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Quantifying the Speed-Up from Non-Reversibility in MCMC Tempering Algorithms Gareth O. Roberts and Jeffrey S. Rosenthal University of Warwick University of Toronto (January, 2025) 1 Introduction Markov chain Monte Carlo (MCMC) algorithms are extremely important for sampling from complicated high-dimensional densities, ... | https://arxiv.org/abs/2501.16506v1 |
C (2,−)(2,+) C C BAAB BAAB BAAB BAAB BAAB BAAB Figure 1: Diagram of the double-birth-death Markov chain. This chain can be viewed as a “lifting” of a symmetric walk on Z. That is, if states ( i,+) and ( i,−) are combined into a single state ifor each i∈Z, with the chain equally likely to be at ( i,+) or ( i,−), then th... | https://arxiv.org/abs/2501.16506v1 |
1, this corresponds to expanding space by a constant factor of ℓ >0, i.e. regarding the adjacent points as being a distance ℓ apart rather than having unit distance. 3 In this context, the transition probabilities A=A(ℓ) and B=B(ℓ) and C=C(ℓ) also become functions of ℓ(still summing to 1 for each ℓ). The value of vin T... | https://arxiv.org/abs/2501.16506v1 |
to a “hottest” distribution which facilitates easy mixing between modes). Typically, we define inverse temperatures 0 ≤βN< β N−1< . . . < β 1< β 0= 1, and let πβ(x)∝[π(x)]βbe a power of the target density π(x). Simulated Tempering (ST) augments the original state space with a one-dimensional component indicating the cu... | https://arxiv.org/abs/2501.16506v1 |
these relative efficiency curves very closely. Theorem 4. Consider a tempering algorithm under the assumptions of [1, 22] as above. Then in the limit as the dimension d→ ∞ , the efficiency measure eff(ℓ)is related to the acceptance rate acc(ℓ)as follows: (i) In the reversible case, eff(ℓ) = acc( ℓ)4 c2[Φ−1(acc(ℓ)/2))]2... | https://arxiv.org/abs/2501.16506v1 |
ℓvalues is 1 .73/2.38.= 0.73, corresponding to a 27% decrease in proposal scaling standard deviation for the non-reversible versus reversible case. More importantly, the ratio of optimal efficiency functions is 1 .89/1.33.= 1.42, corresponding to a 42% increase in efficiency for the non-reversible versus reversible cas... | https://arxiv.org/abs/2501.16506v1 |
relative efficiency of different tempering MCMC algorithms, do indeed provide useful information about the practical information of these algorithms to achieve round-trips between the coldest and hottest temperatures. 6 Proof of Theorem 1 Finally, we now prove Theorem 1. For notation, let Geom( C) be the probability di... | https://arxiv.org/abs/2501.16506v1 |
t >0, the process W(M) :=X⌊Mt⌋ has regenerative increments at times {Tn/t}. Then, it follows from [13, Theorem 1.4] that asM→ ∞ with t >0 fixed, we have W(M)/√ M≡X⌊Mt⌋/√ M→N(0, v), where the corresponding volatility parameter vis computed (using Lemma 9) to be: v=Var[XTn−XTn−1] E[Tn−Tn−1]=2[(A−B)2/C2] + 2[( A+B)/C] 2/C... | https://arxiv.org/abs/2501.16506v1 |
and S. Janson (1983), The Limiting Behaviour of Certain Stopped Sums and Some Applications. Scand. J. Stat. 10(4) , 281–292. 13 [12] W.K. Hastings (1970), Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109. [13] S. Janson (2023), On a central limit theorem in renewal theory. ... | https://arxiv.org/abs/2501.16506v1 |
arXiv:2501.16517v1 [math.ST] 27 Jan 2025SymmetricPerceptrons, Number Partitioning andLa/t_tices Neekon Vafa MIT nvafa@mit.eduVinod Vaikuntanathan MIT vinodv@mit.edu Abstract /T_hesymmetric binary perceptron (SBP)u1D705) problem with parameter )u1D705 ∶ ℝ ≥1→ [0,1] is an average-casesearchproblemdefinedasfollows: givenar... | https://arxiv.org/abs/2501.16517v1 |
is established for the case of matrices )u1D400with i.i.d. Rademacher entries. Nevertheless, [ GKPX22] conjecturethatthe same guaranteeremains true forthe caseof i .i.d. standard normalentries. 2 Wereferthereaderto[ GKPX22]foranextensivediscussionoftherationalebehindtheirconjec- ture. AsurveyofGamarnikontheoverlapgappr... | https://arxiv.org/abs/2501.16517v1 |
list; sorts it; replaces the largest and second large st element with their absolute difference; and repeats until a single element is le/f_t which the algorithm o utputs as the discrep- ancy of the set of numbers. (An informed reader might already have o bserved the analogy of Karmarkar-Karpwith the Blum-Kalai-Wasserma... | https://arxiv.org/abs/2501.16517v1 |
called the asymmetric Ising perceptron problem, which for )u1D400∼(0,1)/u1D45B×/u1D45A, asks to find )u1D431 ∈ {−1,1}/u1D45Asuchthat )u1D400)u1D431≥/u1D705(/u1D45A//u1D45B)√/u1D45A⋅)u1D7CF, in the sensethatfor everyrow )u1D41A/u1D457∈ ℝ/u1D45Afor/u1D457∈ [/u1D45B],we want )u1D41A⊤ )u1D457)u1D431≥/u1D705(/u1D45A//u1D45B... | https://arxiv.org/abs/2501.16517v1 |
footsteps. Explicitly, it is the Chinese remainder theorem : for distinct primes/u1D45D1,…,/u1D45D)u1D4⋯Band/u1D45E=∏)u1D4⋯6∈[)u1D4⋯B]/u1D45D)u1D4⋯6, thereis agroup isomorphism /u1D711∶/uni2A01.size1 )u1D4⋯6∈[)u1D4⋯B]ℤ//u1D45D)u1D4⋯6ℤ⟶ℤ//u1D45Eℤ. Wewill scalethis isomorphismsothat /uni0303.size1 /u1D711∶/uni2A01.size1 ... | https://arxiv.org/abs/2501.16517v1 |
/u1D70Emax()u1D400)to denote the spectral 9 norm, or maximumsingular value,of )u1D400. Explicitly, /u1D70Emax()u1D400) =max )u1D42F≠)u1D7CE‖)u1D400)u1D42F‖2 ‖)u1D42F‖2. For a matrix )u1D400∈ℝ)u1D4⋯B×)u1D4⋯A, we o/f_ten write )u1D400by its columns, as in )u1D400= [)u1D41A1,)u1D41A2,⋯,)u1D41A)u1D4⋯A]for)u1D41A)u1D4⋯6∈ℝ)u... | https://arxiv.org/abs/2501.16517v1 |
(SubexponentialHardnessof ApproximateWorst-caseLa/t_ticeProbl ems).Forall constants )u1D700 > 0, at least one of SIVP)u1D6FE,GapCRP)u1D6FE, orGDD)u1D6FErequires time 2)u1D714()u1D4⋯B1/2−)u1D700)to solve, where )u1D6FE()u1D4⋯B) = 2)u1D4⋯B1/2−)u1D700. 2.3 Continuous and Discrete Gaussian Measures For)u1D707∈ℝand)u1D70E∈ℝ... | https://arxiv.org/abs/2501.16517v1 |
Weemphasizethatthepolynomialinthe poly()u1D447) run-timeof )u1D440(⋅)maydependonthenon-negligiblefunction )u1D707. Since/u1D465isworst-caseand )u1D434∈FNP, standardamplificationappliestomakethe successprobability of )u1D440exponentiallyclose to 1. 3 Reduction to Symmetric Binary Perceptrons /T_heorem 3. Suppose)u1D705(/... | https://arxiv.org/abs/2501.16517v1 |
=poly(/u1D4⋯B,/u1D4⋯A)times, and since we can efficientlyverifywhetherthe SBP)u1D70⋯solver succeeded,thereduction willstill gothrough. 3.2 Variants and Generalizations We mention a few variants of SBPfor which the reduction in /T_heorem 3would also apply. As theyarenot criticaltoour mainresult, for simplicity,we onlysket... | https://arxiv.org/abs/2501.16517v1 |
for numberpartitioning. Corollary 3. Suppose there is a polynomial time algorithm for NPP)u1D70⋯(on average) for )u1D70⋯()u1D4⋯A) = 2−log2+)u1D700)u1D4⋯Afor some constant )u1D700 >0that succeedswith non-negligibleprobability. /T_hen, there are ran- domized2)u1D442)parenleft(size2)u1D4⋯B1 1+)u1D700)parenright(size2-time... | https://arxiv.org/abs/2501.16517v1 |
or equivalently, /u1D70⋯(/u1D4⋯A) ≤√/u1D4⋯A/(/u1D4⋯→ln/u1D4⋯A). Wehave √/u1D4⋯A /u1D4⋯→ln/u1D4⋯A=25)u1D4⋯B11+)u1D700 /u1D4⋯→10ln(2)/u1D4⋯B1 1+)u1D700≥25)u1D4⋯B11+)u1D700 10ln(2)(320 /u1D4⋯B/u1D4⋯A))u1D4⋯B/u1D4⋯B1 1+)u1D700 =25)u1D4⋯B11+)u1D700 10ln(2)(320 /u1D4⋯B))u1D4⋯B210)u1D4⋯B2+)u1D700 1+)u1D700/u1D4⋯B1 1+)u1D700 ≥... | https://arxiv.org/abs/2501.16517v1 |
of the National Academy of Sciences , 118(41):e2108492118, 2021. 3 [GJ79] M. R. Garey and David S. Johnson. Computers and Intractability: A Guide to the /T_heoryofNP-Completeness . W.H. Freeman,1979. 4 [GK21] DavidGamarnikandErenC.Kızılda ˘g. Algorithmicobstructionsintherandomnum- ber partitioning problem, 2021. 4 [GKP... | https://arxiv.org/abs/2501.16517v1 |
to average-ca se reductions based ongaussianmeasures. SIAMJ.Comput. ,37(1):267–302, 2007. 3,6,7,8,11,12,13,14, 18,19 [PX21] Will PerkinsandChangji Xu. Frozen 1-rsbstructure of the s ymmetricising percep- tron. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC ’21: 53rd Annual ACM SIGACT Symposium on /T_... | https://arxiv.org/abs/2501.16517v1 |
Ancestral Inference and Learning for Branching Process in Random Environments Xiaoran Jiang Department of Statistics George Mason University Fairfax, VA, 22030Anand N. Vidyashankar Department of Statistics George Mason University Fairfax, VA, 22030 Abstract Ancestral inference for branching processes in random environm... | https://arxiv.org/abs/2501.16526v1 |
let {ξn,j,k:n≥0, j≥1, k≥1}denote a collection of random variables which are i.i.d. conditioned on a ζn,jwith distribution p(r|ζn,j). The random variables {ζn,j:n≥0, j≥1}are i.i.d. and assume values on Bwhich is equipped with the Borel σ-fieldB(B) generated by the topology of weak convergence (see Athreya and Ney (1972)... | https://arxiv.org/abs/2501.16526v1 |
and develop new technical tools to address the analytical challenges. We now turn to another motivating example where the proposed model and the method arise naturally. When modeling the growth of disease spread such as COVID-19 and smallpox, most research concentrated on where and when the disease began to spread. On ... | https://arxiv.org/abs/2501.16526v1 |
the amount of DNA molecules. Thus, as in PCR literature, we assume that Fn,j=cZn,j, where c depends on the amplicon size and the calibration factor, which represents the number of nanograms of double-stranded DNA per fluorescence unit. More details concerning the experiment and scientific details can be found in Hanlon... | https://arxiv.org/abs/2501.16526v1 |
mirrors the three phases in a PCR dataset, though for different reasons. Consequently, we utilize data from the period when the growth rate remained stable before it began to decline to estimate the initial number of infected subjects. 02468 0 10 20 30 40 WeekLog patientsGroup Large Pop Small PopLog of Weekly Patients ... | https://arxiv.org/abs/2501.16526v1 |
involves an estimate of the marginal offspring mean, ˆmn. Theorem 2.2 provides the limiting distribution for the centered ˆ mnholds using the scaling (Jn(n−τn))1 2with the limiting variance coinciding with the anticipated marginal variance. A natu- ral next question concerns the joint distribution of the ˆ mnand ˆmA,n.... | https://arxiv.org/abs/2501.16526v1 |
the average number of children produced by the (l−1)thgeneration parents. Evidently, the asymptotic properties of (1) will depend on the behavior ofN∗(−1) nPn l=τZl,j. A key observation is that the variance of N∗(−1) nPn l=τZl,jgrows as nincrease leading to a comparison of the marginal mean behavior to that of the marg... | https://arxiv.org/abs/2501.16526v1 |
the total number of parents”; that is, ˆm(0) τ,n,J=PJ j=1Pn l=τ+1Zl,jPJ j=1Pn l=τ+1Zl−1,j. (6) Note that we can rewrite (6) as ˆm(0) τ,n,J=JX j=1w(B) jnX l=τ+1w(W) lZl,j Zl−1,j,where (7) 11 w(B) j=Pn l=τ+1Zl−1,jPJ j′=1 Pn l=τ+1Zl−1,j′and w(W) l=Zl−1,jPn l′=τ+1Zl′−1,j. In (7), note that w(B) jrepresents the between rep... | https://arxiv.org/abs/2501.16526v1 |
cases(ii), where σ2 Iandσ2 F(δ)is specified in Remark 2.1. Furthermore, as δm→ ∞ ,σ2 F(δm)→σ2 I.σ2 F(0) = D m∗ 2, which is the asymptotic variance of ˜mA,n. The details of the proof of this lemma are outlined in Appendix C.4. Remark 2.2 (Comparison to GW case) .Under the GW process, the offspring distributions are i.i.... | https://arxiv.org/abs/2501.16526v1 |
in this case. That is, ˆm′ τ1,τ2,n=1 JnJnX j=11 n−τ2nX l=τ2+1Zl,j Zl−1,j,ˆm′ A,τ1,τ2,n=1 JnˆN′τ1,τ2,nJnX j=1τ2X l=τ1Zl,j, 14 and the corresponding estimate of the mean ofPJn j=1Pτ2 l=τ1Zl,jis ˆN′ τ1,τ2,n=ˆm′τ1τ1,τ2,n( ˆm′τ2−τ1+1 τ1,τ2,n−1) ˆm′τ1,τ2,n−1. The next Theorem describes our result under this setting. Theorem ... | https://arxiv.org/abs/2501.16526v1 |
N 0,R2 4 σ2 τT m2 A,T+σ2 τC m2 A,C , where σ2 τTandσ2 τChas the same form in Remark 2.2. This expression for the limiting variance also aligns with the result of Theorem 2 in Hanlon and Vidyashankar (2011). 3.2 Dynamics of COVID-19 growth Turning to the dynamics of COVID-19 evolution, as described in the introducti... | https://arxiv.org/abs/2501.16526v1 |
element by Yl. Using the conditional independence ofξl,n, we have for any τn, V ar nX l=τn+1Yl =V ar n−1X l=τn+1Yl+E[Yn|Fn] +E1 Z2nV ar[Zn+1|Fn] =V ar n−1X l=τn+1Yl +E1 Znσ2 n =V ar n−1X l=τn+1Yl +γ∗2E1 Zn , where the penultimate equality follows by conditioning on F(n−1). By iteration and using L... | https://arxiv.org/abs/2501.16526v1 |
proof of Theorem 2.1 requires the asymptotic property of ˆ mn, we begin with the proof of Theorem 2.2. We begin by recalling that Yl,j=Z−1 l,jZl+1,j−ml,jand ˆmn−m∗=1 JnJnX j=1 1 n−τnn−1X l=τnYl,j +1 JnJnX j=1 1 n−τnn−1X l=τn(ml,j−m∗) =S1,n+S2,n. (19) 5.1 Proof of Theorem 2.2 Proof. To prove the Theorem, first,... | https://arxiv.org/abs/2501.16526v1 |
l=τn+1Y2,l,j 2 ≤c′ τ,ifτn≡τis fixed (case (i)), and lim n→∞E nX l=τn+1Y2,l,j 2 →0,ifτn→ ∞ (case (ii) and (iii)). Proof. Notice that E nX l=τn+1Y2,l 2 =V ar nX l=τn+1Y2,l + E nX l=τn+1Y2,l 2 . (24) We will prove the theorem by showing the limits of the two terms on the RHS of the above equation ex... | https://arxiv.org/abs/2501.16526v1 |
S2,n, using (A4) , since E[m2 0,1−m∗ 2]2:=ρ∗2 4<∞, we have P(|S′ 2,n|> ϵ)≤V ar(S′ 2,n) ϵ2=ρ∗2 4 Jn(n−τn)ϵ2→0,asn→ ∞ . This concludes the proof of the Theorem. Remark 5.1 (Estimation of σ2∗).Since σ2∗=m∗ 2−m∗2, by Theorem 2.2 and Theorem 5.2, deriving a consistent estimator for σ2∗is straightforward, given by ˆm2,n−ˆm2 ... | https://arxiv.org/abs/2501.16526v1 |
using Lemma 5.1 andn2 Jn(n−τn)→0. For the second term in (35), because E[m0,1]4<∞, setting E[m2 0,1−m∗ 2]2=ρ∗2 4we have P S′ 2,n>ϵm∗ 2 2n =P 1 JnJnX j=11 n−τnn−1X l=τn m2 l,j−m∗ 2 >ϵm∗ 2 2n ≤V ar 1 JnJnX j=11 n−τnn−1X l=τnY2,l,j ϵm∗ 2 2n−2 =n2 Jn(n−τn)ρ∗2 4 ϵm∗ 2→0,(sincen2 Jn(n−τn)→0). (37) Now by combi... | https://arxiv.org/abs/2501.16526v1 |
show the second term converges to mAin probability. From Lemma 2.4, we note that V ar 1 rn∗1 N∗nnX l=τnZl,j <∞. 28 Because by Chebyshev’s inequality, for any ϵ >0, P 1 N∗n 1 JnJnX j=1nX l=τnZl,j −mA > ϵ ≤V arh 1 N∗n 1 JnPJn j=1Pn l=τnZl,ji ϵ =r∗2n JnV arh 1 rn∗1 N∗nPn l=τnZl,1i ϵ→0, where the convergence... | https://arxiv.org/abs/2501.16526v1 |
(42), while the second term converges to 0 in probability by Lemma 5.5. Now we show the proof of Theorem 2.3. Proof. Recall by (11) and (12) that V ar(Zn,1) =m∗(n−1) 2Dn+m∗2nσ2 AandDn↗D<∞. By Chebyshev’s inequality, for any ϵ >0, P 1 JnJnX j=1Zn,j m∗n−mA > ϵ ≤V ar 1 JnPJn j=1Zn,j m∗n ϵ2=r∗2n JnDn m∗ 2+σ2 A →0,... | https://arxiv.org/abs/2501.16526v1 |
Tn(2) + Tn(3)p− →0. Now using Theorem 4.3, Theorem C.5, and Chebyshev’s inequality, it follows that Tn(1)p− →D m∗ 2, hence we have Tn(1) + Tn(2) + Tn(3)p− →D m∗ 2. The details of the above calculation are in Appendix C.6. Turning to the second term on the RHS of (47), it follows from Theorem 2.3, that 1 r∗2n1 JnJnX j=1... | https://arxiv.org/abs/2501.16526v1 |
distributions are modeled as 1+Bernoulli( pl,j), where pl,jare i.i.d. following a Beta (90,10) distribution. In the second setting, the number of ancestors is modeled using the probability model 4+ a negative binomial distribution with parameters r= 4 and p= 0.4. The offspring distribution are conditionally 1+Poisson( ... | https://arxiv.org/abs/2501.16526v1 |
it is evident that for a smaller number of replicates J, the confidence interval (CI) utilizing the t-distribution exhibits a higher coverage rate than the CI using the Gaussian distribution. 7.3 Large offspring variance experiment By Theorem 2.3, we understand that the variance of our estimator diverges to infinity at... | https://arxiv.org/abs/2501.16526v1 |
first τ1toτ2generations (Theorem 3.1), and blue represents the variance of the estimate of ˆ mA,nwhere we learn m∗from τ2tongenerations (Theorem 3.2). The results clearly illustrate the results from Theorem 2.1, Theorem 3.1, and Theorem 3.2, namely that variance is minimized in the last case (blue). We find that using ... | https://arxiv.org/abs/2501.16526v1 |
t−p) and for i= 1,2,ρi= (ρi,0, ρi,1, . . . , ρ i,p). The transitioning logistic function is given by G(qt, γL, cL) =1 1 + exp( −γL(qt−cL). The quantity prepresents the order of the auto-regression and qtrepresents the threshold variable, which allows for a gradual transition between regimens one and two. The threshold ... | https://arxiv.org/abs/2501.16526v1 |
denoted as SV1 and SV2. The SV2 41 product was obtained by diluting the SV1 product by a factor of R= 10. Using section (4.1.2) in Vidyashankar and Li (2019), we set mc= 1.55 to determine the starting and ending cycles of the exponential phase. For the LSTAR model, we will use the entire dataset because the positive po... | https://arxiv.org/abs/2501.16526v1 |
is the average population in the particular group. ˆ mnrepresents the marginal rate of growth of COVID-19 infections, and ˆ mA,nrepresents the estimated initial number of subjects infected with COVID-19. Counties Pop Mean Pop ˆmn(SE) ˆmA,n(SE) Group 1 8 89k - 1.1m 98k 1.1788 (0.0110) 24.7939 (0.1709) Group 2 8 37k - 41... | https://arxiv.org/abs/2501.16526v1 |
methods. The method can be applied to other situations, as illustrated by our application to COVID-19 evolution. A natural extension is to extend our approach to the BPRE model with immigration, given that the growth in the number of cases includes both internal infections and imported cases. Overall, our methods show ... | https://arxiv.org/abs/2501.16526v1 |
using real-time quantitative pcr and the 2[delta][delta]ct method. Methods , 25(4):402–408. Nielsen, M. K., Peterson, D. S., Monrad, J., Thamsborg, S. M., Olsen, S. N., and Kaplan, R. M. (2008). Detection and semi-quantification of strongylus vulgaris dna in equine faeces by real-time quantitative pcr. International Jo... | https://arxiv.org/abs/2501.16526v1 |
¯Z0 10.067 10.028 10.005 9.976 10.012 9.991 10.013 10 var(Z0) 2.002 1.68 1.246 1.004 0.498 0.333 0.251 0.199 (Z0) G CR 0.875 0.887 0.912 0.911 0.93 0.942 0.947 0.941 (Z0) G ML 5.202 4.823 4.219 3.815 2.729 2.242 1.949 1.738 (Z0) t CR 0.95 0.948 0.95 0.944 0.945 0.951 0.954 0.945 (Z0) t ML 7.369 6.325 5.09 4.403 2.915 2... | https://arxiv.org/abs/2501.16526v1 |
1.343 1.112 0.971 0.874 Table 17: Numerical experiment result Relative Quantitation ( n= 30, τ= 15) Let the ancestors generated by 1+Poisson random variables with mean of target group mA,T= 3000 and calibrator mA,C= 100 hence R= 30 53 n= 30, τ= 15: J 20 30 40 50 ˆR 30.014 29.985 30.002 30.01 ˆΛR,τ,n,J 1.117 0.745 0.561... | https://arxiv.org/abs/2501.16526v1 |
have for any n, Cov(T′ τ1,τ2,n, U′ n,1) =√δm Jnr∗τ2Cov JnX j=11 δmnX l=τ2+1(Zl,j Zl−1,j−m∗),JnX j=1 1 N∗τ1,τ2τ2X l=τ1Zl,j−mA =1 r∗τ2√δmE nX l=τ2+1(Zl,1 Zl−1,1−m∗) 1 N∗τ1,τ2τ2X l=τ1Zl,1−mA =1 r∗τ2√δmE 1 N∗τ1,τ2τ2X l=τ1Zl,1−mA E nX l=τ2+1(Zl,1 Zl−1,1−m∗) Fτ2 = 0. This complet... | https://arxiv.org/abs/2501.16526v1 |
converges to zero expo- nentially fast: n(a1 2)n. 59 Lemma C.3. Under conditions and settings of Lemma 5.1, there exists c∗∗ δ>0, and 0< a < 1, Ln,τn≤c∗∗ δn(a1 2)n. Proof. LetLn,τn= 2Ln,τn,1+Ln,2+Ln,3where Ln,τn,1=Cov[Pn−1 l=τn+1Y2,l, Z−1 nσ2 n],Ln,2=V ar[Z−1 nσ2 n], andLn,3=E[V ar(Y2,n|Fn)]. For Ln,τn,1, we have Ln,τn... | https://arxiv.org/abs/2501.16526v1 |
on the RHS of (61) converges to zero, implying that 1 JnPJn j=1Xnj−Cnj p− →0. Also, from (49), since Cnj→D m∗ 2asn→ ∞ we have that Tn(1)p− →D m∗ 2asn→ ∞ . Hence, Tn(1) + Tn(2) + Tn(3)p− →D m∗ 2. C.7 Results from literature The following results are cited from different papers. Since they are necessary for the proof of ... | https://arxiv.org/abs/2501.16526v1 |
COMPARISON THEOREMS FOR THE MINIMUM EIGENVALUE OF A RANDOM POSITIVE-SEMIDEFINITE MATRIX JOEL A. TROPP Abstract. This paper establishes a new comparison principle for the minimum eigenvalue of a sum of independent random positive-semidefinite matrices. The principle states that the minimum eigenvalue of the matrix sum i... | https://arxiv.org/abs/2501.16578v1 |
(dotted red), described by (1.2). The left-hand panel shows the densities; the right-hand panel shows the cumulative distributions. 1.2.Random psd matrices. This paper demonstrates that the same phenomena persist in the matrix setting. Consider a sum of iid random psd matrices, either real or complex: 𝒀=∑︁𝑛 𝑖=1𝑾𝑖w... | https://arxiv.org/abs/2501.16578v1 |
( 𝔽=ℂ) setting; we often suppress the field. For a natural number 𝑑∈ℕ, the set 𝕄𝑑(𝔽)is the linear space of 𝑑×𝑑square matrices with entries in the field 𝔽. The set ℍ𝑑(𝔽)denotes the real-linear space of self-adjoint(i.e., conjugate symmetric) matrices in 𝕄𝑑(𝔽). As usual, Tr[·]returns the (unnormalized) trace... | https://arxiv.org/abs/2501.16578v1 |
random variables; e.g., see [BH24, Sec. 2.1.2]. A self-adjoint Gaussian matrix 𝒁∈ℍ𝑑is uniquely determined by its expectation 𝔼[𝒁]and variance function Var[𝒁]. Given a self-adjoint matrix 𝚫∈ℍ𝑑and a positive quadratic form V:ℍ𝑑→ℝ+, we write normal(𝚫,V)for the unique Gaussian distribution on self-adjoint matrices... | https://arxiv.org/abs/2501.16578v1 |
the (unnormalized) Gaussian orthogonal ensemble (GOE); see Section 3.7.3 for details. Exploiting standard facts about the GOE matrix [AS17, Exer. 6.48], we find that the minimum eigenvalue and the weak variance satisfy 𝔼𝜆min(𝒁)≥𝑛−2√ 𝑑𝑛and𝜎2 ∗(𝒁)≤3𝑛. Therefore, Theorem 2.3 yields the explicit, nonasymptotic bou... | https://arxiv.org/abs/2501.16578v1 |
the second moments of the random matrix. We can obtain a coarse version of Theorem 9.1 by incorporating the estimates (2.11),(2.12), and (2.13). For instance, the expectation bound (2.4) implies that 𝔼𝜆min(𝒀)≥𝜆min(𝔼𝒁)−2√︁ 2𝜎2(𝒁)log(2𝑑). (2.14) In fact, an improvement of the estimate (2.14)follows from simpler ... | https://arxiv.org/abs/2501.16578v1 |
concentration. To understand the first term, recall from (2.12)that the scale for the minimum eigenvalue of 𝔼𝜆min(𝒁′−𝔼𝒁′)is the statistic 𝜎(𝒁′). In some examples, the factor 𝑅1/6log𝑑submerges the first term below this level. TheresultsofBrailovskaya&vanHandelarepowerfulandwideranging. Forsomeoftheapplications ... | https://arxiv.org/abs/2501.16578v1 |
(Theorem 5.4). 3.Gaussian random matrices To take advantage of the Gaussian comparison method, we must be able to construct the comparison model and determine its spectral properties. This section outlines some facts about Gaussian random matrices that will play a role in the applications and the proofs of the main the... | https://arxiv.org/abs/2501.16578v1 |
form on ℍ𝑑(ℂ), we can construct a Gaussian matrix 𝑿∼normal(0,V′−V)that is independent from 𝒁. By linearity of expectation, 𝔼[𝒁+𝑿]=𝚫. By additivity of the variance function (3.1), Var[𝒁+𝑿]=Var[𝒁]+Var[𝑿]=V+(V′−V)=V′. Therefore, the sum 𝒁+𝑿∼normal(𝚫,V′)follows the same distribution as 𝒁′. To obtain the conv... | https://arxiv.org/abs/2501.16578v1 |
variance and weak variance coincide: 𝜎2(𝑺)=𝜎2 ∗(𝑺)=1. 3.7.2.Diagonal Gaussian matrix. Next, consider the diagonal matrix: 𝑫B∑︁𝑑 𝑖=1𝛾𝑖E𝑖𝑖∈ℍ𝑑(ℂ). The extreme eigenvalues satisfy −𝔼𝜆min(𝑫)=𝔼𝜆max(𝑫)=𝔼max{𝛾𝑖:𝑖=1,...,𝑑}≤√︁ 2 log𝑑. This expectation bound is asymptotically sharp. The variance function o... | https://arxiv.org/abs/2501.16578v1 |
this problem is due to Rudelson [Rud99]; see Section 4.2.1 for a proof sketch. 14 COMPARISON FOR RANDOM PSD MATRICES Fact 4.1 (Sampling: Projective 1-design) .Consider a complex projective 1-designUB(𝒖1,...,𝒖𝑛) consisting of 𝑛unit-norm vectors in ℂ𝑑. For a parameter 1≤𝑠≤𝑛, construct a random subsystem U′ with an... | https://arxiv.org/abs/2501.16578v1 |
ℝ(0,1)are independent. Using the inequality (3.12) for the GUE eigenvalues, 𝔼𝜆min(𝒁)=𝛽+√︂ 𝛽 𝑑+1·𝔼 𝜆min(𝑮gue)+𝛾 ≥𝛽−2√︂ 𝛽𝑑 𝑑+1>𝛽−2√︁ 𝛽. Meanwhile, the weak variance satisfies 𝜎2 ∗(𝒁)≤𝛽 𝑑+1[𝜎2 ∗(𝑮gue)+𝜎2 ∗(𝛾I)]<2𝛽 𝑑. Instantiating Theorem 2.1, we arrive at the probability inequality ℙn 𝜆min(𝒀... | https://arxiv.org/abs/2501.16578v1 |
high-dimensional statistics, in a form due to Oliveira [Oli16]. When the second and fourth moments of a random vector are comparable, then the sample complexity of the variance detection problem is proportional to the dimension. The proof appears below in Section 5.2.2. Theorem 5.2 (Sample covariance matrices: Four mom... | https://arxiv.org/abs/2501.16578v1 |
ℙn 𝜆min(b𝑲𝑛)≥1−√︁ 12𝛽2𝑑/𝑛−𝜀/√ 12o ≤2𝑑·e−𝑛𝜀2/(24𝛽2). Introduce the stated bound (5.5)for the sample complexity 𝑛and simplify the expression. The Rayleigh variational formula for the minimum eigenvalue yields the conclusion (5.6). □ 5.2.3.Proof of Lemma 5.3. Without loss, we may assume that 𝒁is centered. The... | https://arxiv.org/abs/2501.16578v1 |
A natural comparison model for 𝒘𝒘ᵀis the Gaussian random matrix 𝑿∈ℍ𝑑(ℝ)with statistics 𝔼[𝑿]=𝔼[𝒘𝒘ᵀ]=I𝑑; Var[𝑿](𝑴)=2∥𝑴∥2 F+(Tr𝑴)2+𝐶𝑑 𝜁∑︁𝑑 𝑖=1𝑚2 𝑖𝑖≥Mom[𝒘𝒘ᵀ](𝑴). Referring back to Section 3.7, we quickly determine the Gaussian matrix with these statistics: 𝑿=I𝑑+𝑮goe+𝛾I𝑑+√︁ 𝐶𝑑/𝜁𝑫. The matri... | https://arxiv.org/abs/2501.16578v1 |
𝚽is a subspace injection for the range of 𝑸withhighprobability. Quantitatively, we need to understand how the contraction factor 𝛼depends on the embedding dimension 𝑘. In view of (6.2), it suffices to establish a lower bound on the minimum eigenvalue of the random psd matrix 𝒀B(𝚽𝑸)ᵀ(𝚽𝑸)=1 𝑘∑︁𝑘 𝑖=1𝑸ᵀ(𝝋𝑖𝝋... | https://arxiv.org/abs/2501.16578v1 |
𝑖=1∑︁𝑛 𝑗=1𝜉𝑖𝑗𝜓𝑖𝑗E𝑖𝑗𝑸where𝜉𝑖𝑗∼bernoulli(𝜁/𝑘)iid; 𝜓𝑖𝑗∼uniform{±1}iid. For these parameter choices, the remaining question is whether we can reach the bound 𝔼𝛽≤Const. 6.3.1.Prior work and discussion. Sarlós [Sar06] introduced the definition of a subspace embedding (Remark 6.2). Clarkson & Woodruff [C... | https://arxiv.org/abs/2501.16578v1 |
6.3, the subspace coherence 𝜇(𝑸)does not appear in this result. As compared with Chennakod et al. [CDD24], we have reduced the dependence on log𝑑significantly, at the cost of a worse dependence on 𝜀. 6.3.2.Proof of Theorem 6.3. To apply the Gaussian comparison theorem, we need to construct an appropriate comparison... | https://arxiv.org/abs/2501.16578v1 |
an alternative treatment that has more potential for generalization. The results in this section will reappear in the proof of the matrix comparison inequalities. 7.1.Completely monotone functions. Our approach takes advantage of a special feature of the decaying exponential that is encapsulated in the next definition.... | https://arxiv.org/abs/2501.16578v1 |
real line that is completelymonotonetoorderfour. For each nonnegative real random variable𝑊, Cov(𝑊,𝑓′(𝑊))=𝔼 (𝑊−𝔼𝑊)𝑓′(𝑊) ≤𝔼[𝑊2]·𝔼[𝑓′′(𝑊)]. The bound is valid when the expectations are finite. Proof.According to Definition 7.2, the derivatives of a completely monotone function 𝑓alternate sign. We only r... | https://arxiv.org/abs/2501.16578v1 |
Consider an independent family(𝑊1,...,𝑊𝑛)ofsquare-integrable, nonnegative real random variables: 𝑊𝑖≥0and𝔼𝑊2 𝑖<+∞. Form the random psd matrix 𝑿B∑︁𝑛 𝑖=1𝑊𝑖𝑨𝑖. (8.1) Wewillcomparetherandompsdmatrix 𝑿withanappropriateGaussianmodel. Definethedeterministic self-adjoint matrices 𝑯𝑖B(𝔼𝑊2 𝑖)1/2𝑨𝑖for𝑖=1,..... | https://arxiv.org/abs/2501.16578v1 |
𝑤↦→Tr(𝑤𝑨+𝑩)𝑝are nonnegative for all 𝑝∈ℕ. This link with real algebraic geometry ignited a new stage of research, based on sum-of-squares hierarchies and semidefinite programming, that generated new evidence supporting the BMV conjecture. Finally, in 2012, Herbert Stahl [Sta13] established Fact 8.2 using classic m... | https://arxiv.org/abs/2501.16578v1 |
−−1 2√ 1−𝑠𝔼Tr 𝒁e−𝒀𝑠 C➀−➁. (8.11) We start with the second term ➁, involving the Gaussian random matrix, as it offers a template for how to bound the first term ➀. 30 COMPARISON FOR RANDOM PSD MATRICES To handle the term ➁from(8.11), we invoke Gaussian integration by parts. In preparation, expand the random matri... | https://arxiv.org/abs/2501.16578v1 |
𝒁∼normal(0,V)where VB𝑘·Mom[𝑾]. (9.2) With these definitions, we can state another comparison theorem. Theorem 9.1 (Gaussian comparison: Sum of iid psd matrices) .Fix an arbitrary self-adjoint matrix 𝚫∈ℍ𝑑. Introduce the random 𝑑-dimensional matrices 𝒀and𝒁from(9.1)and(9.2). Then 𝔼𝜆min(𝒀−𝔼𝒀+𝚫)≥𝔼𝜆min(𝒁+𝚫)... | https://arxiv.org/abs/2501.16578v1 |
and uniformly at random in 𝑛bins. Recall that 𝛿𝑖∼binomial(1/𝑛,𝑘)and∑︁𝑛 𝑖=1𝛿𝑖=𝑘. Because of the coupling between the distributions, b𝒀𝑛=∑︁𝑘 𝑗=1b𝑾𝑗=∑︁𝑛 𝑖=1𝛿𝑖𝑨𝑖. (9.4) Let us emphasize that the coefficient vector 𝜹is independent from the sample A𝑛. 9.4.Step 3: Poissonization. The coefficients 𝛿𝑖i... | https://arxiv.org/abs/2501.16578v1 |
Tr e−𝜃𝒁𝑛 A𝑛 . (9.8) It remains to relate the random matrices b𝒀𝑛and𝒁𝑛to the target models 𝒀and𝒁. 9.6.Step 5: Limits. At this stage, we can unfreeze the random sample A𝑛and take limits. This process will produce the bound 𝔼Tr e−𝜃(𝒀−𝔼𝒀)≤2𝔼Tr e−𝜃𝒁. (9.9) The random matrices 𝒀and𝒁are defined in (9.1)a... | https://arxiv.org/abs/2501.16578v1 |
𝑛. (9.11) Conditional on the event D𝑛occurring, the distribution of the empirical approximation b𝒀𝑛is the same as the distribution 𝒀. More precisely, for each Borel set B⊆ℍ𝑑, ℙn b𝒀𝑛∈B D𝑛o =ℙn∑︁ 𝑖∈supp(𝜹(𝑛))𝑨𝑖∈B D𝑛o =ℙn∑︁𝑘 𝑗=1𝑾𝑗∈Bo =ℙ{𝒀∈B}. Indeed, each sample 𝑨𝑖is an independent draw from the dist... | https://arxiv.org/abs/2501.16578v1 |
that 𝔼∥𝑾∥2is finite. Recall that a sequence of random matrices converges weakly if and only if the characteristic functions converge pointwise to a limit that is continuous at the origin [Dud02, Thm. 9.8.2]. Therefore, to prove the weak convergence statement (9.13), it suffices to verify that 𝜒𝒁𝑛(𝑴)→𝜒𝒁(𝑴)for e... | https://arxiv.org/abs/2501.16578v1 |
Lemma A.2 (Exponential Paley–Zygmund: log-mgf bound) .Let𝑾be a random psd matrix with two finite moments. For 𝜃≥0, we have the semidefinite relation log𝔼e−𝜃𝑾≼𝜃(𝔼𝑾)+1 2𝜃2(𝔼𝑾2). Proof.Recall the numerical inequality e−𝑎≤1−𝑎+𝑎2/2, valid for𝑎≥0. By the transfer rule [Tro15, Prop. 2.1.4], the inequality exten... | https://arxiv.org/abs/2501.16578v1 |
Shao. Hashing embeddings of optimal dimension, with applications to linear least squares. 2021. arXiv: 2105.11815 [math.NA] . [Cha07] S. Chatterjee. “Stein’s method for concentration inequalities”. Probab. Theory Related Fields 138.1-2 (2007), pages 305–321. doi:10.1007/s00440-006-0029-y . [CGS11] L. H. Y. Chen, L. Gol... | https://arxiv.org/abs/2501.16578v1 |
on the field with an eye to software”. 2023. arXiv: 2302.11474 [math.NA] . [NN13] J. Nelson and H. L. Nguyen. “OSNAP: Faster numerical linear algebra algorithms via sparser subspace embeddings”. In: FOCS’13: Proc. 2013 IEEE 54th Ann. Symp. Foundations of Computer Science . IEEE Computer Soc., Los Alamitos, CA, 2013, pa... | https://arxiv.org/abs/2501.16578v1 |
Consistent support recovery for high-dimensional diffusions Dmytro Marushkevych∗Francisco Pina†Mark Podolskij‡ January 29, 2025 Abstract Statistical inference for stochastic processes has advanced significantly due to applications in diverse fields, but challenges remain in high-dimensional settings where parameters ar... | https://arxiv.org/abs/2501.16703v1 |
growing importance of high-dimensional data has introduced new complexities to statistical modeling. Researchers have explored scenarios where the number of model parameters far exceeds the available observations or where most pa- rameters exhibit specific asymptotic behavior, departing from the classical approach that... | https://arxiv.org/abs/2501.16703v1 |
demonstrated the adaptive Lasso’s ability to achieve variable selection and asymptotic normality under specific conditions. Research on the adaptive Lasso in the context of diffusion processes remains relatively sparse. While prior studies, such as [14] and [22], investigate variable selection within multidimensional f... | https://arxiv.org/abs/2501.16703v1 |
sign( θi) = 1 if θi>0, sign(θi) =−1 ifθi<0, and sign( θi) = 0 if θi= 0. For a set S ⊂ 1, . . . , d and a vector x∈Rd, we denote xS∈Rcard(S), which contains only the components xjforj∈ S;SCdenotes the complement of S. Given a matrix M∈ M a×b(R),M⋆denotes its transpose, Mjthej-th row, and Mithei-th column. For a subset G... | https://arxiv.org/abs/2501.16703v1 |
the relevant functions ϕjin (1.3) that correspond to the non-zero coefficients θj 0. Conversely, CSCS Tcaptures the correlation between the relevant and irrelevant functions ϕjin the model. To quantify these terms, we define the following quantities: L:= max 1≤j≤p−s (CSCS ∞)j 2, M:= max 1≤j≤p Cjj ∞ , with their roles d... | https://arxiv.org/abs/2501.16703v1 |
θ0. 6 We consider the following assumptions: Assumption ( B1):The initial estimator eθisrT-consistent for the estimation of certain unknown vector of constants η0depending on θ0, i.e. rT∥eθ−η0∥∞=OP(1), T→ ∞ , and for some constants M1,TandM2,Tthe vector η0satisfies max j /∈S|ηj 0| ≤ M 2,Tand X j∈S1 |ηj 0|+M2,T |ηj 0|2... | https://arxiv.org/abs/2501.16703v1 |
a standard condition in this context, serving to control the behavior of the noise. This restriction is crucial for deriving the required concentration inequalities. For further details, including examples that guarantee the validity of this assumption and guidance on selecting the constant Kbased on the specific model... | https://arxiv.org/abs/2501.16703v1 |
Assume (A1)-(A3),(B1)-(B3)and(C)hold, and let s2 T:=α⋆(CSS ∞)−1αfor any vector α∈Rssatisfying ∥α∥2≤1. Then √ Ts−1 Tα⋆(bθS−θ0,S) =√ Ts−1 Tα⋆(CSS ∞)−1ϵT,S+oP(1)→DN(0,1), (2.7) where oP(1)is a term that converges to zero in probability uniformly with respect to α, if λ2TM2 1,T s2 Tτ2 min+slogs√ TsT√ K+M+sK τ4 min →0. (2... | https://arxiv.org/abs/2501.16703v1 |
of support recovery guarantees and error bounds. While the linear drift assumption is a cornerstone of this work, the methodology could be extended to accommodate more complex nonlinear drift structures, akin to the framework explored in [13]. In the nonlinear setting, the key idea would involve employing a localized l... | https://arxiv.org/abs/2501.16703v1 |
is relevant as a pre- estimator for the adaptive Lasso defined in (2.1). It is rT-consistent for θ0, and the proxy η0is identical to θ0. Consequently, Assumption ( B1) holds with M2,T= 0, and, as noted in Remark 2.2, the adaptive irrepresentable condition in Assumption ( B2) is automatically satisfied. However, assumin... | https://arxiv.org/abs/2501.16703v1 |
ε >0, P rT∥eθ−η0∥∞> ε ≤P rTmax 1≤j≤p|sX i=1θi 0 Cij T−Cij ∞ |> ε/2! +P rTmax 1≤j≤p|ϵj T|> ε/2 ≤pmax 1≤j≤pP rT|sX i=1θi 0 Cij T−Cij ∞ |> ε/2! +pmax 1≤j≤pP rT|ϵj T|> ε/2 ≤pmax 1≤i,j≤pP Cij T−Cij ∞ >ε 2sθmax 0,SrT! +pmax 1≤j≤pP |ϵj T|>ε 2rT ≤6pexp −Tε2 144Ks2(θmax 0,S)2r2 T! + 2pexp −Tε2 8(√ K+M)r2 T! + 6pexp... | https://arxiv.org/abs/2501.16703v1 |
in proving that the adaptive Lasso accurately identifies the true support of the parameter in high-dimensional diffusion settings. Several key results regarding concentration inequalities for additive functionals in dif- fusion processes are discussed in works such as [1, 4, 18, 43, 44]. These results are estab- lished... | https://arxiv.org/abs/2501.16703v1 |
broader scope of Assumption ( C′). For Ornstein-Uhlenbeck (OU) type processes, an alternative route to concentration inequalities involves Malliavin calculus techniques. These rely on the properties of the Malliavin derivative, which satisfies the classical chain rule and offers a rigorous framework for studying the re... | https://arxiv.org/abs/2501.16703v1 |
asymptotic normality of the adaptive Lasso estimator across different time horizons. Inspired by the numerical study in [4] and using the YUIMA package [30], we simulate ad-dimensional diffusion process as described in (1.1), where the linear drift function is defined by: bθ0(x) = 3 sx+pX i=1θi 0cos (( i+ 1)x), (5.1) w... | https://arxiv.org/abs/2501.16703v1 |
parameter dimension is set at p= 30. The estimators are evaluated over varying observation times across 25 iterations. The results indicate that the MLE consistently exhibits a higher error rate, showing minimal improvement in support recovery as the observation time increases. Conversely, the Lasso-based techniques de... | https://arxiv.org/abs/2501.16703v1 |
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