audio scene classification

A Unifying View of Multiple Kernel Learning

Recent research on multiple kernel learning has lead to a number of approaches for combining kernels in regularized risk minimization. The proposed approaches include different formulations of objectives and varying regularization strategies. In this paper we present a unifying general optimization criterion for multiple kernel learning and show how existing formulations are subsumed as special cases. We also derive the criterion's dual representation, which is suitable for general smooth optimization algorithms. Finally, we evaluate multiple kernel learning in this framework analytically using a Rademacher complexity bound on the generalization error and empirically in a set of experiments

A Riemannian Trust Region Method for the Canonical Tensor Rank Approximation Problem

The canonical tensor rank approximation problem (TAP) consists of approximating a real-valued tensor by one of low canonical rank, which is a challenging non-linear, non-convex, constrained optimization problem, where the constraint set forms a non-smooth semi-algebraic set. We introduce a Riemannian Gauss-Newton method with trust region for solving small-scale, dense TAPs. The novelty of our approach is threefold. First, we parametrize the constraint set as the Cartesian product of Segre manifolds, hereby formulating the TAP as a Riemannian optimization problem, and we argue why this parametrization is among the theoretically best possible. Second, an original ST-HOSVD-based retraction operator is proposed. Third, we introduce a hot restart mechanism that efficiently detects when the optimization process is tending to an ill-conditioned tensor rank decomposition and which often yields a quick escape path from such spurious decompositions. Numerical experiments show improvements of up to three orders of magnitude in terms of the expected time to compute a successful solution over existing state-of-the-art methods

A software framework for computing Newton polytopes of resultants and (reduced) discriminants

We present a new software for computing Newton polytopes of resultant and discriminant polynomials. We illustrate its use with a number of examples