We consider the problem of robustly fitting a model to data that includes outliers by formulating a percentile optimization problem. This problem is non-smooth and non-convex, hence hard to solve. We derive properties that the minimizers of such problems must satisfy. These properties lead to methods that solve the percentile formulation both for general residuals and for convex residuals. The methods fit the model to subsets of the data, and then extract the solution of the percentile formulation from these partial fits. As illustrative simulations show, such methods endure higher outlier percentages, when compared with standard robust estimates. Additionally, the derived properties provide a broader and alternative theoretical validation for existing robust methods, whose validity was previously limited to specific forms of the residuals.

This paper proposes a simple thermal noise cancellation technique to effectively reduce the output noise in predictive integrators. By combining a low-gain two-stage amplifier and a small auxiliary noise-cancelling capacitor, the output noise is highly attenuated. Simulation results for a classic Nagaraj-Ki predictive integrator, considering real MOS switches, demonstrate that the output thermal noise can be reduced up to 87%.

The augmented Lagrangian method (ALM) is a classical optimization tool that solves a given “difficult” (constrained) problem via finding solutions of a sequence of “easier” (often unconstrained) subproblems with respect to the original (primal) variable, wherein constraints satisfaction is controlled via the so-called dual variables. ALM is highly flexible with respect to how primal subproblems can be solved, giving rise to a plethora of different primal-dual methods. The powerful ALM mechanism has recently proved to be very successful in various large-scale and distributed applications. In addition, several significant advances have appeared, primarily on precise complexity results with respect to computational and communication costs in the presence of inexact updates and design and analysis of novel optimal methods for distributed consensus optimization. We provide a tutorial-style introduction to ALM and its variants for solving convex optimization problems in large-scale and distributed settings. We describe control-theoretic tools for the algorithms' analysis and design, survey recent results, and provide novel insights into the context of two emerging applications: federated learning and distributed energy trading.

Processing data collected by a network of agents often boils down to solving an optimization problem. The distributed nature of these problems calls for methods that are, themselves, distributed. While most collaborative learning problems require agents to reach a common (or consensus) model, there are situations in which the consensus solution may not be optimal. For instance, agents may want to reach a compromise between agreeing with their neighbors and minimizing a personal loss function. We present DJAM, a Jacobi-like distributed algorithm for learning personalized models. This method is implementation-friendly: it has no hyperparameters that need tuning, it is asynchronous, and its updates only require single-neighbor interactions. We prove that DJAM converges with probability one to the solution, provided that the personal loss functions are strongly convex and have Lipschitz gradient. We then give evidence that DJAM is on par with state-of-the-art methods: our method reaches a solution with error similar to the error of a carefully tuned alternating direction method of multipliers (ADMM) in about the same number of single-neighbor interactions.

We consider learning an environment by a team of spatially distributed agents, each agent with ongoing local measurements and the team of agents linked by a sparse directed communication network. Agents exploit their local measurements and exchange messages with neighbors in the communication network to quickly learn which environment-from a given finite set of presumed ones-is active. Works on such distributed learning setup assumed that the local measurements taken by the agents are uncorrelated over time, a convenient assumption that leaves, however, many practical setups excluded. In this letter, via modifying a recent distributed learning algorithm, we expand distributed learning to environments in which the time correlation of the local measurements is arbitrary. Since correlated measurements invalidate the previous proofs, we need a new proof guaranteeing that the modified distributed learning algorithm succeeds. Our new proof technique is simple, relies only on basic tools such as the Perron-Frobenius and the ergodic Markov chain theorems, and even covers the case of mismatched assumptions-the active environment is not in the finite set of presumed ones. A numerical example confirms the novel theoretical findings.

In today's large social and technological networks, since it is unfeasible to observe all the nodes, the source of diffusion is determined based on the observations of a subset of nodes. The probability of source localization error depends on the particular choice of observer nodes. We propose a criterion for observer node selection based on the minimal pairwise Chernoff distance between distributions of different source candidates. The proposed approach is optimal for the fastest error decay with vanishing noise. Although suboptimal for non-negligible noise, through simulation, we demonstrate its applicability in achieving low error probability. We also analyze the effect of network topology on the resulting error by bounding the smallest Chernoff distance for some specific networks.

We consider distributed optimization where N nodes in a generic, connected network minimize the sum of their individual, locally known, convex costs. Existing literature proposes distributed gradient-like methods that are attractive due to computationally cheap iterations and provable resilience to random inter-node communication failures, but such methods have slow theoretical and empirical convergence rates. Building from the centralized Nesterov gradient methods, we propose accelerated distributed gradient-like methods and establish that they achieve strictly faster rates than existing distributed methods. At the same time, our methods maintain cheap iterations and resilience to random communication failures. Specifically, for convex, differentiable local costs with Lipschitz continuous and bounded derivative, we establish (with respect to the cost function optimality) convergence in probability and convergence rates in expectation and in second moment.

This work considers the problem of locating a single source from noisy range measurements to a set of nodes in a wireless sensor network. We propose two new techniques that we designate as Source Localization with Nuclear Norm (SLNN) and Source Localization with l1-norm (SL-l1), which extend to arbitrary real dimensions our prior work on 2D source localization formulated in the complex plane. Our approach is based on formulating a Maximum-Likelihood (ML) estimation problem, and then using convex relaxation techniques to obtain a semidefinite program (SDP) that can be globally and efficiently solved. SLNN directly approximates the Gaussian ML solution, and the relaxation is shown to be tighter than in other methods in the same class. We present an analysis of the convexity properties of the constraint set for the 2D complex version of SLNN (SLCP) to justify the observed tightness of the relaxation. We propose the SL-l1algorithm to address the Laplacian noise case, which models the presence of outliers in range measurements. We overcome the non-differentiability of the Laplacian likelihood function by rewriting the ML problem as an exact weighted version of the Gaussian case. In terms of accuracy of localization, the proposed algorithms globally outperform state-of-the-art optimization-based methods in different noise scenarios, while exhibiting moderate computational complexity.

We propose a distributed, decentralized algorithm for solving separable optimization problems over a connected network of compute nodes. In a separable problem, each node has its own private function and its own private constraint set. Private means that no other node has access to it. The goal is to minimize the sum of all nodes' private functions, constraining the solution to be in the intersection of all the private sets. Our algorithm is based on the alternating direction method of multipliers (ADMM) and requires a coloring of the network to be available beforehand. We perform numerical experiments of the algorithm, applying it to compressed sensing problems. These show that the proposed algorithm requires in general less iterations, and hence less communication between nodes, than previous algorithms to achieve a given accuracy.

We address the problem of 6D pose estimation of a textureless and shiny object from single-view 2D images, for a bin-picking task. For a textureless object like a mechanical part, conventional visual feature matching usually fails due to the absence of rich texture features. Hierarchical template matching assumes that few templates can cover all object appearances. However, the appearance of a shiny object largely depends on its pose and illumination. Furthermore, in a bin-picking task, we must cope with partial occlusions, shadows, and inter-reflections.

This paper presents a unified approach to solve different bilinear factorization problems in computer vision in the presence of missing data in the measurements. The problem is formulated as a constrained optimization where one of the factors must lie on a specific manifold. To achieve this, we introduce an equivalent reformulation of the bilinear factorization problem that decouples the core bilinear aspect from the manifold specificity. We then tackle the resulting constrained optimization problem via Augmented Lagrange Multipliers. The strength and the novelty of our approach is that this framework can seamlessly handle different computer vision problems. The algorithm is such that only a projector onto the manifold constraint is needed. We present experiments and results for some popular factorization problems in computer vision such as rigid, non-rigid, and articulated Structure from Motion, photometric stereo, and 2D-3D non-rigid registration.

We show that distributed detection over random networks, or using a random protocol, e.g., of the gossip type, is asymptotically optimal, if the rate of information flow across the random network is large enough. Asymptotic optimality is in the sense of Chernoff information; in other words, we determine when the exponential rate of decay of the error probability for distributed detection is the best possible and equal to the rate of decay of the best centralized detector. The rate of information flow is defined by |log r|, where r is the second largest eigenvalue of the second moment of the random, consensus weight matrix. We quantify interesting tradeoffs in distributed detection, between the rate of information flow and the achievable detection performance.

Basis Pursuit (BP) finds a minimum ℓ1-norm vector z that satisfies the underdetermined linear system Mz = b, where the matrix M and vector b are given. Lately, BP has attracted attention because of its application in compressed sensing, where it is used to reconstruct signals by finding the sparsest solutions of linear systems. In this paper, we propose a distributed algorithm to solve BP. This means no central node is used for the processing and no node has access to all the data: the rows of M and the vector b are distributed over a set of interconnected compute nodes. A typical scenario is a sensor network. The novelty of our method is in using an optimal first-order method to solve an augmented Lagrangian-based reformulation of BP. We implemented our algorithm in a computer cluster, and show that it can solve problems that are too large to be stored in and processed by a single node.

This paper considers the problem of filter design with secrecy constraints, where two legitimate parties (Alice and Bob) communicate in the presence of an eavesdropper (Eve), over a Gaussian multiple-input multiple-output (MIMO) wiretap channel. This problem involves the design of transmit and receive filters which minimize the mean-square error (MSE) between the legitimate parties, whilst assuring that the eavesdropper MSE remains above a certain level. We characterize the form of the optimal transmit filter when both the legitimate receiver and the eavesdropper employ Zero-Forcing (ZF) filters. By capitalizing on the dual problem, we also show that the original matrix optimization problem can be reduced to a simple scalar optimization problem, whose solution can be readily computed by employing a simple bisection method. Numerical results illustrate the main conclusions.

This paper considers the problem of Alter design with secrecy constraints, where two legitimate parties, Alice and Bob, communicate in the presence of an eavesdropper, Eve, over multiple-input multiple-output (MIMO) Gaussian channels. In particular, we consider the design of transmit and receive filters that minimize the mean-squared error (MSE) between the legitimate parties subject to a certain eavesdropper MSE level, in the situation where the eavesdropper MIMO channel is a degraded version of the main MIMO channel. We characterize the form of the optimal receive filters as well as the form of optimal transmit filter in different scenarios. We also put forth an iterative algorithm to obtain the optimal values of the transmit and receive filters. Finally, we present a set of numerical results that illustrate the conclusions.

We consider the weight design problem for the consensus algorithm under a finite time horizon. We assume that the underlying network is random where the links fail at each iteration with certain probability and the link failures can be spatially correlated. We formulate a family of weight design criteria (objective functions) that minimize n, n = 1, …,N (out of N possible) largest (slowest) eigenvalues of the matrix that describes the mean squared consensus error dynamics. We show that the objective functions are convex; hence, globally optimal weights (with respect to the design criteria) can be efficiently obtained. Numerical examples on large scale, sparse random networks with spatially correlated link failures show that: 1) weights obtained according to our criteria lead to significantly faster convergence than the choices available in the literature; 2) different design criteria that corresponds to different n, exhibits very interesting tradeoffs: faster transient performance leads to slower long time run performance and vice versa. Thus, n is a valuable degree of freedom and can be appropriately selected for the given time horizon.

This paper addresses the problem of locating a single source from noisy range measurements in wireless sensor networks. An approximate solution to the maximum likelihood location estimation problem is proposed, by redefining the problem in the complex plane and relaxing the minimization problem into semidefinite programming form. Existing methods solve the source localization problem either by minimizing the maximum likelihood function iteratively or exploiting other semidefinite programming relaxations. In addition, using squared range measurements, exact and approximate least squares solutions can be calculated. Our relaxation for source localization in the complex plane (SLCP) is motivated by the near-convexity of the objective function and constraints in the complex formulation of the original (non-relaxed) problem. Simulation results indicate that the SLCP algorithm outperforms existing methods in terms of accuracy, particularly in the presence of outliers and when the number of anchors is larger than three.

This paper deals with the reconstruction of smooth, flexible, flat surfaces isometrically embedded in 3D when observed from uncalibrated camera images. It provides a cost function which captures the global geometry of the problem. The input data consists of a sequence of matched feature points (i.e. known correspondence), but the algorithm allows for missing data so that all points do not need to be visible in all images. Its performance under these conditions is analyzed. A way of inferring the bend radius at each point is provided which can be used back in the cost function, providing a higher order fitting. The performance is validated using both synthetic data and real image examples, where the reconstruction is compared to ground truth and reference points.

This paper describes a new algorithm for recovering the 3D shape and motion of deformable and articulated objects purely from uncalibrated 2D image measurements using an iterative factorization approach. Most solutions to non-rigid and articulated structure from motion require metric constraints to be enforced on the motion matrix to solve for the transformation that upgrades the solution to metric space. While in the case of rigid structure the metric upgrade step is simple since the motion constraints are linear, deformability in the shape introduces non-linearities. In this paper we propose an alternating least-squares approach associated with a globally optimal projection step onto the manifold of metric constraints. An important advantage of this new algorithm is its ability to handle missing data which becomes crucial when dealing with real video sequences with self-occlusions. We show successful results of our algorithms on synthetic and real sequences of both deformable and articulated data.

This paper deals with the reconstruction of smooth, flexible, isometrically embedded flat surfaces in 3D, such as a sheet of paper or a flag waving in the wind, from a set of 2D projected observations such as camera images. To solve the problem, a set of matched features of the waving object at different poses is needed, which are then applied to the reconstruction algorithm here described. The complete algorithm consists of 2 phases, the first obtaining an initial approximation from local features, the second uses this result to iterate a global cost function, trying to achieve a better estimate. To validate the algorithm, synthetic data with noise is generated, reconstructed and compared to ground truth data. Also, a second experiment consisting of real images of a sheet of paper is shown.

We present an algorithm able to register a known 3D deformable model to a set of 2D matched points extracted from a single image. Unlike previous approaches, the problem is solved simultaneously for both the rigid and non-rigid parameters of the model. The key advantage of our approach is the projection of an initial affine estimation of the motion parameters into the motion manifold corresponding to the exact parametrization of the problem. This projection is formulated as theminimization of the distance between the affine solution and the surface of the manifold. Such optimization results in a quadratically constrained quadratic minimization problem that can be efficiently solved with standard optimization tools. Synthetic and real tests demonstrate the effectiveness of the approach.

From the recovery of structure from motion to the separation of style and content, many problems in computer vision have been successfully approached by using bilinear models. The reason for the success of these models is that a globally optimal decomposition is easily obtained from the singular value decomposition (SVD) of the observation matrix. However, in practice, the observation matrix is often incomplete, the SVD can not be used, and only suboptimal solutions are available. The majority of these solutions are based on iterative local refinements of a given cost function, and lack any guarantee of convergence to the global optimum. In this paper, we propose a globally optimal solution, for particular patterns of missing entries. To achieve this goal, we re-formulate the problem as the minimization of the spectral norm of the matrix of residuals, i.e., we seek the completion of the observation matrix such that the largest singular value of its difference to a low rank matrix is the smallest possible. The class of patterns of missing entries we deal with is known as the Young diagram, which includes, as particular cases, many relevant situations, such as the missing of an entire submatrix. We describe experiments that illustrate how our globally optimal solution has impact in practice.

Many applications require a computer representation of 2D shape, usually described by a set of 2D points. The challenge of this representation is that it must not only capture the characteristics of the shape but also be invariant to relevant transformations. Invariance to geometric transformations, such as translation, rotation and scale, has received attention in the past, usually under the assumption that the points are previously labeled, i.e., that the shape is characterized by an ordered set of landmarks. However, in many practical scenarios the landmarks are obtained from an automatic process, e.g., edge/corner detection, thus without natural ordering. In this paper, we represent 2D shapes in a way that is invariant to the permutation of the landmarks. Within our framework, a shape is mapped to an analytic function on the complex plane, leading to what we call its analytic signature (ANSIG). We show that different shapes lead to different ANSIGs but that shapes that differ by a permutation of the landmarks lead to the same ANSIG, i.e., that our representation is a maximal invariant with respect to the permutation group. To store an ANSIG, it suffices to sample it along a closed contour in the complex plane. We further show how easy it is to factor out geometric transformations when comparing shapes using the ANSIG representation. We illustrate the ANSIG capabilities in shape-based image classification.

Widely used SVD-based matrix factorization approaches to the recovery of 3D rigid structure from motion (SFM), require a set of feature points to be visible in a set of images. When there is occlusion, several feature points disappear, the observation matrix misses some entries, there is not equivalent to the SVD, and only suboptimal solutions have been proposed to exploit rigidity. In this paper, we propose a method to complete the trajectories that correspond to a rigid scene, in an optimal way. Our algorithm is not iterative (thus avoiding problems like sensitivity to initialization and local optima); it rather computes in a finite number of steps the globally optimal completion of the observation matrix. We describe experiments that illustrate the gain in accuracy of SFM.

In this paper we address two closely related problems. The first is the object detection problem, i.e., the automatic decision of whether a given image represents a known object or not. The second is the correspondence problem, i.e., the automatic matching of points of an object in two views. In the first problem, we assume object rigidity and model the distortions by a linear shape model. To solve the decision problem, we derive the uniformly most powerful (UMP) hypothesis test that is invariant to the linear shape model. We use the UMP statistic to formulate the correspondence problem in a model invariant framework. We show that it is equivalent to a quadratic maximization on the space of permutation matrices. We derive LASIC, an iterative computationally feasible solution to the quadratic maximization problem for the particular case where the linear shape model is the affine model. Simulations benchmark LASIC against two standard algorithms.

We address two-dimensional shape-based classification, considering shapes described by arbitrary sets of unlabeled points, or landmarks. This is relevant in practice because, in many applications, the points describing the shapes come from automatic processes, e.g., edge detection, thus without labels. Rather than attempting to compute point correspondences (a quagmire, when dealing with nontrivial shapes), we use what we call the analytic signature (ANSIG) of the shapes, a representation that has the key feature of being invariant to point labeling. Geometric transformations, such as translation, rotation, and scale, and different cardinality of point sets, are also dealt with by this representation. We demonstrate the capabilities of our representation with several shape classification experiments.

We investigate the non-coherent single-user MIMO channel in the low signal-to-noise (SNR) regime from two viewpoints: capacity and probability of error analysis. The novelty in both viewpoints is that an arbitrary correlation structure is allowed for the Gaussian observation noise. First, we look at the capacity of the spatially correlated Rayleigh fading channel. We investigate the impact of channel and noise correlation on the mutual information for the on-off and Gaussian signaling schemes. Our results establish that, in the low SNR regime, mutual information is maximized when the transmit antennas are fully correlated (the same holds for the receive array). Then, we consider the deterministic channel setup and perform a pairwise error probability (PEP) analysis for the GLRT receiver. This leads to a codebook design criterion on which we base the construction of new space-time constellations. Their performance is assessed by computer simulations and, as a byproduct, we show that our codebooks are also of interest for Bayesian receivers which decode constellations with non-uniform priors.

This paper addresses the problem of estimating the position and attitude of a rigid body when the available measurements consist only of distances (or ranges) between a set of body fixed beacons and a set of earth fixed landmarks. To this effect, a maximum likelihood (ML) estimator is derived by solving an optimization problem on the special Euclidean group SE(n);n = 2,3 using intrinsic gradient and Newton-like algorithms. The theoretical tools used borrow from optimization theory on Riemannian manifolds. Supported by recent results on performance bounds for estimators on Riemannian manifolds, the intrinsic variance lower bound (IVLB) is derived for the problem at hand. Simulation results are presented to illustrate the estimator performance and to validate the tightness of the IVLB in a wide range of signal to noise ratio scenarios

We address the problem of space-time codebook design for non-coherent communications in multiple-antenna wireless systems. The channel matrix is assumed deterministic (no stochastic model assumed) and unknown at both the receiver and the transmitter. In contrast with other approaches, the Gaussian observation noise has an arbitrary correlation structure, known by the transmitter and the receiver. To handle the unknown deterministic space-time channel, a GLRT receiver is implemented. We propose a new methodology for space-time codebook design under this non-coherent setup. This optimizes the probability of error of the receiver's detector in the high SNR regime, thus solving a high-dimensional nonlinear non-smooth optimization problem in a two-step approach: (i) firstly, a convex SDP relaxation yields a rough estimate of the optimal codebook; (ii) this is then refined through a geodesic descent optimization algorithm that exploits the Riemannian geometry imposed by the power constraints on the space-time codewords. Computer simulations demonstrate that, for the specific case of spatio-temporal white observation noise, our codebooks are marginally better than those provided by state-of-art known solutions. However, the most relevant conclusion is that, for correlated noise environments, our method provides codes that significantly outperform other known codes

We address the problem of computing the Riemannian centroid of a constellation of points in a naturally reductive homogeneous manifold. We note that many interesting manifolds used in engineering (such as the special orthogonal group, Grassman, sphere, positive definite matrices) possess this structure. We develop an intrinsic Newton scheme for the centroid computation. This is achieved by exploiting a formula that we introduce for obtaining the Hessian of the squared Riemannian distance on naturally reductive homogeneous spaces. Some results of finding the centroid of a constellation of points in these spaces are presented, which evidence the quadratic convergence of the Newton method derived herein. These computer simulation results show that, as expected, the Newton method has a faster convergence rate than the usual gradient-based approaches

The autoregressive (AR) process is fundamental to linear signal processing and is commonly used to model the behaviour of an object evolving on Euclidean space. In real life, there are myriad examples of objects evolving not on flat spaces but on curved spaces such as the surface of a sphere. For instance, wind-direction studies in meteorology and the estimation of relative rotations of tectonic plates based on observations on the Earth's surface deal with spherical data, while subspace tracking in signal processing is actually inference on the Grassmann manifold. This paper considers how to extend the AR process to one evolving on a curved space, or in a general, a manifold. Doing so is non-trivial, and in fact, several different extensions are proposed, along with their advantages and disadvantages. Algorithms for estimating the parameters of these generalized AR processes are derived

We propose a solution to the n-frame correspondence problem under the factorization framework. During the matching process, our algorithm takes explicitly into account the geometrical constraints associated to the reconstruction process. To this end, a rank constraint is imposed on the measurement matrix. Since our method relies solely on geometric constraints, it is not dependent on corners as image features and can consequently match generic points (e.g. contours). Outlier rejection is integrated as a part of the actual matching process. In general the problem is formulated as combinatorial, but we develop a method, which can provide a solution with a low computational complexity. Because of this, our algorithm is able to handle high-dimensional matching problems that are common in real-life applications.

We consider parametric statistical models in which the parameter space /spl theta/ is a connected Riemannian manifold. This mathematical structure on the parameter space subsumes, as special cases, submanifolds of Euclidean spaces appearing in parametric estimation scenarios with a priori smooth deterministic constraints, and quotient spaces (such as Grassmann manifolds) which arise in certain parametric estimation scenarios with ambiguities. The Riemannian structure on the parameter space /spl theta/ turns it into a metric space and the associated Riemannian distance is used here to quantify estimation errors. We present the intrinsic variance lower bound (IVLB) which places a lower limit on the accuracy, measured in terms of the mean-square Riemannian distance, of unbiased estimators taking values in /spl theta/. The IVLB depends both on the curvature of the parameter space and a coordinate-free extension of the well-known Fisher information matrix (FIN). We show that for flat Euclidean spaces, the IVLB collapses to the Cramer-Rao bound (CRB). In this sense, we may interpret the IVLB as a generalization of the CRB for curved parameter spaces. Computer simulations illustrating the application of the IVLB are included.

A feature detection system has been developed for real-time identification of lines, circles and people legs from laser range data. A new method suitable for arc/circle detection is proposed: the Inscribed Angle Variance (IAV). Lines are detected using a recursive line fitting method. The people leg detection is based on geometrical relations. The system was implemented as a plugin driver in Player, a mobile robot server. Real results are presented to verify the effectiveness of the proposed algorithms in indoor environment with moving objects.

Many parametric statistical models suffer from "intrinsic ambiguities" in the sense that the distribution of the observation vector is invariant to smooth, structured changes in the model's parameters. The fact that certain members of the parametric statistical family are locally indistinguishable makes the Fisher information matrix (FIM) associated with the given statistical model singular. We examine such degenerate deterministic parameter estimation problems from a Riemannian geometric perspective. We start by replacing the original (ambiguous) parameter set by a lower-dimensional Riemannian (non-ambiguous) parameter set. The new parameter set comes in the form of a quotient space and is obtained by identifying equivalent family members in the initial parameterization. We specialize recently developed extensions of the Cramer-Rao bound (CRB) for the Riemannian setup to this particular setting. This offers a re-interpretation of the CRB inequality involving the pseudo-inverse of the FIM. Also, we present a lower bound on the variance (computed with respect to the geodesic distance) of unbiased estimators taking values in the quotient space. Geometrically, this corresponds to a fundamental limit on the capability of these estimators in discriminating adjacent parameter equivalence classes in the original problem parameterization. A numerical example involving the blind identification of single-input multiple-output (SIMO) channels driven by a Gaussian source is worked out.

We study signal reconstruction with the aid of second order statistics (SOS) in some Bayesian multi-user setups. Under a Bayesian framework, the unknown mixing channel matrix is a random object. Furthermore, it is well known that the SOS of the received data can solve the channel matrix up to an orthogonal factor, which now becomes a random object. That is, the prior for the original channel matrix contracts to another prior for the residual orthogonal mixing matrix. This paper shows how to exploit this new prior over the orthogonal group in two different applications: blind separation of binary users and semi-blind channel identification. Computer simulations assess the benefits of our proposed solutions over some classical "textbook" solutions.

In this paper we tackle the joint source symbol detection and multi-channel acquisition problem in the context of wireless digital flat-fading links with space diversity. We propose a simple parametric statistical channel model which decouples the time dynamics of the channel vector in amplitude and direction. We implement the corresponding maximum a posteriori (MAP) joint estimator for the amplitude and direction of the channel vector and the source information symbols. We derive a computationally attractive iterative scheme to solve the optimization problem associated with the MAP estimator. This scheme is based on differential-geometric concepts and fully exploits the curvature of the surface constraint. Preliminary computer simulations assessing the good performance of our proposed solutions are included.

We consider statistical models parameterized over connected Riemannian manifolds. We present a lower bound on the mean-square distance of unbiased estimators about their mean values. The derived bound depends both on the curvature of the parameter manifold and a coordinate-free extension of the classical Fisher information matrix. Our study can be applied in estimation problems with smooth parametric constraints, and in statistical models indexed over coset spaces. Illustrative examples concerning inferences on the unit-sphere and the complex projective space are worked out.

The CFC/sub 2/ algorithm is a recently introduced analytical solution for the blind MIMO channel identification problem, provided a certain spectral diversity holds for the stochastic inputs of the MIMO system. Here, we develop a theoretical study to derive the asymptotic performance of the CFC/sub 2/ algorithm, in terms of mean-square error. Asymptotic normality of the MIMO channel estimate is proved, and the asymptotic error covariance matrix derived. Computer simulation results are included to validate the theoretical expressions.

We introduce a new technique to separate a linear convolutive mixture of discrete-time sources, emitting uncorrelated data samples. The proposed approach works in the reduced dimension space of the (channel) whitened data samples, where the data matrix is highly structured: it is the product of an orthogonal and generalized Toeplitz matrices embedded in additive Gaussian noise. By itself, this factorization does not unambiguously determine the sources (even in the absence of noise), but uniqueness is restored if short fragments (inadequate for training) of the emitted messages are known beforehand. We present a locally convergent iterative algorithm which implements the joint maximum likelihood (ML) estimator of both the orthogonal mixing matrix and the user signals, subject to the known side information. We also discuss a simple (sub-optimal) adaptation of the proposed algorithm to the class of finite-alphabet (FA) sources. Computer simulations show that the proposed technique outperforms an alternative subspace approach, specially in low signal-to-noise ratio (SNR) scenarios.

We study blind equalization of noisy MIMO-FIR systems driven by white sources. We present a new second order statistics (SOS) based approach which does not require the knowledge of the channel order. This technique blindly transforms a convolutive mixture of users into an instantaneous one. Thus, in the special case of a single user (SIMO systems), an estimate of the input signal is readily obtained. Computer simulations results illustrate the promising performance of the proposed technique. We compare our method with the multistep prediction (MSP) approach (in the context of SIMO systems), and evaluate the algorithm capability in globally nulling the intersymbol interference (ISI) for MIMO systems.

We present a closed-form algorithm for blind identification of multiple-input/multiple-output (MIMO) finite-impulse response (FIR) systems driven by digital sources. The algorithm is based on second-order statistics and yields an asymptotically exact estimate of the MIMO channel. We assign distinct spectral signatures to each user through transmitter correlative filters, and exploit this spectral asymmetry to derive the closed-form solution. Simulation results illustrate the good performance of the proposed approach. We compare the mean-square error (MSE) of the MIMO channel estimate against the Cramer-Rao bound, and assess the algorithm capability in rejecting inter-user crosstalk interference.

We address the problem of blind identification of multiuser multiple-input multiple-output (MIMO) finite-impulse response (FIR) digital systems. This problem arises in spatial division multiple access (SDMA) architectures for wireless communications. We present a closed-form, i.e., noniterative, consistent estimator for the MIMO channel based only on second-order statistics. To obtain this closed form we introduce spectral/correlation asymmetry between the sources by filtering each source output with adequate correlative filters. Our algorithm uses the closed form MIMO channel estimate to cancel the intersymbol interference (ISI) due to multipath propagation and to discriminate between the sources at the wireless base station receiver. Simulation results show that, for single-user channels, this technique yields better channel estimates in terms of mean-square error (MSE) and better probability of error than a well-known alternative method. Finally, we illustrate its performance for MIMO channels in the context of the global system for mobile communications (GSM) system.

We address the problem of the blind reconstruction of binary sequences transmitted by P sources, sharing simultaneously the same frequency channel. This is the situation in space division multiple access (SDMA) systems, where an array receiver is used to discriminate the spatial signatures of the sources. We exploit the geometry induced by the finite nature of the sources alphabet in order to derive a blind multichannel inversion algorithm. This algorithm minimizes a generalized constant modulus (GCM) cost function, subject to the constraints imposed by the signal space geometry (geometric priors). We define the GCM cost function for the case of additive Gaussian noise channels. We show that, except for a sign ambiguity, the minimization of the GCM criterion has a unique solution. Therefore, convergence to the global minimum is guaranteed. Computer simulations illustrate the performance achieved with the proposed method.