{"title": "Convolutional Phase Retrieval", "book": "Advances in Neural Information Processing Systems", "page_first": 6086, "page_last": 6096, "abstract": "We study the convolutional phase retrieval problem, which asks us to recover an unknown signal ${\\mathbf x} $ of length $n$ from $m$ measurements consisting of the magnitude of its cyclic convolution with a known kernel $\\mathbf a$ of length $m$. This model is motivated by applications to channel estimation, optics, and underwater acoustic communication, where the signal of interest is acted on by a given channel/filter, and phase information is difficult or impossible to acquire. We show that when $\\mathbf a$ is random and $m \\geq \\Omega(\\frac{ \\| \\mathbf C_{\\mathbf x}\\|^2}{ \\|\\mathbf x\\|^2 }  n \\mathrm{poly} \\log n)$, $\\mathbf x$ can be efficiently recovered up to a global phase using a combination of spectral initialization and generalized gradient descent. The main challenge is coping with dependencies in the measurement operator; we overcome this challenge by using ideas from decoupling theory, suprema of chaos processes and the restricted isometry property of random circulant matrices, and recent analysis for alternating minimizing methods.", "full_text": "Convolutional Phase Retrieval\n\nQing Qu\n\nColumbia University\nqq2105@columbia.edu\n\nYonina C. Eldar\n\nTechnion\n\nyonina@ee.technion.ac.il\n\nYuqian Zhang\n\nColumbia University\nyz2409@columbia.edu\n\nJohn Wright\n\nColumbia University\njw2966@columbia.edu\n\nAbstract\n\nWe study the convolutional phase retrieval problem, which considers recov-\nery of an unknown signal x 2 Cn from m measurements consisting of the\nmagnitude of its cyclic convolution with a known kernel a of length m. This\nmodel is motivated by applications to channel estimation, optics, and un-\nderwater acoustic communication, where the signal of interest is acted on\nby a given channel/\ufb01lter, and phase information is di\ufb03cult or impossible to\nacquire. We show that when a is random and m is su\ufb03ciently large, x can\nbe e\ufb03ciently recovered up to a global phase using a combination of spec-\ntral initialization and generalized gradient descent. The main challenge is\ncoping with dependencies in the measurement operator; we overcome this\nchallenge by using ideas from decoupling theory, suprema of chaos pro-\ncesses and the restricted isometry property of random circulant matrices,\nand recent analysis for alternating minimizing methods.\n\nIntroduction\n\n1\nWe study the problem of recovering a unknown signal x 2 Cn from measurements y =\nja \u229b xj, which consist of the magnitude of the convolution of x and a given \ufb01lter a 2 Cm,\n(1)\nwhere \u229b denotes cyclic convolution. Let Ca 2 Cm(cid:2)m be a circulant matrix generated by a,\nand let A 2 Cm(cid:2)n be a matrix formed by the \ufb01rst n columns of Ca. Then the convolutional\nphase retrieval problem can be rewritten in the common matrix-vector form\n\ns.t. y = ja \u229b zj ;\n\n\ufb01nd z;\n\ns.t. y = jAzj :\n\n\ufb01nd z;\n\n(2)\nThis problem is motivated by applications like channel estimation [37, 1], (non)coherent op-\ntical communication [14, 24], and underwater acoustic communication [31]. For example, in\nmillimeter-wave (mm-wave) wireless communications for 5G networks [27], one important\nproblem is to reconstruct signal angle of arrival (AoA) from measurements, which are taken\nby the convolution of signal AoA and the antenna pattern.\nBecause of technical di\ufb03culties that the phase measurements are either very noisy and unre-\nliable, or expensive to acquire, it is preferred to only take measurements of signal magnitude\nand the phase information is lost.\nMost known results on the exact solution of phase retrieval problems [8, 29, 10, 38, 36, 35]\npertain to generic random matrices, where the entries of A are independent subgaussian ran-\ndom variables. However, in practice it is almost impossible to design purely random mea-\nsurement matrices: in many cases as we mentioned above, the measurement is much more\n\n31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.\n\n\fstructured \u2013 generated by passing a signal through a manually designed \ufb01lter. Moreover,\nthe structured measurements often admit more e\ufb03cient numerical methods: by using the\nfast Fourier transform for matrix-vector products, the benign structure of the convolutional\nmodel (1) allows us to design methods with O(m) memory and O(m log m) computation\ncost per iteration. While for generic measurements, the cost is around O(mn).\nIn this work, we study the convolutional phase retrieval problem (1) under the assumption\nthat the kernel a = [a1;(cid:1)(cid:1)(cid:1) ; am]\n\u22a4\n\nis random, with each entry i.i.d. complex Gaussian,\n\n0; 1\n2 I\n\n(3)\nCompared to the generic random measurement, as we can see, the random convolution\nmodel we study here is far more structured: it is parameterized by only O(m) independent\ncomplex normal random variables, whereas the generic model involves O(mn) ones. Since\nthe rows and columns of A are probabilistically dependent, standard techniques (based on\nconcentration of functions of independent random vectors) do not apply.\nWe propose and analyze a local1 gradient descent type method, minimizing a weighted,\nnonconvex and nonsmooth objective\n\n:\n\na = u + iv; u; v (cid:24)iid N(\n\n)\n\n(cid:13)(cid:13)(cid:13)b1=2 \u2299 (y (cid:0) jAzj)\n(cid:13)(cid:13)(cid:13)2\n\nmin\nz2Cn\n\nf (z) =\n\n1\n2m\n\n;\n\n(4)\n\n{\n\n)\n\n\u2225x\u22252 n poly log n\n\n(\u2225Cx\u22252\n\n++ is a weighting vector. Our result can\n\n}\nxei\u03d5 j \u03d5 2 [0; 2(cid:25))\n\nwhere \u2299 denotes the Hadamard product and b 2 Rm\nbe informally summarized as follows.\nTheorem 1.1 (Informal) When m (cid:21) \u2126(n poly log n), with high probability, spectral initializa-\ntion [25, 5] produces an initialization z(0) that is O(1= poly log n) close to the optimum. Moreover,\nwhen m (cid:21) \u2126\n, with high probability, a certain gradient descent method based\non (4) converges linearly from this initialization to the set X =\nof points that\ndi\ufb00er from the true signal x only by a global phase.\nHere, Cx 2 Cm(cid:2)m denotes the circulant matrix corresponding to cyclic convolution with a\nlength m zero padding of x, and poly log n denotes a polynomial in log n. A dependence of\nthe sample complexity m on \u2225Cx\u2225 seems inevitable2 and is corroborated by experiments.\nOur proof is based on ideas from decoupling theory [11], the suprema of chaos processes and\nrestricted isometry property of random circulant matrices [26, 20], and a new iterative analy-\nsis of alternating minimizing methods [35]. Our analysis draws connections between the\nconvergence properties of gradient descent and the classical alternating direction method.\nThis allows us to avoid the need to argue that high-degree polynomials in the structured\nrandom matrix A concentrate uniformly, as would be required by a straightforward trans-\nlation of existing analysis to this new setting. Instead, we control the bulk e\ufb00ect of phase\nerrors uniformly in a neighborhood around the ground truth. This requires us to develop\nnew decoupling and concentration tools for controlling nonlinear phase functions of circu-\nlant random matrices, which could be potentially useful for analyzing other random circu-\nlant convolution problems such as blind deconvolution [40], and convolutional dictionary\nlearning [18].\n\nPrior art for phase retrieval. The challenge of developing e\ufb03cient, guaranteed methods\nfor phase retrieval has attracted substantial interest over the past decade [28, 19]. For the\ngeneralized phase retrieval problem in which the sensing matrix A is i.i.d. random, the \ufb01rst\nresult on global recovery is based on semide\ufb01nite programming (SDP) [8, 3, 36]. However,\nthe computational cost of SDP limits its practicality. Nonconvex methods can be more ef-\n\ufb01cient. [25] showed that the alternating minimization method provably converges to the\n\n1It would be nicer to characterize the global geometry of the problem as in [15, 33, 34, 32]. However,\nthe nonhomogeneity of \u2225Cx\u2225 over the space causes tremendous di\ufb03culties for concentration with\nm (cid:21) \u2126(n poly log n) samples.\nn).\nFor instance, \u2225Cx\u2225 = 1 when x is a standard basis vector; and \u2225Cx\u2225 =\n\np\n2The operator norm of Cx is nonhomogeneous over x 2 CSn(cid:0)1, ranging from constant to O(\n\np\nn when x = 1p\n\nn 1.\n\n2\n\n\ftruth, when initialized using a spectral method and provided with fresh samples at each\niteration. Candes et al. [5] showed with the same initalization, gradient descent for the\nnonconvex least squares objective,\n\n(cid:13)(cid:13)(cid:13)y2 (cid:0) jAzj2\n\n(cid:13)(cid:13)(cid:13)2\n\n;\n\n1\n2m\n\nmin\nz2Cn\n\nf1(z) =\n\n(5)\nprovably recovers the ground truth, with near-optimal sample complexity m (cid:21) \u2126(n log n).\nThe work [10, 39, 38] further reduce the sample complexity to m (cid:21) \u2126(n) by using di\ufb00erent\nnonconvex objectives and truncation techniques. Moreover, [34] reveals that the nonconvex\nobjective (5) has a benign global geometry: with high probability, it has no bad critical points\nwith m (cid:21) \u2126(n log3 n) samples3.\nStructured random measurements. The study of structured random measurement in sig-\nnal processing [21] includes the study of random Fourier measurements [7, 9, 12] and partial\nrandom convolutions [26, 20] in compressed sensing [6]. However, the study of structured\nrandom measurement for phase retrieval is still quite limited. In particular, [17] and [4] stud-\nied the performance of SDP methods with t-designs and random masked Fourier transform\nmeasurements. The authors in [5, 2] show that the phase retrieval problem with random\ncoded di\ufb00raction and STFT measurements can be solved by minimizing nonconvex objec-\ntives, while [5] requires resampling for the initialization, and in [2] the contraction radius\nis not large enough for initialization. In addition, the motivation of these measurement\nschemes are quite di\ufb00erent from ours. For more detailed review of this subject, we refer the\nreaders to Section 4 of [21].\nNotations. We use ((cid:1))\n(cid:3) to denote the real and Hermitian transpose, respectively.\nWe use CSn(cid:0)1 to denote a n dimensional complex sphere. Let \u211c((cid:1)) and \u2111((cid:1)) denote the real\nand imaginary parts of a complex variable, respectively. Throughout the paper, we assume\nthe optimal solution is x 2 Cn. Because the solution is only optimal to a global phase shift,\nwe de\ufb01ne the optimal solution set as X =\n, and de\ufb01ne the distance from\na point z 2 Cn to the set X as\n\nxei(cid:18) j (cid:18) 2 [0; 2(cid:25))\n\n\u22a4 and ((cid:1))\n\n{\n\n}\n(cid:13)(cid:13) :\n\n(cid:13)(cid:13)z (cid:0) xei(cid:18)\n\ndist(z;X )\n\n:\n= inf\n\n(cid:18)2[0;2(cid:25))\n\nFor any z 2 C with jzj \u0338= 0, we use \u03d5(z) to denote the phase of z, that is, ei\u03d5(z) = z=jzj.\n\n2 Algorithm\n\nWe develop an approach to convolutional phase retrieval based on local nonconvex opti-\nmization. Our proposed algorithm has two components: (1) a careful initialization using\nthe spectral method; (2) local re\ufb01nement by (generalized) gradient descent. We introduce\nthe two steps in reverse order.\n\n2.1 Minimization of a nonconvex and nonsmooth objective\nWe consider minimizing a weighted nonconvex and nonsmooth objective\n\n(cid:13)(cid:13)(cid:13)2\n(cid:13)(cid:13)(cid:13)b1=2 \u2299 (y (cid:0) jAzj)\n\nf (z) =\n\n1\n2m\n\n:\n\n(6)\n\nThe introduction of the positive weights b facilitates our analysis, by enabling us to compare\ncertain functions of the dependent random matrix A to functions involving more indepen-\ndent random variables. We will substantiate this claim in the next section.\nAlthough the function (4) is not complex-di\ufb00erentiable, if one identi\ufb01es Cn with R2n and\ntreats f (z) as a function in the real domain, f is still di\ufb00erentiable in the real sense. Thus,\nwe adopt the Wirtinger calculus [22], which can be thought of as a clean way of organizing\nthe real partial derivatives [29, 34].\n\n3[30] tightened the sample complexity to m (cid:21) \u2126(n log n) by using advanced probability tools.\n\n3\n\n\fOn the other hand, it should also be noted that the absolute value j(cid:1)j is nonsmooth at 0 and\nhence f (z) is not di\ufb00erentiable everywhere even in the real sense. Similar to [38], for any\ncomplex number u 2 C, if we uniquely de\ufb01ne its phase \u03d5(u) at 0 by\n\n{\n\nexp (i\u03d5(u))\n\n:\n=\n\nu=juj\n1\n\nifjuj \u0338= 0;\notherwise;\n\nthen the Wirtinger gradient of (4) can be uniquely determined as\n\n@\n@z\n\nf (z) =\n\n1\nm\n\n(cid:3)\n\ndiag (b) [Az (cid:0) y \u2299 exp (i\u03d5(Az))] :\n\nA\n\n(7)\n\nStarting from some initialization z(0), we minimize the objective (6) by gradient descent\n\nz(r+1) = z(r) (cid:0) (cid:28)\n\n@\n@z\n\nf (z(r));\n\n(8)\n\nwhere (cid:28) > 0 is the stepsize. Indeed, @\nthe real case; this method is also referred to as amplitude \ufb02ow [38].\n\n@z f (z) can be interpreted as the gradient of f (z) as in\n\n2.2\n\nInitialization via spectral method\n\nSimilar to [25, 29], we compute the initialization z(0) via a spectral method, detailed in [29,\nAlgorithm 1]. More speci\ufb01cally, z(0) is a scaled leading eigenvector of\n\n(\n\n)\n\nm\u2211\n\nk=1\n\nY =\n\n1\nm\n\n(cid:3)\ny2\nkaka\nk =\n\n(cid:3)\n\nA\n\n1\nm\n\ndiag\n\ny2\n\nA;\n\n(9)\n\nwhich is constructed from the knowledge of the sensing vectors and observations. The\nleading eigenvector of Y can be e\ufb03ciently computed via the power method. Note that\nE [Y ] = \u2225x\u22252 I + xx\n(cid:3), so the leading eigenvector of E [Y ] is proportional to the optimal so-\nlution x. Under the random convolutional model of A, by using probability tools from [21],\n(cid:3)E [Y ] v for all v 2 CSn(cid:0)1 whenever\nwe show that v\nm (cid:21) \u2126(n poly log n), ensuring the initialization z(0) close to the optimal set X .\n\nY v concentrates to its expectation v\n\n(cid:3)\n\n3 Main Result and Analysis\n\n)\n\nIn this section, we describe our main theoretical result, which shows that with high proba-\nbility, the algorithm described in the previous section succeeds.\nTheorem 3.1 (Main Result) Whenever m (cid:21) C0n log31 n, the spectral method [29, Algorithm 1]\n(\nproduces an initialization z(0) that satis\ufb01es\n(cid:0)6 n\u2225x\u2225\nz(0);X\ndist\n(\n(cid:0)c2. Suppose b = (cid:16)(cid:27)2 (y), where\n\nwith probability at least 1 (cid:0) c1m\n\n(cid:16)(cid:27)2 (t) = 1 (cid:0) 2(cid:25)(cid:27)2(cid:24)(cid:27)2 (t);\n{\n\n(10)\nwith (cid:27)2 > 1=2. Starting from z(0), with (cid:27)2 = 0:51 and stepsize (cid:28) = 2:02, whenever m (cid:21)\n(cid:0)c4 for all iterate z(r)(r (cid:21) 1)\n)\nC1\nde\ufb01ned in (8), we have\n\n}\n, with probability at least 1 (cid:0) c3m\n(\n\n\u2225Cx\u22252\n\u2225x\u22252 max\n\n1\n2(cid:25)(cid:27)2 exp\n\nlog17 n; n log4 n\n\n(cid:20) c0 log\n\n(cid:0) t2\n2(cid:27)2\n\n(cid:24)(cid:27)2(t) =\n\n;\n\nt > 0;\n\n)\n\nz(r);X\n\n(cid:20) (1 (cid:0) \u03f1)r dist\n\nz(0);X\n\ndist\n\n;\n\n(11)\n\n)\n\n(\n\nholds for a small scalar \u03f1 2 (0; 1). Here, c0; c1; c2; c3; c4; C0; C1 > 0 are numerical constants.\n\n4\n\n\f)\n\n(\n\nRemark: Our result shows that by initializing the problem O(1=polylog(n))-close to the\noptimum via spectral method, the gradient descent (8) converges linearly to the optimal so-\nlution. As we can see, the sample complexity here also depends on \u2225Cx\u2225, which is quite dif-\nferent from the i.i.d. case. For a typical x 2 CSn(cid:0)1 (e.g., x is drawn uniformly random from\nCSn(cid:0)1), \u2225Cx\u2225 remains as O(log n), the sample complexity m (cid:21) \u2126 (n poly log n) matches the\ni.i.d. case up to log factors. However, \u2225Cx\u2225 is nonhomogeneous over x 2 CSn(cid:0)1: if x is\nsparse in the Fourier domain (e.g., x = 1p\nn 1), the sample complexity can be as large as\nm (cid:21) \u2126\n. Such a behavior is also demonstrated in the experiments of Section\n4. We believe the (very large!) number of logarithms in our result is an artifact of our anal-\nysis, rather than a limitation of the method. We expect to reduce the sample complexity\nto m (cid:21) \u2126\nby a tighter analysis, which is left for future work. The choices\nof the weighting b 2 Rm in (10), (cid:27)2 = 0:51, and the stepsize (cid:28) = 2:02 are purely for the\npurpose of analysis. In practice, the algorithm converges with b = 1 and a choice of small\nstepsize (cid:28), or by using backtracking linesearch for the stepsize (cid:28).\nIn the following, we brie\ufb02y highlight some major challenges and novel proo\ufb01ng ideas be-\nhind the analysis. The details can be found in our full paper.\n\n(\u2225Cx\u22252\n\n\u2225x\u22252 n log6 n\n\nn2 poly log n\n\n)\n\n3.1 Proof sketch of iterative contraction\nOur analysis is largely inspired by the recent analysis of alternating direction method (ADM)\n[35]. In this following, we draw connections between the gradient descent method (8) and\nADM, and sketch basic ideas of convergence analysis.\n\nADM iteration. ADM is a classical method for solving phase retrieval problems [16, 25,\n35], which can be considered as a heuristic method that solves the problem\n\n\u2225Az (cid:0) y \u2299 u\u22252 :\n\n1\n2\n\n(cid:3)\n\nA\n\nA\n\nm\n\n5\n\n(\n\n;\n\n(\n\n(cid:3)\n\nA)\n\n;\n\n;\n\n1\n2\n\nz\n\nwhere A\n\ny\n\ny\n\ny \u2299 exp\n\nmin\n\nz2Cn;juj=1\n\n(cid:0)1 A\n)\n\nc(r+1) = y \u2299 exp\n\n= (A\n\n(bz(r+1);X\n\nwhich leads to the following update\n\n(cid:13)(cid:13)(cid:13)Az (cid:0) c(r+1)\n\nAt every iteratebz(r), ADM proceeds in two steps:\n\n(cid:13)(cid:13)(cid:13)2\n(\n)\nAbz(r)\nbz(r+1) = arg min\n(\n))\nbz(r+1) = A\nAbz(r)\n(cid:13)(cid:13)(cid:13)bz(r) (cid:0) xei(cid:18)\n(cid:3) is the pseudo-inverse of A. Letb(cid:18)r = arg min(cid:18)\ndistance betweenbz(r+1) and X is bounded by\n))(cid:13)(cid:13)(cid:13) :\ny(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Axeib(cid:18)r (cid:0)\n(cid:13)(cid:13)(cid:13) (cid:20)(cid:13)(cid:13)A\n(cid:13)(cid:13)(cid:13)bz(r+1) (cid:0) xeib(cid:18)r+1\n(\nAbz(r)\n(cid:13)(cid:13)(cid:13), with stepsize (cid:28) = 1. The distance between\n(cid:13)(cid:13)(cid:13)z(r) (cid:0) xei(cid:18)\n(cid:13)(cid:13)(cid:13)(cid:13)I (cid:0) 1\n(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)z(r) (cid:0) xei(cid:18)r\n(cid:13)(cid:13)(cid:13)\n(cid:13)(cid:13)(cid:13) (cid:20)\n(cid:13)(cid:13)(cid:13)z(r+1) (cid:0) xei(cid:18)r+1\n(cid:13)(cid:13)(cid:13)Axei(cid:18)r (cid:0) y \u2299 exp\n(\n(cid:13)(cid:13)(cid:13)(cid:13) = o(1);\n(cid:13)(cid:13)(cid:13)(cid:13)I (cid:0) 1\ny(cid:13)(cid:13) (cid:25) 1=\n\nGradient descent with b = 1. For simplicity, let us consider the gradient descent update\n(8) with b = 1. Let (cid:18)r = arg min(cid:18)\nthe iterate z(r) and the optimal set X is bounded by\n\nTowards iterative contraction. By measure concentration, it can be shown that\n\n(cid:13)(cid:13)(cid:13). The\n\n\u2225A\u2225 (cid:25) p\n\n)(cid:13)(cid:13)(cid:13) :\n\nz(r+1);X\n\n(cid:13)(cid:13)A\n\nm\n\u2225A\u2225\n\ny \u2299 exp\n\np\n\nm;\n\ni\u03d5(Az(r))\n\n+\n\n1\nm\n\n)\n\n=\n\ndist\n\n=\n\n(13)\n\n(14)\n\n(\n\ndist\n\n(cid:3)\n\nA\n\nA\n\nm;\n\n(12)\n\n\fholds with high probability whenever m (cid:21) \u2126 (n poly log n). Therefore, to show iterative\ncontraction of both methods, based on (12) and (13), it is su\ufb03cient to show that\n\n(cid:13)(cid:13)Axei(cid:18) (cid:0) y \u2299 exp (i\u03d5(Az))\n\n(cid:13)(cid:13) (cid:20) (1 (cid:0) (cid:17))\n\nm\n\n(15)\nfor some constant (cid:17) 2 (0; 1), where (cid:18) = arg min(cid:18)2[0;2(cid:25))\nzj.\n(cid:3)\nBy similar ideas of controlling (15) for the ADM method [35], this observation provides a\nnew way of analyzing the gradient descent method. As an attempt to show (15) for the\nrandom circulant matrix A, we invoke the following lemma, which controls the error in a\n\ufb01rst order approximation to exp(i\u03d5((cid:1))).\nLemma 3.2 (Lemma 3.2, [35]) For any (cid:26) > 0, and for any z; z\n\n\u2032 2 C, we have\n\nz=jx\n\n(cid:3)\n\n(cid:13)(cid:13) ;\n\n(cid:13)(cid:13)z (cid:0) xei(cid:18)\n(cid:13)(cid:13)(cid:13) such that ei(cid:18) = x\n\np\n\n(cid:13)(cid:13)(cid:13)z (cid:0) xei(cid:18)\n\nLet us decompose z = (cid:11)x + (cid:12)w, where w 2 CSn(cid:0)1 with w ? x, and (cid:11); (cid:12) 2 C. Note that\n\u03d5((cid:11)) = (cid:18). Then by Lemma 3.2, for any (cid:26) 2 (0; 1), we have\n\njexp (i\u03d5(z\n\n\u2032\n\n+ z)) (cid:0) exp (i\u03d5(z\n\n(cid:13)(cid:13)Axei(cid:18) (cid:0) y \u2299 exp (i\u03d5(Az))\n\n(cid:13)(cid:13)(cid:13)jAxj \u2299 \u001fj (cid:12)\n{z\n|\n\n(cid:11)\n\n(cid:20)\n\n(cid:13)(cid:13) =\n(cid:13)(cid:13)(cid:13)\n}\n\njjAwj(cid:21)(cid:26)jAxj\nT1\n\n+\n\n)j :\n\n\u2032\n\n\u2032\n\n(cid:0)1 j\u2111 (z=z\n))j (cid:20) 2\u001fjzj(cid:21)(cid:26)jz\u2032j + (1 (cid:0) (cid:26))\n(cid:13)(cid:13)(cid:13)(cid:13)jAxj \u2299\n(\n(\n[\n(cid:12)(cid:12)(cid:12)(cid:12)\u2225\u2111 ((Aw) \u2299 exp ((cid:0)i\u03d5(Ax)))\u2225\n(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)\nexp (i\u03d5 (Ax)) (cid:0) exp\n{z\n|\n}\n\n1\n1 (cid:0) (cid:26)\n\nAx +\n\ni\u03d5\n\n(cid:11)\n\nAw\n\n(cid:12)\n(cid:11)\n\n:\n\n))](cid:13)(cid:13)(cid:13)(cid:13)\n\nT2\n\nThe \ufb01rst term T1 can be bounded using the restricted isometry property of random circulant\nmatrices [20], together with some auxiliary analysis.\nThe second term T2 involves a nonlinear function exp ((cid:0)i\u03d5(Ax)) of the random circulant\nmatrix A. Controlling this nonlinear, highly dependent random process T2(w) for all w\nis a nontrivial task. Next, we explain why controlling T2 is technically challenging, and\nsketch the key ideas about how to control a smoothed variant of T2, by using the weighting\nb = (cid:16)(cid:27)2(y) introduced in (10). We also provide intuition for why the weighting b is helpful.\n3.2 Controlling the phase term T2\n{\nAs elaborated above, the major challenge of showing iterative contraction is bound-\ning the suprema of the nonlinear, dependent random process T2(w) over the set S =\nw 2 CSn(cid:0)1 j w ? x\n\n. By using the fact that \u2111(u) = 1\n\n}\n\n(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)w\n|\n\n2i (u (cid:0) u) for any u 2 C, we have\n{z\n\ndiag ( (Ax)) Aw\n\n}\n\nL(a;w)\n\n(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ;\n\nm, the major task left is to show that\n\njL(a; w)j < (1 (cid:0) (cid:17)\n\n\u2032\n\n)m\n\n(16)\n\nT 2\n\n2\n\n(cid:20) 1\n\n2\n\nsup\nw2S\n\n\u2225A\u22252 + 1\n\n2 sup\nw2S\n\n\u22a4\n\n\u22a4\n\nA\n\nwhere (t)\n\n= exp ((cid:0)2i\u03d5(t)). As from (14), \u2225A\u2225 (cid:25) p\n\n:\n\n\u2032 2 (0; 1).\nfor some constant (cid:17)\nWhy decoupling? Let A =\n\n[\n\nsup\nw2S\n\n]\n\n(cid:3)\na\n1(cid:1)(cid:1)(cid:1)\n(cid:3)\na\nm\n\n. The term\n\nL(a; w) = w\n\n\u22a4\n\nA\n\n\u22a4\n\ndiag ( (Ax)) Aw =\n\nm\u2211\n\nk=1\n\n|\n\n(cid:3)\n (a\nkx)w\n\n{z\n\n\u22a4\n\n}\n\n\u22a4\nk w\n\naka\n\ndependence across k\n\nis a summation of dependent random variables, for which our probability tools are very\nlimited. To overcome this problem, we deploy ideas from decoupling [11]. Informally, decou-\npling allows us to compare moments of the original random function to functions of more\nindependent random variables, which are usually easier to analyze. The book [11] provides\n\n6\n\n\f\u22a4\n\n; w) = w\n\nQL\n\u2032\ndec(a; a\n\na beautiful introduction to this area. In our problem, notice that the random vector a oc-\ncurs twice in the de\ufb01nition of L(a; w) \u2013 one in the phase term (Ax) = exp((cid:0)2i\u03d5(Ax)),\nand another in the quadratic term. The general spirit of decoupling is to seek to replace one\n\u2032 of the same random vector, yielding a random process with\na with an independent copy a\nfewer dependencies. Here, we seek to replace L(a; w) with\n\u2032\n\n\u22a4\n(17)\nThe usefulness of this new, decoupled form QL\n; w), is that it introduces extra ran-\ndomness \u2014 QL\n\u2032\n\u2032. This makes analyz-\n; w) is now a chaos process of a conditioned on a\ndec(a; a\ning supw2S QL\n\u2032\n; w) amenable to existing analysis of suprema of chaos processes for ran-\ndec(a; a\ndom circulant matrices [21]. However, achieving the decoupling requires additional work;\nthe most general existing results on decoupling pertain to tetrahedral polynomials, which are\npolynomials with no monomials involving any power larger than one of any random vari-\nable. By appropriately tracking cross terms, these results can also be applied to more gen-\neral (non-tetrahedral) polynomials in Gaussian random variables [23]. However, our ran-\ndom process L(a; w) involves a nonlinear phase term (Aw) which is not a polynomial,\nand hence is not amenable to a direct appeal to existing results.\n\n\u2032\ndec(a; a\n\ndiag ( (A\n\nx)) Aw:\n\nA\n\nDecoupling is \u201crecoupling\u201d. Existing results [23] for decoupling polynomials of Gaus-\nsian random variables are derived from two simple facts: (i) orthogonal projections of Gaus-\nsian variables are independent, and (ii) Jensen\u2019s inequality. Indeed, for a (cid:24) CN (0; I), let us\nintroduce an independent vector (cid:14) (cid:24) CN (0; I). Write\n\ng1 = a + (cid:14);\n\ng2 = a (cid:0) (cid:14):\n\nw2S\n\n7\n\n:\n\n(\n\nE(cid:14)\n\n[\n\nw2S \u03c6\nsup\n\n]\n\n[\n\n= E(cid:14)\n\n(cid:20) Ea;(cid:14)\n\nsup\nw2S \u03c6\n\n= Eg1;g2\n\nsup\nw2S \u03c6\n\n= Ea\n\nE(cid:14)\n\nw2S \u03c6\nsup\n\n\u2032\ndec(a; a\n\ndec(g1; g2; w)\n\n:\n\ndec(g1; g2; w)\n\n)]\n(cid:12)(cid:12)QL\n\ndec(a + (cid:14); a (cid:0) (cid:14); w)\n\ndec(a + (cid:14); a (cid:0) (cid:14); w)\n\n= bL(a; w):\n])]\n\nBecause of Fact (i), these are independent CN (0; 2I) vectors. By conditional expectation,\n\n]\n(18)\nThus, we can see that the key idea of decoupling L(a; w) into QL\n; w), is essentially\n\u201crecoupling\u201d QL\nviewed as an approximation of L(a; w). Notice that by Fact (ii), for any convex function \u03c6,\nEa\n\n[QL\ndec(g1; g2; w) via conditional expectation \u2013 the \u201crecoupled\u201d term bL can be\n)]\n(bL(a; w)\n[\n)]\n(QL\nThus, by choosing \u03c6(t) = jtjp, we can control moments of supw2S bL(a; w) via\n(cid:13)(cid:13)(cid:13)(cid:13) sup\nFor tetrahedral polynomials, bL = L, so the approximation is exact. As the tail bound of\n(cid:12)(cid:12)(cid:12) can be controlled via its moments bounds [13, Chapter 7.2], this allows\n\n[QL\n[QL\n(cid:13)(cid:13)(cid:13)(cid:13) sup\n\n(QL\n(cid:12)(cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13)\n(cid:12)(cid:12)(cid:12)bL(a; w)\n\n[\ndec(a + (cid:14); a (cid:0) (cid:14); w)\n\nsupw2S\nus to directly control the object L(a; w) of interest. The reason that this control obtains is\nbecause the conditional expectation operator E(cid:14) [(cid:1) j a] \u201crecouples\u201d QL\n; w) back to\nthe target L(a; w). In slogan form, (Gaussian) decoupling is recoupling.\n\u201cRecoupling\u201d is Gaussian smoothing. A distinctive feature in convolutional phase re-\ntrieval is that L is not a polynomial. Hence, it may be challenging to posit a QL\ndec which\n\u201crecouples\u201d back to L. In other words, in the existing form, we need to tolerate an approxi-\n\nmation error as bL \u0338= L. By the triangle inequality,\n(cid:12)(cid:12)(cid:12) + sup\n(cid:12)(cid:12)(cid:12)bL(a; w)\n(cid:12)(cid:12)(cid:12) can be sharply controlled via its moments bound\n(cid:12)(cid:12)(cid:12)bL(a; w)\n\nAs discussed above, the supw2S\nin (19). Now the bound (20) is useful to derive tight control for L(a; w), if L(a; w) is very\n\n(cid:12)(cid:12)(cid:12) :\n(cid:12)(cid:12)(cid:12)bL(a; w) (cid:0) L(a; w)\n\njL(a; w)j (cid:20) sup\nw2S\n\n(cid:12)(cid:12)(cid:12)bL(a; w)\n\ndec(g1; g2; w)\n\n\u2032\ndec(a; a\n\n(cid:20)\n\nLp\n\n(cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13)\n\n:\n\nLp\n\nw2S\n\nw2S\n\nsup\nw2S\n\n(19)\n\n(20)\n\n\fclose to bL(a; w) uniformly. The question is: for what L is it possible to \ufb01nd a \u201cwell-behaved\u201d\nmechanism that links Qdec back to bL is the conditional expectation operator E(cid:14) [(cid:1) j a]. For\n\nQL\ndec for which the approximation error is small? To understand this question, recall that the\n\nour case, from (18) orthogonality leads to\n\nbL(a; w) = wA\n\n\u22a4\n\nThus, by combining the results in (20) and (21), we have\n\ndiag (h(Ax)) Aw;\n\ns(cid:24)CN (0;\u2225x\u22252) [ (t + s)] :\n\n(cid:12)(cid:12)(cid:12)bL(a; w)\n\n:\n= E\n\nh(t)\n\n(cid:12)(cid:12)(cid:12) + \u2225h (cid:0) \u2225\n{z\n\n|\n\n}\n\nL1\n\n\u2225A\u22252 :\n\napproximation error\n\njL(a; w)j (cid:20) sup\nw2S\n\nsup\nw2S\n\n(21)\n\n(22)\n\nTherefore, L(a; w) is a poor target for approximation with a smooth function bL = E(cid:14)[QL\n\nNote that the function h is not exactly , but generated by convolving with a multivariate\nGaussian pdf : indeed, recoupling is Gaussian smoothing. The Fourier transform of a multivari-\nate Gaussian is again a Gaussian; it decays quickly with frequency. So, in order to admit a\nsmall approximation error, the target L must be smooth. However, in our case, the function\n (t) = exp((cid:0)2i\u03d5(t)) is discontinuous at t = 0; it changes extremely rapidly in the vicinity of\nt = 0, and hence its Fourier transform (appropriately de\ufb01ned) does not decay quickly at all.\ndec].\nFrom Fig. 1, the di\ufb00erence between h and increases as jtj \u2198 0. The poor approximation\nerror \u2225h (cid:0) \u2225\nDecoupling and convolutional phase retrieval. The key idea to reduce the approxima-\ntion error \u2225 (cid:0) h\u2225\nL1 = 1 is to smooth . More speci\ufb01cally, we introduce a new objective\n(6) with Gaussian weighting b = (cid:16)(cid:27)2 (y) in (10), replacing the analyzing target T2 with\n\nL1 = 1 results in a trivial bound for supw2S jL(a; w)j instead of (16).\n\n\u2111 ((Aw) \u2299 exp ((cid:0)i\u03d5(Ax)))\nConsequently, we obtain a smoothed variant Ls(a; w) of L(a; w),\n\nb1=2\n\nA\n\nLs(a; w) = w\n\ndiag ((cid:16)(cid:27)2(y) \u2299 (Ax)) Aw:\n\n\u22a4\nNow the approximation error \u2225h (cid:0) \u2225\nL1. As\nobserved from Fig. 1, the function (cid:16)(cid:27)2(t) smoothes (t) especially near the vicinity of t =\n0, such that the new approximation error \u2225f (t) (cid:0) (cid:16)(cid:27)2(t) (t)\u2225\nL1 is signi\ufb01cantly reduced.\nThus, by using similar ideas as above, we can prove a desired bound supw2S jLs(a; w)j <\n(1 (cid:0) (cid:17)s)m. Finally, because the new weighting b = (cid:16)(cid:27)2 (y), the overall analysis needs to be\nslightly modi\ufb01ed correspondingly. We refer the readers to our full paper for more details.\n\n\u22a4\nL1 in (22) is replaced by \u2225h(t) (cid:0) (cid:16)(cid:27)2(t) (t)\u2225\n\n(\n\n(cid:13)(cid:13)(cid:13)diag\n\nbT2 =\n\n)\n\n(cid:13)(cid:13)(cid:13) :\n\nFigure 1: Plots of functions (cid:16)(cid:27)2(t),\nf (t) and (t) for t 2 R+.\n\nFigure 2: Phase transition for recov-\nering the signal x 2 CSn(cid:0)1 with dif-\nferent \u2225Cx\u2225.\n\n4 Experiments\nDependence of sample complexity on \u2225Cx\u2225. First, we investigate the dependence of the\nsample complexity m on \u2225Cx\u2225. We assume the ground truth x 2 CSn(cid:0)1, and consider three\n\n8\n\n\fcases: (1) x = e1 with \u2225Cx\u2225 = 1, where e1 the standard basis vector; (2) x is uniformly ran-\ndom generated from CSn(cid:0)1; (3) x = 1p\nn. For each case, we \ufb01x the signal\nlength n = 1000 and vary the ratio m=n. For each ratio m=n, we randomly generate the\nkernel a (cid:24) CN (0; I) and repeat the experiment for 100 times. We initialize the algorithm\nby the spectral method [29, Algorithm 1] and run the gradient descent (8). Given the al-\n(cid:0)5. From Fig. 2, we can see that the larger the \u2225Cx\u2225, the more samples are needed\n\ngorithm output bx, we judge the success of recovery by inf \u03d52[0;2(cid:25))\n\nn 1, with \u2225Cx\u2225 =\n\n\u03f5 = 10\nfor exact recovery.\n\np\n\n(cid:13)(cid:13)bx (cid:0) xei\u03d5\n\n(cid:13)(cid:13) (cid:20) \u03f5, where\n\nFigure 3: Experiment on real images.\n\nExperiments on real image. Finally, we run the experiment on some real dataset to demon-\nstrate the e\ufb00ectiveness and the e\ufb03ciency of the proposed method. We choose an image of\nsize 200 (cid:2) 300 as in Fig. 4, we use m = 5n log n samples for reconstruction. The kernel\na 2 Cm is randomly generated as complex Gaussian CN (0; I). We run power method\nfor 100 iterations for initialization, and stop the algorithm once the error is smaller than\n1 (cid:2) 10\n(cid:0)4. It takes 197:08s to reconstruct all the RGB channels. Experiment using general\nGaussian measurements A 2 Cm(cid:2)n could easily run out of memory on a personal computer\nfor problems of this size.\n\nFigure 4: Experiment with real antenna pattern.\n\nExperiments on signal AoA phase recovery for 5G communications. Finally, we demon-\nstrate the e\ufb00ectiveness of the proposed method on a problem arising in 5G communication,\nas we mentioned in the introduction. Fig. 4 (left) shows an antenna pattern a 2 C361 ob-\ntained from Bell labs. We observe the modulus of the convolution of this pattern with the\nsignal of interest. For three di\ufb00erent types of signals with length n = 20, (1) x = e1 , (2) x is\nuniformly random generated from CSn(cid:0)1, (3) x = 1p\nn 1, our result in Fig. 4 shows that we\ncan achieve almost perfect recovery.\n\n9\n\n\f5 Acknowledgement\n\nThis work was partially supported by the grants NSF CCF 1527809 and NSF IIS 1546411,\nthe grants from the European Unions Horizon 2020 research and innovation program un-\nder grant agreement No. 646804-ERCCOGBNYQ, and the grant from the Israel Science\nFoundation under grant no. 335/14. QQ thanks the generous support of the Microsoft\ngraduate research fellowship. We would like to thank Shan Zhong for the helpful discus-\nsion for real applications and providing the antenna data for experiments, and we thank Ju\nSun and Han-wen Kuo for helpful discussion and input regarding the analysis of this work.\n\nReferences\n[1] Sercan \u00d6. Arik and Joseph M. Kahn. Direct-detection mode-division multiplexing in modal basis\n\nusing phase retrieval. Opt. Lett., 41(18):4265\u20134268, Sep 2016.\n\n[2] T. Bendory, Y. C. Eldar, and N. Boumal. Non-convex phase retrieval from stft measurements.\n\nIEEE Transactions on Information Theory, PP(99):1\u20131, 2017.\n\n[3] Emmanuel J. Cand\u00e8s, Yonina C. Eldar, Thomas Strohmer, and Vladislav Voroninski. Phase re-\n\ntrieval via matrix completion. SIAM Journal on Imaging Sciences, 6(1), 2013.\n\n[4] Emmanuel J. Cand\u00e8s, Xiaodong Li, and Mahdi Soltanolkotabi. Phase retrieval from coded di\ufb00rac-\n\ntion patterns. Applied and Computational Harmonic Analysis, 39(2):277\u2013299, 2015.\n\n[5] Emmanuel J. Cand\u00e8s, Xiaodong Li, and Mahdi Soltanolkotabi. Phase retrieval via wirtinger \ufb02ow:\n\nTheory and algorithms. Information Theory, IEEE Transactions on, 61(4):1985\u20132007, April 2015.\n\n[6] Emmanuel J Cand\u00e8s, Justin Romberg, and Terence Tao. Robust uncertainty principles: Exact\nsignal reconstruction from highly incomplete frequency information. IEEE Transactions on infor-\nmation theory, 52(2):489\u2013509, 2006.\n\n[7] Emmanuel J Candes, Justin K Romberg, and Terence Tao. Stable signal recovery from incomplete\nand inaccurate measurements. Communications on pure and applied mathematics, 59(8):1207\u20131223,\n2006.\n\n[8] Emmanuel J. Cand\u00e8s, Thomas Strohmer, and Vladislav Voroninski. Phaselift: Exact and stable\nsignal recovery from magnitude measurements via convex programming. Communications on\nPure and Applied Mathematics, 66(8):1241\u20131274, 2013.\n\n[9] Emmanuel J Candes and Terence Tao. Near-optimal signal recovery from random projections:\n\nUniversal encoding strategies? IEEE transactions on information theory, 52(12):5406\u20135425, 2006.\n\n[10] Yuxin Chen and Emmanuel J. Cand\u00e8s. Solving random quadratic systems of equations is nearly\n\nas easy as solving linear systems. arXiv preprint arXiv:1505.05114, 2015.\n\n[11] Victor De la Pena and Evarist Gin\u00e9. Decoupling: from dependence to independence. Springer, 1999.\n\n[12] Yonina C Eldar and Gitta Kutyniok. Compressed sensing: theory and applications. Cambridge Uni-\n\nversity Press, 2012.\n\n[13] Simon Foucart and Holger Rauhut. A mathematical introduction to compressive sensing. Springer,\n\n2013.\n\n[14] Robert M Gagliardi and Sherman Karp. Optical communications. New York, Wiley-Interscience,\n\n1976. 445 p., 1, 1976.\n\n[15] Rong Ge, Furong Huang, Chi Jin, and Yang Yuan. Escaping from saddle points\u2014online stochas-\ntic gradient for tensor decomposition. In Proceedings of The 28th Conference on Learning Theory,\npages 797\u2013842, 2015.\n\n[16] R. W. Gerchberg and W. Owen Saxton. A practical algorithm for the determination of the phase\n\nfrom image and di\ufb00raction plane pictures. Optik, 35:237\u2013246, 1972.\n\n[17] David Gross, Felix Krahmer, and Richard Kueng. A partial derandomization of phaselift using\n\nspherical designs. arXiv preprint arXiv:1310.2267, 2013.\n\n10\n\n\f[18] Felix Heide, Wolfgang Heidrich, and Gordon Wetzstein. Fast and \ufb02exible convolutional sparse\ncoding. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages\n5135\u20135143, 2015.\n\n[19] Kishore Jaganathan, Yonina C. Eldar, and Babak Hassibi. Phase retrieval: An overview of recent\n\ndevelopments. Chapter V, Optical Compressive Imaging, 2016.\n\n[20] Felix Krahmer, Shahar Mendelson, and Holger Rauhut. Suprema of chaos processes and the\nrestricted isometry property. Communications on Pure and Applied Mathematics, 67(11):1877\u20131904,\n2014.\n\n[21] Felix Krahmer and Holger Rauhut. Structured random measurements in signal processing.\n\nGAMM-Mitteilungen, 37(2):217\u2013238, 2014.\n\n[22] Ken Kreutz-Delgado. The complex gradient operator and the CR-calculus.\n\narXiv:0906.4835, 2009.\n\narXiv preprint\n\n[23] Stanislaw Kwapien. Decoupling inequalities for polynomial chaos. The Annals of Probability,\n\npages 1062\u20131071, 1987.\n\n[24] Antonio Mecozzi, Cristian Antonelli, and Mark Shtaif. Kramers\u2013kronig coherent receiver. Optica,\n\n3(11):1220\u20131227, Nov 2016.\n\n[25] Praneeth Netrapalli, Prateek Jain, and Sujay Sanghavi. Phase retrieval using alternating mini-\n\nmization. In Advances in Neural Information Processing Systems, pages 2796\u20132804, 2013.\n\n[26] Holger Rauhut. Compressive sensing and structured random matrices. Theoretical foundations\n\nand numerical methods for sparse recovery, 9:1\u201392, 2010.\n\n[27] Arash Shahmansoori, Gabriel E Garcia, Giuseppe Destino, Gonzalo Seco-Granados, and Henk\nWymeersch. 5g position and orientation estimation through millimeter wave mimo. In Globecom\nWorkshops (GC Wkshps), 2015 IEEE, pages 1\u20136. IEEE, 2015.\n\n[28] Yoav Shechtman, Yonina C. Eldar, Oren Cohen, Henry N. Chapman, Jianwei Miao, and\nMordechai Segev. Phase retrieval with application to optical imaging: A contemporary overview.\nSignal Processing Magazine, IEEE, 32(3):87\u2013109, May 2015.\n\n[29] Mahdi Soltanolkotabi. Algorithms and theory for clustering and nonconvex quadratic programming.\n\nPhD thesis, Stanford University, 2014.\n\n[30] Mahdi Soltanolkotabi. Structured signal recovery from quadratic measurements: Breaking sam-\n\nple complexity barriers via nonconvex optimization. CoRR, abs/1702.06175, 2017.\n\n[31] Milica Stojanovic, Josko A Catipovic, and John G Proakis. Phase-coherent digital communica-\ntions for underwater acoustic channels. IEEE Journal of Oceanic Engineering, 19(1):100\u2013111, 1994.\n[32] Ju Sun, Qing Qu, and John Wright. Complete dictionary recovery over the sphere. arXiv preprint\n\narXiv:1504.06785, 2015.\n\n[33] Ju Sun, Qing Qu, and John Wright. When are nonconvex problems not scary? arXiv preprint\n\narXiv:1510.06096, 2015.\n\n[34] Ju Sun, Qing Qu, and John Wright. A geometric analysis of phase retreival. arXiv preprint\n\narXiv:1602.06664, 2016.\n\n[35] Ir\u00e8ne Waldspurger. Phase retrieval with random gaussian sensing vectors by alternating projec-\n\ntions. arXiv preprint arXiv:1609.03088, 2016.\n\n[36] Ir\u00e8ne Waldspurger, Alexandre d`Aspremont, and St\u00e9phane Mallat. Phase recovery, maxcut and\n\ncomplex semide\ufb01nite programming. Mathematical Programming, 149(1-2):47\u201381, 2015.\n\n[37] P. Walk, H. Becker, and P. Jung. OFDM channel estimation via phase retrieval. In Asilomar 2015,\n\n2015.\n\n[38] G. Wang, G. B. Giannakis, and Y. C. Eldar. Solving systems of random quadratic equations via\n\ntruncated amplitude \ufb02ow. IEEE Transactions on Information Theory, PP(99):1\u20131, 2017.\n\n[39] Huishuai Zhang and Yingbin Liang. Reshaped wirtinger \ufb02ow for solving quadratic systems of\n\nequations. arXiv preprint arXiv:1605.07719, 2016.\n\n[40] Yuqian Zhang, Yenson Lau, Han-wen Kuo, Sky Cheung, Abhay Pasupathy, and John Wright. On\nthe global geometry of sphere-constrained sparse blind deconvolution. In Proceedings of the IEEE\nConference on Computer Vision and Pattern Recognition, 2017.\n\n11\n\n\f", "award": [], "sourceid": 3092, "authors": [{"given_name": "Qing", "family_name": "Qu", "institution": "Columbia University"}, {"given_name": "Yuqian", "family_name": "Zhang", "institution": "Columbia University"}, {"given_name": "Yonina", "family_name": "Eldar", "institution": "Israel Institute of Technology"}, {"given_name": "John", "family_name": "Wright", "institution": "Columbia University"}]}