Although the traits emerged in a mass gathering are often non-deliberative, the act of mass impulse may lead to irre- vocable crowd disasters. The two-fold increase of carnage in crowd since the past two decades has spurred significant advances in the field of computer vision, towards effective and proactive crowd surveillance. Computer vision stud- ies related to crowd are observed to resonate with the understanding of the emergent behavior in physics (complex systems) and biology (animal swarm). These studies, which are inspired by biology and physics, share surprisingly common insights, and interesting contradictions. However, this aspect of discussion has not been fully explored. Therefore, this survey provides the readers with a review of the state-of-the-art methods in crowd behavior analysis from the physics and biologically inspired perspectives. We provide insights and comprehensive discussions for a broader understanding of the underlying prospect of blending physics and biology studies in computer vision.

Martin Jaggi Dept. of Computer Science ETH Zürich, Switzerland The Frank-Wolfe (FW) optimization algorithm has lately regained popularity thanks in particular to its ability to nicely handle the structured constraints appearing in machine learning applications. However, its convergence rate is known to be slow (sublinear) when the solution lies at the boundary. A simple lessknown fix is to add the possibility to take'away steps' during optimization, an operation that importantly does not require a feasibility oracle. In this paper, we highlight and clarify several variants of the Frank-Wolfe optimization algorithm that have been successfully applied in practice: away-steps FW, pairwise FW, fully-corrective FW and Wolfe's minimum norm point algorithm, and prove for the first time that they all enjoy global linear convergence, under a weaker condition than strong convexity of the objective. The constant in the convergence rate has an elegant interpretation as the product of the (classical) condition number of the function with a novel geometric quantity that plays the role of a'condition number' of the constraint set. We provide pointers to where these algorithms have made a difference in practice, in particular with the flow polytope, the marginal polytope and the base polytope for submodular optimization. The Frank-Wolfe algorithm [9] (also known as conditional gradient) is one of the earliest existing methods for constrained convex optimization, and has seen an impressive revival recently due to its nice properties compared to projected or proximal gradient methods, in particular for sparse optimization and machine learning applications. On the other hand, the classical projected gradient and proximal methods have been known to exhibit a very nice adaptive acceleration property, namely that the the convergence rate becomes linear for strongly convex objective, i.e. that the optimization error of the same algorithm after t iterations will decrease geometrically with O((1 ρ)

The paradigm of multi-task learning is that one can achieve better generalization by learning tasks jointly and thus exploiting the similarity between the tasks rather than learning them independently of each other. While previously the relationship between tasks had to be user-defined in the form of an output kernel, recent approaches jointly learn the tasks and the output kernel. As the output kernel is a positive semidefinite matrix, the resulting optimization problems are not scalable in the number of tasks as an eigendecomposition is required in each step. Using the theory of positive semidefinite kernels we show in this paper that for a certain class of regularizers on the output kernel, the constraint of being positive semidefinite can be dropped as it is automatically satisfied for the relaxed problem. This leads to an unconstrained dual problem which can be solved efficiently. Experiments on several multi-task and multi-class data sets illustrate the efficacy of our approach in terms of computational efficiency as well as generalization performance.

We consider the problem of recovering a complete (i.e., square and invertible) matrix $\mathbf A_0$, from $\mathbf Y \in \mathbb R^{n \times p}$ with $\mathbf Y = \mathbf A_0 \mathbf X_0$, provided $\mathbf X_0$ is sufficiently sparse. This recovery problem is central to the theoretical understanding of dictionary learning, which seeks a sparse representation for a collection of input signals, and finds numerous applications in modern signal processing and machine learning. We give the first efficient algorithm that provably recovers $\mathbf A_0$ when $\mathbf X_0$ has $O(n)$ nonzeros per column, under suitable probability model for $\mathbf X_0$. In contrast, prior results based on efficient algorithms provide recovery guarantees when $\mathbf X_0$ has only $O(n^{1-\delta})$ nonzeros per column for any constant $\delta \in (0, 1)$. Our algorithmic pipeline centers around solving a certain nonconvex optimization problem with a spherical constraint, and hence is naturally phrased in the language of manifold optimization. To show this apparently hard problem is tractable, we first provide a geometric characterization of the high-dimensional objective landscape, which shows that with high probability there are no "spurious" local minima. This particular geometric structure allows us to design a Riemannian trust region algorithm over the sphere that provably converges to one local minimizer with an arbitrary initialization, despite the presence of saddle points. The geometric approach we develop here may also shed light on other problems arising from nonconvex recovery of structured signals.

Bayesian Optimization (BO) is a data-efficient method for global black-box optimization of an expensive-to-evaluate fitness function. BO typically assumes that computation cost of BO is cheap, but experiments are time consuming or costly. In practice, this allows us to optimize ten or fewer critical parameters in up to 1,000 experiments. But experiments may be less expensive than BO methods assume: In some simulation models, we may be able to conduct multiple thousands of experiments in a few hours, and the computational burden of BO is no longer negligible compared to experimentation time. To address this challenge we introduce a new Dimension Scheduling Algorithm (DSA), which reduces the computational burden of BO for many experiments. The key idea is that DSA optimizes the fitness function only along a small set of dimensions at each iteration. This DSA strategy (1) reduces the necessary computation time, (2) finds good solutions faster than the traditional BO method, and (3) can be parallelized straightforwardly. We evaluate the DSA in the context of optimizing parameters of dynamic models of microalgae metabolism and show faster convergence than traditional BO.

This monograph presents the main complexity theorems in convex optimization and their corresponding algorithms. Starting from the fundamental theory of black-box optimization, the material progresses towards recent advances in structural optimization and stochastic optimization. Our presentation of black-box optimization, strongly influenced by Nesterov's seminal book and Nemirovski's lecture notes, includes the analysis of cutting plane methods, as well as (accelerated) gradient descent schemes. We also pay special attention to non-Euclidean settings (relevant algorithms include Frank-Wolfe, mirror descent, and dual averaging) and discuss their relevance in machine learning. We provide a gentle introduction to structural optimization with FISTA (to optimize a sum of a smooth and a simple non-smooth term), saddle-point mirror prox (Nemirovski's alternative to Nesterov's smoothing), and a concise description of interior point methods. In stochastic optimization we discuss stochastic gradient descent, mini-batches, random coordinate descent, and sublinear algorithms. We also briefly touch upon convex relaxation of combinatorial problems and the use of randomness to round solutions, as well as random walks based methods.

Numerous applications in data mining and machine learning require recovering a matrix of minimal rank. Robust principal component analysis (RPCA) is a general framework for handling this kind of problems. Nuclear norm based convex surrogate of the rank function in RPCA is widely investigated. Under certain assumptions, it can recover the underlying true low rank matrix with high probability. However, those assumptions may not hold in real-world applications. Since the nuclear norm approximates the rank by adding all singular values together, which is essentially a $\ell_1$-norm of the singular values, the resulting approximation error is not trivial and thus the resulting matrix estimator can be significantly biased. To seek a closer approximation and to alleviate the above-mentioned limitations of the nuclear norm, we propose a nonconvex rank approximation. This approximation to the matrix rank is tighter than the nuclear norm. To solve the associated nonconvex minimization problem, we develop an efficient augmented Lagrange multiplier based optimization algorithm. Experimental results demonstrate that our method outperforms current state-of-the-art algorithms in both accuracy and efficiency.

We present a convex approach to probabilistic segmentation and modeling of time series data. Our approach builds upon recent advances in multivariate total variation regularization, and seeks to learn a separate set of parameters for the distribution over the observations at each time point, but with an additional penalty that encourages the parameters to remain constant over time. We propose efficient optimization methods for solving the resulting (large) optimization problems, and a two-stage procedure for estimating recurring clusters under such models, based upon kernel density estimation. Finally, we show on a number of real-world segmentation tasks, the resulting methods often perform as well or better than existing latent variable models, while being substantially easier to train. 1 Introduction In this paper, we consider the tasks of time series segmentation and modeling. Formally, suppose that we observe a sequence ofT input/output pairs, represented as (x 1,y 1), (x 2,y 2),..., (x T,y T) (1) forx t R n (which can even include functions of past outputs of the time series to capture scenarios such as autoregressive models) andy t R p (though we can also consider other possible forms of the output vector, such as categorical variables).

We present a semi-supervised learning algorithm for learning discrete factor analysis models with arbitrary structure on the latent variables. Our algorithm assumes that every latent variable has an "anchor", an observed variable with only that latent variable as its parent. Given such anchors, we show that it is possible to consistently recover moments of the latent variables and use these moments to learn complete models. We also introduce a new technique for improving the robustness of method-of-moment algorithms by optimizing over the marginal polytope or its relaxations. We evaluate our algorithm using two real-world tasks, tag prediction on questions from the Stack Overflow website and medical diagnosis in an emergency department.

Marginal MAP inference involves making MAP predictions in systems defined with latent variables or missing information. It is significantly more difficult than pure marginalization and MAP tasks, for which a large class of efficient and convergent variational algorithms, such as dual decomposition, exist. In this work, we generalize dual decomposition to a generic power sum inference task, which includes marginal MAP, along with pure marginalization and MAP, as special cases. Our method is based on a block coordinate descent algorithm on a new convex decomposition bound, that is guaranteed to converge monotonically, and can be parallelized efficiently. We demonstrate our approach on marginal MAP queries defined on real-world problems from the UAI approximate inference challenge, showing that our framework is faster and more reliable than previous methods.