text string | source string |
|---|---|
the leading order term satisfies Eq. 40, g−1 ∞=z−σ2g∞. 67 Exercise 2: The isolated eigenvalue and eigenvector projection The goal of this exercise is to derive the expressions for the isolated eigenvalue and for the eigen- vector projection, Eqs (45) and (46). (i) Show that if Ais a matrix and v,uare vectors, then (A+u... | https://arxiv.org/abs/2502.14084v1 |
Mezard and Andrea Montanari. Information, physics, and computation . Oxford University Press, 2009. [5]Iain M Johnstone. On the distribution of the largest eigenvalue in principal components analysis. The Annals of statistics , 29(2):295–327, 2001. [6]Larry Wasserman. All of statistics: a concise course in statistical ... | https://arxiv.org/abs/2502.14084v1 |
microscopy of kardar- parisi-zhang superdiffusion. Science , 376(6594):716–720, 2022. [25]Quentin Fontaine, Davide Squizzato, Florent Baboux, Ivan Amelio, Aristide Lemaître, Martina Morassi, Isabelle Sagnes, Luc Le Gratiet, Abdelmounaim Harouri, Michiel Wouters, et al. Kardar–parisi–zhang universality in a one-dimensio... | https://arxiv.org/abs/2502.14084v1 |
of the spherical sherrington–kirkpatrick model. Journal of Statistical Mechanics: Theory and Experiment , 2021(7):073301, 2021. [46]Anthony Perret and Gregory Schehr. The density of eigenvalues seen from the soft edge of random matrices in the gaussian β-ensembles. Acta Physica Polonica B , 46(9), 2015. [47]Pedro H Pim... | https://arxiv.org/abs/2502.14084v1 |
Applied Mathematics , 72(11):2282–2330, 2019. 73 [68]Valentina Ros, Felix Roy, Giulio Biroli, Guy Bunin, and Ari M Turner. Generalized lotka-volterra equa- tions with random, nonreciprocal interactions: The typical number of equilibria. Physical Review Letters , 130(25):257401, 2023. [69]Valentina Ros, Felix Roy, Giuli... | https://arxiv.org/abs/2502.14084v1 |
parisi’s tree. Journal de Physique I , 5(3):265–286, 1995. [91]Véronique Gayrard. Aging in metropolis dynamics of the rem: a proof. Probability Theory and Related Fields , 174(1):501–551, 2019. [92]Bernard Derrida. Random-energy model: An exactly solvable model of disordered systems. Physical Review B, 24(5):2613, 1981... | https://arxiv.org/abs/2502.14084v1 |
Conformal Prediction under Lévy–Prokhorov Distribution Shifts: Robustness to Local and Global Perturbations Liviu Aolaritei1, Zheyu Oliver Wang2, Julie Zhu2, Michael I. Jordan1,3, and Youssef Marzouk2 1Department of Electrical Engineering and Computer Sciences, UC Berkeley, USA liviu.aolaritei@berkeley.edu, jordan@cs.b... | https://arxiv.org/abs/2502.14105v2 |
broader scenarios, including discrete and transport-based perturbations [7]. For example, LP metrics can cap- ture local shifts such as minor variations in image textures or sensor readings, as well as global shifts like changes in population demographics. This dual capability makes LP metrics particularly suited for r... | https://arxiv.org/abs/2502.14105v2 |
framework for modeling uncertainty in the data distribution. For instance, [9] used an 𝑓-divergence ambiguity set around the training distribution to derive worst-case coverage guarantees and adjusted prediction sets. This work is most closely related to ours, and while their analysis inspired our approach, we rely on... | https://arxiv.org/abs/2502.14105v2 |
, (1) where Γ(P,Q)is the set of all joint probability distributions over 𝒵×𝒵, with marginals PandQ, often called transportation plans or couplings [39]. Moreover, the Total Variation (TV) distance is defined as TV(P,Q) := inf 𝛾∈Γ(P,Q)∫︁ 𝒵×𝒵1{‖𝑧1−𝑧2‖>0}d𝛾(𝑧1, 𝑧2). (2) At first sight, definition (2) might seem ... | https://arxiv.org/abs/2502.14105v2 |
where we study the propagation of the ambiguity set B𝜀,𝜌(P)thorough the scoring function 𝑠, showing that the LP distribution shift can be directly considered in the one-dimensional nonconformity scores. To provide more insights into the LP ambiguity set, we begin by presenting an alternative repre- sentation that de... | https://arxiv.org/abs/2502.14105v2 |
ambiguity set in the one-dimensional scores, due to its immediate relationship with the cumulative distribution functions and quantiles. The following proposition shows that the result of the propagation of B𝜀,𝜌(P)through the scoring function 𝑠is again captured by a an LP ambiguity set, allowing us to effectively re... | https://arxiv.org/abs/2502.14105v2 |
(Case 𝜌≥1−𝛽).If𝜌≥1−𝛽, then QuantWC 𝜀,𝜌(𝛽;P) =Quant (1;P). Intuitively, the LP ambiguity set B𝜀,𝜌(P)allows to displace 𝜌mass from the distribution Pand move it arbitrarily in R. 7 <latexit sha1_base64="yGTaCchiS2V8hdiUb4BomnzfbDA=">AAAB9HicbVDLSgMxFL1TX7W+qi7dBIvgqsyIqMuCIC4r2Ae0Q8mkt21oJjMmmUIZ+h1uXCji1o9x59+... | https://arxiv.org/abs/2502.14105v2 |
coverage and the size of the prediction set: it shifts the quantile level from 1−𝛼to1−𝛼+𝜌, and appears subtractively in the coverage bound. This change in quantile level often has a more pronounced effect on the size of the prediction set than the additive 𝜀 term, particularly when the score distribution is light-t... | https://arxiv.org/abs/2502.14105v2 |
a small ResNet architecture from scratch; and for iWildCam, we adopt the pre-trained ResNet-50 model provided by [6]. The code to reproduce all experiments is available at our GitHub repository.1 5.1 Data-Space Distribution Shift: MNIST and ImageNet Following the split conformal procedure, we partition the hold-out val... | https://arxiv.org/abs/2502.14105v2 |
uses a fixed Lipschitz-based estimate for 𝜀, and LPest 𝜀, which estimates (𝜀, 𝜌) directly from data—consistently maintain valid coverage across all settings. They also achieve compa- rable prediction set sizes, demonstrating the effectiveness of data-driven parameter estimation. Among the remaining baselines, only ... | https://arxiv.org/abs/2502.14105v2 |
with limited test data. Taken together, these results support two key takeaways: (1) LP ambiguity sets flexibly model real distribution shifts, delivering valid coverage across a broad region of the parameter space, and (2) the estimated (𝜀, 𝜌)pair performs comparably to the best grid-tuned pair, both in coverage and... | https://arxiv.org/abs/2502.14105v2 |
PMLR, 2023. [16] A. Gendler, T.-W. Weng, L. Daniel, and Y. Romano. Adversarially robust conformal prediction. InInternational Conference on Learning Representations , 2021. [17] S. Ghosh, Y. Shi, T. Belkhouja, Y. Yan, J. Doppa, and B. Jones. Probabilistically robust conformal prediction. In Uncertainty in Artificial In... | https://arxiv.org/abs/2502.14105v2 |
Tchetgen Tchetgen. Prediction sets adaptive to unknown covariate shift.Journal of the Royal Statistical Society Series B: Statistical Methodology , 85(5):1680–1705, 2023. 15 [36] K. Rahmani, R. Thapa, P. Tsou, S. C. Chetty, G. Barnes, C. Lam, and C. F. Tso. Assessing the effects of data drift on the performance of mach... | https://arxiv.org/abs/2502.14105v2 |
to smaller robust prediction sets Figure 5: ImageNet (𝜀, 𝜌)estimation . Each point in the 20-point grid corresponds to a candidate (𝜀, 𝜌)pair, where 𝜀∈(0.5,1.5)and𝜌is estimated using one-dimensional optimal transport between the empirical calibration and test score distributions, each constructed from 1000 sample... | https://arxiv.org/abs/2502.14105v2 |
respect the desired 1−𝛼coverage level. RemarkA.1 (Sensitivityto 𝜀and𝜌).Itisnaturaltoaskhowsensitivethemethodistomisspecification of the shift parameters (𝜀, 𝜌). While both influence the prediction set, their effects are asymmetric. The parameter 𝜀appears additively in the worst-case quantile and controls the widt... | https://arxiv.org/abs/2502.14105v2 |
estimated via a separately trained classifier. 5.Randomly Smoothed Conformal Prediction [16]: Quant(︁ (1−𝛼)(2 + 𝑛)/(1 +𝑛);̃︀P𝑛)︁ +𝛿/𝜎,̃︀P𝑛:=1 𝑛𝑛∑︁ 𝑖=1𝛿˜𝑠(𝑋𝑖,𝑌𝑖). Here, ˜𝑠denotes the smoothed nonconformity score based on an existing score function 𝑠[16], under which adversarial noise with ‖𝜖‖2≤𝛿is pr... | https://arxiv.org/abs/2502.14105v2 |
. Finally, thelastequalityfollowsfromthefactthat 𝒜𝑐={‖𝑧1−𝑧2‖> 𝜀}. ThisshowsthatTV (̃︀P,Q)≤𝜌, and concludes the proof. Proof of Corollary 2.2. Assertion (i) follows from (7) by setting 𝜀to zero, resulting iñ︀P=P. Moreover, assertion (ii) follows from (7) by setting 𝜌= 0, resulting iñ︀P=Q. Proof of Proposition ... | https://arxiv.org/abs/2502.14105v2 |
of Section 3 Proof of Proposition 3.4. We prove the proposition in two steps. First, we show that the right-hand side in (12) is an upper bound on the 𝛽-quantile of any distribution in B𝜀,𝜌(P). Second, we prove that there exists a sequence of distributions Q𝑛∈B𝜀,𝜌(P), whose 𝛽-quantiles converge to it. Step 1. We... | https://arxiv.org/abs/2502.14105v2 |
that this leads to LP𝜀(P,̃︀Q) = inf 𝛾∈Γ(P,̃︀Q)∫︁ R×R1{|𝑧1−𝑧2|> 𝜀}d𝛾(𝑧1, 𝑧2)> 𝜌. From the inequality in (23), we know that there exists Δ>0such that 𝐹̃︀Q(𝑞)≤𝐹P(𝑞−𝜀)−(𝜌+ Δ) . Meanwhile, for any coupling 𝛾∈Γ(P,̃︀Q), we have 𝜌+ Δ≤𝐹P(𝑞−𝜀)−𝐹̃︀Q(𝑞) =∫︁𝑞−𝜀 −∞∫︁∞ −∞𝑑𝛾(𝑧1, 𝑧2)−∫︁∞ −∞∫︁𝑞 −∞𝑑𝛾(𝑧1, �... | https://arxiv.org/abs/2502.14105v2 |
arXiv:2502.14407v1 [math.ST] 20 Feb 2025Sharp Phase Transitions in Estimation with Low-Degree Polynomials Youngtak Sohn∗1and Alexander S. Wein†2 1Division of Applied Mathematics, Brown University 2Department of Mathematics, UC Davis Abstract High-dimensional planted problems, such as finding a hidden dense su bgraph wit... | https://arxiv.org/abs/2502.14407v1 |
. . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.4 Example: Planted Submatrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.5 Other Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... | https://arxiv.org/abs/2502.14407v1 |
. . . . . . . . . . . . . . 23 4 Planted Dense Subgraph 24 4.1 Orthonormal Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.2 Removing “Bad” Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.3 Constructing u. . . . . . . . . . . . . . . . . . . . . . . . ... | https://arxiv.org/abs/2502.14407v1 |
. . . . . . . . . . . . . . . . 45 7.2 Planted Dense Subgraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 7.3 Spiked Wigner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 7.4 Stochastic Block Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... | https://arxiv.org/abs/2502.14407v1 |
and each pair of communities can have a different connection probability. In each of these models we will be assuming an asymptotic regi me wheren→ ∞, and the other parameters (such as ρ,p0,p1,m) may scale with nin some prescribed way or may be designated as fixed “constants” (such as π,q) that do not depend on n. Asympt... | https://arxiv.org/abs/2502.14407v1 |
be directly related to its vector analogu e; see Section 2.2. If the quantity MMSE ≤Dis “small” (appropriately defined), we will say degree- Dpolynomials succeed at the estimation task, and if MMSE ≤Dis “large” then we will say degree- Dpolynomials fail. The degree Dserves as a measure of an algorithm’s complexity, and ... | https://arxiv.org/abs/2502.14407v1 |
a reliable predicti on for the true computational threshold, as discussed in, e.g., [YZZ24]. Our work is the first to simult aneously capture sharp thresholds and rule out super-logarithmic degree in estimation. Specifica lly, our results will rule out polynomials of degreenδfor a constant δ>0, sometimes for the optimalc... | https://arxiv.org/abs/2502.14407v1 |
using a de tection-to-recovery reduction. 5 1.1 Our Contributions 1.1.1 Example: Planted Submatrix We now illustrate our results in more detail, focusing on the planted submatrix problem. To recap the setting, a principal submatrix of size roughly ρn×ρnwith elevated mean λis planted in an n×nsymmetric Gaussian matrix. ... | https://arxiv.org/abs/2502.14407v1 |
these models. Existing low-degree lowe r bounds show hardness of the asso- ciated hypothesis testing problem below these thresholds [ HS17, KWB19, BBK+21], but no such results were known for estimation and this was stated as an op en problem by Hopkins and Steurer in one of the first papers on the low-degree framework [... | https://arxiv.org/abs/2502.14407v1 |
bound on Corr≤D, and the main difficulty is the factor E[f(Y)2] in the denominator, since Ydoes not have independent entries. We will make use of the und erlying independent random variables from which Yis generated, for instance Zandθin the planted submatrix problem. Choose a basis {φα}α∈IforR[Y]≤D(polynomials in the en... | https://arxiv.org/abs/2502.14407v1 |
polynomials in the underlying ind ependent random variables. Compared to [Wei23], our approach is a generalization (the ψβmay not form a basis) and simplification (we only need to construct usolvingu⊤M=c⊤rather than a left-inverse for M). While [Wei23] studies a different problem from us (tensor decomposition, wh ich doe... | https://arxiv.org/abs/2502.14407v1 |
if there exists some edge ( i,j), possibly a self-loop ( i,i). We call αconnected if every pair of vertices in V(α) has a path connecting them. The empty graph α= 0 is considered to be connected. We write α=1(i,j)for the graph consisting of a single edge between vertices iandj. As above, we may abuse notation and ident... | https://arxiv.org/abs/2502.14407v1 |
We will improve on this by using both the signal and noise. S pecifically, we will set the free variables in such a way to zero out the term in parentheses in (3). As a result, we will have simply uα0=cαfor allα∈ˆI. This avoids some difficult-to-control recursive blowup of theuvalues that is present in [SW22]. Our specific... | https://arxiv.org/abs/2502.14407v1 |
will be called x. We will state our results in terms of Corr≤D, as defined in Section 1.2.1: Corr≤D:= sup f∈R[Y]≤DE[f(Y)·x]/radicalbig E[f(Y)2]·E[x2]= sup f∈R[Y]≤D/vextendsingle/vextendsingleE[f(Y)·x]/vextendsingle/vextendsingle /radicalbig E[f(Y)2]·E[x2]∈[0,1]. To show low-degree hardness of estimation, our goal will b... | https://arxiv.org/abs/2502.14407v1 |
away from the t hreshold as the degree increases beyond that. 13 In the other regime, γ <1/2, the AMP threshold can be beaten by a very simple algorithm: thresholding the diagonal entries of the observed matrix gi ves exact recovery of θ(with high prob- ability) provided λ≫1 (where throughout we use ≫to hide a polylog(... | https://arxiv.org/abs/2502.14407v1 |
a constant C≡C(ǫ)>0for which the following holds. 14 (a) (Lower Bound) If λ≤(1−ǫ)(ρ√en)−1/radicalbig 1−ρandD≤λ−2/C then Corr≤D≤C/radicalbiggρ 1−ρ. (b) (Upper Bound) If λ≥(1+ǫ)(ρ√en)−1, ρ=ω(n−1log12n), ρ=o(log−12n), p0=ω(n−1log24n) then Corr≤Clogn= 1−o(1)asn→ ∞. Note that this matches our result for planted submatrix (T... | https://arxiv.org/abs/2502.14407v1 |
and estimation problems coincide in diffi culty, both transitioning from hard to easy at ρ≈1/√n. More precisely, there is an AMP algorithm for exact recovery above ρ= 1/√en[DM15], but there are in fact better poly-time algorithms th at reach ρ=ǫ/√nfor an arbitrary constant ǫ >0 [AKS98]; so there is not actually a sharp t... | https://arxiv.org/abs/2502.14407v1 |
first studied in random matrix th eory, where a sharp phase transition at λ= 1 was discovered in the eigenvalues and eigenvectors of Y[P´ ec06, FP07, Ma¨ ı07, CDF09, BN11]. This is known as the “BBP” transition, by analo gy to the similar phase transition discovered by Baik, Ben Arous, and P´ ech´ e in the related spike... | https://arxiv.org/abs/2502.14407v1 |
case using self-avoiding walks. We expect that our estimator can be mad e into a poly-time algorithm using the “color coding” trick [AYZ95], as in [HS17]. Alternative ly, poly-time weak recovery based on eigenvectors might be deduced from the spectral analysis by [Hua18]. More sophisticated poly-time algorithms that ai... | https://arxiv.org/abs/2502.14407v1 |
nD/summationdisplay t=1(dλ2)t. Consequently, if dλ2≤1andD≤nδ, thenCorr≤D=o(1). (b) (Upper Bound [AS18, HS17]) If q,π,Qare fixed with dλ2>1, then for large enough nwe haveCorr≤Clogn≥ηfor some constants C≡C(q,π,Q)>0andη≡η(q,π,Q)>0. Remark 2.9. Although Theorem 2.8 is stated only for the sparse SBM with a co nstant number ... | https://arxiv.org/abs/2502.14407v1 |
and Steurer in one of the first papers on the low-degree framework [HS17]. We resolve this, proving that degree- nδpolynomials fail to achieve weak recovery below the sharp KS threshold dλ2≤1. We show this for a particular constantδ>0 but we expect the result to hold for any constant δ <1, as suggested by [DHSS25], and ... | https://arxiv.org/abs/2502.14407v1 |
uβγ= 0 for allβγ /∈ˆJ. For the second condition, fix α /∈ˆI. The corresponding ˆ αis the connected component of vertex 1 in α. (If vertex 1 is isolated in αthen ˆα= 0.) The value µis equal to E[Hα−ˆα(Y)]. Using independence between components, cα=E[Hˆα(Y)·Hα−ˆα(Y)·x] =E[Hα−ˆα(Y)]·E[Hˆα(Y)·x] =µcˆα. Ifβγ∈ˆJis such that β... | https://arxiv.org/abs/2502.14407v1 |
on the terms that do not involve k. We will use the standard bound/parenleftbign k/parenrightbig ≤/parenleftbigen k/parenrightbigkfor allk≥1, as well as the assumption λ≤(1−ǫ)(˜ρ√en)−1. Forv≥2, bv:=/parenleftbiggn−1 v−1/parenrightbigg vv−2λ2(v−1)˜ρ2(v−1)≤/parenleftbigge(n−1) v−1/parenrightbiggv−1 (vλ2˜ρ2)v−1≤/parenleft... | https://arxiv.org/abs/2502.14407v1 |
the second condition, fix α /∈ˆI. The corresponding ˆ αis the connected component of vertex 1 in α. (If vertex 1 is isolated in αthen ˆα= 0.) The value µis equal to E[φα−ˆα]. Using independence between components, cα=E[φˆα·φα−ˆα·x] =E[φα−ˆα]·E[φˆα·x] =µcˆα. Here, we used the fact that V(α−ˆα)∩(V(ˆα)∪{1}) =∅in the second... | https://arxiv.org/abs/2502.14407v1 |
we will not need to use the full power of our new framework but can instead use the method of [SW22] as a starting point. In bounding the resulting formula, we will need to use some ar guments that are somewhat more delicate than those appearing in [SW22], in order to capture the sharp threshold. 27 5.1 Cumulant Formul... | https://arxiv.org/abs/2502.14407v1 |
conclusion holds for all goodβ/lessnotequalα. First, focus on one term in (11) and apply the induction hypo thesis: Tα,β:=/parenleftbiggλ n/parenrightbigg(|α|−|β|)/2 E|π|δ(α−β)·|κβ| ≤/parenleftbiggλ n/parenrightbigg(|α|−|β|)/2 E|π|δ(α−β)·/parenleftbiggλ n/parenrightbigg(|β|+1)/2 f(β)·M(δ(¯β)) =/parenleftbiggλ n/parenri... | https://arxiv.org/abs/2502.14407v1 |
some existing vertex is added, and again de gree-2 vertices can be placed along the two edges; for counting purposes, we think of this as a cas e of the previous operation where one of the two endpoints of the new edge is not an existing vertex b ut rather one of the future vertices to be added along the new edge. We c... | https://arxiv.org/abs/2502.14407v1 |
D≤nδwhereδ≡δ(c,ν)>0. Finally note that E[x2] =λm/nto complete the proof in the case m= 1. 32 5.3 General m Proof of Theorem 2.6(a) for general m.Now write κ(m) αto denote the cumulants for the rank- m case. Above, we have bounded κ(1) α. Recallκ(m) αis the joint cumulant (see Section 2.5 of [SW22]) κ(m) α=κ/parenleftBi... | https://arxiv.org/abs/2502.14407v1 |
step is to apply Lemma 1.4. We define ˆIandˆJas follows. Definition 6.3 (“Good” graphs for SBM) .We define the set ˆI ⊆ {0,1}([n] 2)of “good”αfor the SBM as follows. A graph αis considered “good” if either α=∅or ifαsatisfies all of the following: •1,2∈V(α), •¯α:=α+1(1,2)is connected, •for everyv∈V(α),deg¯α(v)≥2. We define ˆ... | https://arxiv.org/abs/2502.14407v1 |
uαγ=Mαγ,α/summationtext γ′∈[q]V(α)M2 αγ′,αdα, γ∈[q]V(α). (16) The intuition behind this choice of uαγis that given ( uβγ)βγ∈ˆJ:|β|<|α|, the choice (16) minimizes/summationtext γ∈[q]V(α)u2 αγby the Cauchy–Schwarz inequality. Consequently, by Propos ition 1.3, Corr≤D(k0,ℓ0)2≤ /bar⌈blu/bar⌈bl2=/summationdisplay α∈ˆI:|α|≤D... | https://arxiv.org/abs/2502.14407v1 |
=E/bracketleftBig E/bracketleftbig φα−β/vextendsingle/vextendsingleσ⋆ V(β)/bracketrightbig ·E/bracketleftbig φβψβγ/vextendsingle/vextendsingleσ⋆ V(β)/bracketrightbig/bracketrightBig , where the last equality holds because φα−βis conditionally independent from φβψβγgivenσ⋆ V(β). Note that letting W=V(α−β)∩V(β), we have ... | https://arxiv.org/abs/2502.14407v1 |
(i,j)∈α−1eQσ⋆ i,σ⋆ j/bracketrightBig . Sinceα−1eis another tree with one fewer edge, the desired conclusion h olds by induction. With Lemma 6.12 in hand, we lower bound/summationtext γM2 αγ,αas follows. Lemma 6.13. Consider any non-empty graph α∈ {0,1}([n] 2), and letKbe its number of connected components. Writing Qmin... | https://arxiv.org/abs/2502.14407v1 |
that if αis a path with end points W={i1,i2}, the sum in the RHS of Eq. (25) equals (B|α|)i1,i2. To deal with a general tree α, we consider the set of branching points Vb:={v∈V(α) : degα(v)≥3}. Forv∈Vb∪W\{ρ}, consider the uniqueshortest path starting from vand endingat ¯ p(v)∈Vb∪W, ¯p(v)/\⌉}a⊔io\slash=v, where ¯p(v) is... | https://arxiv.org/abs/2502.14407v1 |
πmin/parenrightbigg2|α|−2|V(α)|+|W|+1 , which concludes the proof. Proof of Proposition 6.9. Letα1,...,α Kbe the connected components of αand letWi≡W∩ V(αi). By assumption, Wi/\⌉}a⊔io\slash=∅. Since (φαi(Y),σ⋆ Wi)i∈[K]are independent, we have max τ∈[q]W/vextendsingle/vextendsingleE/bracketleftbig φα(Y)/vextendsingle/ve... | https://arxiv.org/abs/2502.14407v1 |
2n2k−w (2k−w)!(2k)! 2(k!)2(k+1)2(k−1)/bracketrightbigg2 ≤nw/bracketleftbiggn2k+2−w 4(2k)w (k!)2(k+1)2(k−1)/bracketrightbigg2 =/parenleftbigg4k2 n/parenrightbiggwn4k+4 16(k!)−4(k+1)4(k−1) = (1+o(1))/parenleftbigg4k2 n/parenrightbiggwn4k+4 161 4π2k6e4k+4 = (1+o(1))/parenleftbigg4k2 n/parenrightbiggw (64π2k6)−1(en)4k+4, 4... | https://arxiv.org/abs/2502.14407v1 |
(1+o(1))e5nρ2(enλρ)2D/summationdisplay ℓ,m,b/parenleftbiggηρ+1 enλ2ρ2/parenrightbiggℓ/parenleftbigge2(D+2)3ρ enρ2/parenrightbiggm/parenleftbigge2(D+2)3ρ(η+1) ηρ+1/parenrightbiggb−m ≤(1+o(1))Ne5k6nρ/summationdisplay ℓ,m,b(1−δ)ℓ/parenleftbigge(D+2)3 nρ/parenrightbiggm/parenleftbigge2(D+2)3ρ(η+1) ηρ+1/parenrightbiggb−m fo... | https://arxiv.org/abs/2502.14407v1 |
0 p0(1−p0)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleθi= 1,∀i∈˜B =ρ|V(α△β)|λ|α△β|E /productdisplay (i,j)∈α∩β/parenleftbig (1−2p0)ηθiθj+1/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleθi= 1,∀i∈˜B . Taking absolute values on both sides, we can bound |... | https://arxiv.org/abs/2502.14407v1 |
variable s (1Ut=Ut+1)1≤t≤|γ|−1are independent and distributed according to Bernoulli(1 /m). Therefore, the RHS equals m|γ|(1+(K−1)/m)|γ|−1, which concludes the proof. 52 Lemma 7.5. Letα,β∈ SDand denote v:=|V(α)∩V(β)|andℓ:=|α∩β|. Then, 0≤E[Yα+β]≤(Km)v−ℓ−1·/parenleftbigg 1+Kmλ n/parenrightbiggℓ/parenleftbiggλ n/parenrigh... | https://arxiv.org/abs/2502.14407v1 |
First, we choose the two paths in ( α∩β)+that contain vertex 1 and vertex 2 respectively (these are distin ct, sinceℓ<D). Givenα, these paths are determined by their lengths, which are non-negative and sum to at most ℓ, so the number of ways to choose the paths is at most/parenleftbigℓ+2 2/parenrightbig . Second, we ch... | https://arxiv.org/abs/2502.14407v1 |
n/parenrightbiggD ·∞/summationdisplay ℓ=0(ℓ+1)/parenleftbigg1 1+η/2/parenrightbiggℓ = (1+o(1))Km/parenleftbiggλ n/parenrightbiggD ·(1+2η)2 η2. Thus, using these bounds in (37) shows It follows that E[f(Y)2]≤/parenleftbig 1+o(1)/parenrightbig m/parenleftbiggλ n/parenrightbiggD/parenleftBigg K(1+2η)2 η2+n m/parenleftbigg... | https://arxiv.org/abs/2502.14407v1 |
(JACM), 42(4):844–856, 1995. [BABP16] Jo¨ elBun,RomainAllez, Jean-PhilippeBouchaud, andMarcPotters. Rotational invari- ant estimator for general noisy matrices. IEEE Transactions on Information Theory , 62(12):7475–7490, 2016. [BB20] Matthew Brennan and Guy Bresler. Reducibility and st atistical-computational gaps from... | https://arxiv.org/abs/2502.14407v1 |
in Mathematics , 227(1):494–521, 2011. [BWZ23] G´ erard Ben Arous, Alexander S Wein, and Ilias Zadik . Free energy wells and overlap gap property in sparse PCA. Communications on Pure and Applied Mathematics , 76(10):2410–2473, 2023. [CDF09] Mireille Capitaine, Catherine Donati-Martin, and Delphine F´ eral. The largest... | https://arxiv.org/abs/2502.14407v1 |
P´ ech´ e. The largest ei genvalue of rank one deformation of large Wigner matrices. Communications in mathematical physics , 272:185–228, 2007. [FR18] Alyson K Fletcher and Sundeep Rangan. Iterative reco nstruction of rank-one matrices in noise. Information and Inference: A Journal of the IMA , 7(3):531–562, 2018. [Ga... | https://arxiv.org/abs/2502.14407v1 |
limit theorems for indecomposable multidi- mensional Galton-Watson processes. Ann. Math. Statist. , 37:1463–1481, 1966. [KWB19] Dmitriy Kunisky, Alexander S Wein, and Afonso S Band eira. Notes on computational hardness of hypothesis testing: Predictions using the low- degree likelihood ratio. In ISAAC Congress (Interna... | https://arxiv.org/abs/2502.14407v1 |
clique problem to ran k-one perturbations of gaussian tensors. Advances in Neural Information Processing Systems , 28, 2015. [MSS23] Elchanan Mossel, Allan Sly, and Youngtak Sohn. Exac t phasetransitions for stochastic block models and reconstruction on trees. In Proceedings of the 55th Annual ACM Symposium on Theory o... | https://arxiv.org/abs/2502.14407v1 |
Community detection in sparse random networks. Annals of Applied Probability , 25(6):3465–3510, 2015. 63 [Wei23] Alexander S Wein. Average-case complexity of tenso r decomposition for low-degree polynomials. In Proceedings of the 55th Annual ACM Symposium on Theory of Com- puting, pages 1685–1698, 2023. [WEM19] Alexand... | https://arxiv.org/abs/2502.14407v1 |
NADARAYA-WATSON TYPE ESTIMATOR OF THE TRANSITION DENSITY FUNCTION FOR DIFFUSION PROCESSES NICOLAS MARIE†AND OUSMANE SACKO† †Université Paris Nanterre, CNRS, Modal’X, 92001 Nanterre, France. Abstract. This paper deals with a nonparametric Nadaraya-Watson (NW) estimator of the transition densityfunctioncomputedfromindepe... | https://arxiv.org/abs/2502.14498v1 |
bfℓ(x)1bfℓ(x)>m 2; (x, y)∈R2, where m∈(0,1], bsh,t(x, y) :=1 N(T−t0)NX i=1ZT t0Qh(Xi s−x, Xi s+t−y)ds andbfℓ(x) =1 N(T−t0)NX i=1ZT t0Kℓ(Xi s−x)ds. In this paper, risk bounds are established on bph,ℓ,tand on the adaptive estimator bpbh,bℓ,t(x, y) =bsbh,t(x, y) bfbℓ(x)1bfbℓ(x)>m 2; (x, y)∈R2, wherebh(resp.bℓ) is selected... | https://arxiv.org/abs/2502.14498v1 |
T], the joint density of (Xs, Xs+t)is denoted by ps,s+t, and since X is a homogeneous Markov process, (10) pt(ξ, ζ) =ps,s+t(ξ, ζ) ps(x0, ξ);∀(ξ, ζ)∈R2. Then, E(bsh,t(x, y)) =1 T−t0ZT t0Z R2Kh1(ξ−x)Kh2(ζ−y)ps,s+t(ξ, ζ)dξdζds =Z∞ −∞Kh1(ξ−x)(Kh2⋆ pt(ξ,·))(y)f(ξ)dξ − − − − → h2→0Z∞ −∞Kh1(ξ−x)pt(ξ, y)f(ξ)dξ− − − − → h1→0pt(... | https://arxiv.org/abs/2502.14498v1 |
where Nh0⩾1. Moreover, consider h0= (h0, h0), (13) bh=bh(t) := arg min h∈H2 N{∥bsh,t−bsh0,t∥2+ pen( h)} with pen(h) =2 (T−t0)2N2NX i=1*ZT t0Qh(Xi s− ·, Xi s+t− ·)ds, ZT t0Qh0(Xi s− ·, Xi s+t− ·)ds+ ;∀h∈ H2 N, and (14) bℓ:= arg min ℓ∈HN{∥bfℓ−bfh0∥2+ pen†(ℓ)} with pen†(ℓ) =2 (T−t0)2N2NX i=1*ZT t0Kℓ(Xi s− ·)ds,ZT t0Kh0(Xi... | https://arxiv.org/abs/2502.14498v1 |
(CIR): Xi t=∥Ui t∥2 2,dwith d= 6andr=γ= 1. This is the so-called Cox-Ingersoll- Ross model. Here, the transition density function is given by p(3) t(x, y) =ctexp(−ct(xe−rt+y))y xe−rtd 4−1 2Id 2−1,2ctp xye−rt , where ct:=2r γ2(1−e−rt), andI(p, x)is the modified Bessel function of the first kind of order pat point x(... | https://arxiv.org/abs/2502.14498v1 |
References [1] Aït-Sahalia, Y. (1999). Transition Densities for Interest Rate and Other Nonlinear Diffusions. The Journal of Finance LIV, 1361-1395. [2] Comte, F. (2017). Estimation non-paramétrique. Spartacus IDH. [3] Comte, F. and Genon-Catalot, V. (2020). Nonparametric Drift Estimation for I.I.D. Paths of Stochastic... | https://arxiv.org/abs/2502.14498v1 |
Rosier, A. (2023). Nadaraya-Watson Estimator for I.I.D. Paths of Diffusion Processes. Scandinavian Journal of Statistics 50, 2, 589-637. [20] Massart, P. (2007). Concentration Inequalities and Model Selection. Springer, Berlin-Heidelberg. [21] Menozzi, S., Pesce, A. and Zhang, X. (2021). Density and Gradient Estimates ... | https://arxiv.org/abs/2502.14498v1 |
and (17), and since T−t0⩾1, for any (ξ, ζ)∈R2, Z∞ −∞f(y)pt(h1ξ+x, h2ζ+y)2dy⩽∥f∥∞c2 1 tZ∞ −∞exp −2m1(h2ζ+y−h1ξ−x)2 t dy ⩽2c3 1 tZ∞ −∞exp −2m1y2 t dy ⩽c2 t0with c2= 2c3 1Z∞ −∞exp −2m1y2 T dy. So, by Marie and Rosier [19], Corollary 1, b1(ξ, ζ)⩽c2 t0Z∞ −∞(f(h1ξ+x)−f(x))2dx⩽c3 tq+1 0h2 1(ξ2+|ξ|3), where c3andqare pos... | https://arxiv.org/abs/2502.14498v1 |
constant mΦ>0, not depending ont, such that: (1) For every h= (h1, h2)∈(0,1]2andφ∈C0([0, T]), ∥Φh,t(φ;·)∥2⩽mΦ h1h2. (2) For every h,l∈(0,1]2, E(⟨Φh,t(X1;·),Φl,t(X2;·)⟩2)⩽mΦsl,t, where sl,t:=E(∥Φl,t(X;·)∥2). N-W ESTIMATOR OF THE TRANSITION DENSITY FUNCTION FOR DIFFUSION PROCESSES 15 (3) For every h∈(0,1]2andφ∈L2(R2), E(... | https://arxiv.org/abs/2502.14498v1 |
A.3, for any λ >0andθ∈(0,1), with probability larger than 1−5.4|HN|2e−λ, |ψ1(h)|⩽θ 2Nsh,t+2cA.3(1 +λ)3 θN and|ψ1(bh)|⩽θ 2Nsbh,t+2cA.3(1 +λ)3 θN. •For any h,l∈ H2 N, consider ψ2(h,l) :=1 NNX i=1⟨Φh,t(Xi;·), sl,t⟩. By Proposition A.2.(4), |ψ2(h,l)|⩽mΦ. Moreover, since st∈L2(R2)by Inequality (7), |⟨sh,t, sh0,t⟩|⩽∥Qh⋆ st∥ ... | https://arxiv.org/abs/2502.14498v1 |
four kernel type properties of the map Φh,t,h∈ (0,1]2, stated in Proposition A.2. First of all, note that by Inequalities (5) and (6), (24) ∥st∥∞=∥ptf∥∞⩽c1:=mp(t0, T)mf(t0, T). (1) For every h= (h1, h2)∈(0,1]2andφ∈C0([0, T]), by Jensen’s inequality and the change of variable formula, ∥Φh,t(φ;·)∥2=Z R2 1 T−t0ZT t0Qh(φ(s... | https://arxiv.org/abs/2502.14498v1 |
Quantum computer formulation of the FKP-operator eigenvalue problem for probabilistic learning on manifolds Christian Soizea,∗, Lo¨ıc Joubert-Doriola, Artur F. Izmaylovb aUniversit ´e Gustave Eiffel, MSME UMR 8208, 5 bd Descartes, 77454 Marne-la-Vall ´ee, France bUniversity of Toronto Scarborough, Department of Physica... | https://arxiv.org/abs/2502.14580v1 |
the 1980s, numerous papers have been published, covering all aspects necessary for developing quantum computing techniques to address large-scale simulations in engineering science, including data science. This effort requires the development of suitable quantum computing algorithms, software advancements, quantum erro... | https://arxiv.org/abs/2502.14580v1 |
7. Finally, Section 8 addresses the construction of eigenstates and the extraction of overlaps through quantum measurements, evaluated using universal quantum gates. Key steps in the proposed methodology. Hereinafter, we present methodological aspects. Although presented in the context of the FKP-operator eigenvalue pr... | https://arxiv.org/abs/2502.14580v1 |
PY(dy)on the measurable set (Rm,BRm)(we will simply say on Rm). The Lebesgue measure on Rmis noted dyand when PY(dy)is written as pY(y)dy,pYis the probability density function (pdf) on RmofPY(dy)with respect to dy. Finally, Edenotes the mathematical expectation operator that is such that E{Y}=R RmyPY(dy). 2. Probabilis... | https://arxiv.org/abs/2502.14580v1 |
has a covariance matrix equal to the identity matrix. Under these conditions, the available information is only the training dataset of H. Using this dataset, the probability density function pHofHis constructed by using the modification [27] of the GKDE method [28]. Thus, the only available information is represented ... | https://arxiv.org/abs/2502.14580v1 |
where Φ(y)=−log(ξ(y))is such that Φ(y)=−log1 ndndX j=1exp −1 2ˆs2∥y−ˆs sηj∥2!,∀y∈Rν. (2.8) 3. igenvalue problem of the FKP operator and a representation adapted to quantum computing algorithm This section summarizes essential results concerning the Fokker-Planck (FKP) operator and its associated eigen- ... | https://arxiv.org/abs/2502.14580v1 |
defined by Eqs. (3.13) and (3.14) can be rewritten in qas ˆLFKP(q)=λq, (3.17) with the condition at infinity, lim ∥y∥→+∞pH(y)∥∇(pH(y)−1/2q(y))∥=0. (3.18) 6 3.4. Schr ¨odinger-type formulation of the FKP operator ˆLFKP Using Eq. (2.7), which shows that pH(y)−1∇H(y)=−∇Φ, and using Eqs. (3.9) and (3.16), it can be seen th... | https://arxiv.org/abs/2502.14580v1 |
Rν. Such a construction is not straightforward for the following reasons: (i) The problem is set in high dimensions, meaning that νis large. (ii) The sequence of polynomials chosen from the selected subset of all the polynomials on Rνmust be complete in the adapted vector space, and the convergence must be fast with re... | https://arxiv.org/abs/2502.14580v1 |
function, which is the transient solution of the Fokker-Plank equation defined by Eq. (3.8) and estimated by solving the It ˆo stochastic differential equation defined by Eqs. (3.1) and (3.2). It should be noted that in this construction, we introduced a scaling change in the transient anisotropic kernel to connect the... | https://arxiv.org/abs/2502.14580v1 |
Rνand the function h belongs to H, E{|h(Y)|2}=∥h∥2 H<+∞. (4.7) PROOF . (Proposition 1). The continuity of functions f,g, and his simple to be prove. (i) Let pH(y)=cνξ(y)be defined by Eq. (2.7) where ξ(y)=1 ndPnd j=1exp(−1 2bs2∥y−bs sηj∥2). For all yinRν, we have 0< ξ(y)≤1and(logpH(y))2=(logcν+logξ(y))2≤2(log cν)2+2(log... | https://arxiv.org/abs/2502.14580v1 |
(4.11) We have hα(0)=E{h(Y)}and∥h∥2 H=P α∈Nνh2 α<+∞. PROOF . (Proposition 2). We use simultaneously Proposition 1 and Definition 1. (i) Since fis inHand since{ψα,α∈Nν}is a Hilbert basis of H, this yields (i). (ii) Since gjis inH, for all j∈{1,...,ν},fhas a partial derivative with respect to yjinH. Consequently, gj= ∂jf... | https://arxiv.org/abs/2502.14580v1 |
Eq. (3.30) of the representation gj=P α∈Nνgj,αψαandh=P α∈Nνhαψαestablished in Proposition 2- (ii) and (iii). Here, gj,αandhα, which are defined by Eqs. (4.10) and (4.11), are derived from the coefficients fαof the polynomial chaos expansion f=P α∈Nνfαψαestablished in Proposition 2-(i). Consequently, we introduce the tr... | https://arxiv.org/abs/2502.14580v1 |
α)1/2.(4.23) 4.4. Algebraic formulas for computation and convergence analyses In this Section, we propose a numerical method for the effective calculation of the coefficients fαof the polyno- mial chaos expansion. We summarize the calculation process for gj,α,hα, and gαβ, and address the questions related to convergenc... | https://arxiv.org/abs/2502.14580v1 |
right-hand side of Eq. (4.29) as a sum of monomials, which is straightforward. Letψαbe the normalized Hermite polynomial of multi-index αdefined by Eq. (4.8), where αbelongs to A1,µor A2,µas defined by Eq. (4.16). For jfixed in{1,...,ν}, letαjbe the component jofα=(α1,...,αν), and the Hermite polynomial Hαj(yj)for the ... | https://arxiv.org/abs/2502.14580v1 |
α)2}1/2. (5.4) This choice of functions defined by Eqs. (5.2) to (5.4) allows us to control the convergence of all the coefficients of the polynomial chaos expansions. For this Gaussian reference case, we will also use the GKDE representation. In that case, these functions also depend on nd, and we will consider these ... | https://arxiv.org/abs/2502.14580v1 |
Has a function of ndfor a fixed value of ν. We know that the larger νis, the larger ndmust be. However, in the context of the probabilistic learning on manifolds approach, the eigenfunctions associated with the first eigenvalues of the Fokker-Planck operator that we are looking for are only used to construct a vector b... | https://arxiv.org/abs/2502.14580v1 |
thereby representing wavefunctions and operations as qubits. While much of the research community is focused on electronic structure problems [50], new efforts are emerging to adapt other quantum prob- lems for quantum computation [13, 14, 51]. For completeness, we depict here the general approach as follows: (i) choic... | https://arxiv.org/abs/2502.14580v1 |
k−a+ ka− j=δjk, one can express∇2and all monomials in the so-called normal ordering (with creation operators on the left of the annihilation operators) as follows, ∇2=νX j=1ωj 2 (a+ j)2+(a− j)2−2a+ ja− j−1 , (6.2) yα=νY j=1αj!q (2ωj)αj⌊αj/2⌋X s=0αj−2sX p=0(2s−1)!! (αj−2s−p)! (2s)!p!(a+ j)αj−2s−p(a− j)p, (6.3) 18 wher... | https://arxiv.org/abs/2502.14580v1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.