We consider the problem of testing whether a complex-valued vector random sequence is proper. Past work on this problem is limited to a sequence of independent Gaussian random vectors whereas we allow an arbitrary stationary vector sequence that can be non-Gaussian. A binary hypothesis testing approach is formulated and a generalized likelihood ratio test (GLRT) is derived using the power spectral density estimator of an augmented sequence. An asymptotic analytical solution for calculating the test threshold is provided. The results are illustrated via simulations.

We consider the recovery of a low rank real-valued matrix $M$ given a subset of noisy discrete (or quantized) measurements. Such problems arise in several applications such as collaborative filtering, learning and content analytics, and sensor network localization. We consider constrained maximum likelihood estimation of $M$, under a constraint on the entry- wise infinity-norm of $M$ and an exact rank constraint. We provide upper bounds on the Frobenius norm of matrix estimation error under this model. Previous theoretical investigations have focused on binary (1-bit) quantizers, and been based on convex relaxation of the rank. Compared to the existing binary results, our performance upper bound has faster convergence rate with matrix dimensions when the fraction of revealed observations is fixed. We also propose a globally convergent optimization algorithm based on low rank factorization of $M$ and validate the method on synthetic and real data, with improved performance over previous methods.

We consider the problem of determining the location of sensor nodes in a wireless sensor ad hoc network when only connectivity information is available, i.e., one only knows if a pair of nodes is within a fixed radio range distance of each other but does not have access to exact or even approximate distance information. We propose a maximum likelihood based reconstruction algorithm to reconstruct the node positions in a d-dimensional Euclidean space. For an n-node network, the constrained maximum likelihood estimation problem is non-convex in both the d × n node position matrix X and in the Gram matrix QX=XTX. We derive an upperbound on the average Frobenius norm of the estimation error in Q, which is of the order of the reciprocal of square root of n for a fixed radio range. We present a set of algorithms for finding the maximum likelihood estimate of X by first embedding d ×n X into m × n Y, factorizing QY= YTY, d ≤ m ≤ n, and then optimizing in Y. We relate local minima in Y to the global minimum of a relaxed convex formulation in QYto provide global convergence guarantees, despite the nonconvexity of the negative log-likelihood function in Y. We demonstrate that our algorithm is empirically successful for both uniform and irregular networks, using only a few anchor nodes. Finally, numerical experiments demonstrate improved performance of the proposed algorithm relative to the MDS algorithm and a variant.

In this paper, we consider the recovery of a low rank matrix M given a subset of noisy quantized (or ordinal) measurements. We consider a constrained maximum likelihood estimation of M, under a constraint on the entry-wise infinity-norm of M and an exact rank constraint. We provide an upper bound on the Frobenius norm of the matrix estimation error under this model. Past theoretical investigations have been restricted to binary quantizers, and based on convex relaxation of the rank. We propose a globally convergent optimization algorithm exploiting existing work on low rank matrix factorization, and validate the method on synthetic data, with improved performance over past methods.

We consider the problem of determining the location of sensor nodes in a wireless sensor ad hoc network when only connectivity information is available, i.e., one only knows if a pair of nodes are within a fixed radio range (distance) of each other but does not have access to exact or even approximate distance information. We propose a maximum likelihood based reconstruction algorithm to reconstruct the node positions in a d-dimensional Euclidean space. The original problem formulation results in constrained minimization of a negative log-likelihood with respect to n × n symmetric positive semidefinite matrices Q for an n-node network, and is a convex optimization problem. We recast this problem into a non-convex optimization problem with respect to X where Q = XTX. We link our algorithm to existing results which confirm that the nonconvex optimization problem with m × n X provides a global solution to its convex equivalent for certain choices of m. We demonstrate that our algorithm is empirically successful for both uniform and irregular networks, using only a few anchor nodes. Finally, numerical experiments demonstrate improved performance of the proposed algorithm relative to the MDS algorithm and a variant.

Unsupervised feature learning has been shown to be effective at learning representations that perform well on image, video and audio classification. However, many existing feature learning algorithms are hard to use and require extensive hyperparameter tuning. In this work, we present sparse filtering, a simple new algorithm which is efficient and only has one hyperparameter, the number of features to learn. In contrast to most other feature learning methods, sparse filtering does not explicitly attempt to construct a model of the data distribution. Instead, it optimizes a simple cost function -- the sparsity of L2-normalized features -- which can easily be implemented in a few lines of MATLAB code. Sparse filtering scales gracefully to handle high-dimensional inputs, and can also be used to learn meaningful features in additional layers with greedy layer-wise stacking. We evaluate sparse filtering on natural images, object classification (STL-10), and phone classification (TIMIT), and show that our method works well on a range of different modalities.