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asymptotically Gaussian. Furt hermore, we demonstrate that the errors introduced when transitioning from this theoretica l estim-actor to the computable esti- mators are negligible. As a result, the desirable asymptoti c properties of ˜θNextend to the actual estimators. To the best of our knowledge, in all the aforemen... | https://arxiv.org/abs/2502.06514v2 |
insufficient to derive our result s. It is also crucial to prove that the Malliavin derivatives of the interacting particles con verge, asN→ ∞, to the Malliavin derivatives of the independent particles. More precisely, the core tool that allows us to derive our mai n results is the propagation of chaos for the Malliavin ... | https://arxiv.org/abs/2502.06514v2 |
some k,l= 0,1,... |f(x1,µ1)−f(x2,µ2)| ≤C(|x1−x2|+W2(µ1,µ2))(1+|x1|k+|x2|k+Wl 2(µ1,δ0)+Wl 2(µ2,δ0)). (6) Wedenoteby θ0thetruevalueoftheparametervector andwesuppressthedep endenceofseveral objects on the true parameter θ0. In particular, we write P:=Pθ0,E:=Eθ0Xi,N t:=Xθ0,i,N t, ¯Xi t:=¯Xθ0,i t,µN t:=µθ0,N tand ¯µt:= ¯µθ0... | https://arxiv.org/abs/2502.06514v2 |
interactions, one c an easily derive the following: DsXt=σ∂x0Xt ∂x0Xs1s≤t=σexp/parenleftbigg/integraldisplayt s/an}⌊ra⌋ketle{tθ0,∂xb(Xr)/an}⌊ra⌋ketri}htdr/parenrightbigg 1s≤t, (10) wherex0is the initial condition. However, when interactions are introduced, the situation b ecomes significantly more com- plex. Controlling... | https://arxiv.org/abs/2502.06514v2 |
we have |bm(0,δ0)| ≤c, whereδ0denotes the Kronecker delta. Observe that Assumptions 1and2together imply the linear growth of bm,m= 1,...,p, in the following sense: ∀x∈R,∀µ∈ P2(R); there exists a constant c>0 such that |bm(x,u)| ≤c(1+|x|+W2(µ,δ0)). (13) Assumption 3. (Boundedness moments) For all k≥1 Ck:=/integraldispla... | https://arxiv.org/abs/2502.06514v2 |
are considering continuous observations ,σand Hcan be assumed known in our manuscript. Toestablishourmainresults, wewillprimarilyrelyonMall iavincalculusandthepropagation of chaos. A brief introduction to Malliavin calculus is prov ided in Section 6, while the specifics of the propagation of chaos are discussed in the s... | https://arxiv.org/abs/2502.06514v2 |
for fra ctional Brownian motion will play a central role in establishing our main results. Consequent ly, we will frequently rely on the fact that the Malliavin derivative of an interacting partic le is well approximated by that of the independent particle, similar to the propagation of cha os discussed earlier. This fi... | https://arxiv.org/abs/2502.06514v2 |
conv ergence rate of N−q, which surpasses the classical rate typically associated with pro pagation of chaos. A reader familiar with Malliavin calculus might attribute this to the regular izing effect of Malliavin derivatives (see, e.g., [ 62]), which could facilitate a faster convergence in the propa gation of chaos th... | https://arxiv.org/abs/2502.06514v2 |
Ψ Nat a rate of1 N, as detailed in the following proposition. Proposition 3.3. Assume that Assumptions 1-3hold and that g:R×P2(R)→Ris a locally Lipschitz function according to (6). Then, the following convergence in probability holds: 1 NN/summationdisplay i=1/integraldisplayT 0g(Xi,N s,µN s)dsP− →/integraldisplayT 0E[... | https://arxiv.org/abs/2502.06514v2 |
shows that, in the Brownian setting and for interaction functions bthat are linear in the measure component, an accelerated rate of convergence can indeed be achieved. T his result has since been extended to general additive Gaussian noise in [ 63]. We were quite surprised (and excited!) to discover that the i mproved ... | https://arxiv.org/abs/2502.06514v2 |
that, for any N≥N0, /vextendsingle/vextendsingleDi sb(Xi,N t,µN t)−σ∂xb(Xi,N t,µN t)exp/parenleftbig/integraldisplayt s/an}⌊ra⌋ketle{tθ0,∂xb(Xi,N r,µN r)/an}⌊ra⌋ketri}htdr/parenrightbig 1s≤t/vextendsingle/vextendsingle≤c N. 16 However, the proposition above does not fully resolve the is sue, as the proposed approxi- ma... | https://arxiv.org/abs/2502.06514v2 |
consistent in probability: ˆθ(2) N,εP− →θ0asN→ ∞. 2. Ifε=o/parenleftBig 1√ N/parenrightBig , thenˆθ(2) N,εis asymptotically normal: √ N/parenleftBig ˆθ(2) N,ε−θ0/parenrightBigL− → N(0,˜Σ2), with˜Σ2as defined in (21). The proof of Theorem 3.8is presented in Section 7. It builds on the asymptotic properties of the fake-st... | https://arxiv.org/abs/2502.06514v2 |
N,n−1). The conditions ensuring that FNis a contraction also guarantee the convergence of the itera tive estimator to the fixed-point solution as n→ ∞. Moreover, the favorable properties of the iterative estimator are summarized in the following coroll ary. Corollary 3.11. LetH≥1 2. Assume that Assumptions 1,2,3,4,5,6(b... | https://arxiv.org/abs/2502.06514v2 |
to approximate the sample bias and RMSE. To simulate the trajectories on the interval [0,1], we generate inde- pendent fractional Brownian motions, and then use the Euler scheme. The first model we consider is a simple linear model, namely Xi,N t=/integraldisplayt 0θ0 Xi,N s−1 NN/summationdisplay j=1Xj,N s ds+BH,i t... | https://arxiv.org/abs/2502.06514v2 |
Tmax. The ratio estimator is significantly faster than the fixed-po int estimator, as it requires one fewer integral approximation. However, in our simulations , the fixed-point estimator provides more precise results: in particular, the RMSE improves by a f actor of 10 compared to the ratio 21 estimator. The relatively p... | https://arxiv.org/abs/2502.06514v2 |
n, as introduced in ( 26). This would not only address the discrete observation sett ing but also 22 allow for a generalization beyond linearity in the paramete r, overcoming another limitation of our current framework. Second, the efficiency of the proposed estimators remains an o pen question. While the fixed- point est... | https://arxiv.org/abs/2502.06514v2 |
of fBm. Specifically, Ω = C0(R+;RN) is the set of continuous functions from R+toRN, equipped with the uniform topology on compact intervals. Fis the Borel σ-algebra, andPis the probability measure on (Ω ,F), such that the process BH t(ω) is an fBm with Hurst parameterH∈(0,1). 23 Next, we recall some background material ... | https://arxiv.org/abs/2502.06514v2 |
of probability measures We study a stochastic differential equation dependent on a mea sure, and to derive our result, we focus on the differentiability of the associated stochastic fl ow. This requires a concept of differ- entiation for functions defined on spaces of probability mea sures. The notion of differentiability we... | https://arxiv.org/abs/2502.06514v2 |
P2(R), and let˜udenote its lifting to L2(Ω,F,P;R). Then, for any µ∈ P2(R), there exists a measurable functionξ:R→Rsuch that for all X∈L2(Ω,F,P;R)withL(X) =µ, it holds that D˜u(X) = ξ(X)almost surely. Bystatingthat uiscontinuously L-differentiable, wemeanthattheFr´ echet derivative D˜u(X) of the lifting ˜ uis a continuou... | https://arxiv.org/abs/2502.06514v2 |
its empirical projec- tionuNis differentiable on RN, and for all i∈ {1,...,N}, ∂xiuN(x1,...,x N) =1 N∂µu 1 NN/summationdisplay j=1δxj (xi). This result will prove particularly useful when we compute t he Malliavin derivative of the interacting particle system (see Point 1 of Lemma 6.5below). 6.3 Technical results In... | https://arxiv.org/abs/2502.06514v2 |
H⊗H=/integraldisplayT 0/integraldisplayT 0/integraldisplayT 0/integraldisplayT 0E[DugsDvgt]φ(u,v)φ(s,t)dudvdtds, (34) withφas in (9). This recall will be useful for the following technical res ult, whose proof is provided in Section 8. We require this result because, in the proof of consistency , we will frequently app... | https://arxiv.org/abs/2502.06514v2 |
last inequality follows from q≥2. Combining everything, we obtain: E/bracketleftBigg sup t∈[0,T]|Xi,N t−¯Xi t|q/bracketrightBigg ≤c/integraldisplayT 0E/bracketleftBig |Xi,N t−¯Xi t|q/bracketrightBig dt+cN−1 2, ≤c/integraldisplayT 0E/bracketleftbigg sup s≤t|Xi,N s−¯Xi s|q/bracketrightbigg dt+cN−1 2 29 which, after apply... | https://arxiv.org/abs/2502.06514v2 |
bo unds similar in nature to those used in the previous proof. 7.1.3 Proof of Corollary 2.6 Proof.Point 1. From ( 17) we know that, for j/ne}ationslash=i,Dj sf(¯Xi t,¯µt) = 0, so that the result follows from the boundedness of the derivatives of f, as in Assumption 6(f), and Point 2 of Lemma 6.5. Point 2. In the follow... | https://arxiv.org/abs/2502.06514v2 |
N2N/summationdisplay i,j=1E[(Zi,N m−¯Zi,N m)(Zj,N m−¯Zj,N m)] ≤1 N2N/summationdisplay i,j=1/⌊ard⌊lZi,N m−¯Zi,N m/⌊ard⌊lL2(Ω)/⌊ard⌊lZj,N m−¯Zj,N m/⌊ard⌊lL2(Ω)=/⌊ard⌊lZ1,N m−¯Z1,N m/⌊ard⌊l2 L2(Ω), because, thanks to the symmetry of the definition, all Zi,N m−¯Zi,N mare identically distributed. Thus, it suffices to show that... | https://arxiv.org/abs/2502.06514v2 |
fractional Wick–Itˆ o–Skorohod (fWIS)integral, whichcoincides with ourdivergence-type integral for H >1 2 (see also Section 3.12 of [ 14] for further details). Let us begin by proving ( 40). Using the empirical projection as explained in Remark 2.1, we can writeZN masσ/integraltextT 0BN m(Xs)dBH s, where BN m(x) = (b1,... | https://arxiv.org/abs/2502.06514v2 |
. The termI3 Nvanishes asymptotically. This follows directly from Corol lary2.6, which shows that |Dj ub(Xi,N t,µN t)| ≤c Nfori/ne}ationslash=j. Hence, I3 N≤σ2 N/summationdisplay i/ne}ationslash=j/integraldisplayT 0/integraldisplayT 0/integraldisplayT 0/integraldisplayT 0/parenleftBigc N/parenrightBig2 φ(v,s)φ(u,t)dvdu... | https://arxiv.org/abs/2502.06514v2 |
of the Malliavin derivatives in ( 16), together with ( 46), imply |Di sXi,N t−Zi s,t| ≤/integraldisplayt s|∂xb(Xi,N r,µN r)||Di sXi,N r−Zi s,r|dr +/integraldisplayt s1 NN/summationdisplay k=1|∂µb(Xi,N r,µN r)(Xk,N r)||Di sXk,N r|dr ≤c/integraldisplayt s|Di sXi,N r−Zi s,r|dr+c N, where we employed the boundedness of the... | https://arxiv.org/abs/2502.06514v2 |
0Xi,xi 0s≥1 2}∩{1 ǫ(Xi,xi 0+ǫ t−Xi,xi 0 t)<1}. Using Markov’s inequality and Point 2 of Lemma 6.9, we have: P/parenleftbig {∂xi 0Xi,xi 0s≥1 2}∩{1 ǫ(Xi,xi 0+ǫ t−Xi,xi 0 t)<1}/parenrightbig ≤P(ǫ|∂2 xi 0Xxi 0+τǫ t|>1 2)≤2ǫE[|∂2 xi 0Xxi 0+τǫ t|]≤cǫ. Together with ( 50), this ensures: E[E1]≤c N+cǫ. Combining these results, ... | https://arxiv.org/abs/2502.06514v2 |
∂xb(¯Xt,¯µt)∂xb(¯Xr,¯µr)exp/parenleftBig/integraldisplayt sθ0∂xb(¯Xr,¯µr)dr/parenrightBig/bracketrightbig σφ(t,s)drdsdt, asN→ ∞. Therefore, VNP− →Ψ−1˜V=:V, (56) asN→ ∞. In particular, this implies that VNis of order one. We now replace ( 54) into (51), obtaining (ˆθ(fp) N−θ0)(1−VN) =˜RN−RN+Ψ−1 NZN. 41 We claim that ˜RN... | https://arxiv.org/abs/2502.06514v2 |
t|q]≤cE[|Xi,N 0|q]+c/⌊ard⌊lθ0/⌊ard⌊ltq−1/integraldisplayt 0E[/⌊ard⌊lb(Xi,N s,µN s)/⌊ard⌊lq]ds+cE[|Bi,H t|q]. By Assumption 3, we know that E[|Xi,N 0|q]<∞. Additionally, Assumptions 1and2ensure that: /⌊ard⌊lb(Xi,N s,µN s)/⌊ard⌊l ≤c/parenleftbig 1+|Xi,N s|+W2(µN s,δ0)/parenrightbig . (59) 43 Applying Jensen’s inequality ... | https://arxiv.org/abs/2502.06514v2 |
1. Point 2. Observe that |Di s¯Xi t| ≤σ+c/integraldisplayt s/⌊ard⌊l∂xb(¯Xi r,¯µr)/⌊ard⌊l|Di s¯Xi r|dr≤σ+cK/integraldisplayt s|Di s¯Xi r|dr, whereKis the constant from Assumption 6(b). The Gr¨ onwall lemma now yields the desired result. As for the interacting case, using the dynamics of Dj sXi,N t, we have |Dj sXi,N t| ... | https://arxiv.org/abs/2502.06514v2 |
NN/summationdisplay i=1|Dj uXi,N t−Dj vXi,N t| ≤c N|u−v|, (69) which can now be substituted into ( 68) to obtain the desired result. Finally, observe that, from Point 2, we have directly: |Dj uXi,N t−Dj uXi,N s| ≤c/integraldisplayt s/bracketleftBigg |Dj uXi,N r|+1 NN/summationdisplay k=1|Dj uXk,N r|/bracketrightBigg dr... | https://arxiv.org/abs/2502.06514v2 |
0/bracketleftbigg (1{i=j}+1 N)2+2c NN/summationdisplay k=1(1{k=j}+1 N)(1{i=j}+1 N) +c N2N/summationdisplay k,˜k=1(1{k=j}+1 N)(1{˜k=j}+1 N)+c|∂2 xj 0Xi,xj 0+τ r|+c NN/summationdisplay k=1|∂2 xj 0Xk,xj 0+τ r|/bracketrightbigg dr ≤c(1{i=j}+1 N)2+c N(1i=j+1 N)+c N2 +c/integraldisplayt 0|∂2 xj 0Xi,xj 0+τ r|dr+/integraldispl... | https://arxiv.org/abs/2502.06514v2 |
2(9), 1154-1158. [16] Boltzmann, L. (1970). Weitere studien ¨ uber das w¨ arme gleichgewicht unter gasmolek¨ ulen. Kinetische Theorie II, 115-225. [17] Bossy, M., Jabir, J.-F., & Mart´ ınez Rodr´ ıguez, K., In stantaneous turbulent kinetic energy modelling based on lagrangian stochastic approach in cfd an d application... | https://arxiv.org/abs/2502.06514v2 |
models of collective d ynamics and self-organization. In Proceedings of the International Congress of Mathematicia ns: Rio de Janeiro 2018 (pp. 3925-3946). 51 [37] Del Moral, P. (2013). Mean field simulation for Monte Car lo integration. Monographs on Statistics and Applied Probability, 126, 26. [38] Della Maestra, L., ... | https://arxiv.org/abs/2502.06514v2 |
absolute stock returns. Journa l of empirical finance, 11(3), 399-421. [55] Grassi, S., & Pareschi, L. (2021). From particle swarm o ptimization to consensus based optimization: stochastic modeling and mean-field limit. Ma thematical Models and Methods in Applied Sciences, 31(08), 1625-1657. [56] Haress, E. M., & Hu, Y. ... | https://arxiv.org/abs/2502.06514v2 |
preprint arXiv:1509.00003 . [75] Liu, M., & Qiao, H. (2022). Parameter estimation of path -dependent McKean-Vlasov stochastic differential equations. Acta Mathematica Scient ia 42, 876-886. [76] Marie, N. (2024). On a computable Skorokhod’s integral -based estimator of the drift pa- rameter in fractional SDE. Scandinavi... | https://arxiv.org/abs/2502.06514v2 |
Mean field ana lysis of neural networks: A law of large numbers. SIAM Journal on Applied Mathematics, 80(2), 725-752. [99] Sznitman, A.-S. (1991). Topics in propagation of chaos . In Ecole d’et´ e de probabilit´ es de Saint-Flour XIX-1989 (pp. 165-251). Springer, Berlin, Hei delberg. [100] Tanaka, H. (1984). Limit theor... | https://arxiv.org/abs/2502.06514v2 |
arXiv:2502.06564v2 [cs.DS] 12 Apr 2025Nearly Optimal Robust Covariance and Scatter Matrix Estimation Beyond Gaussians Gleb Novikov/asterisk.math April 15, 2025 Abstract We study the problem of computationally efficient robust estimation of the covariance/scatter matrix of elliptical distributions—that is, affine transfor m... | https://arxiv.org/abs/2502.06564v2 |
of samples (and quasi-polynomial running time). We show that the class of elliptical distributions is, in some sense, the right generalization of Gaussian distributions for robust covariance estimation. We hope that our work will motivate further research on elliptical distributions within algor ithmic high-dimensional... | https://arxiv.org/abs/2502.06564v2 |
need a notion of the effective rank . Definition 1.3. LetΣbe a positive definite matrix. Its effective rank is erk(Σ):=TrΣ /⌊ar⌈⌊lΣ/⌊ar⌈⌊l. Note that erk(Σ)is scale-invariant, and hence is the same for all scatter mat rices. For Σ∈ ℝ/u1D451×/u1D451, 1/lessorequalslanterk(Σ)/lessorequalslant/u1D451. In addition, erk(Σ)/grea... | https://arxiv.org/abs/2502.06564v2 |
/u1D442/parenleftBig /u1D>00log1−Ω(1)(1//u1D>00)/parenrightBig would imply the same error for robust covariance estimation in rel ative spectral norm, which means that our scatter matrix estimation error bound is likely not impr ovable for estimators computable in time poly(/u1D451). To the best of our knowledge, prior... | https://arxiv.org/abs/2502.06564v2 |
ariance. Hence we need some moment assumptions if we want to estimate the scaling factor. A reasonable strategy to estimate the scaling factor is to ro bustly estimate /four.supTr(Σ)= /u1D53C/⌊ar⌈⌊l/u1D465/⌊ar⌈⌊l2 (in this subsection we denote the covariance of /u1D465byΣand the scatter matrix that we accurately estima... | https://arxiv.org/abs/2502.06564v2 |
/u1D>09have to /six.supWe call a distribution /u∞D49ℱ/u1D442(1)-sub-Gaussian if for all /u1D462∈ℝ/u1D451and all even /u1D45D∈ℕ,(/u1D53C/u1D465∈/u∞D49ℱ/a\}⌊ra⌋k⌉tl⌉{t/u1D462,/u1D465−/u1D>0>/a\}⌊ra⌋k⌉tri}ht/u1D45D)1//u1D45D/lessorequalslant /u1D442(√/u1D45D)/parenleftBig /u1D53C/u1D465∈/u∞D49ℱ/a\}⌊ra⌋k⌉tl⌉{t/u1D462,/u1D4... | https://arxiv.org/abs/2502.06564v2 |
e section 4.3.2 of [ DK23 ]. /eight.supWe can always reduce the problem to this case by considering (/u1D465/u1D456−/u1D465⌊/u1D45B/2⌋+/u1D456)/√ 2. The resulting samples also come from (an /u1D442(/u1D>00)-corruption of) an elliptical distribution with the same sc atter matrix. 8 In this paper we use the sphere of rad... | https://arxiv.org/abs/2502.06564v2 |
0,Id). The term that corresponds to /u1D458=0 is 1. The term that corresponds to/u1D458=1 is /u1D53C/u1D4<4.alt2 /u1D457(1−1 /u1D451/u1D451/summationdisplay.1 /u1D456=1/u1D>06/u1D456/u1D4<4.alt2 /u1D456)=1−1 /u1D451/summationdisplay.1 /u1D456≠/u1D457/u1D>06/u1D456−3/u1D>06/u1D457 /u1D451=1−1 /u1D451/u1D451/summationdis... | https://arxiv.org/abs/2502.06564v2 |
can focus on dealing with the fourth moment. Recall that the covariance filtering algorithm with nearly op timal error uses the (/u1D4512×/u1D4512)-dimensional covariance of the distribution in the isotrop ic position: /u1D447= /u1D53C/parenleftBig Cov(/u1D466)−1/2/u1D466/u1D466⊤Cov(/u1D466)−1/2−Id/u1D451/parenrightBig⊗... | https://arxiv.org/abs/2502.06564v2 |
a similar way to the Gaussian c ase: It assigns weights to the samples, in the case of the covariance estimation transform s them /one.sup/three.sup, and checks whether the value /u1D>06:=max/u1D443∈/u∞D4Aℬ/barex/barex/a\}⌊ra⌋k⌉tl⌉{t/u1D443,/u1D444−ˆ/u1D444(/u1D461)/a\}⌊ra⌋k⌉tri}ht/barex/barexis too large (where ˆ/u1D4... | https://arxiv.org/abs/2502.06564v2 |
KKM18 ,BP21 ,LN24 ], robust sparse and tensor PCA [ dNNS23 ], robust clustering [HL18b ,BDH+20b,BDJ+22,DBT+24]. 3.2 Future Directions As was mentioned before, one promising direction is optimiz ing the sample complexity of robust PCA for elliptical distributions. Another promising direction is to determine whether it i... | https://arxiv.org/abs/2502.06564v2 |
STOC 2021, Association for Compu ting Machinery, 2021, p. 102–115. 13 [CDG19] Yu Cheng, Ilias Diakonikolas, and Rong Ge, High-dimensional robust mean estimation in nearly-linear time , Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, U SA, January 6-9, 201... | https://arxiv.org/abs/2502.06564v2 |
of the American Statistical Association 113(2018), no. 521, 252–268. 2,5,9 [HL18b] Samuel B. Hopkins and Jerry Li, Mixture models, robustness, and sum of squares proofs , Proceedings of the 50th Annual ACM SIGACT Symposium on Theor y of Comput- ing (New York, NY, USA), STOC 2018, Association for Computin g Machinery, 2... | https://arxiv.org/abs/2502.06564v2 |
0 and scatter matrix Σ. Then we project /u1D4671,...,/u1D467 /u1D45Bonto the sphere of radius√ /u1D451. We can assume that the projected samples are spsign (/u1D466′ 1),...spsign(/u1D466′ /u1D45B), where/u1D466′ 1,.../u1D466′ /u1D45Bis an/u1D>00-corruption of /u1D4661,...,/u1D466 /u1D45Biid∼/u∞D4A9( 0,Σ) (and spsign is... | https://arxiv.org/abs/2502.06564v2 |
spectral norm from Theorem 1.2 from [ KS17 ] on the projection onto the sphere of the first part. We can do that, since ℱ(/u1D466)is/u1D442(1)-sub-Gaussian. Indeed, for all /u1D462∈ℝ/u1D451, (/a\}⌊ra⌋k⌉tl⌉{t/u1D462,ℱ(/u1D466)/a\}⌊ra⌋k⌉tri}ht/u1D45D)1//u1D45D/lessorequalslant1.1(/a\}⌊ra⌋k⌉tl⌉{t/u1D462,/u1D466/a\}⌊ra⌋k⌉tr... | https://arxiv.org/abs/2502.06564v2 |
/u1D456/u1D4<4.alt2 /u1D457/parenleftBigg 1 /u1D451/u1D451/summationdisplay.1 /u1D45A=1/u1D>06/u1D45A(1−/u1D4<4.alt2 /u1D45A)/parenrightBigg3 +/u1D442(/u1D451−10). Note that, /u1D53C/u1D4<4.alt2 /u1D456/u1D4<4.alt2 /u1D457/parenleftBigg 1 /u1D451/u1D451/summationdisplay.1 /u1D45A=1/u1D>06/u1D45A(1−/u1D4<4.alt2 /u1D45A)... | https://arxiv.org/abs/2502.06564v2 |
LetΣ(/u1D461+1)←1/summationtext.1/u1D45B /u1D456=1/u1D464(/u1D461+1) /u1D456/summationtext.1/u1D45B /u1D456=1/u1D464(/u1D461+1) /u1D456(/u1D465(/u1D461+1) /u1D456−/u1D>0>(/u1D461+1))(/u1D465(/u1D461+1) /u1D456−/u1D>0>(/u1D461+1))⊤. Update/u1D461←/u1D461+1. end while return/u1D>0>(/u1D461) Theorem C.4. Let/u1D436>0be a ... | https://arxiv.org/abs/2502.06564v2 |
/lessorequalslant0.1|/a\}⌊ra⌋k⌉tl⌉{t/u1D443∗,/u1D444−Σ/u1D464/a\}⌊ra⌋k⌉tri}ht|. Therefore, /summationdisplay.1 /u1D456∈/u1D435∩[/u1D441]/u1D464/u1D456/u1D>0F/u1D456/greaterorequalslant|/a\}⌊ra⌋k⌉tl⌉{t/u1D443∗,/u1D444−Σ/u1D464/a\}⌊ra⌋k⌉tri}ht|/2. /square Lemma C.5 follows from Lemma C.6 and Lemma C.7 due to the bound on... | https://arxiv.org/abs/2502.06564v2 |
set with respect to Id/u1D451 and2Id/u1D4512, where/u1D>00/lessorequalslant/u1D6FF/lessorsimilar1and/u1D45F/lessorequalslant/u1D442(1). Let{/u1D4671,...,/u1D467 /u1D45B}be an/u1D>00-corruption of{/parenleftBig (Σ′)1/2/u1D>011/parenrightBig/parenleftBig (Σ′)1/2/u1D>011/parenrightBig⊤ ,...,/parenleftBig (Σ′)1/2/u1D>01/u1... | https://arxiv.org/abs/2502.06564v2 |
in/u1D437/u1D452/u1D453/u1D456/u1D45B/u1D456/u1D461/u1D456/u1D45C/u1D45B /u1D435. 1. Then, with high probability, ℱ(/u1D4661)ℱ(/u1D4661)⊤,...,ℱ(/u1D466/u1D45B)ℱ(/u1D466/u1D45B)⊤is an(/u1D>00,/u1D6FF,/u1D45F,ℬ1,ℬ1⊗ℬ 1)-stable set with respect to /u1D440= /u1D53Cℱ(/u1D466)ℱ(/u1D466)⊤and/u1D446= /u1D53C/u1D4<4.alt∼/u1D441... | https://arxiv.org/abs/2502.06564v2 |
/u1D451/parenrightBig/parenrightBig /u1D6FE˜Σ−Id/bardblex/bardblex/bardblex F /lessorequalslant/⌊ar⌈⌊l/u1D6FE˜Σ−Id/⌊ar⌈⌊lF+/u1D442/parenleftBig /u1D>00log(1//u1D>00)/√ /u1D451/parenrightBig/bardblex/bardblex/u1D6FE˜Σ/bardblex/bardblex F /lessorequalslant/u1D442/parenleftbig/u1D>00log(1//u1D>00)/parenrightbig+/u1D442/pa... | https://arxiv.org/abs/2502.06564v2 |
Covariates-Adjusted Mixed-Membership Estimation: A Novel Network Model with Optimal Guarantees∗ Jianqing Fan†Jiawei Ge†Jikai Hou† Abstract This paper addresses the problem of mixed-membership estimation in networks, where the goalistoefficientlyestimatethelatentmixed-membershipstructurefromtheobservednetwork. Recognizi... | https://arxiv.org/abs/2502.06671v1 |
B Local geometry 24 C Properties of the nonconvex iterates 25 D Properties of debiased nonconvex estimator 26 E Proofs of Section B 28 F Proofs of Section C 37 F.1 Proofs of Lemma C.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 F.2 Proofs of Lemma C.2 . . . . . . . . . . . . . . . . . . .... | https://arxiv.org/abs/2502.06671v1 |
Theorem H.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 H.3 Proofs of Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 I Proofs of Proposition 3.4 and Theorem 3.7 101 I.1 Proofs of Proposition 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... | https://arxiv.org/abs/2502.06671v1 |
estimator (MLE), which takes the following form: min H,Γ/summationdisplay i̸=j/parenleftbig log(1 +ePij)−AijPij/parenrightbig +λ∥Γ∥∗ s.t.Pij=z⊤ iHzj+ Γij, PZΓ = 0,ΓPZ= 0, whereZ:= [z1,...,zn]⊤∈Rn×pandPZ:=Z(Z⊤Z)−1Z⊤represents the projection onto the columnspaceof Z. Theobjectivefunctionincludesastandardlogisticlossandar... | https://arxiv.org/abs/2502.06671v1 |
nodes’ connections. Recently, researchers have started to modify the classical models to incorporate covariates information. Based on the relationship between covariates, community membership, and network structure, these modified models are generally divided into two categories: covariates- adjusted models and covaria... | https://arxiv.org/abs/2502.06671v1 |
dimensions, we define vec A1 ... Ak = vec(A1) ... vec(Ak) . Finally,f(n)≲g(n)orf(n) =O(g(n))means|f(n)| |g(n)|≤Cfor some constant C > 0whennis sufficiently large; f(n)≳g(n)means|f(n)| |g(n)|≥Cfor some constant C > 0whennis sufficiently large; andf(n)≍g(n)if and only if f(n)≲g(n)andf(n)≳g(n). 5 2 Problem Set... | https://arxiv.org/abs/2502.06671v1 |
the structure of the covariates, the incoherence properties of the latent membership matrix Γ∗, and the characteristics of the Hessian matrix in the corresponding nonconvex optimization problem. These conditions form the basis for establishing the statistical guarantees presented in the following sections. Assumption 2... | https://arxiv.org/abs/2502.06671v1 |
nearly necessary and sufficient for the convex and nonconvex solutions to be equivalent. While this assumption may not seem intuitive at first, it is typically easy to satisfy in practical applications, with the upper bound 1−ϵoften being quite small. Specifically, Assumption 7 holds in common settings such as stochast... | https://arxiv.org/abs/2502.06671v1 |
to Jin et al. (2017), Lemma 3.1, the successive projection method is an efficient VH algorithm. Align with Jin et al. (2017), we make the following assumptions. Assumption 8. We assume the following conditions hold. 1. Letθ∗ max:= max 1≤i≤nθ∗ i,θ∗ min:= min 1≤i≤nθ∗ iand¯θ∗ 2:=/parenleftbig1 n/summationtextn i=1(θ∗ i)2/... | https://arxiv.org/abs/2502.06671v1 |
Fort= 0,...,t 0−1, we compute: Ht+1 Xt+1 Yt+1 = Ht−η∇Hf(Ht,Xt,Yt) P⊥ Z(Xt−η∇Xf(Ht,Xt,Yt)) P⊥ Z(Yt−η∇Yf(Ht,Xt,Yt)) . We can show, with high probability, that there exists a sequence of rotation matrices {Rt}t0 t=0 such that: ∥Ht−H∗∥F,∥XtRt−X∗∥2,∞,∥YtRt−Y∗∥2,∞≲1√n for all 0≤t≤t0. See Lemma C.4 for more details. ... | https://arxiv.org/abs/2502.06671v1 |
ˆXˆT⊤−Γ∗. To achieve this, we define the corresponding debiased ‘estimator’ as: vec ˆHd−ˆH ˆXd−ˆX ˆYd−ˆY :=−(PˆDP)†P∇L(ˆH,ˆX,ˆY), (6) where ˆD:=/summationdisplay i̸=jeˆPij (1 +eˆPij)2 vec ziz⊤ j 1 neie⊤ jˆY 1 neje⊤ iˆX vec ziz⊤ j 1 neie⊤ jˆY 1 neje⊤ iˆX ⊤ . This debiased estimator can be viewed a... | https://arxiv.org/abs/2502.06671v1 |
to one of the two pure community types, (1,0)or(0,1). The overall matrix Γis then computed as ΘΠWΠ⊤Θ, capturing the combined effects of both individual node attributes and community structures on connectivity. The covariate matrix Zis constructed to lie in the null space of ΘΠ, ensuring that it satisfies the orthogonal... | https://arxiv.org/abs/2502.06671v1 |
analysis. After preprocessing, we retain n= 492companies in the network. For each firm, we compute the mean values of the relevant financial metrics over the given period. For example, when calculating the PE ratio, we first compute the mean values of price and earnings separately. If both mean values are positive, we ... | https://arxiv.org/abs/2502.06671v1 |
by the covariates. By the definition of R2, we have R2=/vextenddouble/vextenddoubleZHZ⊤/vextenddouble/vextenddouble2 F ∥P−mean (vec(Γ))∥2 F=/vextenddouble/vextenddoubleZHZ⊤/vextenddouble/vextenddouble2 F ∥ZHZ⊤∥2 F+∥Γ−mean (vec(Γ))∥2 F. 18 Pluggingourestimated ˆHandˆΓ, wegetR2= 0.586, whichmeansthecovariatesexplainasign... | https://arxiv.org/abs/2502.06671v1 |
zero. We then compute the objective function for both the full model (including all covariates) and the restricted model (with the null constraint applied). The objective function reflects the likelihood of the observed network under the model, regularized by the nuclear norm of Γ. The test statistic is calculated asλs... | https://arxiv.org/abs/2502.06671v1 |
linearly for phase retrieval and matrix completion. In International Conference on Machine Learning , pages 3345–3354. PMLR. Ma, Z., Ma, Z., and Yuan, H. (2020). Universal latent space model fitting for large networks with edge covariates. Journal of Machine Learning Research , 21(4):1–67. 21 Mazumder, R., Hastie, T., ... | https://arxiv.org/abs/2502.06671v1 |
≤λmax /summationdisplay 1≤i,j≤nvec(ziz⊤ j)vec(ziz⊤ j)⊤ ≤c. Proof of Proposition A.1. Note that vec(ziz⊤ j)vec(ziz⊤ j)⊤= (zj⊗zi)(zj⊗zi)⊤= (zj⊗zi)(z⊤ j⊗z⊤ i) = (zjz⊤ j)⊗(ziz⊤ i). 23 Thus, we have /summationdisplay 1≤i,j≤nvec(ziz⊤ j)vec(ziz⊤ j)⊤=/summationdisplay 1≤i,j≤n(zjz⊤ j)⊗(ziz⊤ i) = (Z⊤Z)⊗(Z⊤Z). Consequently, ... | https://arxiv.org/abs/2502.06671v1 |
that PZ(ˆXd) =PZ(ˆYd) = 0. Similarly, let D∗:=/summationdisplay i̸=jeP∗ ij (1 +eP∗ ij)2 vec ziz⊤ j 1 neie⊤ jY∗ 1 neje⊤ iX∗ vec ziz⊤ j 1 neie⊤ jY∗ 1 neje⊤ iX∗ ⊤ . We define vec ¯H−H∗ ¯X−X∗ ¯Y−Y∗ :=−(PD∗P)†P∇L(H∗,X∗,Y∗), (21) which then satisfies P ∇L(H∗,X∗,Y∗) +D∗vec ¯H−H∗ ¯X−X∗ ¯Y−Y∗ = 0... | https://arxiv.org/abs/2502.06671v1 |
ii≤19 100∥A∥2 F,∀A∈T, as long asn≫κ2µr. Lemma E.2. It holds that /parenleftigσmin 4−10(c3+c4)√nσmax/parenrightig/parenleftbig ∥∆X∥2 F+∥∆Y∥2 F/parenrightbig ≤∥∆XYT+X∆T Y∥2 F+1 4∥∆T XX+XT∆X−∆T YY−YT∆Y∥2 F ≤16σmax/parenleftbig ∥∆X∥2 F+∥∆Y∥2 F/parenrightbig . Proof.To show the upper bound, note that ∥∆XYT+X∆T Y∥2 F≤(∥Y∥∥... | https://arxiv.org/abs/2502.06671v1 |
the observations) ≤cσmin n5/2∥∆X∆T Y∥F (as long as c2√ c+3c3√σmax n≤cσmin n2for somec) ≤cσmin n5/2/parenleftbig ∥∆X∥2 F+∥∆Y∥2 F/parenrightbig (ab≤a2+b2) For the second term, note that (eP∗ ij 1+eP∗ ij−Aij)(∆X∆T Y)ijis mean-zero|(∆X∆T Y)ij|2-subgaussian variable. By the independency, it holds that /summationdisplay i̸=j... | https://arxiv.org/abs/2502.06671v1 |
+e2cP)2∥∆H∥2 F+e2cPσmin 8n2(1 +e2cP)2/parenleftbig ∥∆X∥2 F+∥∆Y∥2 F/parenrightbig −e2cPσmin 100n2(1 +e2cP)2/parenleftbig ∥∆X∥2 F+∥∆Y∥2 F/parenrightbig ≥ce2cP 2(1 +e2cP)2∥∆H∥2 F+e2cPσmin 10n2(1 +e2cP)2/parenleftbig ∥∆X∥2 F+∥∆Y∥2 F/parenrightbig . Plugging this in (25) we get vec(∆)T∇2faug(H,X,Y )vec(∆)≥ce2cP 2(1 +e2cP)2∥... | https://arxiv.org/abs/2502.06671v1 |
C.1-Lemma C.5 hold for the t-th iteration. In the following, we prove Lemma C.1 for the (t+ 1)-th iteration. By the gradient decent update, we have vec/bracketleftbiggHt+1 Ft+1/bracketrightbigg =Pvec/bracketleftbiggHt−η∇Hf(Ht,Ft) Ft−η∇Ff(Ht,Ft)/bracketrightbigg , which then gives /vextenddouble/vextenddouble/vextenddou... | https://arxiv.org/abs/2502.06671v1 |
Ft+1Rt−Ft+1,(m)Ot,(m)/bracketrightbigg =Pvec/bracketleftbigg(Ht−η∇Hf(Ht,Ft))−/parenleftbig Ht,(m)−η∇Hf(m)(Ht,(m),Ft,(m))/parenrightbig (FtRt−η∇Ff(Ht,FtRt))−(Ft,(m)Ot,(m)−η∇Ff(m)(Ht,(m),Ft,(m)Ot,(m)))/bracketrightbigg , which further implies that /vextenddouble/vextenddouble/vextenddouble/vextenddouble/bracketleftbiggHt... | https://arxiv.org/abs/2502.06671v1 |
Lemma C.1-Lemma C.5 hold for the t-th iteration. In the following, we prove Lemma C.3 for the (t+ 1)-th iteration. Note that ∇XL(m)(H,X,Y ) =1 n/summationdisplay i̸=j i,j̸=m/parenleftbiggePij 1 +ePij−Aij/parenrightbigg eieT jY +1 n/summationdisplay i̸=m/parenleftbiggePim 1 +ePim−eP∗ im 1 +eP∗ im/parenrightbigg eieT mY+... | https://arxiv.org/abs/2502.06671v1 |
(e). We denote (1) := Xt,(m) m,·Rt,(m)−X∗ m,·−η 1 n/summationdisplay i̸=m/parenleftigg ePt,(m) mi 1 +ePt,(m) mi−eP∗ mi 1 +eP∗ mi/parenrightigg eT iYt,(m)+λXt,(m) m,· Rt,(m) = Ir−η n2/summationdisplay i̸=meci (1 +eci)2(Y∗ i,·)(Y∗ i,·)⊤ /parenleftig Xt,(m)Rt,(m)−X∗/parenrightig m,· −η n2(a)−η n(b)−η ... | https://arxiv.org/abs/2502.06671v1 |
n3/2c11c31√µrσmax+1 n√σmaxczc11+1 n2√µrc11σmax+λ/radicalbiggµrσmax n+1 n√µrσmaxc11/parenrightbigg . Similarly, we have /parenleftig Ft+1,(m)Rt+1,(m)−F∗/parenrightig m+n,· = Ir−η n2/summationdisplay i̸=meci (1 +eci)2(X∗ i,·)(X∗ i,·)⊤ /parenleftig Yt,(m)Rt,(m)−Y∗/parenrightig m,·+r2, (43) where ∥r2∥2 ≲η/parenleft... | https://arxiv.org/abs/2502.06671v1 |
iteration. In the following, we prove Lemma C.4 for the (t+ 1)-th iteration. Since Lemma C.1-Lemma C.5 hold for the t-th, we know Lemma C.1 holds for the t+ 1-th iteration, which implies ∥Ht+1−H∗∥F≤c11√n. For1≤m≤n, we have /vextenddouble/vextenddouble/vextenddouble/parenleftbig Ft+1Rt+1−F∗/parenrightbig m,·/vextenddoub... | https://arxiv.org/abs/2502.06671v1 |
By Lemma C.1, this further implies that f(H∗,F∗)−f(Ht0,Ft0) ≲(cz/radicalbig plogn+λ√µrσmax)·c11√n+Cc2 11n ≲n2 61 as long as (cz√plogn+λ√µrσmax)·c11√n+Cc2 11n≪n2. Thus, we have 2 ηt0/parenleftbig f(H∗,F∗)−f(Ht0,Ft0)/parenrightbig ≲n2 ηt0≤n−10 as long asηt0≥n12(which holds as long as t0large enough). Together with (51), ... | https://arxiv.org/abs/2502.06671v1 |
(53), (54) and (55), we have ∥PˆDP−PD∗P∥ 64 ≲(1 +ecP)2 ecP·/parenleftbiggczc11√n+√µrσmax n3/2c41/parenrightbigg ·cD∗+czc11+/radicalbig µrσmax/n2c11√n ≲1√n(1 +ecP)2 ecP/parenleftbigg czc11cD∗+√µrσmax n(c41cD∗+c11)/parenrightbigg =ˆc√n. By Weyl’s inequality, we have |λi(PˆDP)−λi(PD∗P)|≲∥PˆDP−PD∗P∥≲ˆc√n. SincecD∗≤λmin(PD∗... | https://arxiv.org/abs/2502.06671v1 |
dt(V∗−ˆV)/parenrightbigg + (PD∗P)†P/parenleftbig ∇2L(V∗)−D∗/parenrightbig (V∗−ˆV) +/parenleftbig (PD∗P)†(PD∗P)−I/parenrightbig (V∗−ˆV) + (V∗−ˆV). Consequently, we have /vextenddouble/vextenddouble/vextenddouble/vextenddouble/bracketleftbiggˆXd−¯X ˆYd−¯Y/bracketrightbigg/vextenddouble/vextenddouble/vextenddouble/vextend... | https://arxiv.org/abs/2502.06671v1 |
i∆Hzi)2+(∆Γ)2 ii n2/parenrightbigg ≥ecP (1 +ecP)2/parenleftbigg c∥∆H∥2 F+1 n2∥∆Γ∥2 F/parenrightbigg −c2 z 2n∥∆H∥2 F−1 2n/summationdisplay i=1(∆Γ)2 ii n2 71 ≥cecP 2(1 +ecP)2/parenleftbigg ∥∆H∥2 F+1 n2∥∆Γ∥2 F/parenrightbigg −1 2n/summationdisplay i=1(∆Γ)2 ii n2(62) as long as n≥(1 +ecP)2c2 z/(cecP). To control/summationt... | https://arxiv.org/abs/2502.06671v1 |
term, recall the definition of ∆′′, we have ∇Lc(c(ˆV))T/parenleftig c(¯V)−c(ˆV)/parenrightig =/summationdisplay i̸=j/parenleftigg eˆPij 1 +eˆPij−Aij/parenrightigg/parenleftbigg ⟨∆′′ H,zizT j⟩+1 n⟨¯X¯YT−ˆXˆYT,eieT j⟩/parenrightbigg . Note that ⟨¯X¯YT−ˆXˆYT,eieT j⟩=⟨∆′′ XˆYT+ˆX∆′′T Y,eieT j⟩+⟨∆′′ X∆′′T Y,eieT j⟩. Thu... | https://arxiv.org/abs/2502.06671v1 |
(Cauchy-Schwarz) ≲√ c n/parenleftig ∥ˆ∆X∥2 F+∥ˆ∆Y∥2 F/parenrightig/radicalbigg ∥∆′′ H∥2 F+1 n2∥∆′′ Γ∥2 F. (by Assumption 3) Similarly, we have /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 n/summationdisplay i̸=jeˆPij (1 +eˆPij)2·/parenleftbigg ⟨ˆ∆H,zizT j⟩+1 n⟨ˆXdˆYdT−ˆXˆYT,ei... | https://arxiv.org/abs/2502.06671v1 |
we then have ∇Lc(c(V∗))T/parenleftig c(¯V)−c(ˆVd)/parenrightig =−/parenleftig c(¯V)−c(ˆVd)/parenrightigT ∇2Lc(c(V∗))/parenleftbig c(¯V)−c(V∗)/parenrightbig + (r), (75) where (r) =2 n/summationdisplay i̸=jeP∗ ij (1 +eP∗ ij)2·/parenleftbigg ⟨∆′′′ H,zizT j⟩+1 n⟨¯X¯YT−X∗Y∗T,eieT j⟩/parenrightbigg ⟨∆′′′ X∆′′′T Y,eieT j⟩... | https://arxiv.org/abs/2502.06671v1 |
13D∗ 14 D∗ 23D∗ 24/bracketrightbigg ∆Y=X∗. (77) Based on the specific form of X∗andY∗, it can be seen that the solutions must have the following form:Why??? ∆X= ∆Y=vec a b ...... a b c d ...... c d . (78) The matrix on the RHS has its first nrows being [a,b]and lastnrows being [c,d]for somea,b,c,d. ... | https://arxiv.org/abs/2502.06671v1 |
Z∇ΓLc(H,XY⊤)⊤X+λY. 85 By the definition of ∇Xf(H,X,Y )and∇Yf(H,X,Y ), we have max{∥B1∥F,∥B2∥F}= max{∥P⊥ Z∇Xf(H,X,Y )∥F,∥P⊥ Z∇Yf(H,X,Y )∥F} ≤∥P (∇f(H,X,Y ))∥F. In addition, the definition of B1andB2allow us to obtain ∥X⊤X−Y⊤Y∥F=1 λ∥X⊤(B1−P⊥ Z∇ΓLc(H,XY⊤)Y)−(B2−P⊥ Z∇ΓLc(H,XY⊤)⊤X)⊤Y∥F =1 λ∥X⊤B1−B⊤ 2Y∥F ≤1 λ∥X∥∥B1∥F+1 λ∥B2∥... | https://arxiv.org/abs/2502.06671v1 |
M∗ ij= eP∗ ij (1+eP∗ ij)2i̸=j 0i=j=/braceleftigg ℓ′′ ij(P∗ ij)i̸=j 0i=j. We then have for i̸=j: |M∗ ij−Hij|≤/integraldisplay1 0/vextendsingle/vextendsingleℓ′′ ij/parenleftbig P∗ ij+τ(Pij−P∗ ij)/parenrightbig −ℓ′′ ij(P∗ ij)/vextendsingle/vextendsingledτ≤1 4|Pij−P∗ ij|, where the last inequality follows from the mea... | https://arxiv.org/abs/2502.06671v1 |
long as√µrσmax≫c11√n. Combine the above results with (97), we have /vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble ˆ∆H−∆H ˆ∆X−∆X ˆ∆Y−∆Y /vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble F≲ˆc c2 D∗√n·λ√µrσmax+1 cD∗·... | https://arxiv.org/abs/2502.06671v1 |
as (ˆHcon,ˆΓcon) := arg min PZΓ=0,ΓPZ=0, ∥H−H∗∥F≤cn,∥Γ−Γ∗∥∞≤cnfc(H,Γ), (105) wherefcis the convex objective defined in (13). By (17), (ˆH,ˆXˆY⊤)is feasible for the constraint of (105). By the optimality of (ˆHcon,ˆΓcon), we have Lc(ˆHcon,ˆΓcon) +λ∥ˆΓcon∥∗≤Lc(ˆH,ˆXˆY⊤) +λ∥ˆXˆY⊤∥∗. (106) We denote ∆con H:=ˆHcon−ˆH,∆con Γ... | https://arxiv.org/abs/2502.06671v1 |
Γ)∥∗(114) Combine (113) and (114) we get C′/parenleftigg c 2∥∆con H∥2 F+1 n2∥∆con Γ∥2 F−2 n2n/summationdisplay i=1(∆con Γ)2 ii/parenrightigg ≤−∇HLc(ˆH,ˆXˆY⊤)⊤∆con H+∥PT(R)∥F∥PT(∆con Γ)∥F. (115) In the sequel, we deal with/summationtextn i=1(∆con Γ)2 ii. One can see that n/summationdisplay i=1(∆con Γ)2 ii=n/summationd... | https://arxiv.org/abs/2502.06671v1 |
(120), we getW=˜W. The rest of the proof is the same as (Jin et al., 2023, Proof of Proposition A.1), and we finally reach (Θ,Π,W) = ( ˜Θ,˜Π,˜W). That is to say, the DCMM model Γ = ΘΠWΠ⊤Θis identifiable under the conditions in Proposition 3.4. I.2 Proofs of Theorem 3.7 In this section, we will frequently using (Jin et ... | https://arxiv.org/abs/2502.06671v1 |
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