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: ($1$) $\ell_2$-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

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

Formulation and evaluation of orally disintegrating tablet of Rizatriptan using natural superdisintegrant

Introduction: Rizatriptan benzoate is a potent and selective 5-HT1B/1D receptor agonist and is effective for the treatment of acute migraine. Difficulty in swallowing is common among all age groups, especially elderly and pediatrics. Orally disintegrating tablets may constitute an innovative dosage form that overcome the problem of swallowing and provides a quick onset of action. This study was aimed to formulate and evaluate an Orally Disintegrating Tablet (ODT) containing Rizatriptan while using semi-synthetic and natural superdisintegrants. Methods: Orodispersible tablets were prepared by direct compression using natural superdisntegrant (Plantago ovata mucilage) and semi-synthetic superdisntegrant (crospovidone). The prepared tablets were evaluated for hardness, friability, thickness, drug content uniformity, water absorption and wetting time. A 32 factorial design was used to investigate the effect of independent variables (amount of crospovidone and Plantago ovata mucilage) on dependent variables disintegration time, wetting time and Q5 (cumulative amount of drug release after 5 minutes). A counter plot was also presented to graphically represent the effect of independent variable on the disintegration time, wetting time and Q5. The check point batch was also prepared to prove the validity of the evolved mathematical model. The systematic formulation approach helped in understanding the effect of formulation processing variable. Results: According to the results of optimized batches, the best concentration of superdisintegrant were as follows: 9.4 mg Psyllium mucilage and 8.32 mg crospovidone gave rapid disintegration in 35sec and showed 99% drug release within 5 minutes. Conclusion: Plantago ovata mucilage, a natural superdisintegrant, gives a rapid disintegration and high release when used with synthetic superdisntegrant in formulation of orally disintegrating tablet of Rizatriptan.</p

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

A Precise Analysis of PhaseMax in Phase Retrieval

Recovering an unknown complex signal from the magnitude of linear combinations of the signal is referred to as phase retrieval. We present an exact performance analysis of a recently proposed convex-optimization-formulation for this problem, known as PhaseMax. Standard convex-relaxation-based methods in phase retrieval resort to the idea of “lifting” which makes them computationally inefficient, since the number of unknowns is effectively squared. In contrast, PhaseMax is a novel convex relaxation that does not increase the number of unknowns. Instead it relies on an initial estimate of the true signal which must be externally provided. In this paper, we investigate the required number of measurements for exact recovery of the signal in the large system limit and when the linear measurement matrix is random with iid standard normal entries. If n denotes the dimension of the unknown complex signal and m the number of phaseless measurements, then in the large system limit, m/n > 4/cos^2(θ) measurements is necessary and sufficient to recover the signal with high probability, where θ is the angle between the initial estimate and the true signal. Our result indicates a sharp phase transition in the asymptotic regime which matches the empirical result in numerical simulations

The Impact of Regularization on High-dimensional Logistic Regression

Logistic regression is commonly used for modeling dichotomous outcomes. In the classical setting, where the number of observations is much larger than the number of parameters, properties of the maximum likelihood estimator in logistic regression are well understood. Recently, Sur and Candes have studied logistic regression in the high-dimensional regime, where the number of observations and parameters are comparable, and show, among other things, that the maximum likelihood estimator is biased. In the high-dimensional regime the underlying parameter vector is often structured (sparse, block-sparse, finite-alphabet, etc.) and so in this paper we study regularized logistic regression (RLR), where a convex regularizer that encourages the desired structure is added to the negative of the log-likelihood function. An advantage of RLR is that it allows parameter recovery even for instances where the (unconstrained) maximum likelihood estimate does not exist. We provide a precise analysis of the performance of RLR via the solution of a system of six nonlinear equations, through which any performance metric of interest (mean, mean-squared error, probability of support recovery, etc.) can be explicitly computed. Our results generalize those of Sur and Candes and we provide a detailed study for the cases of ℓ²₂-RLR and sparse (ℓ₁-regularized) logistic regression. In both cases, we obtain explicit expressions for various performance metrics and can find the values of the regularizer parameter that optimizes the desired performance. The theory is validated by extensive numerical simulations across a range of parameter values and problem instances