Dual-arm manipulation tasks can be prescribed to a robotic system in terms of desired absolute and relative motion of the robot's end-effectors. These can represent, e.g., jointly carrying a rigid object or performing an assembly task. When both types of motion are to be executed concurrently, the symmetric distribution of the relative motion between arms prevents task conflicts. Conversely, an asymmetric solution to the relative motion task will result in conflicts with the absolute task. In this work, we address the problem of designing a control law for the absolute motion task together with updating the distribution of the relative task among arms. Through a set of numerical results, we contrast our approach with the classical symmetric distribution of the relative motion task to illustrate the advantages of our method.

Machine learning techniques have been used in the past using Monte Carlo samples to construct predictors of the dynamic stability of power systems. In this paper we move beyond the task of prediction and propose a comprehensive approach to use predictors, such as Decision Trees (DT), within a standard optimization framework for pre- and post-fault control purposes. In particular, we present a generalizable method for embedding rules derived from DTs in an operation decision-making model. We begin by pointing out the specific challenges entailed when moving from a prediction to a control framework. We proceed with introducing the solution strategy based on generalized disjunctive programming (GDP) as well as a two-step search method for identifying optimal hyper-parameters for balancing cost and control accuracy. We showcase how the proposed approach constructs security proxies that cover multiple contingencies while facing high-dimensional uncertainty with respect to operating conditions with the use of a case study on the IEEE 39-bus system. The method is shown to achieve efficient system control at a marginal increase in system price compared to an oracle model.

The low-rank matrix approximation problem with respect to the entry-wise $\ell_{\infty}$-norm is the following: given a matrix $M$ and a factorization rank $r$, find a matrix $X$ whose rank is at most $r$ and that minimizes $\max_{i,j} |M_{ij} - X_{ij}|$. In this paper, we prove that the decision variant of this problem for $r=1$ is NP-complete using a reduction from the problem `not all equal 3SAT'. We also analyze several cases when the problem can be solved in polynomial time, and propose a simple practical heuristic algorithm which we apply on the problem of the recovery of a quantized low-rank matrix.

In the pursuit of real-time motion planning, a commonly adopted practice is to compute a trajectory by running a planning algorithm on a simplified, low-dimensional dynamical model, and then employ a feedback tracking controller that tracks such a trajectory by accounting for the full, high-dimensional system dynamics. While this strategy of planning with model mismatch generally yields fast computation times, there are no guarantees of dynamic feasibility, which hampers application to safety-critical systems. Building upon recent work that addressed this problem through the lens of Hamilton-Jacobi (HJ) reachability, we devise an algorithmic framework whereby one computes, offline, for a pair of "planner" (i.e., low-dimensional) and "tracking" (i.e., high-dimensional) models, a feedback tracking controller and associated tracking bound. This bound is then used as a safety margin when generating motion plans via the low-dimensional model. Specifically, we harness the computational tool of sum-of-squares (SOS) programming to design a bilinear optimization algorithm for the computation of the feedback tracking controller and associated tracking bound. The algorithm is demonstrated via numerical experiments, with an emphasis on investigating the trade-off between the increased computational scalability afforded by SOS and its intrinsic conservativeness. Collectively, our results enable scaling the appealing strategy of planning with model mismatch to systems that are beyond the reach of HJ analysis, while maintaining safety guarantees.

We present a numerical framework for approximating unknown governing equations using observation data and deep neural networks (DNN). In particular, we propose to use residual network (ResNet) as the basic building block for equation approximation. We demonstrate that the ResNet block can be considered as a one-step method that is exact in temporal integration. We then present two multi-step methods, recurrent ResNet (RT-ResNet) method and recursive ReNet (RS-ResNet) method. The RT-ResNet is a multi-step method on uniform time steps, whereas the RS-ResNet is an adaptive multi-step method using variable time steps. All three methods presented here are based on integral form of the underlying dynamical system. As a result, they do not require time derivative data for equation recovery and can cope with relatively coarsely distributed trajectory data. Several numerical examples are presented to demonstrate the performance of the methods.

Reinforcement learning (RL) constitutes a promising solution for alleviating the problem of traffic congestion. In particular, deep RL algorithms have been shown to produce adaptive traffic signal controllers that outperform conventional systems. However, in order to be reliable in highly dynamic urban areas, such controllers need to be robust with the respect to a series of exogenous sources of uncertainty. In this paper, we develop an open-source callback-based framework for promoting the flexible evaluation of different deep RL configurations under a traffic simulation environment. With this framework, we investigate how deep RL-based adaptive traffic controllers perform under different scenarios, namely under demand surges caused by special events, capacity reductions from incidents and sensor failures. We extract several key insights for the development of robust deep RL algorithms for traffic control and propose concrete designs to mitigate the impact of the considered exogenous uncertainties.

This paper deals with the convergence time analysis of a class of fixed-time stable systems with the aim to provide a new non-conservative upper bound for its settling time. Our contribution is fourfold. First, we revisit the well-known class of fixed-time stable systems, given in (Polyakov et al.,2012}, while showing the conservatism of the classical upper estimate of the settling time. Second, we provide the smallest constant that uniformly upper bounds the settling time of any trajectory of the system under consideration. Third, introducing a slight modification of the previous class of fixed-time systems, we propose a new predefined-time convergent algorithm where the least upper bound of the settling time is set a priori as a parameter of the system. At last, predefined-time controllers for first order and second order systems are introduced. Some simulation results highlight the performance of the proposed scheme in terms of settling time estimation compared to existing methods.

As all physical adaptive quantum-enhanced metrology schemes operate under noisy conditions with only partially understood noise characteristics, so a practical control policy must be robust even for unknown noise. We aim to devise a test to evaluate the robustness of AQEM policies and assess the resource used by the policies. The robustness test is performed on QEAPE by simulating the scheme under four phase-noise models corresponding to normal-distribution noise, random-telegraph noise, skew-normal-distribution noise, and log-normal-distribution noise. Control policies are devised either by an evolutionary algorithm under the same noisy conditions, albeit ignorant of its properties, or a Bayesian-based feedback method that assumes no noise. Our robustness test and resource comparison method can be used to determining the efficacy and selecting a suitable policy.

Traditional Input-Output Feedback Linearization (IOFL) requires full knowledge of system dynamics and assumes no disturbance at the input channel and no system's uncertainties. In this paper, a model-free Active Input-Output Feedback Linearization (AIOFL) technique based on an Improved Active Disturbance Rejection Control (IADRC) paradigm is proposed to design feedback linearization control law for a generalized nonlinear system with known relative degree. The Linearization Control Law(LCL) is composed of a scaled generalized disturbance estimated by an Improved Nonlinear Extended State Observer (INLESO) with saturation-like behavior and the nominal control law produced by an Improved Nonlinear State Error Feedback (INLSEF). The proposed AIOFL cancels in real-time fashion the generalized disturbances which represent all the unwanted dynamics, exogenous disturbances, and system uncertainties and transforms the system into a chain of integrators up to the relative degree of the system, the only information required about the nonlinear system. Stability analysis has been conducted based on Lyapunov functions and revealed the convergence of the INLESO and the asymptotic stability of the closed-loop system. Verification of the outcomes has been achieved by applying the proposed AIOFL technique on the Flexible Joint Single Link Manipulator (SLFJM). The simulations results validated the effectiveness of the proposed AIOFL tool based on IADRC as compared to the conventional ADRC based AIOFL and the traditional IOFL techniques.

The safety of mobile robots in dynamic environments is predicated on making sure that they do not collide with obstacles. In support of such safety arguments, we analyze and formally verify a series of increasingly powerful safety properties of controllers for avoiding both stationary and moving obstacles: (i) static safety, which ensures that no collisions can happen with stationary obstacles, (ii) passive safety, which ensures that no collisions can happen with stationary or moving obstacles while the robot moves, (iii) the stronger passive friendly safety in which the robot further maintains sufficient maneuvering distance for obstacles to avoid collision as well, and (iv) passive orientation safety, which allows for imperfect sensor coverage of the robot, i. e., the robot is aware that not everything in its environment will be visible. We complement these provably correct safety properties with liveness properties: we prove that provably safe motion is flexible enough to let the robot still navigate waypoints and pass intersections. We use hybrid system models and theorem proving techniques that describe and formally verify the robot's discrete control decisions along with its continuous, physical motion. Moreover, we formally prove that safety can still be guaranteed despite sensor uncertainty and actuator perturbation, and when control choices for more aggressive maneuvers are introduced. Our verification results are generic in the sense that they are not limited to the particular choices of one specific control algorithm but identify conditions that make them simultaneously apply to a broad class of control algorithms.

This paper considers the problem of solving systems of quadratic equations, namely, recovering an object of interest $\mathbf{x}^{\natural}\in\mathbb{R}^{n}$ from $m$ quadratic equations/samples $y_{i}=(\mathbf{a}_{i}^{\top}\mathbf{x}^{\natural})^{2}$, $1\leq i\leq m$. This problem, also dubbed as phase retrieval, spans multiple domains including physical sciences and machine learning. We investigate the efficiency of gradient descent (or Wirtinger flow) designed for the nonconvex least squares problem. We prove that under Gaussian designs, gradient descent --- when randomly initialized --- yields an $ε$-accurate solution in $O\big(\log n+\log(1/ε)\big)$ iterations given nearly minimal samples, thus achieving near-optimal computational and sample complexities at once. This provides the first global convergence guarantee concerning vanilla gradient descent for phase retrieval, without the need of (i) carefully-designed initialization, (ii) sample splitting, or (iii) sophisticated saddle-point escaping schemes. All of these are achieved by exploiting the statistical models in analyzing optimization algorithms, via a leave-one-out approach that enables the decoupling of certain statistical dependency between the gradient descent iterates and the data.

Recently, deep reinforcement learning (RL) methods have been applied successfully to multi-agent scenarios. Typically, these methods rely on a concatenation of agent states to represent the information content required for decentralized decision making. However, concatenation scales poorly to swarm systems with a large number of homogeneous agents as it does not exploit the fundamental properties inherent to these systems: (i) the agents in the swarm are interchangeable and (ii) the exact number of agents in the swarm is irrelevant. Therefore, we propose a new state representation for deep multi-agent RL based on mean embeddings of distributions. We treat the agents as samples of a distribution and use the empirical mean embedding as input for a decentralized policy. We define different feature spaces of the mean embedding using histograms, radial basis functions and a neural network learned end-to-end. We evaluate the representation on two well known problems from the swarm literature (rendezvous and pursuit evasion), in a globally and locally observable setup. For the local setup we furthermore introduce simple communication protocols. Of all approaches, the mean embedding representation using neural network features enables the richest information exchange between neighboring agents facilitating the development of more complex collective strategies.

In this work we present a new framework for the analysis and design of randomized gossip algorithms for solving the average consensus problem. We show how classical randomized iterative methods for solving linear systems can be interpreted as gossip algorithms when applied to special systems encoding the underlying network and explain in detail their decentralized nature. Our general framework recovers a comprehensive array of well-known gossip algorithms as special cases, including the pairwise randomized gossip algorithm and path averaging gossip, and allows for the development of provably faster variants. The flexibility of the new approach enables the design of a number of new specific gossip methods. For instance, we propose and analyze novel block and the first provably accelerated randomized gossip protocols, and dual randomized gossip algorithms. From a numerical analysis viewpoint, our work is the first that explores in depth the decentralized nature of randomized iterative methods for linear systems and proposes them as methods for solving the average consensus problem. We evaluate the performance of the proposed gossip protocols by performing extensive experimental testing on typical wireless network topologies.

Cellular networks are promising to support effective wireless communications for unmanned aerial vehicles (UAVs), which will help to enable various long-range UAV applications. However, these networks are optimized for terrestrial users, and thus do not guarantee seamless aerial coverage. In this paper, we propose to overcome this difficulty by exploiting controllable mobility of UAVs, and investigate connectivity-aware UAV path planning. To explicitly impose communication requirements on UAV path planning, we introduce two new metrics to quantify the cellular connectivity quality of a UAV path. Moreover, aerial coverage maps are used to provide accurate locations of scattered coverage holes in the complicated propagation environment. We formulate the UAV path planning problem as finding the shortest path subject to connectivity constraints. Based on graph search methods, a novel connectivity-aware path planning algorithm with low complexity is proposed. The effectiveness and superiority of our proposed algorithm are demonstrated using the aerial coverage map of an urban section in Virginia, which is built by ray tracing. Simulation results also illustrate a tradeoff between the path length and connectivity quality of UAVs.

We give algorithms for designing near-optimal sparse controllers using policy gradient with applications to control of systems corrupted by multiplicative noise, which is increasingly important in emerging complex dynamical networks. Various regularization schemes are examined and incorporated into the optimization by the use of gradient, subgradient, and proximal gradient methods. Numerical experiments on a large networked system show that the algorithms converge to performant sparse mean-square stabilizing controllers.

Brain-controlled unmanned aerial vehicle (uav) is a uav that can analyze human brain electrical signals through BCI to obtain flight commands. The research of brain-controlled uav can promote the integration of brain-computer and has a broad application prospect. At present, BCI still has some problems, such as limited recognition accuracy, limited recognition time and small number of recognition commands in the acquisition of control commands by analyzing eeg signals. Therefore, the control performance of the quadrotor which is controlled only by brain is not ideal. Based on the concept of Shared control, this paper designs an assistant controller using fuzzy PID control, and realizes the cooperative control between automatic control and brain control. By evaluating the current flight status and setting the switching rate, the switching mechanism of automatic control and brain control can be decided to improve the system control performance. Finally, a rectangular trajectory tracking control experiment of the same height is designed for small quadrotor to verify the algorithm.

A very simple event frequency approximation algorithm that is sensitive to event timeliness is suggested. The algorithm iteratively updates categorical click-distribution, producing (path of) a random walk on a standard $n$-dimensional simplex. Under certain conditions, this random walk is self-similar and corresponds to a biased Bernoulli convolution. Algorithm evaluation naturally leads to estimation of moments of biased (finite and infinite) Bernoulli convolutions.

Features in machine learning problems are often time-varying and may be related to outputs in an algebraic or dynamical manner. The dynamic nature of these machine learning problems renders current higher order accelerated gradient descent methods unstable or weakens their convergence guarantees. Inspired by methods employed in adaptive control, this paper proposes new algorithms for the case when time-varying features are present, and demonstrates provable performance guarantees. In particular, we develop a unified variational perspective within a continuous time algorithm. This variational perspective includes higher order learning concepts and normalization, both of which stem from adaptive control, and allows stability to be established for dynamical machine learning problems where time-varying features are present. These higher order algorithms are also examined for provably correct learning in adaptive control and identification. Simulations are provided to verify the theoretical results.

Given the increasingly serious air pollution problem, the monitoring of air quality index (AQI) in urban areas has drawn considerable attention. This paper presents ImgSensingNet, a vision guided aerial-ground sensing system, for fine-grained air quality monitoring and forecasting using the fusion of haze images taken by the unmanned-aerial-vehicle (UAV) and the AQI data collected by an on-ground three-dimensional (3D) wireless sensor network (WSN). Specifically, ImgSensingNet first leverages the computer vision technique to tell the AQI scale in different regions from the taken haze images, where haze-relevant features and a deep convolutional neural network (CNN) are designed for direct learning between haze images and corresponding AQI scale. Based on the learnt AQI scale, ImgSensingNet determines whether to wake up on-ground wireless sensors for small-scale AQI monitoring and inference, which can greatly reduce the energy consumption of the system. An entropy-based model is employed for accurate real-time AQI inference at unmeasured locations and future air quality distribution forecasting. We implement and evaluate ImgSensingNet on two university campuses since Feb. 2018, and has collected 17,630 photos and 2.6 millions of AQI data samples. Experimental results confirm that ImgSensingNet can achieve higher inference accuracy while greatly reduce the energy consumption, compared to state-of-the-art AQI monitoring approaches.

In this paper we introduce a unified analysis of a large family of variants of proximal stochastic gradient descent ({\tt SGD}) which so far have required different intuitions, convergence analyses, have different applications, and which have been developed separately in various communities. We show that our framework includes methods with and without the following tricks, and their combinations: variance reduction, importance sampling, mini-batch sampling, quantization, and coordinate sub-sampling. As a by-product, we obtain the first unified theory of {\tt SGD} and randomized coordinate descent ({\tt RCD}) methods, the first unified theory of variance reduced and non-variance-reduced {\tt SGD} methods, and the first unified theory of quantized and non-quantized methods. A key to our approach is a parametric assumption on the iterates and stochastic gradients. In a single theorem we establish a linear convergence result under this assumption and strong-quasi convexity of the loss function. Whenever we recover an existing method as a special case, our theorem gives the best known complexity result. Our approach can be used to motivate the development of new useful methods, and offers pre-proved convergence guarantees. To illustrate the strength of our approach, we develop five new variants of {\tt SGD}, and through numerical experiments demonstrate some of their properties.

The new financial European regulations such as PSD2 are changing the retail banking services. Noticeably, the monitoring of the personal expenses is now opened to other institutions than retail banks. Nonetheless, the retail banks are looking to leverage the user-device authentication on the mobile banking applications to enhance the personal financial advertisement. To address the profiling of the authentication, we rely on tensor decomposition, a higher dimensional analogue of matrix decomposition. We use Paratuck2, which expresses a tensor as a multiplication of matrices and diagonal tensors, because of the imbalance between the number of users and devices. We highlight why Paratuck2 is more appropriate in this case than the popular CP tensor decomposition, which decomposes a tensor as a sum of rank-one tensors. However, the computation of Paratuck2 is computational intensive. We propose a new APproximate HEssian-based Newton resolution algorithm, APHEN, capable of solving Paratuck2 more accurately and faster than the other popular approaches based on alternating least square or gradient descent. The results of Paratuck2 are used for the predictions of users' authentication with neural networks. We apply our method for the concrete case of targeting clients for financial advertising campaigns based on the authentication events generated by mobile banking applications.

This paper establishes a strict mathematical relationship between an arbitrary continuous function on a compact set and its global minima, like the well-known first order optimality condition for convex and differentiable functions. By introducing a class of nascent minima distribution functions that is only related to the target function and the given compact set, we construct a sequence that monotonically converges to the global minima on that given compact set. Then, we further consider some various sequences of sets where each sequence monotonically shrinks from the original compact set to the set of all global minimizers, and the shrink rate can be determined for continuously differentiable functions. Finally, we provide a different way of constructing the nascent minima distribution functions.

Machine learning (ML) problems are often posed as highly nonlinear and nonconvex unconstrained optimization problems. Methods for solving ML problems based on stochastic gradient descent are easily scaled for very large problems but may involve fine-tuning many hyper-parameters. Quasi-Newton approaches based on the limited-memory Broyden-Fletcher-Goldfarb-Shanno (BFGS) update typically do not require manually tuning hyper-parameters but suffer from approximating a potentially indefinite Hessian with a positive-definite matrix. Hessian-free methods leverage the ability to perform Hessian-vector multiplication without needing the entire Hessian matrix, but each iteration's complexity is significantly greater than quasi-Newton methods. In this paper we propose an alternative approach for solving ML problems based on a quasi-Newton trust-region framework for solving large-scale optimization problems that allow for indefinite Hessian approximations. Numerical experiments on a standard testing data set show that with a fixed computational time budget, the proposed methods achieve better results than the traditional limited-memory BFGS and the Hessian-free methods.

High-dimensional data classification is a fundamental task in machine learning and imaging science. In this paper, we propose a two-stage multiphase semi-supervised classification method for classifying high-dimensional data and unstructured point clouds. To begin with, a fuzzy classification method such as the standard support vector machine is used to generate a warm initialization. We then apply a two-stage approach named SaT (smoothing and thresholding) to improve the classification. In the first stage, an unconstraint convex variational model is implemented to purify and smooth the initialization, followed by the second stage which is to project the smoothed partition obtained at stage one to a binary partition. These two stages can be repeated, with the latest result as a new initialization, to keep improving the classification quality. We show that the convex model of the smoothing stage has a unique solution and can be solved by a specifically designed primal-dual algorithm whose convergence is guaranteed. We test our method and compare it with the state-of-the-art methods on several benchmark data sets. The experimental results demonstrate clearly that our method is superior in both the classification accuracy and computation speed for high-dimensional data and point clouds.

Manifold learning techniques seek to discover structure-preserving mappings of high-dimensional data into low-dimensional spaces. While the new sets of coordinates specified by these mappings can closely parameterize the data, they are generally complicated nonlinear functions of the original variables. This makes them difficult to interpret physically. Furthermore, in data-driven model reduction applications the governing equations may have structure that is destroyed by nonlinear mapping into coordinates on an inertial manifold, creating a computational bottleneck for simulations. Instead, we propose to identify a small collection of the original variables which are capable of uniquely determining all others either locally via immersion or globally via embedding of the underlying manifold. When the data lies on a low-dimensional subspace the existing discrete empirical interpolation method (DEIM) accomplishes this with recent variants employing greedy algorithms based on pivoted QR (PQR) factorizations. However, low-dimensional manifolds coming from a variety of applications, particularly from advection-dominated PDEs, do not lie in or near any low-dimensional subspace. Our proposed approach extends DEIM to data lying near nonlinear manifolds by applying a similar pivoted QR procedure simultaneously on collections of patches making up locally linear approximations of the manifold, resulting in a novel simultaneously pivoted QR (SimPQR) algorithm. The immersion provided by SimPQR can be extended to an embedding by applying SimPQR a second time to a modified collection of vectors. The SimPQR method for computing these `nonlinear DEIM' (NLDEIM) coordinates is successfully applied to real-world data lying near an inertial manifold in a cylinder wake flow as well as data coming from a viscous Burgers equation with different initial conditions.

We present a dynamical system framework for understanding Nesterov's accelerated gradient method. In contrast to earlier work, our derivation does not rely on a vanishing step size argument. We show that Nesterov acceleration arises from discretizing an ordinary differential equation with a semi-implicit Euler integration scheme. We analyze both the underlying differential equation as well as the discretization to obtain insights into the phenomenon of acceleration. The analysis suggests that a curvature-dependent damping term lies at the heart of the phenomenon. We further establish connections between the discretized and the continuous-time dynamics.

Diverse fault types, fast re-closures, and complicated transient states after a fault event make real-time fault location in power grids challenging. Existing localization techniques in this area rely on simplistic assumptions, such as static loads, or require much higher sampling rates or total measurement availability. This paper proposes a faulted line localization method based on a Convolutional Neural Network (CNN) classifier using bus voltages. Unlike prior data-driven methods, the proposed classifier is based on features with physical interpretations that improve the robustness of the location performance. The accuracy of our CNN based localization tool is demonstrably superior to other machine learning classifiers in the literature. To further improve the location performance, a joint phasor measurement units (PMU) placement strategy is proposed and validated against other methods. A significant aspect of our methodology is that under very low observability (7% of buses), the algorithm is still able to localize the faulted line to a small neighborhood with high probability. The performance of our scheme is validated through simulations of faults of various types in the IEEE 39-bus and 68-bus power systems under varying uncertain conditions, system observability, and measurement quality.

In this paper, we present libDirectional, a MATLAB library for directional statistics and directional estimation. It supports a variety of commonly used distributions on the unit circle, such as the von Mises, wrapped normal, and wrapped Cauchy distributions. Furthermore, various distributions on higher-dimensional manifolds such as the unit hypersphere and the hypertorus are available. Based on these distributions, several recursive filtering algorithms in libDirectional allow estimation on these manifolds. The functionality is implemented in a clear, well-documented, and object-oriented structure that is both easy to use and easy to extend.

This paper investigates the autonomous control of massive unmanned aerial vehicles (UAVs) for mission-critical applications (e.g., dispatching many UAVs from a source to a destination for firefighting). Achieving their fast travel and low motion energy without inter-UAV collision under wind perturbation is a daunting control task, which incurs huge communication energy for exchanging UAV states in real time. We tackle this problem by exploiting a mean-field game (MFG) theoretic control method that requires the UAV state exchanges only once at the initial source. Afterwards, each UAV can control its acceleration by locally solving two partial differential equations (PDEs), known as the Hamilton-Jacobi-Bellman (HJB) and Fokker-Planck-Kolmogorov (FPK) equations. This approach, however, brings about huge computation energy for solving the PDEs, particularly under multi-dimensional UAV states. We address this issue by utilizing a machine learning (ML) method where two separate ML models approximate the solutions of the HJB and FPK equations. These ML models are trained and exploited using an online gradient descent method with low computational complexity. Numerical evaluations validate that the proposed ML aided MFG theoretic algorithm, referred to as MFG learning control, is effective in collision avoidance with low communication energy and acceptable computation energy.

We investigate the fundamental conditions on the sampling pattern, i.e., locations of the sampled entries, for finite completability of a low-rank tensor given some components of its Tucker rank. In order to find the deterministic necessary and sufficient conditions, we propose an algebraic geometric analysis on the Tucker manifold, which allows us to incorporate multiple rank components in the proposed analysis in contrast with the conventional geometric approaches on the Grassmannian manifold. This analysis characterizes the algebraic independence of a set of polynomials defined based on the sampling pattern, which is closely related to finite completion. Probabilistic conditions are then studied and a lower bound on the sampling probability is given, which guarantees that the proposed deterministic conditions on the sampling patterns for finite completability hold with high probability. Furthermore, using the proposed geometric approach for finite completability, we propose a sufficient condition on the sampling pattern that ensures there exists exactly one completion for the sampled tensor.