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statistic Ψmeasuring some dicrepancy betweenXandY, and reject the null if and only if Ψ(X,Y)>tfor some threshold t, which can also depend onX,Y and whose choice is part of the definition of the test. To evaluate the e ffectiveness of such a test, we consider the errors it may incur in distinguishing between the null an... | https://arxiv.org/abs/2502.13570v2 |
kernel Hilbert space (RKHS) with reproducing kernel κ:Z×Z→ Rand canonical feature map φ(x):=κ(x,·). We will impose the following assumption on the kernel κ. 4 Assumption 2.1: The kernel is bounded with supx∈Zκ(x,x) =∥κ∥∞<∞and measurable. It is a common assumption in the kernel testing literature (see, e.g., Choi et al.... | https://arxiv.org/abs/2502.13570v2 |
den- sity ratio (Kanamori et al. 2011) and kernel Stein discrepancies for goodness-of-fit tests (Huggins et al. 2018; Kalinke et al. 2024). Efficient kernel-based tests The main disadvantage of kernel-based tests is that computing the MMD scales quadratically with the number of samples n. In their seminal paper, Gretto... | https://arxiv.org/abs/2502.13570v2 |
aggregating multiple tests (Biggs et al. 2023; Fromont et al. 2012; Schrab et al. 2023, 2022) or using a Bayesian formalism (Zhang et al. 2022). In this work, we focus on the computational e fficiency of the test, yet our approach could easily be combined with such ideas when adaptation is needed. 3 An approximate MMD ... | https://arxiv.org/abs/2502.13570v2 |
we consider using multiplicative approximations of these scores. Definition 3.1 (AKRLS): Letδ∈(0,1],λ0>0 andz∈[1,∞). The scores ( ˆℓλ(i))i∈[n]∈Rnare said to be (z,λ0,δ)-approximate kernel ridge leverage scores (AKRLS) of Xif with probability at least 1−δ, for allλ≥λ0,i∈[1,...,n ]it holds1 zℓλ(i)≤ˆℓλ(i)≤zℓλ(i). Efficien... | https://arxiv.org/abs/2502.13570v2 |
under the assumption that the data are exchangeable under the null hypothesis. It is satisfied in particular when the data are i.i.d., as is the case here. Lemma 4.1: For anyα∈(0,1), the test described in Algorithm 3.1 with input level αhas exact levelα. (→Proof) By “exact”, we mean here that the inequality in (2) is a... | https://arxiv.org/abs/2502.13570v2 |
can be interpreted as a smooth estimate of the number of eigenvalues of CPthat are greater than λ. Under Assumptions 2.1 and 4.4, there exists a constantcγdepending on aγ,γand∥κ∥∞such that it holds max( deff(P,λ),deff(Q,λ))≤cγλ−γfor any λ>0 (see Chatalic et al. 2023, Lemma F.1). We now have the following result regardi... | https://arxiv.org/abs/2502.13570v2 |
and written in Python2. 2https://anonymous.4open.science/r/nystrom-mmd-0F7B/ 11 (a)Correlated Gaussians ,ρ2= 0.63 (b) Susy ,n= 16000 (c)Higgs ,n= 60000 (d) Correlated Gaussians ,n= 5000 (e)Susy (f)Higgs Figure 1: Power against computation time (top row) and power against correlation coe fficientρ2and total number of sa... | https://arxiv.org/abs/2502.13570v2 |
of the MMD as the test statistic. We provided a bound on the (non-asymptotic) power of the resulting test and demonstrated that it matches the minimax MMD-separation rates. Our procedure is simple to implement and compares favorably with existing state-of-the-art approaches, both theoretically and empirically. We leave... | https://arxiv.org/abs/2502.13570v2 |
Kernel Two-Sample Testing with Random Fourier Features”. In: arXiv preprint arXiv:2407.08976 . Chwialkowski, Kacper P ., Aaditya Ramdas, Dino Sejdinovic, and Arthur Gretton (2015). “Fast Two- Sample Testing with Analytic Representations of Probability Measures”. In: Advances in Neural Information Processing Systems 28.... | https://arxiv.org/abs/2502.13570v2 |
2003.13208 [math, stat] .url:http://arxiv.org/abs/2003.13208 . Pre- published. Kim, Ilmun and Antonin Schrab (2024). Differentially Private Permutation Tests: Applications to Kernel Methods . arXiv: 2310 . 19043 [cs, math, stat] .url:http : / / arxiv . org / abs / 2310 . 19043 . Pre-published. K¨ubler, Jonas M., Wittaw... | https://arxiv.org/abs/2502.13570v2 |
Aggregated Two-Sample Test”. In: Journal of Machine Learning Research 24.194, pp. 1–81. issn: 1533-7928. Schrab, Antonin, Ilmun Kim, Benjamin Guedj, and Arthur Gretton (2022). “E fficient Aggregated Ker- nel Tests Using Incomplete U-statistics”. In: Advances in Neural Information Processing Systems 35, pp. 18793–18807.... | https://arxiv.org/abs/2502.13570v2 |
in the proof. ( →Proof) Proof of Lemma A.1: We apply Rudi et al. (2015, Lemma 7) for Xwith probability δ/4. We get P ∥P⊥ ˜XC1/2 P∥≤p 3λX ≥1−δ/2 (13) and pickλX=19∥κ∥∞log(32nX/δ) nXandℓX= (nX)γ78cγz2(log32nX δ)1−γ (19∥κ∥∞)γ . These parameters correspond to the choice made in (Chatalic et al. 2023, Th.4.6) and thus sat... | https://arxiv.org/abs/2502.13570v2 |
≥1−α, (26) i.e. the r.h.s. of the above equation is an upper bound for the quantile q1−α,ˆUσ(X,Y). (→Proof) Proof of Lemma B.1: Let Σ2 nX,nY:=1 n2 X(nX−1)2X 1≤i,i′≤nX⟨ϕ(Xi),ϕ(X′ i)⟩2 +X 1≤j,j′≤nY⟨ϕ(Yj),ϕ(Yj′)⟩2+ 2X 1≤i≤nX,1≤j≤nY⟨ϕ(Xi),ϕ(Yj)⟩2(27) (28) 20 Given that ˜κ(x,y)2=⟨PZφ(x),PZφ(y)⟩2≤∥φ(x)∥2∥φ(y)∥2asPZis a pro... | https://arxiv.org/abs/2502.13570v2 |
Evaluability of paired comparison data in stochastic paired comparison models: Necessary and sufficient condition László Gyarmati, Csaba Mihálykó, Éva Orbán-Mihálykó, András Mihálykó February 20, 2025 Department of Mathematics, University of Pannonia, 8200 Veszprém, Hungary email: gyarmati.laszlo@phd.mik.uni-pannon.hu ... | https://arxiv.org/abs/2502.13617v1 |
from axiomatic prospects. The number of options was increased from 2 to 3 [31, 32], allowing equal decisions as well. The case of more than three options is considered in [33] applying least squares parameter estimation methods and in [34, 30] applying maximum likelihood estimation. Maximum likelihood estimation is a v... | https://arxiv.org/abs/2502.13617v1 |
in 3.2 the new result, i.e. a necessary and sufficient condition, which characterize the data set from the aspect of evaluability. The proof can be found in Section 6, Appendix. The proof of sufficiency is a development of the previously used lines of reasoning. In the proof of necessity, graph-theoretical and analytic... | https://arxiv.org/abs/2502.13617v1 |
Assuming independent decisions, the likelihood function is L(A|m1, ..., m n) =2Y k=1n−1Y i=1nY j=i+1pAi,j,k i,j,k. (5) The log-likelihood function is the logarithm of the above, logL(A|m1, m2, ..., m n) =2X k=1n−1X i=1nX j=i+1Ai,j,k·log(pi,j,k) =2X k=1nX i,j=1,i̸=j0.5·Ai,j,k·log(pi,j,k).(6) One can see that the likelih... | https://arxiv.org/abs/2502.13617v1 |
value of (17); that is (bm,bd) = arg max m∈Rn,m1=0,0<dlogL(A|m, d). (18) 5 Naturally, the maximal value is not always necessarily attained or the argument is not unique. Nev- ertheless, some conditions for the data can guarantee the existence of a maximum and uniqueness of its argument. Supposing logistic distributed c... | https://arxiv.org/abs/2502.13617v1 |
models pi,j,1=P(ξi−ξj<−d) =F(−d) =π π+π+ν·√π·π= = 1−π π+π+ν·√π·π=π+ν·√π·π π+π+ν·√π·π= 1−F(d),(35) which can be satisfied if and only if ν= 0. This contradicts 0< ν. Therefore, D3 does not coincide with any Thurstone motivated model. 3 Conditions of the existence and uniqueness of the maximizer In this section we presen... | https://arxiv.org/abs/2502.13617v1 |
Subsection 3.2 by C-NS. 3.1.3 Davidson’s model In [37], Davidson formulated the following set of conditions to ensure the existence and uniqueness of MLE (33). Historically, this is the first condition for the existence and uniqueness of MLE in 3-option models. The method of the proof was a direct method based on the p... | https://arxiv.org/abs/2502.13617v1 |
a directed cycle in GR(3) dir(see Definitions 4) and 5), in which the number of the directed ’better’ edges exceeds the number of the bi-directional ’equal’ edges. The proof of Theorem 1 can be found in Appendix Section A. 4 Efficiency of the necessary and sufficient condition In this section, we show that the necessar... | https://arxiv.org/abs/2502.13617v1 |
11 32 comparisons, we cannot detect more than a quarter of the evaluable data sets. These results show that knowing the necessary and sufficient condition (NS) is very useful, particularly when it is needed to determine whether the data set can be evaluated with a small number of comparisons. 5 Summary In this paper, w... | https://arxiv.org/abs/2502.13617v1 |
incomplete pairwise comparison matrices. Mathematical and Computer Modelling, 52(1-2), 318-333. [14] Csató, L. (2019). A characterization of the logarithmic least squares method. European Journal of Operational Research, 276(1), 212-216. [15] Kazibudzki, P. T. (2016). Redefinition of triad’s inconsistency and its impac... | https://arxiv.org/abs/2502.13617v1 |
MM algorithms for generalized Bradley-Terry models. The Annals of Statistics, 32(1), 384-406. [36] Ford Jr, L. R. (1957). Solution of a ranking problem from binary comparisons. The American Math- ematical Monthly, 64(8P2), 28-33. https://doi.org/10.1080/00029890.1957.11989117 . [37] Davidson, R.R. (1970). On Extending ... | https://arxiv.org/abs/2502.13617v1 |
−∞ if−d−(mis−mis+1)−→ ∞ .(41) This can be justified by noticing the fact that d−(mis−mis+1)−→ ∞ ,if−d−(mis−mis+1)−→ ∞ . (42) Therefore, there exists a constant value Kis,is+1such that ifKis,is+1<−d−(mis−mis+1)then Ais,is+1,2·(log(F(d−(mis−mis+1))−F(−d−(mis−mis+1)))< logL 0.(43) Consequently, the maximum has to be withi... | https://arxiv.org/abs/2502.13617v1 |
i l)⊂GRdir(3),(is, is+1)s= 1,2, ..., lare edges in the graph GR(3) dirand by assumption C-NS the number of ‘better’ edges is larger than the number of ’equal’ edges in the cycle CY(D). If the edge ( is, is+1)is a ‘better’ edge, then investigate (50). Ais,is+1,3·log(pis,is+1,3)−→ −∞ ifb+mis+1−mis−→ ∞ . (54) Therefore, t... | https://arxiv.org/abs/2502.13617v1 |
Rn+1, andthelog-likelihoodfunctioniscontinuous; thereforethemaximalvalueisattained. The uniqueness of the argument is again the consequence of the strictly concave property of the log- likelihood function (51). One can check that the Hesse matrices of the functions (48), (49) and (50), as a function of x=mi−mjandbare n... | https://arxiv.org/abs/2502.13617v1 |
(13),(14)and(15)as follows: p(1) i,j,1=F(−d(1)−(m(1) i−m(1) j)), (65) 19 p(1) i,j,2=F(d(1)−(m(1) i−m(1) j))−F(−d(1)−(m(1) i−m(1) j)) (66) p(1) i,j,3= 1−F(d(1)−(m(1) i−m(1) j)). (67) The following inequalities hold: •Suppose iandjare connected by an ‘equal’ edge, i.e. 0< A i,j,2, If|y−z| ≤x,then Ai,j,2·log(pi,j,2)≤Ai,j,... | https://arxiv.org/abs/2502.13617v1 |
other vertex in GWif and only if GWis conservatively weighted, that is, if there is no negative circle in GW [46] (a negative circle is a list of edges (i1, i2, ..., i l, i1), the sum of their weights is negative). Lemma 5. Negation of Condition C-NS is equivalent to the conservative property of the weighted graph GW. ... | https://arxiv.org/abs/2502.13617v1 |
1. (78) The following inequalities hold: •Suppose iandjare connected by an ‘equal’ edge, i.e. 0< A i,j,2, If|y−z| ≤x,then Ai,j,2·log(pi,j,2)≤Ai,j,2·log(p(1) i,j,2). (79) If|y−z|< x,then Ai,j,2·log(pi,j,2)< A i,j,2·log(p(1) i,j,2). (80) If|y−z|> x,then Ai,j,2·log(p(1) i,j,2)< A i,j,2·log(pi,j,2). (81) •Suppose that iand... | https://arxiv.org/abs/2502.13617v1 |
arXiv:2502.13662v2 [cs.LG] 25 Feb 2025Generalization error bound for denoising score matching under relaxed manifold assumption Konstantin Yakovlev* Nikita Puchkin† Abstract We examine theoretical properties of the denoising score ma tching estimate. We model the density of observations with a nonparametric Gaussian mi... | https://arxiv.org/abs/2502.13662v2 |
feed-forward neural networks with ReLU activations as a class of admissib le scores. Under the condition that p∗ 0is supported on the cube [−1,1]Dand bounded away from zero on this set, Oko et al. [2023 ] derived approx- imation and generalization error bounds. Unfortunately, t he rates of convergence in Sriperumbudur ... | https://arxiv.org/abs/2502.13662v2 |
classes, such as multi-modal d istributions or mixtures with well-separated components.” • Both Tang and Yang [2024 ] and Azangulov et al. [2024 ] refer to [ Oko et al. ,2023 , Theorem C.4] when derive the estimation error bound. However, Theorem C. 4 in [ Oko et al. ,2023 ] has a critical flaw. To be more precise, the ... | https://arxiv.org/abs/2502.13662v2 |
s∗(Xt,t)is unknown, Vincent [2011 ] suggested to replace the objective ( 4) byEX0ℓ(s,X0), where ℓ(s,X0) =T/integraldisplay t0E/bracketleftBig /⌊a∇⌈⌊ls(Xt,t)−∇Xtlogp∗ t(Xt|X0)/⌊a∇⌈⌊l2/vextendsingle/vextendsingleX0/bracketrightBig dt andp∗ t(Xt|X0)stands for the conditional density of XtgivenX0. In contrast to ( 4), the ... | https://arxiv.org/abs/2502.13662v2 |
samples Y1,...,Y nare i.i.d. copies of a random element X0∈RDgenerated from the model X0=g∗(U)+σdataξ, whereU∼Un([0,1]d)andξ∼ N(0,ID)are independent. In the caseσdata>0,X0has a density with respect to the Lebesgue measure in RDgiven by p∗ 0(y) = (√ 2πσdata)−D/integraldisplay [0,1]dexp/parenleftbigg −/⌊a∇⌈⌊ly−g∗(u)/⌊a∇⌈... | https://arxiv.org/abs/2502.13662v2 |
clipping is justified as it does not limit the application of gradient-based learning methods due to the non-differentiable nature of the operati on. This is also the case for neural networks that use the ReLU activation function. Distinct from the app roach in [ Chen et al. ,2023b ], our definition of the score estimato... | https://arxiv.org/abs/2502.13662v2 |
which may be as large as O(eD).Tang and Yang [2024 ] do not clarify whether one can get better upper bounds on alki’s thanO(eD). The same concerns the coefficients al1,l2,k,s,i on page 26 in [ Tang and Yang ,2024 ]. 3.2 Score estimation in a general case It this section, we present a sample complexity bound for the scor... | https://arxiv.org/abs/2502.13662v2 |
in all parameters excluding n, with the same confidence level one has 1 T−t0T/integraldisplay t0EXt/bracketleftbig /⌊a∇⌈⌊l/hatwides(Xt,t)−s∗(Xt,t)/⌊a∇⌈⌊l2/bracketrightbig dt/lessorsimilar1 t0/parenleftbigg n−2−2δ(n) d+5+Dn−d+3 d+5/parenrightbigg log3n, whereδ(n) =dloglogn logn. From the comparison of the rates above we ... | https://arxiv.org/abs/2502.13662v2 |
that σdata>0and suppose that the conditions of Theorem 3.4hold fort0=n−2β 6β+d andT≍logn+logD. Then, for any δ∈(0,1), with probability at least 1−δwe have TV(/hatwideZT−t0,X0)/lessorsimilarD5+(d+⌊β⌋ d)n−β 6β+d σdata·log3/2(n/δ)(logn)10+2(d+⌊β⌋ d)logD. The proof of Theorem 3.7is postponed to Appendix D. Theorem 3.7estab... | https://arxiv.org/abs/2502.13662v2 |
to approximate a surrogate score function s◦induced by a local polynomial approximation of g∗. This is essential for our subsequent steps. Our technical fin dings reveal that T/integraldisplay t0EXt∼p∗ t/⌊a∇⌈⌊ls◦(Xt,t)−s∗(Xt,t)/⌊a∇⌈⌊l2/lessorequalslantDH2ε2β 4(σ2 data+e2t0−1)/parenleftBigg d⌊β⌋ ⌊β⌋!/parenrightBigg2 . 12... | https://arxiv.org/abs/2502.13662v2 |
loss functions are uniformly bounded on a compact s etK0, we invoke a Bernstein-like concentration inequality [ Chen et al. ,2023b , Lemma 15] to provide a high-probability bound for the term (A). We deduce that with probability at least 1−δ/3 (A)−(2+a)τ/lessorsimilar(1+a−1)T(D+R2)log(σ−2 t0)SL nσ2 t0log/parenleftbiggD... | https://arxiv.org/abs/2502.13662v2 |
models from corrupted observations. In The Thirty-eighth Annual Conference on Neural Information Processing Systems , 2024. D. Bakry, I. Gentil, and M. Ledoux. Analysis and geometry of Markov diffusion operators , volume 348 of Grundlehren der mathematischen Wissenschaften (Fundamen tal Principles of Mathematical Scien... | https://arxiv.org/abs/2502.13662v2 |
2021. L. Dinh, J. Sohl-Dickstein, and S. Bengio. Density estimati on using real NVP. In International Conference on Learning Representations , 2017. R. M. Dudley. The speed of mean Glivenko-Cantelli convergen ce.Annals of Mathematical Statistics , 40: 40–50, 1968. K. Gatmiry, J. Kelner, and H. Lee. Learning mixtures of... | https://arxiv.org/abs/2502.13662v2 |
2017. P. Vincent. A connection between score matching and denoisi ng autoencoders. Neural computation , 23(7): 1661–1674, 2011. M. J. Wainwright. High-Dimensional Statistics: A Non-Asymptotic Viewpoint . Cambridge Series in Statis- tical and Probabilistic Mathematics. Cambridge Universit y Press, 2019. A. Wibisono, Y .... | https://arxiv.org/abs/2502.13662v2 |
latter term in the right-hand si de is negligible. Lemma A.2. Fix an arbitrary t∈[t0,T]and letKt⊂RDbe as defined above in (21). Then the density p∗ t given by (9)satisfies /integraldisplay RD\Ktp∗ t(y)dy/lessorequalslantexp/braceleftBigg −1 16/parenleftBigg R2 t−D/tildewideσ2 t D/tildewideσ2 t∧/radicalbig R2 t−D/tildewid... | https://arxiv.org/abs/2502.13662v2 |
set C∗ [t0,T]and consider Vj,0(y,t)first. We represent Vj,0(y,t)in the following form: Vj,0(y,t) =/⌊a∇⌈⌊ly−mtg◦ j(uj)/⌊a∇⌈⌊l2 2/tildewideσ2 t=/⌊a∇⌈⌊ly/⌊a∇⌈⌊l2 2/tildewideσ2 t−mty⊤g◦ j(uj) /tildewideσ2 t+m2 t/⌊a∇⌈⌊lg◦ j(uj)/⌊a∇⌈⌊l2 2/tildewideσ2 t. (30) The terms in the right-hand side can be approximated by small neural... | https://arxiv.org/abs/2502.13662v2 |
neural network /tildewideΥj(y,t)NN(LΥ,WΥ,SΥ,BΥ), which approximates Υj(y,t)within the accuracy O(ε′ε)with respect to the L∞-norm on C∗ [t0,T]: /vextenddouble/vextenddouble/vextenddoubleΥj−/tildewideΥj/vextenddouble/vextenddouble/vextenddouble L∞(C∗ [t0,T])/lessorsimilarε′ε. The network/tildewideΥ(y,t)has a depth LΥ/les... | https://arxiv.org/abs/2502.13662v2 |
we obtain that the neural network /tildewideR(Pl(y,t),Q(y,t))approximates the l-th component of /summationtext j∈{1,...,N}d/integraltext Ujg◦ j(u)exp/braceleftBigg −/⌊ard⌊ly−mtg◦ j(u)/⌊ard⌊l2 2/tildewideσ2 t/bracerightBigg du /slashBig /summationtext j∈{1,...,N}d/integraltext Ujexp/braceleftBigg −/⌊ard⌊ly−mtg◦ j... | https://arxiv.org/abs/2502.13662v2 |
. Hence, it holds that /integraldisplay RD\Ktp∗ t(y)dy/lessorequalslantexp/braceleftBigg −1 16/parenleftBigg R2 t−D/tildewideσ2 t D/tildewideσ2 t∧/radicalbig R2 t−D/tildewideσ2 t /tildewideσt/parenrightBigg/bracerightBigg/integraldisplay [0,1]ddu = exp/braceleftBigg −1 16/parenleftBigg R2 t−D/tildewideσ2 t D/tildewideσ... | https://arxiv.org/abs/2502.13662v2 |
Lemma A.6. For this purpose, we prove the following result in Appendix A.7. Lemma A.7. With the notation introduced above, it holds that /⌊a∇⌈⌊lVj,0/⌊a∇⌈⌊lL∞(C∗ [t0,T])/lessorsimilarDlog2/parenleftbiggε−2β D/parenrightbigg +m2 t0 /tildewideσ2 t0,/⌊a∇⌈⌊lV/⌊a∇⌈⌊lL∞([t0,T])/lessorsimilarm2 t0 2/tildewideσ2 t0 and /⌊a∇⌈⌊lV... | https://arxiv.org/abs/2502.13662v2 |
weights of magnitude BΨ, where logBΨ/lessorsimilarlog2(1/ε′)+log2(/tildewideσ−2 t0)+log2D. Step 4: final step. Let us recall that our goal is to approximate the product Υj(y,t) =e−Vj,0(y,t)Ψj(y,t). We already have approximated Vj,0(y,t)andΨj(y,t)by the neural networks /tildewideVj,0(y,t)and/tildewideΨj(y,t), respec- tiv... | https://arxiv.org/abs/2502.13662v2 |
for anyy′satisfying the inequality |y−y′|/lessorequalslant2−2Kεwe havegk(y′) = 0 for allk∈ {1,...,K−1}such that |k−k∗|>2. In addition, for any k∈ {1,...,K−1}fulfilling |k−k∗|/lessorequalslant2it holds that y∈[t(k−2)∨0,t(k+2)∧K]. Hence, the sensitivity analysis from ( 52), the property thath2(gk(y′),qk(x′,y′)) = 0 for|k−... | https://arxiv.org/abs/2502.13662v2 |
[t0,T]√ Dmt/⌊a∇⌈⌊ly−mtg∗(uj)/⌊a∇⌈⌊l /tildewideσ2 t. Using the identical argument as above, we deduce that /⌊a∇⌈⌊lVj,k/⌊a∇⌈⌊lL∞(C∗ [t0,T])/lessorsimilarsup (t,y)∈C∗ [t0,T]√ Dmt /tildewideσ2 tinf u∈[0,1]d(/⌊a∇⌈⌊ly−mtg∗(u)/⌊a∇⌈⌊l+mt/⌊a∇⌈⌊lg∗(u)−g∗(uj)/⌊a∇⌈⌊l) /lessorsimilarsup t∈[t0,T]√ Dmt /tildewideσ2 t(Rt+mt) /lessorsi... | https://arxiv.org/abs/2502.13662v2 |
we let /hatwideEX0[ℓ(s,X0)] =1 nn/summationdisplay i=1ℓ(s,Yi). Next, we write down the oracle inequality for an empirical ri sk minimizer /hatwides∈arg min s∈S(L,W,S,B )/hatwideEX0[ℓ(s,X0)] and some 0<a/lessorequalslant1 EX0[ℓ(/hatwides,X0)] =EX0[ℓ(/hatwides,X0)]−(1+a)/hatwideEX0[ℓ(/hatwides,X0)]+(1+a)/hatwideEX0[ℓ(/ha... | https://arxiv.org/abs/2502.13662v2 |
Now from ( 67) we conclude that sup s∈SEXt|X0=x/bracketleftbig /⌊a∇⌈⌊ls(Xt,t)/⌊a∇⌈⌊l2/bracketrightbig /lessorsimilarsup σ∈[0,1]/parenleftBigg EXt|x[/⌊a∇⌈⌊lXt/⌊a∇⌈⌊l2] (m2 tσ2+σ2 t)2+m2 t (m2 tσ2+σ2 t)2/parenrightBigg /lessorsimilarm2 t σ4 t(1+/⌊a∇⌈⌊lx/⌊a∇⌈⌊l2). (77) In addition, using the mean value theorem we obtain t... | https://arxiv.org/abs/2502.13662v2 |
Dσ2 data∧R σdata/parenrightbigg/parenrightbigg . Hence, setting R/greaterorequalslant16·σdata√ Dlog(3n/δ) (91) guarantees that with probability at least 1−δ/3, it holds that/hatwideEX0[ℓ(¯s,X0)] =/hatwideEX0[ℓtr(¯s,X0)]. Hence, by invoking Lemma B.2, specifically result ( 73), with the degenerate function class defined a... | https://arxiv.org/abs/2502.13662v2 |
following technical lemma derived in Append ixC.1. Lemma C.1. Leth:RD×[t0,T]→RDandh′:RD×[t0,T]→RDbe any Borel functions such that /⌊a∇⌈⌊lh/⌊a∇⌈⌊lL∞(RD×[t0,T])/lessorequalslant2and/⌊a∇⌈⌊lh′/⌊a∇⌈⌊lL∞(RD×[t0,T])/lessorequalslant2. Consider the corresponding scores s(y,t) =−y σ2 t+mt σ2 th(y,t)ands′(y,t) =−y σ2 t+mt σ2 th′... | https://arxiv.org/abs/2502.13662v2 |
t+mt σ2 tclip2(f(y,t)) :f∈ Fτ/bracerightbigg . Due to the union bound and ( 96), there is an event E,P(E)/greaterorequalslant1−δ, such that /vextendsingle/vextendsingle/vextendsingleEX0/parenleftbig ℓ(s,X0)−ℓ(s∗,X0)/parenrightbig −/hatwideEX0/parenleftbig ℓ(s,X0)−ℓ(s∗,X0)/parenrightbig/vextendsingle/vextendsingle/vexte... | https://arxiv.org/abs/2502.13662v2 |
t0)−2β 2β+dlog(2/δ)L(t0,n) holds with probability at least 1−δ. Here we introduced L(t0,n) =L′(t0,n−1). Finally, substituting the optimizedεinto the configuration outlined in ( 102) completes the proof. 58 C.1 Proof of Lemma C.1 It holds that ℓ(s,x)−ℓ(s′,x) =T/integraldisplay t0EXt|X0=x/bracketleftBigg/vextenddouble/vex... | https://arxiv.org/abs/2502.13662v2 |
/tildewideσ2 t0/parenrightbigg −D+(1−mt0)2/⌊a∇⌈⌊lg∗(U)/⌊a∇⌈⌊l2 σ2 data+D/tildewideσ2 t0 σ2 data/bracketrightBigg . Recall that /⌊a∇⌈⌊lg∗(U)/⌊a∇⌈⌊l/lessorequalslant1and also/tildewideσ2 t0=m2 t0σ2 data+σ2 t0. Using the observation that for t0/lessorequalslant1it holds that σ2 t0≍t0, we obtain TV(Xt0,X0)/lessorsimilar/ra... | https://arxiv.org/abs/2502.13662v2 |
network /tildewidef∈NN(L,W,S,1) with depth L= 8+(m+5)(1+ ⌈log2(r∨α)⌉, width W= 6(r∨⌈α⌉)N, and with at most S/lessorequalslant141(r+α+1)3+rN(m+6) non-zero parameters such that /⌊a∇⌈⌊l/tildewidef−f/⌊a∇⌈⌊lL∞([0,1]r)/lessorequalslant6r(2H+1)(1+r2+α2)N2−m+3αHN−α/r. Remark F.5. Assume that Hfrom Theorem F .4is at least 1and ... | https://arxiv.org/abs/2502.13662v2 |
2DCfor allj∈ {1,...,D}and allyj∈[−M,M]. More importantly, we can use φmult to approximate the product /⌊a∇⌈⌊ly/⌊a∇⌈⌊l2/(2/tildewideσ2 t). Indeed, note that for all /⌊a∇⌈⌊ly/⌊a∇⌈⌊l∞/lessorequalslantMand allt∈[t0,T]both/⌊a∇⌈⌊ly/⌊a∇⌈⌊l2/2and/tildewideσ−2 tbelong to [−C,C]. Let us take the neural network χ0,ε0defined in Lem... | https://arxiv.org/abs/2502.13662v2 |
arXiv:2502.13692v1 [cs.LG] 19 Feb 2025Tight Generalization Bounds for Large-Margin Halfspaces Kasper Green Larsen Natascha Schalburg {larsen,n.schalburg }@cs.au.dk Computer Science, Aarhus University Abstract We prove the first generalization bound for large-margin hal fspaces that is asymptotically tight in the tradeof... | https://arxiv.org/abs/2502.13692v1 |
was later improved by Bartlett and Mendelson [2002 ] using Rademacher complexity arguments, replacing the ln2(n)term in ( 2) by1. Here, and throughout the paper, we refer to Lγ S(w)as the (empirical) margin loss . First-Order Bounds. The first work to interpolate between the hard-margin and sof t-margin bounds was due t... | https://arxiv.org/abs/2502.13692v1 |
n/parenrightigg . While one might argue that our improvement is small in magnit ude, this finally pins down the exact generalization performance of a classic learning model. Furthermore, our p roof of Theorem 2brings several novel ideas that we hope may find further applications in generalization bounds. We next proceed... | https://arxiv.org/abs/2502.13692v1 |
by plugging in x=wand settingε= 1, we can also deduce that /bardblhA,t(w)/bardbl2≤2except with probability exp(−k/c). Simple counting arguments show that there are only exp(ck) many vectors of norm at most 2with all coordinates integer multiples of k−1/2. That is, except with probability exp(−k/c),hA,t(w)belongs to a fi... | https://arxiv.org/abs/2502.13692v1 |
into accounts, we get the tighter bounds LD(w) =Lγ/2 AD(hA,t(w))+P(x,y)∼D[y/a\}bracketle{tw,x/a\}bracketri}ht≤0∧y/a\}bracketle{thA,t(w),Ax/a\}bracketri}ht>γ/2] −P(x,y)∼D[y/a\}bracketle{tw,x/a\}bracketri}ht>0∧y/a\}bracketle{thA,t(w),Ax/a\}bracketri}ht≤γ/2], and Lγ S(w) =Lγ/2 AS(hA,t(w))−P(x,y)∼S[y/a\}bracketle{tw,x/a\}b... | https://arxiv.org/abs/2502.13692v1 |
write PA,t[y/a\}bracketle{thA,t(w),Ax/a\}bracketri}ht> γ/2|y/a\}bracketle{tw,x/a\}bracketri}ht= 0] to denote the probability PA,t[y/a\}bracketle{thA,t(w),Ax/a\}bracketri}ht>γ/2]for an arbitrary x,w∈Sd−1andy∈{−1,1}withy/a\}bracketle{tw,x/a\}bracketri}ht= 0asy/a\}bracketle{tw,x/a\}bracketri}htcompletely determines this p... | https://arxiv.org/abs/2502.13692v1 |
n/parenrightbigg +ln(eγ2n) γ2n+ln(e/δ) n/parenrightigg . (16) Note that Theorem 2follows as an immediate corollary since margins change by a cγfactor in our transformation of the input distribution. Since cγis a constant, this disappears in the constant factor cin Theorem 2(note that for marginsγ∈[n−1/2,c−1 γn−1/2)in ... | https://arxiv.org/abs/2502.13692v1 |
3. So let0<δ<1, and fix a pair (Γi,Lj). Following the proof outline in Section 2, we now consider the following random discretization of hypo theses inH(Γi,Lj): Letk=k(i,j)be an integer parameter to be determined. Sample a random k×dmatrixAwith each entryN(0,1/k)distributed as well as k random offsets t= (t1,...,tk)all ... | https://arxiv.org/abs/2502.13692v1 |
w∈H(Γi,Lj)LD(w)−Lγ S(w)≤sup w∈H(Γi,Lj)/vextendsingle/vextendsingle/vextendsingleEA,t[Lγi/2 D(hA,t(w))−Lγi/2 S(hA,t(w))]/vextendsingle/vextendsingle/vextendsingle (21) + sup w∈H(Γi,Lj)/vextendsingle/vextendsingleE(x,y)∼D[φ(y/a\}bracketle{tw,x/a\}bracketri}ht)]−E(x,y)∼S[φ(y/a\}bracketle{tw,x/a\}bracketri}ht)]/vextendsing... | https://arxiv.org/abs/2502.13692v1 |
of α. The simplest case is the following Lemma 11. There is a constant c>0such that the Lipschitz constants of φandρ, when0<α≤γi, are less than: cexp(−kγ2 i+1/c) γi+1. Proof. Sinceφis linear when 0<α≤γi, its Lipschitz constant equals the slope of the line, i.e. 1 γi·PA,t[y/a\}bracketle{thA,t(w),Ax/a\}bracketri}ht>γi/2|... | https://arxiv.org/abs/2502.13692v1 |
density function of Mandt. Let us now define Nisuch that Yi=/radicalig 1−α2 kNiand letN= (N1,...,Nk). ThenNi∼N(0,1)and the event α√ k/a\}bracketle{tX′,M/a\}bracketri}ht+/a\}bracketle{tX′,Y/a\}bracketri}ht>γi/2, 13 may be rewritten as α√ k/a\}bracketle{tX′,M/a\}bracketri}ht+/a\}bracketle{tX′,Y/a\}bracketri}ht>γi/2⇐⇒/a\}... | https://arxiv.org/abs/2502.13692v1 |
i).We simply upper bound the exponential term in ( 25) by1and use the assumption that /bardblM/bardbl2 2≤9 10kto conclude exp/parenleftigg −(√ kγi/2−α/a\}bracketle{tX′,M/a\}bracketri}ht)2 2/bardblX′/bardbl2 2/parenrightigg/parenleftig√ kγi+/bardblM/bardbl2/parenrightig ≤2√ k. Now since Mis multivariate standard nor... | https://arxiv.org/abs/2502.13692v1 |
a single grid is insufficient. Ins tead, we need an infinite sequence of grids. For this, let G0denote the set of all vectors in 4Bk 2whose coordinates are of the form(1/2)(10√ k)−1+z(10√ k)−1for integerz. More generally, let Gifori >0denote the set of all vectors in (2i·4Bk 2)whose coordinates are of this form. Since /b... | https://arxiv.org/abs/2502.13692v1 |
AS(hA,t(w))]/vextendsingle/vextendsingle/vextendsingle≤ ∞/summationdisplay h=02h+8EA,t/bracketleftbigg /bardblhA,t(w)/bardbl2·/parenleftbigg/radicaligg 8Lγi/2 AD(w)(k+ln(1/δ)) n+ 2(k+ln(1/δ)) n/parenrightbigg |A∈Kh(S)/bracketrightbigg PA[A∈Kh(S)]≤ ∞/summationdisplay h=02h+8/radicalig EA,t[/bardblhA,t(w)/bardbl2 2|A∈K... | https://arxiv.org/abs/2502.13692v1 |
AS(w)−L(7/8)γi AD(w)/vextendsingle/vextendsingle/vextendsingle>t/n/bracketrightig <exp/parenleftigg −1 2t2 nL(7/8)γi AD(w)+1 3t/parenrightigg . Setting t=n·/parenleftigg L(7/8)γi AD(w) 2+Z/parenrightigg 22 withZ= 16ln(1/α)/ngives PS/bracketleftigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleL(7/8)γi AS... | https://arxiv.org/abs/2502.13692v1 |
necessarily reflect tho se of the European Union or the European Research Council. Neither the European Union nor the granting author ity can be held responsible for them. References N. Alon and B. Klartag. Optimal compression of approximate i nner products and dimension reduction. In 58th IEEE Annual Symposium on Found... | https://arxiv.org/abs/2502.13692v1 |
this section, we prove a number of auxiliary results used t hroughout the paper. For this, we need the following concentration inequality: Theorem 17 (Wainwright [2019 ], example 2.11) .LetY∼χ2 k, then for any x∈(0,1)it holds that P/bracketleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleY k−1/vextendsin... | https://arxiv.org/abs/2502.13692v1 |
2for sufficiently large c>0in (16). Restatement of Lemma 6.There is a constant c >0, such that for any integer k≥1,w∈H,x∈X and any γ∈(0,1], it holds that PA,t[|/a\}bracketle{thA,t(w),Ax/a\}bracketri}ht−/a\}bracketle{tw,x/a\}bracketri}ht|>γ]<cexp(−γ2k/c). Proof. We start by observing that /bardblAw/bardbl2 2,/bardblAx/ba... | https://arxiv.org/abs/2502.13692v1 |
Claim 4.Letw1,x1,y1be such that α1:=y1/a\}bracketle{tw1,x1/a\}bracketri}htand letw2,x2,y2be such that α2:=y2/a\}bracketle{tw2,x2/a\}bracketri}ht. Con- sider sampling Xi,Yi∼N(0,1/k)independently. Also sample offsets t′ 1,...,t′ kuniformly and independently in [0,1]and letX′ ibeXirounded based on t′ ias above. Let Z1=Y=α... | https://arxiv.org/abs/2502.13692v1 |
arXiv:2502.13711v1 [math.ST] 19 Feb 2025On noncentral Wishart mixtures of noncentral Wisharts and their use for testing random effects in factorial design mode ls Christian Genesta, Anne MacKayb, Fr´ ed´ eric Ouimeta,b,∗ aDepartment of Mathematics and Statistics, McGill Universi ty, Montr´ eal (Qu´ ebec) Canada H3A 0B9 ... | https://arxiv.org/abs/2502.13711v1 |
Analysis and Applications February 20, 2025 Our result extends to arbitrary dimension d≥1 an earlier finding of Jones & Marchand [10], who showed that a noncentral chi-square mixture of noncentral c hi-square distributions with the same degrees of freedom is a noncentral chi-square distribution. The ext ension of this c... | https://arxiv.org/abs/2502.13711v1 |
[13, Theorem 10.3.2]. 2 The lemma below extends Theorem 10.3.5 of Muirhead [13] from the matrix-variate normal setting to the Wishart setting. This result is most likely known but we w ere unable to find an explicit statement, so it is given below for completeness. Lemma 2.1. Letν∈(d−1,∞),C,Σ∈ Sd ++andΘ∈Rd×dbe given suc... | https://arxiv.org/abs/2502.13711v1 |
for every T∈ Sdsatisfying A−1−2T∈ Sd ++and (Σ H)−1−2/tildewideT∈ Sd ++, one has MX(T) =etr/bracketleftbigTA{Id−2(Id+ Σ H)TA}−1∆H/bracketrightbig |Id−2(Id+ Σ H)TA|ν/2 =etr/bracketleftbigTA{(Id+ Σ H)A1/2}{Id−2T A1/2(Id+ Σ H)A1/2}−1{(Id+ Σ H)A1/2}−1A−1/2A1/2∆H/bracketrightbig |Id−2T A1/2(Id+ Σ H)A1/2|ν/2 =etr/bracketleftb... | https://arxiv.org/abs/2502.13711v1 |
matrix Σ ∈ Sd ++. To ensure that the model parameters are identifiable, the following constraints are imposed for all (i, j)∈ {1, . . . , a } × { 1, . . . , b }: α•=β•=(αβ)i•=(αβ)•j=(αβ)••=0d. This is accomplished by absorbing all means into µ. The overline symbol vover a vector v(or an array) stands for averaging and t... | https://arxiv.org/abs/2502.13711v1 |
for ν1, ν2∈(d−1,∞). It is also known as the matrix-variate Fdistribution and corresponds to the distribution ofS−1/2 2S1S−1/2 2, assuming that S1∼ W d(ν1, Id) andS2∼ W d(ν2, Id) are independent; see, e.g., Gupta & Nagar [5, Theorem 5.2.5]. Remark 4.1. If some of the factors have fixed effects, the same test statist ics i... | https://arxiv.org/abs/2502.13711v1 |
fai led to be rejected at the 5% level, providing further numerical evidence for th e validity of (4.4). 8 Reproducibility TheRcode used to conduct the simulations in Section 4.1 is public ly available in the GitHub repository of Genest et al. [4]. Acknowledgments We thank Donald Richards (Penn State University) for en... | https://arxiv.org/abs/2502.13711v1 |
B., & Barnett, I. 2022. Autoregressive mixture mod els for clustering time series. J. Time Series Anal. ,43(6), 918–937. MR4524855. [15] Shakya, S., Batool, N., ¨Ozarslan, E., & Knutsson, H. 2017. Multi-fiber reconstructi on using prob- abilistic mixture models for diffusion MRI examinations of t he brain. Pages 283–308 ... | https://arxiv.org/abs/2502.13711v1 |
arXiv:2502.13813v1 [cs.IT] 19 Feb 20251 Optimal Overlap Detection of Shotgun Reads Nir Luria and Nir Weinberger The Viterbi Faculty of Electrical and Computer Engineering Technion - Israel Institute of Technology Technion City, Haifa 3200004, Israel nir.luria@campus.technion.ac.il, nirwein@technion.ac .il. Abstract We ... | https://arxiv.org/abs/2502.13813v1 |
address a distilled version of this setting: Given a single pair of reads from a long sequence, what are the fundamental limit s of the detection of the overlap length, or the relative alignme nt, between these reads? To address this question, we develop the optimal Bayesian detector for this problem, and derive exact ... | https://arxiv.org/abs/2502.13813v1 |
algorithm is greedy in its nature, and keeps track of a set of contigs – a contiguous fragment of overlapping reads. At first, the se t of contigs is initialized to the set of sequenced reads. Then, at each of m−1steps, the algorithm finds a pair of contigs with maximal overlap and merges them into a single contig. The a... | https://arxiv.org/abs/2502.13813v1 |
First, the assumption on the input process generating the sequence was generalized f rom a memoryless process to a Markov process 2In metagenomics, even the identifiability of the sequences f rom the reads is a challenging question [11]. 4 (assumed for simplicity, to be a first-order Markov process) , in which the second... | https://arxiv.org/abs/2502.13813v1 |
error event, whose probability is ptype-I(ˆT) :=P[ˆT∝\⌉}atio\slash= 0|T= 0] , and the type-II error be the misdetection or 3To be formally defined later on. 4We distinguish between a positive and negative overlaps, to wit, between the case that the first read appears before the se cond one and vice-versa. 5 erroneous-ove... | https://arxiv.org/abs/2502.13813v1 |
s the probability to err from one detection problem to 6 another (that is, a wrong alignment value), is negligible co mpared to this probability. While this intuition may appear to be just a bounding technique, it is actually also us ed to obtain a matching lower bound. In a nutshell, the optimal Bayesian detector will... | https://arxiv.org/abs/2502.13813v1 |
e rror probability of φ≤β∧1 H2(X). In the noisy setting, the type-I error probability pertains to the proba bility that a sum of independent random variables (r.v.’s) crosses a threshold. Here, an application of the Ch ernoff bound does not suffice, despite the fact that it is exponentially tight. We thus turn to strong... | https://arxiv.org/abs/2502.13813v1 |
1 α−1log/parenleftbig/summationtext x∈XPα X(x)Q1−α X(x)/parenrightbig , α∝\⌉}atio\slash= 1 /summationtext x∈XPX(x)logPX(x) QX(x), α = 1(12) where we notice that D1(PX||QX)is the Kullback–Leibler divergence. We denote the total var iation distance betweenPXandQXby ∝⌊ar⌈⌊lP−Q∝⌊ar⌈⌊lTV:=1 2/summationdisplay x∈X|P(x)−Q(x... | https://arxiv.org/abs/2502.13813v1 |
less cumbersome. Concretely, we assume that I(1)∼Uniform[n](or even arbitrary chosen in [n]), and that I(2)|I(1)∼ I(1)+ Uniform{−ℓ+1,−ℓ+2,...,n−ℓ},1≤I(1)≤ℓ−1 Uniform[n], ℓ ≤I(1)≤n−ℓ+1 I(1)+ Uniform{ℓ−n,ℓ−n+1,...,ℓ−1}, n−ℓ+2≤I(1)≤n(16) So, the distribution of I(2)is always uniform over a window of length n... | https://arxiv.org/abs/2502.13813v1 |
1(1). The two reads may be analogously partitioned when t∈−[ℓ−1]. We will assume throughout that the read length scales as ℓ=βlogn, whereβ >0is a fixed parameter (in fact, the interesting regime is in which β >log|X|= 1). We will focus on two specific settings: Definition 1 (Noiseless setting) .Xis a stationary and ergodi... | https://arxiv.org/abs/2502.13813v1 |
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