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Sensitivity for µ3 Ours (full) 81.1 42.3 78.0 58.4 59.5 83.4 45.0 83.7 56.6 54.6 78.2 33.8 73.2 40.8 44.0 Ours (partial) 88.9 63.8 85.2 69.4 69.2 91.6 65.0 89.5 63.2 63.6 90.6 60.7 86.5 51.0 55.4 Ca2018 78.3 37.8 82.1 74.9 79.5 83.6 30.2 85.4 81.1 84.2 86.2 18.5 87.2 84.3 87.2 Ho2018 100.0 99.5 100.0 100.0 99.9 100.0 9... | https://arxiv.org/abs/2502.18038v1 |
100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 Panel C: Specificity for µ3 Ours (full) 86.1 93.5 86.5 91.1 88.6 86.8 96.1 86.6 93.1 90.9 86.7 97.1 86.1 94.6 92.9 Ours (partial) 74.7 77.2 75.2 79.3 79.0 89.4 91.1 87.4 91.5 88.9 95.5 95.9 95.1 97.8 97.0 Ca2018 99.9 99.8 99.8 99.7 99.7 99.9 99.9 99.9 99.8 99.8 100.... | https://arxiv.org/abs/2502.18038v1 |
66.5 69.8 94.2 48.6 91.1 64.4 65.7 97.0 51.5 95.7 62.5 63.6 Ca2018 77.6 33.8 82.2 74.9 79.8 84.1 25.9 85.5 81.0 84.6 85.9 18.7 87.1 84.1 87.6 Ho2018 100.0 99.2 100.0 100.0 99.8 100.0 99.9 100.0 100.0 94.7 100.0 99.9 99.4 100.0 100.0 We2024 3.0 0.5 3.5 2.8 3.4 1.5 0.2 1.8 1.5 1.8 0.7 0.1 0.9 0.8 0.9 Ma2015 0.0 0.0 0.0 0... | https://arxiv.org/abs/2502.18038v1 |
A Unified Bayesian Perspective for Conventional and Robust Adaptive Filters Leszek Szczecinskia,∗, Jacob Benestya, Eduardo Vinicius Kuhnb aINRS–Institut National de la Recherche Scientific, Montreal, QC, H5A-1K6, Canada. bLAPSE–Electronics and Signal Processing Laboratory, Department of Electronics Engineering, Federal... | https://arxiv.org/abs/2502.18325v1 |
of the observational noise in the state-space model, we derive an entire class of new robust adaptive algorithms, which generalize those already known in the literature. By specializing the proposed algorithms to Laplacian noise (which is a particular case of a generalized Gaussian distribution), we derive algorithms w... | https://arxiv.org/abs/2502.18325v1 |
(6) using Gaussian distributions (denoted by tilde), i.e., f(θt|y1:t)≈˜f(θt|y1:t) =N(θt;wt,Vt), (10) as then we can easily calculate (7) as follows: ˜f(θt|y1:t−1) =Z f(θt|θt−1)˜f(θt−1|y1:t−1) dθt−1 (11) =N(θt;wt−1,Vt), (12) where Vt=Vt−1+εI. (13) Then, we can use (11) to approximate (6): ˜f(θt|y1:t) =P ϕ(θt|y1:t) =N(... | https://arxiv.org/abs/2502.18325v1 |
we can use, e.g., w0=0, and V0=v0I, where v0– the prior variance of elements of θ0is a parameter to be set. 6 2.3. Differences with the conventional Kalman filter The presentation of the problem (20) points to two issues that do not appear in the con- ventional Kalman filter derived from the assumption that the noise η... | https://arxiv.org/abs/2502.18325v1 |
average of the diagonal elements inVt, i.e., vt=Tr(Vt) M=1Tvt M, (42) where Tr(·)is the trace of the matrix. Proof : see [3, Appendix A]. Therefore, to obtain the approximations defined in (40), we can find the unconstrained solution (17) and apply (41) or (42). 8 If the projection is carried out via (18)-(19), we can ... | https://arxiv.org/abs/2502.18325v1 |
[8, Sec. VII], which generalizes the regularized NLMS in (55) by using the time-variable regularization factor vη/vt. For completeness, we show also the update for the vKF algorithm: wt=wt−1+vt⊙xtet vη+vT tx2 t, (57) where x2 t=xt⊙xt, andvtis calculated as shown in the first column of (46). It is easily seen to be a ge... | https://arxiv.org/abs/2502.18325v1 |
course, it disappears in the Gaussian case, i.e., for β= 2,ht=1 vη. We may now use (63) in the KF, sKF, fKF, or SG algorithms, shown in Sec. 2.4, which will produce a new family of robust adaptive filters parameterized with β∈[1,2]. For β= 2, we recover, of course, the conventional solutions we have already shown at th... | https://arxiv.org/abs/2502.18325v1 |
where we use a= 0.9 and utis a zero-mean white Gaussian noise with variance vu. The noise ηtis 15 Figure 2: Impulse response, ht, used in numerical examples. generated as a zero-mean, white generalized Gaussian variables with shape parameter βand variance vη. We define the output signal-to-noise ratio (SNR) as SNR = 10... | https://arxiv.org/abs/2502.18325v1 |
already mentioned in Sec. 2.5, in the conventional case, i.e., for β= 2.0, the SG algorithm corresponds to the LMS algorithm, the fKF algorithm is the same as the regularized NLMS, while the sKF corresponds to the broadband Kalman filter. On the other hand, among robust filters, the SG algorithm corresponds to the sign... | https://arxiv.org/abs/2502.18325v1 |
( β= 0.2) for (a) conventional filters ( β= 2.0), and (b) robust filters ( β= 1.0); SNR = 5 dB and target misalignment m∞=−20 dB. 19 a)m∞=−15 dB b)m∞=−20 dB c)m∞=−25 dB Figure 4: Convergence of robust algorithms ( β= 1.0) in heavy-tailed noise ( β= 0.2) for target misalignment a)m∞=−15 dB, b) m∞=−20 dB, and c) m∞=−25 d... | https://arxiv.org/abs/2502.18325v1 |
is remarkable is that these robust algorithms are derived effortlessly from the general Bayesian filtering formulations. This includes a robust Kalman filter, which is straightforwardly derived for a non-Gaussian noise. •We show that, due to the non-Gaussian nature of the noise, the filtering operation may be enhanced ... | https://arxiv.org/abs/2502.18325v1 |
by definition, writing explicitly the minorization condition: q(e)≤ℓ(e), (97) −1 2βeβ−2 te2+eβ t(1 2β−1)≤ −eβ, (98) u(x) =xβ(1 2β−1)≤1 2βxβ−2−1 =p(x), (99) where we set x=et/e≥1. 23 Figure 5: Minorization example for β= 1 and et= 1.5, which is the value where ℓ(et) =q(et). Both sides of inequality (99) are equal for x=... | https://arxiv.org/abs/2502.18325v1 |
Testing Thresholds and Spectral Properties of High-Dimensional Random Toroidal Graphs via Edgeworth-Style Expansions Samuel Baguley Andreas Göbel Marcus Pappik Leon Schiller Hasso Plattner Institute, University of Potsdam {firstname.lastname}@hpi.de Abstract We study high-dimensional random geometric graphs (RGGs) of e... | https://arxiv.org/abs/2502.18346v1 |
. . 11 2.3 The Need for Specialized Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Main Theorem—Approximating the Density of zabove its Ground State . . . . . . . . . 12 2.5 Technical Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.6 Appli... | https://arxiv.org/abs/2502.18346v1 |
of Theorem 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.3.1 Small Values of t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.3.2 Large Values of t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.3.3 Bounding the Higher-Order Correction... | https://arxiv.org/abs/2502.18346v1 |
. . . . . . . . . . . . . . . 65 8.2.1 Bounding the Signed Weight of Cycles and Chains . . . . . . . . . . . . . . . . . . 65 8.2.2 Using The Trace Method (Again) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 8.3 Lower Bound: Finding Explicit Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . .... | https://arxiv.org/abs/2502.18346v1 |
connection probability. Depending on the parametrization of the model, RGGs can differ significantly from the classical random graph model of Erd ˝os–Rényi graphs (denoted by G(n,p)), where edges are sampled indepen- dently with probability p. For example, when the underlying metric space is not homogeneous (e.g., ad-d... | https://arxiv.org/abs/2502.18346v1 |
probability p. The first to study high-dimensional RGGs with ( Td,Lq) as underlying space were Friedrich, et. al. [16] who established that RGG q(n,d,p)and Erd ˝os–Rényi graphs converge in total variation asd→ ∞ (and their inhomogeneous variants), for all fixed values of q∈Z>0∪{∞}—this includes the maximum norm L∞. In ... | https://arxiv.org/abs/2502.18346v1 |
the edge {i,j}. With this, we prove the following. Theorem 1.1. For any fixed 1Éq< ∞,dÊnγand anyα nÉpÉ1−εfor constants ϵ,α>0, the number of signed triangles distinguishes G1∼RGG q(n,d,p)andG2∼G(n,p)with probability 1−o(1) whenever d=o(n3p3), and fails at doing so whenever d=˜ω(n3p3). Precisely, ¯¯¯¯E G∼RGG q(n,d,p)[T(G... | https://arxiv.org/abs/2502.18346v1 |
more direct integration that makes heavy use of the spherical symmetry. We generalize both of these results to a wider class of possible distance functions while relying on the same underlying way of quantifying dependencies between edges for both our upper- and lower-bound. Spectral Properties of RGG q(n,d,p)and RGG ∞... | https://arxiv.org/abs/2502.18346v1 |
arc are all set to 1, while coordinate corresponding to the second arc 7 are−1. Since vertices within the same arc have a slightly larger probability of being adjacent than other pairs, this yields the desired properties. To analyze how these slightly perturbed connection probabilities, we also use (univariate) Edgewor... | https://arxiv.org/abs/2502.18346v1 |
the context of robust testing for RGGs on the sphere [4]. Finally, it would be interesting to improve upon the gap between information-theoretic lower bounds and algorithmic upper bounds in case of p=o(1). This is open even for spherical RGGs, and resolving it would require a better understanding of ‘average-case’ vers... | https://arxiv.org/abs/2502.18346v1 |
cumulants. ‘Pure’ and ‘mixed’ cumulants. There is a simple intuition for why certain cumulants charac- terize the dependencies between the entries of z, which is obtained by comparing the individual (univariate) cumulants of the entries of zto the joint cumulants of z. If the entries of zwere all independent, then the ... | https://arxiv.org/abs/2502.18346v1 |
forces all 10 mixed cumulants of order <kand all cumulants of order kexcept for κ(1,1,...,1)(z) to be 0. The same holds if Hrepresents a chain with endpoints u,v, conditional on the fixed positions of u,v. This is captured in the following observation. Observation 2.3 (Mixed cumulants for cycles and chains) .Assuming t... | https://arxiv.org/abs/2502.18346v1 |
dependence between the coordinates of z, provided that the parameter sis chosen sufficiently large. However, classical Edge- worth expansions turn out to be impractical for quantifying the influence of mixed cumulants on top ofz’s ‘ground state’ dictated by all the pure cumulants. This is for two reasons. First, in cla... | https://arxiv.org/abs/2502.18346v1 |
condition (Definition 3.6) for some ε(δ). Then, for any choice of integer valued parameters bpureÊ3,bmixÊk,apure,amixÊ1, there is a sequence of vectors¡ αj¢apurebpure+amixbmix j=k+1with 3Specifically, summands are i.i.d., the CFs are integrable, and all terms satisfy Cramér’s condition. 12 αj∈Rkjsuch that the density f... | https://arxiv.org/abs/2502.18346v1 |
z that can be Fourier inverted, eventually yielding Theorem 2.4. 2.5 Technical Challenges The two major obstacles in the method described in Section 2.4 are: (1) the individual zihave no density, and (2) a prerequisite of the inversion theorem (Theorem 3.5) is that the CF of zis integrable, which is not guaranteed a pr... | https://arxiv.org/abs/2502.18346v1 |
constant. Furthermore, any constant ηonly leads to a ‘blow-up’ that is polynomial in din Section 4.3.2, which is compensated by a factor exponentially small in d, arising due to Cramér’s condition. 2.6 Applications to RGG q(n,d,p) We continue by describing how the expansions from Theorem 2.4 are helpful for understandi... | https://arxiv.org/abs/2502.18346v1 |
following technical lemma, which we prove in Appendix A.3. Lemma 2.7. Consider a triangle C3with vertices v1,v2,v3and edges e1,e2,e3. For all Lq-norms with qÊ1, we have that E£ γ(e1)γ(e2)γ(e3)¤ <0where γ({u,v}):= |xu−xv|q C−E£ |xu−xv|q C¤ and xv1,xv2,xv3∼ Unif ([−1/2,1/2])are the positions of v1,v2,v3in a fixed dimensi... | https://arxiv.org/abs/2502.18346v1 |
However, this analysis breaks down in the case of general pas the locally-tree-like properties are lost. In this case, their approach based on ‘concentration via optimal transport’ yields the same bound as in Theorem 1.2 (up to polylogarithmic factors). Closing the gap between algorithmic upper bounds and information-t... | https://arxiv.org/abs/2502.18346v1 |
us to treat edges inside and outside of H◦separately. Crucially, the core H◦is still a Eulerian multigraph but it now has minimum degree 4, limiting the number of vertices it can contain to m/2. Moreover, conditional on the positions of vertices in H◦, any removed cycle or contracted chain appears independently and its... | https://arxiv.org/abs/2502.18346v1 |
edges in the core H◦and edges on degenerate cycles (i.e. cycles with only two edges). Our bounds on Eh tr³¡ A−p11T¢m´i1/m further allow us to make a statement about the number of large eigenvalues in a RGG q(n,d,p)since we are able to show that Eh tr³³ A−p11T´m´i É eO³ d³ npp d´m´ ifd≪nplog2(n) eO¡ npnpm¢ ifd≫nplog... | https://arxiv.org/abs/2502.18346v1 |
cumulants to mixed moments by invoking a Möbius inversion over the partition lattice κ(X1,X2,...,Xk)=X π(|π|−1)!(−1)|π|−1Y B∈πE" Y j∈BXj# (5) where the sum goes over all partitions πof [k]={1,...,k}and|π|denotes the number of blocks of said partition. We refer the interested reader to [29, Chapter 3] for more details a... | https://arxiv.org/abs/2502.18346v1 |
with the desired marginal edge probability p. Moreover, given a arbitrary set of edges H, we define the signed weight of Has the random variable SW(H):=Y e∈H(1(e)−p). 3.2 RGGs and the Concrete Random Vectors of Interest Let1(E)denote the indicator random variable of the event E. Arandom geometric graph (RGG) is the fol... | https://arxiv.org/abs/2502.18346v1 |
r:=dX i=1∆i−µp dσandη:=dX i=1ηi ζp dσ∼Nµ 0,d−2η ζ2σ2Ik¶ . (8) The following lemma quantifies the influence of the Gaussian noise on the probability of the event that a subset of entries of zand∆are all at most x. Lemma 3.7 (Quantifying the influence of Gaussian noise) .There is some absolute constant C>0 such that for ... | https://arxiv.org/abs/2502.18346v1 |
that when we expand κdandκsaccording to (5), if at least one sj=0 then all the mixed moments that appear factorize due to Observation 3.9. We therefore obtain identical expansions for both κsandκs, implying that κs=κs=0 as desired. Lemma 3.10 implies in particular that the first (i.e. lowest order) mixed non-zero cumul... | https://arxiv.org/abs/2502.18346v1 |
C>0, there are constants C1,C2>0such that for all xÉC, C1φ(x)ÉΦ(x)ÉC2φ(x). Proof. We have Φ(x)=1p 2πZx −∞e−t2/2dtÉ1p 2πZx −∞−te−t2/2dt=1p 2πe−x2/2=φ(x) where the inequality holds since xÉ −1. Similarly, Φ(x)=1p 2πZx −∞e−t2/2dtÊ1p 2πZx 2xe−t2/2dt Ê1p 2π2|x|Zx 2x−te−t2/2dt =1 2|x|¡ φ(x)−φ(2x)¢ =1 2|x|p 2πe−x2/2(1−e−3x2/2... | https://arxiv.org/abs/2502.18346v1 |
positions of the endpoints v1,vk+1, whose position determine κ. Ifk=2, we further have the technical problem that not all zjare guaranteed to satisfy Cramér’s condition. However, we can still prove that each zjsatisfies Cramér’s condition with a constant probability over the draw of positions of v1,vk+1. For our error ... | https://arxiv.org/abs/2502.18346v1 |
is our main technical result. Theorem 2.4 (Main theorem, the joint density of zabove its ground state) .LetHbe a cycle or a chain of length kÊ2. Let fdenote the density of zand assume that a constant fraction of the zisatisfy Cramér’s condition (Definition 3.6) for some ε(δ). Then, for any choice of integer valued para... | https://arxiv.org/abs/2502.18346v1 |
correction terms (i.e. our bound on ∥αk+j∥∞) in Section 4.3.3. The proof follows a similar high-level ideas as used in [15] and [23], based on bounding the error Err(x):=(2π)k|f(x)−˜f(x)| =¯¯¯¯Z Rke−ixTt(C(t)−˜C(t))dt¯¯¯¯ by defining the set B(δ)=n t∈Rk| ∥t∥ Éδp do for a small constant δ>0, and then splitting the integ... | https://arxiv.org/abs/2502.18346v1 |
Now, we can expand the right-hand term in (11) by applying an apure-th order Taylor series to exp³ K†bpure (pure)(t)´ and an amix-th order series to³ exp³ K†bmix (mix)(t)´ −1´ separately: exp³ K†bpure (pure)(t)´ =s1X ℓ=01 ℓ!³ K†bpure (pure)(t)´ℓ | {z } =:P1(t)+³ K†bpure (pure)(t)´s1+1 exp³ cK†bpure (pure)(t)´ | {z } =:... | https://arxiv.org/abs/2502.18346v1 |
for sufficiently small δ>0and all t∈B(δ), we have |K†bpure (pure)(t)| ÉCp d(tTt)3 2and|K†bmix (mix)(t)| É |κ|µ1p d¶k−2 (tTt)k 2+Cµ1p d¶k−1 (tTt)k+1 2 Proof. This follows from the previous lemma by noting that for any jand any ℓ<j, we have (tTt)j 2µ1p d¶j−2 =(tTt)ℓ 2µ1p d¶ℓ−2 (tTt)j−ℓ 2µ1p d¶j−ℓ =(tTt)ℓ 2µ1p d¶ℓ−2µtTt d... | https://arxiv.org/abs/2502.18346v1 |
(13), we can prove our claim also for K†bpure (pure),r(pure) ,andr(mix) by relying on the bounds from Corollary 4.11 and Lemma 4.12. The above sequence of lemmas can be stacked together to prove Claim 4.9. 37 Proof of Claim 4.9. We only show our claim for |r1(t)|since the arguments for the other terms are analogous. We... | https://arxiv.org/abs/2502.18346v1 |
with |S| Êcdsuch that |Czi(t)| É1−ε(δ) for all i∈S. 39 Usind this, we bound Err 1(t)=Z B(δ)|C(t)|dt=Z B(δ)¯¯¯¯Cz1µtp d¶¯¯¯¯dY j=2¯¯¯¯Czjµtp d¶¯¯¯¯dt É(1−ε(δ))cd−1Z Rk¯¯¯¯Cz1µtp d¶¯¯¯¯dt =(1−ε(δ))cd−1dk/2Z Rk¯¯Cz1(s)¯¯ds É(1−ε(δ))cd−1dk/2³ dηp 2πσ´k =exp(−Ω(d)+O(log(d)))=exp(−Ω(d)). Bounding Err 2(x).To bound Err 2(x)=Z... | https://arxiv.org/abs/2502.18346v1 |
we assume C1Ê1 in the last inequality. Further, from (16) and the multinomial theorem, we get ∥γj∥∞ÉCj k+1 1maxn |κ|j kd−jk−2 2k,d−jk−1 2(k+1)oℓ/kX j=1ℓj ℓ!j! and by setting C:=Cj k+1 1Pj/k ℓ=1ℓj ℓ!j!(note that jÉb2s2is bounded by a constant) we have ∥γj∥∞ÉCmaxn |κ|j kd−j(k−2) 2k,d−j(k−1) 2(k+1)o =Cmaxn |κ|d−(k−2) 2|κ|... | https://arxiv.org/abs/2502.18346v1 |
breaks down in the general case. Lemma 5.1. For a cycle or a chain Hof length kwith edges e1,...,ek, we have E[SW(H)]=E" Y e∈H1(e)# −pk. Proof. First of all, note that multiplying out S W(H)=Q e∈H(1(e)−p) yields that E[SW(H)]=X S⊆H(−1)k−|S|E" Y e∈S1(e)# pk−|S|=E" Y e∈H1(e)# +X S⊂H(−p)k−|S|p|S|, 43 where we used that E£... | https://arxiv.org/abs/2502.18346v1 |
k\ i=1(z(i)Ɉτ)! =Pà k\ i=1(z(i)∈D)! +PÃà k[ i=1(z(i)< −log(n))! ∩à k\ i=1(z(i)Ɉτ)!! ÉPà k\ i=1(z(i)∈D)! +Pà k[ i=1(z(i)< −log(n))! ÉPà k\ i=1(z(i)∈D)! +kn−ω(1) by the tail bound from Lemma 3.8. To bound the remaining probability, we apply Theorem 2.4 and choose the parameters apure,amix,bpure,bmixall large enough suc... | https://arxiv.org/abs/2502.18346v1 |
not rely on signed triangles but instead uses an entropy-based argument built upon an explicit ε-net of Td with specific properties, which has the drawback of not giving a computationally efficient test. We analyze the signed triangle count T(G):=X i<j<k∈[n]Ti jkwhere Ti jk:=(Gi j−p)(Gik−p)(Gjk−p) yielding a more power... | https://arxiv.org/abs/2502.18346v1 |
note that by Lemma 3.7, we have E[1(e1)1(e2)1(e3)]=Pà 3\ i=1(r(i)Ɉτ)! ÊPà 3\ i=1(z(i)Ɉτ)! −Oµlog(n) dη−1¶ ÊPà 3\ i=1(z(i)Ɉτ)! −Oµp3 d¶ , since |˜τ| Élog(n) by Proposition 3.15 and since we can choose ηto be Ê1+ε/γsuch thatlog(n) dη−1Ép3 d, asdÊnγand pÊn−O(1). Now, we use our approximation ˜fto the density of zfrom T... | https://arxiv.org/abs/2502.18346v1 |
p3P(G12G13G23=1)−1=Oµlog(n)p d¶ , we can conclude the proof. With this, we prove Theorem 1.1. Proof of Theorem 1.1. Noting that E G∼G(n,p)[T(G)]=0 and Var G∼G(n,p)[T(G)]=¡n 3¢ p3(1−p)3, and using Lemma 6.4 as well as Lemma 6.5, ¯¯¯¯E G∼RGG q(n,d,p)[T(G)]−E G∼G(n,p)[T(G)]¯¯¯¯=Ωµn3p3 p d¶ while max(r Var G∼RGG q(n,d,p)[T... | https://arxiv.org/abs/2502.18346v1 |
k=0logà Ex,y"µ 1+γ(x,y) p(1−p)¶k#! , with γ(x,y)=Ez[(1e(x,z)−p)(1e(y,z)−p)], where 1e(x,z) denotes the indicator random variable for the event that ∥x−z∥qÉτand that x,y,z∼ Unif ([−1/2,1/2])d, independently. Expanding the above expectation, we obtain Ex,y"µ 1+γ(x,y) p(1−p)¶k# =kX j=0à k j! 1 pj(1−p)jEx,yh γ(x,y)ji =kX j... | https://arxiv.org/abs/2502.18346v1 |
yields P(|κ| Êt)É2expà −t2 2M2d+2 3Mt! É2expà max( −t2 4M2d,−t 4 3M)! so setting t=C′log(d/p2)p kdfor a suitable constant C′>0 yields that P(|κ| Êt)É2expà max( −C′2log2(d/p2)k 4M2,−C′log(d/p2)p kd 4 3M)! ɵp2 d¶2k since we assume that ω(nlog(n)) and kÉnsop kdÊk. This shows that |κ| Éβwith probability 1−³ p2 d´2k . Furt... | https://arxiv.org/abs/2502.18346v1 |
É1/a=o(1) (19) and we are left with bounding the above expectation. Conveniently, tr³¡ A−p11T¢m´ has a combina- torial interpretation. Namely, it can be expressed as a sum over all directed, closed walks of length mon the complete graph with vertex set [ n], where each term in the sum corresponds to the product of the ... | https://arxiv.org/abs/2502.18346v1 |
proceed by making our ideas formal. Constructing the Core H◦.The following definitions formalize which chains/cycles of vertices can be contracted/removed. Definition 8.2 (Chains) .Consider a Eulerian multigraph Hwith ℓedges. For every kÊ3, we call a sequence v1,v2,...,vkofdistinct vertices from V(H) achain if{vi,vi+1}... | https://arxiv.org/abs/2502.18346v1 |
algorithm, i.e., if and only if all its edges are part of some cycle in L. For this case, we define the skeleton eHofHas the version of Hwere all edges in degenerate cycles on vertices u,vare replaced by exactly two edges between u,v. It is furthermore easy to see that |E[SW(H)]| É |E£ SW(eH)¤ |since |E[SW(H)]| =Y e∈L|... | https://arxiv.org/abs/2502.18346v1 |
arbitrary additional edge from EUclosing a cycle whenever this is possible. Denote the number of edges in Fbytand define for every i∈{0,1,...,t}the event Eito be the event that the vertices of H◦are arranged such that exactly iedges in E(F) are present. We further denote by X=(x1,x2,...,x|V(H◦)|) and X=(x1,x2,...,x|V(H... | https://arxiv.org/abs/2502.18346v1 |
degenerate and non-degenerate cycles, respectively, we immediately get from Theorem 2.6 that E[SW(H)]ÉE£ SW(eH)¤ ÉpaY e∈LnCp|Ce|µlog(n)p d¶|Ce|−2 where we accounted for the contribution of degenerate cycles using the factor of pasince two edges between two vertices u,vineHthat form a degenerate cycle refer to the same ... | https://arxiv.org/abs/2502.18346v1 |
the following two sub-cases. Either Uitself is a tree (possibly containing some multi-edges), in which case t=v−1. But we also know that cÊ1 as otherwise contracting cycles would have eliminated all edges and H◦would be an isolated vertex, contradicting the assumption that H◦is non-trivial. This implies t+cÊvas desired... | https://arxiv.org/abs/2502.18346v1 |
all cases for Hwith trivial core. On the other hand, for those Hwith non-trivial core, we use Snon-triv :=mX ℓ=2ℓ/2X s=0ℓ−2sX r=4max{r/2,1}X v=1X [H]∈Unon-triv (ℓ,r,s,v)#[H]|E[SW([H])]| where we used that the number of cycles sis at most ℓ/2, and that the number of remaining edges r in the core H◦is at most ℓ−2ssince e... | https://arxiv.org/abs/2502.18346v1 |
that the above is O³ d³ nplog(n)p d´´ . To further ensure that mis even, we require mÊ2 ε+4. Then, if there were more than eO(dam)eigenvalues of magnitude Ênp ap d, this would lead to a contradiction. 8.2 A Different Threshold for L∞-norm We complement the results from Section 8.1 by studying the spectrum of G∼RGG ∞(n,... | https://arxiv.org/abs/2502.18346v1 |
weight of general graphs, these are not strong enough for our purposes since they do not yield tight bounds in the special case of chains. We prove the following while partly (i.e. in the case of cycles) relying on the bounds from [2]. Lemma 8.12. Consider G∼RGG ∞(n,d,p)with1 nÉpÉ1−ε. Then, for a cycle Ckof length kÊ3o... | https://arxiv.org/abs/2502.18346v1 |
Ép1/d±ξk=p1/d³ 1±p−1/dξk´ . Finally, applying the Taylor series of ξ=1−p1/dfrom (24) and using further that p−1/d=1+o(1) since pÊn−O(1)andd=ω(log(n)), we get that E G∼RGG ∞(n,1,p1/d)|G1" Y e∈H1(e)# Épk/dà 1+µ3log(1/ p) d¶k! . 67 Bad dimensions. For all bad dimensions, we use the same strategy but rely on a slightly wea... | https://arxiv.org/abs/2502.18346v1 |
(Bounding the m-th power of the Centered Adjacency Matrix (Again)) .There is a con- stant C>0such that for all even m∈N, Eh tr³³ A−p11T´m´i É (Clog(n)m)4m+4³ d2³ nplog(1/ p) d´m +npnpm´ ifd≪pnplog(1/ p) (Clog(n)m)4m+2pnpmifd≫pnplog(1/ p). Proof. We define Strivand Snon-triv as in the proof of Lemma 8.10. Applying L... | https://arxiv.org/abs/2502.18346v1 |
the position of all vertices in dimension i. Now, fix a sufficiently small constant c>0 and define the set Aas the set of all vertices which have distance at most cfrom the origin in dimension i, and similarly the set Bas the set of all vertices with distance at least1 2−cfrom the origin. More formally, we set A={v| |x... | https://arxiv.org/abs/2502.18346v1 |
get that µd d−1¶j/2 =µ 1+1 d−1¶j/2 =1+j 2(d−1)±Oµ1 d2¶ . Hence, for every sÊ2 there is a constant C>0 such that fq d d−1¯r(x)ÊÃ φ(x)+φ(x)s−2X j=1pj(x) dj/2! −φ(x)s−2X j=1C|x|3j d1+j/2−oµ1 d(s−2)/2¶ Êfr(x)−φ(x)s−2X j=1C|x|3j d1+j/2−oµ1 d(s−2)/2¶ , (27) where we noted that the expression in the parentheses is just the s-... | https://arxiv.org/abs/2502.18346v1 |
pi, while pi=(1+o(1))pby Lemma 8.18. Hence, Xi takes a value of 1 −piw.p. piand a value of −piw.p. 1 −pi. This implies that E G∼RGG q(n,d,p)|x(1)£ X2 i¤ Épi(1−pi)2+(1−pi)p2 iÉ2piÉ3p. Thus, by a Bernstein bound (Theorem 3.14), we have P G∼RGG q(n,d,p)|x(1)µ¯¯¯¯deg(u→A)− E G∼RGG q(n,d,p)|x(1)[deg(u→A)]¯¯¯¯Ê10p nplog(n)¶ ... | https://arxiv.org/abs/2502.18346v1 |
see that—since the length of our circular arcs is only ξ/2=Θ(log(1/ p)/d)—we can define Θ(d) orthogonal such yfor each dimension, such that in the end, we end up with Θ(d2) vectors in total. To prove that the vectors constructed this way are indeed approximate eigenvectors, we again show that inter-cluster edges are le... | https://arxiv.org/abs/2502.18346v1 |
we have a good x(1). Note that we also applied the fact that d≪pnpabove. An entirely analogous calculation shows that deg( u→B)=Ω¡np d¢ holds as well. Noting that ∥˜y1∥2=O¡n d¢ for good x(1), and that x(1)is good w.p. 1 −1/n2, we get that yT 1Ay1ÊΩ³np d´ holds with probability 1 −2/n2. Since this is true for each yi, w... | https://arxiv.org/abs/2502.18346v1 |
SIAM Journal on Discrete Mathematics , 38(2):1943–2000, 2024. 4, 5 [17] T. Friedrich, A. Göbel, M. Katzmann, and L. Schiller. Real-world networks are low-dimensional: theoretical and practical assessment. In Proceedings of the Thirty-Third International Joint Conference on Artificial Intelligence , pages 2036–2044, 202... | https://arxiv.org/abs/2502.18346v1 |
but constant γ>0. Setting a= −log(n) and applying the above, we thus get that for any b>0 sufficiently small, P¡ r(1)ÉΦ−1(p)−b¢ =P(r(1)Éa)+ZΦ−1(p)−b ag(x)dx. We immediately get that the first term is Én−ω(1)by Lemma 3.8. The second term can be bounded as ZΦ−1(p)−b ag(x)dxÉZΦ−1(p)−b aà 1+s−2X j=1pj(x) dj/2! φ(x)dx+Oµlog... | https://arxiv.org/abs/2502.18346v1 |
of i.i.d. RVs with mean 0 and absolute value bounded by some constant Ma.s. Therefore, applying Theorem 3.14 yields that P(r(1)Êclog(n))=P³p dr(1)Êcp dlog(n)´ Éexpà −c2dlog(n)2 2(dσ2+1 3Mcp dlog(n))! Éexpà −min( c2log(n)2 4M2,cp dlog(n) 4M)! =n−ω(1). Doing the same for the lower tail finishes the proof of the first par... | https://arxiv.org/abs/2502.18346v1 |
fx1,x3,λ(x) the density of Yx1,x3,λ=gx1,x3(x2,λ) for x2∼Unif (I2). Then, there exist constants M,ssuch that sup x1∈I1,x3∈I3 λ∈[0,1],x∈Rfx1,x3,λ(x)ÉM and sup x1∈I1,x3∈I3 λ∈[0,1]Var£ Yx1,x3,λ¤ És 9In case of a chain v1=vk+1. 84 With the above claim, our result follows using the following proposition from [6]. Proposition... | https://arxiv.org/abs/2502.18346v1 |
by the same εfor all j∈[d]. To prove that our statement holds for the same εover allj∈[d], we therefore derive an explicit upper bound on°°∂K(mix)(zj)°°using the relationship be- tween cumulants and mixed moments combined with mixed moments that signal rjunderlying z is bounded almost surely. Before this, however, we n... | https://arxiv.org/abs/2502.18346v1 |
Let us prove that E£ γ(e1)γ(e2)γ(e3)¤ <0 for all Lq-norms. Lemma A.10. Consider a triangle C3with vertices v1,v2,v3and edges e1,e2,e3. For all Lq-norms with qÊ1, we have that E£ γ(e1)γ(e2)γ(e3)¤ <0where γ({u,v}):= |xu−xv|q C−E£ |xu−xv|q C¤ andxv1,xv2,xv3∼ Unif ([−1/2,1/2])are the positions of v1,v2,v3in a fixed dimensi... | https://arxiv.org/abs/2502.18346v1 |
can split E[γ(e1)γ(e2)γ(e3)]=2Z1/2 0f1(y)f2(y)dy =2 Zy2 0f1(y)f2(y)dy |{z } =:I1+Zy1 y2f1(y)f2(y)dy |{z } =:I2+Z1/2 y1f1(y)f2(y)dy |{z } =:I3 . Now, I1É0 since f1(y)É0 and f2(y)Ê0 for y∈[0,y2] by monotonicity. Furthermore, we have f1(y)f2(y)Éf1(y)f2(y1) for all y∈[y2,1/2] because f2(y1) is the smallest (’most... | https://arxiv.org/abs/2502.18346v1 |
Learning sparse generalized linear models with binary outcomes via iterative hard thresholding Namiko Matsumoto Arya Mazumdar February 26, 2025 Abstract In statistics, generalized linear models (GLMs) are widely used for modeling data and can expressively capture potential nonlinear dependence of the model’s outcomes o... | https://arxiv.org/abs/2502.18393v1 |
(McCulloch 1997; Hardin and Hilbe 2007), where the estimates can be obtained through techniques such as iterative weighted least-squares methods (Nelder and Wedderburn 1972; Firth 1992; Hardin and Hilbe 2007), the Newton-Raphson method (Jin et al. 2022; Hardin and Hilbe 2007), and the Gauss-Newton method (Wedderburn 19... | https://arxiv.org/abs/2502.18393v1 |
this present work is motivated by the connection of binary GLMs to 1-bit compressed sensing, a topic within compressed sensing where the entries of the compressed signal representations are quantized to single bits: the ±signs of the unquantized values. The next section, Section 1.2, briefly introduces 1-bit compressed... | https://arxiv.org/abs/2502.18393v1 |
Matsumoto and Mazumdar (2024a) improves the sample complexity to the theoretically order-wise optimal (up to logarithmic factors) sample complexity: ˜O(k ϵlog(d k)p log(1 ϵ) +k ϵlog3/2(1 ϵ)), matching lower bound on the sample complexity for recovery in 1-bit compressed sensing established by Jacques et al. (2013b). On... | https://arxiv.org/abs/2502.18393v1 |
sample complexity were known only for the high noise regime, e.g., Plan et al. (2017), and the noiseless regime Matsumoto and Mazumdar (2024a), while no such optimal algorithms were known for the intermediate regimes1. Thus, to the best of our knowledge, BIHT is the first computationally efficient algorithm with the op... | https://arxiv.org/abs/2502.18393v1 |
to hold for Gaussian covariate matrices with high probability. Towards this, the quantity in Equation (3) will be shown to describe a notion of deviation of the random vector wˆθaround its mean in the sense that θ∗−wˆθ ∥wˆθ∥2 2= wˆθ ∥wˆθ∥2−E[wˆθ] ∥E[wˆθ]∥2 2.Furthermore, it can be shown that this deviation is roughly p... | https://arxiv.org/abs/2502.18393v1 |
are closer to the true parameter, θ∗. In essence, since the number of covariates—and hence also the variance—involved in the evaluation of hf;Jat a pair of points decreases as the distance between the points decreases, the improvement of the approximation in one iteration of BIHT leads to even better control over hf;J,... | https://arxiv.org/abs/2502.18393v1 |
the results in this manuscript extend to the dense regime by taking k=dandΘ = Sd−1. The covariates are d-variate i.i.d. standard Gaussian random vectors, written as x1, . . . ,xn∼ N(0,Id), which are stacked up into the covariate matrix, X= (x1···xn)T∈Rn×d. The unknown parameter vector which is being estimated is denote... | https://arxiv.org/abs/2502.18393v1 |
at z∈R by p(z) =1 1 +e−βz. (10) Definition 3.3. For the probit model with signal-to-noise ratio (SNR) β >0, the function p:R→[0,1]is given at z∈Rby p(z) =1√ 2πZu=βz u=−∞e−1 2u2du. (11) Note that, equivalently, pis simply the distribution function of a standard Gaussian random variable composed with multiplication by β.... | https://arxiv.org/abs/2502.18393v1 |
constants b1, b2>0can be different for the logistic and probit cases. Remark 4.1 . In Corollary 4.3, Algorithm 1 achieves the order-wise optimal sample complexity (up to logarithmic factors) for parameter estimation in logistic regression under the Gaussian design. See, Hsu and Mazumdar (2024) for the establishment of ... | https://arxiv.org/abs/2502.18393v1 |
relation is point-wise bounded from above to yield the rate of convergence and, consequently, the asymptotic convergence in the limit as t→ ∞ of the approximations produced by BIHT: ∥θ∗−ˆθ(t)∥2≤22−tϵ1−2−t, t∈Z≥0, lim t→∞∥θ∗−ˆθ(t)∥2≤ϵ, completing the proof of the main theorem. 5 Restricted Approximate Invertibility of G... | https://arxiv.org/abs/2502.18393v1 |
Finally, it will be interesting to analyze BIHT (and a stochastic perceptron-like version of it) from a learning theory perspective, especially in the agnostic setting, where data not necessarily comes from a GLM. Other noise models, such as Massart noise, can also be interesting. References J Aitchison and SD Silvey. ... | https://arxiv.org/abs/2502.18393v1 |
models with isotonic regression. Advances in Neural Information Processing Systems , 24, 2011. Adam Klivans and Raghu Meka. Learning graphical models using multiplicative weights. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS) , pages 343–354. IEEE, 2017. Felix Kuchelmeister and Sara van d... | https://arxiv.org/abs/2502.18393v1 |
a perceptron-like algorithm. In Proceedings of the Twelfth Annual Conference on Computational learning theory , pages 296–307, 1999. Pragya Sur and Emmanuel J Cand `es. A modern maximum-likelihood theory for high-dimensional logistic regression. Proceedings of the National Academy of Sciences , 116(29):14516–14525, 201... | https://arxiv.org/abs/2502.18393v1 |
BIHT requires that the GLM has binary outcomes with a mild condition on the link function, g, but otherwise need not know the specific choice of link function, while GLM-tron can learn a larger class of GLMs that only necessitates that the transfer function, g−1, satisfies a certain derivative condition (which indeed h... | https://arxiv.org/abs/2502.18393v1 |
is assumed to have unit norm, but the models incorporate SNR denoted by β >0. Formulating the (sparse) estimation problem as a convex program, Plan and Vershynin (2012) shows that the estimation of θ∗from binary responses is possible with ˜O(klog(d k) min{β2,1}ϵ4)samples under the Gaussian covariate design. Plan et al.... | https://arxiv.org/abs/2502.18393v1 |
considers an alternative (adversarially) noisy setting. As discussed in Section 2, the analysis in this work largely centers around an invertibility condition that uniformly bounds an expression of the form θ∗−ˆθ+hf;J(θ∗,ˆθ) ∥ˆθ+hf;J(θ∗,ˆθ)∥2 2(22) for all ˆθ∈Θand all J⊆[d],|J| ≤k. In contrast, Matsumoto and Mazumdar (... | https://arxiv.org/abs/2502.18393v1 |
(2024). This includes generalized approximate message passing (GAMP), an algorithm first proposed by Rangan (2011). While the error-rate of GAMP is information theoretically optimal for some GLM’s, it falls short of the information theoretical optimum for GLMs in some paradigms (Barbier et al. 2019). In fact, Barbier e... | https://arxiv.org/abs/2502.18393v1 |
+vw, t ∈Z+, f2(t) = 22−t(u2v)1−2−t, t∈Z≥0. Then, f1(t)> f1(t′), t < t′∈Z≥0, f2(t)> f2(t′), t < t′∈Z≥0, f1(t)≤f2(t), t∈Z≥0, lim t→∞f1(t)≤lim t→∞f2(t) =u2v. B.2 Proof of Theorem 4.1 With the above results in Appendix B.1, the convergence of the BIHT approximations, as stated in the main theorem, can now be proved. Proof ... | https://arxiv.org/abs/2502.18393v1 |
induction that for all t∈Z≥0, thetthinductive claim, C(t), holds: ∥θ∗−ˆθ(t)∥2≤f1(t). Therefore, the assumption that Equation (27) holds uniformly—which occurs with probability at least 1−ρ—and Equations (31) and (32) together imply that ∥θ∗−ˆθ(t)∥2≤f1(t)≤f2(t) = 22−tϵ1−2−t for every t∈Z≥0and that lim t→∞∥θ∗−ˆθ(t)∥2≤lim... | https://arxiv.org/abs/2502.18393v1 |
yields: T(J∪J′′)\J′(v) ∥TJ∪J′∪J′′(v)∥2 2=s u−TJ∪J′∪J′′(v) ∥TJ∪J′∪J′′(v)∥2 2 2− u−TJ′(v) ∥TJ∪J′∪J′′(v)∥2 2 2. (42) From Equation (42), it follows that ∥T(J∪J′′)\J′(v)∥2 ∥TJ∪J′∪J′′(v)∥2=s u−TJ∪J′∪J′′(v) ∥TJ∪J′∪J′′(v)∥2 2 2− u−TJ′(v) ∥TJ∪J′∪J′′(v)∥2 2 2(43) ≤ u−TJ∪J′∪J′′(v) ∥TJ∪J′∪J′′(v)∥2 2. (44) Combining Equations (39)... | https://arxiv.org/abs/2502.18393v1 |
and where θ∈ C \ B τ(θ∗)such that ∥θ−ˆθ∥2≤2τandsupp( θ)∪J= supp( ˆθ)∪J(see, Lemma C.2). Per the design of the τ-net,C ⊂Θ, in Step 2, such a point θ∈ Cexists for any choice of ˆθ∈Θ. 5. The three terms on the right-hand-side of Equation (52) can be viewed as bounding (50) by relating it (with appropriate scaling) to the ... | https://arxiv.org/abs/2502.18393v1 |
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