We consider a situation where the state of a system is represented by a real-valued vector. Under normal circumstances, the vector is zero, while an event manifests as non-zero entries in this vector, possibly few. Our interest is in the design of algorithms that can reliably detect events (i.e., test whether the vector is zero or not) with the least amount of information. We place ourselves in a situation, now common in the signal processing literature, where information about the vector comes in the form of noisy linear measurements. We derive information bounds in an active learning setup and exhibit some simple near-optimal algorithms. In particular, our results show that the task of detection within this setting is at once much easier, simpler and different than the tasks of estimation and support recovery.
We mention that the present paper may be seen as a companion paper to @cite_0 which considers the tasks of estimation and support recovery in the same setting.
@cite_0
182164380 2130979540 1986223934 1999136078
In this paper, we study the problem of one-bit compressed sensing (1-bit CS), where the goal is to design a measurement matrix A and a recovery algorithm such that a k-sparse unit vector x* can be efficiently recovered from the sign of its linear measurements, i.e., b = sign(Ax*). This is an important problem for signal acquisition and has several learning applications as well, e.g., multi-label classification (, 2009). We study this problem in two settings: a) support recovery: recover the support of x*, b) approximate vector recovery: recover a unit vector x such that ||x - x*||2 = e. For support recovery, we propose two novel and efficient solutions based on two combinatorial structures: union free families of sets and expanders. In contrast to existing methods for support recovery, our methods are universal i.e. a single measurement matrix A can recover all the signals. For approximate recovery, we propose the first method to recover a sparse vector using a near optimal number of measurements. We also empirically validate our algorithms and demonstrate that our algorithms recover the true signal using fewer measurements than the existing methods. We consider the approximate sparse recovery problem, where the goal is to (approximately) recover a high-dimensional vector xisinRopfn from its lower-dimensional sketch AxisinRopfm. Specifically, we focus on the sparse recovery problem in the L1 norm: for a parameter k, given the sketch Ax, compute an approximation xcirc of x such that the L1 approximation error parx-xcircpar1 is close to minx' parx-x'par1, where x' ranges over all vectors with at most k terms. The sparse recovery problem has been subject to extensive research over the last few years. Many solutions to this problem have been discovered, achieving different trade-offs between various attributes, such as the sketch length, encoding and recovery times. In this paper we provide a sparse recovery scheme which achieves close to optimal performance on virtually all attributes (see Figure 1). In particular, this is the first recovery scheme that guarantees O(k log(n k)) sketch length, and near-linear O(n log (n k)) recovery time simultaneously. It also features low encoding and update times, and is noise-resilient. In the context of reverse-engineering project we designed a use-case specification recovery technique for legacy information systems. With our technique, we can recover the alternative flows of each use-case of the system. It is based on a dynamic (i.e. runtime) analysis of the working of the system using execution traces. But "traditional" execution trace format do not contain enough information for this approach to work. Then we designed a new execution trace format together with the associated tool to get the program's dynamic decision tree corresponding to each of the use-case scenario. These trees are then processed to find the possible variants from the main scenario of each use-case. In this paper we first present our approach to the use-case specification recovery technique and the new trace format we designed. Then the decision tree compression technique is showed with a feasibility study. The contribution of the paper is our approach to the recovery of legacy systems' use-case, the new trace format and the decision tree processing technique. Robust tensor recovery plays an instrumental role in robustifying tensor decompositions for multilinear data analysis against outliers, gross corruptions, and missing values and has a diverse array of applications. In this paper, we study the problem of robust low-rank tensor recovery in a convex optimization framework, drawing upon recent advances in robust principal component analysis and tensor completion. We propose tailored optimization algorithms with global convergence guarantees for solving both the constrained and the Lagrangian formulations of the problem. These algorithms are based on the highly efficient alternating direction augmented Lagrangian and accelerated proximal gradient methods. We also propose a nonconvex model that can often improve the recovery results from the convex models. We investigate the empirical recoverability properties of the convex and nonconvex formulations and compare the computational performance of the algorithms on simulated data. We demonstrate through a number o...