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the two-community stochastic block m odel does not exhibit computational-statistical gaps for partial and exact recovery w hen the edge density scales as Θ(n−1) [MNS12 ]; [MNS13 ]; [ABH16 ]; [Abb17 ]. Another well-studied problem is Gaus- sian biclustering, which has been analyzed in both detection and reco very settin... | https://arxiv.org/abs/2503.19069v2 |
exists between the amount of data required by computationally efficient algorithms and t he data needed for statistically optimal procedures. Rigorous evidence supporting th is computational-statistical gap has been established through various approaches, which can b e broadly classified into the following categories: 1.F... | https://arxiv.org/abs/2503.19069v2 |
of the parameters of the pr oblem at hand and will be reused for different constants. A graph Gis a pair (v(G),e(G)), wherev(G) is the vertex set and e(G)⊆/parenleftbigv(G) 2/parenrightbig is the edge set; |v(G)|and|e(G)|=|G|denote the sizes of thereof. We say H⊆Gis a subgraph of G ifv(H)⊆v(G) ande(H)⊆e(G). We treat sub... | https://arxiv.org/abs/2503.19069v2 |
and weak detection is impossible if limn→∞R⋆ n= 1. Our results will be expressed in terms of the following graph theoret ic measures. We letdmax(Γn) denote the maximum degree in Γ n, and we define the density of Γ nasη(Γn)/defines |e(Γn)|/|v(Γn)|. Finally, we recall the definition of the maximum subgraph density . 8 Defin... | https://arxiv.org/abs/2503.19069v2 |
evidence for the above claim, we use the framework of low- degree polynomials (see, e.g., [ HB18 ,KWB22 ]). We continue with a brief background on the low-degree polynomial (LDP) method, and refer the reader to Sec tion6, for a more detailed exposition. The LDP framework hings on the hypothesis that all poly nomial-tim... | https://arxiv.org/abs/2503.19069v2 |
can be detected in linear time with high proba- bility by counting the total number of edges, or evaluating the maxim um degree in the observed graph. 2. For Γ′swith super-logarithmic density µ(Γn) = Ω(log |v(Γn)|), there are three comple- mentary regimes: •The impossible regime: No test can detect the planted subgraph... | https://arxiv.org/abs/2503.19069v2 |
appear to take place. Specifically, it is sh own in Section 5.4.4 that the behavior of the statistical limits vary dramatically between three main regimes: µ(Γ)>1,µ(Γ) = 1−o(1), andµ(Γ)≤1−δ, for a fixed δ >0. Due to space limitation, we will focus on the case µ(Γ) = 1−o(1), where we observe a sharp phase transition pheno... | https://arxiv.org/abs/2503.19069v2 |
Given the adjacency matrix An∈ {0,1}n×n, define, Tcount(An)/defines/summationdisplay i<jAij, (21) Tdeg(An)/definesmax i∈[n]/summationdisplay j∈[n]Aij, (22) Tscan(An)/definesmax ¯Γ∈SΓmax/summationdisplay (i,j)∈¯ΓAij. (23) Accordingly, the corresponding tests are φcount/defines /BD{Tcount(An)≥τcount},φdeg/defines/BD{Tdeg(... | https://arxiv.org/abs/2503.19069v2 |
least one row i⋆such that/summationtext j∈[n]Ai⋆jis distributed as the independent sum of Binomial (dmax(Γ),p) andBinomial (n−1−dmax(Γ),q). Therefore, by the multiplicative Chernoff’s bound, we get, PH1(φdeg(A) = 0) = PH1/parenleftbigg max i∈[n]Wi(An)<τdeg/parenrightbigg (39) =/summationdisplay Γ0∈SΓPH1/parenleftbigg ma... | https://arxiv.org/abs/2503.19069v2 |
proving general informa tion-theoretic (or sta- tistical) lower bounds on the risk of the detection problem stated in S ection 2. For the sake of clarity and to transparently present the main ideas and steps of our proof, we begin with an overview of the key stages involved in our strategy. The missing d etails and pro... | https://arxiv.org/abs/2503.19069v2 |
the number of connected components of H, which we denote by m(H), and the vertex cover number τ(Γ), i.e., the minimal size of a vertex cover of Γ (see, Definition 4). Specifically, as we show in Lemma 4, PΓ[H⊆Γ]≤[2τ(Γ)]m(H)[dmax(Γ)]|v(H)|−m(H) (n−|v(Γ)|)|v(H)|/definesϑ(m(H),v(H)). (69) Accordingly, when ( 69) is applied ... | https://arxiv.org/abs/2503.19069v2 |
lim n→∞log (τ(Γ)dmax(Γ)) log|e(Γ)|=7 5>1, (77) which implies that there is a gap between the lower bound in Th eorem6and the upper bounds in Theorem 5. As it turns out, however, much more can be proved. To wit, note th at the graphs in the example above can be decomposed into two disjoint vcd-balanced graphs (Γ 1,Γ2), ... | https://arxiv.org/abs/2503.19069v2 |
(1 +λ2 M)|e(Γi∩Γ′ j)|/bracketrightig =/summationdisplay H⊆Γ′ jλ2|H|·PΓi[H⊆Γi], (83) which results in formula similar to the one in ( 68). Consequently, by applying the same ideas as in the proof of Theorem 6, it can be shown that ( 83) is bounded if, roughly speaking, (1 +χ2(p||q))µ(Γi)∧µ(Γj)·max/parenleftig/radicalb... | https://arxiv.org/abs/2503.19069v2 |
take a Fourier-analytic approach and decompose L(G) w.r.t. an or- thonormal polynomial basis. We denote by L2(H0) the Hilbert space of random variables over the probability space on which PH0is defined, with a finite second moment, and equipped with the inner product, /a\}⌊ra⌋k⌉tl⌉{tϕ(G),ψ(G)/a\}⌊ra⌋k⌉tri}htH0/definesEH0... | https://arxiv.org/abs/2503.19069v2 |
subset of edges and denote |H|/defines|e(H)|. Now, let us analyze the r.h.s. of ( 101), beginning with the following chain of equalities, /summationdisplay H⊆([n] 2)λ2|H|·PΓ1[H⊆Γ1]·PΓ2[H⊆Γ2] =/summationdisplay H⊆([n] 2)λ2|H|·PΓ1⊥ ⊥Γ2[H⊆Γ1,H⊆Γ2] (103) =/summationdisplay H⊆([n] 2)λ2|H|·PΓ1⊥ ⊥Γ2[H⊆e(Γ1∩Γ2)] (104) 25 =/sum... | https://arxiv.org/abs/2503.19069v2 |
in this subsection, we consider the dense regime, whereχ2(p||q) = Θ(1) (in particular, δ < p−q <1−δ, for someδ > 0), and provide a complete formal proof for the statistical lower bounds in Theorem 7. We will generalize this proof strategy in Subsection 5.4to other regimes. Theorem 7. Assume that λ2=χ2(p||q) = Θ(1) , an... | https://arxiv.org/abs/2503.19069v2 |
in Observation 1. For all 1 ≤i≤ℓ, let H(i) Udenote the subgraph induced by X1,...,X i, namely, H(i) U/defines(Vi,(Vi×Vi)∩E(HU)), (135) Vi/defines{X1,...,X i}. (136) We next prove that P/bracketleftig H(2m) U⊆Γ′/bracketrightig ≤/parenleftbigg2τd (n−k)2/parenrightbiggm . (137) Indeed, note that by Observation 1, and by... | https://arxiv.org/abs/2503.19069v2 |
(equivalently, connected set s) withℓvertices, one of which is a given vertex. We are now in a position to prove Lemma 5. Proof of Lemma 5.We begin our enumeration by first running over all possible sizes (i.e., number of vertices) of the connected components C1,...,Cm. We observe that any choice of the sizes of these c... | https://arxiv.org/abs/2503.19069v2 |
+λ2)µℓ(163) = 1 +k/summationdisplay ℓ=2ec√ ℓ/parenleftbigg(1 +λ2)µ·ed2 n−k/parenrightbiggℓ⌊ℓ/2⌋/summationdisplay m=1/parenleftbigg2τ2 ed2/parenrightbiggm (164) (b) ≤1 +Cε·k/summationdisplay ℓ=2/parenleftbigg(1 +ε)(1 +λ2)µ·ed2 n−k/parenrightbiggℓ⌊ℓ/2⌋/summationdisplay m=1/parenleftbigg2τ2 ed2/parenrightbiggm , (165) (16... | https://arxiv.org/abs/2503.19069v2 |
Theorem 6, which pro- vides the conditions under which the moment generating function of the intersection of two arbitrary subgraphs is bounded. Lemma 7. LetΓ1= (Γ1,n)nandΓ2= (Γ2,n)nbe two sequences of graphs. If, (1 +λ2)min(µ(Γ1),µ(Γ2)) n−min (|v(Γ1)|,|v(Γ2)|)max/parenleftig/radicalbig τ(Γ1)τ(Γ2)dmax(Γ1)dmax(Γ2),dmax... | https://arxiv.org/abs/2503.19069v2 |
of ΓiinKn. LetC > 0 be the smallest constant for which Theorem 6and Lemma 7hold simultaneously. For i=j, by Theorem 6, we know that, EH0/bracketleftbig (1 +λ2 M)|e(Γi∩Γi)|/bracketrightbig =O(1), (198) provided that, (1 +λ2 M)µimax(τidi,d2 i) n−ki≤C. (199) This, however, is satisfied when ( 177) holds because, (1 +λ2 M)µ... | https://arxiv.org/abs/2503.19069v2 |
i, we have that v∈Sjfor somej < i . In particular, for any edgeewhich includes vwe havee∈Ej, and therefore, e /∈/parenleftigg e(Γ)\i−1/uniondisplay ℓ=1Eℓ/parenrightigg ⊇Ei. (212) In particular v /∈Vi, which is a contradiction. Next, since the smallest degree (in Γ) of an y vertex inSiis at leastdmax(Γ)M−i M, we have,... | https://arxiv.org/abs/2503.19069v2 |
for, e.g., f(n) = logn(17 logn) =o(1) in ( 132). 5.4 Other regimes In this subsection, we will present and prove the most refined vers ion of our lower bounds, and subsequently analyze those bounds in the two distinct regimes t hat garnered the most attention in the literature: 1.Sparse regime , whereχ2(p||q) = Θ(n−α), ... | https://arxiv.org/abs/2503.19069v2 |
·1 |V|(249) (a) ≤e/parenleftbigg2|E| |V|/parenrightbigg|V|−1 ·1 2(|E|/|V|)(250) ≤e(2µ(G))|V|−2, (251) where (a) is because |E| ≤/parenleftbig|V| 2/parenrightbig . We are now in a position to prove Lemmas 8and9. Proof of Lemma 8.Recall that Sm,ℓ,jis the set of all subgraphs H⊆Γ containing no isolated vertices, and ℓvert... | https://arxiv.org/abs/2503.19069v2 |
the connected components . Specifically, we proceed as follows: we consider partitions in Par(ℓ,m), which dictate how the ℓvertices are distributed among the mdifferent connected components. Once a partition pis fixed, we proceed to choose mconnected components with the corresponding sizes in the following manner: First, ... | https://arxiv.org/abs/2503.19069v2 |
sum on the r.h.s. of ( 271) is dominated by (1 + ε)ℓ, and therefore, EH0[L(G)2]≤1 +C′ ε·k/summationdisplay ℓ=2/parenleftbigg2e(1 +ε)2(1 +λ2)µ−1λ2d2µ n−k/parenrightbiggℓ . (276) The above is bounded provided that, (1 +λ2)µ−1λ2d2µ n−k<C, (277) for some universal constant C > 0. Since ( 272) holds exactly when ( 274) domi... | https://arxiv.org/abs/2503.19069v2 |
vertex from each connected component of Γ, and therefore, we have L≤τ. On the other hand, from the definition of τwe haveτ≤k. Combining together, we get, k t≤L≤τ≤k≤t·L, (293) where the first inequity holds for a sufficiently large n, astis bounded with n. 49 We next evaluate ( 96) by running over subgraphs with fixed number... | https://arxiv.org/abs/2503.19069v2 |
+λ2 M)|e(Γi∩Γ′ j)|/bracketrightig1 M2. (313) Accordingly, in order to find the conditions under which ( 313) is bounded it remains to generalize the argument of Lemma 7, for our case. Let us follow the notation used in the proof of Lemma 7, whereτi=τ(Γi),di=dmax(Γi),µi=µ(Γi),ki=|v(Γi)|,¯k= min(k1,k2), and ¯µ= min(µ1,µ2... | https://arxiv.org/abs/2503.19069v2 |
=O(1) if ( 314) holds. Finally, note that for any i,j∈[M], the terms on the l.h.s. of ( 314) and ( 315) are bounded by the term on the l.h.s. of ( 312). This implies that ( 312), with an appropriate choice ofC > 0, is a sufficient condition for EΓi/bracketleftig (1 +λ2 M)|e(Γi∩Γ′ j)|/bracketrightig1 M2≤2, (333) and the... | https://arxiv.org/abs/2503.19069v2 |
the proof of Theorem 7, where we have used the vcd-balanced decomposition argument presented in Subsection 5.3.3. Assume that ( 335) holds, and let us denote, 2ρ/definesβ−min/parenleftbiggα ζ,1 +α 2δ+ζ,2 +α 2ǫ/parenrightbigg >0. (343) Using Proposition 3, consider the decomposition of Γ into Γ 1,..., ΓM, whereM=⌈ρ−1⌉, ... | https://arxiv.org/abs/2503.19069v2 |
to, 2(ǫ−2δ−ζ) 2δ+ζ−2ǫ<ζ δ, (359) and is equivalent to, ǫ>6ζδ+ζ2+ 8δ2 4δ+ 2ζ=(2δ+ζ)2+ 4δ2+ 2ζδ 2(2δ+ζ)=2δ+ζ 2+4δ2+ 2ζδ 2(2δ+ζ)= 2δ+ζ 2.(360) We conclude by noting that the condition ǫ>2δ+ζ 2dominates the condition ǫ>2δ+ζ 2. 57 The condition in ( 352) is satisfied for several non-trivial families of graphs. As an example ... | https://arxiv.org/abs/2503.19069v2 |
is vcd-balanced. Consider the case where Γ nis a clique with nβvertices. The bound in Theorem 9imply that detection is impossible if, β <min/parenleftbigg α,α+ 1 3/parenrightbigg . (370) This bound is loose. Indeed, as was shown in [ HWX15a ], detection is impossible if, β <min/parenleftbigg α,α 4+1 2/parenrightbigg , ... | https://arxiv.org/abs/2503.19069v2 |
the detection problem of a planted bipartite graph was studied. Specifically, it is shown in [RHS24b] that in the balanced case (where both sides of the bipartite graph have approxi- mately the same number of vertices), the computational and s tatistical thresholds for the pos- sibility and impossibility of detection ar... | https://arxiv.org/abs/2503.19069v2 |
(386) for some fixed ε>0. (b) Ifdmax(Γ) =O(1), then: i. Ifσ>2ed2, then weak detection is impossible if, |e(Γ)| ≤n1 2−ε, (387) while strong detection is possible if |e(Γ)| ≥n1 2+ε, (388) for someε>0. ii. Ifσ < 1andµ(Γ)≥1− |v(Γ)|−β, for 0< β≤1, then weak detection is impossible if, |v(Γ)| ≤log/parenleftig ed2 σ/parenrigh... | https://arxiv.org/abs/2503.19069v2 |
have τ2 λ2d2≤σk2 n=o(1), (410) and eλ2d2 n−k≥Ckβ≥(1 +o(1))ed2 σ>1. (411) In particular, the condition in ( 235) is satisfied. Similarly to the scenario of unbounded degrees, the condition in ( 236) is satisfied if, 2 logk−log(n−k) +k(1 + log(1 + ε)−logσ+ 2 logd)→ −∞, (412) which holds if, k≤log/parenleftig ed2 σ/parenri... | https://arxiv.org/abs/2503.19069v2 |
lower and upper bounds coincide. Theorem 11. LetΓ = (Γ n)nbe a sequence of graphs satisfying (418). Then, for all 0≤α<1 µ, weak detection is impossible if, |v(Γ)|=o/parenleftbig ngµ(α)/parenrightbig , (419) while strong detection is possible if, |v(Γ)|=ω/parenleftbig n1−α 2/parenrightbig , (420) where gµ(α)/defines/bra... | https://arxiv.org/abs/2503.19069v2 |
The concepts described below were developed through a fundamental sequence of works in the s um-of-squares optimization literature [ BHK+16,HB18 ,HS17 ,HKP+17]. We begin by outlining the fundamentals of of the LDP framework, adh ering to the nota- tions and definitions established in [ HB18 ,KWB22 ]. Recall that any dis... | https://arxiv.org/abs/2503.19069v2 |
independent copy of H′inKn. We remark that ( 436) also follows as a corollary of Lemma 15. 6.2 LDP analysis for planted subgraphs Our investigation begins with the following fundamental property: a ll planted graphs Γ with super-logarithmic density exhibit a (conjecturally) non-empty har d region, where detection is st... | https://arxiv.org/abs/2503.19069v2 |
nD√ D 2λ2/parenrightbiggm . (457) Clearly, the above expression is bounded provided that, λ2|v(Γ)|2D√ D 2 n<1⇐⇒ |v(Γ)|<√n λD√ D 4. (458) As for the last term on the r.h.s. of ( 454), we have, /summationdisplay √ 2D≤m≤2D/parenleftbigg|v(Γ)|2 n/parenrightbiggm/parenleftbiggm2λ2 2/parenrightbiggD ≤/parenleftbig 2Dλ2/paren... | https://arxiv.org/abs/2503.19069v2 |
we have used the Hardy-Ramanujan f ormula, and the sixth inequality is because we assume without loss of generality that λ2>1; otherwise, we may replaceλ2by 1 and proceed with the analysis. We now separate our analysis into two complementary cases, starting with the case where, 2τ2 ed2≥1 +α, (476) for some fixed α>0, in... | https://arxiv.org/abs/2503.19069v2 |
( 493) is bounded provided that, d2D2λ2 n−k<C. (497) Combining the above conditions, we obtain ( 467), which concludes the proof. 76 6.3 Reduction to the vertex cover-degree balanced case There is a gap between the computational lower bound in Theorem 15and the performance of the efficient algorithms (i.e., the count and... | https://arxiv.org/abs/2503.19069v2 |
Γ 1, then the probability above is clearly zero. We define an equivalence r elation on subgraphs of Γ′ 1, where two subgraphs are considered equivalent if and only if they a re isomorphic. Let [ H] denote the equivalence class of a subgraph H⊆Γ1, and let Pbe the set of all equivalence classes corresponding to graphs wit... | https://arxiv.org/abs/2503.19069v2 |
the condition in ( 531) is satisfied if (534) holds, which implies that /⌊ar⌈⌊lLn,≤D/⌊ar⌈⌊lH0is bounded provided that ( 482) holds. Next, we move to the complementary case where, 2τ1τ2 ed1d2≤1 +o(1), (535) for someo(1) function. Here, for a sufficiently large n, the inner sum on the r.h.s. of ( 530) is dominated by (1 + ε... | https://arxiv.org/abs/2503.19069v2 |
question, which forms the main a rgument in the proof. Proof of Proposition 6.First, we note that by Lemma 15, we have /⌊ar⌈⌊lLn,≤D/⌊ar⌈⌊l2 H0=/summationdisplay H⊆([n] 2) |H|≤Dλ2|H|PΓ[H⊆Γ]2(551) =EΓ⊥ ⊥Γ′ min(D,|e(Γ∩Γ′)|)/summationdisplay m=0/parenleftbigg|e(Γ∩Γ′)| m/parenrightbigg λ2m , (552) where Γ′is an independ... | https://arxiv.org/abs/2503.19069v2 |
≤/productdisplay i,j∈[M] E min(D,|Γi∩Γ′ j|)/summationdisplay m=0/parenleftbigg|Γi∩Γ′ j| m/parenrightbigg λ2m M2 1 M2 . (565) We now analyze the inner expectation on the r.h.s. of ( 565), for any given pair ( i,j). Let ϕ∈[0,1] be defined as ϕ/definesλ2 1+λ2, and as so λ2=ϕ 1−ϕ. Then, min(D,|Γi∩Γ′ j|)/summation... | https://arxiv.org/abs/2503.19069v2 |
regime where |Γi∩Γ′ j| ≥M2D, we havem⋆=cM2D, for some 0 < c≤1 (in fact, sinceM≥2, under the assumption |Γi∩Γ′ j| ≥M2Dwe have that m⋆=M2Dandc= 1). Thus, M2D/summationdisplay m=1/parenleftbiggM2|Γi∩Γ′ j| m/parenrightbigg λ2m≤M2D·/parenleftbiggM2|Γi∩Γ′ j| cM2D/parenrightbigg λ2cM2D(588) ≤M2D·/parenleftbiggeM2|Γi∩Γ′ j| cM2... | https://arxiv.org/abs/2503.19069v2 |
Then, it can be checked that the term on the r.h.s. of ( 607) is larger than the term on the r.h.s. of ( 606) for any, ˜λ2≥λ2 M,4/defines/bracketleftigg c−cDD/parenleftbiggM2 c/parenrightbiggM2 λ2cD M,3/bracketrightigg1 cD =D1 cD/parenleftig M2 c/parenrightigM2 cD cλ2 M,3=O(1). (608) Thus, combining ( 595) and ( 60... | https://arxiv.org/abs/2503.19069v2 |
(624) This, however, is satisfied when ( 498) holds because, ν(M,D)·max (τidi,d2 i) n−ki≤max 1≤j≤Mν(M,D)·max/parenleftbig τjdj,d2 j/parenrightbig n−kj≤C. (625) Next, fori/\⌉}atio\slash=j, we note that, ν(M,D) max/parenleftbig/radicalbig τidiτjdj,didj/parenrightbig n−min (ki,kj)≤ν(M,D) max(/radicalbig τiτjdidj,didj) n−mi... | https://arxiv.org/abs/2503.19069v2 |
an ( ǫ,δ,ζ )-polynomial sequence, and since|v(Γ)|= Θ(nβ) andD=O(log(n)) =no(1)for allℓ, ¯λMD·max/parenleftbig τ(Γℓ)dmax(Γℓ),d2 max(Γℓ)¯λMD/parenrightbig n−|v(Γℓ)|≤n−α 2+o(1)max/parenleftbig nǫβ+ρ,n2δβ−α 2+o(1)/parenrightbig n1−o(1)(644) =nmax(βǫ+ρ−1−α 2,2βδ−1−α)+o(1). (645) Note that by ( 643) we have max/parenleftig ... | https://arxiv.org/abs/2503.19069v2 |
and Wasim Huleihel. Reducibilit y and com- putational lower bounds for problems with planted sparse structu re. InPro- ceedings of the 31st Conference On Learning Theory , volume 75, pages 48–166, 06–09 Jul 2018. [BBH19a] Matthew Brennan, Guy Bresler, and Wasim Huleihel. Univers ality of compu- tational lower bounds fo... | https://arxiv.org/abs/2503.19069v2 |
with few eigenvalues using the schur–horn relaxation. SIAM Journal on Optimization , 28(1):735–759, 2018. [CDMF+09] Mireille Capitaine, Catherine Donati-Martin, Delphine F´ eral, et al. The largest eigenvalues of finite rank deformation of large wigner matrices: con vergence and nonuniversality of the fluctuations. The A... | https://arxiv.org/abs/2503.19069v2 |
on e hypergraph in another. Israel Journal of Mathematics , 105:251–256, 1998. [FK00] Uriel Feige and Robert Krauthgamer. Finding and certifying a large hidden clique in a semirandom graph. Random Structures and Algorithms , 16(2):195– 208, 2000. [FK15] Alan Frieze and Micha/suppress l Karo´ nski. Introduction to Rando... | https://arxiv.org/abs/2503.19069v2 |
Yihong Wu, and Jiaming Xu. Computational low er bounds for community detection on random graphs. In Proceedings of The 28th Conference on Learning Theory , volume 40, pages 899–928, 03–06 Jul 2015. [HWX15b] Bruce Hajek, Yihong Wu, and Jiaming Xu. Computational low er bounds for community detection on random graphs. In ... | https://arxiv.org/abs/2503.19069v2 |
of Proceedings of Machine Learning Research , pages 5573–5577. PMLR, 12–15 Jul 2023. [Mon15] Andrea Montanari. Finding one community in a sparse graph. Journal of Statistical Physics , 161(2):273–299, 2015. [MPW15] Raghu Meka, Aaron Potechin, and Avi Wigderson. Sum-of- squares lower bounds for planted clique. In Procee... | https://arxiv.org/abs/2503.19069v2 |
al. Local algorithms for indep endent sets are half-optimal. The Annals of Probability , 45(3):1543–1577, 2017. [Sas14] Igal Sason. Bounds on f-divergences and related distanc es.CCIT Report , (859), 2014. [SW22] Tselil Schramm and Alexander S. Wein. Computational barrier s to estimation from low-degree polynomials. Th... | https://arxiv.org/abs/2503.19069v2 |
independentlyat random fr om SΓ1andSΓ2, respectively. Assume further that dmax(Γ1) =O(logn)and|e(Γ2)|=O(n1−ε), for someε>0. Then, for any constant c>0, EΓ1⊥ ⊥Γ2/bracketleftbig (1 +λ2)c·|e(Γ1∩Γ2)|/bracketrightbig = 1 +o(1). (656) Proof of Lemma 16.Throughout this proof, let kidenote the number of vertices in Γ iand dide... | https://arxiv.org/abs/2503.19069v2 |
P[Sℓ=sℓ]·1 n−k2·/summationdisplay j≤ℓ {ℓ+1,j}∈e(ΓF 2)d1 (677) =/summationdisplay xℓ∈AℓP[Xℓ=xℓ] P[Sℓ=sℓ]·d1·degp ΓF 2(ℓ+ 1) n−k2(678) =pℓ+1 (679) where (a) follows since conditioned on {Xℓ=xℓ},Zℓ+1= 1 only if Xℓ+1is in the Γ 0- neighborhood of xjfor somej≤ℓsuch thatxj∈[k1]. Namely, there are at most /summationdisplay j≤... | https://arxiv.org/abs/2503.19069v2 |
bounded. While at first sight, this might seem like a promising approach, it is quite lo ose. To illustrate this, let us consider the case of planted path in the dense regime; ou r bounds in this paper show that the detection problem in this case is impossible whenever |v(Γ)|=o(n), which is rather an intuitive result. W... | https://arxiv.org/abs/2503.19069v2 |
and the one that reverses the direction of the path. U sing Lemma 21 |Aut(H)|=ℓ/productdisplay i=12hi·hi! = 2/summationtextm i=1hiℓ/productdisplay i=1hi! = 2mℓ/productdisplay i=1hi!. (705) Next, we turn to evaluate N(H,Γ), i.e., the number of copies of Hin Γ. We count this number of copies as follows. First, we go over... | https://arxiv.org/abs/2503.19069v2 |
Lean Formalization of Generalization Error Bound by Rademacher Complexity Sho Sonoda1,5sho.sonoda@riken.jp Kazumi Kasaura2,5kazumi.kasaura@sinicx.com Yuma Mizuno3,5mizuno.y.aj@gmail.com Kei Tsukamoto4,5milanotukamoto@g.ecc.u-tokyo.ac.jp Naoto Onda2,5naoto.onda@sinicx.com 1RIKEN AIP 2OMRON SINIC X Corporation 3Universit... | https://arxiv.org/abs/2503.19605v2 |
Conventions in Measure-Theoretic Probability Aprobability space (Ω,M, µ)is a triple composed of measurable space Ω,σ-algebra M, and finite positive measure µsatisfying µ(Ω) = 1. Arandom variable X: Ω→ Xis a measurable map from Ωto a measurable space X. Arealization x∈ X of a random variable Xis an image X(ω)ofXat a cer... | https://arxiv.org/abs/2503.19605v2 |
manipulate. For example, in deep learning, the hypothesis class is a set of deep neural networks (DNNs). Alearning algorithm Ais a measurable map Xn→ Fthat describes how to associate datasets with hy- potheses. Regarding the data generation process by concepts as a forward process , learning algorithm A corresponds to ... | https://arxiv.org/abs/2503.19605v2 |
generalization gap ∆(f)by using theRademacher complexity , which estimates the second meaning of generalization error. We note that, as clarified in the remark (Remark 2), an upper bound of the gap L(f)−L(f|X)≤Bcan be trivially turned into the upper bound of the risk L(f)≤L(f|X) +B, which estimates the first meaning of... | https://arxiv.org/abs/2503.19605v2 |
≤b) {t :R} (ht : 0 ≤t) (ht’ : t * b ^ 2 ≤1 / 2) {ε:R} (hε: 0≤ε) : (µn(fun ω7→2·rademacherComplexity n f µX +ε≤ uniformDeviation n f µX (X◦ω))).toReal ≤ (-ε^ 2 * t * n).exp := by Remark 1.In the main theorem, we assume that a hypothesis includes a loss function . Namely, in the example of image recognition, a hypothesis... | https://arxiv.org/abs/2503.19605v2 |
) (hf : ∀i, Measurable (f i ◦X)) {b :R} (hb : 0 ≤b) (hf’: ∀i x, |f i x| ≤b) : µn[fun ω: Fin n →Ω7→uniformDeviation n f µX (X◦ω)]≤ 2·rademacherComplexity n f µX := by Finally, the combination of the estimates (7) and (13) yields the assertion. 5 McDiarmid’s Inequality McDiarmid’s inequality, a.k.a. the bounded differenc... | https://arxiv.org/abs/2503.19605v2 |
in Mathlib.Probability.Moments.SubGaussian at the same time of the first submission of this draft. 7 7 Related Work 7.1 Formalization of Machine Learning Theory Bagnall and Stewart (2019) [3] have also formalized a generalization error bound in Coq via Hoeffding’s inequality, which is limited to finitehypothesis class,... | https://arxiv.org/abs/2503.19605v2 |
pages 5–17, New York, NY, USA, 2021. Association for Computing Machinery. [10] Matus Telgarsky. Deep learning theory lecture notes, 2021. URL: https://mjt.cs.illinois.edu/ dlt/. [11] Koundinya Vajjha, Avraham Shinnar, Barry Trager, Vasily Pestun, and Nathan Fulton. CertRL: for- malizing convergence proofs for value and... | https://arxiv.org/abs/2503.19605v2 |
No-prior Bayesian inference reIMagined: probabilistic approximations of inferential models Ryan Martin∗ March 26, 2025 Abstract When prior information is lacking, the go-to strategy for probabilistic inference is to combine a “default prior” and the likelihood via Bayes’s theorem. Objective Bayes, (generalized) fiducia... | https://arxiv.org/abs/2503.19748v1 |
all of these approaches as no-prior Bayes solutions . A difficulty with Bayesian-like probabilistic inference in this no-prior setting concerns the interpretation of the posterior probabilities themselves. When genuine prior informa- tion is available, then the Bayesian posterior probabilities are the unique coherent u... | https://arxiv.org/abs/2503.19748v1 |
herits some—but necessarily not all—of the original possibilistic IM’s strong reliability properties: in particular, the corresponding credible sets for the full parameter are exact confidence sets. There’s just only so much that can be achieved by constructing default priors and checking if the corresponding posterior... | https://arxiv.org/abs/2503.19748v1 |
Z. The set C(Π) is called the credal set and all (coherent) upper probabilities have one. Aside from being relatively simple, a key advantage to possibilistic uncertainty quantification is that the associated credal set has a statistically oriented characterization, which is important for the statistical developments b... | https://arxiv.org/abs/2503.19748v1 |
(6), the test “reject HifΠx(H)≤α” controls the Type I error probability at level α. Third, the above property ensures that the possibilistic IM is safe from false confidence (Balch et al. 2019; Martin 2019, 2024b), unlike default-prior Bayes and fiducial solutions. Further insights about IMs are presented in Appendix A... | https://arxiv.org/abs/2503.19748v1 |
Q⋆ x-credible sets are also confidence sets, by definition/construction. What does it take to satisfy (9)? From the characterization (8) in Theorem 1, it is relatively easy to see that equality in (9) amounts to a choice of kernel Kβ xthat is fully supported on ∂Cβ(x), the boundary of Cβ(x), and marginal Mx=Unif(0,1). ... | https://arxiv.org/abs/2503.19748v1 |
before the inner probabilistic approximation was available more generally, so my goal 7 there was different than it is here. But the result established there states that the stan- dard no-prior Bayes solution—namely, the Bayes posterior with respect to the invariant right Haar prior—is precisely the inner probabilistic... | https://arxiv.org/abs/2503.19748v1 |
contour is πx(θ) =Pθ{R(X, θ)≤R(x, θ)}= 1−FD (x−θ)⊤v−1(x−θ) , θ∈RD. This is exactly the Gaussian possibility contour with mean xand covariance matrix v. More generally, by the famous theorem of Wilks (1938), the relative likelihood R(Xn, θ), as a function of iid samples Xn= (X1, . . . , X n) from Pθ, will be asymptoti... | https://arxiv.org/abs/2503.19748v1 |
theorem says. 4.3 Marginalization risks, or lack thereof Except for the impossibly rare cases where a genuine prior distribution is available, reli- able statistical inference is inherently imprecise—those familiar test and confidence proce- dures that control error rates all have an imprecise probabilistic characteriz... | https://arxiv.org/abs/2503.19748v1 |
above, quantifying uncertainty about Θ, given X=x. One way—but not the only way, see Section 6—to carry out possibilistic marginalization is by applying the general extension operation (e.g., Zadeh 1975a,b, 1978). Extension yields a new marginal IM Πm xfor Φ = m(Θ) with possibility contour πm x(ϕ) = sup θ:m(θ)=ϕπx(θ), ... | https://arxiv.org/abs/2503.19748v1 |
is, in fact, the inner probabilistic approximation Q⋆ xofΠxdefined in Section 3 above. Therefore, the same computational strategy for (approximately) sampling from Q⋆ xadvanced in Martin (2025) can be reIMagined here as an implementation the new no-prior Bayes solution. The strategy in Martin (2025) is, roughly, to sti... | https://arxiv.org/abs/2503.19748v1 |
the goal is marginal inference on the difference Φ = m(Θ) = Θ 21−Θ11of the two means. The problem is straightforward if the two variances are known or if their ratio is known, but the fully unknown variance case has remained elusive, i.e., lots of candidate solutions are available but, despite its apparent simplicity a... | https://arxiv.org/abs/2503.19748v1 |
marginal IM implies, e.g., that credible intervals for Φ based on this “posterior” are exact confidence intervals. The only wrinkle is that the relative profile likelihood doesn’t have a closed-form expression in the Behrens–Fisher problem, and its distribution depends on a nuisance parameter; this is precisely what ma... | https://arxiv.org/abs/2503.19748v1 |
that prioritizes reliability, insisting on calibration of its data-dependent degrees of belief. The kind of calibration IMs require makes them incompatible with probabilistic Bayesian inference, and it’s for this reason that the IM’s output possibilistic, i.e., takes the mathematical form of a possibility measure. Howe... | https://arxiv.org/abs/2503.19748v1 |
hypothesis Hsuch that H̸∋Θ and PΘ{QX(H)> τ}> ρ. This is called false confidence because, with τlarge and ρnot small, the posterior tends (at non-negligible rate ρ) to be confident (high probability τ) in the truthfulness of H even though His actually false. There’s a sense in which this result is trivial: take H 16 equ... | https://arxiv.org/abs/2503.19748v1 |
a mathematically rigorous framework of uncer- tainty quantification are those that can be described by an imprecise probability of some sort. Possibility theory is arguably one of the simplest imprecise probability theories and, since the possibilistic IM framework that I’ve been advancing has close connections to p-va... | https://arxiv.org/abs/2503.19748v1 |
the main paper and, therefore, the equivalent uniform calibration property in (17) above. The point is that, if ΩX({Θ}) is bounded away from 1 for almost all X, then it can’t be stochastically no less than Unif(0,1). Moreover, if one was able to find a non-possibilistic IM (Ωx,Ωx) that does satisfy the validity propert... | https://arxiv.org/abs/2503.19748v1 |
a new coordinate system on Xwhich will be useful for us in what follows. Identify x∈Xwith ( gx, ux), where ux∈Udenotes the label of orbit Gxandgx∈Gdenotes the position of xon the orbit Gx. Assumptions. Let{pθ:θ∈T}be a family of densities invariant with respect to a locally compact topological group Gin the sense of (18... | https://arxiv.org/abs/2503.19748v1 |
in Martin (2022b) says the IM construction should condition on the observation U=u. Write πg|u(θ) for the possibilistic IM’s contour, where the subscript is meant to emphasize that the conditional distribution of X≡(G, U), given the observed value of u, is used in the validification step (2). Then the IM contour is def... | https://arxiv.org/abs/2503.19748v1 |
are square ν-integrable. Assumptions. The parameter space Tis open and there exists a vector ˙ sθ(x) ={˙sθ,d(x) : d= 1, . . . , D }, with ˙ sθ,dan element of L2(ν), such that the following conditions hold: A1. the maps θ7→˙sθ,dfromTtoL2(ν) are continuous for each d= 1, . . . , D ; A2. at each θ∈T, Z sθ+u(x)−sθ(x)−u⊤˙sθ... | https://arxiv.org/abs/2503.19748v1 |
construct a confidence disc Cα(x) and a confidence distribution Q⋆ x= N2(x, I). Consider marginalization via the familiar non-linear function m(θ) =∥θ∥2. I say “familiar” because this is an infamous example going to back at least to Stein (1959) 22 0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0 Confidence levelPosterio... | https://arxiv.org/abs/2503.19748v1 |
a root to the function gα(ξ) := max θ∈∂Cξ α(x)πx(θ)−α. Design of an iterative algorithm to find this root requires care, primarily because evaluat- ingπxis expensive; so the goal is to evaluate gα(ξ) with as few πxevaluations as possible. Cella and Martin (2024) propose to represent the boundary of Cξ α(x) by 2 D-many ... | https://arxiv.org/abs/2503.19748v1 |
suggested in Martin (2025) and Section 5 above, based on the variational approximation strategy developed in Cella and Martin (2024). A relevant feature of the exact IM solution is that, when Θ is relatively close the extremes, i.e., Θ ≈ ±1, the IM’s possibility contour tends to be very asymmetric. Conse- quently, appr... | https://arxiv.org/abs/2503.19748v1 |
5: Comparison of Jeffreys-prior Bayes and the possibilistic IM’s inner probabilistic approximation. In Panel (a), the solid line is the Bayes posterior density and the bars are based on samples from the inner probabilistic approximation. In Panel (b), the black and gray lines correspond to the Bayes and inner probabili... | https://arxiv.org/abs/2503.19748v1 |
distribution functions of UXfor each of the two methods. Being above the diagonal line indicates that the elliptical credible sets derived from the method’s posterior samples have frequentist coverage probability below the nominal level; similarly, being on or below the line indi- cates that the credible sets have freq... | https://arxiv.org/abs/2503.19748v1 |
(1973). Marginalization paradoxes in Bayesian and structural inference. J. Roy. Statist. Soc. Ser. B , 35:189–233. With discussion and reply by the authors. De Cooman, G. (1997). Possibility theory. I. The measure- and integral-theoretic ground- work. Internat. J. Gen. Systems , 25(4):291–323. De Cooman, G. and Aeyels,... | https://arxiv.org/abs/2503.19748v1 |
J.-L., editors, Past, Present, and Future of Statistical Science , chapter 22. Chapman & Hall/CRC Press. Fraser, D. A. S., B´ edard, M., Wong, A., Lin, W., and Fraser, A. M. (2016). Bayes, reproducibility and the quest for truth. Statist. Sci. , 31(4):578–590. Fraser, D. A. S., Reid, N., and Wong, A. (1997). Simple and... | https://arxiv.org/abs/2503.19748v1 |
Martin, R. (2021a). An imprecise-probabilistic characterization of frequentist statistical inference. arXiv:2112.10904 . Martin, R. (2021b). Inferential models and the decision-theoretic implications of the validity property. arXiv:2112.13247 . Martin, R. (2022a). Valid and efficient imprecise-probabilistic inference w... | https://arxiv.org/abs/2503.19748v1 |
models. J. Roy. Statist. Soc. Ser. B , 44(3):322–352. With discussion. Shafer, G. (1987). Belief functions and possibility measures. In Bezdek, J. C., editor, The Analysis of Fuzzy Information, Vol. 1: Mathematics and Logic , pages 51–84. CRC. Staicu, A.-M. and Reid, N. M. (2008). On probability matching priors. Canad.... | https://arxiv.org/abs/2503.19748v1 |
Interpretable Deep Regression Models with Interval-Censored Failure Time Data Changhui Yuan School of Mathematics, Jilin University and Shishun Zhao School of Mathematics, Jilin University and Shuwei Li School of Economics and Statistics, Guangzhou University and Xinyuan Song Department of Statistics, Chinese Universit... | https://arxiv.org/abs/2503.19763v1 |
a nuclear-norm-based imputation method, extending the work of Katzman et al. (2018) to handle missing covariates. Recently, motivated by DNN’s powerful approximation ability, nonparametric and par- tially linear regression models have revived in survival analysis (Xie and Yu , 2021; Zhong et al., 2022; Sun and Ding, 20... | https://arxiv.org/abs/2503.19763v1 |
popular partially linear PH and proportional odds (PO) models as specific cases and is highly explanatory while maintaining sufficient flexibility. We em- ploy monotone splines and DNN to approximate the cumulative baseline hazard function and other nonparametric components. We resort to sieve maximum likelihood estima... | https://arxiv.org/abs/2503.19763v1 |
data, where the exact failure time of interest Tcannot be obtained but is only known to lie in a specific interval. More specifically, let ( L, R] be the interval that brackets Twith L < R . Clearly, Tis left censored ifL= 0 and right censored when R=∞. Define δL= 1 if Tis left censored, δR= 1 when Tis right censored, ... | https://arxiv.org/abs/2503.19763v1 |
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