arXiv subjects

Ali Pezeshki

Publications and source records attributed to Ali Pezeshki.

At least 37 records · Page 2Linked to original sources

arXiv2020

Single-Pixel Fluorescent Diffraction Tomography

Optical diffraction tomography is an indispensable tool for studying objects in three-dimensions due to its ability to accurately reconstruct scattering objects. Until now this technique has been limited to coherent light because spatial phase information is required to solve the inverse scattering problem. We introduce a method that extends optical diffraction tomography to imaging spatially incoherent contrast mechanisms such as fluorescent emission. Our strategy mimics the coherent scattering process with two spatially coherent illumination beams. The interferometric illumination pattern encodes spatial phase in temporal variations of the fluorescent emission, thereby allowing incoherent fluorescent emission to mimic the behavior of coherent illumination. The temporal variations permit recovery of the propagation phase, and thus the spatial distribution of incoherent fluorescent emission can be recovered with an inverse scattering model.

physics.optics↗

arXiv2020

Bayesian Learning of Occupancy Grids

Occupancy grids encode for hot spots on a map that is represented by a two dimensional grid of disjoint cells. The problem is to recursively update the probability that each cell in the grid is occupied, based on a sequence of sensor measurements from a moving platform. In this paper, we provide a new Bayesian framework for generating these probabilities that does not assume statistical independence between the occupancy state of grid cells. This approach is made analytically tractable through the use of binary asymmetric channel models that capture the errors associated with observing the occupancy state of a grid cell. Binary-valued measurement vectors are the thresholded output of a sensor in a radar, sonar, or other sensory system. We compare the performance of the proposed framework to that of the classical formulation for occupancy grids. The results show that the proposed framework identifies occupancy grids with lower false alarm and miss detection rates, and requires fewer observations of the surrounding area, to generate an accurate estimate of occupancy probabilities when compared to conventional formulations.

eess.SP↗

arXiv2020

A General Framework for Bounding Approximate Dynamic Programming Schemes

For years, there has been interest in approximation methods for solving dynamic programming problems, because of the inherent complexity in computing optimal solutions characterized by Bellman's principle of optimality. A wide range of approximate dynamic programming (ADP) methods now exists. It is of great interest to guarantee that the performance of an ADP scheme be at least some known fraction, say $β$, of optimal. This paper introduces a general approach to bounding the performance of ADP methods, in this sense, in the stochastic setting. The approach is based on new results for bounding greedy solutions in string optimization problems, where one has to choose a string (ordered set) of actions to maximize an objective function. This bounding technique is inspired by submodularity theory, but submodularity is not required for establishing bounds. Instead, the bounding is based on quantifying certain notions of curvature of string functions; the smaller the curvatures the better the bound. The key insight is that any ADP scheme is a greedy scheme for some surrogate string objective function that coincides in its optimal solution and value with those of the original optimal control problem. The ADP scheme then yields to the bounding technique mentioned above, and the curvatures of the surrogate objective determine the value $β$ of the bound. The surrogate objective and its curvatures depend on the specific ADP.

math.OC↗

arXiv2020

Coordinating Complementary Waveforms for Suppressing Range Sidelobes in a Doppler Band

We present a general method for constructing radar transmit pulse trains and receive filters for which the radar point-spread function in delay and Doppler (radar cross-ambiguity function) is essentially free of range sidelobes inside a Doppler interval around the zero-Doppler axis. The transmit and receive pulse trains are constructed by coordinating the transmission of a pair of Golay complementary waveforms across time according to zeros and ones in a binary sequence $P$. In the receive pulse train filter, each waveform is weighted according to an element from another sequence $Q$. We show that the spectrum of essentially the product of $P$ and $Q$ sequences controls the size of the range sidelobes of the cross-ambiguity function. We annihilate the range sidelobes at low Doppler by designing the $(P,Q)$ pairs such that their products have high-order spectral nulls around zero Doppler. We specify the subspace, along with a basis, for such sequences, thereby providing a general way of constructing $(P,Q)$ pairs. At the same time, the signal-to-noise ratio (SNR) at the receiver output, for a single point target in white noise, depends only on the choice of $Q$. By jointly designing the transmit-receive sequences $(P,Q)$, we can maximize the output SNR subject to achieving a given order of the spectral null. The proposed $(P,Q)$ constructions can also be extended to sequences consisting of more than two complementary waveforms; this is done explicitly for a library of Golay complementary quads. Finally, we extend the construction of $(P,Q)$ pairs to multiple-input-multiple-output (MIMO) radar, by designing transmit-receive pairs of paraunitary waveform matrices whose matrix-valued cross-ambiguity function is essentially free of range sidelobes inside a Doppler interval around the zero-Doppler axis.

cs.IT↗

arXiv2019

Submodular Optimization Problems and Greedy Strategies: A Survey

The greedy strategy is an approximation algorithm to solve optimization problems arising in decision making with multiple actions. How good is the greedy strategy compared to the optimal solution? In this survey, we mainly consider two classes of optimization problems where the objective function is submodular. The first is set submodular optimization, which is to choose a set of actions to optimize a set submodular objective function, and the second is string submodular optimization, which is to choose an ordered set of actions to optimize a string submodular function. Our emphasis here is on performance bounds for the greedy strategy in submodular optimization problems. Specifically, we review performance bounds for the greedy strategy, more general and improved bounds in terms of curvature, performance bounds for the batched greedy strategy, and performance bounds for Nash equilibria.

math.OC↗

arXiv2018

Improved Bounds for the Greedy Strategy in Optimization Problems with Curvature

Consider the problem of choosing a set of actions to optimize an objective function that is a real-valued polymatroid function subject to matroid constraints. The greedy strategy provides an approximate solution to the optimization problem, and it is known to satisfy some performance bounds in terms of the total curvature. The total curvature depends on the value of objective function on sets outside the constraint matroid. If we are given a function defined only on the matroid, the problem still makes sense, but the existing bounds involving the total curvature do not apply. This is puzzling: If the optimization problem is perfectly well defined, why should the bounds no longer apply? This motivates an alternative formulation of such bounding techniques. The first question that comes to mind is whether it is possible to extend a polymatroid function defined on a matroid to one on the entire power set. This was recently shown to be negative in general. Here, we provide necessary and sufficient conditions for the existence of an \emph{incremental} extension of a polymatroid function defined on the uniform matroid of rank $k$ to one defined on the uniform matroid of rank $k+1$, together with an algorithm for constructing the extension. Whenever a polymatroid objective function defined on a matroid can be extended to the entire power set, the greedy approximation bounds involving the total curvature of the extension apply. However, these bounds still depend on sets outside the constraint matroid. Motivated by this, we define a new notion of curvature called \emph{partial curvature}, involving only sets in the matroid. We derive necessary and sufficient conditions for an extension of the function to have a total curvature that is equal to the partial curvature. Moreover, we prove that the bounds in terms of the partial curvature are in general improved over the previous ones......

math.OC↗

arXiv2017

Performance Bounds for Nash Equilibria in Submodular Utility Systems with User Groups

In this paper, we consider variations of the utility system considered by Vetta, in which users are grouped together. Our aim is to establish how grouping and cooperation among users affect performance bounds. We consider two types of grouping. The first type is from \cite{Zhang2014}, where each user belongs to a group of users having social ties with it. For this type of utility system, each user's strategy maximizes its social group utility function, giving rise to the notion of \emph{social-aware Nash equilibrium}. We prove that this social utility system yields to the bounding results of Vetta for non-cooperative system, thus establishing provable performance guarantees for the social-aware Nash equilibrium. For the second type of grouping, the set of users is partitioned into $l$ disjoint groups, where the users within a group cooperate to maximize their group utility function, giving rise to the notion of \emph{group Nash equilibrium}. In this case, each group can be viewed as a new user with vector-valued actions, and a 1/2 bound for the performance of group Nash equilibrium follows from the result of Vetta. But as we show tighter bounds involving curvature can be established. By defining the group curvature $c_{k_i}$ associated with group $i$ with $k_i$ users, we show that if the social utility function is nondecreasing and submodular, then any group Nash equilibrium achieves at least $1/(1+\max_{1\leq i\leq l}c_{k_i})$ of the optimal social utility, which is tighter than that for the case without grouping. As a special case, if each user has the same action space, then we have that any group Nash equilibrium achieves at least $1/(1+c_{k^*})$ of the optimal social utility, where $k^*$ is the least number of users among the $l$ groups. Finally, we present an example of a utility system for database assisted spectrum access to illustrate our results.

math.OC↗

arXiv2017

Performance Bounds with Curvature for Batched Greedy Optimization

The batched greedy strategy is an approximation algorithm to maximize a set function subject to a matroid constraint. Starting with the empty set, the batched greedy strategy iteratively adds to the current solution set a batch of elements that results in the largest gain in the objective function while satisfying the matroid constraints. In this paper, we develop bounds on the performance of the batched greedy strategy relative to the optimal strategy in terms of a parameter called the total batched curvature. We show that when the objective function is a polymatroid set function, the batched greedy strategy satisfies a harmonic bound for a general matroid constraint and an exponential bound for a uniform matroid constraint, both in terms of the total batched curvature. We also study the behavior of the bounds as functions of the batch size. Specifically, we prove that the harmonic bound for a general matroid is nondecreasing in the batch size and the exponential bound for a uniform matroid is nondecreasing in the batch size under the condition that the batch size divides the rank of the uniform matroid. Finally, we illustrate our results by considering a task scheduling problem and an adaptive sensing problem.

math.OC↗

arXiv2016

Performance Bounds for the $k$-Batch Greedy Strategy in Optimization Problems with Curvature

The $k$-batch greedy strategy is an approximate algorithm to solve optimization problems where the optimal solution is hard to obtain. Starting with the empty set, the $k$-batch greedy strategy adds a batch of $k$ elements to the current solution set with the largest gain in the objective function while satisfying the constraints. In this paper, we bound the performance of the $k$-batch greedy strategy with respect to the optimal strategy by defining the total curvature $α_k$. We show that when the objective function is nondecreasing and submodular, the $k$-batch greedy strategy satisfies a harmonic bound $1/(1+α_k)$ for a general matroid constraint and an exponential bound $\left(1-(1-α_k/{t})^t\right)/α_k$ for a uniform matroid constraint, where $k$ divides the cardinality of the maximal set in the general matroid, $t=K/k$ is an integer, and $K$ is the rank of the uniform matroid. We also compare the performance of the $k$-batch greedy strategy with that of the $k_1$-batch greedy strategy when $k_1$ divides $k$. Specifically, we prove that when the objective function is nondecreasing and submodular, the $k$-batch greedy strategy has better harmonic and exponential bounds in terms of the total curvature. Finally, we illustrate our results by considering a task-assignment problem.

math.OC↗

arXiv2016

Bounding the Greedy Strategy in Finite-Horizon String Optimization

We consider an optimization problem where the decision variable is a string of bounded length. For some time there has been an interest in bounding the performance of the greedy strategy for this problem. Here, we provide weakened sufficient conditions for the greedy strategy to be bounded by a factor of $(1-(1-1/K)^K)$, where $K$ is the optimization horizon length. Specifically, we introduce the notions of $K$-submodularity and $K$-GO-concavity, which together are sufficient for this bound to hold. By introducing a notion of \emph{curvature} $η\in(0,1]$, we prove an even tighter bound with the factor $(1/η)(1-e^{-η})$. Finally, we illustrate the strength of our results by considering two example applications. We show that our results provide weaker conditions on parameter values in these applications than in previous results.

math.OC↗

arXiv2015

Subspace selection for projection maximization with matroid constraints

Suppose that there is a ground set which consists of a large number of vectors in a Hilbert space. Consider the problem of selecting a subset of the ground set such that the projection of a vector of interest onto the subspace spanned by the vectors in the chosen subset reaches the maximum norm. This problem is generally NP-hard, and alternative approximation algorithms such as forward regression and orthogonal matching pursuit have been proposed as heuristic approaches. In this paper, we investigate bounds on the performance of these algorithms by introducing the notions of elemental curvatures. More specifically, we derive lower bounds, as functions of these elemental curvatures, for performance of the aforementioned algorithms with respect to that of the optimal solution under uniform and non-uniform matroid constraints, respectively. We show that if the elements in the ground set are mutually orthogonal, then these algorithms are optimal when the matroid is uniform and they achieve at least $1/2$-approximations of the optimal solution when the matroid is non-uniform.

cs.IT↗

arXiv2015

Threshold Effects in Parameter Estimation from Compressed Data

In this paper, we investigate threshold effects associated with swapping of signal and noise subspaces in estimating signal parameters from compressed noisy data. The term threshold effect refers to a sharp departure of mean-squared error from the Cramer-Rao bound when the signal-to-noise ratio falls below a threshold SNR. In many cases, the threshold effect is caused by a subspace swap event, when the measured data (or its sample covariance) is better approximated by a subset of components of an orthogonal subspace than by the components of a signal subspace. We derive analytical lower bounds on the probability of a subspace swap in compressively measured noisy data. These bounds guide our understanding of threshold effects and performance breakdown for parameter estimation using compression. As a case study, we investigate threshold effects in maximum likelihood (ML) estimation of directions of arrival of two closely-spaced sources using co-prime subsampling. Our results show the impact of compression on threshold SNR. A rule of thumb is that every doubling of compression ratio brings a penalty in threshold SNR of 3 dB.

cs.IT↗

arXiv2015

String Submodular Functions with Curvature Constraints

The problem of objectively choosing a string of actions to optimize an objective function that is string submodular has been considered in [1]. There it is shown that the greedy strategy, consisting of a string of actions that only locally maximizes the step-wise gain in the objective function achieves at least a (1-e^{-1})-approximation to the optimal strategy. This paper improves this approximation by introducing additional constraints on curvatures, namely, total backward curvature, total forward curvature, and elemental forward curvature. We show that if the objective function has total backward curvature σ, then the greedy strategy achieves at least a \frac{1}σ(1-e^{-σ})-approximation of the optimal strategy. If the objective function has total forward curvature ε, then the greedy strategy achieves at least a (1-ε)-approximation of the optimal strategy. Moreover, we consider a generalization of the diminishing-return property by defining the elemental forward curvature. We also consider the problem of maximizing the objective function subject to general a string-matroid constraint. We investigate an applications of string submodular functions with curvature constraints.

cs.DS↗

arXiv2015

Modal Analysis Using Sparse and Co-prime Arrays

Let a measurement consist of a linear combination of damped complex exponential modes, plus noise. The problem is to estimate the parameters of these modes, as in line spectrum estimation, vibration analysis, speech processing, system identification, and direction of arrival estimation. Our results differ from standard results of modal analysis to the extent that we consider sparse and co-prime samplings in space, or equivalently sparse and co-prime samplings in time. Our main result is a characterization of the orthogonal subspace. This is the subspace that is orthogonal to the signal subspace spanned by the columns of the generalized Vandermonde matrix of modes in sparse or co-prime arrays. This characterization is derived in a form that allows us to adapt modern methods of linear prediction and approximate least squares, such as iterative quadratic maximum likelihood (IQML), for estimating mode parameters. Several numerical examples are presented to demonstrate the validity of the proposed modal estimation methods, and to compare the fidelity of modal estimation with sparse and co-prime arrays, versus SNR. Our calculations of Cramér-Rao bounds allow us to analyze the loss in performance sustained by sparse and co-prime arrays that are compressions of uniform linear arrays.

cs.IT↗

arXiv2015

Analysis of Fisher Information and the Cramér-Rao Bound for Nonlinear Parameter Estimation after Compressed Sensing

In this paper, we analyze the impact of compressed sensing with complex random matrices on Fisher information and the Cramér-Rao Bound (CRB) for estimating unknown parameters in the mean value function of a complex multivariate normal distribution. We consider the class of random compression matrices whose distribution is right-orthogonally invariant. The compression matrix whose elements are i.i.d. standard normal random variables is one such matrix. We show that for all such compression matrices, the Fisher information matrix has a complex matrix beta distribution. We also derive the distribution of CRB. These distributions can be used to quantify the loss in CRB as a function of the Fisher information of the non-compressed data. In our numerical examples, we consider a direction of arrival estimation problem and discuss the use of these distributions as guidelines for choosing compression ratios based on the resulting loss in CRB.

math.ST↗

arXiv2014

Guaranteed Bounds for General Approximate Dynamic Programming

In this paper, we will develop a systematic approach to deriving guaranteed bounds for approximate dynamic programming (ADP) schemes in optimal control problems. Our approach is inspired by our recent results on bounding the performance of greedy strategies in optimization of string-submodular functions over a finite horizon. The approach is to derive a string-submodular optimization problem, for which the optimal strategy is the optimal control solution and the greedy strategy is the ADP solution. Using this approach, we show that any ADP solution achieves a performance that is at least a factor of $β$ of the performance of the optimal control solution, which satisfies Bellman's optimality principle. The factor $β$ depends on the specific ADP scheme, as we will explicitly characterize. To illustrate the applicability of our bounding technique, we present examples of ADP schemes, including the popular rollout method.

math.OC↗

arXiv2013

Hypothesis Testing in Feedforward Networks with Broadcast Failures

Consider a countably infinite set of nodes, which sequentially make decisions between two given hypotheses. Each node takes a measurement of the underlying truth, observes the decisions from some immediate predecessors, and makes a decision between the given hypotheses. We consider two classes of broadcast failures: 1) each node broadcasts a decision to the other nodes, subject to random erasure in the form of a binary erasure channel; 2) each node broadcasts a randomly flipped decision to the other nodes in the form of a binary symmetric channel. We are interested in whether there exists a decision strategy consisting of a sequence of likelihood ratio tests such that the node decisions converge in probability to the underlying truth. In both cases, we show that if each node only learns from a bounded number of immediate predecessors, then there does not exist a decision strategy such that the decisions converge in probability to the underlying truth. However, in case 1, we show that if each node learns from an unboundedly growing number of predecessors, then the decisions converge in probability to the underlying truth, even when the erasure probabilities converge to 1. We also derive the convergence rate of the error probability. In case 2, we show that if each node learns from all of its previous predecessors, then the decisions converge in probability to the underlying truth when the flipping probabilities of the binary symmetric channels are bounded away from 1/2. In the case where the flipping probabilities converge to 1/2, we derive a necessary condition on the convergence rate of the flipping probabilities such that the decisions still converge to the underlying truth. We also explicitly characterize the relationship between the convergence rate of the error probability and the convergence rate of the flipping probabilities.

cs.IT↗

arXiv2012

Learning in Hierarchical Social Networks

We study a social network consisting of agents organized as a hierarchical M-ary rooted tree, common in enterprise and military organizational structures. The goal is to aggregate information to solve a binary hypothesis testing problem. Each agent at a leaf of the tree, and only such an agent, makes a direct measurement of the underlying true hypothesis. The leaf agent then makes a decision and sends it to its supervising agent, at the next level of the tree. Each supervising agent aggregates the decisions from the M members of its group, produces a summary message, and sends it to its supervisor at the next level, and so on. Ultimately, the agent at the root of the tree makes an overall decision. We derive upper and lower bounds for the Type I and II error probabilities associated with this decision with respect to the number of leaf agents, which in turn characterize the converge rates of the Type I, Type II, and total error probabilities. We also provide a message-passing scheme involving non-binary message alphabets and characterize the exponent of the error probability with respect to the message alphabet size.

cs.SI↗