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0 0 0 RW-TRAE LossDyDg DeepHit 5 0.176 0.175 0.175 0.015 0.014 0.021 RW-TRAE SUMO DeepHit 5 0.301 0.301 0.3 0.0033 0.0033 0.0034 RW-TRAE Standard GBSA 5 0.15 0.15 0.15 0.06 0.06 0.06 RW-TRAE Standard RSF 5 0.191 0.191 0.19 0.03 0.034 0.038 RW-TRAE LossDyDg SurvSurf 5 0.155 0.155 0.155 0 0 0 RW-Property Standard CoxPH 1... | https://arxiv.org/abs/2504.04997v1 |
in the price of existing detached properties at various locations since 2015. The color bar is centered at 20%, capped at 10% and 30%. Fig. 11: Test-set prediction: predicted probability for reaching a maximum percentage price increase of 20% by 1, 2, and 5 years since 2015. Results shown are for existing detached prop... | https://arxiv.org/abs/2504.04997v1 |
with the ability to handle missing intermediate events, makes SurvSurf compatible to many real-world applications. Finally, this study only used baseline (i.e., not time-dependent) features for prediction. The structure of our model, however, does not prohibit the inclusion of time-dependent features. Extensions of thi... | https://arxiv.org/abs/2504.04997v1 |
concluded that∂z1 ∂t≥0. Similarly, as Ak is point-wise non-negative (so are αkandγk),∂zk+1 ∂t≥0 when k= 1 and hence for allk >1 as well. Therefore∂Mraw ∂t≥0.□ Theorem 3 (monotonicity in g)∂Mraw ∂g≤0 Proof This proof proceeds in a similar way to that for Theorem 2. By the chain rule,∂zk ∂g=∂σk ∂uk·∂uk ∂gwhere uk=αkt+γk⋄... | https://arxiv.org/abs/2504.04997v1 |
of Statistical Software , 50(14):1 – 24, 2012. doi: 10.18637/jss.v050.i14. URL https://www.jstatsoft. org/index.php/jss/article/view/v050i14 . H. Ghasemi, E. Shahrabi Farahani, M. Fotuhi-Firuzabad, P. Dehghanian, A. Ghasemi, and F. Wang. Equipment failure rate in electric power distribution networks: An overview of con... | https://arxiv.org/abs/2504.04997v1 |
URL https://onlinelibrary.wiley.com/doi/abs/10.1155/2015/793161 . P. Warwick. Economic trends and government survival in west european parliamentary democracies. American Political Science Review , 86(4):875–887, 1992. ISSN 0003-0554. doi: 10.2307/1964341. URL https://www. cambridge.org/core/product/E838D3BA59914C79EAE... | https://arxiv.org/abs/2504.04997v1 |
same model trained by initializing with different random seeds. Points closer to the bottom-left corner represent models with better performance and less monotonicity violation. The denser set of gconsists of the observed gs in each trajectory and equally spaced gs between the maximum observed gin each trajectory and t... | https://arxiv.org/abs/2504.04997v1 |
0.00 0.00 -0.02 0.01 0.18 0.07 0.07 0.08 0.10 0.07 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -0.05 -0.26 -0.38 -0.42 -0.29 -0.23 -0.11 -0.02 0.01 0.08 0.06 0.11 0.07 0.10 0.15 0.11 0.10 0.10 0.10 0.10 0.00 0.00 0.00 0.00 0.00 0.00 -0.05 -0.09 -0.16 -0.18 -0.18 -0.18 -0.17 -0.1... | https://arxiv.org/abs/2504.04997v1 |
Detecting relevant dependencies under measurement error with applications to the analysis of planetary system evolution Patrick Bastian Ruhr-University BochumNicolai Bissantz Ruhr-University Bochum Abstract Exoplanets play an important role in understanding the mechanics of plane- tary system formation and orbital evol... | https://arxiv.org/abs/2504.05055v1 |
of Jupiter and extremely short orbital periods lasting only a few days as they typically orbit their parent star at rather short distances. While the possibility of the existence of such planets had already been considered by Struve (1952) they have not been predicted by planet system formation models, thereby pointing... | https://arxiv.org/abs/2504.05055v1 |
ways, each accounting for a deficiency of the previous setup. First we note that one often only observes a noisy version Z=X+ϵof the desired quantities, here ϵis a noise term with a known distribution, the details of which we discuss further below. Ignoring the measurement error results in unreliable inference and so f... | https://arxiv.org/abs/2504.05055v1 |
are a straightforward but very cumbersome matter and are therefore omitted as they shed little additional insight into the nature of the problem we consider. Further Related Literature: We first give some general references on deconvolution and its theoretical properties. Several nonparametric estimators for fare avail... | https://arxiv.org/abs/2504.05055v1 |
regarding a parameter θthat can be expressed as the expected value of a U-statistic of the latent sample (( X1, Y1), ...,(Xn, Yn)). For the sake of notational brevity we will restrain ourselves to U-statistics of order 2, extension of the results to higher orders is a straightforward but cumbersome matter. Recall that ... | https://arxiv.org/abs/2504.05055v1 |
unknown density fwhile the second part is satisfied whenever ∥(X, Y)∥has finite (2 + δ) moments for some δ >0. Equation (8) in Assumption (A4) is similar to Assumption 2 in Bissantz et al. (2007) and can therefore be considered as a technical refinement of (7). It holds, 8 for instance, for the Laplace distribution. As... | https://arxiv.org/abs/2504.05055v1 |
(A1) to (A5) as well as (B1) and (B2). Then lim n→∞P(ˆTn,∆≥q∗ 1−α) = 0 θ <∆ α θ = ∆ 1 θ >∆(9) Note that the quantile q∗ 1−αdoes not depend on the choice of ∆, combined with the fact that the statistic Tn,∆is a monotone function of ∆ we therefore have that for ∆1>∆2the implication ˆTn,∆1> q∗ 1−α=⇒ˆTn,∆2> q∗ 1−α ho... | https://arxiv.org/abs/2504.05055v1 |
1.11.4 Virtanen et al. (2020) with a grid size of 512x512 for model 1 and 1024x1024 for model 2, where model 2 requires a larger grid size due to its more complicated shape and the large difference of the variances in the covariance matrix T2. 12 2. Naively calculating the 4 dimensional integral in the definitions of ˆ... | https://arxiv.org/abs/2504.05055v1 |
estimators starts overfitting. We therefore choose the bandwidth as follows: Consider a log-spaced sequence h1, ..., h m of bandwidths and define ˆhopt= arg min 1≤hi≤m−1 ˆfn(hi)−ˆfn(hi+1) 2| {z } =:Di(hi) Figure 2 illustrates empirically that this choice coincides fairly well with the minimum of the global mean square ... | https://arxiv.org/abs/2504.05055v1 |
Bandwidths shown are for the simulations under H0(on which CI coverage rates are also based) andHA 1,HB 1, respectively. The method thus performs very well for the symmetric model and seems to have a slightly inflated size for the more complex radially asymmetric model as can been in particular from the H0column in tab... | https://arxiv.org/abs/2504.05055v1 |
determine whether or not further study might be worthwhile. As already mentioned in the introduction Hot Jupiters have fairly large masses com- bined with a small orbit around their parent star. These characteristic make them ideal targets for radial-velocity based detection methods as the induced variability of the ra... | https://arxiv.org/abs/2504.05055v1 |
sample. We instead opted to use the optimal regularization parameter for n= 100 in model 2, which assumes the same error distribution but assumes that the latent data are normally distributed. For real data the data distri- bution can be very irregular, either due to insufficient sample size or true physical characteri... | https://arxiv.org/abs/2504.05055v1 |
R2ky(x)(fn(x)−E[fn(x)])dx+oP(1) Lemma 5.4 and 5.5 then yield 2√nh1+βZ R2ky(x)(fn(x)−E[fn(x)])dx= 2√nhS n+oP(1). The proof is finished by combining the previous equations with Theorem 5.7. Lemma 5.1. Assume that (A1) to (A5) hold. Then |E[ˆθn]−θ|≲O(n−1h−2−2β n +h2) Proof. We first note that fn(x)fn(y) =1 nh2nX k=1Kn((x−... | https://arxiv.org/abs/2504.05055v1 |
empirical measure Fpo nhas the property that nFpo nis a poisson process on a plane, in particular we have the following properties (compare Rosenblatt (1975)). nkE(d(Fpo n−F))2k=(2k)! k! 2k(dF)k+k−1X j=1a(2k) j,n(dF)j, (13) n(2k+1)/2E(d(Fpo n−F))2k+1=kX j=1a(2k+1) j,n (dF)j, where a(s) j,n=O n−((s/2)−j) for each fixe... | https://arxiv.org/abs/2504.05055v1 |
fpo,∗ n=h−1(˜K∞⋆ F∗ n) Fpo,∗ n=N nF∗ N V∗ jk=Z Ijkky,n(x)Z R2˜K∞x−y h d(Fpo,∗ n−Fn)(y)dx S∗ n=Z R2ky,n(x)(fpo,∗(x)−E∗[fpo,∗(x)])dx 27 We also define for any measurable set Aa shorthand for the conditional probability and expectation given Z1, ..., Z nas follows P∗(A) =P(A|Z1, ..., Z n) E∗[A] =E[A|Z1, ..., Z n]. Proof... | https://arxiv.org/abs/2504.05055v1 |
consequence of the dominated convergence theorem. This yields the desired statement along the subsequence we took. We can argue like this along a subsub- sequence of any subsequence which yields the desired result by the metrizability of weak convergence. References Bastian, P., Dette, H., and Heiny, J. (2024). Testing... | https://arxiv.org/abs/2504.05055v1 |
deconvolution. Ann. Statist. , 27:2033–2053. Qiu, P. (2005). Image Processing and Jump Regression Analysis . Wiley-Interscience. Rosenblatt, M. (1975). A Quadratic Measure of Deviation of Two-Dimensional Den- sity Estimates and A Test of Independence. The Annals of Statistics , 3(1):1 – 14. Russo, F. and and, J. W. (20... | https://arxiv.org/abs/2504.05055v1 |
DDPM Score Matching and Distribution Learning Sinho Chewi Alkis Kalavasis Anay Mehrotra Omar Montasser Yale University Abstract Score estimation is the backbone of score-based generative models (SGMs), and particularly denoising diffusion probabilistic models (DDPMs). A fundamental theoretical result in this area is th... | https://arxiv.org/abs/2504.05161v1 |
. . . 3 1.2.1 DDPM is an asymptotically efficient parameter estimator . . . . . . . . . . . 4 1.3 Applications to density estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.1 PAC density estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.2 Minimax optimal density estimati... | https://arxiv.org/abs/2504.05161v1 |
. . . . . . . . . . . . . 22 2.3 Early stopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4 Application to estimating the differential entropy . . . . . . . . . . . . . . . . . . . . 23 3 DDPM is an asymptotically efficient parameter estimator 23 3.1 Implications of the likeliho... | https://arxiv.org/abs/2504.05161v1 |
. 54 B.3.2 CLWE decision problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 B.3.3 Hardness of CLWE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 C Auxiliary lemmas 56 C.1 Standard facts about sub-Gaussianity . . . . . . . . . . . . . . . . . . . . . . . . . . 56 C.2 Integral es... | https://arxiv.org/abs/2504.05161v1 |
0),(1) where C(P, X 0) is a constant that does not depend on st.1Since we can freely generate Zt(and thusXt) given X0, the right-hand side of the identity above is readily turned into an empirical loss that can be minimized over the choice of st. Despite a number of recent works investigating its efficacy [KLV24; KV24;... | https://arxiv.org/abs/2504.05161v1 |
cations for parameter recovery and density estimation. We introduce the notion of DDPM score estimation in Definition 1 and provide relevant background in Appendix A. Given the above landscape of computationally efficient reductions from density estimation and parameter estimation to score estimation, we can obtain sev... | https://arxiv.org/abs/2504.05161v1 |
work of Koehler, Heckett, and Risteski [KHR23] investigated the performance of a variant of score matching known as implicit score matching (ISM): Given i.i.d. samples X(1), . . . , X(n)from Pθ⋆, for some parameter θ⋆∈Θ, the ISM estimator is bθISM n:= arg min θ∈Θ1 nXn i=1 ∥∇logPθ(X(i))∥2+ 2 ∆ log Pθ(X(i)) . Under appr... | https://arxiv.org/abs/2504.05161v1 |
t)∥2+ ∇logPθ,t(X(i) t),2Z(i) t√ 1−e−2t X(i) 0i dt and for each i∈[n]andt∈[0, T], we draw Z(i) t∼N(0,Id)independently from X(i) 0and define the noised sample X(i) t:=e−tX(i) 0+√ 1−e−2tZ(i) t. Our main result for this application is that, under mild regularity assumptions on the distribution familyP(essentially the same ... | https://arxiv.org/abs/2504.05161v1 |
of points (according to the probability mass of the target distribution P), with high probability. Definition 3 (PAC density estimation algorithm) .LetPbe a class of distributions over Rd. An (ε, δ)-PAC density estimation algorithm for Pis an algorithm which, for any P∈P,given ε, δ > 0 andpoly(d,1/ε,1/δ)i.i.d. samples ... | https://arxiv.org/abs/2504.05161v1 |
to our work, such a result had only implications to generation; the next result converts such a guarantee to a density estimation result. We mention that for this section, we actually achieve a stronger guarantee compared to PAC density estimation: we give minimax optimal rates for estimation in the standard L1risk usi... | https://arxiv.org/abs/2504.05161v1 |
tools from algebraic geometry (tailored to GMMs) and it does not seem to extend to the more general distribution class of Gaussian location mixtures. Even providing a generator for such a general problem with qualitatively similar runtime when specialized to the GMM setting is a surprising algorithmic result. However, ... | https://arxiv.org/abs/2504.05161v1 |
this result, it suffices to apply our blueprint for GMMs: we must show that PAC density estimation for GMMs is computationally intractable under some standard complexity assumption. Following Bruna, Regev, Song, and Tang [BRST21], we show that CLWE reduces to it. To complete the reduction, we have to show that our “har... | https://arxiv.org/abs/2504.05161v1 |
Rd, and let ε >0be the desired accuracy. Assume access to a score estimation oracle with early stopping τand error ε∗. 1.Assume that Phas second moment bounded by M2≤poly(d)andL-Lipschitz score function. There is an algorithm that outputs a function bP(in the form of an evaluation oracle) such that Z E logbP(x0) P(x0) ... | https://arxiv.org/abs/2504.05161v1 |
score estimation, while the bound on the second moment is used to convert the integrated score to a PAC density estimator. For the proof of Item 2, it suffices to apply Item 1 with Pequal to Pτ, since its score is sub-Gaussian with parameter 1 /(1−e−2τ) (see Section 2.3). 1.6 Other related work Score estimation for gen... | https://arxiv.org/abs/2504.05161v1 |
ncan be done efficiently), and has a comparable statistical efficiency to MLE, while the MLE objective is intractable to optimize using gradient-based methods. Meanwhile, Chen, Kontonis, and Shah [CKS24] and Gatmiry, Kelner, and Lee [GKL24] used score estimation to establish new algorithmic results for generating sampl... | https://arxiv.org/abs/2504.05161v1 |
ζ≍ε/√ d, which trivializes the runtime guarantee, so it seems that new ideas are needed. Open problem 4. Can one boost the coverage probability δin the definition of PAC density estimation? A weakness of our reduction is that through the use of Markov’s inequality in our ( ε, δ)-PAC density estimation guarantee, εscale... | https://arxiv.org/abs/2504.05161v1 |
logPt(Xt) = ∂tlogPt(Xt)− ⟨∇logPt(Xt), Xt⟩+ ∆ log Pt(Xt) dt+√ 2⟨∇logPt(Xt),dBt⟩ = ∥∇logPt(Xt)∥2+ 2 ∆ log Pt(Xt) +d dt+√ 2⟨∇logPt(Xt),dBt⟩. Integrating over time and taking expectations, for ε >0, E logPT(XT)−logPε(Xε) =d(T−ε) +ZT εE ∥∇logPt(Xt)∥2+ 2 ∆ log Pt(Xt) dt , where we used the fact that {Rt ε⟨∇logPs(Xs),dBs... | https://arxiv.org/abs/2504.05161v1 |
about the sub-Gaussianity of the score along the OU process. This result is a key technical ingredient in our subsequent reduction and likely of independent interest. Next, in Section 2.2.3, we show that under mild assumptions onP, the score estimation oracle can be efficiently transformed to an integrated score estima... | https://arxiv.org/abs/2504.05161v1 |
y)P(x, y)µ(dx)R P(x, y)µ(dx)=E[∇logP(X,·)|Y=y]. Lemma 6 (Sub-Gaussianity of the score of a mixture) .In the setting of Lemma 5, suppose that for eachx∈X,∇logP(x,·)isσ2-sub-Gaussian under P(x,·). Then,∇logµPis also σ2-sub-Gaussian under µP. Proof. For any vector v∈Rd, Jensen’s inequality implies Eexp⟨v,∇logµP(Y)⟩=Eexp⟨v... | https://arxiv.org/abs/2504.05161v1 |
ti)∥ ∥bsti(xi ti)∥+∥∇logPti(xi ti)∥ P(dx0) ≤T−τ mZ EX i∈[m]∥bsti(xi ti)−∇logPti(xi ti)∥2P(dx0) +2 (T−τ) mZ EX i∈[m]∥bsti(xi ti)−∇logPti(xi ti)∥∥∇logPti(xi ti)∥P(dx0) ≤T−τ mEX i∈[m]ε2 ti+2 (T−τ) mEX i∈[m]εtiZ ∥∇logPti(xi ti)∥2Pti(dxi ti)1/2 ≲ε2 ∗+(T−τ)√ d mEX i∈[m]εtip Lti≲ε2 ∗+p L∗d ε∗. The second inequality follows ... | https://arxiv.org/abs/2504.05161v1 |
P(dx0) ≤2 (T−τ)√mZ EhsZ ∥∇logPt1(xt1)∥4Qt1|0(dxt1|x0) ×sZ ∥∇logQt1|0(xt1|x0)∥4Qt1|0(dxt1|x0)i P(dx0)1/2 ≲T−τ√m Ehd 1−exp(−2t1)ZsZ ∥∇logPt1(xt1)∥4Qt1|0(dxt1|x0)P(dx0)i1/2 ≤T−τ√m Ehd 1−exp(−2t1)sZZ ∥∇logPt1(xt1)∥4Qt1|0(dxt1|x0)P(dx0)i1/2 ≲(T−τ)d√m EhLt1 1−exp(−2t1)i1/2 ≲dp L∗,3T√m. In the above, the second and th... | https://arxiv.org/abs/2504.05161v1 |
with probability at least 9/10over the samples, it holds thatRT τ∥st−∇logPt∥2 L2(Pt)dt≤10ε2 ∗. Conditioned on this event, we can apply Theorem 2.2 and Markov’s inequality to deduce that EP{x∈Rd:bP(x)/∈[e−2ε/δP(x), e2ε/δP(x)]} ≤δ , i.e., it yields a (2 ε/δ, δ )-PAC density estimator. 2.2.5 Completing the reduction Combi... | https://arxiv.org/abs/2504.05161v1 |
focus on applications of our reduction from PAC density estimation. However, here we briefly mention that our guarantee immediately implies that a score estimation oracle can be used to estimate the differential entropy of the distribution P, which is also a well-studied problem (see [HJWW20] and references therein). T... | https://arxiv.org/abs/2504.05161v1 |
is well-defined. 3.2 The proof of Informal Theorem 1 We now proceed with the proof of Informal Theorem 1 regarding the asymptotic efficiency of DDPM score matching. Let Qt|0(· |x0) denote the transition density of the OU process run until time t started at time 0 at x0. Step 1 (Likelihood identity). As a first step, Le... | https://arxiv.org/abs/2504.05161v1 |
n ) ≤bRDDPM n (bθMLE n) +cd,T+Pnerr(bθDDPM n ) =−PnmbθMLEn+Pn[err(bθDDPM n )−err(bθMLE n)], where cd,Tis a constant and err(θ, x):=KL(QT|0(· |x)∥Pθ,T). Since erris non-negative, it yields PnmbθDDPMn≥PnmbθMLEn−Pnerr(bθDDPM n ). Since the DDPM estimator is consistent, Lemma 7 and our assumptions imply Pθ⋆err(bθDDPM n )≤2... | https://arxiv.org/abs/2504.05161v1 |
risk is defined as follows. Given an estimator bPusing nsamples and a probability density P, we define the L1risk Rn(bP, P):=Z [−1,1]EP|bP(x0)−P(x0)|dx0. Note that if bPwere a probability density on [ −1,1], this would correspond to twice the total variation distance. Here, we use the subscript on EPto indicate that th... | https://arxiv.org/abs/2504.05161v1 |
1/ε∗),i.e.,ε≍r∗ n√logn. Hence, we have the guarantee Z EP logbPτ(x0) Pτ(x0) Pτ(dx0)≲r∗ np logn . Recall that EPcorresponds to the expectation over the ni.i.d. samples used for the estimator bPτ. Since Pτ≳1 on [−1,1] (see [DKXZ24, Lemma 11]), it implies Z [−1,1]EP logbPτ(x0) Pτ(x0) dx0≲r∗ np logn . Since 1 /C′≤Pτ≤C′on [... | https://arxiv.org/abs/2504.05161v1 |
said to be (k, R, D, w min)-local if the following hold: •For every point xin the support of Q,Q(B(x, R))≥wmin. •There exist points x1, x2, . . . , x ksuch that the support of Pis a subset ofSk i=1B(xi, R). •Q(B(0, D)) = 1 . However, they left the problem of obtaining a density estimation algorithm for this family open... | https://arxiv.org/abs/2504.05161v1 |
ℓ, R, D and sample access to M, draws N= (d)O(ℓ)+O(log (dD)/ε)7samples from P, runs in poly(N)time, and outputs an (ε/δ, δ)-PAC density estimator for Mfor any coverage parameter δ∈(0,1). As noted in [GKL24], this is a family of distributions for which diffusion models can perform density estimation while standard metho... | https://arxiv.org/abs/2504.05161v1 |
in L2(Mti) for ζ2 i=ζ2(σ2+t+1) log(tN+1),i.e., ∥sti−∇logMti∥2 L2(Mti)≤ζ2 i. They needed the following properties on the time sequence ( t1, . . . , t N): 1.P1 (Start and end points): t1≍ζ2σ2 2√ dandtN≍d+M2 ζ2. 2.P2 (Recurrence): For each 1 ≤i≤N−1, tk+ 1 = ( tk+1+ 1)·max e−2α,(tk+1+ 1)−α where α:=ζ2 M2+dlogT+ 1. The ab... | https://arxiv.org/abs/2504.05161v1 |
all Ccalls implies that with probability 1 −δ, the construction in Theorem 2.1 outputs a valid ( ε, T)-integrated score estimation oracle as required. It remains to bound the total number of samples from Mused. Since each call to the score estimation in Theorem 5.2 requires dlogN δO (log1 ζ)7+(R σlog1 ζ)4 samples ... | https://arxiv.org/abs/2504.05161v1 |
the reader to Appendix B.3. In this section, we modify this reduction and show that PAC density estimation also implies an algorithm for hCLWE ; note that this is not immediate from the reduction in [BRST21] since PAC density estimation is an easier task than density estimation. We say that an algorithm Ais a ( c, ε, δ... | https://arxiv.org/abs/2504.05161v1 |
has the following properties: 35 1. If P=P0, then the tester outputs H0with probability at least c(1−δ)m. 2.IfP∈H1,dTV(P0, P)≥1/10,ε≤1/160, and δ≤1/80, then the tester outputs H1with probability at least c(1−(79/80)m). In particular, if c > 1/2and we take m,1/ε, and 1/δto be sufficiently large absolute constants, then ... | https://arxiv.org/abs/2504.05161v1 |
ratio. Let us assume that we have an ( ε, δ)-PAC density estimator bPforH(k)and write bP(x) Hw,β,γ(x)=bP(x) H(k)(x)·H(k)(x) Hw,β,γ(x). Using the above calculations, we know that dTV(H(k), Hw,β,γ)≤2 exp(−g(d))≤2 exp(−4π). Letε0∈(0,1) and B:={x∈Rd:|H(k)(x)/Hw,β,γ(x)−1| ≤ε0}. We know that Z H(k)(x) Hw,β,γ(x)−1 Hw,β,γ(dx)≤... | https://arxiv.org/abs/2504.05161v1 |
PAC density estimator for GMMs can be used to efficiently solve the CLWE problem. However, from [BRST21], we know that there is a polynomial-time quantum reduction from standard lattice problems such as GapSVP andSIVP [Reg09]10toCLWE . Note that we can only exclude eO(1/√ d)-accurate score estimation oracles for GMMs a... | https://arxiv.org/abs/2504.05161v1 |
that realizes the above reduction satisfies the assumptions of our density-to-score reduction with parameters L, M 2that arepoly(d, k).The key observation of Bruna, Regev, Song, and Tang [BRST21] is that hCLWE has a natural interpretation as an instance of mixtures of Gaussians. This mixture has infinitely many compone... | https://arxiv.org/abs/2504.05161v1 |
parameters. Hence, our Theorem 6.1 then implies that score estimation for Gaussian mixtures with eΩ(√ d) components and error smaller than 1/√dlogdwould 40 imply the existence of an efficient quantum algorithm that approximates worst-case lattice problems within polynomial factors, which is believed to be hard. The fol... | https://arxiv.org/abs/2504.05161v1 |
number of components k=d1/(2α)logd. 11Moreover, the result of Gupte, Vafa, and Vaikuntanathan [GVV22] provides hardness of CLWE under the classical (instead of quantum) worst-case hardness of GapSVP . 12Sparsity with parameter ∆ means that the secret vector of LWE orCLWE has exactly ∆ non-zero entries. 41 Acknowledgmen... | https://arxiv.org/abs/2504.05161v1 |
596–629 (cit. on p. 30). [BRST21] Joan Bruna, Oded Regev, Min Jae Song, and Yi Tang. “Continuous LWE”. In: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing . 2021, pp. 694–707 (cit. on pp. 6, 8, 11, 34, 35, 37–41, 53–55, 59). [BS10] Mikhail Belkin and Kaushik Sinha. “Polynomial learning of dis... | https://arxiv.org/abs/2504.05161v1 |
Ed. by Andreas Krause, Emma Brunskill, Kyunghyun Cho, Barbara Engelhardt, Sivan Sabato, and Jonathan Scarlett. Vol. 202. Proceedings of Machine Learning Research. PMLR, 2023, pp. 4735–4763 (cit. on p. 10). [CLT22] Tianrong Chen, Guan-Horng Liu, and Evangelos A. Theodorou. “Likelihood train- ing of Schr¨ odinger bridge ... | https://arxiv.org/abs/2504.05161v1 |
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between score matching and denoising autoencoders”. In:Neural Computation 23.7 (2011), pp. 1661–1674 (cit. on p. 4). [VJ08] Suriyanarayanan Vaikuntanathan and Christopher Jarzynski. “Escorted free energy simulations: improving convergence by reducing dissipation”. In: Physical Review Letters 100.19 (2008), p. 190601 (c... | https://arxiv.org/abs/2504.05161v1 |
process arising from OU. The forward process arising from the OU process is the following stochastic differential equation (SDE): dXt=−Xtdt+√ 2 dBt, X 0∼P , (10) where ( Bt)t≥0is a standard Brownian motion in Rd. The forward process transforms samples from the data distribution Pinto standard Gaussian noise N(0,1). Nam... | https://arxiv.org/abs/2504.05161v1 |
Definition 12 (LWE).Letε >0be the advantage13. For dimension d, number of samples m, modulus q=q(d)≥2, and error parameter σ=σ(d)>0, the average-case decision problem LWE q,σ is to distinguish with advantage εthe following two distributions over Zd q×R/qZusing mi.i.d. samples: (1) the LWE distribution As,σfor some unif... | https://arxiv.org/abs/2504.05161v1 |
(CLWE ), the continuous analogue of LWE. In what follows, we let ρσ(x) = exp( −π∥x∥2/σ2). (13) Following Bruna, Regev, Song, and Tang [BRST21], we also let ρ(x) = exp( −π∥x∥2). B.3.1 CLWE distributions We first define the CLWE distribution. Definition 16 (CLWE distribution [BRST21]) .For unit vector w∈Rdand parameters ... | https://arxiv.org/abs/2504.05161v1 |
Tang [BRST21] also show the hardness of hCLWE by reducing from CLWE . The idea of the reduction is to transform CLWE samples to hCLWE samples using rejection sampling. Theorem B.2 (Hardness of hCLWE [BRST21]) .For any β=β(d)∈(0,1)andγ=γ(d)≥2√ dsuch thatγ/βis polynomially bounded, there is a polynomial-time quantum redu... | https://arxiv.org/abs/2504.05161v1 |
a+ℓ/3],y∈[a+ 2ℓ/3, a+ℓ] such that|DkP(x)| ∨ |DkP(y)| ≤C′. Now, if |Dk+1P|is always large on [ a, a+ℓ], say Dk+1P≥C′′, this leads to a contradiction for C′′>6C′/ℓ, and thus the inductive step holds. Now we perform backward induction on kand argue that in fact |DkP| ≤C′on all of [ −1,1]. When sis an integer and k=s, this... | https://arxiv.org/abs/2504.05161v1 |
one of the most well-studied parametric distribution families for density estimation in particular and, also in statistics more broadly, with a history going back to the work of Pearson [Pea94] (also see the survey by Titterington, Smith, and Makov [TSM85] for applications for GMMs in the sciences). The study of statis... | https://arxiv.org/abs/2504.05161v1 |
(1) learning mixtures models and (2) robust statistics (see e.g., the book by Diakonikolas and Kane [DK23] and the works of [KMV10; MV10; HL18; BK20; DHKK20; LL22; LM22; BS23; ABBKS24]) and, as a result, is robust to a small number of outliers. Lower bounds for GMM density estimation. Regarding lower bounds, Diakonikol... | https://arxiv.org/abs/2504.05161v1 |
arXiv:2504.05300v1 [cs.LG] 7 Apr 2025Dimension-free convergence of diffusion models for approximate Gaussian mixtures Gen Li∗†Changxiao Cai∗‡Yuting Wei§ April 8, 2025 Abstract Diffusion models are distinguished by their exceptional gen erative performance, particularly in pro- ducing high-quality samples through iterativ... | https://arxiv.org/abs/2504.05300v1 |
Relating to score estimation error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.5 Step 4: Controlling GMM approximation error . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5 Discussion 15 ∗The authors contributed equally. Corresponding author: Yu ting Wei. †Department of Statistics, The Chine... | https://arxiv.org/abs/2504.05300v1 |
steps required for accu rate sample generation should scale at least linearly with the data dimension ( Chen et al. ,2022;Benton et al. ,2024) in order to learn the distribution accurately. While various acceleration schemes have been p roposed in literature (see, e.g. Li and Cai (2024); Li et al. (2024a );Li and Jiao ... | https://arxiv.org/abs/2504.05300v1 |
s⋆ t(x) :=∇logpXt(x),∀x∈Rd. (3) Given the complexity of developing a comprehensive end-to- end theory, a divide-and-conquer approach — pioneered by ( Chen et al. ,2022) — has become standard, separating the score learning phase (i.e., estimating score functions reliably from training data) from the gener ative sampling... | https://arxiv.org/abs/2504.05300v1 |
some logarithmic factors. Moreover, our result is rob ust to score estimation errors: the TV distance between the learne d distribution and the target distribution scales proportionally to the score estimation error, modulo logar ithmic factor. This leads to a surprising insight: Even in ultra-high-dimensional settings... | https://arxiv.org/abs/2504.05300v1 |
1.4 Notation For any a, the Dirac delta function δa(x)is defined as δa(x) =∞ifx=aandδa(x) = 0 otherwise. For positive integer N >0, let[N]:={1,···,N}. In addition, given any matrix A, we use /ba∇dblA/ba∇dbl,tr(A), and det(A)to denote the spectral norm, trace, and determinant of the ma trix, respectively. Next, we recall... | https://arxiv.org/abs/2504.05300v1 |
these score estimates in hand, the renown DDPM algorit hmHo et al. (2020) serves as a stochastic sampler that recursively generates s amples using the following update rule. Starting fromYT∼ N(0,Id), DDPM computes Yt−1via Yt−1=1√αt/parenleftbig Yt+(1−αt)/hatwidest(Yt)/parenrightbig +√1−αtZt, t=T,...,2. (9) 5 Here,Z2,..... | https://arxiv.org/abs/2504.05300v1 |
/hatwidest:=clip{st}for every t∈[T]. Here, the truncation function clip{x}:Rd→Rdis defined as clip{x}:=/braceleftBigg x,if/ba∇dblx/ba∇dbl2≤Cclip/radicalBig dlog(dT) 1−αt, 0,otherwise ,(13) withCclip>0being a sufficiently large absolute constant. We remark that t his truncation step is only introduced for technical conveni... | https://arxiv.org/abs/2504.05300v1 |
functions are concerned. Noteworthily, a recent line of literature (e .g.Li et al. (2023);Li and Yan (2024b );Li et al. (2024b )) enriches the toolbox of analyzing diffusion models by prov iding a framework of directly working with the TV distance. Intuition for efficient sampling of GMMs. Let us first provide some intuiti... | https://arxiv.org/abs/2504.05300v1 |
convergence results on DDPM. The runtime and sample complexity of the res ulting algorithms scale quasi-polynomially withK/εorlog(K/ε)depending on the covariance assumptions. Notably, the numb er of diffusion steps used in these two works still scales linearly with d. Our result serves as a complementary contribution to... | https://arxiv.org/abs/2504.05300v1 |
Th ese sequences, together with YTimplemented practically, form a Markov chain with the following transit ion structure: YT→Y− T→YT→Y− T−1→YT−1→ ··· → Y− 1→Y1→Y− 0→Y0. (29) •Initialization. Fort=T, we define Y− T:=/braceleftBigg YT,ifYT∈ ET, ∞,otherwise .(30a) The density of Y− Tsatisfies pY− T(y) =pYT(y)1/braceleftbig y... | https://arxiv.org/abs/2504.05300v1 |
control each term separately. 4.3 Step 2: Bounding discretization error In this section, we proceed to bound TV(pXGMM 1, pY1). Let us first define function ∆t(x) :Rd→R, where for eacht= 1,...,T : ∆t(x):=pXGMM t(x)−pYt(x),∀x∈Rd. (39) In view of relation ( 32), one has ∆t(x)≥0for allt≥0andx∈Rd. Applying the formula for the... | https://arxiv.org/abs/2504.05300v1 |
allx∈Rd; (ii) uses ( 47); in (iii) we define εGMM score,t:=/radicalBigg/integraldisplay Rd/vextenddouble/vextenddouble/hatwidest(xt)−s⋆GMM t(xt)/vextenddouble/vextenddouble2 2pXGMM t(xt)dxt; (50a) εGMM score:=/radicaltp/radicalvertex/radicalvertex/radicalbt1 TT/summationdisplay t=2(1−αt)/parenleftbig εGMM score,t/parenr... | https://arxiv.org/abs/2504.05300v1 |
the εdependence can not be further improved, it would be interesting to derive a matching lower bound to rigorously confirm the sharpness of our result. Finally, our analysis primarily addresses the s ampling phase, leaving open the question of how score estimation efficiency is affected by the structure of GMM s. It remai... | https://arxiv.org/abs/2504.05300v1 |
pXt−1(xt−1)−∆t−1(xt−1)+∆t→t−1(xt−1)≥pY− t−1(xt−1)+∆t→t−1(xt−1). (61) Here, we use the fact that pYt−1(xt−1) =pXGMM t−1(xt−1)∧pY− t−1(xt−1)sincepXGMM t−1(xt−1)> pYt−1(xt−1). To further control the right hand side, recall the definition of ∆t(x)in (39) and the constructed transition kernel ofpY− t−1|Ytin (33c). For any xt... | https://arxiv.org/abs/2504.05300v1 |
the right-hand-side of the above bound is controll ed by Lemma 6below. Lemma 6. Recall the definition of Etin(26). For any t∈[T], one has /integraldisplay Ec tpXGMM t(xt)dxt/lessorsimilarT−3. (70) Proof. See Appendix A.5. Putting everything together completes the proof of Claim ( 68). A.3 Proof of Lemma 4 Recall the defi... | https://arxiv.org/abs/2504.05300v1 |
81) together, we obtain (1−αt)/parenleftbig εGMM score,t/parenrightbig2/lessorsimilar(1−αt)ε2 score,t+εapprxdlog(dT)+T−1. Then the claim ( 54) in Lemma 4immediately follows from the definition of εGMM scorein (50). Proof of Claim (76).Define the set Hx:=/braceleftBig x0∈Rd:/vextenddouble/vextenddoublex−√αtx0/vextenddoubl... | https://arxiv.org/abs/2504.05300v1 |
2/vextenddouble/vextenddoublext−√αtµk/vextenddouble/vextenddouble2 2+1−α2 t 2α2 tK/summationdisplay i=1π(t) i/vextenddouble/vextenddoublext−√αtµi/vextenddouble/vextenddouble2 2−1−α2 t 2α2 t/vextenddouble/vextenddoubles⋆ t(xt)/vextenddouble/vextenddouble2 2+ζ(t) k(xt) (ii)=1 2/vextenddouble/vextenddoublext−√αtµk/vextend... | https://arxiv.org/abs/2504.05300v1 |
k=1πkN(√αtµk,Id), it is easily seen that P/braceleftBig tr/parenleftbig Id+Jt(XGMM t)/parenrightbig > C1log(KT)/bracerightBig =K/summationdisplay k=1πkP/braceleftBig tr/parenleftbig Id+Jt(Zk)/parenrightbig > C1log(KT)/bracerightBig ≤k/summationdisplay k=1πkP/braceleftBig tr/parenleftbig Id+Jt(Zk)/parenrightbig > C1log(... | https://arxiv.org/abs/2504.05300v1 |
2/parenrightBig +1−αt α2 t/parenleftbigg −x+K/summationdisplay i=1π(t) i√αtµi/parenrightbigg⊤/parenleftbiggK/summationdisplay i=1π(t) i√αt(µi−µk)/parenrightbigg =1−α2 t 2α2 tK/summationdisplay i=1π(t) i/parenleftBig −1 2αt/ba∇dblµi−µk/ba∇dbl2 2+(x−√αtµk)⊤√αt(µi−µk)/parenrightBig −1−αt α2 tK/summationdisplay i=1π(t) i/p... | https://arxiv.org/abs/2504.05300v1 |
=1 (1−αt)4E/bracketleftBig/vextenddouble/vextenddoubleE/bracketleftbig XGMM t−√αtXGMM 0|XGMM t/bracketrightbig/vextenddouble/vextenddouble4 2/bracketrightBig (i) ≤1 (1−αt)4E/bracketleftBig E/bracketleftbig/vextenddouble/vextenddoubleXGMM t−√αtXGMM 0/vextenddouble/vextenddouble4 2|XGMM t/bracketrightbig/bracketrightBig ... | https://arxiv.org/abs/2504.05300v1 |
arXiv preprint arXiv:2208.05314 . Dhariwal, P. and Nichol, A. (2021). Diffusion models beat gan s on image synthesis. Advances in neural information processing systems , 34:8780–8794. Diakonikolas, I. and Kane, D. M. (2020). Small covers for nea r-zero sets of polynomials and learning latent variable models. In 2020 IEE... | https://arxiv.org/abs/2504.05300v1 |
. Hyvärinen, A. (2005). Estimation of non-normalized statis tical models by score matching. Journal of Machine Learning Research , 6(4). Jiang, W. and Zhang, C.-H. (2009). General maximum likeliho od empirical Bayes estimation of normal means. The Annals of Statistics , 37(4):1647 – 1684. Kalai, A. T., Moitra, A., and ... | https://arxiv.org/abs/2504.05300v1 |
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