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which it can be applied in Section 1.5. 4 1.3 Logistic regression for supervised classification of Gaussian mixtures Let us explore our results in the context of one of the most canonical supervised classification tasks namely, logistic regression for a k-component Gaussian mixture model, which is as follows. In this s... | https://arxiv.org/abs/2502.15655v2 |
k))≤q:=C+k . (1.6) For a summary statistic matrix G, define its unique positive semi-definite square root√ G⪰0. The map from Gto√ Gis continuous, so if we have a sequence G(d)→G, then√ G(d)→√ G. In both the Hessian and Gradient matrix cases, there is a coupled i.i.d. family of Gaussians ⃗ g such that the entry Dℓℓwith ... | https://arxiv.org/abs/2502.15655v2 |
point xwith summary statistics Gin parameter space are given by the effective bulk νGand the effective outliers. To state this we need a little more notation. In the following, let dBLdenote the Bounded Lipschitz metric on M1(R). For each α∈[C], letˆνH,(d)andˆνG,(d)denote the empirical spectral measures of the (α, α)-b... | https://arxiv.org/abs/2502.15655v2 |
vary, we now relate this to the spectra seen over the course of training by following the effective bulk and effective outliers as Gevolves per the limiting evolution of summary statistics under SGD. 1.3.2 The effective spectrum at initialization and along the training trajectory The (online) SGD with respect to the lo... | https://arxiv.org/abs/2502.15655v2 |
and its BBP-type spectral transitions (in λ, α) are exactly solvable. Corollary 1.5. At the initialization x≡0, asn, d→ ∞withn/d=ϕ, and µ1, ..., µ korthonormal with (pj)k j=1= (1 k, ...,1 k), we have the following characterization of the effective spectra: Hessian matrix . The empirical Hessian bHαα’s effective bulk sp... | https://arxiv.org/abs/2502.15655v2 |
recall are the n, d→ ∞limit of G(x⌊tn⌋)over training. To demonstrate this, we describe a result on the evolution of the bulk and outliers along the SGD trajectory in the early stages of training. Namely, we will implicitly differentiate the effective bulk and effective outlier equations (1.9)–(1.11) 10 Figure 1.3: From... | https://arxiv.org/abs/2502.15655v2 |
one varies the training time. Emergence of outlier eigenvalues over the course oftrainingshouldbeevenmorepronounced(withdelayedonsets)inmorecomplicatedproblems, like single-indexmodelswhosefirstnon-zeroHermiteexponentsareatleast 2,describedinSubsection1.5. The above types of emergence of effective outliers as the SNR v... | https://arxiv.org/abs/2502.15655v2 |
identity, and more generally, where Dis positive semi-definite but independent ofA, have a long history in the random matrix literature on empirical covariance matrices . See e.g., the book of [7], in particular the chapter on sample covariance matrices, for more. The key distinction for us is that Dis coupled to A; al... | https://arxiv.org/abs/2502.15655v2 |
of the effective bulk and outliers are dimension independent, only depending on the relevant vectors xand the mean vectors µ1, ..., µ kthrough their summary statistic matrix G∈Rq×qforq=C+k=O(1). We finally introduce a stronger assumption on the entries of Dℓℓthat allows us to pass to limits asd→ ∞in the cases where the... | https://arxiv.org/abs/2502.15655v2 |
In general, consider the case that the means are µ1, ..., µ k(i.e., the data distribution is still PYof (1.3) and the labels are assigned to subsets of [k]which are the classes, I1, ..., I Cpartitioning [k]the distribution of Yis the same as above, and the label yis the class among I1, ..., I Cthat the mean belongs to.... | https://arxiv.org/abs/2502.15655v2 |
line. To extend the result to settings like ReLU activation, the zero-entries in Dwould have to be handled separately since the invertibility of Dis used at several points in the proofs. In the fixed-width K=O(1)regime, the second layer matrices are K×Kin the n, d→ ∞ limit, so there are not good notions of bulk and out... | https://arxiv.org/abs/2502.15655v2 |
applies and Theorem 1.12. Remark 1.17. The boundedness requirement on the link function may seem strong (in particular it rules out the example of phase retrieval, where g(x) =|x|or more smoothly g(x) =|x|2), but without it discussion of outlier eigenvalues may not be especially pertinent at high enough d, since 1 nADA... | https://arxiv.org/abs/2502.15655v2 |
des chaires de recherche du Canada. 2 Spectral theory for self-coupled empirical matrices InthissectionweestablishourgeneralrandommatrixtheoryresultwhichwillimplyTheorems1.10– 1.12. Rather than working directly with (1/n)ADA⊤from (1.2), it helps to work instead with the rescaling (λ/d)ADA⊤. Throughout this section, we ... | https://arxiv.org/abs/2502.15655v2 |
G(d)≡Gfor all d, then Dℓℓare i.i.d. over ℓandd by Assumption 1. Therefore, the almost sure weak convergence follows from the strong law of large numbers for the empirical spectral measure. The strong convergence for the empirical measure for i.i.d. samples is similarly straightforward. 19 2.2 Estimates for the Stieltje... | https://arxiv.org/abs/2502.15655v2 |
≤Zk! ϵk+1µ⊠νmp(dt)≤k! ϵk+1, as desired. For the lower bound, observe that by (2.13), we have that for z∈B(K)∩Rand t∈supp µ, |t−z|≲ϕτ−1+K≤(1 +K)/τ, from which it follows that |∂(k) zSµ(z)| ≥k!τk+1 (1 +K)k+1. Then we can extend the above estimate off the real line by observing that |∂(k) zSµ(z)| ≥ |∂(k) zSµ(Re[z])| − |Im... | https://arxiv.org/abs/2502.15655v2 |
is proved under the assumption that ˆρ(d)→ρweakly a.s. This is the exactly what is guaranteed by Assumption 3. We now turn to showing that σdhas no outliers. Here and in the rest of this section, let Eϵ=Eϵ(νG). We also have here the following result whose proof follows a modification of a classical result of Bai–Silver... | https://arxiv.org/abs/2502.15655v2 |
As this proof is somewhat lengthy, we only describe here what changes in the argument. In the remainder of this section, we then extend this bound all the way up to R. Theorem 2.9. Fix any τ >0independent of n, d. Suppose that Dis invertible and its empirical spectral measure is τ-regular. Let Wdenote an independent d×... | https://arxiv.org/abs/2502.15655v2 |
local law (2.20) follows from the entry-wise local law (2.27) and [32, Lemma 6.2]. We now extend this result to the real line for our choice of D. We start with the following basic estimates. Recall that we let σddenote spectrum of (1/n)WDWandB(K)is the ball of radius KinC. Lemma 2.10. Assume that σd∩Eϵ/2(ν) =∅and that... | https://arxiv.org/abs/2502.15655v2 |
orthonormal and satisfy the property that L⊤[x1···xCµ1···µk] =p G(d). To see that such a matrix exists, simply compute the compact singular value decomposition of this matrix as [x1···xCµ1···µk] =UΛV⊤, (2.33) where Uis ad×qmatrix with orthonormal columns, Λis diagonal with non-negative entries, andV⊤is aq×qorthogonal m... | https://arxiv.org/abs/2502.15655v2 |
the second term satisfies, for z∈Eϵ,K,τ, 1 nX D4 ℓℓ (1 +eSDℓℓ)41{(|1 +eSDℓℓ| ≥δ/4)}≲K,ϵ,ϕ,τ1 δ4 eventually almost surely. Combining this with the above we see 1 n||˜Λ(d) δ−¯Λ(d) δ||2→0 (2.41) 29 uniformly on Eϵ,K,τfor each K > 0almost surely. Now we relate this to Λ(d) δ. To this end, first observe that by definition o... | https://arxiv.org/abs/2502.15655v2 |
a function of i.i.d. Gaussians and the summary statistic matrix G(d), Dℓℓ=fa(ℓ) ⃗ gℓ/√ λ;G(d) , (2.45) where ⃗ gℓ∼ N(0, Iq)andfyis as in (1.19). Moreover, recalling (2.34), we have that, conditionally on the hidden labels,√ dRℓ=⃗ gℓ+√ λ(L⊤µa(ℓ)), for the same ⃗ g. Notice that√ dRℓis a Gaussian random vector with mean... | https://arxiv.org/abs/2502.15655v2 |
orthogonal matrix,1 dUWDW⊤U⊤has the same eigenvalues as1 dWDW⊤. As shown in Proposition 2.7, the limit of the empirical eigenvalue distribution of1 dWDW⊤is given by ν, whose Stieltjes transform solves (2.2). This establishes the “bulk" distribution (2.4). 2.10 Proof of (2.5) We now turn to studying the outliers. We beg... | https://arxiv.org/abs/2502.15655v2 |
space Span (x1, ..., x C, µ1, ..., µ k). To see this observe that if uis the eigenvector of interest then v=Uuand, by construction of Ufrom (2.34), we have v1=L⊤u. 34 Let us now take z′=z(d)be the eigenvalue as in the statement of the theorem and let us denote v(d)= (v(d) 1, v(d) 2)the corresponding eigenvector and w(d... | https://arxiv.org/abs/2502.15655v2 |
invariant, namely O⊤⃗ gand⃗ ghave the same law, we have (OF(z)O⊤)ij=Fij(z;O). In particularly, rotating√ GbyOdoes not change the solutions of (1.11) det(zIq−F(z)) = det( zIq−OF(z)O⊤) = det( zIq−F(z;O)) = 0 (2.59) and transforms the solutions to (1.12)-(1.13) in the natural way: if we let Udenote the space of solutions ... | https://arxiv.org/abs/2502.15655v2 |
follows from (1.7) and the fact that φα∈[0,1]. To complete Assumption 2, it remains to establish the uniform O(ϵ)bound on the probability of fj falling within ϵof0. Fix any λand any Gand consider P(|fj(λ−1/2⃗ g,G)|< ϵ)which since fjis non-negative a.s., is the same as P(fj(λ−1/2⃗ g,G)∈[0, ϵ)). Beginning with the Hessia... | https://arxiv.org/abs/2502.15655v2 |
because the law of fjwill no longer be a delta-mass. Lemma 3.3. Consider the k-GMM with loss function (1.4), with orthonormal means, equal weights pc≡1 k, and initialization x∼ N (0, Id/d). The empirical spectral distribution of the Hessian at initialization converges to νmp⊠νremwhere νremis the law of πrem(1−πrem)for ... | https://arxiv.org/abs/2502.15655v2 |
l k, but these were taken to be exactly the mean vectors µ1, ..., µ k. Therefore, by Theorem 1.4, at finite d, any outlier that is o(1)away from z∗has a choice of eigenvector whose projection into Span(µ1, ..., µ k)iso(1)away from cjµjfor a constant cjgoverned by (1.13). To calculate how large that constant cjis, per(1... | https://arxiv.org/abs/2502.15655v2 |
λEhΠ(1−Π) λϕ+S(z)Π(1−Π)g1i , cz=ϕEhΠ(1−Π) λϕ+S(z)Π(1−Π)g1g2i , d z=ϕEhΠ(1−Π) λϕ+S(z)Π(1−Π)(g2 1−g1g2)i , capturing different correlations between the REM-measure Πand its constituent Gaussians. Using Schur complement, the solutions to det(zI−F) = 0can be found to be at solutions of det(( z−az)Ik) = 0 ,or det (z−dz)IC−... | https://arxiv.org/abs/2502.15655v2 |
they exist), and λc,1≤λc,2. 44 3.6 Spectral transitions over course of training We now turn to the spectral transitions (i.e., emergence and splitting of outliers) that arise over the courseoftrainingbySGD.Themainresulthereisanunderstandingofthedriftsoftheeffectivebulk and effective outliers over the first Ω(n)steps of... | https://arxiv.org/abs/2502.15655v2 |
l̸=jel+X l(λ−1/2X m∈[k]ξδlmgm+ξ′(1−δlm)gm)el)⟩(1−2φα) =⟨∇φα, ξej+ξ′X l̸=jel+X lλ−1/2(ξgl+X m̸=lξ′gm)el⟩(1−2φα). (3.24) Note that at t= 0, we have that φα= 1/k, and∇φα=φαeα−φαP βφβ. Thus taking the limit as first cη→0then t→0of (3.21), we see that ˙Sis well defined at t= 0. Observe that at x= 0,fJ=ξwhere ξ:=1 k(1−1 k). ... | https://arxiv.org/abs/2502.15655v2 |
we are using the consistent choice of (li)i∈[k]= (µi)k i=1. At the end of the proof, we will show that the error from higher order terms in the Taylor expansion to the effective outliers are o(t), implying that the derivative at initialization is indeed captured by this linearization. By (1.10), the derivative ∂tF↾t=0s... | https://arxiv.org/abs/2502.15655v2 |
zero moves to the right when adding a negative function to an increasing function) and to the left of the time zero effective outlier by an order t amount when ζ∈ {ξ′,c−}. It remains to establish that this implies the effective eigenvectors at time tare within o(t)of the ones solving (3.25) to conclude the proof. By di... | https://arxiv.org/abs/2502.15655v2 |
+E" fj λϕ+Sλ(z)fj λ−1/2Sym(⃗ g⊗√ G[q]j) +λ−1⃗ g⊗2#! , where the subscript λis to clarify the λdependence. Using that f∞ jis deterministic for the first term, and Cauchy–Schwarz and (2.5) for the latter two terms, for z /∈supp( νH G), we get F=λϕX jpjf∞ j λϕ+Sλ(z)f∞ j e⊗2 j+O(λ−1/2) =X jpjf∞ j 1−Sλ(z)f∞ j λϕ+··· ... | https://arxiv.org/abs/2502.15655v2 |
yα,byα∈[0,1], as long as g′andg′′are bounded functions, for fixed values of vwe get that the coefficients of Y⊗2are uniformly compactly supported on R. Next consider the probability that the coefficient of Y⊗2in (4.2)–(4.3) are less than ϵin absolute value. 51 For (4.2), for vα i̸= 0, this is bounded by the probability... | https://arxiv.org/abs/2502.15655v2 |
each Xc∈Rd. (∇cL)⊗2(X) = 2(g(XY)−y)∂cg(XY)2 Y⊗2, ∇2 ccL(X) = 2∂cg(XY)2+ 2(g(XY)−y)∂ccg(XY) Y⊗2. Taking the empirical Hessian or empirical Gradient matrix, this is of the form of (1.2) where Y∼ N (0, Id)(which is a trivial case of the GMM distribution where the mean is zero and the signal-to-noise is 1). Notice that... | https://arxiv.org/abs/2502.15655v2 |
if ∥G(d)−G∥=o(1/logd), then by Lemma 3.1, one gets Assumption 3 if the coefficients ofY⊗2above are uniformly continuous as functions of the inner products WY,Θ∗Y; this evidently holds if the coefficients have bounded derivatives, which follows if the link function gis bounded differentiable, so that hnnis, and if the a... | https://arxiv.org/abs/2502.15655v2 |
edge of stability. In International Conference on Learning Representations , 2021. [17] Elizabeth Collins-Woodfin, Courtney Paquette, Elliot Paquette, and Inbar Seroussi. Hitting the high-dimensional notes: an ode for sgd learning dynamics on glms and multi-index models. Information and Inference: A Journal of the IMA ... | https://arxiv.org/abs/2502.15655v2 |
BackProp , pages 9–48. Springer Berlin Heidelberg, Berlin, Heidelberg, 2012. [35] Xinyan Li, Qilong Gu, Yingxue Zhou, Tiancong Chen, and Arindam Banerjee. Hessian based analysis of SGD for Deep Nets: Dynamics and Generalization , pages 190–198. 2020. [36] ZhenyuLiaoandMichaelW.Mahoney. Hessianeigenspectraofmorerealisti... | https://arxiv.org/abs/2502.15655v2 |
spectrum of the fisher information matrix of a single-hidden-layer neural network. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, Advances in Neural Information Processing Systems , volume 31. Curran Associates, Inc., 2018. [53] Maria Refinetti, Sebastian Goldt, Florent K... | https://arxiv.org/abs/2502.15655v2 |
arXiv:2502.15717v1 [math.ST] 26 Jan 2025On interpolation problem for multidimensional harmonizable stable sequences with noise observations Mikhail Moklyachuk1∗ 25th February 2025 Abstract We consider the problem of optimal linear estimation of the f unctional AN/vectorξ=N/summationdisplay j=0(/vector a(j))⊤/vectorξ(j)... | https://arxiv.org/abs/2502.15717v1 |
Moklyach uk (2019 – 2024). In papers by Moklyachuk and Ostapenko (2015, 2016) minimax-robust inte rpolation problems are studied for harmonizable stable sequences. In this paper the problem of optimal estimation is investiga ted for the linear functional AN/vectorξ=N/summationdisplay j=0(/vector a(j))⊤/vectorξ(j) that ... | https://arxiv.org/abs/2502.15717v1 |
¯z<αβ>, •(xy)<α>=x<α>y<α>, •(zα)<β>= (z<β>)α, •(z<α>)β= (zβ)<α>, •|z<α>|β=|z|αβ, •(x+y)<α>= ¯x|x+y|α−1+ ¯y|x+y|α−1. 3 Definition 2 (symmetric α-stable stochastic sequence) .A stochastic sequence {ξ(n),n∈Z} is called symmetric α-stable,SαS, if all linear combinations/summationtextl m=1amξ(nm)areSαSrandom variables. A vec... | https://arxiv.org/abs/2502.15717v1 |
3 Interpolation problem. Observations with noise. Projection approach Consider the problem of the optimal estimation of the linear functional AN/vectorξ=N/summationdisplay j=0(/vector a(j))⊤/vectorξ(j) =/integraldisplayπ −π(AN(eiθ))⊤d/vectorZξ(θ), where AN(eiθ) =N/summationdisplay j=0/vector a(j)eijθ, that depends on t... | https://arxiv.org/abs/2502.15717v1 |
, stochastic sequences {ξ(n),n∈Z}and {η(n),n∈Z}, have absolutely continuous spectral measures and the spec tral densities f(θ), g(θ),satisfying the minimality condition (15). The optimal line ar estimate ˆAN/vectorξof the func- tional AN/vectorξ=N/summationdisplay j=0(/vector a(j))⊤/vectorξ(j) =N/summationdisplay j=0T−... | https://arxiv.org/abs/2502.15717v1 |
theorem holds true. 8 Theorem 3. Let/vectorξ(j) ={ξk(j)}T k=1, j∈Zbe a vector-valued harmonizable symmetric α- stable,HSαS , stochastic sequence which has absolutely continuous spec tral measure µ(θ) and the spectral density f(θ),satisfying the minimality condition (23). The optimal line ar estimate ˆAN/vectorξof the f... | https://arxiv.org/abs/2502.15717v1 |
and |eiθ+d|4=r−2e−2iθ+r−1e−iθ+r0+r1eiθ+r2e2iθ, (35) where r−2=d2, r−1= 2d+2d3, r0= 1+4d2+d4, r1= 2d+2d3, r2=d2. (36) It follows from (33) – (36) that the spectral characteristic h(θ)of the optimal estimate of the functional is of the form h(θ) =h−3e−3iθ+h−2e−2iθ+h−1e−iθ+h0+h1eiθ+h2e2iθ+h3e3iθ+h4e4iθ, where h−3=−b−1r−2,... | https://arxiv.org/abs/2502.15717v1 |
={ξk(j)}T k=1, from observations of the sequence /vectorξ(j)at points j∈Z\{0,1,...,N}in the particular case where α= 2. In this case the harmonizable symmetric α-stable stochastic sequences /vectorξ(j) ={ξk(j)}T k=1,j∈ Z}and/vector η(j) ={ηk(j)}T k=1,j∈Z}are stationary sequences and we have the problem of the optimal e... | https://arxiv.org/abs/2502.15717v1 |
system is of the form cN=B−1 NDNaN, (46) that is the components of the vector cN=/vector cNare calculated by the formula /vector c(j) = (B−1 NDN/vector aN)(j), j= 0,1,2...,N. (47) Here vectors aN=/vector aN= (/vector a(j),j= 0,1,2...,N); cN=/vector cN= (/vector c(j),j= 0,1,2...,N); matrices BN={B(k,j)}N k,j=0,DN={D(k,j... | https://arxiv.org/abs/2502.15717v1 |
variable A1/vectorζis of the form ˆA1/vectorζ=A/bracketleftbig (α+β)(1+b2)+b(γ+δ)/bracketrightbig ζ1(−1)+ +A/bracketleftbig (γ+δ)(1+b2)+b(α+β)/bracketrightbig ζ1(2). 16 The value of the mean-square error equals ∆(F) =/a\}bracketle{t/vector c1, /vector a1/a\}bracketri}ht= =AP1 2πb/bracketleftbig (α+β)2(1+b2)+(γ+δ)2(1+b2... | https://arxiv.org/abs/2502.15717v1 |
α= =/integraldisplayπ −π/parenleftig AN(eiθ)−h(θ)/parenrightig⊤ f(θ)/parenleftig AN(eiθ)−h(θ)/parenrightig<α−1> dθ+ +/integraldisplayπ −π(h(θ))⊤g(θ)(h(θ))<α−1>dθ(53) The constrained optimization problem (52) is equivalent to the unconditional extremum problem ∆D(f,g) =−∆(h(f0,g0);f,g)+δ(f,g|Df×Dg)→inf, (54) 18 wher... | https://arxiv.org/abs/2502.15717v1 |
minimality con dition (23), equation (29) and condition f0∈D0, that is/integraltextπ −πf0(θ)dθ=P, is the least favorable density f0∈D0 ffor the optimal linear estimation of the functional ANξ=/summationtextN j=0a(j)ξ(j). The minimax-robust spectral characteristic h(f0)of the optimal estimate of the functional ANξis det... | https://arxiv.org/abs/2502.15717v1 |
determined by the (27) with f(θ) =f0(θ). 4.4 Least favorable spectral densities in the class D−1 f Consider the problem of optimal linear estimation of the fun ctionalANξ=/summationtextN j=0a(j)ξ(j)that depends on the unknown values ξ(j),j= 0,1,...,N , of a random sequence {ξ(k),k∈Z},from observations of the sequence {... | https://arxiv.org/abs/2502.15717v1 |
a set of admissible spectral densities is available, relations which determine least fa vorable densities and the minimax- robust spectral characteristics for different classes of sp ectral densities are found. 23 References [1] Cambanis, S., 1983. Complex stable variables and proces ses, Contributions to Statistics: Es... | https://arxiv.org/abs/2502.15717v1 |
olation of harmonizable se- quences. Theor. Probability and Math. Statist. 92, 135-146 . [23] Pshenichnyj, B.N., 1971. Necessary conditions for an e xtremum. Pure and Applied math- ematics. 4. New York: Marcel Dekker, 230. [24] Pourahmadi, M., 1984. On minimality and interpolation of harmonizable stable processes. SIAM... | https://arxiv.org/abs/2502.15717v1 |
BANKRUPTCY ANALYSIS USING IMAGES AND CONVOLUTIONAL NEURAL NETWORKS (CNN) Luiz Wanderley Tavares1 José Afonso Mazzon2 Francisco Carlos Paletta3 Fábio Meletti de Oliveira Barros4 ABSTRACT The marketing departments of financial institutions strive to craft products and services that cater to the diverse needs of businesse... | https://arxiv.org/abs/2502.15726v1 |
factors transcending financial regulations and oversight mechanisms. The implications of fragmentation are multifaceted, affecting the efficiency of financial service delivery, impinging upon transparency, and impacting consumer and investor safeguards; it also bears significance for global financial equilibria (Claess... | https://arxiv.org/abs/2502.15726v1 |
status, duration of residence, occupational role, and monthly income, among others. Altman (1968) innovated in this domain by analyzing the annual financial statements of corporations and deriving a set of financial ratios from these records. He initiated his research with a cohort of sixty-six enterprises across diver... | https://arxiv.org/abs/2502.15726v1 |
accuracy of 72.5%. In a comparative analysis of machine learning techniques, Aktan (2011) examined the efficacy of eight machine learning algorithms—Naive Bayes, Bayesian Network, k-NN, ANN, SVM, C4.5, CHAID, and CART—in the context of financial distress. Utilizing a dataset of fifty-three financial indices, the study ... | https://arxiv.org/abs/2502.15726v1 |
of the model. Regarding time-variant data, time series analysis is traditionally employed. However, recent investigations have incorporated neural network methodologies, including convolutional neural networks (CNNs) as noted by Jin et al. (2020), recurrent neural networks (RNNs) as explored by Madan and Mangipudi (201... | https://arxiv.org/abs/2502.15726v1 |
the financial state of the company. 3. DATA SOURCE This study accessed archival data from 580 accounting firms, encompassing comprehensive financial records of approximately 105,000 small and medium-sized enterprises (SMEs) across various sectors of the Brazilian economy. This extensive dataset included over 20 million... | https://arxiv.org/abs/2502.15726v1 |
the foundational framework of accounting information systems within various organizations and industries. These charts are essentially compendiums of account titles and their associated numerical codifications, which are instrumental in systematically recording financial transactions, including revenues and expenses, a... | https://arxiv.org/abs/2502.15726v1 |
facilitate the calculation of financial ratios, enabling month-to-month comparative analysis of a company’s financial trajectory against its industry counterparts. The development of these 208 accounts was informed by this requirement and mirrors the contemporary fiscal status of the entities within the research databa... | https://arxiv.org/abs/2502.15726v1 |
are most frequently implicated in financial indices and that possess a significant capacity to reflect the fiscal health of a business. Calculations were performed to determine the relative contributions of each account, as delineated by horizontal and vertical analyses. The accounts incorporated into this vector, whic... | https://arxiv.org/abs/2502.15726v1 |
at conveying the financial dynamics that transpired in preceding periods (Krugman, 1999), thereby limiting the fidelity of year-over-year comparative analyses due to a lack of intermediate temporal data. However, the corpus of this study is enriched with monthly data, thus offering a more granular temporal perspective ... | https://arxiv.org/abs/2502.15726v1 |
Therefore, an inflation scenario of 0.54% monthly and an annual rate of 7.02064% (June 1997's figure) translates to the pixel (100, 132, 151). Opting for a 24x24 format enriches the image with a greater density of information, with two rows allocated to the period representation. In the financial ratios-based images, b... | https://arxiv.org/abs/2502.15726v1 |
indices enhance the model's analytical capabilities. For the analysis of test images, the confusion matrix is a pivotal tool, offering a structured overview of the correctly and incorrectly classified instances. Given the binary nature of the classification in this study, the matrix delineates the counts of true positi... | https://arxiv.org/abs/2502.15726v1 |
explanation for this phenomenon became apparent upon analyzing the accounting practices of the SMEs within the dataset. Unlike their larger, publicly listed counterparts, these smaller entities do not always adhere to the stringent accounting standards that facilitate precise financial ratio calculations. Such an infor... | https://arxiv.org/abs/2502.15726v1 |
divisions When assessed separately, Divisions 47 and 86 yielded contrasting results. Division 47 developed a high-caliber model utilizing images based on accounting account data, yet this was not replicated with images derived from financial indices, which resulted in a weaker model. For Division 86, the models were su... | https://arxiv.org/abs/2502.15726v1 |
barrier to developing cost-effective financial products tailored for smaller enterprises. The challenge is further compounded by the difficulties these companies face in securing capital. When SMEs do access funds, the risk is often assessed superficially, resulting in higher rates than those offered to larger, publicl... | https://arxiv.org/abs/2502.15726v1 |
of a uniform chart of accounts, coupled with the application of the Universal Sentence Encoder to detect similarities between accounting nomenclature, is an innovative strategy with broad applicability. Specifically, this methodology empowers the system to process accounting data across languages by leveraging multilin... | https://arxiv.org/abs/2502.15726v1 |
USP de Controladoria e Contabilidade (Vol. 4). https://doi.org/10.1590/s1415-65552007000600005 Krugman, P. (1999). Balance sheets, the transfer problem, and financial crises. International tax and public finance, 6, 459-472. https://doi.org/10.1007/978-94-011-4004-1_2 LeCun, Y., Bengio, Y., & Hinton, G. (2015). Deep le... | https://arxiv.org/abs/2502.15726v1 |
Universality of High-Dimensional Logistic Regression and a Novel CGMT under Dependence with Applications to Data Augmentation Matthew Esmaili Mallory *†, Kevin Han Huang *‡, Morgane Austern† Abstract . Over the last decade, a wave of research has characterized the exact asymptotic risk of many high-dimensional models i... | https://arxiv.org/abs/2502.15752v2 |
2025 10 20 30 400.00.10.20.30.40.5 No aug Uniform Gamma Exponential Student’s tGaussian rperm= 0.8 rperm= 1.0 10 20 30 400.0250.0500.0750.1000.1250.150 No augUniform Gamma Exponential Student’s tGaussian rperm= 0.8 rperm= 1.0Training risk (cross-entropy) Test risk (excess 0-1 loss relative to β∗) number of augmentation... | https://arxiv.org/abs/2502.15752v2 |
e.g. in the EM algorithm. 2 a true signal β∗∈Rpsuch that P yi=1|X=σ X⊺ iβ∗, σ (t) :=(1+e−t)−1. (2) The signal is estimated via a penalized and weighted logistic regression: ˆβ(X)Barg min β∈Sp1 nXn i=1ωi log 1+eX⊺ iβ −yiX⊺ iβ +λ 2n∥β∥2, (3) where (ωi)∈[0,1]Nare deterministic weights, and Spis a convex subset of Rp... | https://arxiv.org/abs/2502.15752v2 |
the asymptotical risk is the same as in the independent setting. Hence previously derived results for logistic regression still hold (see Section 7). To tackle the case where the data is correlated, we propose a novel CGMT result. 3 (ii)CGMT . We introduce a novel extension of the CGMT for Gaussian matrices with a “low... | https://arxiv.org/abs/2502.15752v2 |
state the various assumptions we place on our data generating process and the model. We postpone the discussion of those assumptions to Section 3.1 after the result is stated. 4 Assumption 1 (Block-dependence) .There exists k≥1 such that ( Xi,yi) is independent of ( Xj,yj) whenever j<Bi=n k⌊i−1 k⌋+1,..., k⌊i−1 k⌋+ko . ... | https://arxiv.org/abs/2502.15752v2 |
the Gaussian universality of the testing risk, in addition to the training risk, it is necessary to introduce further assumptions, specifically Assumptions 6 and 7. Assumption 6 closely resembles Assumption 5, but applies to Xnewrather than our original data. Note that we did not require Xnewto share the same distribut... | https://arxiv.org/abs/2502.15752v2 |
we are ready to state our extended universality result for m-dependence and specific mixing processes. Theorem 2 (Universality under m-dependence or mixing) .Let Xi,yi(Xi)n i=1and Gi,yi(Gi)n i=1 be generated under Assumptions 2-4, where each Gi∼N 0,Var(Xi). Assume in addition that Assumptions 8 and 9 hold. Then dH m... | https://arxiv.org/abs/2502.15752v2 |
a more gen- eral CGMT framework that accommodates a “low-rank assumption” on the dependence structure ofH. Assumption 10 (Low-rank Dependence) .There exist M∈Nand symmetric positive semi- definite matrices ( Σ(l),˜Σ(l))l≤M, with Σ(l)∈Rp×pand˜Σ(l)∈Rn×n, such that Cov[ Hji,Hj′i′]=XM l=1Σ(l) j j′˜Σ(l) ii′ for all i,i′≤nan... | https://arxiv.org/abs/2502.15752v2 |
[2] generalizes the block diagonal setup to allow non-identity subblocks, which is a special case of our Assumption 10, but they also allow for transforming wandu, which we do not address here. 6 Applications to Data Augmentation (DA) As an example application of Theorems 1 and 3, we analyze the e ffect of data augment... | https://arxiv.org/abs/2502.15752v2 |
and ρ∗may be known or unknown. This motivates the use of random sign flipping to shrink the estimate ˆβat locations where the entries of β∗may be zero. We fix some ⌈rflipp⌉entries of p, where rflip∈[0,1] is a parameter chosen by users. Each ϕiis a random diagonal matrix, generated by drawing the fixed ⌈rflipp⌉entries o... | https://arxiv.org/abs/2502.15752v2 |
at least within our model, exploiting the full set of permutation invariance is critical for obtaining noticeable improvements. On the other hand, the sparsity setup for cropping and sign flipping does not allow the knowledge of the full structure by design, as it would otherwise imply that we know exactly which coordi... | https://arxiv.org/abs/2502.15752v2 |
its critical role, a number of studies have investigated its theoretical properties (e.g. Chen et al. [17], Hanin and Sun [32], Huang et al. [36], Lin et al. [45]). The 11 first work that applies CGMT to the study of data augmentation is Dhifallah and Lu [25], which examines the impact of noise injection on logistic re... | https://arxiv.org/abs/2502.15752v2 |
di fferently transformed data Cov[ ϕ1(Z1),ϕ2(Z1)]. As a result, this satisfies the low-rank dependence assumption of our CGMT (Assumption 10) with M=2. How- ever, the actual application of the CGMT is more subtle since the logistic regression (3) is not a priori in the form of the primary optimization (9). Similar to D... | https://arxiv.org/abs/2502.15752v2 |
establish the universality of the global minima , but do not answer whether the training trajectories to reach these minima are universal, since the latter question is specific to the optimization methods employed. In Fig. 4 in the appendix, we observe that with the same learning rate, the training loss curves di ffer ... | https://arxiv.org/abs/2502.15752v2 |
[19] N. Cressie. Statistics for spatial data . John Wiley & Sons, 1993. [20] J. D. Cryer. Time series analysis , volume 286. Duxbury Press Boston, 1986. [21] Y . Dandi, L. Stephan, F. Krzakala, B. Loureiro, and L. Zdeborov ´a. Universality laws for Gaussian mixtures in generalized linear models. Advances in Neural Info... | https://arxiv.org/abs/2502.15752v2 |
34:18970–18983, 2021. [42] S. B. Korada and A. Montanari. Applications of the lindeberg principle in communications and statistical learn- ing. IEEE transactions on information theory , 57(4):2440–2450, 2011. [43] S. Lahiry and P. Sur. Universality in block dependent linear models with applications to nonparametric reg... | https://arxiv.org/abs/2502.15752v2 |
Sons, 2008. [60] R. T. Rockafellar. Convex analysis . Princeton Mathematical Series. Princeton University Press, Princeton, N. J., 1970. [61] N. Ross. Fundamentals of Stein’s method. Probability Surveys , 8:210 – 293, 2011. [62] M. Rudelson and R. Vershynin. Hanson-wright inequality and sub-gaussian concentration, 2013... | https://arxiv.org/abs/2502.15752v2 |
A. Tewari. Lasso guarantees for β-mixing heavy-tailed time series. The Annals of Statistics , 48(2):1124–1142, 2020. [84] M. Wu and J. H. Ware. On the use of repeated measurements in regression analysis with dichotomous responses. Biometrics , pages 513–521, 1979. [85] J. Yang, R. Walters, N. Dehmamy, and R. Yu. Genera... | https://arxiv.org/abs/2502.15752v2 |
locally dependent, as defined in [61]. Definition 6. Let ( Xi)i≤p∈Rpbe a random vector. We say that it is locally dependent if for all i≤pthere exists a subset Ni⊂[p] such that Xiis independent from ( Xk)k<Ni. We callNithe dependency neighborhood of Xi. A similar definition can be made for random arrays: Definition 7. ... | https://arxiv.org/abs/2502.15752v2 |
We state the main result here. Theorem 8 (Effect of data augmentation on the test risk) .Letˆβ(X,XΦ)be the estimator fitted via (OO) with S =Sp. Assume that Assumptions 1 – 6 hold, that the minimizer-maximizers of (DO) 18 are within the interior of the domain of optimization and Assumption 11 holds. Then |Rtest(ˆβ(X,XΦ... | https://arxiv.org/abs/2502.15752v2 |
minimizer-maximizers of (DO) are within the interior of the domain of optimization, so the converged risk of (DO) changes by Θ(ϵ2) depending on whether the optimization domain of βrequires|(β⊺Σnewβ)1/2− ( ¯χr,θ,σ,τ 2)1/2|> ϵ. This verifies Assumption 7 and proves the universality of the test risk. The deterministic app... | https://arxiv.org/abs/2502.15752v2 |
sum of the block, which can be utilized for situations such as repeated measurements and peer e ffects. C Simulation Details We present some additional simulation details on top of the setups described in Section 6 here. The regularization parameter is held at λ=0.01, and the test loss is computed as the di fference be... | https://arxiv.org/abs/2502.15752v2 |
are again drawn i.i.d. from ˜t3. Cropping and sign flipping are always performed on the bottom ⌈s0(1−ρ∗)p⌉coordinates, as well as also on⌈r⌉−⌈s0(1−ρ∗)p⌉of the remaining coordinates, where r=rflip=rcrop=0.2. Data are generated coordinate-wise i.i.d. according to N. We also remark that even with knowledge of the coordina... | https://arxiv.org/abs/2502.15752v2 |
first introduce the various terminology and techniques that are used throughout, from smoothing the labels and minimum function to the continuous Lindeberg interpolation. D.3.A. Smoothing the Labels First we will define the way in which we smooth our labels and subsequently the risk function. To do so, let us define th... | https://arxiv.org/abs/2502.15752v2 |
Ut:=sin(t)X+cos(t)G,t∈[0,π 2]. 25 By the fundamental theorem of calculus, since U0=GandUπ/2=X, we may bound Eh fδ(X)−h fδ(G) ≤Zπ/2 0 Eh ∂th fδ(Ut)i dt. Using the chain rule we may expand ∂th fδ(U)=−h′ fδ(U) nnX i=1⟨˜U⊺ iDi(β)⟩ where we set ˜Ut:=∂tUt=cos(t)X−sin(t)G. Using Lemma 19 we obtain that lim n→∞ Eh fδ(... | https://arxiv.org/abs/2502.15752v2 |
≤Cβ1/15 mix 27 Proof. For ease, let us refer to the quantity of interest as dH min βˆRn(β;X),min βˆRn(β;G)! =(⋆). Letα,δ,γ,τ> 0. We may first bound (⋆)≤dH min βˆRn(β;X),min βˆRγ n(β;X)! +dH min βˆRγ n(β;X),min βˆRγ n(β;G)! +dH min βˆRγ n(β;G),min βˆRn(β;G)! (i) ≤2C1√γq max(E∥X∥2 Sp,E∥G∥2 Sp) √n+dH min βˆRγ n(β;X),min β... | https://arxiv.org/abs/2502.15752v2 |
n−1∥U∥2 Sp1/8+max i∥˜UT iβ∗∥3r E n−1∥U∥2 Spo . where (i) is a consequence of the fact that I(|˜UT iβ|≥L)≤|˜UT iβ|1/4 L1/4, (ii) comes from Jensen inequal- ity combined with H ¨older inequality. Hence using the bound in Corollary 15 under Assump- tion 8(ii) and using Assumption 3 we have that there exists a constant... | https://arxiv.org/abs/2502.15752v2 |
for an arbitrary ˜ mgives us the desired result. □ G Proof of Theorem 1 (6)and Theorem 2 (8): Test risk universality In this section, we prove the second equation of both Theorems 1 and 2 concerning test risk universality, as both share the same proof once training risk universality is proved. We focus on presenting th... | https://arxiv.org/abs/2502.15752v2 |
and Wright [15]), there is an absolute constant C′′>0 such that, almost surely, P (VG)1∈[−δ,δ) ˆβ+P (VG)2∈[−δ,δ) ˆβ, εnew≤C′′δ1 ˆβ⊺Σnewˆβ+1 β∗⊺Σnewβ∗ . Meanwhile by Assumption 7, for every ϵ >0, P(Dϵ(G)>0)→1, where Dϵ(G)B min β∈Sp,|(β⊺Σnewβ)1/2−¯χ|>ϵˆRn(β;G)−min β∈SpˆRn(β;G), and by the universality of the training... | https://arxiv.org/abs/2502.15752v2 |
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