arXiv subjects

Emmanuel Candes

Publications and source records attributed to Emmanuel Candes.

33 records · Page 2Linked to original sources

arXiv2012

A single-photon sampling architecture for solid-state imaging

Advances in solid-state technology have enabled the development of silicon photomultiplier sensor arrays capable of sensing individual photons. Combined with high-frequency time-to-digital converters (TDCs), this technology opens up the prospect of sensors capable of recording with high accuracy both the time and location of each detected photon. Such a capability could lead to significant improvements in imaging accuracy, especially for applications operating with low photon fluxes such as LiDAR and positron emission tomography. The demands placed on on-chip readout circuitry imposes stringent trade-offs between fill factor and spatio-temporal resolution, causing many contemporary designs to severely underutilize the technology's full potential. Concentrating on the low photon flux setting, this paper leverages results from group testing and proposes an architecture for a highly efficient readout of pixels using only a small number of TDCs, thereby also reducing both cost and power consumption. The design relies on a multiplexing technique based on binary interconnection matrices. We provide optimized instances of these matrices for various sensor parameters and give explicit upper and lower bounds on the number of TDCs required to uniquely decode a given maximum number of simultaneous photon arrivals. To illustrate the strength of the proposed architecture, we note a typical digitization result of a 120x120 photodiode sensor on a 30um x 30um pitch with a 40ps time resolution and an estimated fill factor of approximately 70%, using only 161 TDCs. The design guarantees registration and unique recovery of up to 4 simultaneous photon arrivals using a fast decoding algorithm. In a series of realistic simulations of scintillation events in clinical positron emission tomography the design was able to recover the spatio-temporal location of 98.6% of all photons that caused pixel firings.

cs.IT↗

arXiv2012

Compressive Fluorescence Microscopy for Biological and Hyperspectral Imaging

The mathematical theory of compressed sensing (CS) asserts that one can acquire signals from measurements whose rate is much lower than the total bandwidth. Whereas the CS theory is now well developed, challenges concerning hardware implementations of CS-based acquisition devices---especially in optics---have only started being addressed. This paper presents an implementation of compressive sensing in fluorescence microscopy and its applications to biomedical imaging. Our CS microscope combines a dynamic structured wide-field illumination and a fast and sensitive single-point fluorescence detection to enable reconstructions of images of fluorescent beads, cells and tissues with undersampling ratios (between the number of pixels and number of measurements) up to 32. We further demonstrate a hyperspectral mode and record images with 128 spectral channels and undersampling ratios up to 64, illustrating the potential benefits of CS acquisition for higher dimensional signals which typically exhibits extreme redundancy. Altogether, our results emphasize the interest of CS schemes for acquisition at a significantly reduced rate and point out to some remaining challenges for CS fluorescence microscopy.

stat.AP↗

arXiv2012

Adaptive Restart for Accelerated Gradient Schemes

In this paper we demonstrate a simple heuristic adaptive restart technique that can dramatically improve the convergence rate of accelerated gradient schemes. The analysis of the technique relies on the observation that these schemes exhibit two modes of behavior depending on how much momentum is applied. In what we refer to as the 'high momentum' regime the iterates generated by an accelerated gradient scheme exhibit a periodic behavior, where the period is proportional to the square root of the local condition number of the objective function. This suggests a restart technique whereby we reset the momentum whenever we observe periodic behavior. We provide analysis to show that in many cases adaptively restarting allows us to recover the optimal rate of convergence with no prior knowledge of function parameters.

math.OC↗

arXiv2012

Simple Bounds for Recovering Low-complexity Models

This note presents a unified analysis of the recovery of simple objects from random linear measurements. When the linear functionals are Gaussian, we show that an s-sparse vector in R^n can be efficiently recovered from 2s log n measurements with high probability and a rank r, n by n matrix can be efficiently recovered from r(6n-5r) with high probability. For sparse vectors, this is within an additive factor of the best known nonasymptotic bounds. For low-rank matrices, this matches the best known bounds. We present a parallel analysis for block sparse vectors obtaining similarly tight bounds. In the case of sparse and block sparse signals, we additionally demonstrate that our bounds are only slightly weakened when the measurement map is a random sign matrix. Our results are based on analyzing a particular dual point which certifies optimality conditions of the respective convex programming problem. Our calculations rely only on standard large deviation inequalities and our analysis is self-contained.

cs.IT↗

arXiv2010

Stable Principal Component Pursuit

In this paper, we study the problem of recovering a low-rank matrix (the principal components) from a high-dimensional data matrix despite both small entry-wise noise and gross sparse errors. Recently, it has been shown that a convex program, named Principal Component Pursuit (PCP), can recover the low-rank matrix when the data matrix is corrupted by gross sparse errors. We further prove that the solution to a related convex program (a relaxed PCP) gives an estimate of the low-rank matrix that is simultaneously stable to small entrywise noise and robust to gross sparse errors. More precisely, our result shows that the proposed convex program recovers the low-rank matrix even though a positive fraction of its entries are arbitrarily corrupted, with an error bound proportional to the noise level. We present simulation results to support our result and demonstrate that the new convex program accurately recovers the principal components (the low-rank matrix) under quite broad conditions. To our knowledge, this is the first result that shows the classical Principal Component Analysis (PCA), optimal for small i.i.d. noise, can be made robust to gross sparse errors; or the first that shows the newly proposed PCP can be made stable to small entry-wise perturbations.

cs.IT↗

arXiv2009

NESTA: A Fast and Accurate First-order Method for Sparse Recovery

Accurate signal recovery or image reconstruction from indirect and possibly undersampled data is a topic of considerable interest; for example, the literature in the recent field of compressed sensing is already quite immense. Inspired by recent breakthroughs in the development of novel first-order methods in convex optimization, most notably Nesterov's smoothing technique, this paper introduces a fast and accurate algorithm for solving common recovery problems in signal processing. In the spirit of Nesterov's work, one of the key ideas of this algorithm is a subtle averaging of sequences of iterates, which has been shown to improve the convergence properties of standard gradient-descent algorithms. This paper demonstrates that this approach is ideally suited for solving large-scale compressed sensing reconstruction problems as 1) it is computationally efficient, 2) it is accurate and returns solutions with several correct digits, 3) it is flexible and amenable to many kinds of reconstruction problems, and 4) it is robust in the sense that its excellent performance across a wide range of problems does not depend on the fine tuning of several parameters. Comprehensive numerical experiments on realistic signals exhibiting a large dynamic range show that this algorithm compares favorably with recently proposed state-of-the-art methods. We also apply the algorithm to solve other problems for which there are fewer alternatives, such as total-variation minimization, and convex programs seeking to minimize the l1 norm of Wx under constraints, in which W is not diagonal.

math.OC↗

arXiv2008

A Fast Butterfly Algorithm for the Computation of Fourier Integral Operators

This paper is concerned with the fast computation of Fourier integral operators of the general form $\int_{\R^d} e^{2\piıΦ(x,k)} f(k) d k$, where $k$ is a frequency variable, $Φ(x,k)$ is a phase function obeying a standard homogeneity condition, and $f$ is a given input. This is of interest for such fundamental computations are connected with the problem of finding numerical solutions to wave equations, and also frequently arise in many applications including reflection seismology, curvilinear tomography and others. In two dimensions, when the input and output are sampled on $N \times N$ Cartesian grids, a direct evaluation requires $O(N^4)$ operations, which is often times prohibitively expensive. This paper introduces a novel algorithm running in $O(N^2 \log N)$ time, i. e. with near-optimal computational complexity, and whose overall structure follows that of the butterfly algorithm [Michielssen and Boag, IEEE Trans Antennas Propagat 44 (1996), 1086-1093]. Underlying this algorithm is a mathematical insight concerning the restriction of the kernel $e^{2\piıΦ(x,k)}$ to subsets of the time and frequency domains. Whenever these subsets obey a simple geometric condition, the restricted kernel has approximately low-rank; we propose constructing such low-rank approximations using a special interpolation scheme, which prefactors the oscillatory component, interpolates the remaining nonoscillatory part and, lastly, remodulates the outcome. A byproduct of this scheme is that the whole algorithm is highly efficient in terms of memory requirement. Numerical results demonstrate the performance and illustrate the empirical properties of this algorithm.

math.NA↗

arXiv2008

The Dantzig selector: Statistical estimation when $p$ is much larger than $n$

In many important statistical applications, the number of variables or parameters $p$ is much larger than the number of observations $n$. Suppose then that we have observations $y=Xβ+z$, where $β\in\mathbf{R}^p$ is a parameter vector of interest, $X$ is a data matrix with possibly far fewer rows than columns, $n\ll p$, and the $z_i$'s are i.i.d. $N(0,σ^2)$. Is it possible to estimate $β$ reliably based on the noisy data $y$? To estimate $β$, we introduce a new estimator--we call it the Dantzig selector--which is a solution to the $\ell_1$-regularization problem \[\min_{\tilde{\b eta}\in\mathbf{R}^p}\|\tildeβ\|_{\ell_1}\quad subject to\quad \|X^*r\|_{\ell_{\infty}}\leq(1+t^{-1})\sqrt{2\log p}\cdotσ,\] where $r$ is the residual vector $y-X\tildeβ$ and $t$ is a positive scalar. We show that if $X$ obeys a uniform uncertainty principle (with unit-normed columns) and if the true parameter vector $β$ is sufficiently sparse (which here roughly guarantees that the model is identifiable), then with very large probability, \[\|\hatβ-β\|_{\ell_2}^2\le C^2\cdot2\log p\cdot \Biggl(σ^2+\sum_i\min(β_i^2,σ^2)\Biggr).\] Our results are nonasymptotic and we give values for the constant $C$. Even though $n$ may be much smaller than $p$, our estimator achieves a loss within a logarithmic factor of the ideal mean squared error one would achieve with an oracle which would supply perfect information about which coordinates are nonzero, and which were above the noise level. In multivariate regression and from a model selection viewpoint, our result says that it is possible nearly to select the best subset of variables by solving a very simple convex program, which, in fact, can easily be recast as a convenient linear program (LP).

math.ST↗

arXiv2006

Sparsity and Incoherence in Compressive Sampling

We consider the problem of reconstructing a sparse signal $x^0\in\R^n$ from a limited number of linear measurements. Given $m$ randomly selected samples of $U x^0$, where $U$ is an orthonormal matrix, we show that $\ell_1$ minimization recovers $x^0$ exactly when the number of measurements exceeds \[ m\geq \mathrm{Const}\cdotμ^2(U)\cdot S\cdot\log n, \] where $S$ is the number of nonzero components in $x^0$, and $μ$ is the largest entry in $U$ properly normalized: $μ(U) = \sqrt{n} \cdot \max_{k,j} |U_{k,j}|$. The smaller $μ$, the fewer samples needed. The result holds for ``most'' sparse signals $x^0$ supported on a fixed (but arbitrary) set $T$. Given $T$, if the sign of $x^0$ for each nonzero entry on $T$ and the observed values of $Ux^0$ are drawn at random, the signal is recovered with overwhelming probability. Moreover, there is a sense in which this is nearly optimal since any method succeeding with the same probability would require just about this many samples.

math.ST↗

arXiv2006

Fast Computation of Fourier Integral Operators

We introduce a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation and general hyperbolic equations. The problem is to evaluate numerically a so-called Fourier integral operator (FIO) of the form $\int e^{2πi Φ(x,ξ)} a(x,ξ) \hat{f}(ξ) \mathrm{d}ξ$ at points given on a Cartesian grid. Here, $ξ$ is a frequency variable, $\hat f(ξ)$ is the Fourier transform of the input $f$, $a(x,ξ)$ is an amplitude and $Φ(x,ξ)$ is a phase function, which is typically as large as $|ξ|$; hence the integral is highly oscillatory at high frequencies. Because an FIO is a dense matrix, a naive matrix vector product with an input given on a Cartesian grid of size $N$ by $N$ would require $O(N^4)$ operations. This paper develops a new numerical algorithm which requires $O(N^{2.5} \log N)$ operations, and as low as $O(\sqrt{N})$ in storage space. It operates by localizing the integral over polar wedges with small angular aperture in the frequency plane. On each wedge, the algorithm factorizes the kernel $e^{2 πi Φ(x,ξ)} a(x,ξ)$ into two components: 1) a diffeomorphism which is handled by means of a nonuniform FFT and 2) a residual factor which is handled by numerical separation of the spatial and frequency variables. The key to the complexity and accuracy estimates is that the separation rank of the residual kernel is \emph{provably independent of the problem size}. Several numerical examples demonstrate the efficiency and accuracy of the proposed methodology. We also discuss the potential of our ideas for various applications such as reflection seismology.

math.NA↗

arXiv2006

Near Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?

Suppose we are given a vector $f$ in $\R^N$. How many linear measurements do we need to make about $f$ to be able to recover $f$ to within precision $ε$ in the Euclidean ($\ell_2$) metric? Or more exactly, suppose we are interested in a class ${\cal F}$ of such objects--discrete digital signals, images, etc; how many linear measurements do we need to recover objects from this class to within accuracy $ε$? This paper shows that if the objects of interest are sparse or compressible in the sense that the reordered entries of a signal $f \in {\cal F}$ decay like a power-law (or if the coefficient sequence of $f$ in a fixed basis decays like a power-law), then it is possible to reconstruct $f$ to within very high accuracy from a small number of random measurements.

math.CA↗

arXiv2005

Stable Signal Recovery from Incomplete and Inaccurate Measurements

Suppose we wish to recover an n-dimensional real-valued vector x_0 (e.g. a digital signal or image) from incomplete and contaminated observations y = A x_0 + e; A is a n by m matrix with far fewer rows than columns (n << m) and e is an error term. Is it possible to recover x_0 accurately based on the data y? To recover x_0, we consider the solution x* to the l1-regularization problem min \|x\|_1 subject to \|Ax-y\|_2 <= epsilon, where epsilon is the size of the error term e. We show that if A obeys a uniform uncertainty principle (with unit-normed columns) and if the vector x_0 is sufficiently sparse, then the solution is within the noise level \|x* - x_0\|_2 \le C epsilon. As a first example, suppose that A is a Gaussian random matrix, then stable recovery occurs for almost all such A's provided that the number of nonzeros of x_0 is of about the same order as the number of observations. Second, suppose one observes few Fourier samples of x_0, then stable recovery occurs for almost any set of p coefficients provided that the number of nonzeros is of the order of n/[\log m]^6. In the case where the error term vanishes, the recovery is of course exact, and this work actually provides novel insights on the exact recovery phenomenon discussed in earlier papers. The methodology also explains why one can also very nearly recover approximately sparse signals.

math.NA↗

arXiv2005

Decoding by Linear Programming

This paper considers the classical error correcting problem which is frequently discussed in coding theory. We wish to recover an input vector $f \in \R^n$ from corrupted measurements $y = A f + e$. Here, $A$ is an $m$ by $n$ (coding) matrix and $e$ is an arbitrary and unknown vector of errors. Is it possible to recover $f$ exactly from the data $y$? We prove that under suitable conditions on the coding matrix $A$, the input $f$ is the unique solution to the $\ell_1$-minimization problem ($\|x\|_{\ell_1} := \sum_i |x_i|$) $$ \min_{g \in \R^n} \| y - Ag \|_{\ell_1} $$ provided that the support of the vector of errors is not too large, $\|e\|_{\ell_0} := |\{i : e_i \neq 0\}| \le ρ\cdot m$ for some $ρ> 0$. In short, $f$ can be recovered exactly by solving a simple convex optimization problem (which one can recast as a linear program). In addition, numerical experiments suggest that this recovery procedure works unreasonably well; $f$ is recovered exactly even in situations where a significant fraction of the output is corrupted.

math.MG↗

arXiv2004

Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions

We develop a robust uncertainty principle for finite signals in C^N which states that for almost all subsets T,W of {0,...,N-1} such that |T|+|W| ~ (log N)^(-1/2) N, there is no sigal f supported on T whose discrete Fourier transform is supported on W. In fact, we can make the above uncertainty principle quantitative in the sense that if f is supported on T, then only a small percentage of the energy (less than half, say) of its Fourier transform is concentrated on W. As an application of this robust uncertainty principle (QRUP), we consider the problem of decomposing a signal into a sparse superposition of spikes and complex sinusoids. We show that if a generic signal f has a decomposition using spike and frequency locations in T and W respectively, and obeying |T| + |W| <= C (\log N)^{-1/2} N, then this is the unique sparsest possible decomposition (all other decompositions have more non-zero terms). In addition, if |T| + |W| <= C (\log N)^{-1} N, then this sparsest decomposition can be found by solving a convex optimization problem.

math.CA↗

arXiv2004

Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information

This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal $f \in \C^N$ and a randomly chosen set of frequencies $Ω$ of mean size $τN$. Is it possible to reconstruct $f$ from the partial knowledge of its Fourier coefficients on the set $Ω$? A typical result of this paper is as follows: for each $M > 0$, suppose that $f$ obeys $$ # \{t, f(t) \neq 0 \} \le α(M) \cdot (\log N)^{-1} \cdot # Ω, $$ then with probability at least $1-O(N^{-M})$, $f$ can be reconstructed exactly as the solution to the $\ell_1$ minimization problem $$ \min_g \sum_{t = 0}^{N-1} |g(t)|, \quad \text{s.t.} \hat g(ω) = \hat f(ω) \text{for all} ω\in Ω. $$ In short, exact recovery may be obtained by solving a convex optimization problem. We give numerical values for $α$ which depends on the desired probability of success; except for the logarithmic factor, the condition on the size of the support is sharp. The methodology extends to a variety of other setups and higher dimensions. For example, we show how one can reconstruct a piecewise constant (one or two-dimensional) object from incomplete frequency samples--provided that the number of jumps (discontinuities) obeys the condition above--by minimizing other convex functionals such as the total-variation of $f$.

math.NA↗