Deep Decision Trees for Discriminative Dictionary Learning with Adversarial Multi-Agent Trajectories

With the explosion in the availability of spatio-temporal tracking data in modern sports, there is an enormous opportunity to better analyse, learn and predict important events in adversarial group environments. In this paper, we propose a deep decision tree architecture for discriminative dictionary learning from adversarial multi-agent trajectories. We first build up a hierarchy for the tree structure by adding each layer and performing feature weight based clustering in the forward pass. We then fine tune the player role weights using back propagation. The hierarchical architecture ensures the interpretability and the integrity of the group representation. The resulting architecture is a decision tree, with leaf-nodes capturing a dictionary of multi-agent group interactions. Due to the ample volume of data available, we focus on soccer tracking data, although our approach can be used in any adversarial multi-agent domain. We present applications of proposed method for simulating soccer games as well as evaluating and quantifying team strategies.Comment: To appear in 4th International Workshop on Computer Vision in Sports (CVsports) at CVPR 201

Deliverable 2.2.1: Online Dictionary Learning from Big Data Using Accelerated Stochastic Approximation Algorithms

Applications involving large-scale dictionary learning tasks motivate well online optimization algorithms for generally non-convex and non-smooth problems. In this big data context, the present paper develops an online learning framework by jointly leveraging the stochastic approximation paradigm with first-order acceleration schemes. The generally non-convex objective evaluated online at the resultant iterates enjoys quadratic rate of convergence. The generality of the novel approach is demonstrated in two online learning applications: (i) Online linear regression using the total least-squares approach; and, (ii) a semi-supervised dictionary learning approach to network-wide link load tracking and imputation of real data with missing entries. In both cases, numerical tests highlight the potential of the proposed online framework for big data network analytics

Online dictionary learning from big data using accelerated stochastic approximation algorithms

Applications involving large-scale dictionary learning tasks motivate well online optimization algorithms for generally non-convex and non-smooth problems. In this big data context, the present paper develops an online learning framework by jointly leveraging the stochastic approximation paradigm with first-order acceleration schemes. The generally non-convex objective evaluated online at the resultant iterates enjoys quadratic rate of convergence. The generality of the novel approach is demonstrated in two online learning applications: (i) Online linear regression using the total least-squares approach; and, (ii) a semi-supervised dictionary learning approach to network-wide link load tracking and imputation of real data with missing entries. In both cases, numerical tests highlight the potential of the proposed online framework for big data network analytics

A Visual Tracking Study and A Proposal of Modifications

On-line visual tracking of a specified target in motion throughout frames of video clips faces challenges in robust identification of the target in the current frame based on the past frames. Three approaches for tracking the target image patch are described and compared. These approaches utilize particle filtering and principal component analysis (PCA) to identify the most likely location of the target in the current frame and a low dimensional subspace representation of the patches of images to be kept as the templates in the dictionary for the identification. By using a combination of methods and compare the result of each, a new model based is proposed. The goal is to achieve a more robust and accurate tracking of a target throughout the video and continue updating the identification templates to adapt the target changes, such as apparences in lighting, angle, scale and occlusions. The challenges in tracking are to introduction of the "right" templates into the identification templates in the dictionary and identify the most accurate particle image patch while tracking the target with the right tracking patch scaling. The first approach considered and on which the structure of the visual tracker is based is the "Incremental Learning for Robust Visual Tracking" by D. Ross et al., which is a computationally fast tracker that utilizes a method of low dimensional subspace for the identification template dictionary and incremental PCA for its tracking. The tracker has a simple rule in accepting the patches of images to be in the identification template dictionary after the image patch has gone through a singular value decomposition (SVD), where it eliminates singular values are smaller than $10^{-6}$ of the sum of squared sinuglar values and the corresponding bases are also eliminated. This elimination scheme has very limited robustness in tracking, therefore, more selective processes in accepting identification templates in the dictionary are explored and introduced on top of the existing method in comparison and to address the challenges in on-line video tracking. The second approach is the "Least Soft-Threshold Squares Tracking" proposed by D. Wang et al. solves the least soft-threshold squares distance problem to identify the distances of the particles to the templates in the dictionary, which greatly improves the tracking accuracy. This method is also computationally cheap in comparison to the first approach, and its accuracy is also better than the first approach, but it would sometimes fail to track in some applications. Finally, the third approach reviewed is the "Robust Visual Tracking and Vehicle Classification via Sparse Representation" by X. Mei et al. is to weight each particles when selecting the most likely target patch so the best patch has a highest weighted probability which ensures it being selected and introduced to the template dictionary. This approach performs well in comparison to the first and the second approaches in tracking accuracy and robustness, but this approach is extremely computationally expensive. Three new components are proposed in an effort to mitigate some of the limitations that the three approaches exhibit. One such component is to simply reject the image patches that exhibit too great of difference to the current template dictionary, which resulted in improved tracking robustness. This method is computationally cheap and easy to implement. Another component introduced is a second set of dictionary that is composed of admitted image patches, which is used for tracking when the image patches appears to be too dissimilar to the dictionary with low dimensional representation. It is expected that with more well defined and stronger features, it forces the tracking to identify the target. Finally, the third component introduced is the to prevent shrinkage of the target boundary box by weighting the particles drawn with the ratio of area change so that more weight is placed on particles with less arial change. This increases the likelihood of recovering the target again if tracking loses the target, and instead of shrinking the boundary box, the tracking is biased to staying with the image patch of the same size. The resulting performance of the proposed tracking scheme has not been noticeably improved, part of the reason is because the metrics available to identify a noisy image patch from the good image patches are not always indicative of the noisy-good image patch divide

Dictionary Learning under Symmetries via Group Representations

The dictionary learning problem can be viewed as a data-driven process to learn a suitable transformation so that data is sparsely represented directly from example data. In this paper, we examine the problem of learning a dictionary that is invariant under a pre-specified group of transformations. Natural settings include Cryo-EM, multi-object tracking, synchronization, pose estimation, etc. We specifically study this problem under the lens of mathematical representation theory. Leveraging the power of non-abelian Fourier analysis for functions over compact groups, we prescribe an algorithmic recipe for learning dictionaries that obey such invariances. We relate the dictionary learning problem in the physical domain, which is naturally modelled as being infinite dimensional, with the associated computational problem, which is necessarily finite dimensional. We establish that the dictionary learning problem can be effectively understood as an optimization instance over certain matrix orbitopes having a particular block-diagonal structure governed by the irreducible representations of the group of symmetries. This perspective enables us to introduce a band-limiting procedure which obtains dimensionality reduction in applications. We provide guarantees for our computational ansatz to provide a desirable dictionary learning outcome. We apply our paradigm to investigate the dictionary learning problem for the groups SO(2) and SO(3). While the SO(2)-orbitope admits an exact spectrahedral description, substantially less is understood about the SO(3)-orbitope. We describe a tractable spectrahedral outer approximation of the SO(3)-orbitope, and contribute an alternating minimization paradigm to perform optimization in this setting. We provide numerical experiments to highlight the efficacy of our approach in learning SO(3)-invariant dictionaries, both on synthetic and on real world data.Comment: 29 pages, 2 figure

Online Learning Discriminative Dictionary with Label Information for Robust Object Tracking

A supervised approach to online-learn a structured sparse and discriminative representation for object tracking is presented. Label information from training data is incorporated into the dictionary learning process to construct a robust and discriminative dictionary. This is accomplished by adding an ideal-code regularization term and classification error term to the total objective function. By minimizing the total objective function, we learn the high quality dictionary and optimal linear multiclassifier jointly using iterative reweighed least squares algorithm. Combined with robust sparse coding, the learned classifier is employed directly to separate the object from background. As the tracking continues, the proposed algorithm alternates between robust sparse coding and dictionary updating. Experimental evaluations on the challenging sequences show that the proposed algorithm performs favorably against state-of-the-art methods in terms of effectiveness, accuracy, and robustness