arXiv subjects

Christos Thrampoulidis

Publications and source records attributed to Christos Thrampoulidis.

At least 91 records · Page 5Linked to original sources

arXiv2019

Linear Stochastic Bandits Under Safety Constraints

Bandit algorithms have various application in safety-critical systems, where it is important to respect the system constraints that rely on the bandit's unknown parameters at every round. In this paper, we formulate a linear stochastic multi-armed bandit problem with safety constraints that depend (linearly) on an unknown parameter vector. As such, the learner is unable to identify all safe actions and must act conservatively in ensuring that her actions satisfy the safety constraint at all rounds (at least with high probability). For these bandits, we propose a new UCB-based algorithm called Safe-LUCB, which includes necessary modifications to respect safety constraints. The algorithm has two phases. During the pure exploration phase the learner chooses her actions at random from a restricted set of safe actions with the goal of learning a good approximation of the entire unknown safe set. Once this goal is achieved, the algorithm begins a safe exploration-exploitation phase where the learner gradually expands their estimate of the set of safe actions while controlling the growth of regret. We provide a general regret bound for the algorithm, as well as a problem dependent bound that is connected to the location of the optimal action within the safe set. We then propose a modified heuristic that exploits our problem dependent analysis to improve the regret.

cs.LG↗

arXiv2019

A simple bound on the BER of the MAP decoder for massive MIMO systems

The deployment of massive MIMO systems has revived much of the interest in the study of the large-system performance of multiuser detection systems. In this paper, we prove a non-trivial upper bound on the bit-error rate (BER) of the MAP detector for BPSK signal transmission and equal-power condition. In particular, our bound is approximately tight at high-SNR. The proof is simple and relies on Gordon's comparison inequality. Interestingly, we show that under the assumption that Gordon's inequality is tight, the resulting BER prediction matches that of the replica method under the replica symmetry (RS) ansatz. Also, we prove that, when the ratio of receive to transmit antennas exceeds $0.9251$, the replica prediction matches the matched filter lower bound (MFB) at high-SNR. We corroborate our results by numerical evidence.

cs.IT↗

arXiv2019

Near-Optimal Coded Apertures for Imaging via Nazarov's Theorem

We characterize the fundamental limits of coded aperture imaging systems up to universal constants by drawing upon a theorem of Nazarov regarding Fourier transforms. Our work is performed under a simple propagation and sensor model that accounts for thermal and shot noise, scene correlation, and exposure time. Focusing on mean square error as a measure of linear reconstruction quality, we show that appropriate application of a theorem of Nazarov leads to essentially optimal coded apertures, up to a constant multiplicative factor in exposure time. Additionally, we develop a heuristically efficient algorithm to generate such patterns that explicitly takes into account scene correlations. This algorithm finds apertures that correspond to local optima of a certain potential on the hypercube, yet are guaranteed to be tight. Finally, for i.i.d. scenes, we show improvements upon prior work by using spectrally flat sequences with bias. The development focuses on 1D apertures for conceptual clarity; the natural generalizations to 2D are also discussed.

eess.IV↗

arXiv2018

Lifting high-dimensional nonlinear models with Gaussian regressors

We study the problem of recovering a structured signal $\mathbf{x}_0$ from high-dimensional data $\mathbf{y}_i=f(\mathbf{a}_i^T\mathbf{x}_0)$ for some nonlinear (and potentially unknown) link function $f$, when the regressors $\mathbf{a}_i$ are iid Gaussian. Brillinger (1982) showed that ordinary least-squares estimates $\mathbf{x}_0$ up to a constant of proportionality $μ_\ell$, which depends on $f$. Recently, Plan & Vershynin (2015) extended this result to the high-dimensional setting deriving sharp error bounds for the generalized Lasso. Unfortunately, both least-squares and the Lasso fail to recover $\mathbf{x}_0$ when $μ_\ell=0$. For example, this includes all even link functions. We resolve this issue by proposing and analyzing an alternative convex recovery method. In a nutshell, our method treats such link functions as if they were linear in a lifted space of higher-dimension. Interestingly, our error analysis captures the effect of both the nonlinearity and the problem's geometry in a few simple summary parameters.

stat.ML↗

arXiv2018

The Generalized Lasso for Sub-gaussian Measurements with Dithered Quantization

In the problem of structured signal recovery from high-dimensional linear observations, it is commonly assumed that full-precision measurements are available. Under this assumption, the recovery performance of the popular Generalized Lasso (G-Lasso) is by now well-established. In this paper, we extend these types of results to the practically relevant settings with quantized measurements. We study two extremes of the quantization schemes, namely, uniform and one-bit quantization; the former imposes no limit on the number of quantization bits, while the second only allows for one bit. In the presence of a uniform dithering signal and when measurement vectors are sub-gaussian, we show that the same algorithm (i.e., the G-Lasso) has favorable recovery guarantees for both uniform and one-bit quantization schemes. Our theoretical results, shed light on the appropriate choice of the range of values of the dithering signal and accurately capture the error dependence on the problem parameters. For example, our error analysis shows that the G-Lasso with one-bit uniformly dithered measurements leads to only a logarithmic rate loss compared to the full-precision measurements.

cs.IT↗

arXiv2018

Phase Retrieval via Polytope Optimization: Geometry, Phase Transitions, and New Algorithms

We study algorithms for solving quadratic systems of equations based on optimization methods over polytopes. Our work is inspired by a recently proposed convex formulation of the phase retrieval problem, which estimates the unknown signal by solving a simple linear program over a polytope constructed from the measurements. We present a sharp characterization of the high-dimensional geometry of the aforementioned polytope under Gaussian measurements. This characterization allows us to derive asymptotically exact performance guarantees for PhaseMax, which also reveal a phase transition phenomenon with respect to its sample complexity. Moreover, the geometric insights gained from our analysis lead to a new nonconvex formulation of the phase retrieval problem and an accompanying iterative algorithm, which we call PhaseLamp. We show that this new algorithm has superior recovery performance over the original PhaseMax method. Finally, as yet another variation on the theme of performing phase retrieval via polytope optimization, we propose a weighted version of PhaseLamp and demonstrate, through numerical simulations, that it outperforms several state-of-the-art algorithms under both generic Gaussian measurements as well as more realistic Fourier-type measurements that arise in phase retrieval applications.

cs.IT↗

arXiv2018

Revealing hidden scenes by photon-efficient occlusion-based opportunistic active imaging

The ability to see around corners, i.e., recover details of a hidden scene from its reflections in the surrounding environment, is of considerable interest in a wide range of applications. However, the diffuse nature of light reflected from typical surfaces leads to mixing of spatial information in the collected light, precluding useful scene reconstruction. Here, we employ a computational imaging technique that opportunistically exploits the presence of occluding objects, which obstruct probe-light propagation in the hidden scene, to undo the mixing and greatly improve scene recovery. Importantly, our technique obviates the need for the ultrafast time-of-flight measurements employed by most previous approaches to hidden-scene imaging. Moreover, it does so in a photon-efficient manner based on an accurate forward model and a computational algorithm that, together, respect the physics of three-bounce light propagation and single-photon detection. Using our methodology, we demonstrate reconstruction of hidden-surface reflectivity patterns in a meter-scale environment from non-time-resolved measurements. Ultimately, our technique represents an instance of a rich and promising new imaging modality with important potential implications for imaging science.

stat.AP↗

arXiv2017

Analysis and Optimization of Aperture Design in Computational Imaging

There is growing interest in the use of coded aperture imaging systems for a variety of applications. Using an analysis framework based on mutual information, we examine the fundamental limits of such systems---and the associated optimum aperture coding---under simple but meaningful propagation and sensor models. Among other results, we show that when thermal noise dominates, spectrally-flat masks, which have 50% transmissivity, are optimal, but that when shot noise dominates, randomly generated masks with lower transmissivity offer greater performance. We also provide comparisons to classical pinhole cameras.

eess.IV↗

arXiv2017

Symbol Error Rate Performance of Box-relaxation Decoders in Massive MIMO

The maximum-likelihood (ML) decoder for symbol detection in large multiple-input multiple-output wireless communication systems is typically computationally prohibitive. In this paper, we study a popular and practical alternative, namely the Box-relaxation optimization (BRO) decoder, which is a natural convex relaxation of the ML. For iid real Gaussian channels with additive Gaussian noise, we obtain exact asymptotic expressions for the symbol error rate (SER) of the BRO. The formulas are particularly simple, they yield useful insights, and they allow accurate comparisons to the matched-filter bound (MFB) and to the zero-forcing decoder. For BPSK signals the SER performance of the BRO is within 3dB of the MFB for square systems, and it approaches the MFB as the number of receive antennas grows large compared to the number of transmit antennas. Our analysis further characterizes the empirical density function of the solution of the BRO, and shows that error events for any fixed number of symbols are asymptotically independent. The fundamental tool behind the analysis is the convex Gaussian min-max theorem.

eess.SP↗

arXiv2017

Exploiting Occlusion in Non-Line-of-Sight Active Imaging

Active non-line-of-sight imaging systems are of growing interest for diverse applications. The most commonly proposed approaches to date rely on exploiting time-resolved measurements, i.e., measuring the time it takes for short light pulses to transit the scene. This typically requires expensive, specialized, ultrafast lasers and detectors that must be carefully calibrated. We develop an alternative approach that exploits the valuable role that natural occluders in a scene play in enabling accurate and practical image formation in such settings without such hardware complexity. In particular, we demonstrate that the presence of occluders in the hidden scene can obviate the need for collecting time-resolved measurements, and develop an accompanying analysis for such systems and their generalizations. Ultimately, the results suggest the potential to develop increasingly sophisticated future systems that are able to identify and exploit diverse structural features of the environment to reconstruct scenes hidden from view.

eess.IV↗

arXiv2017

Phase Retrieval via Linear Programming: Fundamental Limits and Algorithmic Improvements

A recently proposed convex formulation of the phase retrieval problem estimates the unknown signal by solving a simple linear program. This new scheme, known as PhaseMax, is computationally efficient compared to standard convex relaxation methods based on lifting techniques. In this paper, we present an exact performance analysis of PhaseMax under Gaussian measurements in the large system limit. In contrast to previously known performance bounds in the literature, our results are asymptotically exact and they also reveal a sharp phase transition phenomenon. Furthermore, the geometrical insights gained from our analysis led us to a novel nonconvex formulation of the phase retrieval problem and an accompanying iterative algorithm based on successive linearization and maximization over a polytope. This new algorithm, which we call PhaseLamp, has provably superior recovery performance over the original PhaseMax method.

cs.IT↗

arXiv2016

Phaseless super-resolution in the continuous domain

Phaseless super-resolution refers to the problem of superresolving a signal from only its low-frequency Fourier magnitude measurements. In this paper, we consider the phaseless super-resolution problem of recovering a sum of sparse Dirac delta functions which can be located anywhere in the continuous time-domain. For such signals in the continuous domain, we propose a novel Semidefinite Programming (SDP) based signal recovery method to achieve the phaseless superresolution. This work extends the recent work of Jaganathan et al. [1], which considered phaseless super-resolution for discrete signals on the grid.

cs.IT↗

arXiv2016

Improved Bounds on the Epidemic Threshold of Exact SIS Models on Complex Networks

The SIS (susceptible-infected-susceptible) epidemic model on an arbitrary network, without making approximations, is a $2^n$-state Markov chain with a unique absorbing state (the all-healthy state). This makes analysis of the SIS model and, in particular, determining the threshold of epidemic spread quite challenging. It has been shown that the exact marginal probabilities of infection can be upper bounded by an $n$-dimensional linear time-invariant system, a consequence of which is that the Markov chain is "fast-mixing" when the LTI system is stable, i.e. when $\fracβδ<\frac{1}{λ_{\max}(A)}$ (where $β$ is the infection rate per link, $δ$ is the recovery rate, and $λ_{\max}(A)$ is the largest eigenvalue of the network's adjacency matrix). This well-known threshold has been recently shown not to be tight in several cases, such as in a star network. In this paper, We provide tighter upper bounds on the exact marginal probabilities of infection, by also taking pairwise infection probabilities into account. Based on this improved bound, we derive tighter eigenvalue conditions that guarantee fast mixing (i.e., logarithmic mixing time) of the chain. We demonstrate the improvement of the threshold condition by comparing the new bound with the known one on various networks and epidemic parameters.

cs.SI↗

arXiv2016

Precise Error Analysis of Regularized M-estimators in High-dimensions

A popular approach for estimating an unknown signal from noisy, linear measurements is via solving a so called \emph{regularized M-estimator}, which minimizes a weighted combination of a convex loss function and of a convex (typically, non-smooth) regularizer. We accurately predict the squared error performance of such estimators in the high-dimensional proportional regime. The random measurement matrix is assumed to have entries iid Gaussian, only minimal and rather mild regularity conditions are imposed on the loss function, the regularizer, and on the noise and signal distributions. We show that the error converges in probability to a nontrivial limit that is given as the solution to a minimax convex-concave optimization problem on four scalar optimization variables. We identify a new summary parameter, termed the Expected Moreau envelope to play a central role in the error characterization. The \emph{precise} nature of the results permits an accurate performance comparison between different instances of regularized M-estimators and allows to optimally tune the involved parameters (e.g. regularizer parameter, number of measurements). The key ingredient of our proof is the \emph{Convex Gaussian Min-max Theorem} (CGMT) which is a tight and strengthened version of a classical Gaussian comparison inequality that was proved by Gordon in 1988.

cs.IT↗

arXiv2015

BER Analysis of the box relaxation for BPSK Signal Recovery

We study the problem of recovering an $n$-dimensional vector of $\{\pm1\}^n$ (BPSK) signals from $m$ noise corrupted measurements $\mathbf{y}=\mathbf{A}\mathbf{x}_0+\mathbf{z}$. In particular, we consider the box relaxation method which relaxes the discrete set $\{\pm1\}^n$ to the convex set $[-1,1]^n$ to obtain a convex optimization algorithm followed by hard thresholding. When the noise $\mathbf{z}$ and measurement matrix $\mathbf{A}$ have iid standard normal entries, we obtain an exact expression for the bit-wise probability of error $P_e$ in the limit of $n$ and $m$ growing and $\frac{m}{n}$ fixed. At high SNR our result shows that the $P_e$ of box relaxation is within 3dB of the matched filter bound MFB for square systems, and that it approaches MFB as $m $ grows large compared to $n$. Our results also indicates that as $m,n\rightarrow\infty$, for any fixed set of size $k$, the error events of the corresponding $k$ bits in the box relaxation method are independent.

cs.IT↗

arXiv2015

Isotropically Random Orthogonal Matrices: Performance of LASSO and Minimum Conic Singular Values

Recently, the precise performance of the Generalized LASSO algorithm for recovering structured signals from compressed noisy measurements, obtained via i.i.d. Gaussian matrices, has been characterized. The analysis is based on a framework introduced by Stojnic and heavily relies on the use of Gordon's Gaussian min-max theorem (GMT), a comparison principle on Gaussian processes. As a result, corresponding characterizations for other ensembles of measurement matrices have not been developed. In this work, we analyze the corresponding performance of the ensemble of isotropically random orthogonal (i.r.o.) measurements. We consider the constrained version of the Generalized LASSO and derive a sharp characterization of its normalized squared error in the large-system limit. When compared to its Gaussian counterpart, our result analytically confirms the superiority in performance of the i.r.o. ensemble. Our second result, derives an asymptotic lower bound on the minimum conic singular values of i.r.o. matrices. This bound is larger than the corresponding bound on Gaussian matrices. To prove our results we express i.r.o. matrices in terms of Gaussians and show that, with some modifications, the GMT framework is still applicable.

cs.IT↗

arXiv2015

The Gaussian min-max theorem in the Presence of Convexity

Gaussian comparison theorems are useful tools in probability theory; they are essential ingredients in the classical proofs of many results in empirical processes and extreme value theory. More recently, they have been used extensively in the analysis of non-smooth optimization problems that arise in the recovery of structured signals from noisy linear observations. We refer to such problems as Primary Optimization (PO) problems. A prominent role in the study of the (PO) problems is played by Gordon's Gaussian min-max theorem (GMT) which provides probabilistic lower bounds on the optimal cost via a simpler Auxiliary Optimization (AO) problem. Motivated by resent work of M. Stojnic, we show that under appropriate convexity assumptions the (AO) problem allows one to tightly bound both the optimal cost, as well as the norm of the solution of the (PO). As an application, we use our result to develop a general framework to tightly characterize the performance (e.g. squared-error) of a wide class of convex optimization algorithms used in the context of noisy signal recovery.

cs.IT↗

arXiv2015

Asymptotically Exact Error Analysis for the Generalized $\ell_2^2$-LASSO

Given an unknown signal $\mathbf{x}_0\in\mathbb{R}^n$ and linear noisy measurements $\mathbf{y}=\mathbf{A}\mathbf{x}_0+σ\mathbf{v}\in\mathbb{R}^m$, the generalized $\ell_2^2$-LASSO solves $\hat{\mathbf{x}}:=\arg\min_{\mathbf{x}}\frac{1}{2}\|\mathbf{y}-\mathbf{A}\mathbf{x}\|_2^2 + σλf(\mathbf{x})$. Here, $f$ is a convex regularization function (e.g. $\ell_1$-norm, nuclear-norm) aiming to promote the structure of $\mathbf{x}_0$ (e.g. sparse, low-rank), and, $λ\geq 0$ is the regularizer parameter. A related optimization problem, though not as popular or well-known, is often referred to as the generalized $\ell_2$-LASSO and takes the form $\hat{\mathbf{x}}:=\arg\min_{\mathbf{x}}\|\mathbf{y}-\mathbf{A}\mathbf{x}\|_2 + λf(\mathbf{x})$, and has been analyzed in [1]. [1] further made conjectures about the performance of the generalized $\ell_2^2$-LASSO. This paper establishes these conjectures rigorously. We measure performance with the normalized squared error $\mathrm{NSE}(σ):=\|\hat{\mathbf{x}}-\mathbf{x}_0\|_2^2/σ^2$. Assuming the entries of $\mathbf{A}$ and $\mathbf{v}$ be i.i.d. standard normal, we precisely characterize the "asymptotic NSE" $\mathrm{aNSE}:=\lim_{σ\rightarrow 0}\mathrm{NSE}(σ)$ when the problem dimensions $m,n$ tend to infinity in a proportional manner. The role of $λ,f$ and $\mathbf{x}_0$ is explicitly captured in the derived expression via means of a single geometric quantity, the Gaussian distance to the subdifferential. We conjecture that $\mathrm{aNSE} = \sup_{σ>0}\mathrm{NSE}(σ)$. We include detailed discussions on the interpretation of our result, make connections to relevant literature and perform computational experiments that validate our theoretical findings.

math.ST↗