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s among the different possibilities t∈ −[ℓ−1],t= 0 andt∈[ℓ]. First, let us compare the posterior probabilities only amo ng overlaps in t∈[ℓ]. Using (32) from Lemma 3, it holds for t∈[ℓ]under the assumption that Xis a memoryless process that argmax t∈[ℓ]f(xℓ 1(1),xℓ 1(2);t) = argmax t∈[ℓ] /BD/bracketleftBig xℓ ℓ−t+1(1) ... | https://arxiv.org/abs/2502.13813v1 |
case with memory is well approximated by the likelihood in Prop. 4. Indeed, for T=t∈[ℓ]the conditioning on past symbols X0 −(ℓ−t−1)=xℓ−t 1(1)will only affect a relatively small number of symbols at the b eginning of the read, and so f/parenleftBig xℓ 1(1),xℓ 1(2);t/parenrightBig ≈ /BD/bracketleftBig xℓ ℓ−t+1(1) =xt 1(2... | https://arxiv.org/abs/2502.13813v1 |
process is aperiodic: R(X) := sup s∈N+P[X1=X1+s]<1. (52) Then, P∗ error≤2[1+on(1)]·/braceleftbigg β∧1 H1(X)/bracerightbigg ·logn n. (53) The proof of Prop. 4, is deferred to Appendix A, as all the othe r proofs in this section. We next discuss the result of Theorem 6. The dependence of the error probability on β:The ro... | https://arxiv.org/abs/2502.13813v1 |
0<|ˆT|< tMDO, wheretMDO∼logn H−∞(X), because for 0<|T|< tMDO, the posterior evidence of any substring common to both reads is not strong enough compa red to the prior knowledge that there is no overlap between the reads. This has two implications. First , the error probability is at least the probability that 0<|T|< tM... | https://arxiv.org/abs/2502.13813v1 |
we now turn to consider the type-II error probability. Due to the symmetry between positive and negat ive overlaps, let us upper bound the contribution ofT >0, given by /summationdisplay t∈[ℓ]P[T=t]·ptype-II(ˆT;t) =1 n/summationdisplay t∈[ℓ]ptype-II(ˆT;t). (61) We may then double it to include also the contribution of ... | https://arxiv.org/abs/2502.13813v1 |
for general processes, possibly with memory. The second and third cases , to wit, an error to ˆT > T or toˆT <−Tcan be similarly bounded, and so we only discuss the second case. Specifically, let us focus on an error from T=˜t > t∗toˆT=t >˜t. According to the detection rule of ˆT((41) in Prop. 4), such an error occurs w... | https://arxiv.org/abs/2502.13813v1 |
overlaps actually occur the det ector err. This is the dominating error event. Now, suppose that we would like to further reduce the c ontribution of the type-I error probability, to make it decay faster than o(logn n). This can be achieved by modifying the detection rule (41) of Prop. 4 by replacing nℓ∼nwithnµfor some... | https://arxiv.org/abs/2502.13813v1 |
under the true joint probability of the suffix /prefix of the reads. We circumvent this challenging aspect by upper bound the proba bility of the event {Xℓ ℓ−t+1(1) =Xt 1(2)}, with the probability of a slightly shorter match {Xℓ ℓ−t+1+τ(1) =Xt τ(2)}for some τ≡τ(n). This results a distance ofτtime points between the symbo... | https://arxiv.org/abs/2502.13813v1 |
of Theorem 6 tri vially hold (H−∞(X) =−logpmin<∞,d(s)≡0for alls∈N, andR(X) =|X|−H2(PX1)<1). Also, the entropy rate exists, and so P∗ error≤2[1+on(1)]·/braceleftbigg β∧1 H1(PX1)/bracerightbigg ·logn n. (70) Example 8 (First-order Markov Process (Markov chains)) .IfXis a first-order Markov process, which forms an irreduci... | https://arxiv.org/abs/2502.13813v1 |
only reveals to the detector that the overlap is ei therT= 0 orT=t∗. If the overlap is T= 0, then the genie reveals to the detector that the true value is eitherT= 0 or some random T=t, chosen uniformly in [t∗]. This reduces the operation of the genie-aided detector to t hat of[t∗]binary detectors. The average error pr... | https://arxiv.org/abs/2502.13813v1 |
λ(y,˜y) :=PY˜Y(y,˜y) PY(y)P˜Y(˜y). (87) We may note that λmax:= max (y,˜y)∈Y⊗2λ(y,˜y)∈[1,∞), (88) and, in addition, since PY˜Y≪PY⊗PYthen λ(Y,˜Y)≥λmin:= min/braceleftbig λ(y,˜y):(y,˜y)∈supp(PY˜Y)/bracerightbig >0 (89) with probability 1. A. The MAP Detection Rule Our first step in deriving the MAP detection rule is to ob... | https://arxiv.org/abs/2502.13813v1 |
o(logn n), and so taking into account the union bound over Θ(logn)terms, we need to prove that the probability in (96) is o(1 n). To this end, if we deploy 27 the standard Chernoff bound, then we only obtain O(1 n), which does not suffice for our purpose. Therefore, we deploy a stronger large-deviations bound, similar t... | https://arxiv.org/abs/2502.13813v1 |
X). One of the challenges in proving such a result, beyond the additional anticipated technical effort, is tha t even if Xis a first-order Markov process and the 28 reading kernel is memoryless, each read is a hidden Markov process, for which the likelihood more complicated than the cases we considered. In a similar spi... | https://arxiv.org/abs/2502.13813v1 |
from T∝\⌉}atio\slash= 0 toˆT= 0 (the bound (A.2)). 29 Lemma 14. LetX:={Xi}i∈Zbe a stationary ergodic process for which H1(X)<∞. Letρ∈(0,1)be given and{τ(n)}n∈Nbe a sequence of non-negative integers such that τ(n) =ωn(1)andτ(n) =o(logn). Then, for anyη >0 max t>(1+η) H1(X)logn/summationdisplay xt 1∈X⊗tP2 Xt 1(xt 1)· /BD... | https://arxiv.org/abs/2502.13813v1 |
0≤i≤r−1|t−i−ti−1|, (A.28) as well as d:=dmin pmin. (A.29) Ifd≤1 2then P/bracketleftbig Xt0|X−t1,...,X −tr,Xt1,...,X tr/bracketrightbig ≤P[Xt0]·e2d(r+1). (A.30) Proof: To ease the notation, let us denote Vi:=Xti. Then, P/bracketleftbig V0|V−r,...,V −1,V1,...,V r/bracketrightbig (a)=P/bracketleftbig V−r,...,V −1,V0,V1,..... | https://arxiv.org/abs/2502.13813v1 |
whereI⊂{0}∪[k]is a retained set of indices. Specifically, we would like to gu arantee anytwo symbols that are a part of the definition the even ˜Eto wit,{X1+iτ,X1+iτ+s}i∈Iare at least τtime points apart. This can be seen to be achieved by the following choice. First, we incl ude the events{Ej}j∈{0,1,...,m−1}, but exclude... | https://arxiv.org/abs/2502.13813v1 |
H1(X), which is simple to analyze. Indeed, 36 consider a trivial detector that always outputs T= 0. This detector will err if T∝\⌉}atio\slash= 0, which happens with probability2ℓ−1 n=2βlogn−1 n. Hence, P∗ error≤2ℓ n= 2β·logn n. (A.68) The rest of the proof is devoted to the case that β >1 H1(X)and to prove the bound P∗... | https://arxiv.org/abs/2502.13813v1 |
t∈[ℓ]\[t∗ η]−ℓ/summationdisplay k=−nP/bracketleftbigg/braceleftBig Xℓ+k ℓ−t+k+τ=Xt τ/bracerightBig ∩/braceleftbigg1 PXt 1(Xt 1)≥nℓ/bracerightbigg/bracketrightbigg (A.80) (b) ≤1 n/summationdisplay t∈[ℓ]\[t∗ η]−ℓ/summationdisplay k=−nP/bracketleftbigg/braceleftBig Xℓ+k ℓ−t+k+τ=Xt τ/bracerightBig ∩/braceleftbigg1 PXt τ(xt... | https://arxiv.org/abs/2502.13813v1 |
positive overlap T∈[ℓ]\[˜t]. (3) An error to a longer negative overlap T∈−{[ℓ−1]\[t∗ η]}. We next upper bound the probability of each of these error ev ents, and then use a union bound to upper bound P[T∝\⌉}atio\slash=T|T=˜t]. First, the error probability of an error from T=˜ttoT= 0, is upper bounded as P/bracketleftbi... | https://arxiv.org/abs/2502.13813v1 |
of err or 8 Suppose that we modify the detection rule (41) of Prop. 4 and m ake the condition of overlap detection more stringent, with the goal of reducing the type-I error probab ility too(logn nq)for some degree q >1. We can achieve this by replacing nℓ∼nin the detection rule (41) of Prop. 4 by nµfor some µ >1. In t... | https://arxiv.org/abs/2502.13813v1 |
s >1, it is easy to verify that Ks=as·11⊤+bs·I, (A.140) wherea1=ǫ,b1= (1−|X|ǫ), and the recursive formulas hold: as+1=as+bsǫ (A.141) bs+1=bs·(1−|X|ǫ). (A.142) Thus,bs= (1−|X|ǫ)sand as=ǫ·s−1/summationdisplay i=0(1−|X|ǫ)i=ǫ·1−(1−|X|ǫ)s |X|ǫ=1 |X|[1−(1−|X|ǫ)s]. (A.143) Due to the symmetry between the letters, the mixing c... | https://arxiv.org/abs/2502.13813v1 |
q∈(0,1)there exists a setB1∈F such that P[B1]≤q/2 and the convergence is uniform onBc 1. Thus, for any ν >0there exists n0(ν,q)and such that sup ω∈Bc1 m+klogP[Xm+k 1(ω)] P[Xk 1(ω)]·P[Xm+k k+1(ω)]≤ν (A.158) for alln≥n0sufficiently large so that m(n)≥m(n0)andk(n)≥k(n0). Usingm=ℓ−tandk=ℓand noting that for any θ0logn≤t≤θ1i... | https://arxiv.org/abs/2502.13813v1 |
may lower bound the second sum in the r.h.s. of ( A.169) as 1 nℓn/summationdisplay i2=i1+ℓP/bracketleftBig/braceleftBig Xℓ 1=xℓ 1(1)/bracerightBig ∩/braceleftBig Xi2−i1+ℓ i2−i1+1=xℓ 1(2)/bracerightBig/bracketrightBig (a)=1 nℓn−i1/summationdisplay s=ℓP/bracketleftBig/braceleftBig Xℓ 1=xℓ 1(1)/bracerightBig ∩/braceleftBi... | https://arxiv.org/abs/2502.13813v1 |
the detector the exact value of T. Otherwise, it reveals to the detector that the overlap is eit her zero or a positive overlap in [t1]\[t0], where one of these options is the true overlap. We next lower bound the Bay esian error probability of anygenie-aided detector. First, conditioned on the event T∈{1,...,t0}∪{t1+ ... | https://arxiv.org/abs/2502.13813v1 |
[25, Th. 14.9]. 52 following derivation, which holds for all nsufficiently large: P/bracketleftbigg log1 PXt 1(Xt 1)>logξ(n)−νlogn+logπ0 πt/vextendsingle/vextendsingle/vextendsingle/vextendsingleT=t/bracketrightbigg (i) ≤P/bracketleftbigg log1 PXt 1(Xt 1)>logξ(n)+(1−ζ(n)−ν)logn/bracketrightbigg (A.213) =P/bracketleftbig... | https://arxiv.org/abs/2502.13813v1 |
mas. The first lemma will be used in the analysis of the type-I error probability. It is a reminder fr om [24, Lemma 47], and is commonly used in finite blocklength information theory. This result is based on a change-of-mea sure argument combined with Berry– Esseen central limit theorem (e.g. [8, Th. 2, Chapter XVI.5] )... | https://arxiv.org/abs/2502.13813v1 |
Y˜Y(y,˜y) Pν Y(y)Pν Y(˜y)logPY˜Y(y,˜y) PY(y)PY(˜y)/parenrightBig2 /parenleftBig/summationtext (y,˜y)∈Y⊗2Pν Y˜Y(y,˜y)P1−ν Y(y)P1−ν Y(˜y)/parenrightBig2(B.23) =Eν/bracketleftbigg log2PY˜Y(y,˜y) PY(y)PY(˜y)/bracketrightbigg −E2 ν/bracketleftbigg logPY˜Y(y,˜y) PY(y)PY(˜y)/bracketrightbigg (B.24) =Vν/bracketleftbigg logPY˜Y... | https://arxiv.org/abs/2502.13813v1 |
probability: We begin by analyzing ptype-I(ˆT). By the union bound and Prop. 10 ptype-I(ˆT) =P/bracketleftBig ˆT >0|T= 0/bracketrightBig (B.40) =P/bracketleftBiggℓ/uniondisplay t=tMDO/braceleftBiggt/summationdisplay i=1logλ(Yℓ−t+i(1),Yi(2))≥lognℓ/bracerightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/ve... | https://arxiv.org/abs/2502.13813v1 |
ˆT= 0, it holds that P/bracketleftBig ˆT= 0|T=t/bracketrightBig (a)=P/bracketleftBiggt/summationdisplay i=1logλ(Yℓ−t+i(1),Yi(2))≤lognℓ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleT=t/bracketrightBigg (B.65) (b)=P/bracketleftBiggt/summationdisplay i=1logλ(Yi,˜Yi)≤lognℓ/bracketrightBigg (B.66) 61... | https://arxiv.org/abs/2502.13813v1 |
and each distribution of joint letters is PY˜Y. Under the alternative the joint distribution of every pair of lett ers isPY⊗PY. Finally, (c)holds for any t∈[t1]\[t0] due to the following derivation: P/bracketleftBiggt/summationdisplay i=1logλ(Yi,˜Yi)>logπ0 πt/bracketrightBigg (i) ≤P/bracketleftBiggt/summationdisplay i=... | https://arxiv.org/abs/2502.13813v1 |
, 2(3):231–239, 1988. 2, 3, 28 [17] E. Le Chatelier, T. Nielsen, J. Qin, E. Prifti, F. Hildeb rand, G. Falony, M. Almeida, M. Arumugam, J-M Batto, and S. Ke nnedy. Richness of human gut microbiome correlates with metabolic markers. Nature , 500(7464):541–546, 2013. 3 [18] D. A. Levin and Y . Peres. Markov chains and mi... | https://arxiv.org/abs/2502.13813v1 |
arXiv:2502.14069v2 [math.ST] 22 Feb 2025Finite sample bounds for barycenter estimation in geodesic spaces Victor-Emmanuel Brunel∗and Jordan Serres† Abstract: We study the problem of estimating the barycenter of a distribution given i.i.d. data in a geodesic space. Assumin g an upper curvature bound in Alexandrov’s sens... | https://arxiv.org/abs/2502.14069v2 |
obtain relevan t statistical error bounds. The main question that we are concerned with is the following . Given a probability distribu- tionµon(M,d)andnindependent, identically distributed (i.i.d) random poin tsX1,...,X nwith distribution µ(n≥1), how likely is a barycenter ˆbnofX1,...,X n(we callˆbnanempirical barycen... | https://arxiv.org/abs/2502.14069v2 |
closely related to the one we give below. Under th e hugging condition mentioned above, they prove (Theorem 12), a nearly sub-Gaussian tail bound th e empirical barycenter of i.i.d sub- Gaussian random variables, with a residual term that decays exponentially fast with n. [ACLGP20 ] obtains concentration inequalities f... | https://arxiv.org/abs/2502.14069v2 |
IfXhas two moments, we define its total variance as inf x∈ME[d(X,x)2]. For a random variable X(resp. a probability measure µ) in(M,d)with two moments, a barycenter of X(resp.µ) is any minimizer b∈MofE[d(b,X)2](resp.∫µd(b,x)2dµ(x)). As mentioned above, one could define barycenters of random vari ables or distributions wit... | https://arxiv.org/abs/2502.14069v2 |
that in Mκ, any side of a triangle with perimeter less than 2 Dκmust be of length less thanDκ, so the geodesics connecting the vertices are unique. Moreo ver, given any positive numbers a,b,cwitha≤b+c,b≤a+c,c≤a+banda+b+c<2Dκ, there exists a unique triangle in Mκ with side lengths given by a,bandc, up to rigid transform... | https://arxiv.org/abs/2502.14069v2 |
2 distance between Nd(0,A)and Nd(0,B), thed-variate centered Gaussian distributions with respective covariance matrices AandB. It can be shown that d 3(A,B)=min{/parall⟩l.alt1M−N/parall⟩l.alt1F∶M,N∈Rd×d,MM⊺=A,NN⊺= B}=minU∈O(d)/parall⟩l.alt1A1/slash.left2−UB1/slash.left2/parall⟩l.alt1F, whereO(d)is the set of d×dorthogo... | https://arxiv.org/abs/2502.14069v2 |
if Cwas, say, a ball of radius larger than Dκ/slash.l⟩ft4. The following lemma appears in [ BH13, Proposition II.2.4] for κ≤0 and in [ EFL09, Proposition 3.5] forκ=1 (and hence, via rescaling the metric d, for any κ>0). BARYCENTER ESTIMATION 7 Lemma 2 (Metric projection onto a convex domain) .LetMbe a CAT(κ)space and C... | https://arxiv.org/abs/2502.14069v2 |
was obtained in [ RBS21, Theorem 2 and Lemma 1] in the case of Riemannian manifolds. Theorem 1.Let(M,d)be a CAT(κ)space for some κ∈Rand letCbe a convex domain. Then, for all integers n≥1, the function ˆBnisL/slash.l⟩ftn-Lipschitz on Cnwith respect to d(n) 1, where L=⎧⎪⎪⎨⎪⎪⎩1ifκ≤0,independently of C 2/slash.l⟩ft(ε1/slas... | https://arxiv.org/abs/2502.14069v2 |
concentration of measure phenomenon was highlighted in the 1970’s by V. Milman in the context of the asymptotics of Banach spaces. It was then very studied through its deep connections with a lot of mathematical objects, such as isoperimetry, Ma rkov relaxation time, spectrum of diffusion operators and large deviation t... | https://arxiv.org/abs/2502.14069v2 |
•Definition 5is stronger than the standard definition of sub-Gaussian ran dom variables in Euclidean spaces. Indeed, if Xis a random variable in Rp(p≥1),Xis said to be K2-sub-Gaussian in the Euclidean, standard sense, if it satis fies Definition 5only with linear 1-Lipschitz functions (see [ Ver18, Section 2.5]), that is, ... | https://arxiv.org/abs/2502.14069v2 |
Proposition 4 (Composition with Lipschitz functions) .Let(M(1),d(1))and(M(2),d(2))be metric spaces and let Xbe a random variable in M1. LetK,L>0. IfXisK2-sub-Gaussian and Φ∶M(1)→M(2)isL-Lipschitz, then Φ(X)is(L2K2)-sub-Gaussian. Proof. By using Lemma 5, for allλ≥0, ΛΦ(X)(λ)≤ΛX(λL)≤eλ2L2K2/slash.l⟩ft2. Let usconcludethi... | https://arxiv.org/abs/2502.14069v2 |
[Oht07b])∀r≥0,µ(B(x0,r)≤ ∫r 0sK/par⟩nl⟩ft.alt2t√ N−1/par⟩nright.alt2N−1 dt, withs0(t)=tandsK(t)=1√ −Ksinh(√ −Kt),K<0. The proof ends with the integral being controlled on balls of large diameters, usin g Bishop-Gromov inequality. 3.3 Sub-Gamma random variables Definition 6.Letσ2>0andc>0. A random variable Xin(M,d)is ca... | https://arxiv.org/abs/2502.14069v2 |
barycenter, where t=(t2,...,tn)∈(0,1)n−1is a deterministic sequence to be specified later. We do not specify the dependence of ˜bnon the choice of the sequence tin our notation for the sake of simplicity. The estimator ˆbnwill be referred to as the empirical barycenter ofX1,...,X n and˜bnas theiriterated barycenter . Ou... | https://arxiv.org/abs/2502.14069v2 |
α=2 here) 2E[d(ˆbn,b∗)2]≤1 nn /summation.disp i=1E[d(Xi,X′ i)d(ˆbn,ˆb(i) n)] ≤1 n2n /summation.disp i=1E[d(Xi,X′ i)2] =E[d(X1,X′ 1)2] n ≤4σ2 n. The second inequality used Theorem 1that states that Bnis 1/slash.l⟩ftn-Lipschitz and the last inequality follows from the fact that E[d(X1,X′ 1)2]≤E[2(d(X1,b∗)2+d(X′ 1,b∗)2)]=... | https://arxiv.org/abs/2502.14069v2 |
consistent with an upper bound in Theorem 5that is larger than when κ≤0. •The dependence on the strong convexity constant α(ε,κ)of the upper bound in Theorem 5is strictly worse than that of Theorem 3for empirical barycenters. Again, we leave the question of optimality open. A key ingredient in the proof of Theorem 5is ... | https://arxiv.org/abs/2502.14069v2 |
some κ∈R. Ifκ≤0, all the random variables X1,...,X nthat are considered in this section are assumed to have two mo ments. If κ>0, they are all assumed to be almost surely contained in one and the same c onvex domain C⊆Mand we let ε>0 be such that Cis contained in a ball of radius 1 /slash.l⟩ft2(Dκ/slash.l⟩ft2−ε). Theor... | https://arxiv.org/abs/2502.14069v2 |
re quire any notion of dimension (e.g., Hausdorff dimension) to be finite, as long as the Xi’s have finite second moment. Now, when κ≤0, we obtain similar results for iterated barycenters ˜bn. However, proving similar tail bounds in the case when κ>0 remains open. Theorem 9.Assume that(M,d)is a CAT(0)space. Let X1,...,X n... | https://arxiv.org/abs/2502.14069v2 |
computing matrixgeometricmeans.Recallthatthegeometricmeanofpo sitivedefinitematrices A1,...,A n∈Sp (n,p≥1) is their barycenter, associated with the metric d(A,B)=/parall⟩l.alt1log(A−1/slash.l⟩ft2BA−1/slash.l⟩ft2)/parall⟩l.alt1F, which makesSpan NPC space [ BH06, Proposition 5]. The geometric mean of two matrices A,B∈Spi... | https://arxiv.org/abs/2502.14069v2 |
connected and that its sectional curvature is unif ormly bounded from above by some κ∈R. By [Cha06, Theorem IX.5.1], this guarantees that (M,d)is a CAT(κ)space. Let us also assume that (M,g)is symmetric around p. That is, there exists an 24 V.-E. BRUNEL AND J. SERRES isometry sp(called symmetry around p) such that sp(p... | https://arxiv.org/abs/2502.14069v2 |
in the definition of ˆb(P) n, empirical barycenters with inductive barycenters and obtain the same guarantee as in Theorem 10(with A=2 andL=1). That is, for all j=1,...,P,Yjmay be replaced with Zj∶=˜B(t) N((Xi)i∈Ij)with t=(1/slash.l⟩ft2,...,1/slash.l⟩ftN)andˆb(P) nmay be replaced with ˜b(P) n=˜B(s) Pwiths=(1/slash.l⟩ft2... | https://arxiv.org/abs/2502.14069v2 |
with gradient given by −2Log⋅(x). LetF(x)=E[d(X1,x)2]andFn(x)=n−1∑n i=1d(Xi,x)2,x∈M, the population and empirical Fr´ echet functions. Then, bot hFandFnareα-strongly convex on B, whereα=2 ifκ≤0 andα=α(ε,κ)otherwise. As usual, let b∗be the population barycenter of X1andˆbnthe empirical barycenter of X1,...,X n. Then,Fan... | https://arxiv.org/abs/2502.14069v2 |
given in these two theorems look worse than the one s we obtained in a more gen- eral framework. Indeed, both dependences on α(ε,κ)(in the small εregime) and in(σ2,K2)are 28 V.-E. BRUNEL AND J. SERRES deteriorated. However, the assumption we made on the distri bution of X1in Theorem 11is less stringent than, say, in Th... | https://arxiv.org/abs/2502.14069v2 |
0/par⟩nl⟩ft.alt4sinh(√−κt) √−κ/par⟩nright.alt4N−1 dt wherecN−1=2πN/slash.left2 Γ(N/slash.l⟩ft2)and where the integral should be understood as rN/slash.l⟩ftNifκ=0. Ifκ<0, we readily obtain the inequality VN,κ(r)=cN−1e(N−1)√−κr (N−1)(−κ)N/slash.l⟩ft2. Now, let us show that for all α>0,I(α)∶=∫Me−αd(x,x0)2dµ(x)is finite. Th... | https://arxiv.org/abs/2502.14069v2 |
Rajendra Bhatia. Positive definite matrices. In Positive Definite Matrices . Princeton university press, 2009. [BL17] Rabi Bhattacharya and Lizhen Lin. Omnibus CLTs for Fr ´ echet means and nonpara- metric inference on non-euclidean spaces. Proceedings of the American Mathematical Society, 145(1):413–428, 2017. [BLB03] S... | https://arxiv.org/abs/2502.14069v2 |
Darina Dvinskikh, Pa vel Dvurechensky, Alexander Gasnikov, andCesar Uribe. Onthecomplexity of approximati ng Wasserstein barycen- ters. InInternational conference on machine learning , pages 3530–3540. PMLR, 2019. [Led01] Michel Ledoux. The concentration of measure phenomenon , volume 89 of Mathe- matical Surveys and M... | https://arxiv.org/abs/2502.14069v2 |
High-dimensional random landscapes: from typical to large deviations Valentina Ros Université Paris–Saclay, CNRS, LPTMS, 91405, Orsay, France We discuss tools and concepts that emerge when studying high-dimensional random land- scapes, i.e., random functions on high-dimensional spaces. As an example, we consider a high... | https://arxiv.org/abs/2502.14084v1 |
(or group to group) of components, both in magnitude and sign. Such interactions are often complicated to write down or to infer from measurements, an occurrence that makes it meaningful to model them as random variables. The complex systems in general evolve by making local moves in configuration space, updating their... | https://arxiv.org/abs/2502.14084v1 |
of systems with rugged energy landscapes, in which opti- mization (that is, equilibration at low /zero temperature) is a hard task. Mean-field models of glasses are associated to energy landscapes with a complicated geometry, with plenty of local minima acting as metastable states for the dynamics, that slow it down an... | https://arxiv.org/abs/2502.14084v1 |
formulas and the Replica Method (RM). In Sec. III C we summarize the main outcomes of the landscape analysis of Case 2. 6 When addressing these problems, our primary focus is on characterizing the structure of the land- scape and the features of the dynamics that occur typically —i.e., those that occur for most realiza... | https://arxiv.org/abs/2502.14084v1 |
NvvT+J, v∈SN(p N), J∈GOE (σ2), (1) where the first term is a rank-one matrix, often called the spike , corresponding to the signal, while the second term is a matrix with random entries corresponding to the noise. Let us discuss these two terms separately. • The signal. The signal is an N-dimensional vector v= (v1,···,... | https://arxiv.org/abs/2502.14084v1 |
subspace of SN(p N)that is orthogonal to the (unknown) vector vasthe equator . We are now in the position to state more precisely the inference problem: we have access to several instances of the noisy matrix M, and we assume knowledge of the signal to noise ratio r/σ, as well as of the form of the distribution of the ... | https://arxiv.org/abs/2502.14084v1 |
to configuration ssuch that qN(s,v) =0: for any fixed v, this region has a surface that for large Nscales exactly as the total surface of the hypersphere, meaning that the overwhelming majority of configurations in SN(p N)are at the equator when N→∞ . In other words, if a vector sis extracted randomly with uniform meas... | https://arxiv.org/abs/2502.14084v1 |
(12) Since the overlap is rotationally invariant and the manifold we are integrating over as well, one can rotate the reference frame and choose a new basis ˆeiinRNin such a way that vcoincides with one of the vectors, v=p Nˆe1=p N(1, 0,···, 0). Then Ess·v N2 =1 NEs s2 1 . (13) By rotational invariance, we could ... | https://arxiv.org/abs/2502.14084v1 |
that the search for sGSrequires exponentially large timescales (in N) to be successful? In particular, we focus here on stochastic optimization dynamics of the form ds(t) dt=−∇⊥Er(s) +vt2 βη⊥(t),E η⊥(t)η⊥(t′) =Iδ(t−t′). (18) 13 The first term corresponds to gradient descent : the configuration evolves following the d... | https://arxiv.org/abs/2502.14084v1 |
of the main ingredients to get such distribution for random quadratic landscapes: Random Matrix Theory (RMT). B. How: Random matrix theory 1. From landscapes back to random matrices Consider a fixed realization of M, which generates a fixed realization of the landscape Er(s). To find the stationary points on the hypers... | https://arxiv.org/abs/2502.14084v1 |
moreover, their properties are encoded in the spectral properties of the matrix M. To characterize these properties statistically, therefore, one has to study the statistics of the eigenvalues and eigenvectors of these type of matrices. RMT comes in handy for this. [B3]Riemannian Hessian and Lagrange multipliers. Let u... | https://arxiv.org/abs/2502.14084v1 |
with the probability (3). The variance of the entries is scaled with Nin such a way to guarantee that the eigenvalues of the matrix are typically ofO(1), meaning that they lie on an interval in the real axis whose width remains bounded when N→∞ . The GOE ensemble is an example of invariant ensemble: the distribution of... | https://arxiv.org/abs/2502.14084v1 |
a distribution: it allows to compute averages over the eigenvalues of the matrix, such as 1 NTr[f(M)] =1 NNX α=1f(λα) =Z RdνN(λ)f(λ). (33) Eigenvalue density and isolated eigenvalues. For fixed matrix Mand finite N, the eigenvalue distribution is a collection of discrete delta peaks on the real axis. The general scenar... | https://arxiv.org/abs/2502.14084v1 |
the resolvent of the matrix M. The function (37) is singular (it has simple poles) when zapproaches the eigenvalues of the matrix, which lie on the real axis. To avoid singularities, we define it on the lower-half complex plane, z=λ−iη, whereη>0. The goal is to analytically continue this function to the real axis and c... | https://arxiv.org/abs/2502.14084v1 |
outliers). As it follows from (37) combined with (34), their contribution to the eigenvalues distribution is of order 1 /N, and it is thus sub-leading with respect to that of the density. When isolated eigenvalues exist, their typical values (36) can also 22 be extracted from the Stieltjes transform gN(z), by computing... | https://arxiv.org/abs/2502.14084v1 |
[16, 17 ]. ■Finite Nfluctuations: small deviations. The results discussed so far describe the spectral prop- erties of matrices when N→∞ ; in this limit, quantities are self-averaging and concentrate around their typical value. For Nlarge but finite, ρN(λ)andλiso Nfluctuate from realization to realization of the disord... | https://arxiv.org/abs/2502.14084v1 |
Section devoted to RMT by briefly discussing some results on the Large Deviations of the maximal eigenvalue λmax N. The results on small deviations show that the maximal eigenvalue has fluctuations that are of O(N−α), where α=1/2 in the supercritical regime and α=2/3 in the subcritical regime: at finite but large N, on... | https://arxiv.org/abs/2502.14084v1 |
eigenvectors uαhave the statistics of vectors uniformly distributed on the hypersphere. In particular, arbitrary vectors wthat are independent of Jhave components in the basis uαthat are statistically equivalent for each element of the basis, with no special direction (i.e., no particular uα) that is significantly more... | https://arxiv.org/abs/2502.14084v1 |
the transition is of second order: the order parameter q∞(sGS,v)grows continuously from zero to a positive value, without any jump, see Fig. 5 ( Left). Let us make a few comments on this inference transition. (i) First, as we discussed in Sec. II A 3, the recovery transition can also be found from the equilibrium forma... | https://arxiv.org/abs/2502.14084v1 |
expression for the index (24) shows that the only stationary point that is a stable minimum is the Ground State: all other stationary points have at least one negative Hessian eigenvalue. When N≫1, the variable αlabeling eigenvalues can be thought of as a continuous variable and most of the eigenvalues have an α=O(N): ... | https://arxiv.org/abs/2502.14084v1 |
is in general a challenge, as it requires to go beyond the mean-field limit. Langevin dynamic for quadratic Gaussian landscape is one of the rare examples for which the dynamics can be characterized both at short and at large timescales, and it exhibits interesting features in both cases. We begin by quantifying what w... | https://arxiv.org/abs/2502.14084v1 |
timescale distinguishing between mean-field and non mean-field dynamics is logarithmic in N[33, 34 ]. Finally, the statistics of the gap in the critical regime is, as far as we know, still an open problem in RMT . In summary, τcross(N)∼1 gN∼ O(N2 3) subcritical regime (Tracy-Widom) ? critical regime O(N0) =... | https://arxiv.org/abs/2502.14084v1 |
respect to the Ground State energy density, we therefore have: ε∞(t) = lim N→∞E[∆εN(t)] = vMF(t)t≫1∼ 3σ 8tr≤rc(σ), 3σ 8t−σ−εGSr>rc(σ).(70) The power-law decay of the energy density is an indicator of slow dynamics in the system. •Out-of-equilibrium and aging. In this mean-field regime, the dynamics of the system ... | https://arxiv.org/abs/2502.14084v1 |
limit N→∞ , the equilibrium regime of the dynamics ( t,t′∼τeq) lies outside the regimes of timescales captured by the mean-field formalism. To characterize how the system reaches equilibration, and in general to characterize the dynamics beyond the crossover timescales (68), one needs to go beyond mean-field and study ... | https://arxiv.org/abs/2502.14084v1 |
(r ≈rc).In this case, the average excess energy should be computed from (72), but the explicit form of the gap distribution in the critical regime is (as far as we know) unknown. Recent numerical studies [47]suggest that P(gN)behaves a power law for gNsmall, with an r-dependent coefficient, and they suggest that E[∆εN(... | https://arxiv.org/abs/2502.14084v1 |
∂Er,p(s,λ) ∂si=X i2≤i3···≤ipMii2···ipsi2···sip−λsi s∗,λ∗=0, ∂Er,p(s,λ) ∂λ=X is2 i−N s∗,λ∗=0.(83) Once more, multiplying the first equation by si, summing over iand using the second equation, we obtain: λ∗=−∇Er,p(s∗)·s∗ N=−pEr,p(s∗) N=−pεN(s∗), (84) which fixes the value of the Lagrange multiplier λas being proportional... | https://arxiv.org/abs/2502.14084v1 |
analytic continuation of the resulting expressions from n∈Nton∈R, in order to take the limit n→0. The annealed complexity (89), in contrast, requires to determine only the first moment of the random variable, n=1. Because of the concavity of the logarithm, the annealed complexity is always an upper bound to the quenche... | https://arxiv.org/abs/2502.14084v1 |
∂fi(x) ∂xj i j , (95) where now the Jacobian term involves the calculation of the determinant of a matrix of derivatives. Let us apply this formula to our counting problem: we are interested in the number of solutions of the non-linear equation ∇⊥Er,p(s) =0, which corresponds to f(x)→∇⊥Er,p(s)andy→0. In addition, 42 w... | https://arxiv.org/abs/2502.14084v1 |
i ∇⊥E=0 E=Nε≡Eh det∇2 ⊥E(s) i E=Nε. (99) This equality implies that for this model, the statistics of the Hessian at a stationary point are the same as at any arbitrary point of the same energy density. F2. When Nis large, the (N−1)×(N−1)matrix∇2 ⊥E(s)conditioned to E(s) =Nεhas exactly the same statistics as random mat... | https://arxiv.org/abs/2502.14084v1 |
is parametrized in terms of the overlap, and for all configurations ssuch that qN(s,v) =q, we find: P∇⊥E(0)→P(1) N(q) = 2πσ2 (p−1)!−N−1 2 e−N 2(p−1)!(r σ)2q2p−2(1−q2), PE(Nε)→P(2) N(ε,q) =vtp! 2πNσ2e−N p! 2σ2 ε+rqp p!2 .(103) Similarly, the expected value of the determinant (99) is only a function of the parameters... | https://arxiv.org/abs/2502.14084v1 |
annealed complexity of this model has been derived rigorously. From (113), one can check that the complexity vanishes for p→2, con- sistently with the results of Sec. II (see Box [B7]). [B7]The limit of quadratic landscapes: vanishing complexity. For all energy densities ε, the annealed complexity (113) is maximal at q... | https://arxiv.org/abs/2502.14084v1 |
has an O(1)overlap with the direction connecting s∗to the signal vin configuration space: the saddle is geometrically connected to the signal. 3. The quenched complexity: a roadmap The annealed complexity ΣA(ε,q)gives some indications on the distribution of stationary points in the energy landscape, but such indication... | https://arxiv.org/abs/2502.14084v1 |
directions (the signal, and the configuration sa). Despite this, the problem still shows a significant dimensionality reduction, typical of mean-field prob- 50 lems: while in principle the expression (117) depends on Nndistinct variables sa i, in fact when Nis large it can be parametrized in terms of only n(n−1)/2+nqua... | https://arxiv.org/abs/2502.14084v1 |
density of the Ground States is the minimum of this curve, and q∞(sGS,v)the minimizer. For p=3, the numerical value of the recovery threshold is (r/σ)1st≈2.56. The comments we made for the case p=2 extend also to the tensorial case: (i) the order parameter is like a magnetization in the direction of a generalized magne... | https://arxiv.org/abs/2502.14084v1 |
as we shall see also below when discussing the dynamics of this model. •Local minima in the northern hemisphere undergo a stability transition with r. Stationary points withε<εthare such that the Hessian has all eigenvalues positive, except possibly the isolated eigenvalue. Let us now discuss the behavior of the latter... | https://arxiv.org/abs/2502.14084v1 |
to the tensor denoising problem for r=O(N0)isrugged , with an exponentially large number of metastable states, the majority of which are at the equator, uninfor- mative of the signal. One expects therefore the optimization of the landscape to be hard, such that τeq∼eN, when the system is initialized in a configuration ... | https://arxiv.org/abs/2502.14084v1 |
aspects of this solution may not generalize straightforwardly to other glassy energy landscapes [83], this theory has profoundly shaped our understanding of relaxational out-of- equilibrium dynamics in high-dimensional systems, serving as a framework for developing key concepts such as timescales separation, weak ergod... | https://arxiv.org/abs/2502.14084v1 |
HIGH DIMENSION: THE LANDSCAPE PROGRAM MEETING LARGE DEVIATION THEORY We conclude these notes by discussing how the landscape program summarized in Fig. 1 connects with Large Deviation Theory. The strongest connections arise when considering the dynamics at large timescales in rugged landscapes. Indeed, as we mentioned ... | https://arxiv.org/abs/2502.14084v1 |
renewal: once the threshold level is reached, any other trap is accessible (the trap network is fully-connected). At the threshold the system looses memory of the trap it was coming from, and the dynamical process starts again independently of the past history. This model is exactly solvable [89, 90 ], and it shows par... | https://arxiv.org/abs/2502.14084v1 |
curvature: a perturbed GOE ensemble. The connection to LDT is due to the entropic nature of the problem. Below the threshold energy, ε<εth, the energy landscape is overwhelmingly dominated by minima, see Fig. 8 ( right). At those values of energy, the local minima are exponentially more numerous than the saddles of fin... | https://arxiv.org/abs/2502.14084v1 |
of the perturbed GOE ensemble (121) have been analyzed in the works [98, 99, 101 ]. In[98]we have shown that the BBP transition for this ensemble occurs when µ[ε,q|ε0]<−˜σ 1+˜σ′ ˜σ2 , (124) and the typical value of the isolated eigenvalue of H(s)reads λiso ∞=g−1 sc,˜σ gsc,˜σ′(µ) =1 gsc,˜σ′(µ)+˜σ2gsc,˜σ′(µ), (125)... | https://arxiv.org/abs/2502.14084v1 |
partition function over both the 62 noise of the dynamics and the realizations of the random landscape takes the form [104]: Zdyn s0=Z s(t=0)=s0Dste−AN[st]E[·]−→Z DQe−NA[Q(t,t′) ]+o(N), (127) where Q(t,t′)denotes the collections of dynamical order parameters of the theory (among which the correlation function, the ener... | https://arxiv.org/abs/2502.14084v1 |
the large de- viations of the dynamical action are to be interpreted as the effective path emerging from a sequence of local jumps in the landscape. The analysis and classification of instantonic solutions under different constraints remains an open challenge in this field. V . CONCLUSION In these notes, we have discus... | https://arxiv.org/abs/2502.14084v1 |
Eq. 40. The starting point of the calculation is the following Gaussian identity: 1 z1−M i j=1 Z[M]ZNY i=1dψip 2πψiψje−1 2PN k,l=1ψk(zI−M)klψl(A.1) with the normalization Z[M] =ZNY i=1dψip 2πe−1 2PN k,l=1ψk(zI−M)klψl. (A.2) 65 The quantities ψiare auxiliary variables. We wish to take the average of this expression wi... | https://arxiv.org/abs/2502.14084v1 |
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