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[7]. of main in problem analyzed the and to presented introduction brief a in provided are issues above the IV. on Section details More propose. we X loi 7,teatositouetento of notion the introduce authors the [7], in Also Let osdragnrcmatrix generic a Consider h ae sognzda olw.I eto I eprovide we II, Section In follows. as organized is paper The k = T ∗ ujc to subject minimize ersnstencernr of norm nuclear the represents UΣV P ∈ n ( R ,TeSaeUiest fNwJersey New of University State The s, M nodrt eieseiccniin ne which under conditions specific derive to order in , u N I L II. M i × ) T ∈ µ Ω U N ∈ 7 Let [7] from ( ≡ R U OW eteotooa rjcinonto projection orthogonal the be otisalmti oriae corresponding coordinates matrix all contains R ihrsett h tnadbasis standard the to respect with K X k U ) P K X × , ≡ × R ( 1 , P i k ,j i, ∈ L o ANK U ∗ N N ( N r r [ i M u ∈ eoea nrws apigof sampling entrywise an denote r + = ) ≡ N i i 1 M sup sup σ ∈ ∈ r + ) U u M N N i R ysligtecne program convex the solving by M 2 N + N + ( cnandin (contained and r ATRIX and M M . . . k k ⊆ X ( ∈ P ) ,j i, N ∈ n u U R V u r + n i v M e ) N r v ( R C u i i ] epciey pne by spanned respectively, , U i t k X h omldefinition formal The . K ∈ OMPLETION i ∈ easbpc spanned subspace a be 2 2 n ihclm and column with and ( ∈ × . ∀ R ,k i, R L R N L ( P ,j i, × N frank of )) × ( 1 × r 2 M o ) 1 . n let and i o ∈ ∈ ) U n where and ) i N ∈ Ω r + hnthe Then . N { r subspace , respec- , r + e whose , i Also, . P } i ∈ U M ion N (2) (3) , N + al . Additionally, the following crucial assumptions regarding a t [0, 1] and a probability of failure γ [0, 1]. Then, as the subspaces U and V are of particular importance [7]. long∈ as ∈ 2θ (log (d + 2) log (γ)) N , (6) R −2 A0 max µ (U) ,µ (V ) µ0 ++. ≥ (1 t) { }≤ ∈ − the associated Euclidean distance matrix ∆ with worst case t r R A1 ∈N+ uivi µ1 , µ1 ++. i r ∞ ≤ KL ∈ rank d +2 obeys the assumptions A0 and A1 with r Indeed, ifP the constants µ and µ associated with the θ θ 0 1 µ0 = and µ1 = , (7) singular vectors of a matrix M are known, the following t (d + 2) t√d +2 theorem holds. with probability of success at least 1 γ, where the constant × − Theorem 1. [7] Let M RK L be a matrix of rank r obeying θ (that is, independent of N) is defined as ∈ A0 and A1 and set N , max K,L . Suppose we observe 2 2 4 { } 1+ dc + d c m entries of M with matrix coordinates sampled uniformly at θ , ∗ , (8) λ random. Then there exist constants C, c such that if ∗ with λ , min λ , λ , λ ,m , and where λ , λ , λ are the 2 1/2 1/4 1 2 3 2 1 2 3 m C max µ ,µ µ ,µ N Nrβ log N (4) real and positive{ solutions to the} cubic equation ≥ 1 0 1 0 n o i for some β > 2, the minimizer to the program (2) is unique αiλ =0, (9) −β and equal to M with probability at least 1 cN . For r i∈N3 −1 1/5 − ≤ X µ N this estimate can be improved to 0 with 6/5 m Cµ0N rβ log N, (5) , 3 2 ≥ α0 m4m2 m2 m3 d, (10) − − with the same probability of success. α ,  m3d2+  1 − 2 Of course, the lower the rank of M, the less the required + m3 + m2 + m2 m m m d m , (11) number of observations for achieving exact reconstruction. 2 3 2 − 4 − 4 2 − 2 ,  2 2 2  Regarding the rank the EDM ∆ at hand, one can easily prove α2 m2d + m4 m2 d + m2 + 1 and (12) the following lemma [1]. − α , 1.   (13) × 3 Lemma 1. Let ∆ RN N be an EDM corresponding to the − distances N of points∈ (nodes) in Rd. Then, rank (∆) d+2. Before we proceed with the proof of the theorem, let us ≤ state a well known result from random matrix theory, the Thus, in the most common cases of interest, that is, when matrix Chernoff bound (exponential form - see [8], Remark d equals 2 or 3 and for sufficiently large number of nodes N, 5.3), which will come in handy in the last part of the proof. rank (∆) N and consequently the problem of recovering ∆ from a restricted≪ number of observations is of great interest. Lemma 2. [8] Consider a finite sequence of N Hermitian and statistically independent random matrices × III. THE COHERENCE OF RANDOM EDMS F CK K satisfying i N+ ∈ i∈ N According to Theorem 1, the bound of the minimum number n o of observations required for the exact recovery of ∆ from F 0 and λ (F ) R, i N+ (14) i  max i ≤ ∀ ∈ N (∆), involves the parameters µ and µ . Next, we derive a 0 1 and define the constants generalP result, which provides estimates of these parameters in a probabilistic fashion. ξ , λ E F . (15) Theorem 2. Consider N points (nodes) lying almost surely in min(max) min(max) i  + { } , d i∈N a d-dimensional convex polytope defined as d [a,b] with XN , tH N+   a < 0 < b and let pi [xi1 xi2 ... xid] d,i N , + Define Fs ∈N+ Fi. Then, it is true that ∈ H ∈ N i N denote the position of node i, where xij 1, j d represents its respective -th position coordinate.∈ H Assume∈ that j P ξ each x , (i, j) N+ N+ is drawn independently and P λ (F ) tξ K exp min (1 t)2 , (16) ij ∈ N × d { min s ≤ min}≤ − 2R − identically according to an arbitrary but atomless probability   measure P with finite moments up to order 4. Define t [0, 1], and ∀ ∈ tξmax , E k k e R mk xij x dP < , P λ (F ) tξ K , t e. ≡ ˆ ∞ max s max t n o { ≥ }≤ ∀ ≥ N+ N+ N+   k 4 , (i, j) N d and let, without any loss of Proof of Theorem 2: In order to make it more tractable, generality,∈ ∀m 0∈. Also,× define c , max a , b and pick we will divide the proof into the following subsections. 1 ≡ {| | | |} × A. Characterization of the SVD of ∆ R(d+2) (d+2) is symmetric by definition, the finite dimen- sional spectral theorem implies that it is diagonalizable with To begin with, observe that ∆ admits the rank-1 decompo- T T sition eigendecomposition given by ADA = QΛQ , where R(d+2)×(d+2) T T t t t Q with QQ = Q Q I(d+2) and ∆ = 1 × p 2 x x + p1 × , (17) ∈ (d+2)×(d+2) ≡ N 1 − i i N 1 Λ R is diagonal, containing the eigenvaluese of ∈N+ ∈ T iXd ADA . Thus, we arrive at the expression T where e T T t ∆ = VQΛQ V VQ Λ VQ sign Λ , (28) 2 2 2 RN ≡ | | · p = p1 p2 ... pN + , (18) k k2 k k2 k k2 ∈    T , h t i N+ which constitutese a valid SVDe of ∆, since (VQe ) VQ pi [xi1 xi2 ... xid] d, i N and (19) ≡ t ∈ H ∈ I(d+2) and consequently, by the uniqueness of the singular x , [x x ... x ] , i N+. (20) i 1i 2i Ni ∈ HN ∈ d values of a matrix, Λ Σ. Therefore, we can set U = VQ R1×(d+2)| |≡ N+ Then, we can equivalently express ∆ as and if Vi ,i N denotes the i-th row of V, ∈ ∈ T e ∆ = XDX , (21) N 2 N 2 µ (U)= sup ViQ 2 sup Vi 2 . (29) d +2 ∈N+ k k ≡ d +2 ∈N+ k k where i N i N × 2 N (d+2) B. Bounding sup N+ Vi X , col (X , X ,..., X ) R , (22) i∈ N 2 1 2 N ∈ k k t × Next, consider the hypotheses of the statement of The- X , 1 p p 2 R1 d+2, i N+ and (23) i i k ik2 ∈ ∈ N orem 2. Since the probability measure P is atomless, the , h i R(d+2)×(d+2) columns of X will be almost surely linearly independent. D diag (1, 211×d, 1) . (24) − ∈ Then, rank (A)= d +2 almost surely and consequently Since ∆ is a symmetric matrix, it can be easily shown that X 2 its SVD possesses the special form 2 i 2 N+ Vi k k , i N , (30) T T k k2 ≤ σ2 (A) ∀ ∈ ∆ = U Λ (U sign (Λ)) , UΣU±, (25) min | | · × T where X R1 (d+2),i N+ denotes the i-th row of X. λi (∆) ui sign (λi (∆)) ui , (26) i ∈ ∈ N ≡ + | | Thus, in order to bound from above, it suffices to bound ∈N µ (U) i Xd+2   2 2 Xi 2 from above and σmin (A) from below. Considering that k k 2 where A and sign (A) denote the entrywise absolute value p is bounded almost surely in , we can easily bound X | | K×L i d i 2 and sign operators on the matrix A R , respectively. as H k k In the expressions above, Σ Λ , ∈Λ R(d+2)×(d+2) is 2 2 2 4 N+ Xi 2 1+ dc + d c , i N (31) the diagonal matrix containing≡ the | (at| most∈ d +2) non zero k k ≤ ∀ ∈ , 2 eigenvalues of ∆ in decreasing order of magnitude, denoted as where c max a , b . Regarding σmin (A) T {| | | |} T ≡ N+ λi (∆) ,i d+2, whose absolute values coincide with its sin- λmin A A , observe that the Grammian A A admits the ∈ × gular values, that is, σ (∆) λ (∆) , and U RN (d+2) rank-1 decomposition i ≡ | i | ∈   contains as columns the eigenvectors of ∆ corresponding to T T T T T N+ A A A V VA = X X = Xi Xi. (32) its non zero eigenvalues, denoted as ui,i d+2, which ≡ ∈ ∈N+ essentially coincide with its left singular vectors. iXN Due to this special form of the SVD of ∆, if we denote its In general, it seems impossible to find a deterministic lower T column and row subspaces with U and U±, respectively, it is bound for λ A A . For this reason, one could resort on true that min probabilistic bounds that are generally a lot easier to derive, N 2 providing considerably good estimates. Towards this direction, µ (U±)= sup (sign (λi (∆)) U (i, k)) + d +2 ∈N + below we will employ the matrix Chernoff bound (Lemma 2), i N k∈N Xd+2 which fits perfectly to our bounding problem. N T U 2 RN×N = sup ( (i, k)) µ (U) . (27) Observe that the sequence Xi Xi + con- + ∈N d +2 ∈N ≡ ∈ i N i N ∈N+ k Xd+2 sists of N statistically independentn random Gramians,o which T N+ As a result, at least regarding the Assumption A0, it suffices implies that Xi Xi 0, i N . Also, since all these Gramians are rank-1, it can∀ be∈ trivially shown that the one to study the coherence of only one subspace, say U. It is T and only non zero eigenvalue of X X coincides with X 2 then natural to consider how the SVD of ∆, given by (25), is i i k ik2 related to its alternative representation given by (21). and consequently Consider the thin QR decomposition of X given by X = T 2 2 4 N+ RN×(d+2) T λmax Xi Xi 1+ dc + d c , i N . (33) VA, where V with V V I(d+2) and ≤ ∀ ∈ × ∈ ≡ A R(d+2) (d+2) constitutes an upper triangular matrix. As a result, the hypotheses of Lemma 2 are satisfied and T T T Then,∈ ∆ = VADA V and since the matrix ADA our next task involves specifying the constant ξ . Since ∈ min the coordinates x , (i, j) N+ N+ are independent and Thus, R has an eigenvalue λ , m with multiplicity d 1 ij ∈ N × d d 0 2 − identically distributed, it can be easily shown that for sure, and three additional eigenvalues λ1, λ2, λ3, which of course are the roots to the cubic polynomial appearing in E T E T A A = N X1 X1 (37) and can be computed easily in closed form. Since Rd + ≻ n o n o0 0, d N , all its eigenvalues must be strictly positive and, 1 1×d dm2 ∀ ∈ 0 × m I m 1 × therefore, = N  d 1 2 d 3 d 1  ∗ 2 2 2 ξmin = Nλmin (Rd)= Nλ , (38) dm2 m311×d d m4 m2 + d m2  −  ∗ , R ,     where λ min λ1, λ2, λ3,m2 ++. NRd. (34) { } ∈ C. Putting it altogether Thus, We can now directly apply the matrix Chernoff bound E T ξmin = λmin A A = Nλmin (Rd) (Lemma 2) to upper bound the probability of the event , 2 ∗  n o σmin (A) tNλ , t [0, 1] as and it suffices to characterize the eigenvalues of Rd × Z ≤ ∀ ∈ R(d+2) (d+2). ∈ n o N (1 t)2 1) Positive Definiteness of Rd: We first argue that Rd P ( ) (d + 2)exp − , ǫ (t) , (39) is a positive definite matrix. We can prove this argument Z ≤ − 2θ ! using the strong Law of Large Numbers (LLN). Since A is T where almost surely of full rank, the Gramian A A will be almost 1+ dc2 + d2c4 T θ , ∗ . (40) surely positive definite. Consider the sequence Ai Ai , λ i∈N T consisting of independent realizations of A An. Of course,o The inequality (39) is equivalent to the statement that ∗ T N 2 A with probability at least Ai Ai 0, i . Then, the strong LLN implies that σmin ( ) tNλ 1 ǫ (t) , t ≻ ∀ ∈ [0, 1] . It is≥ natural to select N such that 1 −ǫ (t) [0∀ , 1]∈. 1 T a.s. T − ∈ A A E A A , as N . (35) Since the involved exponential function is strictly decreasing N i i −→ → ∞ i∈N in N, we can choose a γ [0, 1] such that ǫ (t) γ, yielding XN n o ∈ ≤ As a consequence of the fact that the set of positive definite the condition matrices is closed under linear combinations with nonnegative 2θ (log (d + 2) log (γ)) T N − . (41) weights, E A A 0 and, therefore, by (34), R ≥ (1 t)2 ≻ d ≻ − 0, d N+n. o Consequently, as long as (41) holds, the inequality ∗ 2)∀ Characterization∈ of the eigenvalues of R : Next, in σ2 (A) tNλ will hold with probability at least 1 d min ≥ − order to find the eigenvalues of Rd, we would like to solve γ, γ [0, 1]. the equation ∀Hence,∈ since X 2 1 + dc2 + d2c4, i N+ with k ik2 ≤ ∀ ∈ N det (R λI )=0. (36) probability 1, we can write d − d+2 2 2 2 4 Using a well known identity, we can rewrite (36) as N Xi 2 1+ dc + d c µ (U) sup 2k k = ∗ ≤ d +2 ∈N+ σ (A) tλ (d + 2) det (R λI )= i N min d − d+2 − θ det (E λI ) det (H λ) G (E λI ) 1 F = , µ , t [0, 1] , (42) − d+1 − − − d+1 ≡ t (d + 2) 0 ∀ ∈ (1 λ) (m λ)d  − 2 − · holding true with probability at least 1 γ, γ [0, 1] and 2 2 2 − ∀ ∈ 2 2 2 d m2 dm3 with N satisfying (41). d m4 m2 + d m2 λ =0, Finally, regarding the Assumption A1, by a simple argu- · − − − 1 λ − m2 λ!   − − ment involving the Cauchy - Schwarz inequality, it can be where shown that [7]

1 01×d dm2 θ E , , F , , , √ (43) 0 × m I m 1 × µ1 µ0 d +2= , t [0, 1] ,  d 1 2 d  3 d 1 t√d +2 ∀ ∈ , , 2 2 2 G dm2 m311×d , H d m4 m2 + d m2, under the same of course circumstances as µ , therefore − 0 completing the proof. and which, after lots of dull algebra, yields the equation If the coordinates of the nodes are drawn from a symmetric probability distribution, the following more compact theorem − (m λ)d 1 α λi =0, (37) holds. 2 −  i  i∈N X3 Theorem 3. Consider N points (nodes) lying almost surely where the coefficients α  are given by (10), (11), (12) in a d-dimensional hypercube defined as , [ a,a]d with i i∈N3 d and (13), respectively. { } a > 0 and let the rest of the hypothesesH of Theorem− 2 hold, with the additional assumption m3 0. Pick a t [0, 1] which constitutes a special case of the matrix Rd defined in and a probability of failure γ [0, 1]≡. Then, as long∈ as the (34), equals 1/3. One can easily confirm that this is not true. condition (6) holds, the associated∈ Euclidean distance matrix Additionally, there is a minor oversight in both proofs, ∆ with worst case rank d +2 obeys the assumptions A0 and where it is implicitly stated that since a Euclidean distance A1 with constants µ0 and µ1 defined as in (7), respectively, matrix is symmetric, its (real) left singular vectors coincide with probability of success at least 1 γ and with its right ones. This argument would be correct only if one − could prove that all EDMs belong to the cone of symmetric 2 2 4 2 1+ da + d a and positive semidefinite matrices. Further, such an argument θ , ∗ , (44)  λ  is also incorrect since one can easily prove by counterexample where that there is at least one EDM that is not positive semidefinite, i.e., with spectrum containing at least one negative eigenvalue. ∗ In the case of simply symmetric matrices, the SVD takes the λ , min ζ ζ2 4d m m2 , 2m (45) 4 2 2 special form of (25). However, we should note here that this ( − r − − )   mistake by itself does not affect the outcome of the proofs, , 2 2 2 and ζ d m4 m2 + d m2 +1. since the coherence of a matrix essentially depends on the − norm of the rows of the matrices whose columns constitute   The proof of Theorem 3 is almost identical to that of the sets of its left and right singular vectors, respectively. Theorem 2. Therefore, it is omitted. Further, if the node coor- dinates are drawn independently from [ 1, 1], the following V. CONCLUSION U − corollary constitutes a direct consequence of Theorem 2 and To the best of the authors’ knowledge, Corollary 1 provides is also presented without proof. a novel result regarding the coherence of EDMs, for the Corollary 1. Consider N points (nodes) lying almost surely in special case where the node coordinates are independently , d drawn from [ 1, 1]. Furthermore, our main result, Theorem the d-dimensional hypercube defined as d [ 1, 1] and let U − , t N+ H − 1 presented above, provides a substantial generalization to pi [xi1 xi2 ... xid] d,i N denote the position of ∈ H N+∈ Corollary 1, essentially covering any case where the node node i, where xij 1, j d represents its respective j-th ∈ H ∈ + + coordinates are independently drawn according to an arbitrary, position coordinate. Assume that each x , (i, j) N N is ij N d non - singular probability measure. Therefore, the theoretical drawn independently and identically according to∈ the unifo× rm results presented in this paper can provide strong evidence as in [ 1, 1] probability measure, [ 1, 1]. Pick a t [0, 1] well as sufficient conditions under which the EDM completion and− a probability of failure γ U[0−, 1]. Then, as long∈ as the problem can be successfully solved with high probability, a condition (6) holds, the associated∈ Euclidean distance matrix fact that also justifies their direct practical applicability in ∆ with worst case rank d +2 obeys the assumptions A0 and various modern applications which involve EDMs, such as A1 with constants µ and µ defined as in (7), respectively, 0 1 sensor network localization and estimation of the second order with probability of success at least 1 γ and − statistics of channel state information in wireless communica- 90 1+ d + d2 tion networks. θ , , (46)   REFERENCES ζ ζ2 720d − − [1] A. Montanari and S. Oh, “On positioning via distributed matrix com- q where ζ , 5d2 +4d + 45. pletion,” in IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM), 2010, pp. 197 – 200. [2] P. Drineas, A. Javed, M. Magdon-Ismail, G. Pandurangant, R. Vir- IV. DISCUSSION REGARDING RELATED RESULTS rankoski, and A. 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