Learning without the Phase: Regularized PhaseMax Achieves Optimal Sample Complexity

The problem of estimating an unknown signal, x_0 ϵ R^n, from a vector y ϵ R^m consisting of m magnitude-only measurements of the form y_i = |a_ix_o|, where a_i’s are the rows of a known measurement matrix A is a classical problem known as phase retrieval. This problem arises when measuring the phase is costly or altogether infeasible. In many applications in machine learning, signal processing, statistics, etc., the underlying signal has certain structure (sparse, low-rank, finite alphabet, etc.), opening of up the possibility of recovering x_0 from a number of measurements smaller than the ambient dimension, i.e., m < n. Ideally, one would like to recover the signal from a number of phaseless measurements that is on the order of the "degrees of freedom" of the structured x_0. To this end, inspired by the PhaseMax algorithm, we formulate a convex optimization problem, where the objective function relies on an initial estimate of the true signal and also includes an additive regularization term to encourage structure. The new formulation is referred to as regularized PhaseMax. We analyze the performance of regularized PhaseMax to find the minimum number of phaseless measurements required for perfect signal recovery. The results are asymptotic and are in terms of the geometrical properties (such as the Gaussian width) of certain convex cones. When the measurement matrix has i.i.d. Gaussian entries, we show that our proposed method is indeed order-wise optimal, allowing perfect recovery from a number of phaseless measurements that is only a constant factor away from the degrees of freedom. We explicitly compute this constant factor, in terms of the quality of the initial estimate, by deriving the exact phase transition. The theory well matches empirical results from numerical simulations

The Performance Analysis of Generalized Margin Maximizer (GMM) on Separable Data

Logistic models are commonly used for binary classification tasks. The success of such models has often been attributed to their connection to maximum-likelihood estimators. It has been shown that gradient descent algorithm, when applied on the logistic loss, converges to the max-margin classifier (a.k.a. hard-margin SVM). The performance of the max-margin classifier has been recently analyzed. Inspired by these results, in this paper, we present and study a more general setting, where the underlying parameters of the logistic model possess certain structures (sparse, block-sparse, low-rank, etc.) and introduce a more general framework (which is referred to as "Generalized Margin Maximizer", GMM). While classical max-margin classifiers minimize the 2-norm of the parameter vector subject to linearly separating the data, GMM minimizes any arbitrary convex function of the parameter vector. We provide a precise analysis of the performance of GMM via the solution of a system of nonlinear equations. We also provide a detailed study for three special cases: (1) ℓ₂-GMM that is the max-margin classifier, (2) ℓ₁-GMM which encourages sparsity, and (3) ℓ_∞-GMM which is often used when the parameter vector has binary entries. Our theoretical results are validated by extensive simulation results across a range of parameter values, problem instances, and model structures

The Performance Analysis of Generalized Margin Maximizer (GMM) on Separable Data

Logistic models are commonly used for binary classification tasks. The success of such models has often been attributed to their connection to maximum-likelihood estimators. It has been shown that gradient descent algorithm, when applied on the logistic loss, converges to the max-margin classifier (a.k.a. hard-margin SVM). The performance of the max-margin classifier has been recently analyzed. Inspired by these results, in this paper, we present and study a more general setting, where the underlying parameters of the logistic model possess certain structures (sparse, block-sparse, low-rank, etc.) and introduce a more general framework (which is referred to as "Generalized Margin Maximizer", GMM). While classical max-margin classifiers minimize the $2$-norm of the parameter vector subject to linearly separating the data, GMM minimizes any arbitrary convex function of the parameter vector. We provide a precise analysis of the performance of GMM via the solution of a system of nonlinear equations. We also provide a detailed study for three special cases: ($1$) $\ell_2$-GMM that is the max-margin classifier, ($2$) $\ell_1$-GMM which encourages sparsity, and ($3$) $\ell_{\infty}$-GMM which is often used when the parameter vector has binary entries. Our theoretical results are validated by extensive simulation results across a range of parameter values, problem instances, and model structures.Comment: ICML 2020 (submitted February 2020

Sensing and transmission strategies in wireless cognitive radio systems

The main challenge in any cognitive radio system is to maximize secondary users throughput while limiting interference imposed on licensed users. In this regard, finding the optimal sensing and transmission timing strategies and accurate sensing techniques are of great importance in a cognitive radio network. In this thesis, we study a sensing-transmission scheme for secondary user in a cognitive radio system where the secondary user senses every primary channel independently and transmits a signal for a fixed duration if it finds the channel empty and stays idle for another fixed duration if it senses the channel busy. We obtain optimal idle and transmission durations which maximize access opportunity of the secondary user while keeping the interference ratios on the primary channels below some thresholds. Our results show that unless we have an error free perfect channel sensing, adding the idle duration improves the performance of the system. We also study a cooperative spectrum sensing scheme for cognitive radio systems where each sensor transmits multi-bit quantized information to a fusion center where the decision about the availability or occupancy of the channel is made. We compare iii the performance of our proposed multi-bit combining scheme with hard and soft combining schemes and show that with transmission of a few bits of information from each sensor , the system can achieve an error rate very close to the optimal soft combining scheme

COMPARISON OF SERUM VITAMIN A LEVELS BETWEEN NEONATES WITH CONGENITAL HEART DISEASE AND CONTROLS

Objective: Prevention of congenital heart disease (CHD) has been hampered by a lack of information about the known modifiable risk factors for abnormalities in cardiac development. Vitamin A plays an important role in the periods of rapid cellular growth and differentiation, especially during pregnancy. Assuming a link between Vitamin A levels and congenital malformations, hypothetical different levels of Vitamin A were evaluated in neonates with and without CHD, in this study.Methods: In a case–control study that was conducted in 2015 in Mashhad/Iran, serum levels of Vitamin A in 30 neonates with CHD were compared to 30 healthy controls. The cases were diagnosed by echocardiography and recruited by convenience sampling. Demographic and laboratory data including age, sex, and serum Vitamin A level in each group were collected. Data analysis was done in SPSS V 20 software, and descriptive statistics, t-test, and analysis of covariance were used.Results: The mean age in cases was 11±3.4 days and in controls was 12.5±4.8 days. A total of 18 patients (60%) were male. In CHD patients, 10 cases (33.3%) had cyanotic heart disease, and 20 cases (66.7%) had non-cyanotic heart disease. The mean serum Vitamin A values in subjects (11.54±9.56 μg/dL) and controls (21.84±14.3 μg/dL) were significantly different, (p&lt;0.05) and in case group was lower than the normal range.Conclusion: There was a significant difference in serum Vitamin A values in subjects and controls. Therefore, awareness of people about the importance of this vitamin in preventing CHD in children seems necessary