{"title": "The Landscape of Non-convex Empirical Risk with Degenerate Population Risk", "book": "Advances in Neural Information Processing Systems", "page_first": 3507, "page_last": 3517, "abstract": "The landscape of empirical risk has been widely studied in a series of machine learning problems, including low-rank matrix factorization, matrix sensing, matrix completion, and phase retrieval. In this work, we focus on the situation where the corresponding population risk is a degenerate non-convex loss function, namely, the Hessian of the population risk can have zero eigenvalues. Instead of analyzing the non-convex empirical risk directly, we first study the landscape of the corresponding population risk, which is usually easier to characterize, and then build a connection between the landscape of the empirical risk and its population risk. In particular, we establish a correspondence between the critical points of the empirical risk and its population risk without the strongly Morse assumption, which is required in existing literature but not satisfied in degenerate scenarios. We also apply the theory to matrix sensing and phase retrieval to demonstrate how to infer the landscape of empirical risk from that of the corresponding population risk.", "full_text": "The Landscape of Non-convex Empirical Risk with\n\nDegenerate Population Risk\n\nShuang Li, Gongguo Tang, and Michael B. Wakin\n\nDepartment of Electrical Engineering\n\nColorado School of Mines\n\nGolden, CO 80401\n\n{shuangli,gtang,mwakin}@mines.edu\n\nAbstract\n\nThe landscape of empirical risk has been widely studied in a series of machine\nlearning problems, including low-rank matrix factorization, matrix sensing, matrix\ncompletion, and phase retrieval. In this work, we focus on the situation where the\ncorresponding population risk is a degenerate non-convex loss function, namely, the\nHessian of the population risk can have zero eigenvalues. Instead of analyzing the\nnon-convex empirical risk directly, we \ufb01rst study the landscape of the corresponding\npopulation risk, which is usually easier to characterize, and then build a connection\nbetween the landscape of the empirical risk and its population risk. In particular,\nwe establish a correspondence between the critical points of the empirical risk and\nits population risk without the strongly Morse assumption, which is required in\nexisting literature but not satis\ufb01ed in degenerate scenarios. We also apply the theory\nto matrix sensing and phase retrieval to demonstrate how to infer the landscape of\nempirical risk from that of the corresponding population risk.\n\n1\n\nIntroduction\n\nUnderstanding the connection between empirical risk and population risk can yield valuable insight\ninto an optimization problem [1, 2]. Mathematically, the empirical risk f (x) with respect to a\nparameter vector x is de\ufb01ned as\n\nM(cid:88)\n\nm=1\n\nf (x) (cid:44) 1\nM\n\nL(x, ym).\n\nHere, L(\u00b7) is a loss function and we are interested in losses that are non-convex in x in this work.\ny = [y1,\u00b7\u00b7\u00b7 , yM ](cid:62) is a vector containing the random training samples, and M is the total number of\nsamples contained in the training set. The population risk, denoted as g(x), is the expectation of the\nempirical risk with respect to the random measure used to generate the samples y, i.e., g(x) = Ef (x).\nRecently, the landscapes of empirical and population risk have been extensively studied in many\n\ufb01elds of science and engineering, including machine learning and signal processing. In particular, the\nlocal or global geometry has been characterized in a wide variety of convex and non-convex problems,\nsuch as matrix sensing [3, 4], matrix completion [5, 6, 7], low-rank matrix factorization [8, 9, 10],\nphase retrieval [11, 12], blind deconvolution [13, 14], tensor decomposition [15, 16, 17], and so on.\nIn this work, we focus on analyzing global geometry, which requires understanding not only regions\nnear critical points but also the landscape away from these points.\nIt follows from empirical process theory that the empirical risk can uniformly converge to the\ncorresponding population risk as M \u2192 \u221e [18]. A recent work [1] exploits the uniform convergence\nof the empirical risk to the corresponding population risk and establishes a correspondence of their\n\n33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Vancouver, Canada.\n\n\fcritical points when provided with enough samples. The authors build their theoretical guarantees\nbased on the assumption that the population risk is strongly Morse, namely, the Hessian of the\npopulation risk cannot have zero eigenvalues at or near the critical points1. However, many problems\nof practical interest do have Hessians with zero eigenvalues at some critical points. We refer to such\nproblems as degenerate. To illustrate this, we present the very simple rank-1 matrix sensing and\nphase retrieval examples below.\nExample 1.1. (Rank-1 matrix sensing). Given measurements ym = (cid:104)Am, x(cid:63)x(cid:63)(cid:62)(cid:105), 1 \u2264 m \u2264 M,\nwhere x(cid:63) \u2208 RN and Am \u2208 RN\u00d7N denote the true signal and the m-th Gaussian sensing matrix with\nentries following N (0, 1), respectively. The following empirical risk is commonly used in practice\n\nf (x) =\n\n1\n4M\n\nThe corresponding population risk is then\n\nM(cid:88)\n\n(cid:0)\n\nm=1\n\n(cid:1)2\n\n.\n\n(cid:104)Am, xx(cid:62)(cid:105) \u2212 ym\n\ng(x) = Ef (x) =\n\n1\n4(cid:107)xx(cid:62) \u2212 x(cid:63)x(cid:63)(cid:62)(cid:107)2\nF .\n\nElementary calculations give the gradient and Hessian of the above population risk as\nand \u22072g(x) = 2xx(cid:62) \u2212 x(cid:63)x(cid:63)(cid:62) + (cid:107)x(cid:107)2\n\n\u2207g(x) = (xx(cid:62) \u2212 x(cid:63)x(cid:63)(cid:62))x,\n\n2IN .\n\nWe see that g(x) has three critical points x = 0, \u00b1 x(cid:63). Observe that the Hessian at x = 0 is\n\u22072g(0) = \u2212x(cid:63)x(cid:63)(cid:62), which does have zero eigenvalues and thus g(x) does not satisfy the strongly\nMorse condition required in [1]. The conclusion extends to the general low-rank matrix sensing.\nExample 1.2. (Phase retrieval). Given measurements ym = |(cid:104)am, x(cid:63)(cid:105)|2, 1 \u2264 m \u2264 M, where\nx(cid:63) \u2208 RN and am \u2208 RN denote the true signal and the m-th Gaussian random vector with entries\nfollowing N (0, 1), respectively. The following empirical risk is commonly used in practice\n\n(1.1)\n\n(1.2)\n\nf (x) =\n\n1\n2M\n\nThe corresponding population risk is then\n\nM(cid:88)\n\n(cid:0)\n\nm=1\n\n(cid:1)2 .\n\n|(cid:104)am, x(cid:105)|2 \u2212 ym\n\ng(x) = Ef (x) = (cid:107)xx(cid:62) \u2212 x(cid:63)x(cid:63)(cid:62)(cid:107)2\n\nF +\n\n1\n2\n\n((cid:107)x(cid:107)2\n\n2 \u2212 (cid:107)x(cid:63)(cid:107)2\n\n2)2.\n\nElementary calculations give the gradient and Hessian of the above population risk as\n\n\u2207g(x) = 6(cid:107)x(cid:107)2\n\u22072g(x) = 12xx(cid:62) \u2212 4x(cid:63)x(cid:63)(cid:62) + 6(cid:107)x(cid:107)2\n\n2x \u2212 2(cid:107)x(cid:63)(cid:107)2\n\n2x \u2212 4(x(cid:63)(cid:62)x)x(cid:63),\n\n2IN \u2212 2(cid:107)x(cid:63)(cid:107)2\n\n2IN .\n\n1\u221a3(cid:107)x(cid:63)(cid:107)2w with w(cid:62)x(cid:63) = 0 and\nWe see that the population loss has critical points x = 0, \u00b1 x(cid:63),\n(cid:107)w(cid:107)2 = 1. Observe that the Hessian at x = 1\u221a3(cid:107)x(cid:63)(cid:107)2w is \u22072g( 1\u221a3(cid:107)x(cid:63)(cid:107)2w) = 4(cid:107)x(cid:63)(cid:107)2\n2ww(cid:62) \u2212\n4x(cid:63)x(cid:63)(cid:62), which also has zero eigenvalues and thus g(x) does not satisfy the strongly Morse condition\nrequired in [1].\n\nIn this work, we aim to \ufb01ll this gap and establish the correspondence between the critical points of\nempirical risk and its population risk without the strongly Morse assumption. In particular, we work\non the situation where the population risk may be a degenerate non-convex function, i.e., the Hessian\nof the population risk can have zero eigenvalues. Given the correspondence between the critical\npoints of the empirical risk and its population risk, we are able to build a connection between the\nlandscape of the empirical risk and its population counterpart. To illustrate the effectiveness of this\ntheory, we also apply it to applications such as matrix sensing (with general rank) and phase retrieval\nto show how to characterize the landscape of the empirical risk via its corresponding population risk.\n\n1A twice differentiable function f (x) is Morse if all of its critical points are non-degenerate, i.e., its Hessian\nhas no zero eigenvalues at all critical points. Mathematically, \u2207f (x) = 0 implies all \u03bbi(\u22072f (x)) (cid:54)= 0 with\n\u03bbi(\u00b7) being the i-th eigenvalue of the Hessian. A twice differentiable function f (x) is (\u0001, \u03b7)-strongly Morse if\n(cid:107)\u2207f (x)(cid:107)2 \u2264 \u0001 implies mini |\u03bbi(\u22072f (x))| \u2265 \u03b7. One can refer to [1] for more information.\n\n2\n\n\fThe remainder of this work is organized as follows. In Section 2, we present our main results on the\ncorrespondence between the critical points of the empirical risk and its population risk. In Section 3,\nwe apply our theory to the two applications, matrix sensing and phase retrieval. In Section 4, we\nconduct experiments to further support our analysis. Finally, we conclude our work in Section 5.\nNotation: For a twice differential function f (\u00b7): \u2207f, \u22072f, grad f, and hess f denote the gradient\nand Hessian of f in the Euclidean space and with respect to a Riemannian manifold M, respectively.\nNote that the Riemannian gradient/Hessian (grad/hess) reduces to the Euclidean gradient/Hessian\n(cid:80)\n(\u2207/\u22072) when the domain of f is the Euclidean space. For a scalar function with a matrix variable,\ne.g., f (U), we represent its Euclidean Hessian with a bilinear form de\ufb01ned as \u22072f (U)[D,D] =\n\u2202D(i,j)\u2202D(p,q) D(i, j)D(p, q) for any D having the same size as U. Denote B(l) as a compact\nand connected subset of a Riemannian manifold M with l being a problem-speci\ufb01c parameter.2\n2 Main Results\n\n\u22022f (U)\n\ni,j,p,q\n\nIn this section, we present our main results on the correspondence between the critical points of the\nempirical risk and its population risk. Let M be a Riemannian manifold. For notational simplicity, we\nuse x \u2208 M to denote the parameter vector when we introduce our theory3. We begin by introducing\nthe assumptions needed to build our theory. Denote f (x) and g(x) as the empirical risk and the\ncorresponding population risk de\ufb01ned for x \u2208 M, respectively. Let \u0001 and \u03b7 be two positive constants.\nAssumption 2.1. The population risk g(x) satis\ufb01es\n(2.1)\n\nin the set D (cid:44) {x \u2208 B(l) : (cid:107)grad g(x)(cid:107)2 \u2264 \u0001}. Here, \u03bbmin(\u00b7) denotes the minimal eigenvalue (not\n\nthe eigenvalue of smallest magnitude).\n\n|\u03bbmin(hess g(x))| \u2265 \u03b7\n\nAssumption 2.1 is closely related to the robust strict saddle property [19] \u2013 it requires that any point\nwith a small gradient has either a positive de\ufb01nite Hessian (\u03bbmin(hess g(x)) \u2265 \u03b7) or a Hessian with\na negative curvature (\u03bbmin(hess g(x)) \u2264 \u2212\u03b7). It is weaker than the (\u0001, \u03b7)-strongly Morse condition\nas it allows the Hessian hess g(x) to have zero eigenvalues in D,\nprovided it also has at least one suf\ufb01ciently negative eigenvalue.\nAssumption 2.2. (Gradient proximity). The gradients of the\nempirical risk and population risk satisfy\n\nsup\n\nx\u2208B(l)(cid:107)grad f (x) \u2212 grad g(x)(cid:107)2 \u2264\n\n\u0001\n2\n\n.\n\n(2.2)\n\nAssumption 2.3. (Hessian proximity). The Hessians of the\nempirical risk and population risk satisfy\n\nsup\n\nx\u2208B(l)(cid:107)hess f (x) \u2212 hess g(x)(cid:107)2 \u2264\n\n\u03b7\n2\n\n.\n\n(2.3)\n\n(cid:80)M\n\n2M\n\nm=1 a4\n\n2 (x2\u22121)2 and the empirical risk f (x) = 1\n\nTo illustrate the above three assumptions, we use the phase\nFigure 1: Phase retrieval with N = 1.\nretrieval Example 1.2 with N = 1, x(cid:63) = 1, and M = 30. We\npresent the population risk g(x) = 3\nm(x2\u22121)2\ntogether with their gradients and Hessians in Figure 1. It can be seen that in the small gradient region\n(the three parts between the light blue vertical dashed lines), the absolute value of the population\nHessian\u2019s minimal eigenvalue (which equals the absolute value of Hessian here since N = 1) is\nbounded away from zero. In addition, with enough measurements, e.g., M = 30, we do see the\ngradients and Hessians of the empirical and population risk are close to each other.\nWe are now in the position to state our main theorem.\nTheorem 2.1. Denote f and g as the non-convex empirical risk and the corresponding population\nrisk, respectively. Let D be any maximal connected and compact subset of D with a C2 boundary \u2202D.\nUnder Assumptions 2.1-2.3 stated above, the following statements hold:\n2The subset B(l) can vary in different applications. For example, we de\ufb01ne B(l) (cid:44) {U \u2208 RN\u00d7k\n\n(cid:107)UU(cid:62)(cid:107)F \u2264 l} in matrix sensing and B(l) (cid:44) {x \u2208 RN : (cid:107)x(cid:107)2 \u2264 l} in phase retrieval.\nrepresentation of the matrix.\n\n3For problems with matrix variables, such as matrix sensing introduced in Section 3, x is the vectorized\n\n\u2217\n\n:\n\n3\n\n-1.5-1-0.500.511.50.511.522.5RisksPopulationEmpirical-1.5-1-0.500.511.5-10-1110Gradients-1.5-1-0.500.511.50102030Hessians\f(a) D contains at most one local minimum of g. If g has K (K = 0, 1) local minima in D, then f\nalso has K local minima in D.\n(b) If g has strict saddles in D, then if f has any critical points in D, they must be strict saddle\npoints.\n\nThe proof of Theorem 2.1 is given in Appendix A (see supplementary material). In particular, we\nprove Theorem 2.1 by extending the proof of Theorem 2 in [1] without requiring the strongly Morse\nassumption on the population risk. We \ufb01rst present two key lemmas, in which we show that there\nexists a correspondence between the critical points of the empirical risk and those of the population\nrisk in a connected and compact set under certain assumptions, and the small gradient area can be\npartitioned into many maximal connected and compact components with each component either\ncontaining one local minimum or no local minimum. Finally, we \ufb01nish the proof of Theorem 2.1 by\nusing these two key lemmas.\nPart (a) in Theorem 2.1 indicates a one-to-one correspondence between the local minima of the\nempirical risk and its population risk. We can further bound the distance between the local minima of\nthe empirical risk and its population risk. We summarize this result in the following corollary, which\nis proved in Appendix C (see supplementary material).\n\nCorollary 2.1. Let {(cid:98)xk}K\npopulation risk, and Dk be the maximal connected and compact subset of D containing xk and(cid:98)xk.\n\nk=1 denote the local minima of the empirical risk and its\nLet \u03c1 be the injectivity radius of the manifold M. Suppose the pre-image of Dk under the exponential\nmapping Expxk (\u00b7) is contained in the ball at the origin of the tangent space TxkM with radius \u03c1.\nAssume the differential of the exponential mapping DExpxk (v) has an operator norm bounded by \u03c3\nfor all v \u2208 TxkM with norm less than \u03c1. Suppose the pullback of the population risk onto the tangent\nRiemannian distance between(cid:98)xk and xk satis\ufb01es\nspace TxkM has Lipschitz Hessian with constant LH at the origin. Then as long as \u0001 \u2264 \u03b72\n, the\ndist((cid:98)xk, xk) \u2264 2\u03c3\u0001/\u03b7,\n\nk=1 and {xk}K\n\n2\u03c3LH\n\n1 \u2264 k \u2264 K.\n\n2 and (cid:107)x(cid:63)(cid:107)3\n\nIn general, the two parameters \u0001 and \u03b7 used in Assumptions 2.1-2.3 can be obtained by lower bounding\n|\u03bbmin(hess g(x))| in a small gradient region. In this way, one can adjust the size of the small gradient\nregion to get an upper bound on \u0001, and use the lower bound for |\u03bbmin(hess g(x))| as \u03b7. In the case\nwhen it is not easy to directly bound |\u03bbmin(hess g(x))| in a small gradient region, one can also \ufb01rst\nchoose a region for which it is easy to \ufb01nd the lower bound, and then show that the gradient has a\nlarge norm outside of this region, as we do in Section 3. For phase retrieval, note that |\u03bbmin(\u22072g(x))|\nand (cid:107)\u2207g(x)(cid:107)2 roughly scale with (cid:107)x(cid:63)(cid:107)2\n2 in the regions near critical points, which implies\nthat \u03b7 and the upper bound on \u0001 should also scale with (cid:107)x(cid:63)(cid:107)2\n2, respectively. For matrix\nsensing, in a similar way, |\u03bbmin(hess g(U))| and (cid:107)grad g(U)(cid:107)F roughly scale with \u03bbk and \u03bb1.5\nin\nthe regions near critical points, which implies that \u03b7 and the upper bound on \u0001 should also scale with\n\u03bbk and \u03bb1.5\nk , respectively. Note, however, with more samples (larger M), \u0001 can be set to smaller\nvalues, while \u03b7 typically remains unchanged. One can refer to Section 3 for more details on the\nnotation as well as how to choose \u03b7 and upper bounds on \u0001 in the two applications.\nNote that we have shown the correspondence between the critical points of the empirical risk and its\npopulation risk without the strongly Morse assumption in the above theorem. In particular, we relax\nthe strongly Morse assumption to our Assumption 2.1, which implies that we are able to handle the\nscenario where the Hessian of the population risk has zero eigenvalues at some critical points or even\neverywhere in the set D. With this correspondence, we can then establish a connection between the\nlandscape of the empirical risk and the population risk, and thus for problems where the population\nrisk has a favorable geometry, we are able to carry this favorable geometry over to the corresponding\nempirical risk. To illustrate this in detail, we highlight two applications, matrix sensing and phase\nretrieval, in the next section.\n\n2 and (cid:107)x(cid:63)(cid:107)3\n\nk\n\n3 Applications\n\nIn this section, we illustrate how to completely characterize the landscape of an empirical risk from its\npopulation risk using Theorem 2.1. In particular, we apply Theorem 2.1 to two applications, matrix\nsensing and phase retrieval. In order to use Theorem 2.1, all we need is to verify that the empirical\nrisk and population risk in these two applications satisfy the three assumptions stated in Section 2.\n\n4\n\n\f3.1 Matrix Sensing\nLet X \u2208 RN\u00d7N be a symmetric, positive semi-de\ufb01nite matrix with rank r. We measure X with\na symmetric Gaussian linear operator A : RN\u00d7N \u2192 RM . The m-th entry of the observation\ny = A(X) is given as ym = (cid:104)X, Am(cid:105), where Am = 1\n2 (Bm + B(cid:62)m) with Bm being a Gaussian\nrandom matrix with entries following N (0, 1\nM ). The adjoint operator A\u2217 : RM \u2192 RN\u00d7N is\nm=1 ymAm. It can be shown that E(A\u2217A) is the identity operator, i.e.\nE(A\u2217A(X)) = X. To \ufb01nd a low-rank approximation of X when given the measurements y = A(X),\none can solve the following optimization problem:\n\nde\ufb01ned as A\u2217(y) = (cid:80)M\n(cid:101)X\u2208RN\u00d7N\n2 \u2264 k \u2264 r (cid:28) N. By using the Burer-Monteiro type factorization [20, 21],\n\ni.e., letting (cid:101)X = UU(cid:62) with U \u2208 RN\u00d7k, we can transform the above optimization problem into the\n\ns. t. rank((cid:101)X) \u2264 k,(cid:101)X (cid:23) 0.\n\n4(cid:107)A((cid:101)X \u2212 X)(cid:107)2\n\n1\n\nHere, we assume that r\n\n(3.1)\n\nmin\n\n2\n\nfollowing unconstrained one:\n\nmin\nU\u2208RN\u00d7k\n\nf (U) (cid:44) 1\n\n4(cid:107)A(UU(cid:62) \u2212 X)(cid:107)2\n2.\n\n(3.2)\n\nObserve that this empirical risk f (U) is a non-convex function due to the quadratic term UU(cid:62). With\nsome elementary calculation, we obtain the gradient and Hessian of f (U), which are given as\n\n\u2207f (U) = A\u2217A(UU(cid:62) \u2212 X)U,\n\n\u22072f (U)[D, D] =\n\n1\n2(cid:107)A(UD(cid:62) + DU(cid:62))(cid:107)2\n\n2 + (cid:104)A\u2217A(UU(cid:62) \u2212 X), DD(cid:62)(cid:105).\n\nComputing the expectation of f (U), we get the population risk\n\ng(U) = Ef (U) =\n\n1\n4(cid:107)UU(cid:62) \u2212 X(cid:107)2\nF ,\n\n(3.3)\n\nwhose gradient and Hessian are given as\n\n1\n2(cid:107)UD(cid:62) + DU(cid:62)(cid:107)2\n\n\u2207g(U) = (UU(cid:62) \u2212 X)U and \u22072g(U)[D, D] =\n2 + (cid:104)UU(cid:62) \u2212 X, DD(cid:62)(cid:105).\nThe landscape of the above population risk has been studied in the general RN\u00d7k space with k = r\nin [8]. The landscape of its variants, such as the asymmetric version with or without a balanced term,\nhas also been studied in [4, 22]. It is well known that there exists an ambiguity in the solution of (3.2)\ndue to the fact that UU(cid:62) = UQQ(cid:62)U(cid:62) holds for any orthogonal matrix Q \u2208 Rk\u00d7k . This implies\nthat the Euclidean Hessian \u22072g(U) always has zero eigenvalues for k > 1 at critical points, even at\nlocal minima, violating not only the strongly Morse condition but also Assumption 2.1. To overcome\nthis dif\ufb01culty, we propose to formulate an equivalent problem on a proper quotient manifold (rather\nthan the general RN\u00d7k space as in [8]) to remove this ambiguity and make sure Assumption 2.1 is\nsatis\ufb01ed.\n\n3.1.1 Background on the quotient manifold\n\nTo keep our work self-contained, we provide a brief introduction to quotient manifolds in this section\nbefore we verify our three assumptions. One can refer to [23, 24] for more information. We make\nthe assumption that the matrix variable U is always full-rank. This is required in order to de\ufb01ne a\nproper quotient manifold, since otherwise the equivalence classes de\ufb01ned below will have different\ndimensions, violating Proposition 3.4.4 in [23]. Thus, we focus on the case that U belongs to the\nparameterization ambiguity caused by the factorization (cid:101)X = UU(cid:62), we de\ufb01ne an equivalence class\n, i.e., the set of all N \u00d7 k real matrices with full column rank. To remove the\nmanifold RN\u00d7k\nfor any U \u2208 RN\u00d7k\n: VV(cid:62) = UU(cid:62)} = {UQ : Q \u2208 Rk\u00d7k, Q(cid:62)Q = Ik}.\nWe will abuse notation and use U to denote also its equivalence class [U] in the following. Let\nM denote the set of all equivalence classes of the above form, which admits a (unique) differential\nstructure that makes it a (Riemannian) quotient manifold, denoted as M = RN\u00d7k\n/Ok. Here Ok is\nthe orthogonal group {Q \u2208 Rk\u00d7k : QQ(cid:62) = Q(cid:62)Q = Ik}. Since the objective function g(U) in\n\nas [U] (cid:44) {V \u2208 RN\u00d7k\n\n\u2217\n\n\u2217\n\n\u2217\n\n\u2217\n\n5\n\n\f\u2217\n\n\u2217\n\n\u2217\n\n\u2217\n\n\u2217\n\n\u2217\n\n\u2217\n\n\u2217\n\n= RN\u00d7k\n\nof the manifold RN\u00d7k\n\nat any point U \u2208 RN\u00d7k\n\n/Ok, also denoted as g(U).\n\n: HUM (cid:44) {D \u2208 RN\u00d7k\n\n(3.3) (and f (U) in (3.2)) is invariant under the equivalence relation, it induces a unique function on\nthe quotient manifold RN\u00d7k\nNote that the tangent space TURN\u00d7k\nis still\n. We de\ufb01ne the vertical space VUM as the tangent space to the equivalence classes (which\nRN\u00d7k\n\u2217\nare themselves manifolds): VUM (cid:44) {U\u2126 : \u2126 \u2208 Rk\u00d7k, \u2126(cid:62) = \u2212\u2126}. We also de\ufb01ne the\nhorizontal space HUM as the orthogonal complement of the vertical space VUM in the tangent\n,\nspace TURN\u00d7k\n: D(cid:62)U = U(cid:62)D}. For any matrix Z \u2208 RN\u00d7k\nits projection onto the horizontal space HUM is given as PU(Z) = Z \u2212 U\u2126, where \u2126 is a skew-\nsymmetric matrix that solves the following Sylvester equation \u2126U(cid:62)U + U(cid:62)U\u2126 = U(cid:62)Z \u2212 Z(cid:62)U.\nThen, we can de\ufb01ne the Riemannian gradient (grad \u00b7) and Hessian (hess \u00b7) of the empirical risk and\npopulation risk on the quotient manifold M, which are given in the supplementary material.\n3.1.2 Verifying Assumptions 2.1, 2.2, and 2.3\nAssume that X = W\u039bW(cid:62) with W \u2208 RN\u00d7r and \u039b = diag([\u03bb1,\u00b7\u00b7\u00b7 , \u03bbr]) \u2208 Rr\u00d7r is an eigen-\ndecomposition of X. Without loss of generality, we assume that the eigenvalues of X are in\ndescending order. Let \u039bu \u2208 Rk\u00d7k be a diagonal matrix that contains any k non-zero eigenvalues\nof X and Wu \u2208 RN\u00d7k contain the k eigenvectors of X associated with the eigenvalues in \u039bu. Let\n\u039bk = diag([\u03bb1,\u00b7\u00b7\u00b7 , \u03bbk]) be the diagonal matrix that contains the largest k eigenvalues of X and\nWk \u2208 RN\u00d7k contain the k eigenvectors of X associated with the eigenvalues in \u039bk. Q \u2208 Ok is any\northogonal matrix. The following lemma provides the global geometry of the population risk in (3.3),\nwhich also determines the values of \u0001 and \u03b7 in Assumption 2.1.\n\n(cid:44) U\\U (cid:63).\n(cid:27)\n(cid:27)\n\n3\n\u03bb\n2\nk\n\nk Q(cid:62)} \u2286 U, and U (cid:63)\n(cid:27)\n\n\u03bbk \u2265 1 as the condition number of any U(cid:63) \u2208 U (cid:63). De\ufb01ne the following regions:\n\nP\u2208Ok (cid:107)U \u2212 U(cid:63)P(cid:107)F \u2264 0.2\u03ba\u22121(cid:112)\n\n(cid:26)\n(cid:26)\nU\u2208 RN\u00d7k\n(cid:26)\nU\u2208 RN\u00d7k\n(cid:26)\nU\u2208 RN\u00d7k\n(cid:26)\nU\u2208 RN\u00d7k\nU\u2208 RN\u00d7k\n\nDenote \u03ba (cid:44)(cid:113) \u03bb1\nLemma 3.1. De\ufb01ne U (cid:44) {U = Wu\u039b\nR1 (cid:44)\n(cid:44)\nR(cid:48)2\nR(cid:48)(cid:48)2\nR(cid:48)3\nR(cid:48)(cid:48)3\nwhere \u03c3k(U) denotes the k-th singular value of a matrix U \u2208 RN\u00d7k\n, i.e., the smallest singular\nvalue of U. These regions also induce regions in the quotient manifold M in an apparent way. We\nadditionally assume that \u03bbk+1 \u2264 1\n(1) For any U \u2208 U, U is a critical point of the population risk g(U) in (3.3).\n(2) For any U(cid:63) \u2208 U (cid:63), U(cid:63) is a global minimum of g(U) with \u03bbmin(hess g(U(cid:63))) \u2265 1.91\u03bbk. Moreover,\n\n\u03bbk, \u2200 U(cid:63) \u2208 U (cid:63)\n8\n7(cid:107)U(cid:63)U(cid:63)(cid:62)(cid:107)F , (cid:107) grad g(U)(cid:107)F \u2264\n(cid:27)\n8\n7(cid:107)U(cid:63)U(cid:63)(cid:62)(cid:107)F , (cid:107) grad g(U)(cid:107)F >\n8\n7(cid:107)U(cid:63)U(cid:63)(cid:62)(cid:107)F\n\nP\u2208Ok(cid:107)U\u2212U(cid:63)P(cid:107)F > 0.2\u03ba\u22121(cid:112)\n\n1\n2\n1\n2\n1\n2\n: (cid:107)UU(cid:62)(cid:107)F >\n\n\u03bbk, (cid:107)UU(cid:62)(cid:107)F \u2264\n\u03bbk, (cid:107)UU(cid:62)(cid:107)F \u2264\n(cid:27)\n\n12 \u03bbk and k \u2264 r (cid:28) N. Then, the following properties hold:\n\n(cid:112)\n(cid:112)\n(cid:112)\n\n1\n2\n\nu Q(cid:62)}, U (cid:63) (cid:44) {U(cid:63) = Wk\u039b\n\n8\n7(cid:107)U(cid:63)U(cid:63)(cid:62)(cid:107)F\n\n\u03bbk,(cid:107)UU(cid:62)(cid:107)F \u2264\n\n: \u03c3k(U) \u2264\n\n: \u03c3k(U) \u2264\n\n1\n80\n1\n80\n\n: \u03c3k(U) >\n\n\u03bbk, min\n\n: min\n\n\u2217\n\n\u2217\n\n\u2217\n\n\u2217\n\n\u2217\n\n3\n\u03bb\n2\nk\n\n(cid:44)\n\n(cid:44)\n\n(cid:44)\n\n,\n\n,\n\n\u2217\n\n1\n2\n\n,\n\n,\n\ns\n\n,\n\n\u03bbmin(hess g(U)) \u2264 \u22120.06\u03bbk.\n\nR(cid:48)(cid:48)3, we have a large gradient. In particular,\nif U \u2208 R(cid:48)(cid:48)2 ,\nif U \u2208 R(cid:48)3,\nif U \u2208 R(cid:48)(cid:48)3 .\n\n1\n80 \u03bb\n1\n60 \u03ba\u22121\u03bb\nk ,\n84 k 1\nk ,\n4 \u03bb\n\n(cid:107) grad g(U)(cid:107)F >\n\nk ,\n\n3\n2\n\n3\n2\n\n3\n2\n\n5\n\n\uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3\n\n6\n\n(3) For any U(cid:63)\n\ns \u2208 U (cid:63)\n\nMoreover, for any U \u2208 R(cid:48)2, we have\n\ns , U(cid:63)\n\ns is a strict saddle point of g(U) with \u03bbmin(hess g(U(cid:63)\n\ns)) \u2264 \u22120.91\u03bbk.\n\n\u03bbmin(hess g(U)) \u2265 0.19\u03bbk.\n\nfor any U \u2208 R1, we have\n\n(cid:83)\n\n(cid:83)\n\n(4) For any U \u2208 R(cid:48)(cid:48)2\n\nR(cid:48)3\n\n\f3\n2\n\n\u2217\n\n(cid:83)\n\nThe proof of Lemma 3.1 is inspired by the proofs of [8, Theorem 4], [3, Lemma 13] and [4,\nTheorem 5], and is given in Appendix D (see supplementary material). Therefore, we can set\nk and \u03b7 = 0.06\u03bbk. Then, the population risk given in (3.3) satis\ufb01es\n\u0001 \u2264 min{1/80, 1/60\u03ba\u22121}\u03bb\nAssumption 2.1. It can be seen that each critical point of the population risk g(U) in (3.3) is either\na global minimum or a strict saddle, which inspires us to carry this favorable geometry over to the\ncorresponding empirical risk.\nTo illustrate the partition of the manifold RN\u00d7k\nused in the above\nLemma 3.1, we use the purple (x), yellow (y), and green (z) re-\ngions in Figure 2 to denote the regions that satisfy minP\u2208Ok (cid:107)U \u2212\nU(cid:63)P(cid:107)F \u2264 0.2\u03ba\u22121\u221a\u03bbk, \u03c3k(U) \u2264 1\n2\u221a\u03bbk, and (cid:107)UU(cid:62)(cid:107)F \u2264\n7(cid:107)U(cid:63)U(cid:63)(cid:62)(cid:107)F , respectively. It can be seen that R1 is exactly the pur-\n8\nple region, which contains the areas near the global minima ([U(cid:63)]).\nR(cid:48)(cid:48)2 is the intersection of the yellow and green regions.\nR2 = R(cid:48)2\n(cid:83)\nR(cid:48)3 is the part of the green region that does not intersect with the\npurple or yellow regions. Finally, R(cid:48)(cid:48)3 is the space outside of the\ngreen region. Therefore, the union of R1, R2, and R3 = R(cid:48)3\nR(cid:48)(cid:48)3\ncovers the entire manifold RN\u00d7k\nWe de\ufb01ne a norm ball as B(l) (cid:44) {U \u2208 RN\u00d7k\nfollowing lemma veri\ufb01es Assumptions 2.2 and 2.3 under the restricted isometry property (RIP).\nLemma 3.2. Assume r\nsatis\ufb01es the following RIP\n\n7(cid:107)U(cid:63)U(cid:63)(cid:62)(cid:107)F . The\n2 \u2264 k \u2264 r (cid:28) N. Suppose that a linear operator B with [B(Z)]m = (cid:104)Z, Bm(cid:105)\n(3.4)\nfor any matrix Z \u2208 RN\u00d7N with rank at most r + k. We construct the linear operator A by setting\nAm = 1\n\n2 (Bm + B(cid:62)m). If the restricted isometry constant \u03b4r+k satis\ufb01es\n\nFigure 2: Partition of regions in\nLemma 3.1.\n\n: (cid:107)UU(cid:62)(cid:107)F \u2264 l} with l = 8\n\n2 \u2264 (1 + \u03b4r+k)(cid:107)Z(cid:107)2\n\n(1 \u2212 \u03b4r+k)(cid:107)Z(cid:107)2\n\nF \u2264 (cid:107)B(Z)(cid:107)2\n\n\u2217\n\n\u2217\n\nF\n\n.\n\n,\n\n1\n,\n36\n\n1\n2\nF\n\n2(16\n7\n\n\u221ak(cid:107)U(cid:63)U(cid:63)(cid:62)(cid:107)F + 8\n\n\u03b7\n7(cid:107)U(cid:63)U(cid:63)(cid:62)(cid:107)F +(cid:107)X(cid:107)F)\n\n\uf8fc\uf8fd\uf8fe\n\n\uf8f1\uf8f2\uf8f3\n\n(cid:113)8\n\n\u03b4r+k \u2264 min\n2\nthen, we have\n\n\u0001\n\n7 k 1\n4(8\n7(cid:107)U(cid:63)U(cid:63)(cid:62)(cid:107)F +(cid:107)X(cid:107)F)(cid:107)U(cid:63)U(cid:63)(cid:62)(cid:107)\n\nsup\n\nU\u2208B(l)(cid:107) grad f (U) \u2212 grad g(U)(cid:107)F \u2264\n\n\u0001\n2\n\n,\n\nand\n\nsup\n\nU\u2208B(l)(cid:107) hess f (U) \u2212 hess g(U)(cid:107)2 \u2264\n\n\u03b7\n2\n\n.\n\nThe proof of Lemma 3.2 is given in Appendix E (see supplementary material). As is shown in existing\nliterature [25, 26, 27], a Gaussian linear operator B : RN\u00d7N \u2192 RM satis\ufb01es the RIP condition (3.4)\nwith high probability if M \u2265 C(r + k)N/\u03b42\nr+k for some numerical constant C. Therefore, we can\nconclude that the three statements in Theorem 2.1 hold for the empirical risk (3.2) and population\nrisk (3.3) as long as M is large enough. Some similar bounds for the sample complexity M under\ndifferent settings can also be found in papers [8, 4]. Note that the particular choice of l can guarantee\nthat (cid:107) grad f (U)(cid:107)F is large outside of B(l), which is also proved in Appendix E. Together with\nTheorem 2.1, we prove a globally benign landscape for the empirical risk.\n\n3.2 Phase Retrieval\n\n(cid:26)\n(cid:26)\n(cid:26)\n\n1\n2(cid:107)x(cid:63)(cid:107)2\n\nWe continue to elaborate on Example 1.2. The following lemma provides the global geometry of the\npopulation risk in (1.2), which also determines the values of \u0001 and \u03b7 in Assumption 2.1.\nLemma 3.3. De\ufb01ne the following four regions:\nR1 (cid:44)\n, R2 (cid:44)\nR3 (cid:44)\n1\n\u221a3(cid:107)x(cid:63)(cid:107)2w\nR4 (cid:44)\n\u03b3\u2208{1,\u22121}(cid:107)x \u2212 \u03b3x(cid:63)(cid:107)2 >\n\n(cid:27)\n1\n\u03b3\u2208{1,\u22121}(cid:107)x \u2212 \u03b3x(cid:63)(cid:107)2 \u2264\n10(cid:107)x(cid:63)(cid:107)2\n1\n5(cid:107)x(cid:63)(cid:107)2, w(cid:62)x(cid:63) = 0, (cid:107)w(cid:107)2 = 1\n(cid:27)\n1\n5(cid:107)x(cid:63)(cid:107)2, w(cid:62)x(cid:63) = 0, (cid:107)w(cid:107)2 = 1\n\nx \u2208 RN : (cid:107)x(cid:107)2 \u2264\nx \u2208 RN : min\n\u03b3\u2208{1,\u22121}\n1\n2(cid:107)x(cid:63)(cid:107)2, min\nx \u2208 RN : (cid:107)x(cid:107)2 >\n1\n\u221a3(cid:107)x(cid:63)(cid:107)2w\n\n(cid:27)\n(cid:13)(cid:13)(cid:13)(cid:13)x \u2212 \u03b3\n(cid:13)(cid:13)(cid:13)(cid:13)x \u2212 \u03b3\n\n(cid:26)\n(cid:13)(cid:13)(cid:13)(cid:13)2 \u2264\n\nx \u2208 RN : min\n\n1\n10(cid:107)x(cid:63)(cid:107)2,\n\n(cid:27)\n\nmin\n\n(cid:13)(cid:13)(cid:13)(cid:13)2\n\n,\n\n,\n\n\u03b3\u2208{1,\u22121}\n\n>\n\n7\n\nRN\u21e5k\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\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\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\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\fThen, the following properties hold:\n(1) x = 0 is a strict saddle point with \u22072g(0) = \u22124x(cid:63)x(cid:63)(cid:62) \u2212 2(cid:107)x(cid:63)(cid:107)2\n\n\u22126(cid:107)x(cid:63)(cid:107)2\n\n2. Moreover, for any x \u2208 R1, the neighborhood of strict saddle point 0, we have\n\n2IN and \u03bbmin(\u22072g(0)) =\n\n\u03bbmin(\u22072g(x)) \u2264 \u2212\n\n3\n2(cid:107)x(cid:63)(cid:107)2\n2.\n\n(2) x = \u00b1x(cid:63) are global minima with \u22072g(\u00b1x(cid:63)) = 8x(cid:63)x(cid:63)(cid:62) + 4(cid:107)x(cid:63)(cid:107)2\n\n2. Moreover, for any x \u2208 R2, the neighborhood of global minima \u00b1x(cid:63), we have\n\n2IN and \u03bbmin(\u22072g(\u00b1x(cid:63))) =\n\n4(cid:107)x(cid:63)(cid:107)2\n\n\u03bbmin(\u22072g(x)) \u2265 0.22(cid:107)x(cid:63)(cid:107)2\n2.\n\n(3) x = \u00b1 1\u221a3(cid:107)x(cid:63)(cid:107)2w, with w(cid:62)x(cid:63) = 0 and (cid:107)w(cid:107)2 = 1, are strict saddle points with\n2ww(cid:62) \u2212 4x(cid:63)x(cid:63)(cid:62) and \u03bbmin(\u22072g(\u00b1 1\u221a3(cid:107)x(cid:63)(cid:107)2w)) = \u22124(cid:107)x(cid:63)(cid:107)2\n2.\n\n\u22072g(\u00b1 1\u221a3(cid:107)x(cid:63)(cid:107)2w) = 4(cid:107)x(cid:63)(cid:107)2\nMoreover, for any x \u2208 R3, the neighborhood of strict saddle points \u00b1 1\u221a3(cid:107)x(cid:63)(cid:107)2w, we have\n\n\u03bbmin(\u22072g(x)) \u2264 \u22120.78(cid:107)x(cid:63)(cid:107)2\n2.\n\n(4) For any x \u2208 R4, the complement region of R1, R2, and R3, we have (cid:107)\u2207g(x)(cid:107)2 > 0.3963(cid:107)x(cid:63)(cid:107)3\n2.\nThe proof of Lemma 3.3 is inspired by the proof\nof [8, Theorem 3] and is given in Appendix F (see\nsupplementary material). Letting \u0001 \u2264 0.3963(cid:107)x(cid:63)(cid:107)3\n2\nand \u03b7 = 0.22(cid:107)x(cid:63)(cid:107)2\n2, the population risk (1.2) then\nsatis\ufb01es Assumption 2.1. As in Lemma 3.1, we also\nnote that each critical point of the population risk\nin (1.2) is either a global minimum or a strict saddle.\nThis inspires us to carry this favorable geometry over\nto the corresponding empirical risk.\nThe partition of regions used in Lemma 3.3 is illus-\nFigure 3: Partition of regions in Lemma 3.3.\ntrated in Figure 3. We use the purple, green, and blue\nballs to denote the three regions R1, R2, and R3, respectively. R4 is then represented with the light\ngray region. Therefore, the union of the four regions covers the entire RN space.\nDe\ufb01ne a norm ball as B(l) (cid:44) {x \u2208 RN : (cid:107)x(cid:107)2 \u2264 l} with radius l = 1.1(cid:107)x(cid:63)(cid:107)2. This particular\nchoice of l guarantees that (cid:107) grad f (x)(cid:107)2 is large outside of B(l), which is proved in Appendix G.\nwith (cid:101)O denoting an asymptotic notation that hides polylog\nde\ufb01ne h(N, M ) (cid:44) (cid:101)O\nTogether with Theorem 2.1, we prove a globally benign landscape for the empirical risk. We also\n\nfactors. The following lemma veri\ufb01es Assumptions 2.2 and 2.3 for this phase retrieval problem.\nLemma 3.4. Suppose that am \u2208 RN is a Gaussian random vector with entries following N (0, 1).\nIf h(N, M ) \u2264 0.0118, we then have\n\n(cid:113) N\n\n(cid:16) N 2\n\nM +\n\n(cid:17)\n\nM\n\nsup\n\nx\u2208B(l)(cid:107)\u2207f (x) \u2212 \u2207g(x)(cid:107)2 \u2264\n\n\u0001\n2\n\n,\n\nand\n\nsup\n\nx\u2208B(l)(cid:107)\u22072f (x) \u2212 \u22072g(x)(cid:107)2 \u2264\n\n\u03b7\n2\n\nhold with probability at least 1 \u2212 e\u2212CN log(M ).\nThe proof of Lemma 3.4 is given in Appendix G (see supplementary material). The assumption\nh(N, M ) \u2264 0.0118 implies that we need a sample complexity that scales like N 2, which is not\noptimal since x has only N degrees of freedom. This is a technical artifact that can be traced back to\nAssumptions 2.2 and 2.3\u2013which require two-sided closeness between the gradients and Hessians\u2013and\nthe heavy-tail property of the fourth powers of Gaussian random process [12]. To arrive at the\nconclusions of Theorem 2.1, however, these two assumptions are suf\ufb01cient but not necessary (while\nAssumption 2.1 is more critical), leaving room for tightening the sampling complexity bound. We\nleave this to future work.\n\n4 Numerical Simulations\n\nWe \ufb01rst conduct numerical experiments on the two examples introduced in Section 1, i.e., the rank-1\nmatrix sensing and phase retrieval problems. In both problems, we \ufb01x N = 2 and set x(cid:63) = [1 \u2212 1](cid:62).\n\n8\n\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\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\u00b7AAAbknicxdndbuPGFQBgJWmbdNs0m6ZX9Q0Ro0ARbBdWEiBNrmJLtvwjWbJk/dgrZzGkRhTXHJIhR7S9hPoKvW0eLW/TIXk051DDDZqrClhA55szQ2rmcMil7cj3Enlw8PN773/wm9/+7sOPfv/sD3/8+E+fPP/0z5MkXMcOHzuhH8YzmyXc9wI+lp70+SyKORO2z6f2fStvn6Y8TrwwuJZPEb8TzA28pecwqWg8dxahfP18/+DlQfGxzC9N+LLfgM/g9ad/fZgvQmcteCAdnyXJq+ZBJO8yFkvP8fnm2Xyd8Ig598zlr9TXgAme3GXF2W6svylZWMswVv8CaRVKe2RMJMmTsFWmYHKV7LblWNf2ai2X/7zLvCBaSx445YGWa9+SoZX/dGvhxdyR/pP6wpzYU+dqOSsWM0eqCXpWOYxtixdwqBdwOi8iLz9f9ePaXP3omPdU40g1hH42T1ZhLIUXrJNNNs972V6wyQ57I7bJPv/q240afcGX1tx3rfKTzX0WuMVcFQ0xNsTQ8Gwe8AcnFIIFi2yeOnLzqnmXzdXxFkl52P3mZlPNEvKxzCrOYVmTIZPYGKfIdlg54G4+D5Lw13VJuMST2ObsjCrCqMwJIx4zGcZ5jXxRZlowJynzoxWzsuqBCyyOWSbZXDJrNylHkuMyddzdnAJJkq+umgXbSSqRZMlVfrydrAJJ0qNnGaf06JGEYG0mBGuSEK08I0EZzUhqMhKaseC+eaoFkiSu+vhhsJsGTBLjVWiZJ600z4Ek0SkmejepU51p0S5ObDerXT0z0S2mfjeru7Mgoi+4aw5WKMkarGqWZEAnVAySuhQ6o2LkuTU/r1CSNS7nbidrvDujYuDVTOjAI/MpD63tFZgd6o7yCPUItYXaQm2jtlGPUY9RT1BPUDuoHdRT1FPUM9Qz1HPUc9QL1AvULmoXtYfaQ71EvUTto/ZRB6gD1CvUK9Qh6hB1hDpCvUa9Rh2jjlEnqBPUKeoUdYY6Q71BvUG9Rb3F4lGXTn7XyJhOtEFsLQ6Io2UBstDCQbiWJchSiwvialmBrLR4IJ6WNyBvtNyD3JNtGcjHCwdEaAlAAi0hSKglAom0/Ajyo5YYJNaSgCRankCkljXIGk8aJNXyAPKg5RHk0Rj5SctbkLe4qqm5rKm5rqm5sKm5sqm5tKm5tqm5uKm5uqm5vKm5vun/tMCpucKpucSpucapucipucqpucypuc6pudBpzUqbS52aa52ai52aq53WLLfI7wDqqY5s/+IICPd+0QLCjV+0gXDXF8dAuOWLEyDc70UHCDd7cQqEO704A8JtXpwD4R4vLoBwgxddINzdRQ8It3ZxCYT7uugD4aYuBkC4o4srINzOxRAI93IxAsKNXFwD4S4uxkC4hYsJEO7fYgqEm7eYAeHOLW6AcNsWt0Bkz1ZPy/mCqydosuAKjwCPKLYAWxTbgG2Kx4DHFE8ATyh2ADsUTwFPKZ4BnlE8BzyneAF4QbEL2KXYA+xRvAS8pNgH7FMcAA4oXgFeURwCDimOAEcUrwGvKY4BxxQngBOKU8ApxRngjOIN4A3FW0BSIio0S0RFZomoyCwRFZkloiKzRFRkloiKzBJRkVkiKjJLREVmiajILBEVmSWiIrNEVGSWiIrMElGRWSIqMktERWaJqMgsERWZJaIis0RUZJaIiswSUZFZIioyS0RFNSVi28VdI38FYJMise0j5CPCLeQW4TZym/Ax8jHhE+QTwh3kDuFT5FPCZ8hnhM+RzwlfIF8Q7iJ3CfeQe4QvkS8J95H7hAfIA8JXyFeEh8hDwiPkEeFr5GvCY+Qx4QnyhPAUeUp4hjwjfIN8Q/gWuSiryjuvPryx+ULleIEnvLd8U32Pk2nfvLMne6zvufV39mSxK/KXa9V+Sv+V8y92Y4+13Rg+lSXS2n0llaztN9yRc0uGc0snBoF+XxeEwVrYPN42RaH/5IeuMRC4HiLygrQc4of5grkujrBtUC35f+weZfaPJr434E98e2h4y0eewc4W72x7y+Ow2nZAHj2hsXwAJQ1hwC1saOKRsEf+DIM9BPbIG7DH1vH42PZr3ouWDQuPudCQv0zM8lgPJ2Pdq2iUMWliDq9vUge4p73yGCeCO5UhVUw7unTMYR7r1uEvtl6ufZ+Om8e6cRSxgDbmMdaqG1TOSMXY5rmVjnmsG+9xdhwvdny+YInU88plZVQV4xyweNtWLe0Jw1l0wm1ZlwPk8QZXbVkdnbQksrqeCR545QVL+VS+LFsxmc2LGC+JQbhcWvi+efAaBlGskjYf67TAqk0LaNbDKk968BZcHUrfU/+fp7DdlotLa7szi+Lygbbj7bThHXl7h40FucPW3pBqb/87twAo1rp75WE+JdsfdIgzMtp6eZtTHHP6N4D+pnqHpW2HYbQpt75kCb+sGPn18/3m7h+mzC+TL182v3r55dXX+98fwR+tPmrsNT5v/L3RbHzT+L5x2hg0xg2n4TX+3fhP46e9v+x9t3e41ypT338P+nzWqHz2uv8FiajznA==\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\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\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0x?AAAbo3icxdndbuPGFQBgJWnadJu0m7RX9Q0RI0ARbBZWEiDJXWzLln8kS9a/vXKMITWiuOaQDDmiZRPqs/S2faO+TYfkEc8hhxs0VxEQROebM0Nq5nDIpc3AdSJ5cPDf997/4Hcf/v4PH/3xxZ8+/uTPf3n56WeTyF+HFh9bvuuHM5NF3HU8PpaOdPksCDkTpsun5sNx2j6NeRg5vjeSTwG/E8z2nKVjMano/uXf5qbvLqInof6XbLY/zSPJwvuX+wevD7KPoX9pwpf9Bnz695/+/XG+8K214J60XBZFb5oHgbxLWCgdy+XbF/N1xANmPTCbv1FfPSZ4dJdkp781vlCyMJZ+qP7zpJEp7ZEwEaVnqDIFk6uo2pZiXdubtVx+f5c4XrCW3LPyAy3XriF9I50LY+GE3JLuk/rCrNBR52pYKxYyS6oZe1E6jGmKV3CoV3A6rwInPV/141pc/eiQd1XjMJ/JebTyQykcbx1tk3nay3S8bXLYHbJt8vk3P2zV6Au+NOaubeSfZO4yz87mKmsIsSGEhhdzjz9avhDMWyTz2JLbN827hC7gfnO7LWcJucmzsnNY1mTIKNTGybItlg9Yzede5P+6LhGXeBK7nMqowg/yHD/gIZN+mNbIl3mmAXMSMzdYMSMpHzjD7Jh5ksklM6pJKZIcm6njVnMyJEmuuowWrJKUI8mSq/R4lawMSdLGMbRT2jgkwVvrCd6aJAQrR0tQRjOimoyIZiy4q59qhiSJqz6u71XTgEliuPIN/aSVpjmQJNrZRFeT2uWZFq3sxKpZrfKZiU429dWsTmVBRE9wWx8sU5LVX9UsSZ9OqOhHdSl0RsXQsWt+XqYka5zPXSVrXJ1R0XdqJrTvkPmUh8buCkwOi47yCPUI9Rj1GLWF2kI9QT1BPUU9RW2jtlHPUM9Qz1HPUS9QL1AvUS9RO6gd1C5qF/UK9Qq1h9pD7aP2Ua9Rr1EHqAPUIeoQdYQ6Qh2jjlEnqBPUKeoUdYY6Q71BvUG9Rb3F4lGXTnrXSFiRaIKYhVggViELkEUhHIQXsgRZFmKD2IWsQFaFOCBOIW9B3hbyAPJAtmUgFy8cEFGIB+IV4oP4hQQgQSE/g/xcSAgSFhKBRIU8gchC1iBrPGmQuJBHkMdCNiAbbeSnQp5BnnFVY31ZY31dY31hY31lY31pY31tY31xY311Y315Y3194/9rgWN9hWN9iWN9jWN9kWN9lWN9mWN9nWN9oeOaldaXOtbXOtYXO9ZXO65ZbpHeAdRTHdn+xREQ7v3iGAg3ftECwl1fnADhli9OgXC/F20g3OzFGRDu9OIcCLd5cQGEe7y4BMINXnSAcHcXXSDc2sUVEO7rogeEm7roA+GOLq6BcDsXAyDcy8UQCDdyMQLCXVyMgXALFxMg3L/FFAg3bzEDwp1b3ADhti1ugcierZ6W0wVXT9BkwRUeAR5RPAY8ptgCbFE8ATyheAp4SrEN2KZ4BnhG8RzwnOIF4AXFS8BLih3ADsUuYJfiFeAVxR5gj2IfsE/xGvCa4gBwQHEIOKQ4AhxRHAOOKU4AJxSngFOKM8AZxRvAG4q3gKREVKiXiIr0ElGRXiIq0ktERXqJqEgvERXpJaIivURUpJeIivQSUZFeIirSS0RFeomoSC8RFekloiK9RFSkl4iK9BJRkV4iKtJLREV6iahILxEV6SWiIr1EVKSXiIpqSsQ0s7tG+grAJEVimkfIR4SPkY8Jt5BbhE+QTwifIp8SbiO3CZ8hnxE+Rz4nfIF8QfgS+ZJwB7lDuIvcJXyFfEW4h9wj3EfuE75GviY8QB4QHiIPCY+QR4THyGPCE+QJ4SnylPAMeUb4BvmG8C1yVlald149eGPzpcpxPEc4z3xbfo+TFL59Z0+2qe+583f2ZKEt0pdr5X5K/5nyL3Zjm9puDJ/KImlUX0lFa/Mtt+TckP7cKBI9r3hf5/neWpg83DUFvvvk+rY2EHgxROB4cT7ET/MFs20cYdegWtJ/2G1k8lUT3xvwJ747NLzlI89g54t3tj3z0C+3HZBHT2jMH0BJg+9xAxuaeCTskT7DYA+BPdIG7LFzPD62/Zr3onnDwmE2NKQvE5M0LoaTYdEra5QhaWIWr29SB3igvdIYJ4JbpSFVTDvadMxBGhetg19svVq7Lh03jYvGYcA82pjGWKu2VzojFWObY5c6pnHR+ICzYzmh5fIFi2Qxr1yWRlUxzgELd23l0p4wnEXL35V1PkAab3HVluXRSUsky+sZ4YFXjreUT/nLshWTyTyL8ZLo+8ulge+b+/cwiGKVtP2kSPOM2jSPZj2u0qRHZ8HVoYp76m95CrttObu0djuzyC4faDvZTRvekXd32FCQO2ztDan29l+5BUCx1t0rD9Mp2f2gQ5yR4c7z25zikNO/AfS25TssbTv0g22+9UVL+GXZyPcv95vVP0zpXyZfv25+8/rr62/3fzyCP1p91NhrfN74R6PZ+K7xY+Os0W+MG1bjufGvxr8b/9n7Yu9yb7A3ylPffw/6/LVR+uzd/Q8YcfrU-x?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?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-1p3kx?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\f(a)\n\n(b) M = 3\n\n(c) M = 10\n\nFigure 4: Rank-1 matrix sensing: (a) Population risk. (b, c) A realization of empirical risk. In both\nthis and Figure 5, we use the red star and cross to denote the global minima and saddle points of the\npopulation risk, and use blue square, circle, and diamond to denote the global minima, spurious local\nminima, and saddle points of the empirical risk, respectively.\n\n(a)\n\n(b) M = 3\n\n(c) M = 10\n\nFigure 5: Phase retrieval: (a) Population risk. (b, c) A realization of empirical risk.\n\nThen, we generate the population risk and empirical risk based on\nthe formulation introduced in these two examples. The contour plots\nof the population risk and a realization of empirical risk with M = 3\nand M = 10 are given in Figure 4 for rank-1 matrix sensing and\nFigure 5 for phase retrieval. We see that when we have fewer samples\n(e.g., M = 3), there could exist some spurious local minima as is\nshown in plots (b). However, as we increase the number of samples\nFigure 6: Rank-2 matrix sensing.\n(e.g., M = 10), we see a direct correspondence between the local\nminima of empirical risk and population risk in both examples with a much higher probability. We\nalso notice that extra saddle points can emerge as shown in Figure 4 (c), which shows that statement\n(b) in Theorem 2.1 cannot be improved to a one-to-one correspondence between saddle points in\ndegenerate scenarios. We still observe this phenomenon even when M = 1000, which is not shown\nhere. Note that for the rank-1 case, Theorem 2.1 can be applied directly without restricting to full-rank\nrepresentations. Next, we conduct another experiment on general-rank matrix sensing with k = 2,\nr = 3, N = 8, and a variety of M. We set U(cid:63) as the \ufb01rst r columns of an N \u00d7 N identity matrix and\ncreate X = U(cid:63)U(cid:63)(cid:62). The population and empirical risks are then generated according to the model\nintroduced in Section 3.1. As shown in Figure 6, the distance (averaged over 100 trials) between the\nlocal minima of the population and empirical risk decreases as we increase M.\n\n5 Conclusions\n\nIn this work, we study the problem of establishing a correspondence between the critical points of\nthe empirical risk and its population counterpart without the strongly Morse assumption required\nin some existing literature. With this correspondence, we are able to analyze the landscape of an\nempirical risk from the landscape of its population risk. Our theory builds on a weaker condition\nthan the strongly Morse assumption. This enables us to work on the very popular matrix sensing and\nphase retrieval problems, whose Hessian does have zero eigenvalues at some critical points, i.e., they\nare degenerate and do not satisfy the strongly Morse assumption. As mentioned, there is still room to\nimprove the sample complexity of the phase retrieval problem that we will pursue in future work.\n\n9\n\n-2-1012-2-1012-2-1012-2-1012-2-1012-2-1012-2-1012-2-1012-2-1012-2-1012-2-1012-2-1012204060801000.20.250.3Local minima dist\fAcknowledgments\n\nSL would like to thank Qiuwei Li at Colorado School of Mines for many helpful discussions on the\nanalysis of matrix sensing and phase retrieval. The authors would also like to thank the anonymous\nreviewers for their constructive comments and suggestions which greatly improved the quality of this\npaper. This work was supported by NSF grant CCF-1704204, and the DARPA Lagrange Program\nunder ONR/SPAWAR contract N660011824020.\n\nReferences\n[1] S. Mei, Y. Bai, and A. Montanari, \u201cThe landscape of empirical risk for non-convex losses,\u201d\n\narXiv preprint arXiv:1607.06534, 2016.\n\n[2] V. 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