New sufficient conditions of robust recovery for low-rank matrices

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1 New sufficien condiions of ous ecovey fo low-an maices Jianwen Huang a, Jianjun Wang a,, Feng Zhang a, Wendong Wang a a School of Mahemaics and Saisics, Souhwes Univesiy, Chongqing, , China Reseach Cene fo Aificial Inelligence & Educaion Big Daa, Souhwes Univesiy, Chongqing, , China Asac. In his pape we invesigae he econsucion condiions of nuclea nom minimizaion fo low-an maix ecovey fom a given linea sysem of equaliy consains. Sufficien condiions ae deived o guaanee he ous econsucion in ounded l and Danzig seleco noise seings ϵ 0) o exacly econsucion in he noiseless conex ϵ = 0) of all an maices X R m n fom = AX) + z via nuclea nom minimizaion. Fuhemoe, we no only show ha when = 1, he uppe ound of δ is he same as he esul of Cai and Zhang 9], u also demonsae ha he gained uppe ounds concening he ecovey eo ae ee. Finally, we pove ha he esiced isomey popey condiion is shap. Keywods. Low-an maix ecovey; nuclea nom nomalizaion; esiced isomey popey condiion; compessed sensing; convex opimizaion. AMS Classificaion010): 90C5, 90C59, 15A5, 5A41, 49M30, 65K10, 65J 1 Inoducion Suppose ha X R m n is an unnown low an maix, A : R m n R q is a nown linea map, R q is a given osevaion and z R q is measuemen eo. The an minimizaion polem is defined as follows: min X anx) s.. AX) ϵ, 1.1) whee = AX) + z and ϵ sands fo he noise level. Since he polem 1.1) is NP-had in geneal, Rech e al. 1] inoduced a convex elaxaion, which minimizes nuclea nom also nown as he Schaen 1-nom o ace nom) min X X s.. AX) ϵ, 1.) whee X = min{m,n} i σ i X) and σ i X) ae he singula values. The polem 1.) is convex, hus hee ae a lage nume of appoaches which can e used fo solving i. Fo efficien algoihms Coesponding auho. addess: wjj@swu.edu.cn J. Wang). 1

2 solving oad-scale insans, a gea deal of eseaches have developed hem, such as Tanne and Wei ], Zhang and Li 3], Lu e al 4] and Lin e al. 5]. When m = n and he maix X = diagx) x R m ) is diagonal maix, he polems 1.1) and 1.) degeneae o he l 0 -minimizaion and l 1 -minimizaion, especively, which elong o he main opimizaion polems in he compessed sensing CS). In ode o sudy he elaionship eween l 1 -minimizaion and he nuclea nom minimizaion polems, Rech e al. 1] exended he noion of esiced isomey consan poposed y Candès 6] o low-an maix ecovey case. The concep is as follows: Definiion 1.1. Le A : R m n R q e a linea map. Fo any inege 1 min{m, n}), he esiced isomey consan RIC) of ode is defined as he smalles posiive nume δ ha saisfies 1 δ ) X F AX) 1 + δ ) X F 1.3) fo all -an maices X i.e., he an of X is a mos ), whee X F = X, X = T X X) is he Foenius nom of X, which is also equal o he sum of he squae of singula values and he inne poduc in R m n as X, Y = T X Y ) = m n i=1 j=1 X ijy ij fo maices X and Y of he same dimensions. By he afoemenioned definiion, i is easy o see ha if 1, hen δ 1 δ. Alhough i is no easy o examine he esiced isomey popey fo a given linea map, i is one of he cenal noions in low-an maix ecovey. In fac, i has een showed 1] ha Gaussian o su-gaussian andom measuemen map A fulfills he esiced isomey popey wih high poailiy. Rech e al. 1] showed ha fo he noiseless case i.e., ϵ = 0), if δ 5 < 1/10, hen he minimuman soluion o 1.) can e ecoveed y solving a convex opimizaion polem. Candès and Plan 7] poved ha when δ 4 < 1, a low-an maix can e ously ecoveed y nuclea nom minimizaion 1.). Mohan and Fazel 8] impoved he uppe ound of RIC o δ 4 < Cai and Zhang 9] pesened he shap condiion δ < 1/3 δ < 1/3) fo low-an maix spase signal) ecovey. Wang and Li 10] showed ha he uppe ounds δ < 1/3 and δ < / ae opimal. Kong and Xiu 11] oained a unifom ound on RIC δ 4 < 1 fo any p 0, 1] fo low-an maix ecovey via Schaen p-minimizaion. Chen and Li 1] showed ha fo any given δ 4 3/, 1), p 0, 1 δ 4 )] suffices fo he ous ecovey of all -an maices via Schaen p-minimizaion. Cai and Zhang 13] showed ha fo any given 4/3, δ < 1)/ ensues he exac econsucion fo all maices wih an no moe han in he noise-fee case via he consained nuclea nom minimizaion 1.). Fuhemoe, fo any ε > 0, δ < 1)/+ε doesn suffice o mae sue he exac ecovey of all -an maices fo lage. Besides, hey showed ha condiion δ < 1)/ suffices fo ous econsucion of nealy low-an maices in he noisy case. Moivaed y he afoemenioned papes, we fuhe discuss he uppe ounds of δ associaed wih some linea map A as 0 < < 4/3. Sufficien condiions egading δ wih 0 < < 4/3 ae esalished o guaanee he ous econsucion ϵ 0) o ϵ = 0) of all -an maices X R m n saisfying = AX) + z wih z ϵ and A z) ϵ, especively. Theey, comined wih 13], a complee descipion fo shap esiced isomey popey RIP) consans fo all > 0 is esalished o ensue he exac econsucion of all maices wih an no moe han via nuclea nom minimizaion.

3 Peliminaies We egin y inoducing asic noaions. We also gahe a few lemmas needed fo he poofs of main esuls. y Fo any maix X R m n m n), he singula value decomposiion SVD) of X is epesened X = UdiagσX))V, whee U R m m and V R n m ae ohogonal maices, and σx) = σ 1 X),, σ m X)) indicaes he veco of he singula values of X. Assume ha σ 1 X) σ X) σ m X). Consequenly, he es -an appoximaion o he maix X is whee σ X) = σ 1 X),, σ X)). X ) = U diagσ ] X)) 0 V, 0 0 Fo a linea map A : R m n R q, denoe y is adjoin opeao A : R q R m n. Then, fo all X R m n and R q, X, A ) = AX),. Wihou loss of genealiy, le X e he oiginal maix ha we wan o find and X e an opimal soluion o he polem 1.). Le Z = X X. Le SVD of U ZV R m m e povided y U diag σt U ZV = U ZV ) ) ] diag σ T cu ZV ) ) V0 whee U 0, V 0 R m m ae ohogonal maices, σ T U ZV ) = σ 1 U ZV ),, σ U ZV ) ), σ T cu ZV ) = σ +1 U ZV ),, σ m U ZV ) ), and we suppose ha σ1 U ZV ) σ U ZV ) σ +1 U ZV ) σ m U ZV ). Theefoe, he maix Z is decomposed as Z = Z ) + Z ) c, whee and Z ) diag σt U = UU ZV ) ) ] 0 0 V0 V 0 0 ] Z c ) 0 0 = UU 0 0 diag σ T cu ZV ) ) V0 V. I is no had o see ha X ) Z c ) ) = 0 and X ) ) Z c ) = 0. In ode o show he main esuls, we need some elemenay ideniies, which wee given in 14] see Lemma 1). Lemma.1. Give maices {V i : i T } in a maix space V wih inne poduc, whee T denoes he index se wih T =. Selec all suses T wih =, i I and I = ), hen we ge ) 1 V p = V 1 p 1),.1) i I p p T 3

4 and i I p q V p, V q = ) V p, V q )..) p q T Cai and Zhang developed a new elemenay echnique which saes an elemenay geomeic fac: Any poin in a polyope can e epesened as a convex cominaion of spase vecos see Lemma 1.1 in 13]). I gives a cucial echnical ool fo he poof of ou main esuls. I is also he special case p = 1 of Zhang and Li s esul see Lemma. in 15]). Lemma.. Le m e an inege, and α e a posiive eal nume. We can epesen any veco x in he se V = {x R m : x 1 α, x α}, as a convex cominaion of -spase vecos, i.e., x = i λ i u i whee i λ i = 1 wih λ i 0, supu i ), supu i ) supx) and i λ i u i α. Lemma.3. Lemma.3 in 1]) Le X, Y e he maices of same dimensions. If XY = 0 and X Y = 0, hen Lemma.4. We have X + Y = X + Y..3) Z ) c Z ) + X X )..4) Poof. Since X is he opimal soluion o he polem 1.), we ge Applying he evese inequaliy o.5), we ge X X = X Z..5) X Z = X ) Z c ) ) + X X ) Z ) ) X ) Z c ) X X ) Z )..6) By Lemma.3 and he fowad inequaliy, we ge Comining wih.5),.6) and.7), we ge The poof of he lemma is compleed. X ) + Z c ) ) X X ) + Z ) ) X ) + Z c ) X X ) Z )..7) Z c ) X X ) + X X ) + Z ) Z ) + X X ). 4

5 Selec posiive ineges a and saisfying a + = and a. We use, o epesen all possile index se conained in {1,,, } i.e.,, {1,, }) and = a, =, whee i A and j B wih A = a) and B = ). Define Z ) diag σti U = UU ZV ) ) ] 0 0 V0 V, 0 0 and Z ) diag σsj U = UU ZV ) ) ] 0 0 V0 V. 0 0 Hee σ Ti U ZV ) σ Sj U ZV )) denoes he veco ha equals o σ T U ZV ) on ), and zeo elsewhee. Lemma.5. We have Z ) c = µ U, Z ) c = ν V, Z ) c = τ W, whee µ = ν = τ = 1 wih ν, µ, τ 0, U, V, W ae -an, a-an and 1)-an > 1) wih µ U F α,.8) ν V F a α,.9) and Poof. Se By Lemma.4, hen τ W F α = Z) + X X ). 1 α..10) Z c ) α. By he definiion of Z c ), we ge By he decomposiion of Z, we ge σ T cu ZV ) 1 α α..11) σ T cu ZV ) σ T U ZV ) 1 Z) + X X ) 5

6 α α..1) Comining wih Lemma.,.11) and.1), σ T cu ZV ) is decomposed ino he convex cominaion of -spase vecos, i.e., σ T cu ZV ) = µ u wih Define µ u α. ] 0 0 U = UU 0 V0 V. 0 diagu ) I is easy o see ha U is -an. Theefoe, Z ) c µ U F = is decomposed as Z ) c µ u α. = µ U wih Liewise, Z ) c can also e denoed y Z ) c = ν V, Z ) c = τ W, whee V is a-an, W is 1)-an > 1) wih and ν V F a α, τ V F 1 α. One can easily chec ha Lemma.6. We have ha fo 0 < < 1, ρ a, ) a) a ) Sj = = )a Z ), U = 0, Z ), V = 0 and Z ), W = 0. A ) Z ) + Z ) ) ] A Z ) az ) a a AZ ), AZ + a,,.13) and fo 1 < 4/3, ρ a, ) A ) τ Z ) ) ] + 1)W 1)A Z ) W = 3 a 1) ] AZ ), AZ + 4 3) a,,.14) whee ρ a, ) = a + ) a4 ), 6

7 and a, = a a) i A, + a ) j B, µ a A Z ) + ) U A Z ) a U ν A Z ) + a ) V a A Z ) V ) ] ) ]..15) Poof. The poof aes advanage of he ideas fom 9], 14]. By Lemma.1, we ge a, = a ) a AZ ) a ] a) i A ) AZ ) j B + a + a ) a AZ ) + a a) i A ) AZ ), AZ c ) j B = a ) a a) 1 a 1 ) AZ) a ) ) 1 1 ) AZ) aa + ) + a a) 1 a 1 )AZ) + a ) 1 1 )AZ), AZ c ) = a ) a AZ ) + a a AZ ), AZ c ) AZ ), AZ = ρ a, ) AZ ) + a )..16) whee he fis equaliy follows fom Lemma.5, i.e., Z c ) he second equaliy, we used he ideniy.1). As 0 < < 1, y Lemma 14], we ge has he convex decomposiion, and in ) a + LHS = ρ a, ) AZ ) = ρ a, ) AZ )..17) Susiuing.16) o he igh hand side of.13), we ge ] RHS = ρ a, ) AZ ) + a ) AZ ), AZ )a AZ ), AZ =LHS. Accodingly, he ideniy.13) holds. As 1 < 4/3, we ge { LHS = ρ a, ) 1 1) ] AZ ) + 1) AZ ), } τ AW { } = ρ a, ) 1 1) ] AZ ) + 1) AZ ), AZ c ) 7

8 { = ρ a, ) 4 3 ) AZ ) + 1) AZ ), AZ }. We have ] RHS =4 3) ρ a, ) AZ ) + a ) AZ ), AZ 3 a 1) ] AZ ), AZ =4 3 )ρ a, ) AZ ) + { a )4 3) a 1) ] } AZ ), AZ =LHS. Theefoe, he ideniy.14) holds. Lemma.7. Le X, Y e maices wih X, Y R m n. If XY = 0 and X Y = 0, hen he following holds: X + Y F = X F + Y F..18) Poof. Due o XY = 0 and X Y = 0, comining wih he poof of Lemma.3 1], hen hee ae maices U X U Y U Z ] and V X V Y V Z ] such ha he singula value decomposiions of X and Y ae as follows: diagσx)) 0 0 X = U X U Y U Z ] V X V Y V Z ], and Y = U X U Y U Z ] 0 diagσy )) 0 V X V Y V Z ], whee U X U Y U Z ] and V X V Y V Z ] ae ohogonal maices. As a consequence, we have he SVD of X + Y as follows: diagσx)) 0 0 X + Y = U X U Y U Z ] 0 diagσy )) 0 V X V Y V Z ] Theefoe, The desied esul ae deived. X + Y diagσx)) 0 0 F = 0 diagσy )) F = X F + Y F. Lemma.8. I holds ha {a + ) 4a] a + ) a] )δ } Z ) F aδ α ) a,..19) 8

9 Poof. Noe ha he ans of maices U, Z ) ae no moe han, he ans of maices V, Z ) ae a mos a and a + =. By he -ode esiced isomey popey, we ge a, a µ a 1 δ ) a) Z) + U 1 + δ ) Z ) a ] i A, F U F + a ) ν 1 δ ) Z) + V a 1 + δ ) Z) V ]. j B, Since he inne poduc of Z ) Z ) ) and U V ) equals o zeo, y some elemenay calculaion, we ge a, a ) a Z ) F a a) i A ) Z ) F j B a + )δ a Z ) F + a a) ) Z ) F By Lemma.1, we ge a )δ i A i A F j B µ U F a a)δ ν V F..0) Z ) F = 1 a 1 ) Z) F,.1) F and j B Z ) F = 1 1 ) Z) F..) Susiuing.1) and.) ino.0) and comining wih inequaliies.8) and.9), we ge a, a ) a + ) Z ) F a + )δ ) Z ) F a )δ α a a)δ α a = {a + ) 4a] a + ) a] )δ } Z ) F aδ α ). Lemma.9. I holds ha AZ ϵ.3) Poof. Due o he feasiiliy of X, we ge AZ = AX AX AX + AX ϵ. 9

10 Lemma.10. Lemma 4.1 in 9]) Fo all linea maps A : R m n R q and, s, we have Lemma.11. I holds ha fo 0 < < 1, ρ a, ) a) a ) Sj = δ s s 1)δ..4) A ) Z ) + Z ) S a ) ] j A Z ) T a i az ) ρ a, ) )δ ] Z ) F,.5) and fo 1 < 4/3, ρ a, ) A ) τ Z ) ) ] + 1)W 1)A Z ) + W ] } ρ a, ) { ) + )δ Z ) F α δ 1),.6) whee ρ a, ) = a + ) a4 ). Poof. We fis conside he case of 0 < < 1. As equals o even, we can fix a = = /; And as equals o odd, we can se a = + 1 = + 1)/; Fo oh cases, one can easily pove ha ρ a, ) < 0. Since Z ), Z ) ae a-an and -an, especively, y uilizing -ode RIP, we ge ρ a, ) a) a ) ρ a,) a) a = ρ a,) a) a ) Sj = ) Sj = { 1 δ ) 1 a 1 + δ ) = ρ a,) a) a ) 1 + δ ) 1 a A ) Z ) + Z ) 1 δ ) Z ) a ) Z ) i A a a a ) Z ) F + a a i A { 1 δ ) a a ) 1 ) 1 ) j B ) ] A Z ) az ) + Z ) S a j 1 + δ ) F a F + a ) ] Z ) F j B ]} Z ) F a 1 ) + a a 1 ) + a a ) 1 1 ] Z ) ) F } ) 1 1 ) ] Z ) F Z ) az ) = ρ a, ) )δ ] Z ) F,.7) whee we made use of Lemma.1 o he second equaliy. Nex, we discuss he case of 1 < 4/3. Oseve ha Z ), W ae -an and 1)-an, especively. Unde he assumpion of ρ a, ) < 0, comining wih -ode RIP, we ge ρ a, ) A ) τ Z ) ) ] + 1)W 1)A Z ) + W F F F ] 10

11 ρ a, ) τ 1 δ ) Z ) ] + 1)W F 1) 1 + δ ) Z ) + W F = ρ a, ) ] τ {1 δ ) Z ) F + 1) W F )} 1) 1 + δ ) Z ) F + W F { ] = ρ a, ) 1 δ ) 1) 1 + δ ) Z ) F δ 1) } τ W F ] } ρ a, ) { ) + )δ Z ) F α δ 1),.8) whee he fis equaliy follows fom he fac ha Z ), W = 0, and he las inequaliy, we used he inequaliy.10). 3 Main esuls Theoem 3.1. Conside an minimizaion polem = AX + z wih z ϵ. If δ < /4 ) wih 0 < < 4/3, hen he soluion X o he nuclea nom minimizaion polem 1.) fulfils X X F C 1 ϵ + C X X ), 3.1) whee and C = C 1 = 1 + δ )κ 4 δ, 3.) δ δ 4 ) 4 δ 4 δ ) 3.3) wih { κ = max 4, }. 4 Similaly, Conside an minimizaion polem = AX + z wih z such ha A z) ϵ. If δ < /4 ) wih 0 < < 4/3, hen he soluion X o he nuclea nom minimizaion polem min X X s.. A z) ϵ fulfils X X F D 1 ϵ + C X X ), 3.4) whee and C is given y 3.3). D 1 = κ 4 δ, 3.5) 11

12 Rema 3.1. As = 1, he uppe ound δ < 1/3 is coinciden wih Theoems 3.7 and 3.8 of 9]. Fuhemoe, he uppe ounds of eo esimaes X X F X X F ) ae smalle han he esuls of 9]. In heoy, he ecoveed pecision is given y ou esuls is highe han ha of heis. Coollay 3.1. Assume ha X R m n is a -an maix. Le = AX. If δ < /4 ) 3.6) fo 0 < < 4/3, hen he soluion X o he nuclea nom minimizaion polem 1.) in he noiseless case i.e., ϵ = 0) econsucs X exacly. Rema 3.. As = 1, he uppe ound δ < 1/3 is he same as Theoem 3.5 of 9]. The Gaussian noise siuaion is of special inees in saisics and image pocessing. Noe ha he Gaussian andom vaiales ae essenially ounded. The esuls given in Theoem 3.1 egading he ounded noise siuaion ae immediaely applied o he Gaussian noise siuaion, which employs he simila discussion as ha in 16]. Theoem 3.. Assume ha he low-an ecovey model = AX + z wih z N q 0, σ I). δ < /4 ) fo some 0 < < 4/3. Le X epesen he minimize of min X X s.. z σ q + q log q and le X e he minimize of min X X s.. A z) σ log n. We have wih poailiy a leas 1 1/q, X X F 1 + δ )κ 4 δ σ q + q log q ) + δ 1 + δ 4 ) δ 4 δ X X ), and poailiy a leas 1 1/ π log n, X X F 4 κ 4 δ σ log n + δ ) δ 4 ) 4 δ 4 δ X X ), whee κ is defined in Theoem 3.1. Theoem 3.3. Le 1 m/. Thee is a linea map A : R m m R q wih δ < /4 ) + ε wih 0 < < 4/3, ε > 0 such ha fo some -an maices Y 1, Y R m m wih Y 1 Y, AY 1 = AY 1. Hence, hee don exis any appoach o exacly econsuc all -an maices X ased on A, z). Rema 3.3. Theoems 3.1 and 3.3 joinly indicae he condiion δ < /4 ) wih 0 < < 4/3 is shap. 1

13 4 Poofs of main esuls Wih aove pepaaion, we pesen he poof of main esul. Poof of Theoem 3.1. By he definiion of α and noice ha he an of Z ) is a mos, we ge α = Z) + 4 Z ) X X ) + 4 X X ) Z) F + 4 Z) F X X ) + 4 X X), 4.1) whee in he las sep, we used he fac ha fo any X R m n m n) and p 0, 1], wih X p = i σp i X))1/p. In he siuaion of 0 < < 1, y Lemma.10, we have AZ ), AZ AZ ) AZ m 1 p 1 X F X p 4.) 1 + δ Z ) F AZ = 1 + δ 1 ) Z) F AZ ) δ Z ) F AZ 1 + δ Z ) F AZ, 4.3) whee in he fis inequaliy, we used Cauchy-Schwaz inequaliy, and he second inequaliy follows fom RIP of -ode. Plugging.3) o 4.3), i follows ha 1 + AZ ) δ, AZ ϵ Z ) F. 4.4) Comining wih equaion.13) and inequaliies.19),.5) and 4.4), we have 1 + ρ a, ) )δ ] Z ) F + 4aϵ δ ) Z ) F { } {a + ) 4a] a + ) a] )δ } Z ) F aδ α ) 0. Applying inequaliy 4.1) o aove equaliy, we ge ) a ) 4 ) 4 δ Z ) F ϵ ] 1 + δ ) + 4δ X X ) Z ) F 4δ X X ) ] ) 13

14 In he siuaion of 1 < 4/3, due o he monooniciy of RIC δ, i implies ha AZ ), AZ 1 + δ Z ) F AZ I is easy o chec ha a 1 + δ Z ) F AZ ϵ 1 + δ Z ) F. 4.6) ) 1 4 = ) 1 1 ) 4 > 1 ). 4.7) Comining wih equaion.14) and inequaliies.19),.6) and 4.6), i holds ha ] } ρ a, ) { ) + )δ Z ) F α δ 1) + 4ϵ 1 + δ 3 a 1) ] Z ) F { } 4 3) {a + ) 4a] a + ) a] )δ } Z ) F aδ α ) ) Due o inequaliy 4.1), y fundamenal calculaion, we ge ) 1) a] 4 ) 4 δ Z ) F ϵ ] 1 + δ + 4δ X X ) Z ) F 4δ X X ) ] ) Theey, wo second-ode inequaliies concening Z ) F ae esalished. Unde he condiion of δ < /4 ), applying quadaic fomula and some elemenay compue, we have Z ) 1 4δ X X ) F 4 ) 4 δ + ϵ 1 + δ 4 )κ ) 4δ X X ) + + ϵ ) 1 + δ 4 )κ + 16δ X X ) ) ] 1 4 ) ] 4 δ 1 8δ X X ) 4 ) 4 δ + 4ϵ 1 + δ 4 )κ ) + 4 X ] X) 4 ) 4 δ )δ = 1 + δ κ 4 δ ϵ 4δ ) 4 δ )δ 4 ) 4 δ )δ 14 X X) 4.10)

15 wih κ is defined in Theoem 3.1, whee he second inequaliy follows fom he fac ha fo any veco x R n, x x 1. Then, c F = σi U ZV ) Z ) i +1 1/ max {σ iu ZV )} σ i U ZV ) i +1 i +1 = Z c ) 1/ Z c ) 1/ Z) 1/ Z ) + X X ) ) 1/ Z ) F + X X) Z ) F ) 1/, 4.11) 1/ whee in he second inequaliy, we used Lemma.4, and he hid inequaliy follows fom he fac ha fo any -an maix X, X X F. A cominaion of 4.10) and 4.11) implies ha Z F = Z ) c F + Z ) F ) 1/ ) 1/ Z ) F + X X) Z ) F Z ) F + X X) 1 + δ )κϵ 4 δ + 1 δ ) 4 δ 4 δ )δ X X ). In he siuaion of he eo ound A z) ϵ, se Z = X X. I holds ha A AZ = A AX ) A AX ) A AX ) + A AX ) ϵ. Moeove, AZ ), AZ = Z ), A AZ Z ) ϵ ϵ Z ) F. 15

16 The es of seps ae simila wih he siuaion of he eo ound z ϵ. The poof of Theoem 3.1 is compleed. Poof of Theoem 3.3. Le E = diagx) R m m wih Define A : R m m R q as AX = x = 1 1,, 1, 0,, 0). }{{} 4 σx) σx), σe) σe)). Applying he Cauchy-Schwaz inequaliy, fo all -an maices X, we ge and Theefoe, σx), σe) σx) σe) 1 sup σx)) X F, AX = 4 σx) σx), σe) σe), σx) σx), σe) σe) 4 = 4 X 4 F σx), σe) ]. AX 1 + ) X F ) 4 + ε X F. 4.1) Fo > 1/ε, we ge AX ) X F 1 1 ) X F 1 ) X F 1 4 ε ε ) X F. Accodingly, y Definiion 1.1, we oain δ = δ = 4 + ε. Suppose Y 1 = diagy 1 ), Y = diagy ) R m m wih y 1 = 1,, 1, 0,, 0) }{{} 16

17 and y = 0,, 0, 1,, 1, 0,, 0). }{{}}{{} I is easy o veify ha Y 1 and Y ae oh maices of an such ha Y 1 Y N A), i.e., AY 1 = AY. Consequenly, i is no possile o econsuc oh Y 1 and Y ased on z, A). 5 Conclusions In his pape, we esalish sufficien condiions which ensue he sale ecovey o exacly ecovey of any -an maix saisfying a given linea sysem of equaliy consains via solving a convex opimizaion polem, i.e., nuclea nom minimizaion. When he paamee is equal o 1, he ound of RIC δ coincide wih he esul of 9]. Meanwhile, he deived uppe ounds egading he econsucion eo ae ee han hose of 9]. Besides, he esiced isomey popey condiion is poved shap. Acnowledgmens This wo was suppoed y Naual Science Foundaion of China No , , ), Fundamenal Reseach Funds fo he Cenal Univesiies No. XDJK015A007, SWU180900), Science Compuing and Inelligen Infomaion Pocessing of GuangXi highe educaion ey laoaoy No. GXSCIIP0170), Science and Technology Plan Pojec of Guizhou Povince No. LH015]7053, LH015]7055 ), Science and echnology Foundaion of Guizhou povince Qian e he Ji Chu 016]1161), Guizhou povince naual science foundaion in China Qian Jiao He KY 016]55). Refeences 1] Rech, B., Fazel, M., Pailo, P. Guaaneed minimum-an soluions of linea maix equaions via nuclea nom minimizaion. SIAM Rev., 14: ) ] Tanne J, Wei K. Low an maix compleion y alenaing seepes descen mehods. Applied & Compuaional Hamonic Analysis, 016, 40): ] Zhang N, Li Q. On opimal soluions of he consained l 0 egulaizaion and is penaly polem. Invese Polems, 016, 33). 4] Lu C, Tang J, Yan S, e al. Nonconvex Nonsmooh Low-Ran Minimizaion via Ieaively Reweighed Nuclea Nom. IEEE Tans Image Pocess, 015, 5): ] Lin Z, Xu C, Zha H. Rous Maix Facoizaion y Majoizaion Minimizaion. IEEE Tansacions on Paen Analysis & Machine Inelligence, 018, 401): ] Candes, E.J. The esiced isomey popey and is implicaions fo compessed sensing. C. R. Acad. Sci. Pais. Se.1, 346: ) 7] Candes E J, Plan Y. Tigh Oacle Inequaliies fo Low-Ran Maix Recovey Fom a Minimal Nume of Noisy Random Measuemens. IEEE Tansacions on Infomaion Theoy, 011, 574): ] Mohan K, Fazel M. New esiced isomey esuls fo noisy low-an ecovey. In: IEEE Inenaional Symposium on Infomaion Theoy Poceedings. Seale: IEEE, 010,

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