We investigate the two primary categories of structured recovery problems, namely Compressed Sensing (CS) and Low Rank Recovery (LRR). Our focus is on the performance analysis of their two tightest convex relaxation based heuristics, the so-called$\ell _{1}$and the nuclear norm ($\ell _{1}^{*}$) minimizations. We examine two standard types of phase transitions (PTs): 1) general PT, obtained by enforcing sparsity as a fundamental form of structuring, and 2) nonnegative PT, achieved by imposing nonnegativity as an additional form of structuring alongside sparsity. We establish explicit relations between the CS and LRR PTs. Our analysis reveals that the nonnegative PT essentially interpolates between the general and the binary CS PT, in a manner that can be explicitly characterized. Quite surprisingly, although the phase transitions themselves admit fairly complicated mathematical formulations, their relations can be expressed in a very neat and elegant way. This ultimately allows to quickly assess and compare the effects additional presence/absence of the nonnegativity has on$\ell _{1}$and$\ell _{1}^{*}$.

In this paper, we establish a rigorous connection between Causal inference (C-inf) and the low-rank recovery (LRR). Using Random Duality Theory (RDT) and novel mathematical strategies related to free probability theory, we obtain the exact explicit typical (and achievable) worst case phase transitions (PT). These PT precisely separate scenarios where causal inference via LRR is possible from those where it is not. We supplement our mathematical analysis with numerical experiments that confirm the theoretical predictions of PT phenomena, and further show that the two closely match for fairly small sample sizes. We obtain simple closed form representations for the resulting PTs, which highlight direct relations between the low rankness of the target C-inf matrix and the time of the treatment. Hence, our results can be used to determine the range of C-inf’s typical applicability.

In this paper we revisit two groups of the most famous structured recovery problems, the compressed sensing (CS) and the low-rank recovery (LRR). Our particular focus is on two tightest convex relaxation based heuristics, the so-called ℓ1and the nuclear norm $\left({l_1^ * }\right)$ minimizations, and their accompanying phase transitions (PT) phenomena. For concreteness, we consider two standard types of PTs, the first obtained by imposing sparsity and the second obtained by additionally imposing nonnegativity. We establish an explicit relation between these two types of PTs in CS, and the two corresponding types of PTs in the LRR. We also uncover a connection between PTs in CS and LRR. Our results make it possible to compare in a mathematically rigorous way both the level of heuristics’ successes and the effects that different types of structuring can have on the recovery process. Quite surprisingly, although the phase transitions themselves admit fairly complicated mathematical formulations, their relations can be expressed in a very neat and elegant way which allows for a quick assessment of the heuristic approximation quality.

We propose a novel calibration methodology based on the maximum likelihood estimator to recover the parameters of a structural model of credit risk which accounts for potential reporting bias. Such bias is introduced by the managers and it is unobserved by outsider investors which can only estimate it. The calibration is performed using a combination of balance sheet, financial indicators and market prices of equities. We apply the calibration algorithm to Tyco, a real case of reporting bias in the United States history. We show that the calibrated model is able to predict the market stock price with a high degree of accuracy.

We start discussing the main components of credit risk frameworks which require to model default probability, loss given default and their product which generates the credit spreads. We discuss how credit spreads are related to default risk. We review the main approaches to credit risk modeling including structural frameworks, intensity based methods and models with incomplete information which combine the best features of the previous two approaches.

We first show that for any bipartite graph H with at most five vertices there exists an on-line competitive algorithm for the class of H-free bipartite graphs. We then analyze the performance of an on-line algorithm for coloring bipartite graphs on various subfamilies. The algorithm yields new upper bounds for the on-line chromatic number of bipartite graphs. We prove that the algorithm is on-line competitive for $P_7$-free bipartite graphs, i.e., that do not contain an induced path on seven vertices. The number of colors used by the on-line algorithm for $P_6$-free and $P_7$-free bipartite graphs is, respectively, bounded by roughly twice and roughly eight times the on-line chromatic number. In contrast, it is known that there exists no competitive on-line algorithm to color $P_6$-free (or $P_7$-free) bipartite graphs, i.e., for which the number of colors is bounded by any function depending only on the chromatic number.

In this paper we investigate the k-connectivity threshold of distributed dense ad hoc heterogeneous wireless sensor network architecture. We consider the situation when sensors are deployed in the surveillance area according to a uniform distribution perturbed by a Gaussian noise. We derive analytically the minimum detection range which guarantees an emerging structure in the network, namely the connectivity, which becomes larger and larger as the number of sensors in the network increase. This allows the target track to be propagated almost surely throughout the network using the minimum possible amount of prime energy. We report the results of some simulation experiments which further support the theoretical results.

This paper compares two different approaches for sensor selection for distributed tracking: 1) The Frisbee method, and 2) Global Node Selection (GNS). The Frisbee method is based on the proximity of the nodes to the predicted location of the target; GNS is based on minimizing the unbiased Cramer Rao lower bound (CRLB). Both theoretical and experimental results indicate that the Frisbee method is as effective as GNS. Furthermore, the Frisbee method is attractive due to its very light computational load.