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without difficulty, from lpmax norm to lp,qmax norm. This shows the minimum value of 18 is no more than that of 13. The other direction comes with using the reverse construction to turn the solution of the former into the latter. B. Proofs on results in section VI Proof of lemma VI.2.1. 1) An integral expression for th... | https://arxiv.org/abs/2503.02129v1 |
clear to the readers, we divide it into two cases. •m1≥m2. We will take a similar procedure as that for L= 2. For any δ >0, since fis in the closed convex hull of T(G2), we can also find f1, . . . , f n∈ G2such that a convex combination of them f∗=γ1f1+···+γnfnis within δ2distance tof, that is, ∥f−f∗∥ ≤δ2. We can find ... | https://arxiv.org/abs/2503.02129v1 |
g1m2∈Bm2g1 m2 |Bm2|) m2−f H≤LT(R+ϵ) +1√m(R+ϵ) (126) Note that the above coarse estimate is not useful as we have a constant nonzero LT(R+ϵ)gap. This is due to having used the trivial estimate |Bi|>1. Now we give a better estimate by directly estimating a combinatoric expression from 113. If we expand 113, we have 110≤L... | https://arxiv.org/abs/2503.02129v1 |
24 One can see the above manipulation as a pesudo form of bias-variance decomposition inequality (instead of equality as we take a pesudo mean g(·;θ∗)of our estimator), both from the idea and the expressions. By definition we have T1=λ(ν(θ∗)−ν(ˆθ)) = 2 λν(θ∗)−λν(θ∗−ˆθ) (143) Theorem VI.2 gives the bound for T2 T2=1 2∥g... | https://arxiv.org/abs/2503.02129v1 |
the estimate of T3doesn’t depend on width vector ⃗ mcompared to T2. To prove Eq. (64), one way is using method totally analogous to the one in [Wang and Lin, 2023] (Eq.(14) of theorem 2). We refer the readers to the proof therein for details. We give a second proof, based on the estimation of Gaussian complexity which ... | https://arxiv.org/abs/2503.02129v1 |
(181) =Eρsupν(θ)≤F m2X k2=1w3 k2∥w2 k2,k1∥1∥w1 k1∥2nX i=1ρiσ(m1X k1=1vk2,k1σ(uk1xi)) (182) (183) One can then deduce that vk2,k1doesn’t depend on k2by using the same argument as for L= 2 case for the deduction that uk doesn’t depend on k. So we abuse the notation a little bit to write vk2,k1asvk1and continue from the l... | https://arxiv.org/abs/2503.02129v1 |
8, AUGUST 2015 29 Now, conditioning on the event {ˆ∆/(6∥f∗∥WL)∈ F∗(⃗ m,1)}, applying Lemma VIII.1.3 with t= 8p 6elogn/n < 12e yields ∥ˆ∆∥2 2≤ ∥ˆ∆∥2 n+36√nCF∥f∗∥2 WL+ 288∥f∗∥2 WLr 6elogn n(209) with probability at least 1−n−1. By VII.1, with probability at least 1−n−4we have ∥ˆ∆∥2 n=∥ˆf−f∗∥2 n≤C3 H2(⃗ m)∥f∗∥2 WL (210) ... | https://arxiv.org/abs/2503.02129v1 |
L ATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 30 omitting its dependency on ˜xin notation. Thus we can getPm1 k=1|a′j k| ≤1for both j. Repeating the argument in L= 2 case, we have |g(x;θ1)−g(x;θ2)| (223) ≤Lσm1X k=1|a′1 k−a′2 k|+ 2Lσmax 1≤k≤m1∥w1 k−w2 k∥2 (224) For [Zhang et al., 2021] each a′j kit’s a two-layer netwo... | https://arxiv.org/abs/2503.02129v1 |
into supplementary materials in [Wang and Lin, 2023]. JOURNAL OF L ATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 31 Proof of theorem IX.2. The proof follows completely the same steps as the proof of theorem 4 in [Wang and Lin, 2023], modifying appropriately according to lemma IX.0.1 and IX.1.1. Some constants now depen... | https://arxiv.org/abs/2503.02129v1 |
to L(f∗, f∗) = 0 . By Talagrand’s concentration inequality [Wainwright, 2019], P(Zn−EZn≥t)≤2exp −nt2 8eKn+ 4Ut ≤2exp −nt2 32(2 + L2 0)e+ 8L0t (263) Note that −nt2/(32(2 + L2 0)e+ 8L0t)≤ −nt/(16L0)ift≥4e(2 +L2 0)/L0, and−nt2/(64(2 + L2 0)e)otherwise. We then conclude that P Zn≥CF√n+t ≤exp −n (16L0)min(t2 4e(2 +L2... | https://arxiv.org/abs/2503.02129v1 |
2023]. See proposition 3 there for details. If L >2, we first get the output of the first layer, denoted by X2, which is also the input of the second layer, then we do the same step for the second layer as did for the first layer. That is, in the second layer we form p2regions R2= (R2 1, . . . , R2 p2) according to the... | https://arxiv.org/abs/2503.02129v1 |
2˜xi) (281) =sj×1 iL−1× ··· × i211×|SL iL−1|σ(1|SL iL−1|×|SL−1 iL−2|σ(···1|S2 i2|×|S1 i1|σ(|aL l1|wL−1′ 1 (282) ···w3′ 1w2′ 1w1′ 1˜xi+|aL l2|wL−1′ 2···w3′ 2w2′ 2w1′ 2˜xi))) (283) where 1m×nis an all ones m×nmatrix. The collinearity of the parameters in the same region of the same layer of the optimal solution of proble... | https://arxiv.org/abs/2503.02129v1 |
l=L−1, . . . , L −t, and assume on contrary ( ˆatL,ˆwL−t−1 t )Tand ( ˆasL,ˆwL−t−1 s )Tis not collinear. For l=L−t−1, as aforementioned, we can regard it (not equivalently) as a t+ 1-layer neural network with inputs xl. Choose any sequence of cones Qil,il+1,...,iL=Ql il, Ql+1 il+1, . . . , QL iLstarting from Ql ilas spe... | https://arxiv.org/abs/2503.02129v1 |
Jn(ˆθ, λ) =1 2n||y−1X s=02pL−1X jL−1=1···2p2X j2=12p1X j1=1DL−1 jL−1···D2 j2D1 j1Xγj1,j2,...,jL−1(ˆθ)||2 2+λ||Γ(ˆθ)||2,1 (314) Proof. One can expand the error and regularization terms in Eq. (314) explicitly to get the terms in Eq. (9) . So, in particular, ||Γ(θ∗)||2,1=ν(θ∗). Unfortunately, these pidepends on mi, rende... | https://arxiv.org/abs/2503.02129v1 |
An in-depth analysis through the lens of learned feature space. arXiv preprint arXiv:2310.13572 , 2023. Trevor Hastie, Andrea Montanari, Saharon Rosset, and Ryan J. Tibshirani. Surprises in high-dimensional ridgeless least squares interpolation. Ann. Statist. , 50(2):949–986, 2022. Daniel Hsu, Sham Kakade, and Tong Zha... | https://arxiv.org/abs/2503.02129v1 |
representer theorems for neural networks and ridge splines. J. Mach. Learn. Res., 22(43):1–40, 2021. Rahul Parhi and Robert D. Nowak. Near-minimax optimal estimation with shallow ReLU neural networks. IEEE Trans. Inf. Theory , 2022. G. M. Rotskoff and E. Vanden-Eijnden. Trainability and accuracy of artificial neural ne... | https://arxiv.org/abs/2503.02129v1 |
On the Realized Joint Laplace Transform of Volatilities with Application to Test the Volatility Dependence Xinwei Feng Zhongtai Securities Institute for Financial Studies, Shandong University Yu Jiang∗ Department of Mathematics, University of Macau Zhi Liu Department of Mathematics, University of Macau Zhe Meng Zhongta... | https://arxiv.org/abs/2503.02283v1 |
ity has become a concern for statisticians and financial economists. Todorov and Tauchen (2012a) first proposed the realized Laplace transform (RLT) of volatility, which is a consis- tent estimator of the empirical Laplace transform (ELT). The empirical Laplace transform of volatility contains more information than the... | https://arxiv.org/abs/2503.02283v1 |
joint distribution of multivariate volatilities and the relationships among the volatilities of different assets to manage and control financial risk. For instance, we aim to determine whether the distribution of volatility of two assets in different sectors or the same sector across different time periods exhibits sim... | https://arxiv.org/abs/2503.02283v1 |
,0≤t≤T, where the ρt,0≤t≤Tis a deterministic function and take values in the interval [ −1,1]. Actually, the marginal processes WX t, WY tare standard Brownian motions. There exists a standard Brownian motion W⋆ tindependent of WX t, and we can rewrite dWY t=ρtdWX t+p 1−ρ2 tdW⋆ t,0≤t≤T. Besides, ˜ µXand ˜µYare homogene... | https://arxiv.org/abs/2503.02283v1 |
the error between the two terms is asymptotically negligible. Moreover, E" cos √ 2uσX 2i−2∆n 2i−1WX+√ 2vσY 2i−2∆n 2iWY √∆n! F2i−2# =E" cos √ 2uσX 2i−2∆n 2i−1WX √∆n! cos √ 2vσY 2i−2∆n 2iWY √∆n! F2i−2# =e−⟨(u,v),((σX 2i−2)2,(σY 2i−2)2)⟩. The equalities hold because sin( ·) is an odd function and ∆n 2i−1WXis independent o... | https://arxiv.org/abs/2503.02283v1 |
2(v+v′)∆n i+1Y√∆n! . Remark 1.Un(u, v)makes good use of the data compared to Vn(u, v)and naturly have smaller asymptotic variance. Indeed, some straightforward computation can show this, and here we display the plots of two asymptotic variances as a function of (u, v)in Figure 1. In 7 the figure, we let T= 1,σX s= 1,σY... | https://arxiv.org/abs/2503.02283v1 |
conditions of Todorov and Tauchen (2012a), the Assumption 3 is slightly weaker and not restrictive in practice as it can capture a wide variety of volatility models, such as a large class of processes driven by Brownian motion(e.g., Heston (1993)) or the L´ evy-driven Ornstein-Uhlenbeck model of Barndorff-Nielsen and S... | https://arxiv.org/abs/2503.02283v1 |
discrete time points {i∆n, i= 1,2, ..., n},n= 1760, which correspond to monthly observations with 5-minutes frequency. We study the following three commonly used stochastic volatility models. Example 1.We generate the high-frequency data {(Xt, Yt), t∈[0, T]}from the following processes:( dXt= 0.03dt+σX tdWX t, dYt= 0.0... | https://arxiv.org/abs/2503.02283v1 |
histograms of1√ ∆n˜Γn Un(u, v)−R1 0e−⟨(u,v),((σX s)2,(σY s)2)⟩ds . 13 Table 1: Monte Carlo Results of Vn(u, v),Un(u, v) and V′ n(u, v) Vn(u, v)u= 2.5 u= 3.5 u= 4.5 Bias SD MSE Bias SD MSE Bias SD MSE Example 1 v2.75 0.00024 0.02419 0.00059 0.00077 0.02390 0.00057 -0.00050 0.02465 0.00061 3.75 0.00046 0.02415 0.00058 ... | https://arxiv.org/abs/2503.02283v1 |
j{tY j:tY j≤tX i−1}, tY i+= min j{tY j:tY j≥tX i},∆Y′ i,n=tY′ i+−tY′ i−. Mathematically, ∆Y′ i,nis the smallest cover of ∆X i,nin process Yand ∆n iY′is the corresponding increment. The volatility and price processes settings are the same as Example 1, and the results of two other Example settings are similar. The perfo... | https://arxiv.org/abs/2503.02283v1 |
Tand ∆ n, the size and power of the test are exhibited in Table 3, •T= 22(one month), ∆ n= 1/390, •T= 44(two months), ∆ n= 1/390, •T= 44(two months), ∆ n= 1/780, •T= 66(one quarter), ∆ n= 1/780. From Table 3, we see that when the number of days Tincreases and the increment ∆ n decreases, the power of the test becomes c... | https://arxiv.org/abs/2503.02283v1 |
<0.01 <0.01 <0.01 0.01 0.05 0.08 0.14 0.02 0.03 0.04 MSFT 0.02 0.03 0.11 0.02 <0.01 0.01 0.03 0.04 <0.01 <0.01 <0.01 0.01 0.30 <0.01 0.11 0.01 <0.01 0.03 ORCL 0.01 0.03 0.03 0.02 0.03 <0.01 0.15 <0.01 <0.01 <0.01 0.01 <0.01 0.03 0.25 0.17 0.03 0.04 IBM 0.01 0.18 <0.01 0.10 0.24 0.32 <0.01 <0.01 <0.01 <0.01 0.04 <0.01 0... | https://arxiv.org/abs/2503.02283v1 |
i=1ξ(3) 2i−1(u, v)P−→0. Proof. First forP⌊n/2⌋ i=1ξ(3) 2i−1(u, v), note that ξ(3) 2i−1(u, v) can be written asP3 j=1ξ(3,j) 2i−1(u, v), with ξ(3,1) 2i−1(u, v) =∆−1 2nZtn 2i tn 2i−2 k(σX ∗, σY ∗, u, v)−k(σX 2i−2, σY 2i−2, u, v),(σX 2i−2−ˆσX s, σY 2i−2−ˆσY s) ds, ξ(3,2) 2i−1(u, v) =∆−1 2nZtn 2i tn 2i−2 k(σX 2i−2, σY 2i−2,... | https://arxiv.org/abs/2503.02283v1 |
2i−2(σX s−σX 2i−2)dWY s 21 −2√ 2vsin √ 2uσX 2i−2∆n 2i−1WX+√ 2vσY 2i−2∆n 2iWY √∆n!Ztn 2i tn 2i−1(σY s−σY 2i−2)dWY s, where ˜ x1is between√ 2u∆−1/2 nRtn 2i−1 tn 2i−2bX sds+√ 2u∆−1/2 nRtn 2i−1 tn 2i−2σX sdWX sand√ 2u∆−1/2 nbX 2i−2∆n+ Rtn 2i−1 tn 2i−2σX sdWX s; ˜y1is between√ 2v∆−1/2 nRtn 2i tn 2i−1bY sds+√ 2v∆−1/2 nRtn 2i... | https://arxiv.org/abs/2503.02283v1 |
j(u′, v′) Gj−1i P−→Fp(u, v, u′, v′), (20) ⌊n/2⌋X j=1E ζ(2) j(u, v)4 Gj−1 P−→0, (21) ⌊n/2⌋X j=1Eh ζ(2) j(u, v)δjWZ Gj−1i P−→0,forZ=X, Y, (22) ⌊n/2⌋X j=1Eh ζ(2) j(u, v)δjN Gj−1i P−→0, (23) where Nis any bounded G-martingale orthogonal to WXandWY,δjWZ=WZ 2j∆n− WZ 2(j−1)∆nandδjN=N2j∆n−N2(j−1)∆n. To show (20), we have ⌊... | https://arxiv.org/abs/2503.02283v1 |
orthogonal to WXandWYand defined on F, we have Eh ζ(2) j(u, v)δjN Gj−1i = 0, which yields the desired result. We now show the tightness. For the convenience of the following explanation, we denote: ˆSt,1(u, v) =⌊t/2∆n⌋X i=⌊(t−1)/2∆n⌋+1ξ(2) 2i−1(u, v), ˆSt,2(u, v) =⌊t/2∆n⌋X i=⌊(t−1)/2∆n⌋+1 ξ(3,2) 2i−1(u, v) +ξ(1,2) 2i−... | https://arxiv.org/abs/2503.02283v1 |
For the termPNp j=1ζn,p j(u, v), we considerPNp j=1ζn,p′ j(u, v) with ζn,p′ j(u, v) =αp j+1−1X i=αp j+1ξ(2)′ i(u, v), ξ(2)′ i(u, v) = ∆1 2n cos√ 2uσX αp j∆n iWX+√ 2vσY αp j∆n i+1WY √∆n −e−⟨(u,v),((σX αp j)2,(σY αp j)2)⟩ . Under Assumption 2, we can show that NpX j=1 ζn,p j(u, v)−ζn,p′ j(u, v) P−→0. (29) The... | https://arxiv.org/abs/2503.02283v1 |
nn−1X i=1cos √ 2u∆n iX+√ 2v∆n i+1Y√∆n cos √ 2u′∆n iX+√ 2v′∆n i+1Y√∆n −∆nn−1X i=1cos p 2(u+u′)∆n iX+p 2(v+v′)∆n i+1Y√∆n + ∆ nn−2X i=1cos √ 2u∆n i−1X√∆n cos √ 2v∆n i+1Y√∆n cos √ 2u′∆n i+1X√∆n cos √ 2v′∆n i+2Y√∆n −∆nn−1X i=1cos p 2(u+u′)∆n iX+p 2(v+v′)∆n i+1Y√∆n + ∆ nn−2X i=1cos √ 2u′∆n i−1X√∆n cos √ 2v′∆n i+1Y√∆... | https://arxiv.org/abs/2503.02283v1 |
|ˆCi 11−Ci 11| ≤C1 TTX t=1( ˆZx,y t(u, v)−Zx,y t(u, v) + 1 TTX t=1ˆZx,y t(u, v)−µx,y t(u, v) ) ≤C1 TTX t=1( ˆZx,y t(u, v)−Zx,y t(u, v) + 1 TTX t=1ˆZx,y t(u, v)−1 TTX t=1Zx,y t(u, v) + 1 TTX t=1Zx,y t(u, v)−µx,y t(u, v) ) (34) The first and second terms of (34) are Op(√∆n) according to Theorem 2, the magnitude of the la... | https://arxiv.org/abs/2503.02283v1 |
121(10):2416–2454, 2011. 35 Steven L Heston. A closed-form solution for options with stochastic volatility with appli- cations to bond and currency options. The Review of Financial Studies , 6(2):327–343, 1993. Ulrich Hounyo, Zhi Liu, and Rasmus Tangsgaard Varneskov. Bootstrapping Laplace Trans- forms of volatility: Su... | https://arxiv.org/abs/2503.02283v1 |
Noisy Low-Rank Matrix Completion via Transformed L1 Regularization and its Theoretical Properties Kun Zhao1Jiayi Wang1Yifei Lou2 1The University of Texas at Dallas2The University of North Carolina at Chapel Hill Abstract This paper focuses on recovering an under- lying matrix from its noisy partial entries, a problem c... | https://arxiv.org/abs/2503.02289v1 |
of Riemannian geometry, while Chen et al. (2022) introduced a non-convex framework for matrix completion with linearly parameterized factors, further enriching the landscape of low-rank matrix re- covery methods. Despite these numerical advances, matrix factorization remains inherently limited. First, its non-convex na... | https://arxiv.org/abs/2503.02289v1 |
max-norm was first proposed by Srebro et al. (2004) for matrix completion under non-uniform sampling mechanisms. Later, Cai and Zhou (2016) proved its theoretical superiority over the nuclear norm for noisy matrix completion under a general sampling model. Sequentially, Fang et al. (2018) proposed a more flexible estim... | https://arxiv.org/abs/2503.02289v1 |
of whether it is under uniform or non-uniform sampling with noisy data. The effectiveness of TL1 reg- ularization is demonstrated through a comprehensive simulation study under various missing data mecha- nisms, highlighting TL1 regularization’s adaptability and robustness. We also validate its performance using two re... | https://arxiv.org/abs/2503.02289v1 |
we define L2(Π) norm of Aby∥A∥2 L2(Π)=E(⟨A, T⟩2) =Pm1 k=1Pm2 l=1πklA2(k, l). Lastly, our analysis requires the following asymptotic notations. For two non-negative sequences {an}and {bn}, we say an=O(bn) if there exists a constant C such that an≤Cbnandan=Op(bn) if there exists a constant C′such that an≤C′bnwith high pr... | https://arxiv.org/abs/2503.02289v1 |
3. There exists a constant c0>0such that max i=1,...,nE[exp(|ξi|/c0)]≤e, where eis the base of the natural logarithm. These assumptions are commonly used in the literature (Koltchinskii et al., 2011; Klopp, 2014; Cai and Zhou,2016; Klopp et al., 2017). In Assumption 1, a larger value of Lindicates greater imbalance in ... | https://arxiv.org/abs/2503.02289v1 |
is not applicable for TL1 regular- ization, because, unlike the convex nuclear norm, the triangle inequality does not hold for TL1. Instead, we carefully analyze the gradient of the TL1 function to obtain the bound; please refer to Appendix C for the proof of Theorem 2. Theorem 2. Suppose Assumptions 1-3 hold, A0∈ Rm1×... | https://arxiv.org/abs/2503.02289v1 |
4 provides a theoretical control on the rank of the matrix estimated by the TL1 regularization, showing the estimated rank decreases as λincreases for sufficiently small a. Theorem 4 indicates that TL1 regularization controls the rank of estimated matrices to some extent and implies that TL1 regularization Noisy Low-Ra... | https://arxiv.org/abs/2503.02289v1 |
we generate the target ma- trixA0∈Rm1×m2as the product of two matrices of smaller dimensions, i.e., A0=UV⊺, where U∈Rm1×r, V∈Rm2×rwith each entry of UandVindependently sampled from a standard normal distribution N(0,1). As a result, the rank of A0is at most r, which is significantly smaller than min( m1, m2). We adopt ... | https://arxiv.org/abs/2503.02289v1 |
difficulty of Scheme 3 for matrix completion. In short, we conclude that the TL1-regularized approach is robust across various noise levels and sampling dis- tributions. Discussion 1. We explore how the hyper-parameter ain the TL1 regularization affects the performance of the model (5)for low-rank matrix completion. In... | https://arxiv.org/abs/2503.02289v1 |
entries from the original test set. We use the test root mean squared error (TRMSE) restricted on the evaluation set to gauge the recovery performance, as outlined in Wang et al. (2021). We re- port the TRMSE values and the ranks of the estimators in Table 5, showing that TL1 outperforms the other ap- proaches in terms... | https://arxiv.org/abs/2503.02289v1 |
(0.015) 19.19 0.655 (0.016) 16.97 0.833 (0.007) 5.80 0.572 (0.016) 6.40 (10, 0.2) 0.279 (0.016) 19.28 0.267 (0.018) 17.72 0.638 (0.011) 6.58 0.208 (0.024) 7.12 500 (5, 0.1) 0.406 (0.023) 63.43 0.350 (0.023) 61.82 0.764 (0.013) 60.88 0.171 (0.026) 60.26 (5, 0.2) 0.177 (0.008) 53.47 0.159 (0.007) 50.34 0.611 (0.014) 50.1... | https://arxiv.org/abs/2503.02289v1 |
Statistical Association , 112(519):1344– 1353, 2017.S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eck- stein. Distributed optimization and statistical learn- ing via the alternating direction method of multipli- ers.Found. Trends Mach. Learn. , 3(1):1–122, 2011. Jian-Feng Cai, Emmanuel J Cand` es, and Zuowei Shen. A s... | https://arxiv.org/abs/2503.02289v1 |
concave penalized likelihood and its oracle properties. Journal of the American statistical Association , 96 (456):1348–1360, 2001. Ethan X Fang, Han Liu, Kim-Chuan Toh, and Wen-Xin Zhou. Max-norm optimization for robust matrix re- covery. Mathematical Programming , 167:5–35, 2018. Vivek Farias, Andrew A Li, and Tianyi... | https://arxiv.org/abs/2503.02289v1 |
penetrating radar data recon- struction via matrix completion. International Jour- nal of Remote Sensing , 42(12):4607–4624, 2021. Jiangyuan Li, Jiayi Wang, Raymond KW Wong, and Kwun Chuen Gary Chan. A pairwise pseudo- likelihood approach for matrix completion with in- formative missingness. In The Thirty-eighth Annual... | https://arxiv.org/abs/2503.02289v1 |
signal processing. In International Conference on Acoustics, Speech and Signal Process- ing (ICASSP) , pages 2697–2700. IEEE, 2012. Dong Xia and Ming Yuan. Statistical inferences of linear forms for noisy matrix completion. Journal of the Royal Statistical Society Series B: Statistical Methodology , 83(1):58–77, 2021. ... | https://arxiv.org/abs/2503.02289v1 |
Ak+1= arg min ∥A∥∞≤ζ1 n∥Y−T◦A∥2 F+ρ 2∥A−Zk+1 ρWk∥2 F. (18) Without the constraint ∥A∥∞≤ζ,the optimal solution to (18) can be expressed by Ak+1 2:=2 nT◦Y+ρZk−Wk ⊘(2 nT+ρ), (19) where ⊘denotes the elementwise division. Then we project the solution to the constraint [ −ζ, ζ], thus leading to Ak+1= minn max{Ak+1 2, ζ},−ζ... | https://arxiv.org/abs/2503.02289v1 |
Assumptions 1 and 2 hold. Then for any A∈ K(ζ, γ), the inequality 1 nnX i=1⟨Ti, A⟩2≥ ∥A∥2 L2(Π)−ζ2r Llogd n−ζ∥A∥∗√m1m2r LMlogd n holds with probability at least 1−κ d, where κis a constant depending on a universal constants KandLdefined in Assumption 1. Proof of Lemma 2. LetXi=⟨Ti, A⟩2and define V:=nE(X2 i) + 16 nγ√ Mr... | https://arxiv.org/abs/2503.02289v1 |
Llogd n, (31) with probability at least 1−κ+1 d, where κis a constant depending on L. Proof of Lemma 3. It follows from the optimality of the estimator ˆAin (5) that 1 nnX i=1(Yi− ⟨Ti,ˆA⟩)2+λTL1 a(ˆA)≤1 nnX i=1(Yi− ⟨Ti, A0⟩)2+λTL1 a(A0). Replacing Yiwith the trace regression model, we obtain 1 nnX i=1(⟨Ti, A0⟩+σξi− ⟨Ti... | https://arxiv.org/abs/2503.02289v1 |
along with some definitions. For any matrix A∈Rm1×m2, letUA andVAbe the left and right singular matrices of A, and DAis the diagonal matrix with the singular values of A, i.e., the SVD of Ais expressed by A=UADAV⊺ A. We denote rA:=rank(A) and σj(A) is the jth singular values ofA,j= 1, . . . , r A. We define SU(A) and S... | https://arxiv.org/abs/2503.02289v1 |
Then, we have 1 νm1m2∥ˆA−A0∥2 F≤ ∥ˆA−A0∥2 L2(Π) ≲1 nnX i=1⟨Ti,ˆA−A0⟩2+ζ2r Llogd n+ζ∥ˆA−A0∥∗√m1m2r LMlogd n ≲2∥Σ∥∥ˆA−A0∥∗+λTL1 a(A0)−λTL1 a(ˆA) +ζ2r Llogd n+ζ∥ˆA−A0∥∗√m1m2r LMlogd n ≲( 2∥Σ∥+ζ√m1m2r LMlogd n) ∥ˆA−A0∥∗+λTL1 a(A0)−λTL1 a(ˆA) +ζ2r Llogd n ≲(ζ∨σ)√m1m2r Ldlogd n ∥P⊥ A0(ˆA−A0)∥∗+∥PA0(ˆA−A0)∥∗ +λTL1 a(A0) +ζ2... | https://arxiv.org/abs/2503.02289v1 |
a rank-1 matrix, then ∥usv⊺ s∥∗=∥us∥∥v⊺ s∥= 1. We further use (35) to get I1=O(r Ldlogd nm1m2), (50) holds with probability at least 1 −1 d. Noisy Low-Rank Matrix Completion via Transformed L1Regularization and its Theoretical Properties ForI2.Following Lemma 1 and Lemma 2, we take γ= 1/√m1m2in the constraint set (23) ... | https://arxiv.org/abs/2503.02289v1 |
L1Regularization and its Theoretical Properties Hence, there exists a constant C7only depending on c0such that rank( ˆA)≤C7 λ−1Ldlogd n√m1m2 (1 +a)(a+√m1m2) √a a+√m1m21/4+ 1!2 +rank( A0) √a a+√m1m21/4+ 1!) , with high probability. Proof of Corollary 1. Replacing λwith the order of(ζ∨σ)√m1m2a+ζ√m1m2 1+aq Ldlog... | https://arxiv.org/abs/2503.02289v1 |
0.2) 0.174 (0.016) 0.139 (0.022) 0.606 (0.020) 0.063 (0.043) (10, 0.1) 0.480 (0.026) 0.477 (0.029) 0.798 (0.010) 0.505 (0.031) (10, 0.2) 0.217 (0.014) 0.204 (0.026) 0.610 (0.016) 0.138 (0.029) 500 (5, 0.1) 0.265 (0.027) 0.209 (0.022) 0.753 (0.014) 0.048 (0.037) (5, 0.2) 0.144 (0.007) 0.126 (0.006) 0.606 (0.015) 0.006 (... | https://arxiv.org/abs/2503.02289v1 |
Nuclear TL1 1 0.0625 (2488.8) 0.0083 (2458.0) 0.0774 (800.64) 0.0027 (866.07) 2 0.1048 (2364.8) 0.0431 (2350.4) 0.5972 (753.65) 0.0679 (798.64) 3 0.0901 (2382.6) 0.0385 (2372.3) 0.5747 (750.75) 0.0582 (801.10) Table 9: The bias and variance results of the estimators derived from the TL1 and nuclear norm under Scheme 2 ... | https://arxiv.org/abs/2503.02289v1 |
arXiv:2503.02536v1 [math.AC] 4 Mar 2025The Likelihood Correspondence Thomas Kahle, Hal Schenck, Bernd Sturmfels, and Maximilian Wiesmann Abstract An arrangement of hypersurfaces in projective space is SNC i f and only if its Euler discriminant is nonzero. We study the critical loci of all La urent monomials in the equa... | https://arxiv.org/abs/2503.02536v1 |
equal to the symmetric algebra. Equivalently, IAequals thepre-likelihood ideal [14]. We conclude that SNC arrangements are gentle(Corollary 5.4). Catanese et al. [ 2, Theorem 1] gave the following formula for the ML degree , i.e. the num- ber of critical points of ℓA. This number is the coefficient of zn−1in the generati... | https://arxiv.org/abs/2503.02536v1 |
is on determinantal ideals, circuit syzygies, and the Buch sbaum–Rim resolution. In Section 4we determine the multidegree of the generic likelihood corresponden ce, and weshowthatitmatchesthatoftheproposeddeterminantal ideal usingtheGiambelli–Thom– Porteous Formula. Theorem 4.5unifies algebraic and topological approache... | https://arxiv.org/abs/2503.02536v1 |
polynomials f1,f2,...,f mhave degree 1. We setℓ= min(n,L). We have ℓ= 0 in Theorem 2.1, and we have ℓ= 1 in Example 1.1. Theorem 2.3. IfAis SNC then the likelihood ideal IAis minimally generated by ( 2) and/parenleftbigm+n−1−ℓ n−2/parenrightbig of the maximal minors of Qs \1. We have/parenleftbign−ℓ i−ℓ/parenrightbig ·... | https://arxiv.org/abs/2503.02536v1 |
Finally, if |I|=nthen ∆ Iis theresultant fornhypersurfaces inPn−1. See [18] for a recent study in this subject area and many relevant refere nces. 5 Example 2.5 (n= 3).Consider mcurves of degree at least two in P2. Then the Euler discriminant is a product of m+/parenleftbigm 2/parenrightbig +/parenleftbigm 3/parenright... | https://arxiv.org/abs/2503.02536v1 |
Buchsbaum–Rim complex is most typically used when a matrix, or th e module it presents, is close to generic. The matrices Q,Q\1, andQs \1are far from being generic. However, they are “generic enough” for the Buchsbaum–Rim cons truction to be applicable. 6 We recall some material from Eisenbud’s textbook [ 6,§20.7 and §... | https://arxiv.org/abs/2503.02536v1 |
a rank nvector bundle on Pn−1. Our assumption that intersections of the hypersurfaces V(fi) are well-behaved will be crucial. Lemma 3.5. LetAbe SNC, I⊂[m]andXI:=V(fi|i∈I)\/uniontext j/\e}atio\slash∈IV(fj). At any point p∈XI, the rank of the Jacobian JIof{fi:i∈I}satisfies rk(JI(p)) = min(|I|,n). Proof.This follows from t... | https://arxiv.org/abs/2503.02536v1 |
modifi- cations made to Qs \1(p) in order to simplify our local calculation. We conclude that the ideal Im+1(Qs \1) has codimension n−1 inC[s2,...,s m,x1,...,x n] and that V(Im+1(Qs \1)) defines a vector bundle on Pn−1, so is irreducible. Since s1does not appear in Qs \1, adding/summationtextdisito Im+1(Qs \1) yields an ... | https://arxiv.org/abs/2503.02536v1 |
by Lemma 3.11. Proposition 3.12falls just short of being a full proof of Theorem 2.1: whileKisIA- primary, it need not be reduced. The next section introduces the t ools needed to complete the proof of Theorem 2.1. We also recall that Theorem 2.3is just a variant of Theorem 2.1. 10 4 Multidegrees and Euler Discriminant... | https://arxiv.org/abs/2503.02536v1 |
Chern class and∆1 n−1is a double Schur polynomial. In our case, the right hand side is simply the degree n−1 part of c(F)/c(E). The ratio of Chern classes equals c(F) c(E)=(1−τ)n (1−σ)/producttextm i=1(1−diτ)=/parenleftigg/summationdisplay j≥0σj/parenrightigg (1−τ)n (1−d1τ)(1−d2τ)···(1−dmτ). Thedegree n−1 partofthise... | https://arxiv.org/abs/2503.02536v1 |
context. Theorem 4.5. The algebraic Euler discriminant ∇agrees with the topological Euler dis- criminant∇χof the projective hypersurface family complement defined by f=f1f2···fm. Proof.WeapplytheCayley trick(see[ 3,§2], [8, (2.5)])andconsider theauxiliarypolynomial ˜f(x,y) =y1f1(x)+y2f2(x)+···+ymfm(x)∈R[y1,...,y m], whe... | https://arxiv.org/abs/2503.02536v1 |
the rational functions ∂ℓA/∂xj: Qs,∂ \1= 0...0∂f1 ∂x1∂f1 ∂x2...∂f1 ∂xn f2...0∂f2 ∂x1∂f2 ∂x2...∂f2 ∂xn ..................... 0... f m∂fm ∂x1∂fm ∂x2...∂fm ∂xn s2···sm∂ℓA ∂x1∂ℓA ∂x2···∂ℓA ∂xn (16) By (10), these new matrix entries generate the likelihood ideal IAin the localized ring Sf. This implies tha... | https://arxiv.org/abs/2503.02536v1 |
arrangem ents: being tame and being gentle. For SNC arrangements, the modules of logarithmic p-forms with poles along Aare simply exterior powers. Namely, we have Ωp(A) =/logicalandtextpΩ1(A) by [22, Lemma 3.3.27]. Recall (e.g. from [ 4]) that an arrangement Aistameif pdimR(Ωp(A))≤pfor all 0≤p≤n. In the hyperplane sett... | https://arxiv.org/abs/2503.02536v1 |
Planck Institute (MPI-M iS) in Leipzig. TK and HS thank MPI-MiS for its great working atmosphere . TK is supported by the DFG (SPP 2458, 539866293). HS is supported by the NSF (DM S 2006410). BS and MW are supported by the ERC (UNIVERSE+, 101118787). Views and opinions expressed are however those of the authors only an... | https://arxiv.org/abs/2503.02536v1 |
PROOF OF A CONJECTURE OF DRTON, STURMFELS AND SULLIVANT ON THE MAXIMUM LIKELIHOOD DEGREE OF THE GAUSSIAN GRAPHICAL MODEL OF A CYCLE RODICA ANDREEA DINU AND MARTIN VODI ˇCKA Abstract. In this article, we compute the precise value of the maximum likelihood degree of the Gaussian graphical model of a cycle, confirming a c... | https://arxiv.org/abs/2503.02704v1 |
j)̸∈E(Γ). 1arXiv:2503.02704v1 [math.AG] 4 Mar 2025 2 RODICA ANDREEA DINU AND MARTIN VODI ˇCKA The undirected Gaussian graphical model associated with the graph Γ is the family of multivariate normal distributions with covariance matrix Σ such that (Σ−1)ij= 0 for every missing edge ( i, j). The maximum likelihood estima... | https://arxiv.org/abs/2503.02704v1 |
1.3. 2.Fiber over the identity at a glance Setting: Consider the projection π:P(S2(Cn))99KP(S2(Cn)/L⊥ Cn). Consider the graph of this projection restricted to L−1 Cn, i.e. Γn={(A, π(A)) :A∈L−1 Cn\L⊥ Cn} ⊂P(S2(Cn))×P(S2(Cn)/L⊥ Cn). Theorem 2.1. InΓnthere does not exist a point of the form (A,Id)forA∈L⊥ Cn. In other word... | https://arxiv.org/abs/2503.02704v1 |
In other words, we find the non-zero entry that is “closest” to the main diagonal. Without loss of generality, we may assume i= 1. 4 RODICA ANDREEA DINU AND MARTIN VODI ˇCKA Consider a submatrix A({1,2. . . , j},{j, j+ 1, . . . , n, 1}). Since A∈L−1 Cn, the rank of this matrix is at most 2 (all of its 3 ×3 minors are 0... | https://arxiv.org/abs/2503.02704v1 |
invariant with respect to this group action. In addition, the set L−1 Cnis also invariant. Thus, the intersection L−1 Cn∩(Id + L⊥ Cn) is also invariant. Let us consider another group action, namely the action by cyclic shift. For this, we consider the following matrices: N+ n:= 0 1 0 . . . 0 0 0 0 0 1 . . . ... | https://arxiv.org/abs/2503.02704v1 |
minor and evaluate at the identity, it is clear that∂(δ(1,2,3)(1,3,k))) ∂x2,k=−x1,1x3,3evaluated at the identity is -1 and all the other partial derivations evaluated at the identity are 0. Thus, we see that by taking partial derivations of δ(1,2,3)(1,3, k) we obtain a row of the Jacobian matrix which is equal to −e2,k... | https://arxiv.org/abs/2503.02704v1 |
zero vector. Assume that k≥2 and the statement holds for all k′≤k. PROOF 7 We have e1,2k−e1,2k+2−e3,2k+e3,2k+2, e1,2k−e3,2k+2, e1,2k−2−e3,k∈T(Cn). This yields e1,2k−e1,2k+2−e3,2k+e3,2k+2+e1,2k−e3,2k+2−e1,2k−2+e3,2k= 2e1,2k−e1,2k−2−e1,2k+2∈T(Cn). By induction hypothesis, (2k−2)e1,4−2e1,2k,−(k−2)e1,4+e1,2k−2∈T(Cn). There... | https://arxiv.org/abs/2503.02704v1 |
thus we get e1,j+1−e2,j+2∈T((M+ n(z))−1) as well which concludes the first part of the lemma. The proof is completely analogous in the case of the matrix ( M− n(z))−1. We consider the same minors; the only difference is that there will be a change of signs when one of the indices is greater than n. □ Proposition 3.5. T... | https://arxiv.org/abs/2503.02704v1 |
even case n= 2m, we get to the analogous contradiction with the different fac- torization xn−2+ (−1)nPn−2(x) =Pm−1(x)((Pm−1(x)−x2Pm−3(x)).This shows that the point ( M+ n(z))−1is smooth at the intersection L−1 Cn∩(Id + L⊥ Cn). Analogously, we can show that the point ( M− n(z))−1is smooth, the difference will be that in... | https://arxiv.org/abs/2503.02704v1 |
cka, Geometry of the Gaussian graphical model of the cycle , arXiv:2111.02937, preprint 2021. [10] M. Drton, B. Sturmfels, S. Sullivant. Lectures on algebraic statistics , volume 39, Springer Science & Business Media, 2008. [11] S. Ho¸ sten, A. Khetan, B. Sturmfels, Solving the likelihood equations, Foundations of Comp... | https://arxiv.org/abs/2503.02704v1 |
arXiv:2503.02802v1 [math.ST] 4 Mar 2025Computational Equivalence of Spiked Covariance and Spiked Wigner Models via Gram-Schmidt Perturbation Guy Bresler∗Alina Harbuzova Massachusetts Institute of Technology Abstract In this work, we show the first average-case reduction transfor ming the sparse Spiked Co- variance Model... | https://arxiv.org/abs/2503.02802v1 |
Our Contributions 14 4.1 Internal Reductions within Spiked Models . . . . . . . . . . . . . . . . . . . . . . . . 14 4.2 Overview of Existing SpWig→SpCovReductions . . . . . . . . . . . . . . . . . . . . 16 4.3 Our Results: SpCov→SpWigReductions and Equivalence . . . . . . . . . . . . . . . 16 ∗Supported by NSF Grant C... | https://arxiv.org/abs/2503.02802v1 |
. . . . . . . . . . . . . 47 A.2 Average-Case Reductions in Total Variation . . . . . . . . . . . . . . . . . . . . . . . 48 A.3 Implications of Average-Case Reductions for Phase Diag rams. . . . . . . . . . . . . 50 A.4 Average-Case Reductions on the Computational Threshol d. . . . . . . . . . . . . . 51 B Definitions ... | https://arxiv.org/abs/2503.02802v1 |
that their associated statistical and computational phenomena has the same root. In parallel, researchers have sought to analyze the limits o frestricted classes of algorithms , including sum-of-squares [ BHK+16,HKP+17], approximate message passing (AMP) [ ZK16], sta- tistical query [ DKS17,Kea98,FPV15,FGR+13], low-deg... | https://arxiv.org/abs/2503.02802v1 |
a d×dmatrix λuu⊤+W, whereWis symmetric with N(0,1) off-diagonal and N(0,2) diagonal entries, u∈Rdis a sig- nal vector, and λ∈Ris a signal strength parameter. The Spiked Wigner Model capt uresZ2 synchronization [ JMRT16 ] and a Gaussian version of the stochastic block model [ DAM16]. For both models, it is often assumed ... | https://arxiv.org/abs/2503.02802v1 |
Wigner Model, SpWig(d,k,λ).In thed-dimensional Spiked Wigner Model we observe Y=λuu⊤+W, W∼GOE(d), (1) whereu∈Bd(k) (orBd(k)) is ak-sparse unit signal vector, λ∈Ris a signal-to-noise ratio (SNR) parameter, and GOE( d) denotes the distribution of1√ 2(A+A⊤) withA∼N(0,1)⊗d×d [DM14,LKZ15]. We denote the law of YbySpWig(d,k,... | https://arxiv.org/abs/2503.02802v1 |
2E I? αβ (b)SpCov(d,k,θ,n) with 1< γ= logdn <3 and α= logdk,β= logdθ1 210 1−γ 2 −γ 2HE I αβ (c)SpCov(d,k,θ,n) with γ= logdn≥3 and α= logdk,β= logdθ1 211 2 0HE I αβ (d)SpWig(d,k,λ) with α= logdkand β= logdλ Figure 1: Subset of phase diagrams from [BBH18], [BB19] plotted as signal β= logdθ(orβ= logdλ) vs. sparsity α= log... | https://arxiv.org/abs/2503.02802v1 |
for sparse PCA that is different from SpCov. [BB19] showed a reduction to the canonical SpCovmodel via average-case reductions from SpWig, substan- tiating the entire “Hard” regime in case γ∈{1}∪[3,∞] and the part of hard regime α≤γ/6 in case 1<γ <3. ForSpWig, the entire hard region was obtained by reduction from the bi... | https://arxiv.org/abs/2503.02802v1 |
andSpWig(ν) arecomputationally equivalent points , denoted SpCov(µ)≡SpWig(ν), ifSpCov(µ)/√r⌉⌋⌉⌈⌉s⌉qualSpWig(ν) andSpWig(ν)/√r⌉⌋⌉⌈⌉s⌉qualSpCov(µ). Weproposetoaimforabijectiveequivalencebetweenthe SpWigphasediagramand2-dimensional slicesSpCovγof the 3-dimensional SpCovphase diagram. In addition to the bijective equivalen... | https://arxiv.org/abs/2503.02802v1 |
strong implications for both detection and reco very. Theorem 1.2 (Main Result 1: SpCov→SpWigReductions) . 1.(Reductions with Canonical Parameter Correspondence) Forµandνin accordance to(5), i.e,µ= (α,β,γ)↔ν= (α,β+γ/2): 7A toy example illustrates how this can occur. Suppose that th e computational threshold for two pro... | https://arxiv.org/abs/2503.02802v1 |
introduce several novel techn iques for dependence removal and structure manipulation. This section overviews the main id eas. We first use a CLT result for Wishart matrices to show that the t wo models’ distributions are close in total variation if n≫d3, establishing a bijective equivalence between SpCovandSpWigin this... | https://arxiv.org/abs/2503.02802v1 |
but remained algorithmically passive. In contrast, our new approach described next introduces an activeingredient – an orthog- onalization step that directly creates independence – succ eeding in the much wider range n≫d. Our approach makes use of the rotational invariance of Gauss ian distributions: if n≥d, X⊤∼N(0,Id)... | https://arxiv.org/abs/2503.02802v1 |
of Theorem 9.2. Bernoulli Denoising. Following the flipping procedure, we use our novel Bernoulli Denoising step described in Section 6.1, which allows us to remove unwanted perturbations in the mea ns of the entries of Ythat contain the signal. Notice that our flipping procedure involves discretization ofYto±1 values. I... | https://arxiv.org/abs/2503.02802v1 |
and recovery are d eferred to Sec. AandB. 13 4 Reductions for SpCovandSpWig: Known Results and Our Con- tributions In this section we overview existing internal reductions wi thinSpCovandSpWigmodels (Sec- tion4.1.1,4.1.2) as well as reductions from SpWigtoSpCov(Section 4.2). Then in Section 4.3we describe how to combin... | https://arxiv.org/abs/2503.02802v1 |
of 2. 4.1.2 Changing Sparsity in the More Dense Regime k≥√ d The following “reflection cloning” reduction for SpWigwas introduced in [ BBH18] and can be easily generalized to an internal reduction for SpCov. We note that it requires the use of Bd(k) (rather thanBd(k)) as the collection of allowed sparse vectors for the ... | https://arxiv.org/abs/2503.02802v1 |
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