Principal component pursuit (PCP) is a state-of-the-art approach for background estimation problems. Due to their higher computational cost, PCP algorithms, such as robust principal component analysis (RPCA) and its variants, are not feasible in processing high definition videos. To avoid the curse of dimensionality in those algorithms, several methods have been proposed to solve the background estimation problem in an incremental manner. We propose a batch-incremental background estimation model using a special weighted low-rank approximation of matrices. Through experiments with real and synthetic video sequences, we demonstrate that our method is superior to the state-of-the-art background estimation algorithms such as GRASTA, ReProCS, incPCP, and GFL.

In this paper, we introduce Path Integral Networks (PI-Net), a recurrent network representation of the Path Integral optimal control algorithm. The network includes both system dynamics and cost models, used for optimal control based planning. PI-Net is fully differentiable, learning both dynamics and cost models end-to-end by back-propagation and stochastic gradient descent. Because of this, PI-Net can learn to plan. PI-Net has several advantages: it can generalize to unseen states thanks to planning, it can be applied to continuous control tasks, and it allows for a wide variety learning schemes, including imitation and reinforcement learning. Preliminary experiment results show that PI-Net, trained by imitation learning, can mimic control demonstrations for two simulated problems; a linear system and a pendulum swing-up problem. We also show that PI-Net is able to learn dynamics and cost models latent in the demonstrations.

Observable operator models (OOMs) and related models are one of the most important and powerful tools for modeling and analyzing stochastic systems. They exactly describe dynamics of finite-rank systems and can be efficiently and consistently estimated through spectral learning under the assumption of identically distributed data. In this paper, we investigate the properties of spectral learning without this assumption due to the requirements of analyzing large-time scale systems, and show that the equilibrium dynamics of a system can be extracted from nonequilibrium observation data by imposing an equilibrium constraint. In addition, we propose a binless extension of spectral learning for continuous data. In comparison with the other continuous-valued spectral algorithms, the binless algorithm can achieve consistent estimation of equilibrium dynamics with only linear complexity.

The alternating direction method of multipliers (ADMM) is commonly used for distributed model fitting problems, but its performance and reliability depend strongly on user-defined penalty parameters. We study distributed ADMM methods that boost performance by using different fine-tuned algorithm parameters on each worker node. We present a O(1/k) convergence rate for adaptive ADMM methods with node-specific parameters, and propose adaptive consensus ADMM (ACADMM), which automatically tunes parameters without user oversight.

We propose a new reinforcement learning algorithm for partially observable Markov decision processes (POMDP) based on spectral decomposition methods. While spectral methods have been previously employed for consistent learning of (passive) latent variable models such as hidden Markov models, POMDPs are more challenging since the learner interacts with the environment and possibly changes the future observations in the process. We devise a learning algorithm running through episodes, in each episode we employ spectral techniques to learn the POMDP parameters from a trajectory generated by a fixed policy. At the end of the episode, an optimization oracle returns the optimal memoryless planning policy which maximizes the expected reward based on the estimated POMDP model. We prove an order-optimal regret bound with respect to the optimal memoryless policy and efficient scaling with respect to the dimensionality of observation and action spaces.

We present a framework for learning human user models from joint-action demonstrations that enables the robot to compute a robust policy for a collaborative task with a human. The learning takes place completely automatically, without any human intervention. First, we describe the clustering of demonstrated action sequences into different human types using an unsupervised learning algorithm. These demonstrated sequences are also used by the robot to learn a reward function that is representative for each type, through the employment of an inverse reinforcement learning algorithm. The learned model is then used as part of a Mixed Observability Markov Decision Process formulation, wherein the human type is a partially observable variable. With this framework, we can infer, either offline or online, the human type of a new user that was not included in the training set, and can compute a policy for the robot that will be aligned to the preference of this new user and will be robust to deviations of the human actions from prior demonstrations. Finally we validate the approach using data collected in human subject experiments, and conduct proof-of-concept demonstrations in which a person performs a collaborative task with a small industrial robot.

Non-negative matrix factorization is a basic tool for decomposing data into the feature and weight matrices under non-negativity constraints, and in practice is often solved in the alternating minimization framework. However, it is unclear whether such algorithms can recover the ground-truth feature matrix when the weights for different features are highly correlated, which is common in applications. This paper proposes a simple and natural alternating gradient descent based algorithm, and shows that with a mild initialization it provably recovers the ground-truth in the presence of strong correlations. In most interesting cases, the correlation can be in the same order as the highest possible. Our analysis also reveals its several favorable features including robustness to noise. We complement our theoretical results with empirical studies on semi-synthetic datasets, demonstrating its advantage over several popular methods in recovering the ground-truth.

Policy evaluation is a crucial step in many reinforcement-learning procedures, which estimates a value function that predicts states' long-term value under a given policy. In this paper, we focus on policy evaluation with linear function approximation over a fixed dataset. We first transform the empirical policy evaluation problem into a (quadratic) convex-concave saddle point problem, and then present a primal-dual batch gradient method, as well as two stochastic variance reduction methods for solving the problem. These algorithms scale linearly in both sample size and feature dimension. Moreover, they achieve linear convergence even when the saddle-point problem has only strong concavity in the dual variables but no strong convexity in the primal variables. Numerical experiments on benchmark problems demonstrate the effectiveness of our methods.

Underactuation is ubiquitous in human locomotion and should be ubiquitous in bipedal robotic locomotion as well. This chapter presents a coherent theory for the design of feedback controllers that achieve stable walking gaits in underactuated bipedal robots. Two fundamental tools are introduced, virtual constraints and hybrid zero dynamics. Virtual constraints are relations on the state variables of a mechanical model that are imposed through a time-invariant feedback controller. One of their roles is to synchronize the robot's joints to an internal gait phasing variable. A second role is to induce a low dimensional system, the zero dynamics, that captures the underactuated aspects of a robot's model, without any approximations. To enhance intuition, the relation between physical constraints and virtual constraints is first established. From here, the hybrid zero dynamics of an underactuated bipedal model is developed, and its fundamental role in the design of asymptotically stable walking motions is established. The chapter includes numerous references to robots on which the highlighted techniques have been implemented.

Solving symmetric positive definite linear problems is a fundamental computational task in machine learning. The exact solution, famously, is cubicly expensive in the size of the matrix. To alleviate this problem, several linear-time approximations, such as spectral and inducing-point methods, have been suggested and are now in wide use. These are low-rank approximations that choose the low-rank space a priori and do not refine it over time. While this allows linear cost in the data-set size, it also causes a finite, uncorrected approximation error. Authors from numerical linear algebra have explored ways to iteratively refine such low-rank approximations, at a cost of a small number of matrix-vector multiplications. This idea is particularly interesting in the many situations in machine learning where one has to solve a sequence of related symmetric positive definite linear problems. From the machine learning perspective, such deflation methods can be interpreted as transfer learning of a low-rank approximation across a time-series of numerical tasks. We study the use of such methods for our field. Our empirical results show that, on regression and classification problems of intermediate size, this approach can interpolate between low computational cost and numerical precision.

Recent advances in combining deep learning and Reinforcement Learning have shown a promising path for designing new control agents that can learn optimal policies for challenging control tasks. These new methods address the main limitations of conventional Reinforcement Learning methods such as customized feature engineering and small action/state space dimension requirements. In this paper, we leverage one of the state-of-the-art Reinforcement Learning methods, known as Trust Region Policy Optimization, to tackle intersection management for autonomous vehicles. We show that using this method, we can perform fine-grained acceleration control of autonomous vehicles in a grid street plan to achieve a global design objective.

We consider nonlinear stochastic systems that arise in path planning and control of mobile robots. As is typical of almost all nonlinear stochastic systems, the optimally solving problem is intractable. We provide a design approach which yields a tractable design that is quantifiably near-optimal. We exhibit a "separation" principle under a small noise assumption consisting of the optimal open-loop design of nominal trajectory followed by an optimal feedback law to track this trajectory, which is different from the usual effort of separating estimation from control. As a corollary, we obtain a trajectory-optimized linear quadratic regulator design for stochastic nonlinear systems with Gaussian noise.

Multiresolution analysis and matrix factorization are foundational tools in computer vision. In this work, we study the interface between these two distinct topics and obtain techniques to uncover hierarchical block structure in symmetric matrices -- an important aspect in the success of many vision problems. Our new algorithm, the incremental multiresolution matrix factorization, uncovers such structure one feature at a time, and hence scales well to large matrices. We describe how this multiscale analysis goes much farther than what a direct global factorization of the data can identify. We evaluate the efficacy of the resulting factorizations for relative leveraging within regression tasks using medical imaging data. We also use the factorization on representations learned by popular deep networks, providing evidence of their ability to infer semantic relationships even when they are not explicitly trained to do so. We show that this algorithm can be used as an exploratory tool to improve the network architecture, and within numerous other settings in vision.

In this paper we propose an implicit force control scheme for a one-link flexible manipulator that interact with a compliant environment. The controller was based in the mathematical model of the manipulator, considering the dynamics of the beam flexible and the gravitational force. With this method, the controller parameters are obtained from the structural parameters of the beam (link) of the manipulator. This controller ensure the stability based in the Lyapunov Theory. The controller proposed has two closed loops: the inner loop is a tracking control with gravitational force and vibration frequencies compensation and the outer loop is a implicit force control. To evaluate the performance of the controller, we have considered to three different manipulators (the length, the diameter were modified) and three environments with compliance modified. The results obtained from simulations verify the asymptotic tracking and regulated in position and force respectively and the vibrations suppression of the beam in a finite time.

In a spiking neural network (SNN), individual neurons operate autonomously and only communicate with other neurons sparingly and asynchronously via spike signals. These characteristics render a massively parallel hardware implementation of SNN a potentially powerful computer, albeit a non von Neumann one. But can one guarantee that a SNN computer solves some important problems reliably? In this paper, we formulate a mathematical model of one SNN that can be configured for a sparse coding problem for feature extraction. With a moderate but well-defined assumption, we prove that the SNN indeed solves sparse coding. To the best of our knowledge, this is the first rigorous result of this kind.

As we aim to control complex systems, use of a simulator in model-based reinforcement learning is becoming more common. However, it has been challenging to overcome the Reality Gap, which comes from nonlinear model bias and susceptibility to disturbance. To address these problems, we propose a novel algorithm that combines data-driven system identification approach (Gaussian Process) with a Differential-Dynamic-Programming-based robust optimal control method (Iterative Linear Quadratic Control). Our algorithm uses the simulator's model as the mean function for a Gaussian Process and learns only the difference between the simulator's prediction and actual observations, making it a natural hybrid of simulation and real-world observation. We show that our approach quickly corrects incorrect models, comes up with robust optimal controllers, and transfers its acquired model knowledge to new tasks efficiently.

This paper introduces an end-to-end fine-tuning method to improve hand-eye coordination in modular deep visuo-motor policies (modular networks) where each module is trained independently. Benefiting from weighted losses, the fine-tuning method significantly improves the performance of the policies for a robotic planar reaching task.

As robots and other automated systems are introduced to unknown and dynamic environments, robust and adaptive control strategies are required to cope with disturbances, unmodeled dynamics and parametric uncertainties. In this paper, we propose and provide theoretical proofs of a combined $\mathcal{L}_1$ adaptive feedback and iterative learning control (ILC) framework to improve trajectory tracking of a system subject to unknown and changing disturbances. The $\mathcal{L}_1$ adaptive controller forces the system to behave in a repeatable, predefined way, even in the presence of unknown and changing disturbances; however, this does not imply that perfect trajectory tracking is achieved. ILC improves the tracking performance based on experience from previous executions. The performance of ILC is limited by the robustness and repeatability of the underlying system, which, in this approach, is handled by the $\mathcal{L}_1$ adaptive controller. In particular, we are able to generalize learned trajectories across different system configurations because the $\mathcal{L}_1$ adaptive controller handles the underlying changes in the system. We demonstrate the improved trajectory tracking performance and generalization capabilities of the combined method compared to pure ILC in experiments with a quadrotor subject to unknown, dynamic disturbances. This is the first work to show $\mathcal{L}_1$ adaptive control combined with ILC in experiment.

We present a new method of learning control policies that successfully operate under unknown dynamic models. We create such policies by leveraging a large number of training examples that are generated using a physical simulator. Our system is made of two components: a Universal Policy (UP) and a function for Online System Identification (OSI). We describe our control policy as universal because it is trained over a wide array of dynamic models. These variations in the dynamic model may include differences in mass and inertia of the robots' components, variable friction coefficients, or unknown mass of an object to be manipulated. By training the Universal Policy with this variation, the control policy is prepared for a wider array of possible conditions when executed in an unknown environment. The second part of our system uses the recent state and action history of the system to predict the dynamics model parameters mu. The value of mu from the Online System Identification is then provided as input to the control policy (along with the system state). Together, UP-OSI is a robust control policy that can be used across a wide range of dynamic models, and that is also responsive to sudden changes in the environment. We have evaluated the performance of this system on a variety of tasks, including the problem of cart-pole swing-up, the double inverted pendulum, locomotion of a hopper, and block-throwing of a manipulator. UP-OSI is effective at these tasks across a wide range of dynamic models. Moreover, when tested with dynamic models outside of the training range, UP-OSI outperforms the Universal Policy alone, even when UP is given the actual value of the model dynamics. In addition to the benefits of creating more robust controllers, UP-OSI also holds out promise of narrowing the Reality Gap between simulated and real physical systems.

There have been numerous studies on the problem of flocking control for multiagent systems whose simplified models are presented in terms of point-mass elements. Meanwhile, full dynamic models pose some challenging problems in addressing the flocking control problem of mobile robots due to their nonholonomic dynamic properties. Taking practical constraints into consideration, we propose a novel approach to distributed flocking control of nonholonomic mobile robots by bounded feedback. The flocking control objectives consist of velocity consensus, collision avoidance, and cohesion maintenance among mobile robots. A flocking control protocol which is based on the information of neighbor mobile robots is constructed. The theoretical analysis is conducted with the help of a Lyapunov-like function and graph theory. Simulation results are shown to demonstrate the efficacy of the proposed distributed flocking control scheme.

With the huge influx of various data nowadays, extracting knowledge from them has become an interesting but tedious task among data scientists, particularly when the data come in heterogeneous form and have missing information. Many data completion techniques had been introduced, especially in the advent of kernel methods. However, among the many data completion techniques available in the literature, studies about mutually completing several incomplete kernel matrices have not been given much attention yet. In this paper, we present a new method, called Mutual Kernel Matrix Completion (MKMC) algorithm, that tackles this problem of mutually inferring the missing entries of multiple kernel matrices by combining the notions of data fusion and kernel matrix completion, applied on biological data sets to be used for classification task. We first introduced an objective function that will be minimized by exploiting the EM algorithm, which in turn results to an estimate of the missing entries of the kernel matrices involved. The completed kernel matrices are then combined to produce a model matrix that can be used to further improve the obtained estimates. An interesting result of our study is that the E-step and the M-step are given in closed form, which makes our algorithm efficient in terms of time and memory. After completion, the (completed) kernel matrices are then used to train an SVM classifier to test how well the relationships among the entries are preserved. Our empirical results show that the proposed algorithm bested the traditional completion techniques in preserving the relationships among the data points, and in accurately recovering the missing kernel matrix entries. By far, MKMC offers a promising solution to the problem of mutual estimation of a number of relevant incomplete kernel matrices.

A function $f: \mathbb{R}^d \rightarrow \mathbb{R}$ is a Sparse Additive Model (SPAM), if it is of the form $f(\mathbf{x}) = \sum_{l \in \mathcal{S}}ϕ_{l}(x_l)$ where $\mathcal{S} \subset [d]$, $|\mathcal{S}| \ll d$. Assuming $ϕ$'s, $\mathcal{S}$ to be unknown, there exists extensive work for estimating $f$ from its samples. In this work, we consider a generalized version of SPAMs, that also allows for the presence of a sparse number of second order interaction terms. For some $\mathcal{S}_1 \subset [d], \mathcal{S}_2 \subset {[d] \choose 2}$, with $|\mathcal{S}_1| \ll d, |\mathcal{S}_2| \ll d^2$, the function $f$ is now assumed to be of the form: $\sum_{p \in \mathcal{S}_1}ϕ_{p} (x_p) + \sum_{(l,l^{\prime}) \in \mathcal{S}_2}ϕ_{(l,l^{\prime})} (x_l,x_{l^{\prime}})$. Assuming we have the freedom to query $f$ anywhere in its domain, we derive efficient algorithms that provably recover $\mathcal{S}_1,\mathcal{S}_2$ with finite sample bounds. Our analysis covers the noiseless setting where exact samples of $f$ are obtained, and also extends to the noisy setting where the queries are corrupted with noise. For the noisy setting in particular, we consider two noise models namely: i.i.d Gaussian noise and arbitrary but bounded noise. Our main methods for identification of $\mathcal{S}_2$ essentially rely on estimation of sparse Hessian matrices, for which we provide two novel compressed sensing based schemes. Once $\mathcal{S}_1, \mathcal{S}_2$ are known, we show how the individual components $ϕ_p$, $ϕ_{(l,l^{\prime})}$ can be estimated via additional queries of $f$, with uniform error bounds. Lastly, we provide simulation results on synthetic data that validate our theoretical findings.

We consider the problem of generating motion plans for a robot that are guaranteed to succeed despite uncertainty in the environment, parametric model uncertainty, and disturbances. Furthermore, we consider scenarios where these plans must be generated in real-time, because constraints such as obstacles in the environment may not be known until they are perceived (with a noisy sensor) at runtime. Our approach is to pre-compute a library of "funnels" along different maneuvers of the system that the state is guaranteed to remain within (despite bounded disturbances) when the feedback controller corresponding to the maneuver is executed. We leverage powerful computational machinery from convex optimization (sums-of-squares programming in particular) to compute these funnels. The resulting funnel library is then used to sequentially compose motion plans at runtime while ensuring the safety of the robot. A major advantage of the work presented here is that by explicitly taking into account the effect of uncertainty, the robot can evaluate motion plans based on how vulnerable they are to disturbances. We demonstrate and validate our method using extensive hardware experiments on a small fixed-wing airplane avoiding obstacles at high speed (~12 mph), along with thorough simulation experiments of ground vehicle and quadrotor models navigating through cluttered environments. To our knowledge, these demonstrations constitute one of the first examples of provably safe and robust control for robotic systems with complex nonlinear dynamics that need to plan in real-time in environments with complex geometric constraints.

Matrix completion models are among the most common formulations of recommender systems. Recent works have showed a boost of performance of these techniques when introducing the pairwise relationships between users/items in the form of graphs, and imposing smoothness priors on these graphs. However, such techniques do not fully exploit the local stationarity structures of user/item graphs, and the number of parameters to learn is linear w.r.t. the number of users and items. We propose a novel approach to overcome these limitations by using geometric deep learning on graphs. Our matrix completion architecture combines graph convolutional neural networks and recurrent neural networks to learn meaningful statistical graph-structured patterns and the non-linear diffusion process that generates the known ratings. This neural network system requires a constant number of parameters independent of the matrix size. We apply our method on both synthetic and real datasets, showing that it outperforms state-of-the-art techniques.

Nowadays, many applications rely on images of high quality to ensure good performance in conducting their tasks. However, noise goes against this objective as it is an unavoidable issue in most applications. Therefore, it is essential to develop techniques to attenuate the impact of noise, while maintaining the integrity of relevant information in images. We propose in this work to extend the application of the Non-Local Means filter (NLM) to the vector case and apply it for denoising multispectral images. The objective is to benefit from the additional information brought by multispectral imaging systems. The NLM filter exploits the redundancy of information in an image to remove noise. A restored pixel is a weighted average of all pixels in the image. In our contribution, we propose an optimization framework where we dynamically fine tune the NLM filter parameters and attenuate its computational complexity by considering only pixels which are most similar to each other in computing a restored pixel. Filter parameters are optimized using Stein's Unbiased Risk Estimator (SURE) rather than using ad hoc means. Experiments have been conducted on multispectral images corrupted with additive white Gaussian noise and PSNR and similarity comparison with other approaches are provided to illustrate the efficiency of our approach in terms of both denoising performance and computation complexity.

To manage and maintain large-scale cellular networks, operators need to know which sectors underperform at any given time. For this purpose, they use the so-called hot spot score, which is the result of a combination of multiple network measurements and reflects the instantaneous overall performance of individual sectors. While operators have a good understanding of the current performance of a network and its overall trend, forecasting the performance of each sector over time is a challenging task, as it is affected by both regular and non-regular events, triggered by human behavior and hardware failures. In this paper, we study the spatio-temporal patterns of the hot spot score and uncover its regularities. Based on our observations, we then explore the possibility to use recent measurements' history to predict future hot spots. To this end, we consider tree-based machine learning models, and study their performance as a function of time, amount of past data, and prediction horizon. Our results indicate that, compared to the best baseline, tree-based models can deliver up to 14% better forecasts for regular hot spots and 153% better forecasts for non-regular hot spots. The latter brings strong evidence that, for moderate horizons, forecasts can be made even for sectors exhibiting isolated, non-regular behavior. Overall, our work provides insight into the dynamics of cellular sectors and their predictability. It also paves the way for more proactive network operations with greater forecasting horizons.

This paper introduces a potential learning scheme that can dynamically predict the stability of the reconnection of sub-networks to a main grid. As the future electrical power systems tend towards smarter and greener technology, the deployment of self sufficient networks, or microgrids, becomes more likely. Microgrids may operate on their own or synchronized with the main grid, thus control methods need to take into account islanding and reconnecting of said networks. The ability to optimally and safely reconnect a portion of the grid is not well understood and, as of now, limited to raw synchronization between interconnection points. A support vector machine (SVM) leveraging real-time data from phasor measurement units (PMUs) is proposed to predict in real time whether the reconnection of a sub-network to the main grid would lead to stability or instability. A dynamics simulator fed with pre-acquired system parameters is used to create training data for the SVM in various operating states. The classifier was tested on a variety of cases and operating points to ensure diversity. Accuracies of approximately 85% were observed throughout most conditions when making dynamic predictions of a given network.

This papers shows that using separators, which is a pair of two complementary contractors, we can easily and efficiently solve the localization problem of a robot with sonar measurements in an unstructured environment. We introduce separators associated with the Minkowski sum and the Minkowski difference in order to facilitate the resolution. A test-case is given in order to illustrate the principle of the approach.

Stochastic variance reduction algorithms have recently become popular for minimizing the average of a large, but finite, number of loss functions. In this paper, we propose a novel Riemannian extension of the Euclidean stochastic variance reduced gradient algorithm (R-SVRG) to a compact manifold search space. To this end, we show the developments on the Grassmann manifold. The key challenges of averaging, addition, and subtraction of multiple gradients are addressed with notions like logarithm mapping and parallel translation of vectors on the Grassmann manifold. We present a global convergence analysis of the proposed algorithm with decay step-sizes and a local convergence rate analysis under fixed step-size with some natural assumptions. The proposed algorithm is applied on a number of problems on the Grassmann manifold like principal components analysis, low-rank matrix completion, and the Karcher mean computation. In all these cases, the proposed algorithm outperforms the standard Riemannian stochastic gradient descent algorithm.

We consider a robotic vehicle tasked with gathering information by visiting a set of spatially-distributed data sources, the locations of which are not known a priori, but are discovered on the fly. We assume a first-order robot dynamics involving drift and that the locations of the data sources are Poisson-distributed. In this setting, we characterize the performance of the robot in terms of its sensing, agility, and computation capabilities. More specifically, the robot's performance is characterized in terms of its ability to sense the target locations from a distance, to maneuver quickly, and to perform computations for inference and planning. We also characterize the performance of the robot in terms of the amount and distribution of information that can be acquired at each data source. The following are among our theoretical results: the distribution of the amount of information among the target locations immensely impacts the requirements for sensing targets from a distance; performance increases with increasing maneuvering capability, but with diminishing returns; and the computation requirements increase more rapidly for planning as opposed to inference, with both increasing sensing range and maneuvering ability. We provide computational experiments to validate our theoretical results. Finally, we demonstrate that these results can be utilized in the co-design of sensing, actuation, and computation capabilities of mobile robotic systems for an information-gathering mission. Our proof techniques establish novel connections between the fundamental problems of robotic information-gathering and the last-passage percolation problem of statistical mechanics, which may be of interest on its own right.