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2.1. In particular, we show that Algorithm 1 returns, with probability at least 1−δ, a hypothesis hwhose 0-1 error is O(OPT) + ϵ. Proof of Theorem 2.1. Letw∗(1), . . . ,w∗(K)∈Rdandt∗ 1, . . . , t∗ K∈Rbe the parameters of the affine functions of an optimal classifier f∗, i.e., f∗(x) =argmaxi∈[K](w∗(i)·x+t∗ i)and denote ... | https://arxiv.org/abs/2502.09525v2 |
i=1∥(b(i))Vt+1∩⊥Vt∥2. 26 Using the dual characterization of the norm, we can further bound KX i=1∥(b(i))Vt+1∩⊥Vt∥2≥KX i=1 b(i)·(v(t))⊥Vt 2 Note that from Cauchy-Schwarz inequality we have that KX i=1 b(i)·(v(t))⊥Vt 2 ≥ KX i=1(b(i)·w)(b(i)·(v(t))⊥Vt) 2 , where we used thatPK i=1 b(i)·w 2= 1from Pythagorean theorem. Ther... | https://arxiv.org/abs/2502.09525v2 |
in each iteration we add at most poly(K,1/ϵ)vectors to our current vector set Lt, hence the dimension ktofVtsatisfies kt= poly( K,1/ϵ)for all t= 1, . . . , T. Assume that for every t= 1, . . . , Tthe error satisfies Pr(x,y)∼D[ht(x)̸=y]> C·OPT +ϵ. Note that since kt=poly(K,1/ϵ)for all t∈[T], if we choose ϵ′=poly(ϵ/K)for... | https://arxiv.org/abs/2502.09525v2 |
prove it by induction. Note that for n= 1the hypothesis holds trivially. Assume as an inductive hypothesis that pn−1≤4p1/(n−1)2for some n≥2. We will prove that pn≤4p1/n2forn≥2. Fixi∈[n2]and denote by Athe event that s(x)≥γand by B(n) ithe event that s(z(i))≥nγ. Note that Pr A, B(n) i,\ j̸=iB(n) j ≥Pr[A, B(n) i]−n2X... | https://arxiv.org/abs/2502.09525v2 |
conditions that we assume in order to obtain efficient algorithms. Definition 3.2 (Well-Behaved Multi-Index Model) .Letd, K, m ∈Z+, α, ζ, σ, τ ∈(0,1)andY ⊆Z of finite cardinality. We say that a class of functions FfromRd7→ Yis a family of concept classes of(m, ζ, α, K, τ, σ, Γ)-well-behaved Multi-Index Models if, for e... | https://arxiv.org/abs/2502.09525v2 |
that, by leveraging our assumptions, there exists a αfraction of finite width cylinders that exhibit this property. Condition 3(a) of Definition 3.2 further provides an approximation factor that the optimal function within the subspace Vcan achieve. This function can be approximated using a non- parametric regression a... | https://arxiv.org/abs/2502.09525v2 |
a distribution DoverRd× Ywhose x-marginal isN(0,I). Output: A set of unit vectors E. 1. Let t, ηbe sufficiently small polynomials in ϵ, α, σ, 1/K,1/Γ,1/m,1/|Y|. 2. Construct the empirical distribution bDof{(x(1), y1), . . . , (x(N), yN)}. 3.Construct S, anϵ′-approximating partition with respect to V(Definition 2.2). 4.... | https://arxiv.org/abs/2502.09525v2 |
Multi- Index Models according to Definition 3.2. Let Vbe ak-dimensional subspace of Rd, with k≥1, and letSbe an η-approximating partition of Vaccording to Definition 2.2, with η=ϵ3/(C|Y|Γ2k2), for C >0a sufficiently large universal constant. Suppose that for f∈ FwithPr(x,y)∼D[f(x)̸=y]≤ζthere exists no function g:V→ Ysu... | https://arxiv.org/abs/2502.09525v2 |
polynomial p:U→Rof degree at most mand a label y∈ Ysuch that E x,y′[p(xU)1(y′=z)|xV=xV 0]> σ∥p(xU)∥2. However, since the moments of y′are identical when conditioned on each xV 0that belongs to the same region in S, we deduce that if the moment condition holds for some polynomial at a particular point xV 0, then it must... | https://arxiv.org/abs/2502.09525v2 |
(m, ζ+O(ϵ), α, K, τ, σ, Γ)-well-behaved Multi-Index Models as defined in Definition 3.2. Let Vbe ak-dimensional subspace of Rd, with k≥1, and let Sbe an η-approximating partition of V(see Definition 2.2), where η=ϵ3/(C|Y|Γ2k2)andC >0is a sufficiently large universal constant. Let f∈ Fbe a function that depends only on ... | https://arxiv.org/abs/2502.09525v2 |
for each region S∈ Tthe associated set US (see Line 5 of Algorithm 4) contains a vector that correlates with W. 37 Claim 3.9 (Existence of Correlating Vectors) .LetC >0be a sufficiently large universal constant. FixS∈ T. If the number of samples that fall in SisNS≥(dm)Cmlog(|Y|/δ)/ηC, then with probability at least 1−δ... | https://arxiv.org/abs/2502.09525v2 |
λi(ej·u(i))2≥σ2/(4K|US,z|). This implies that (ej·u(i))2≥σ2/(4|US,z|Ktr(M))asMis positive definite. Therefore, by Fact B.3, we have that|US,z|=poly(mK/σ ),tr(M) =O(m)which implies that |ej·u(i)| ≥poly(σ/(mK)). Finally, note that since ej·u(i)̸= 0, we have that u(i)cannot belong in the space ES. Thus, there exists a uni... | https://arxiv.org/abs/2502.09525v2 |
quantifying the improvement at every step, we consider the following potential function Φt=KX i=1 (w∗(i))⊥Vt 2 . Note that from Lines 1 and 3 of Algorithm 3, we perform at most poly(mKΓ|Y|/(ϵασ))iterations. Furthermore, in each iteration, we update the vector set with at most poly(mKΓ|Y|/(ϵασ))vectors (see Proposition ... | https://arxiv.org/abs/2502.09525v2 |
probability 1/2. Therefore, by the mean value theorem we have that (OPT + ϵ2) log(1 /(OPT + ϵ2) + 1) ≤(OPT) log(1 /(OPT) + 1) + ϵ2log(1/OPT + 1) ≤(OPT) log(1 /(OPT) + 1) + ϵ . Hence, by applying Proposition 4.2 we have that the class of intersections of K-halfspaces is a class of (2,OPT +ϵ, ϵ, K, KO (OPT log (1/OPT + 1... | https://arxiv.org/abs/2502.09525v2 |
less than ϵ, i.e., Pr[x∈U]< ϵ. Then, by the definition of the piecewise constant classifier we can write Pr(x,y)∼D[h(x)̸=y] =E xV 0 min i∈{0,1}Pr(x,y)∼D i̸=y|xV=xV 0 . Note that for every xV 0not in U, it holds that min i∈{0,1}Pr(x,y)∼D i̸=y|xV=xV 0 ≤(C−4)K τ Pr(x,y)∼D f(x)̸=y|xV=xV 0 +ϵ. Therefore, by taking... | https://arxiv.org/abs/2502.09525v2 |
1) , T(x, y):=1(f(x)̸=y, y= 1). Note that I1is the second moment of f(x), i.e., the noiseless classifier, and I2is the contribution corresponding to the noise. First, we show the following tail bounds, which will help us bound the contribution corresponding to the noise. Claim 4.5. Letϵ∈(0,1)less than a sufficiently sm... | https://arxiv.org/abs/2502.09525v2 |
to the noise, using Claim 4.5. Claim 4.7. For every unit vector v∈Rd, it holds that |v⊤I2v|=O(OPT log(1 /OPT)). Proof.Letv∈Rdbe a unit vector, and denote by R(x) =E(x,y)∼D[S(x, y)|x]. Note that R(x)∈[0,1]for all x∈Rd, and moreover Ex∼N(0,I)[R(x)]≤OPT. Hence, by Claim 4.5 we have that|E(x,y)∼D[(v·x)2S(x, y)]|=O(OPT log ... | https://arxiv.org/abs/2502.09525v2 |
K, O (ζ+ϵ),poly(ϵ/K), O(Kp log(K)))-well-behaved. As a result, by applying Theorem 4.9, we obtain an efficient algorithm for learning under Random Classification Noise (RCN). Moreover, for the class of multiclass linear classifiers Algorithm 4 performs linear regression under Gaussian marginals (see Line 4). This task ... | https://arxiv.org/abs/2502.09525v2 |
vector must have a significant component in W. Claim 4.11. FixS∈ S. Suppose that the number of samples falling in Ssatisfies NS≥ (dm)Cmlog(|Y/δ|)/ηCfor a sufficiently large universal constant C >0. Then, for any u∈USand any unit vector v∈W⊥we have that |v·u| ≤√ 2Kmη/σ. Proof.LetNS, S∈ S, be the number of samples that l... | https://arxiv.org/abs/2502.09525v2 |
|US|=poly(mK|Y|/σ)by Claim 3.9. Hence, since bUis a symmetric PSD matrix, applying Claim 4.12 we have that for all u∈ Eand unit vectors v∈W⊥it holds that|u·v| ≤η/p poly( ϵσα/ (mKΓ|Y|). As a result, we have that if ηis chosen to be a sufficiently small polynomial of ϵ, σ, α, 1/Γ,1/m,1/K,1/|Y|, then for all u(i)∈ Eand an... | https://arxiv.org/abs/2502.09525v2 |
the problem of learning multiclass linear classifiers under RCN. Before presenting our hardness results, we first provide a short introduction to the SQ model. Basics on SQ Model SQ algorithms are a broad class of algorithms that, instead of having direct access to samples, are allowed to query expectations of bounded ... | https://arxiv.org/abs/2502.09525v2 |
F. For each Q∈ Q, let DQbe the joint distribution of (x, yQ), where x∼ N(0,I). Denote by D={DQ:Q∈ Q}the class of these distributions. Denote by D0the distribution of pairs (x, y0), where x∼ N(0,I)andy0a random variable supported on Ythat depends only on WVsuch that Pr(x,y0)∼D0[y0=i|x⊥U=z⊥U] =Prx∼N(0,I)[y=i|x⊥U=z⊥U], fo... | https://arxiv.org/abs/2502.09525v2 |
MLC RCN) .Any SQ algorithm that learns K-Multiclass Linear Classifiers under the d-dimensional standard normal in the presence of RCN to error OPT + poly(1 /K), requires either 2dΩ(1)queries or at least one query to VSTAT dΩ(K) . SQ-Hard Instances To construct our family of hard instances, we first define a two-dimen... | https://arxiv.org/abs/2502.09525v2 |
y)∈R2:f(x, y) =j}is equivalent toθ∈[2πj/K −π/K, 2πj/K +π/K], where θ=arctan ( y/x). Fix a, b∈Z+, a̸=b, and let θ1,j= 2πj/K −π/K,θ2,j= 2πj/K +π/K,j∈[K]. We have that v(p) j=E (x,y)∼N2[p(x, y)1(f(x, y) =j)] = E (x,y)∼N2[ x2+y2a+b 2eiθ(a−b)1(f(x, y) =j)] 53 =Z∞ 0Zθ2,j θ1,jra+b+1eiθ(a−b)g(r)dθdr =Z∞ 0ra+b+1g(r)dr Zθ2,j... | https://arxiv.org/abs/2502.09525v2 |
. . , K −1}. Hence, ω(k)∈ker(H) =ker(H⊤)forall k∈ {1, . . . , K/ 2−2, K/2, K/2+2, . . . , K −1}. Therefore, by applying Claim 5.6, we conclude the proof of Lemma 5.5. 54 5.2.1 Proof of Theorem 5.4 Proof of Theorem 5.4. Consider the case where Kis divisible by 4and let Hbe a matrix that satisfies the statement of Lemma ... | https://arxiv.org/abs/2502.09525v2 |
hardness result extends to the general case of any K. This completes the proof. References [AAM22] E. Abbe, E. B. Adsera, and T. Misiakiewicz. The merged-staircase property: a necessary and nearly sufficient condition for sgd learning of sparse functions on two-layer neural networks. In Conference on Learning Theory , ... | https://arxiv.org/abs/2502.09525v2 |
PMLR, 2021. [DKK+21b]I. Diakonikolas, D. M. Kane, V. Kontonis, C. Tzamos, and N. Zarifis. Efficiently learning halfspaces with Tsybakov noise. In S. Khuller and V. V. Williams, editors, STOC ’21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, 2021 , pages 88–101. ACM, 2021. [DKK+22]I. Diakonikolas, D. M. Kane... | https://arxiv.org/abs/2502.09525v2 |
2024. [Fel16] V. Feldman. Statistical query learning. In Encyclopedia of Algorithms , pages 2090–2095. 2016. [FGR+13]V. Feldman, E. Grigorescu, L. Reyzin, S. Vempala, and Y. Xiao. Statistical algorithms and a lower bound for detecting planted cliques. In Proceedings of STOC’13 , pages 655–664, 2013. Full version in Jou... | https://arxiv.org/abs/2502.09525v2 |
R. O’Donnell, and R. Servedio. Learning geometric concepts via Gaussian surface area. In Proc. 49th IEEE Symposium on Foundations of Computer Science (FOCS), pages 541–550, 2008. 59 [KSST08] S. M. Kakade, S. Shalev-Shwartz, and A. Tewari. Efficient bandit algorithms for online multiclass prediction. In Proceedings of t... | https://arxiv.org/abs/2502.09525v2 |
[WLT18] R. Wang, T. Liu, and D. Tao. Multiclass learning with partially corrupted labels. IEEE Transactions on Neural Networks and Learning Systems , 29(6):2568–2580, 2018. [Xia08] Y. Xia. A multiple-index model and dimension reduction. Journal of the American Statistical Association , 103(484):1631–1640, 2008. [XTLZ02... | https://arxiv.org/abs/2502.09525v2 |
Gaussian noise sensitivity of a function when we re-randomize its input over a subspace rather than over the entire space Rd. This claim is used in the proof of Claim 3.6 to show that a function with bounded Gaussian surface area remains nearly invariant when its input is re-randomized over sufficiently small cubes. Cl... | https://arxiv.org/abs/2502.09525v2 |
analysis of our general algorithm. Fact B.2 shows that by using a sufficiently small number of samples in polynomial time, one can efficiently obtain a low-degree polynomial whose L2approximation error is within an additive ϵof the best achievable by any polynomial of the same degree. Fact B.3 shows that the subspace s... | https://arxiv.org/abs/2502.09525v2 |
SQ algorithm that solves Bwith probability at least 2/3requires at least s·γ/βqueries to the VSTAT(1/ γ) oracle. In order to construct a large set of nearly uncorrelated hypotheses, we need the following fact: Fact C.4 (see, e.g., Lemma 2.5 in [ DKPZ21 ]).Let0< a, c <1 2andm, n∈Z+such that m≤na. There exists a set Sof2... | https://arxiv.org/abs/2502.09525v2 |
to real-valued MIMs: we say that a function f:RK→Rhas leap complexity Leap(f) =kif every Hermite decomposition of fcan be ordered such that each consecutive term introduces new directions via a polynomial of degree at most k. Since these measures rely solely on the non-negativity of the Hermite coefficients and do not ... | https://arxiv.org/abs/2502.09525v2 |
=y] =E x∼N(0,I)[Hem(w·x)|f(x) =y]2Pr x∼N(0,I)[f(x) =y] = 0 . Otherwise, the generative exponent would be less than k∗. Thus, for every m∈[k∗−1]andy∈ Y either Ex∼N(0,I)[Hem(w·x)|f(x) =y]orPrx∼N(0,I)[f(x) =y] = 0. In either case, this implies thatEx∼N(0,I)[Hem(w·x)1(f(x) =y)] = 0, for all m∈[k∗−1]andy∈ Y. Since any zero-... | https://arxiv.org/abs/2502.09525v2 |
will refer to the example distribution, D, as a distribution over Rd× Yrather than a distribution overRd×2Yas there are |Y|subsets S⊆ Ywhere |S|=|Y| − 1. Moreover, by Definition E.1, we have that there exists a |Y|×|Y| stochastic matrix Hsuch that Pr(x,¯y)∼D[¯y=j|x, f(x) =i] =Hi,j for all x∈Rd, i, j∈ Yand the identifia... | https://arxiv.org/abs/2502.09525v2 |
it is truethat Hisrowstochasticwith Hi,i= 0forall i∈[K]suchthat mini̸=jHi,j= Ω(1 /K3). Moreover, His symmetric, because it holds that Hi,j=hj−imod K=hi−jmod K=Hj,iandhj=hK−j, j∈ [K−1]. This equality is justified by the identity (−1)jcos (2 πj/K )= (−1)K−jcos (2 π(K−j)/K), which holds since Kis an even integer. Hence Hi... | https://arxiv.org/abs/2502.09525v2 |
over the labels. Since, ϵ≤1/Kwe have that we can solve B(D, D)with an additional query of accuracy less than 1/K−ϵ. ForKnot divisible by 4, we add r= 4 mod Kadditional classes whose weight vectors are identical to w(1). We construct the new confusion matrix as follows: first let Hbe the matrix that satisfies the condit... | https://arxiv.org/abs/2502.09525v2 |
the direction v. We show that for any vector wso that w·u≤c1 where c1>0is a sufficiently small absolute constant, then the hypothesis A(w(3)·x,w(4)·x)achieves large L2 2error where w(3)=v/√ 2+w/√ 2andw(4)=v/√ 2−w/√ 2. First, we choose wso that w·u= 0. Note that from Jensen’s inequality it holds that E x∼N(0,I)[(A(w(1)·... | https://arxiv.org/abs/2502.09525v2 |
F.4 To prove Lemma F.4, we first show that there is a function that satisfies our properties up to an error and has exponential pieces. Lemma F.5. Fixk∈Z+andδ≳kandη∈(0,1). There exists a (1/η+1)poly( k)-piecewise-constant function fη:R→ {0,2}such that f(x) = 0for|x| ≥δand for all t≤kit holds that E y∼N(0,1)[ytf(y)]−E y... | https://arxiv.org/abs/2502.09525v2 |
2t+ 12−t⌈t/2⌉X i=1t i2t−2i k E y∼N(0,1)[yt−2i1(|y| ≥δ)] ≲exp(−δ2/2) 2t+ 12−t⌈t/2⌉X i=1t i2t−2i k δt−2i≲exp(−δ2/2) 2t+ 1Pt(δ). 75 Furthermore, using the Corollary 5.4 in [ DKS17], we have that |Pt(δ)| ≤O(δt). Hence, we have that |ai|≲exp(−δ2/2)δk 2i+1. Choosing δ≳k, we have thatPk i=0|ai| ≤ϕ(1)and that completes... | https://arxiv.org/abs/2502.09525v2 |
every η >0there is a point b∈Rk+1such that|M(b)·ei−νi| ≤ηfor all i < k. Thus, from compactness, we have that there exists a point b∗∈Rk+1such that M(b∗) =0. This completes the proof. ThefollowinglemmaisanalogoustoLemma3.8in[ DKZ20]. Weincludeitsproofforcompleteness, as our work considers more general distributions and ... | https://arxiv.org/abs/2502.09525v2 |
a vector ugiven z. We define a differential equation for the function v:R7→Rm−1, as follows: v(0) = b, where b= (b1, . . . , b m−1), and v′(T) =u(v(T))for all T∈R. Ifvis a solution to this differential equation, then we have: d dTM(v(T)) =d dv(T)M(v(T))d dTv(T) =d dv(T)M(v(T))u(v(T)) =0, where we used the chain rule an... | https://arxiv.org/abs/2502.09525v2 |
Fast Tensor Completion via Approximate Richardson Iteration Mehrdad Ghadiri1, Matthew Fahrbach2, Yunbum Kook3, and Ali Jadbabaie1 1Massachusetts Institute of Technology, {mehrdadg,jadbabai}@mit.edu 2Google Research, fahrbach@google.com 3Georgia Institute of Technology, yb.kook@gatech.edu Abstract We study tensor comple... | https://arxiv.org/abs/2502.09534v1 |
that the Kronecker product structure in (3)no longer exists for𝐀Ωin(4), hence fast TD methods do not immediately extend to Ω-masked TC versions. A natural idea to overcome the lack of structure in the Ω-masked updates is to lift (4)to a higher-dimensional problem by introducing variables 𝐛Ω, whereΩis the complement o... | https://arxiv.org/abs/2502.09534v1 |
CP completion. Tomasi and Bro (2005) proposed an ALS algorithm for CP completion that repeats the following two-step process: (1) fill in the missing values using the current CP decomposition, and (2) update one factor matrix. Their algorithm is equivalent to running one iteration of mini-ALS in each step of ALS. As To... | https://arxiv.org/abs/2502.09534v1 |
Xby𝐀, and is expressed elementwise as (X×𝑛𝐀)𝑖1…𝑖𝑛−1𝑗𝑖𝑛+1…𝑖𝑁=𝐼𝑛 ∑ 𝑖𝑛=1𝑥𝑖1𝑖2…𝑖𝑁𝑎𝑗𝑖𝑛. The inner product of two tensors X,Y∈ℝ𝐼1×⋯×𝐼𝑁is ⟨X,Y⟩=𝐼1 ∑ 𝑖1=1𝐼2 ∑ 𝑖2=1⋯𝐼𝑁 ∑ 𝑖𝑁=1𝑥𝑖1𝑖2…𝑖𝑁𝑦𝑖1𝑖2…𝑖𝑁. 3 The Frobenius norm of a tensor Xis‖X‖F=√ ⟨X,X⟩. 2.1 Tensor Decompositions The tensor decom... | https://arxiv.org/abs/2502.09534v1 |
A.3 for details). When ALS updates A(𝑛)with all other TT-cores fixed, it solves A(𝑛)←argmin B∈ℝ𝑅𝑛−1×𝐼𝑛×𝑅𝑛‖‖(𝐀<𝑛⊗𝐀⊤ >𝑛)𝐁⊤ (2)−𝐗⊤ (𝑛)‖‖F, which is equivalent to solving 𝐼𝑛Kronecker regression problems in ℝ𝐼≠𝑛. 3 Approximate Richardson Iteration We now present our main techniques for reducing tensor com... | https://arxiv.org/abs/2502.09534v1 |
the following equivalent manner: (𝐱∗,𝐛∗ Ω)= argmin 𝐱∈ℝ𝑅,𝐛Ω∈ℝ𝐼−|Ω|‖‖‖‖[𝐀−𝐈∶,Ω][𝐱 𝐛Ω]−̃𝐪‖‖‖‖2 2, where𝐈is the𝐼×𝐼identity matrix. Remark 3.3.Problem 8 is not a linear regression problem with (structured) design matrix 𝐀since there are 𝐛Ωvariables in the response. There is, however, enough structure to empl... | https://arxiv.org/abs/2502.09534v1 |
𝜀)2)min 𝐱‖̃𝐏𝐱−̃𝐪‖2 2+𝜀‖̃𝐏(̃𝐏⊤̃𝐏)−1̃𝐏⊤̃𝐪‖2 2, in𝑂(𝛽 1−√ ̂ 𝜀𝛽⋅𝑇log𝛽/𝜀)time. Proof. We show that Algorithm 1 gives the desired output. Suppose approx-least-squares yields𝐱(𝑘+1)for given inputs𝐀,̃𝐏,̃𝐪, and̃𝐪(𝑘)(i.e.,̂𝐱←𝐱(𝑘),𝐟←̃𝐪(𝑘), and𝐱←𝐱(𝑘+1)), which satisfies ‖𝐀𝐱(𝑘+1)−̃𝐪(𝑘)‖2 2≤(1+... | https://arxiv.org/abs/2502.09534v1 |
the approximate solver computes a solution 𝐱∈ℝ𝑅in time𝑂(𝑇̂ 𝜀)such that ‖𝐀𝐱−𝐛‖2 2≤(1+̂ 𝜀) min 𝐱‖𝐀𝐱−𝐛‖2 2. Therefore, for a desired 𝜀1∈(0,1), we set̂ 𝜀=Θ(𝜀1/𝛽2)and use a sufficiently small 𝜀←𝜀2in Theorem 3.7. Putting everything together, Algorithm 1 finds an approximate solution ̃𝐱∈ℝ𝑅in time𝑂(𝛽𝑇𝜀... | https://arxiv.org/abs/2502.09534v1 |
time𝑂(|Ω|𝑅2+𝑅𝜔). 11 To achieve a fast lifted method, we solve the second step of Algorithm 1 using the leverage score sampling-based core tensor update algorithm in (Fahrbach et al., 2022, Theorem 1.2) with running time ̃𝑂(𝑁 ∑ 𝑛=1(𝐼𝑛𝑅𝑛+𝑅𝜔 𝑛𝑁2 𝜀2)+𝑅2−𝜃∗ 𝜀), where𝜃∗>0is an optimizable constant dependi... | https://arxiv.org/abs/2502.09534v1 |
1980). For a fixed 𝐘, we can compute 𝐗by solvingmin𝐱‖(𝐁⊗𝐀)𝐱−vec(𝐄−𝐂𝐘𝐃⊤)‖2. Thus, we can apply an alternating minimization algorithm for computing 𝐗and𝐘. We present the results in Figure 1 and give the full details in Appendix B.1. Figure 1: Coupled matrix results for 𝐄∈ℝ𝑛×𝑛,𝐗,𝐘∈ℝ𝑑×𝑑with𝑛=2000 and𝑑=... | https://arxiv.org/abs/2502.09534v1 |
to a larger structured problem and use fast sketching-based TD algorithms as ALS subroutines, extending their guarantees to the TC setting and establishing novel connections to iterative methods. We prove guarantees for the convergence rate of an approximate version of the Richardson iteration, and we study how these a... | https://arxiv.org/abs/2502.09534v1 |
L. Overton. The convergence of inexact Chebyshev and Richardson iterative methods for solving linear systems. Numerische Mathematik , 53(5):571–593, 1988. G. H. Golub and H. A. van der Vorst. Closer to the solution: Iterative linear solvers. In Institute of Mathematics and its Applications Conference Series , volume 63... | https://arxiv.org/abs/2502.09534v1 |
13(1):1–48, 2019. G. Tomasi and R. Bro. PARAFAC and missing values. Chemometrics and Intelligent Laboratory Systems , 75(2):163–180, 2005. 16 A Missing Details for Section 2 A.1 Least-Squares Linear Regression For a design matrix 𝐀∈ℝ𝑛×𝑑and response 𝐛∈ℝ𝑛, consider the least-squares problem 𝐱∗=argmin 𝐱∈ℝ𝑑‖𝐀𝐱−𝐛... | https://arxiv.org/abs/2502.09534v1 |
tensor. For example, a node without an edge indicates a scalar, one with one dangling edge is a vector, and one with two dangling edges is a matrix. When two dangling edges of two nodes are connected, we say that the two tensors are contracted along that mode (i.e., a mode product of those two tensors). For example, wh... | https://arxiv.org/abs/2502.09534v1 |
direct application of Corollary 4.2. For leverage score sampling, we sample 1%of rows in each iteration of approximate-mini-als , and then we run the algorithm for 7iterations. B.2 CP Completion B.2.1 Synthetic Tensors We run the same set of experiments as in Section 5 on two different random low-rank tensors: •random-... | https://arxiv.org/abs/2502.09534v1 |
arXiv:2502.09832v2 [stat.ML] 20 Apr 2025Algorithmic contiguity from low-degree conjecture and app lications in correlated random graphs Zhangsong Li∗ Abstract In this paper, assuming a natural strengthening of the low-degre e conjecture, we provide evi- dence of computational hardness for two problems: (1) the (par tia... | https://arxiv.org/abs/2502.09832v2 |
the following two spe cial cases, namely the correlated Erd˝ os-R´ enyi model and the correlated stochastic block mo del (SBM). Definition 1.2 (Correlated Erd˝ os-R´ enyi graphmodel) .Given an integer n≥1and two parameters p,s∈(0,1), we generate a triple of correlated random graphs (G,A,B)such that we first generate Gacc... | https://arxiv.org/abs/2502.09832v2 |
when ρ>√αwhereα≈ 0.338 is the Otter’s constant [ Ott48], there is an efficient algorithm that strongly distinguish these two models; on the other hand, it was shown in [ DDL23+ ] that when s <√αthere are evidences suggesting that all algorithms based on low-degree polynomials fail to strongly distinguish these two models... | https://arxiv.org/abs/2502.09832v2 |
1asn→ ∞by efficient algorithms. (3) Assuming the low-degree conjecture (see Conjecture 2.2for its precise meaning), then for the correlated stochastic block models S(n,λ n;k,ǫ;s), whens<min{√α,1 ǫ2λ}it is impossible to strongly distinguish this model and a pair of independent SBMsS(n,λs n;k,ǫ)by efficient algorithms, provi... | https://arxiv.org/abs/2502.09832v2 |
Pwe expect that gis “not large” as each gishould approximate 1{π∗(1)=i}in some sense. Thus, the statistics gaccumulates more signals than all low-degree polynomials, which (non-rigorously speaking) violates the low-degree conjec ture. (4) As for Item (3), denote Pto be the law of correlated SBMs and Qof independent SBM... | https://arxiv.org/abs/2502.09832v2 |
e weak recovery problem in the stochastic block model S(n,λ n;k,ǫ) whenk=no(1)andǫ2λ <1 lies below the KS threshold. Notably, their proof is also based on a “recovery-to-detection reduction” approach, which argues that if there exists an efficient algor ithm that achieves weak recovery, then we can use this algorithm to ... | https://arxiv.org/abs/2502.09832v2 |
theset ofm-cycles ofHanddenote by Cm(H) theset of independent 5 m-cycles ofH. ForH⊂S, we define Cm(S,H) to be the set of independent m-cycles inS whose vertex set is disjoint from V(H). Define C(S,H) =∪m≥3Cm(S,H). •Leaves.A vertexu∈V(H) is called a leaf of H, if the degree of uinHis 1; denote L(H) as the set of leaves of... | https://arxiv.org/abs/2502.09832v2 |
low-degree polynomial framework primarily focus on the following notions on strong and weak detection. Definition 2.1 (Strong/weak detection) .We say an algorithm Athat takes Yas input and outputs either0or1achieves •strong detection , if the sum of type-I and type-II errors Q(A(Y) = 1)+P(A(Y) = 0)tends to0asn→ ∞. •weak... | https://arxiv.org/abs/2502.09832v2 |
with respect to Q, denoted as P✁≤DQ, if noQ- based one-sided testing algorithm runs in time n/tildewideO(D). We say that QandPare degree-Dalgorithmic mutually contiguous, denoted as Q⊲⊳≤DP, if both Q✁≤DPandP✁≤DQhold. Recall that in probability theory we say a sequence of probab ility measure P=Pnis contiguous with resp... | https://arxiv.org/abs/2502.09832v2 |
the next two sections that this conjecture is also useful in performing reductions between different inference tasks. 3 Partial recovery in correlated random graphs In this section, we will use the framework we established in S ection2to show the hardness of partial matching in correlated random graphs, thus justify ing... | https://arxiv.org/abs/2502.09832v2 |
in Proposition 3.3 This subsection is devoted to the proof of Item (1) in Proposi tion3.3. Throughout this subsection, we will denote P∗to be the law of ( π∗,G,A,B) where (G,A,B)∼ G(n,q;ρ). We will also denote Pto be the marginal law of ( A,B). In addition, we assume throughout this subsection that th ere exists a smal... | https://arxiv.org/abs/2502.09832v2 |
n/summationdisplay j=1EP∗/bracketleftBig/parenleftbig 1{π∗(1)=j}−gj/parenrightbig2/bracketrightBig ≤1−cfor some constant c>0. (3.6) Without losing of generality, we may assume that 0 ≤gi≤1, since otherwise we may replace gi with min {max{gi,0},1}, which will only make the left hand side of ( 3.6) smaller. Denote Λ :=/b... | https://arxiv.org/abs/2502.09832v2 |
how to construct the event Ein Item (2) in Proposition 3.3. Definition 3.7. Denote˜λ=λ∨1. Given a graph H=H(V,E), define Υ(H) =/parenleftBig2˜λ2k2n D50/parenrightBig|V(H)|/parenleftBig1000˜λ20k20D50 n/parenrightBig|E(H)| . (3.19) Then we say the graph HisbadifΥ(H)<(logn)−1, and we say a graph Hisself-bad ifHis bad and Υ(... | https://arxiv.org/abs/2502.09832v2 |
Thus µIndER/parenleftbig A(A,B) = 0/parenrightbig = 1−o(1). Thus, this algorithm Astrongly distinguish µCorSBMandµIndER. This contradicts Conjecture 2.2 and the low-degree hardness established in [ CDGL24+ , Theorem 1.3]. The rest part of this section is devoted to the proof of Theore m4.1. For notational simplicity, i... | https://arxiv.org/abs/2502.09832v2 |
(4.18) where the last inequality follows from the fact that for Mu⊤ 0=cwe have ∝a\}bracke⊔le{⊔/hatwidef,c∝a\}bracke⊔ri}h⊔=∝a\}bracke⊔le{⊔/hatwidef,Mu⊤ 0∝a\}bracke⊔ri}h⊔=∝a\}bracke⊔le{⊔/hatwidefM,u0∝a\}bracke⊔ri}h⊔ ≤ ∝bardblu0∝bardbl·∝bardbl/hatwidefM∝bardbl. Regarding ( 4.18), it suffices to show that there exists Mu⊤=ca... | https://arxiv.org/abs/2502.09832v2 |
follows immediately. Now we assume that ( A.4) holds forl. Then we have Eσ∼ν/bracketleftBigl+1/productdisplay i=1ω(σi−1,σi)|σ0,σl+1/bracketrightBig =Eσ∼ν/bracketleftBig ω(σl,σl+1)Eσ∼ν/bracketleftBigl/productdisplay i=1ω(σi−1,σi)|σ0,σl,σl+1/bracketrightBig |σ0,σl+1/bracketrightBig =Eσ∼ν/bracketleftBig ω(σl,σl+1)ω(σ0,σl)... | https://arxiv.org/abs/2502.09832v2 |
inequalit y one gets EP′[f]/radicalbig EQ′[f2]=/summationtext α∈ΛCαEP′[fα(Y)]/radicalbig/summationtext α∈ΛC2α≤/parenleftBigg/summationdisplay α∈ΛEP′[fα(Y)]2/parenrightBigg1/2 , with equality holds if and only if Cα∝EP′[fα]. This yields ( B.7). Now, note that Q= (Q′)⊗Mis a product measure of Q′, there is a natural stand... | https://arxiv.org/abs/2502.09832v2 |
= 1]/vextendsingle/vextendsingleis bounded by 1 (n−1)!/summationdisplay S0,S′ 0:S0∼=S′ 0 S0⊂S1,S′ 0⊂S2#{π∈Sn:π(1) = 1,π(S0) =S′ 0,S0=S1∩π−1(S2)}M(S0,S1,S2).(B.18) For the case when 1 ∝\e}a⊔io\slash∈V(S1)∩V(S2), we must have 1 ∝\e}a⊔io\slash∈V(S0) and thus Enum:= #{π∈Sn:π(1) = 1,π(S0) =S′ 0} (B.19) is upper-bounded by A... | https://arxiv.org/abs/2502.09832v2 |
the following estimation. Again, we say a poly nomialφS1,S2∈ ODis admissible if both S1andS2are admissible graphs. Furthermore, we define O′ D⊂ ODas the set of admissible polynomials in OD, and define P′ D⊂ PDas the linear subspace spanned by polynomials 25 inO′ D. Using [ CDGL24+ , Proposition 4.5], we see that for any ... | https://arxiv.org/abs/2502.09832v2 |
restriction 1 ∈V(H1)∩V(H2), for any fixed H1,H2∈ Hthe enumeration of (H1,H2) such that H1∼=H1,H2∼=H2is upper-bounded by n|V(H1)|−1 Aut(H1)·n|V(H2)|−1 Aut(H2)=n|V(H1)|+|V(H2)| Aut(H1)Aut(H2)×n−2. Thus, after canceling the factor of n−2with the factor of n2above we derive the same upper bound as we obtain on ( B.42). This... | https://arxiv.org/abs/2502.09832v2 |
show that /summationdisplay σ∈[k]n/summationdisplay H⋐Kn,|E(H)|≤Du2 σ,H=/summationdisplay H⋐Kn,L(H)=∅ |E(H)|≤DΞ(H)2(B.44) is bounded by Oδ(1). Using Lemma B.5, we see that ( B.44) is bounded by /summationdisplay m3,...,mD≥0,t≥0 v≥3m3+...+DmD210k+m3+...+mDk10tn−(v+t)(1−δ/2)v+t·ENUM′(m3,...,m D;t),(B.45) 30 where ENUM′(m... | https://arxiv.org/abs/2502.09832v2 |
j=1EP∗/bracketleftbig fj|π∗(i) =j/bracketrightbig ≥c. (C.5) Thus, for all λ∈[0,1] we have n/summationdisplay j=1EP∗/bracketleftBig/parenleftbig 1{π∗(i)=j}−1−λ n−λfj/parenrightbig2/bracketrightBig(C.4),(C.5) ≤1+λ2−2cλ+O/parenleftbig1 n/parenrightbig . Thus, by choosing λ=λ(c) to be a sufficiently small positive constant w... | https://arxiv.org/abs/2502.09832v2 |
Li. Low-degree hardne ss of detection for correlated Erd˝ os- R´ enyi Graphs. arXiv preprint, arXiv:2311.15931 . to appear in Annals of Statistics . [DL22+] Jian Ding and Zhangsong Li. A polynomial time iterative algorith m for matching Gaussian ma- trices with non-vanishing correlation. arXiv preprint, arXiv:2212.1367... | https://arxiv.org/abs/2502.09832v2 |
The power of sum-of-squares for detecting h idden structures. In IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS) , pages 720–731. IEEE, 2017. [HS17] Samuel B. Hopkins and David Steurer. Efficient Bayesian estim ation from few samples: community detection and related problems. In IEEE 58th Annual Symp... | https://arxiv.org/abs/2502.09832v2 |
Information Processing Systems (NIPS), volume 34, pages 22259–22273. Curran Associates, Inc., 2021. [RSS19] Prasad Raghavendra, Tselil Schramm, and David Steurer. High-dimensional estimation via sum- of-squares proofs. In Proceedings of the International Congress of Mathematicia ns (ICM 2018) , pages 3389–3423, 2019. [... | https://arxiv.org/abs/2502.09832v2 |
Improved dependence on coherence in eigenvector and eigenvalue estimation error bounds Hao Yan Keith Levin Department of Statistics University of Wisconsin–Madison Madison, WI 53706 Abstract Spectral estimators are fundamental in low- rank matrix models and arise throughout ma- chinelearningandstatistics, withapplicati... | https://arxiv.org/abs/2502.09840v1 |
Vu, 2024). Re- cent work by Yan and Levin (2024) disproves this con- jecture under the model in Equation (1)whenM⋆and Hare symmetric, the signal matrix M⋆has suitably large eigenvalues and the entries of the noise matrix H are homoscedastic. The present paper aims to improvearXiv:2502.09840v1 [math.ST] 14 Feb 2025 Impr... | https://arxiv.org/abs/2502.09840v1 |
power method literature (see, e.g., Hardt and Price, 2014). That line of work typically assumes that we access only a subsample of the entries of a large (typically Hermitian) matrix, subject to noise. This is in contrast to the setting considered here, in which we assume that we observe the matrix Min its entirety. As... | https://arxiv.org/abs/2502.09840v1 |
example, analyzing Equa- tion(4)can lead to improved estimation methods for multilayer network analysis. This field that has recently attracted a lot of research interest (Levin et al., 2017; Draves and Sussman, 2020; MacDonald et al., 2022; Lei et al., 2023; Xie, 2024), and we intend to explore this direction in futur... | https://arxiv.org/abs/2502.09840v1 |
(6)is reasonable. 2.2 Preliminaries Before presenting our main results, we first establish severalimportantpreliminaryfacts. UnderCondition3, the higher-order moments of Hijare bounded by E|Hij|ℓ≤Bℓ−2EH2 ij≤σ2Bℓ−2, ℓ≥2,(7) which satisfies the Bernstein moment condition (see Equation (2.15) in Wainwright, 2019). Another... | https://arxiv.org/abs/2502.09840v1 |
establish the bound in Equation (10). Full details of the proof are provided in the supplementary materials. Applying a higher-order Markov inequality and a union bound over all integers 2≤k≤20logn, we can trans- late Theorem 2 into a high probability bound. Corollary 1. Under the same setting as Theorem 2, for all int... | https://arxiv.org/abs/2502.09840v1 |
Theorem 3, x⊤Hy ≤c2maxn σp logn,∥x∥∞∥y∥∞Blogno (15) holds with probability at least 1−O(n−40), where c2is the same universal constant as in Equation (14). 3.2 Symmetric noise matrix Our result in Theorem 3 can be directly extended to the case where the noise matrix is symmetric. Theorem 4 and Lemma 4 provide analogues ... | https://arxiv.org/abs/2502.09840v1 |
x⊤Hkyis a key quantity in establishing many technical results, our new results in Theorem 3 have a range of applications. This section highlights a few of them in the context of asymmetric noise. As noted in Remark 4, our results also apply to symmetric noise, but we leave these applications to future work.4.1 Rank-one... | https://arxiv.org/abs/2502.09840v1 |
large constant C2≥0such that λ⋆ max κ≥C2max σq nlog3n, Blog3n .(27) Then for any fixed unit vector a∈Rnand for any l∈[r], with probability at least 1−O(rn−20), one has a⊤ul−rX j=1λ⋆ ju⋆⊤ jul λla⊤u⋆ ≲s κ2rlog4n n× max( µ1 2Blog3n λ⋆ min∥a∥∞, σp nlog3n λ⋆ min!k0) .(28) Remark 8. Similar to the rank-one case in Theorem ... | https://arxiv.org/abs/2502.09840v1 |
set u⋆to be a linear combination 0.7v(1)+ 0.3v(2), normalized to have unit norm. These two approaches both guarantee that with high probability, ∥u∥∞≍p µ/nup to logarithmic factors. 5.1 Gaussian matrix denoising Suppose Hhas independent entries drawn from N(0, σ2 ij), where the σijare generated independently from a uni... | https://arxiv.org/abs/2502.09840v1 |
to inaccuracies caused by logarithmic factors and small values of n. Hao Yan, Keith Levin Figure 3: Eigenvalue estimation error as a function of n. Different colors represent different levels of coher- ence µ∈ {O(1), O(n1/3), O(n2/5)}. The dashed lines indicate the predicted rate from Equation (33). Shaded bands indica... | https://arxiv.org/abs/2502.09840v1 |
Wu, Y. (2013). Sparse PCA: Optimal rates and adaptive estimation. Annals of Statistics , 41(6):3074 – 3110. Cai, T. T. and Zhang, A. (2018). Rate-optimal per- turbation bounds for singular subspaces with ap- plications to high-dimensional statistics. Annals of Statistics , 46(1):60 – 89. Candès, E. and Recht, B. (2012)... | https://arxiv.org/abs/2502.09840v1 |
. MacDonald, P. W., Levina, E., and Zhu, J. (2022). Latent space models for multiplex networks with shared structure. Biometrika , 109(3):683–706. Mao, X., Sarkar, P., and Chakrabarti, D. (2021). Esti- mating mixed memberships with sharp eigenvector deviations. Journal of the American Statistical As- sociation , 116(53... | https://arxiv.org/abs/2502.09840v1 |
mathematical set- ting, assumptions, algorithm, and/or model. [Yes/No/Not Applicable] See Sections 1.1 and 2 (b)An analysis of the properties and complexity (time, space, sample size) of any algorithm. [Yes/No/Not Applicable] Our results are cen- tered on understanding the estimation rate of a certain class of spectral... | https://arxiv.org/abs/2502.09840v1 |
paper and does not use human subjects. (c)The estimated hourly wage paid to partici- pants and the total amount spent on partici- pant compensation. [Yes/No/ Not Applica- ble] This is a theory paper and does not use human subjects. Hao Yan, Keith Levin A PROOF OF THEOREM 2 We first present an outline of the proof of Th... | https://arxiv.org/abs/2502.09840v1 |
3. With the construction of G, we turn our attention back to controlling Equation (A.2). By Lemma 7, two graphs constructed from two different choices of J(i.e., two different expectation terms in the sum in Equation (A.2), corresponding to different instantiations of the expression in Equation (A.3)) share the same up... | https://arxiv.org/abs/2502.09840v1 |
≤pmax(16kp c2 2log3nkp 2 ,(kp)−k4kp c2log3nkp) . For any k≤20logn, choosing p=k−1log3n(here again, we assume that pis an integer without loss of generality), the above display is bounded by pmax(1280 c2 220 log n ,1 (kp)k3202 c2 220 log n) , which is bounded by O(n−40log3n)for some constant c2sufficiently large... | https://arxiv.org/abs/2502.09840v1 |
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