In this paper, we consider the so-called worst-case linear optimization (WCLO) with uncertainties on the right-hand side of the constraints. Such a problem often arises in applications such as in systemic risk estimation in finance and stochastic optimization. We first show that the WCLO problem with the uncertainty set corresponding to the [Formula: see text]p-norm ((WCLOp)) is NP-hard for p ɛ (1,∞). Second, we combine several simple optimization techniques, such as the successive convex optimization method, quadratic convex relaxation, initialization, and branch-and-bound (B&B), to develop an algorithm for (WCLO2) that can find a globally optimal solution to (WCLO2) within a prespecified ε-tolerance. We establish the global convergence of the algorithm and estimate its complexity. We also develop a finite B&B algorithm for (WCLO∞) to identify a global optimal solution to the underlying problem, and establish the finite convergence of the algorithm. Numerical experiments are reported to illustrate the effectiveness of our proposed algorithms in finding globally optimal solutions to medium and large-scale WCLO instances.

Product quantization (PQ) (and its variants) has been effectively used to encode high-dimensional data into compact codes for many problems in vision. In principle, PQ decomposes the given data into a number of lower-dimensional subspaces where the quantization proceeds independently for each subspace. While the original PQ approach does not explicitly optimize for these subspaces, later proposals have argued that the performance tends to benefit significantly if such subspaces are chosen in an optimal manner. Despite such consensus, existing approaches in the literature diverge in terms of which specific properties of these subspaces are desirable and how one should proceed to solve/optimize them. Nonetheless, despite the empirical support, there is less clarity regarding the theoretical properties that underlie these experimental benefits for quantization problems in general. In this paper, we study the quantization problem in the setting where subspaces are orthogonal and show that this problem is intricately related to a specific type of spectral decomposition of the data. This insight not only opens the door to a rich body of work in spectral analysis, but also leads to distinct computational benefits. Our resultant biresolution spectral formulation captures both the subspace projection error as well as the quantization error within the same framework. After a reformulation, the core steps of our algorithm involve a simple eigen decomposition step, which can be solved efficiently. We show that our method performs very favorably against a number of state of the art methods on standard data sets.

Multiple Kernel Learning (MKL) generalizes SVMs to the setting where one simultaneously trains a linear classifier and chooses an optimal combination of given base kernels. Model complexity is typically controlled using various norm regularizations on the vector of base kernel mixing coefficients. Existing methods, however, neither regularize nor exploit potentially useful information pertaining to how kernels in the input set 'interact'; that is, higher order kernel-pair relationships that can be easily obtained via unsupervised (similarity, geodesics), supervised (correlation in errors), or domain knowledge driven mechanisms (which features were used to construct the kernel?). We show that by substituting the norm penalty with an arbitrary quadratic function Q \succeq 0, one can impose a desired covariance structure on mixing coefficient selection, and use this as an inductive bias when learning the concept. This formulation significantly generalizes the widely used 1- and 2-norm MKL objectives. We explore the model’s utility via experiments on a challenging Neuroimaging problem, where the goal is to predict a subject’s conversion to Alzheimer’s Disease (AD) by exploiting aggregate information from several distinct imaging modalities. Here, our new model outperforms the state of the art (p-values << 10−3 ). We briefly discuss ramifications in terms of learning bounds (Rademacher complexity).

Our primary interest is in generalizing the problem of Cosegmentation to a large group of images, that is, concurrent segmentation of common foreground region(s) from multiple images. We further wish for our algorithm to offer scale invariance (foregrounds may have arbitrary sizes in different images) and the running time to increase (no more than) near linearly in the number of images in the set. What makes this setting particularly challenging is that even if we ignore the scale invariance desiderata, the Cosegmentation problem, as formalized in many recent papers (except [1]), is already hard to solve optimally in the two image case. A straightforward extension of such models to multiple images leads to loose relaxations; and unless we impose a distributional assumption on the appearance model, existing mechanisms for image-pair-wise measurement of foreground appearance variations lead to significantly large problem sizes (even for moderate number of images). This paper presents a surprisingly easy to implement algorithm which performs well, and satisfies all requirements listed above (scale invariance, low computational requirements, and viability for the multiple image setting). We present qualitative and technical analysis of the properties of this framework.

We propose a new algorithm for learning kernels for variants of the Normalized Cuts (NCuts) objective - i.e., given a set of training examples with known partitions, how should a basis set of similarity functions be combined to induce NCuts favorable distributions. Such a procedure facilitates design of good affinity matrices. It also helps assess the importance of different feature types for discrimination. Rather than formulating the learning problem in terms of the spectral relaxation, the alternative we pursue here is to work in the original discrete setting (i.e., the relaxation occurs much later). We show that this strategy is useful - while the initial specification seems rather difficult to optimize efficiently, a set of manipulations reveal a related model which permits a nice SDP relaxation. A salient feature of our model is that the eventual problem size is only a function of the number of input kernels and not the training set size. This relaxation also allows strong optimality guarantees, if certain conditions are satisfied. We show that the sub-kernel weights obtained provide a complementary approach for MKL based methods. Our experiments on Caltech101 and ADNI (a brain imaging dataset) show that the quality of solutions is competitive with the state-of-the-art.

In this paper, the general problem of dynamic assignment of sensors to local fusion centers (LFCs) in a distributed tracking framework is considered. With technological advances, a large number of sensors can be deployed for multitarget tracking purposes. However, due to physical limitations such as frequency, power, bandwidth, and fusion center capacity, only a limited number of them can be used by each LFC. The transmission power of future sensors is anticipated to be software controllable within certain lower and upper limits. Thus, the frequency reusability and the sensor reachability can be improved by controlling transmission powers. Then, the problem is to select the sensor subsets that should be used by each LFC and to find their transmission frequencies and powers in order to maximize the tracking accuracies and minimize the total power consumption. The frequency channel limitation and the advantage of variable transmitting power have not been discussed in the literature. In this paper, the optimal formulation for the aforementioned sensor management problem is provided based on the posterior Cramer-Rao lower bound. Finding the optimal solution to the aforementioned NP-hard multiobjective mixed-integer optimization problem in real time is difficult in large-scale scenarios. An algorithm is presented to find a suboptimal solution in real time by decomposing the original problem into subproblems, which are easier to solve, without using simplistic clustering algorithms that are typically used. Simulation results illustrating the performance of sensor array manager are also presented.

We study the so-called Generalized Median graph problem where the task is to to construct a prototype (i.e., a 'model') from an input set of graphs. The problem finds applications in many vision (e.g., object recognition) and learning problems where graphs are increasingly being adopted as a representation tool. Existing techniques for this problem are evolutionary search based; in this paper, we propose a polynomial time algorithm based on a linear programming formulation. We present an additional bi-level method to obtain solutions arbitrarily close to the optimal in non-polynomial time (in worst case). Within this new framework, one can optimize edit distance functions that capture similarity by considering vertex labels as well as the graph structure simultaneously. In context of our motivating application, we discuss experiments on molecular image analysis problems - the methods will provide the basis for building a topological map of all pairs of the human chromosome.

We consider the ensemble clustering problem where the task is to ‘aggregate’ multiple clustering solutions into a single consolidated clustering that maximizes the shared information among given clustering solutions. We obtain several new results for this problem. First, we note that the notion of agreement under such circumstances can be better captured using an agreement measure based on a 2D string encoding rather than voting strategy based methods proposed in literature. Using this generalization, we first derive a nonlinear optimization model to max- imize the new agreement measure. We then show that our optimization problem can be transformed into a strict 0-1 Semidefinite Program (SDP) via novel con- vexification techniques which can subsequently be relaxed to a polynomial time solvable SDP. Our experiments indicate improvements not only in terms of the proposed agreement measure but also the existing agreement measures based on voting strategies. We discuss evaluations on clustering and image segmentation databases.

Spherical k-means is a popular algorithm for document clustering. However, it may still yield poor performance in some circumstances. In this paper, we consider a discrete optimization model for spkmeans. By using the convexity of objective function and specific structure of constraint set, we first reformulate the discrete problem as an equivalent convex maximization problem with linear constraints. Then we characterize the local optimality of relaxed problem. Based on the characteristics, we refine the spherical k-means algorithm by alternatively performing spherical k-means and switching data points between clusters. This strategy guarantees that the refined algorithm can always attain a local optimal solution.

The minimum sum-of-squared error clustering problem is shown to be a concave continuous optimization problem whose every local minimum solution must be integer. We characterize its local minima. A procedure of moving from a fractional solution to a better integer solution is given. Then we adapt Tuy's convexity cut method to find a global optimum of the minimum sum-of-squared error clustering problem. We prove that this method converges in finite steps to a global minimum. Promising numerical examples are reported.