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Boston, 1992, pp. 297–331 (cit. on p. 16). [Rub76] Donald B. Rubin. “Inference and Missing Data”. In: Biometrika 63.3 (Dec. 1976), pp. 581–592 (cit. on p. 16). [TA23] Wai Ming Tai and Bryon Aragam. “Learning Mixtures of Gaussians with Censored Data”. In: Proceedings of the 40th International Conference on Machine Learn... | https://arxiv.org/abs/2504.07133v1 |
more, gives us the following expression ∇2L(W) =∇W W−E x,ymaxE z∼N(W⊤x,I)|z∈P(ymax)h xz⊤i! =Idk−E x,ymax∇W E z∼N(W⊤x,I)h xz⊤|z∈P(ymax)i! . (89) Next, for a fixed (x,ymax), we compute the above gradient. To simplify the calculations, let us define (for fixed xand ymax) f(W) = f(W;z):=∥z−W⊤x∥2 2. For the purpose of this ... | https://arxiv.org/abs/2504.07133v1 |
can compute the gradient as follows ∇WR P(ymax,2 ,imax)(xz⊤)♭e−1 2f(W)dz R P(ymax,2 ,imax)e−1 2f(W)dz =−R P(ymax,2 ,imax)(xz⊤)♭∇f(W)⊤e−1 2f(W)dz 2R P(ymax,2 ,imax)e−1 2f(W)dz +R P(ymax,2 ,imax)(xz⊤)♭e−1 2f(W)dzR P(ymax,2 ,imax)∇f(W)⊤e−1 2f(W)dz 2R P(ymax,2 ,imax)e−1 2f(W)dz2. Substituting ∇f(W) =2(xx⊤W−xz⊤)implie... | https://arxiv.org/abs/2504.07133v1 |
Then, we extend the proof to the special case where viis parallel to wi. Case A ( viis not parallel to wi).Sincebuis a projection of viin a space orthogonal to wi,bu is orthogonal to wi. Further since x∼N(0,I)andbwi,buare orthonormal, ⟨x,bwi⟩and⟨x,bu⟩are independent random variables with distributions ⟨x,bwi⟩ ∼N(0, 1)a... | https://arxiv.org/abs/2504.07133v1 |
verify that Equation (96) holds. Proof of Claim 3. Observe that |ρi,V−ρi,W|= projspan(vi,wi)(x)⊤(vi−wi) =|⟨x,bzi⟩⟨vi−wi,bzi⟩+⟨x,bwi⟩⟨vi−wi,bwi⟩|. Sincebziis orthogonal to bwi(by definition) and vi−wilies in span (zi,wi), it follows that |ρi,V−ρi,W|≥|min{⟨x,bzi⟩,⟨x,bwi⟩}−min{⟨x,bzi⟩,⟨x,bwi⟩}|·∥vi−wi∥. Therefore, conditi... | https://arxiv.org/abs/2504.07133v1 |
to proving Lemmas B.2 and B.3. In the remainder of this section, we prove Lemmas B.2 and B.3. Proof of Lemma B.2. By definition, Φ(µ;σ2) =1 2+sgn(µ)√ 2πσZ|µ| 0e−z2/(2σ2)dz≥1 2+max 0≤z≤|µ|ze−z2/(2σ2) √ 2πσ. SinceΦ(0;ν2) = 1/2, Φ(µ;σ2) Φ(0;ν2)≥1+r 2 πmax 0≤z≤|µ|ze−z2/(2σ2) σ. Finally, asr 2 πmax 0≤z≤|µ|ze−z2/(2σ2) σ<1 2,... | https://arxiv.org/abs/2504.07133v1 |
fact that z+1≥1 2(√ 4+z2+z)forz≥0. Thus, we conclude (w+1)(1−Φ(w))≥ϕ(w)or equivalently, ϕ(w) 1−Φ(w)≤w+1 . Returning to the variable z(with w=−z), we deduce ϕ(z) Φ(z)≤ |z|+1 . C Log-Concave Sampling over Convex Bodies In this section, we review well-known results about sampling from a log-concave density ∝e−f constraine... | https://arxiv.org/abs/2504.07133v1 |
we can take M=O(R2)so that Assumption (S4) is satisfied. Next, we can again use the fact that we sample from polytopes contained in B∞(0,R)to deduce that P∩B∞(0,R)is contained in an ℓ2ball of radius R√ d. Moreover, the facet-complexity of P∩B∞(0,R)is at most φ=φP+log2(R). On the other hand, to handle the inner ball, we... | https://arxiv.org/abs/2504.07133v1 |
A GARMA Framework for Unit-Bounded Time Series Based on the Unit-Lindley Distribution with Application to Renewable Energy Data Guilherme Pumia,∗, Danilo Hiroshi Matsuokaaand Taiane Schaedler Prassa Abstract The Unit-Lindley is a one-parameter family of distributions in (0 ,1) obtained from an appropriate transformatio... | https://arxiv.org/abs/2504.07351v1 |
distri- bution. Understanding how this share fluctuates over time is essential for managing energy resources effectively in an evolving and increasingly renewable-focused energy landscape. The constrained nature of such time series hinders the applicability of traditional Gaussian- based models, calling for specialized... | https://arxiv.org/abs/2504.07351v1 |
versatile unipara- metric distribution supported in (0 ,1) belonging to the canonical exponential family. The systematic component is prescribed through an ARMA-like set of difference equations (see (2)), which can also contain a set of exogenous covariates. This structure is considered, for G. Pumi, D.H. Matsuoka and ... | https://arxiv.org/abs/2504.07351v1 |
the standard Lindley distribution (Lindley, 1958) to the unit. Its probability density function is defined by f(y;µ) =(1−µ)2 µ(1−y)3expy(µ−1) µ(1−y) I(0< y < 1), (1) where µ∈(0,1). We use the notation X∼UL(µ) to say that a random variable Xhas the Unit-Lindley distribution with parameter µ. It easy to see that, under... | https://arxiv.org/abs/2504.07351v1 |
PMLE of γif it exists, it is obtained as a solution of the so-called normal equations, given by the system U(γ) =0, where 0is the null vector in Rp+q+r+1. However, the normal equa- tions cannot be solved analytically. In this case, we have to resort to numerical optimization to approximate the PMLE. 3.2 Conditional inf... | https://arxiv.org/abs/2504.07351v1 |
parameter γ0lies in an open set Ω ⊆Rp+q+1, and the covariates Zt−1are almost surely contained within a compact subset Γ ⊂Rp+q+1, such that P nX t=1Zt−1Z′ t−1>0! = 1. 2. The link function gis twice continuously differentiable, with inverse g−1satisfying ∂g−1(x)/∂x̸= 0. Furthermore, Z′ t−1γlies almost surely in the domai... | https://arxiv.org/abs/2504.07351v1 |
or can be obtained (by forecast- ing, for instance). Let ˆγdenote the PMLE estimated from the sample, in view of (2), starting att= 1, we recursively obtain ˆηt= ˆα+ˆX′ tˆβ+pX i=1ˆϕi g2(ˆYt−i)−ˆX′ t−iˆβ +qX k=1ˆθkˆrt−k, (8) with ˆ µt=g−1 1(ˆηt), for t≥1, ˆrt= (g(ˆYt)−ˆηt)I(1≤t≤n), ˆYt= g−1(0), p > 0, t < 1, Yt,... | https://arxiv.org/abs/2504.07351v1 |
steps sequentially, for k∈ {2,···, h}: 1. Update ˆ µ(m) n+kthrough (8) using the augmented sample y1,···, yn,ˆy(m) n+1,···ˆy(m) n+k−1and error terms ˆ r1,···,ˆrn,ˆr(m) n+1,···,ˆr(m) n+k−1. 2. Sample ˆ y(m) n+kfrom a Unit-Lindley distribution with parameter ˆ µ(m) n+kand update ˆ r(m) n+k= g(ˆy(m) n+k)−g(ˆµ(m) n+k). Aft... | https://arxiv.org/abs/2504.07351v1 |
was generated considering n= 500, α= 0.5,β= 0.5,ϕ=−0.4,θ=−0.2. Table 1 summarizes the simulation results. For each set of parameters, we present the mean (left), median (center, in italics), and standard deviations (in parentheses) calculated from the 1,000 replicas. Even when n= 100, parameters αandβare well estimated... | https://arxiv.org/abs/2504.07351v1 |
(0.082) 0.500 0.501 (0.032) -0.798 -0.799 (0.021) 0.198 0.199 (0.045) n α= 1 β= 0.5 ϕ=−0.4 θ=−0.2 100 0.980 0.981 (0.108) 0.501 0.502 (0.062) -0.383 -0.392 (0.122) -0.223 -0.221 (0.144) 200 0.992 0.990 (0.072) 0.501 0.501 (0.043) -0.393 -0.397 (0.081) -0.209 -0.213 (0.093) 500 0.998 0.998 (0.043) 0.501 0.502 (0.026) -0... | https://arxiv.org/abs/2504.07351v1 |
2 provides a summary of the simulation results. All normality tests demonstrated satisfactory performance, with rejection rates close to the nominal value of 0.05 for most parameters. The only notable exception occurred in the scenario where ϕ=−0.8 and θ= 0.2, where the SF test exhibited slightly higher-than-expected r... | https://arxiv.org/abs/2504.07351v1 |
(n): 100 200 500 01234 0.5 1.0density −1.0−0.50.00.5 −1.0−0.50.00.5 0.5 1.0 α^ φ^ −1.0−0.50.00.5 0.5 1.0 Sample size (n): 100 200 500 0369 0.2 0.4 0.6 0.8density −1.0−0.50.00.5 −1.0−0.50.00.5 0.2 0.4 0.6 0.8 β^ φ^ −1.0−0.50.00.5 0.2 0.4 0.6 0.8 Sample size (n): 100 200 500 01234 0.5 1.0density −1.0−0.50.00.51.0 −1.0−0.... | https://arxiv.org/abs/2504.07351v1 |
patterns without the aid of seasonal components and its superior forecasting capabilities in this scenario. To further highlight that, we shall compare the ULARMA to benchmark models – the KARMA andβARMA models – which are based on bi-parametric conditional distribution and are known to be very flexible. 6.1 Parameter ... | https://arxiv.org/abs/2504.07351v1 |
second, with exactly 7 coefficients for each model, whereas theβARMA presented uniformly more complex models, with 10 to 11 significant coefficients in each model. Considering simple residuals, none of the models rejected the martingale difference null hypothesis considering the Dom´ ınguez-Lobato test. It is interesti... | https://arxiv.org/abs/2504.07351v1 |
parameter was ϕ4andϕ7present in 7 out of the 9 models. Parameter θwas significant for all fitted βARMA models, but for no other. 18 Unit-Lindley Autoregressive Moving Average Table 4: Fitted ULARMA, KARMA and βARMA models: estimated coefficients and respec- tive standard errors (in parenthesis). Model ULARMA KARMA βARM... | https://arxiv.org/abs/2504.07351v1 |
G. Pumi, D.H. Matsuoka and T.S. Prass 19 an indication of overfitting. ULARMA’s forecasted values stayed at lower values, much closer to the observed data. The KARMA and βARMA approach the observed data in the end of the 12-step horizon. Table 5 complements the results, presenting the RMSE, MAPE forecast accuracy mea- ... | https://arxiv.org/abs/2504.07351v1 |
KARMA andβARMA for modeling the proportion of net electricity generated by conventional hydro- electric power in the United States, as presented in the previous section, several conclusions can be drawn. The empirical results emphasize the value of having a diverse repertoire of mod- els available, and illustrate why i... | https://arxiv.org/abs/2504.07351v1 |
the one parameter unit- lindley distribution and its associated regression model for proportion data. Journal of Applied Statistics , 46(4):700–714. Pe˜ na Ram´ ırez, M., Rojas Guerra, R., and Bayer, F. M. (2024). The Rayleigh generalized autoregressive score model for SAR data interpretation. IEEE Geoscience and Remot... | https://arxiv.org/abs/2504.07351v1 |
arXiv:2504.07384v1 [q-bio.PE] 10 Apr 2025Convergence-divergence models: Generalizations of phylogenetic trees modeling gene flow over time Jonathan D. Mitchell1,2*and Barbara R. Holland1,2 1School of Natural Sciences (Mathematics), University of Ta smania, Hobart, TAS, Australia. 2ARC Centre of Excellence for Plant Succ... | https://arxiv.org/abs/2504.07384v1 |
lutionary processes leading to gene flow, for example, introgressive hybridization, horiz ontal gene transfer and recombination. See Kong et al. (2022) for a thorough review of the classes of phylogeneticnetworks.Phylogeneticnetworkshave“hybrid”nod esmodeling geneflow — not necessarily hybridization — between taxa. Howev... | https://arxiv.org/abs/2504.07384v1 |
multiple individuals per taxon; CD Ms can be inferred from datasets with a single individual per taxon. Distinct fr om ABBA-BABA tests,CDMs canbeinferredfromdatasetswithanynumberoftax a.ForABBA-BABA tests, rejection of the null hypothesis — a phylogenetic tree — is as sumed to be due to gene flow, with no explicit model... | https://arxiv.org/abs/2504.07384v1 |
event m ust be modified to represent probabilities of combinations of states on k+1 edges after the event. One of the kedges before the speciation event splits into two edges after the s pe- ciation event. The edge that is split is modeled by the Markov model in t he epoch before the speciation event, with nindependent ... | https://arxiv.org/abs/2504.07384v1 |
a set of leaf taxa, with each edge having one descendant leaf tax on. We call this set of leaf taxa identical. When there is no ambiguity, we simply refer to the leaf taxa as “taxa”. Taxa that are not identical are diverging at that time. After pushing back splitting operators, at an arbitrary time below t he root each... | https://arxiv.org/abs/2504.07384v1 |
the principal tree. We define the root as the node with outdegree 2 that is the most rec ent common ancestor of all leaf taxa. It is useful to define the root as having in degree 1 when considering splitting operators — see Figure 1— and defining the root as having indegree 0 otherwise — see Figure 2. It is of no conseque... | https://arxiv.org/abs/2504.07384v1 |
the leaf taxon set Xis a partition Pof non-empty sets and par- titions, where each set in Pis a strict subset of X, each partition in Pis a partition of a strict subset of Xand each taxon in Xappears in exactly one set or partition in P. On CDMs, decorated partitions correspond to epochs where the re is convergence. Fo... | https://arxiv.org/abs/2504.07384v1 |
ts does not include all possible scenarios of modeling convergence on CDMs. Furthermore , the assumptions that follow in Section 3.2further restrict convergence scenarios on CDMs. For an example of events on a CDM, consider CDM 5 in Figure 2(e). The event at the root and the second and third events are speciation event... | https://arxiv.org/abs/2504.07384v1 |
epoch, as in Sumner et al. (2012). This rate matrix, defined by a continuous-time Markov model, describes all evolutionary processes in the epoch. Recall that in each epoch we consider there to be a one-to-one cor respondence between the edges and the taxa after pushing back splitting opera tors. Thus, for N taxaandasta... | https://arxiv.org/abs/2504.07384v1 |
taxa have the same state have non-zero probabilities. For the algorithms that follow in Sections 6-8, we consider a special type of CDM. Definition 4 A 2-state general convergence-divergence model is a convergence-divergence model with rate matrices from the 2-state general Markov mod el, equal ratios of substitution ra... | https://arxiv.org/abs/2504.07384v1 |
probabilities of combinations of states at the leaves of the principal tree is called the phylogenetic tensor . It is a vector representation of a tensor. The phylogenetic tensor Pis P=r/productdisplay a=1exp(Qata)·Π, where, for epoch a,Qais the rate matrix, tais the epoch length and the product is over the epochs, who... | https://arxiv.org/abs/2504.07384v1 |
space. Formostalgorithms,propositionsandtheoremsofthefollowingsec tions—exclud- ing Section 4— the assumptionsofSection 3.2hold, aswellassomeother assumptions that we describe later, sufficient for consistent inference of many aspects of the CDM. In addition to the assumptions on the CDMs, all random variables are indepe... | https://arxiv.org/abs/2504.07384v1 |
an MSA to a gene or genomic window, then discard ing most sites likely gives poor statistical power. With this in mind and the desira ble statistical propertiesofthe compositelikelihood, weretain the iid assumptionwit hout discarding any random variables. Correctly discovering sister convergence is challenging, typical... | https://arxiv.org/abs/2504.07384v1 |
some Cr∈ C,/producttext a∈Crja= 0,/producttext a∈Cria= 1andia=jafor alla∈X\Cr, βr if for some Cr∈ C,/producttext a∈Cr(1−ja) = 0,/producttext a∈Cr(1−ia) = 1andia=jafor alla∈X\Cr, 0 otherwise if i/\e}atio\slash=j, −/summationtext2N s=1,s/ne}ationslash=j/bracketleftBig Q[C]/bracketrightBig sjifi=j, whereαr,βr>0. 14 See Ap... | https://arxiv.org/abs/2504.07384v1 |
lead to incorrect inference of the topology of the principa l tree or false dis- covery of non-sister convergence groups. Thus, we assume the N-taxon CDM has no sister convergence (Assumption 10). However, N-taxon CDMs with no sister conver- gence may still display 4-taxon CDMs with sister convergence.Again , ignoring ... | https://arxiv.org/abs/2504.07384v1 |
on the right. Parameters are labeled on sections of the edges of CDM 5 CDMs must be “small”. Assuming the generating parameter is a gener ic point in the CDM parameter space, this property along with the nested proper ty of our CDMs guarantees that if one of the 4-taxon CDMs is displayed on the gene rating CDM then it ... | https://arxiv.org/abs/2504.07384v1 |
and rows are taxa. Alter - natively, it could be ancestral/derived states or a gene presence/ absence dataset. The algorithms use criteria, including a multiple comparisons correction, w hen inferring convergence groups on 4-taxon CDMs to avoid overfitting. If con vergence groups are falsely inferred on the 4-taxon CDMs... | https://arxiv.org/abs/2504.07384v1 |
all others — for example, according to the AIC or BIC — are segregate d from those sets where other CDMs have similar goodness of fit to the best fitting CDM . For a given 4- taxonset,when asingleCDM easilyfitsbest, asingletopologyofthe4 -taxonprincipal tree is inferred. Otherwise, for a given4-taxonset, topologiesof 4-ta... | https://arxiv.org/abs/2504.07384v1 |
transformation does not influence inference of any other parts of the CDM, including parameters. Th e metrization is only used to infer the topology of theN-taxon principal tree. Parameters are inferred in later algorithms. The resulting tree metric is a slight modification of the tree metric dRTofRhodes (2019) for roote... | https://arxiv.org/abs/2504.07384v1 |
l selection criterion values below τtoTQ. 3. Use consistent supertree inference method to infer set of top ologies of N-taxon principal trees SfromTQ. 4. Infer consensus tree /hatwideT′fromS, rooting with o. 5. If/hatwideT′is not resolved, either set /hatwideT=/hatwideT′and terminate algorithm or resolve: 5.1. Initiali... | https://arxiv.org/abs/2504.07384v1 |
limit as some epoch lengths convergeto 0 or diverge to ∞. That is, the phylogenetic tensor as a function of the parametersof the CDM is restricted by taking the limit of some of the parameters. o a bc∞ 0 (a)N1o b c a∞ (b)N2o c b a∞ (c)N3 Fig. 44-taxon CDMs N1,N2andN3, with identical sets of possible phylogenetic tensor... | https://arxiv.org/abs/2504.07384v1 |
rs of some convergence groups. Suppose C1={c1,a,c1,b}andC2={c2,a,c2,b}are convergence groups, where c1,a,c1,b,c2,aandc2,bare sets of taxa. If c2,a⊂c1,a, thenC1must be in an epoch beforeC2. Furthermore, since CDMs 4 and 5 of Section 5.2both have two convergence groups in separate epochs, it is possible to infer relative... | https://arxiv.org/abs/2504.07384v1 |
{a,b}is |Xv\XC| N−3=|Xv|−|XC| N−3, where|Xv|and|XC|are the cardinalities of sets XvandXC. See Appendix Ifor the proof. Although we do not attempt to infer sister convergence, it is const ructive to consider a scenario with sister convergence groups. Corollary 10 IfC={c1,c2}is a sister convergence group on CDM N, witha∈... | https://arxiv.org/abs/2504.07384v1 |
with the aid of an indicator variab le and two criteria at each step. We only allow a convergence group on the inferred N-taxon CDM if the set of inferred 4-taxon CDMs is similar to the set of 4-taxon CDMs af ter suppressing sister convergence groups displayed on the inferred N-taxon CDM. When inferring a convergence g... | https://arxiv.org/abs/2504.07384v1 |
on /hatwideThas no polytomies: 2.1.1. Select CDM with 4-taxon principal tree displayed on /hatwideTwith model selection criterion, using multiple comparisons correction, such as in Appendix H, and append to LQ. 3. Compute [ O]ijfor all pairs of taxa i,j. 4. Compute initial sum of squared differences between elements of ... | https://arxiv.org/abs/2504.07384v1 |
recall th at the distance between taxa is the sum of convergence and divergence paramete rs along the shortest path between the two taxa. Thus, the distances on the N-taxon principal tree do not necessarily reflect how similar the random variables are to each othe r. Proposition 14 All edge lengths of the principal tree... | https://arxiv.org/abs/2504.07384v1 |
CDM being the 4-taxon CDM displayed on the generating N-taxon CDM converges to 1. However, for the latter, some pairwise distances between leaf taxa may not be estimated since some 4-taxon sets are not considered. Mat rixXdescribes the edges of principal tree Tthat are traversed to compute pairwise distances between ta... | https://arxiv.org/abs/2504.07384v1 |
gene flow between taxa ov er time that can be applied to large datasets. Convergence-divergence models are generalizations of phylogenetic trees for many-taxon datasets. In contrast to phylogenetic networks, they have a single “principal tree”. A Markovmodel describes indep endent divergence of taxa on the principal tre... | https://arxiv.org/abs/2504.07384v1 |
— from similar se lective pres- sures. This violates the assumptions of phylogenetic trees of indep endent divergence of taxa from common ancestors. Replicated evolution can lead to th e gradual conver- gence of taxa. This process is not appropriately modeled by phyloge netic networks, but can be modeled by our converg... | https://arxiv.org/abs/2504.07384v1 |
Journal of Mathematical Biology , 88(2):17, 2024. doi:10.1007/s00285-023-02038-9 . Daniel Huson, Scott Nettles, Laxmi Parida, Tandy Warnow, and Sh ibu Yooseph. The disk-covering method for tree reconstruction. Proceedings of “Algorithms and Experiments,” ALEX , 98:62–75, 1998. Daniel H Huson, Scott M Nettles, and Tandy... | https://arxiv.org/abs/2504.07384v1 |
Sumner, BR Holland, and PD Jarvis. The algebra of the gene ral markov model on phylogenetic trees and networks. Bulletin of Mathematical Biology , 74(4): 858–880, 2012. doi: 10.1007/s11538-011-9691-z . EB Taylor, JW Boughman, M Groenenboom, M Sniatynski, D Schluter, and JL Gow. Speciation in reverse: morphological and ... | https://arxiv.org/abs/2504.07384v1 |
of diverging and converging sections. Since differences in parameters between contiguous diverging sections cannot be identified, the diverging sections we co nsider are those sections on the principal tree between a node or converging sectio n and another node or convergingsection. Furthermore, since the exact root loca... | https://arxiv.org/abs/2504.07384v1 |
all in the same convergence- divergence group in some epoch. Then from Sumner et al. (2012b), the rate matrix for the epoch is Q[l]=αL[l] α+βL[l] β, whereα,β >0. We first prove that /bracketleftBig Q[l]/bracketrightBig ij= αif/producttextl a=1ia= 1 and/producttextl a=1ja= 0, βif/producttextl a=1(1−ia) = 1 and/prod... | https://arxiv.org/abs/2504.07384v1 |
is obtained from Q[l]⊗I⊗N−lby the permutation σon the slots of the Kronecker products of each term of L[l] α⊗I⊗N−landL[l] β⊗I⊗N−l. Then it follows directly from 2) that /bracketleftBig Q[Cr]/bracketrightBig ij= αif/producttext a∈Cria= 1,/producttext a∈Crja= 0 andia=jafor alla∈[N]\Cr, βif/producttext... | https://arxiv.org/abs/2504.07384v1 |
and/producttextl a=1(1−ja) = 0, 0 otherwise if i/\e}atio\slash=j, ∗otherwise if i=j. Thus, for columns to sum to zero, /bracketleftBig Q[l]/bracketrightBig ij= α if/producttextl a=1ia= 1 and/producttextl a=1ja= 0, β if/producttextl a=1(1−ia) = 1 and/producttextl a=1(1−ja) = 0, 0 otherwise if i/\e}at... | https://arxiv.org/abs/2504.07384v1 |
be non-zero, there must exist some index s, such that /bracketleftBig M[Cv−1]/bracketrightBig is>0 (B1) and /bracketleftBig Pk−v+2/bracketrightBig s>0. (B2) For Equation ( B1) to be true,/braceleftBigg/producttext a∈Cv−1ia= 1 or/producttext a∈Cv−1(1−ia) = 1, ia=safor alla∈[N]\Cv−1.(B3) For Equation ( B2) to be true, by... | https://arxiv.org/abs/2504.07384v1 |
determine any invariants involving multiple variables r0011,r0101, ...,r1111,δ.) In the Macaulay2 file S4.m2 (output file S5.txt) on https://github.com/ jonathanmitchell88/CDMsSI we derive the (reduced) Gr¨ obner basis for this ideal for a particular monomial order for CDM 5. Below we outline how this Gr¨ ob ner basis is... | https://arxiv.org/abs/2504.07384v1 |
of paramete rs is identifiable. We note thatxi∈(0,1) for all i∈ {1,2,...,11}. It follows that r0011,r0101,...,r1111,δ∈(0,1) and yi∈(0,1) for all i∈ {1,2,...,9}. In S2.nb (text version S3.txt), we see that the solutions to the system are y1=δ r0111, y2=r0111√r1001r1010 δ√r0011, y3... | https://arxiv.org/abs/2504.07384v1 |
z4z7z8−z5z9/\e}atio\slash= 0. Thus, for CDM 5, any two CDMs with different leaf labelings are distinguishab le. 51 CDM 4 The proof is identical to that of CDM 5, but with the addition o fy9=z9= 1. Again, see S8.nb (text version S9.txt) and S10.m2 (output file S11.txt) . We obtain z1z2z3z4z5z6z8(1−z6)(1−z7z8) = 0, which a... | https://arxiv.org/abs/2504.07384v1 |
the display ed 4-taxon CDMs. However, it is possible that a displayed 4-taxon CDM does not meet th e assumptions of Section 3.2. Specifically, even if an N-taxon CDM meets the assumptions, some dis- played 4-taxon CDMs may have sister convergence. By assuming th at all convergence 53 parameters of the N-taxon CDM are su... | https://arxiv.org/abs/2504.07384v1 |
>0. The remainder of the proof then follows from Haughton (1988). /square A convergencegroup on the generating N-taxon CDM may be a sister convergence group on some displayed 4-taxon CDMs and a non-sister convergen cegroup on others. Thus, we must assume that all convergence parameters of the ge nerating N-taxon CDM ar... | https://arxiv.org/abs/2504.07384v1 |
the principal tree ofNconsistently and the proof is complete. All that is left to prove is the claim that the probability of t he set of inferred 4-taxon principal trees equalling the set of topologies of principa l trees of 4-taxon CDMs displayed onNconverges to 1. 56 Suppose Aiis the event where the topology of the i... | https://arxiv.org/abs/2504.07384v1 |
values by small amounts. Suppose OandEare the matrices of observed and expected proportions of converging quartets. Suppose C={c1,c2}is an arbitrary convergence group, with c1∪c2⊂[N]. Thena1= max i∈c1,j∈c2/vextendsingle/vextendsingle/vextendsingle[O]ij−[E]ij/vextendsingle/vextendsingle/vextendsingleanda2=1 (N−1)2/summa... | https://arxiv.org/abs/2504.07384v1 |
in an epoch beforeC2. In order to share at least one pair of converging taxa, C2must be nested in C1. How- ever, by Assumption 9of Section 3.2, there can be no convergence groups nested in other convergence groups. /square Appendix K Proof of Proposition 12 We assume that the topology of the principal tree of Nis known... | https://arxiv.org/abs/2504.07384v1 |
principal tree of N4is (o,(b,(a,c))). Suppose that C′=/braceleftbig c′ 1,c′ 2/bracerightbig is one such non-sister convergence group that defines S′, withc′ 1,c′ 2,v′,X′vandXC′as in Proposition 9. Now consider 4-taxon CDM N′ 4, defined by C′and on leaf taxon set {o,a,b,c}, with topology of principal tree ( o,(b,(a,c))). ... | https://arxiv.org/abs/2504.07384v1 |
of whether the 4-taxon CDMs have s ister convergence groups or not, the non-sister convergence group is inferred consiste ntly. 62 Sinceαl=βl,γ= 0 and the transformed phylogenetic tensor for a 4-taxon CDM of Equation ( C6) simplifies to /hatwideP= 1 0 0 r0011 0 r0101 r0110 0 0 r1001 r1010 0 r... | https://arxiv.org/abs/2504.07384v1 |
L3for a graphical depiction of the two CDMs. Again, suppose N4,3has parameters with no apostrophes and N4,4has parameters with apostrophes. ForN4,3(see Mathematica file S12.nb (text version S13.txt) for a der ivation), r0011=x4x5, r0101=x2x3x4, r0110=x2x3x5, r1001=x1x2x4, r1010=x1x2x5, r1100=x... | https://arxiv.org/abs/2504.07384v1 |
the tree is returned. If Nis not a tree, since u= 1, a potential convergence group on Nis only considered if, for all pairs of converging taxa in the conver gence group, the inferred 4-taxon CDMs with that pair of taxa as non-sisters all have the pair co nverging. Thus, asymptotically with probability 1, only convergen... | https://arxiv.org/abs/2504.07384v1 |
convergence group orders. Suppose an arbitrary convergencegroup is Ci={c1,i,c2,i}. On the N-taxon CDM, if|c1,i|>1 and/or |c2,i|>1 orCiis in an epoch before another convergence group, thenCicannot be in the tip epoch. For other convergence groups, wheth er they are in the tip epoch or not must be inferred. For each 4-ta... | https://arxiv.org/abs/2504.07384v1 |
of observed convergence group orders Oas zero matrix, where kis length of list /hatwideG. Initialize k×kmatrixEof expected convergence group orders as convergence group orders defined by P, with [E]ij= 1 if convergence group ibeforejand 0 otherwise. 2. For each 4-taxon set that includes outgroup o, with model selection ... | https://arxiv.org/abs/2504.07384v1 |
of divergence and possibly convergence parameters along the terminal edges whose descendent leaf t axa area,bandcrespectively andlbcis the sum of divergence parameters along the edge whose desc endent leaf taxa are b andc. It follows that all edge lengths are also identifiable for CDM s 1−4 since expressions for the sum... | https://arxiv.org/abs/2504.07384v1 |
estimates of the sums of edge lengths between taxa converge in probability to the values f orN. Now, since the matrix Xhas rank 2 N−3,XTXis invertible. It follows that /hatwidelalso converges in probability to lin step 7 of Algorithm 3. By assumption, for each convergence group of Gthere is at least one 4-taxon CDM dis... | https://arxiv.org/abs/2504.07384v1 |
Adversarial Subspace Generation for Outlier Detection in High-Dimensional Data Jose Cribeiro-Ramallo jose.cribeiro@kit.edu Karlsruhe Institute of Technology Federico Matteucci federico.matteucci@kit.edu Karlsruhe Institute of Technology Paul Enciu paul.enciu@student.kit.edu Karlsruhe Institute of Technology Alexander J... | https://arxiv.org/abs/2504.07522v1 |
2024). These methods assume that the data lies on a single, low-dimensional subspace preserving its properties, such as point distances, topology, or notably, the underlying distribution. As discussed in Example 1, however, a single, low-dimensional subspace might not be enough to characterize the data. It is therefore... | https://arxiv.org/abs/2504.07522v1 |
to arbitrary data types; we will elaborate on this in Section 3.1. (2) Even with a theory able to recognize the subspaces relevant for MV, these latter live in an exponential search space — the power set of the set of features. An exhaustive search is therefore unfeasible. The designed method should be able to find rel... | https://arxiv.org/abs/2504.07522v1 |
can triviallyprojectthe data into each subspace for the desired downstream task. They are popular for outlier detection, as they allow the use of subspaces to create ensembles with off-the-shelf outlier detectors (Aggarwal, 2017). However, as discussed earlier, the main drawbacks of these methods are the cardinality of... | https://arxiv.org/abs/2504.07522v1 |
notion of "important" subspace. This solves the representation problem of subspace search, while not sacrificing the ability to projectthe data into the subspaces. 3 Myopic Subspace Theory In this section, we will discuss the preliminaries for introducing our Subspace Generation method. We will frame our theoretical ba... | https://arxiv.org/abs/2504.07522v1 |
find a U∗that verifies Definition 1 for a given xν. Even assuming that we can perfectly estimate the densities, how to find such a random matrix that |Pxν(p)−PUxν(p)|= 0for almost allpis unclear for the finite sample setting. In the following section, we will propose a general definition that addresses all previous wea... | https://arxiv.org/abs/2504.07522v1 |
class of functionals on E. The Maximum Mean Discrepancy ( MMD) is defined as: MMDκ(p,q) = sup f∈F(Epf−Eqf). (2) 6 InHan RKHS with kernel κis measurable and such that4/integraltext/radicalbig κ(·,·)dp<∞,for all p∈M+ 1(E),andFthe unit ball inH, one can easily prove (Sriperumbudur et al., 2010) that: ∃!µp∈Hsuch that Epf=<... | https://arxiv.org/abs/2504.07522v1 |
such a sequence in MΘ(X) x. Under Theorem 2, we know that such sequence has a limit in MΘ(X) x, and that the limit will also be a global optimum in M+ 1. The usefulness of this is made clear in the following corollary, which also give us the conditions to write Theorem 2 in terms of operators on Θ(X). This corollary wi... | https://arxiv.org/abs/2504.07522v1 |
20). In particular, Generative Moment Matching Networks (MMD-GANs) use the squared sample MMD as their loss function, written as L(θ) =\MMD2 κ(Px,PFθ(z)), withFθ:z∈Z/map∫to−/∫hortrightarrowFθ(z) = ˆx∈Ea generative network. These networks guarantee convergence in distribution of Fθ(z)toxwhen minimizing the loss in terms... | https://arxiv.org/abs/2504.07522v1 |
xandUxhave the same distribution, we need to use the following hypothesis test: /braceleftigg H0:Px=PUx, Ha:Px̸=PUx.(10) As the sample MMD’s asymptotic distribution is tabulated, one can use it for such statistical test. In other words, we can test whether a given operator Uis a lens operator for xby using the MMD tes... | https://arxiv.org/abs/2504.07522v1 |
κϕ=ς◦Eϕ. Here,ςis a Gaussian kernel with the median heuristic bandwidth parameter (Garreau et al., 2018) and Eϕ an encoder trained by kernel learning — see Section 4.1.2. Particularly, we use an upside-down version of the generator’s hidden layers for Eϕ, with the identity function as the output layer. Training We trai... | https://arxiv.org/abs/2504.07522v1 |
al., 2012; Trittenbach and Böhm, 2019) apart from V-GAN. We could not include the subspace selection method CLIQUE (Agrawal et al., 2005) as it does not report a quality metric for the subspaces. As we are using a 3-dimensional dataset, it will be enough to employ a regular Gaussian kernel with the recommended bandwidt... | https://arxiv.org/abs/2504.07522v1 |
subspace, following (Cribeiro-Ramallo et al., 2024, Propositon 1). For FB, this reduces to a simple average. Furthermore, we will evaluate its performance on datasets with and without a myopic distribution, providing insights into both the best-case scenario (where Vis a lens operator) and the worst-case scenario (wher... | https://arxiv.org/abs/2504.07522v1 |
++−− + ++ −− HiCS ++ + ++ −− CLIQUE−− −− −− −− −− −− ELM ++− ++ ++−− GMD ++ ++−− PCA− ++ ++−− UMAP−− −− −− −− −− −− V-GAN ++ ++ ++ ++ ++ ++ ++ ECODCAE −− ++ +−− −− HiCS ++ ++ −− CLIQUE−− −− −− −− −− ELM− ++ −− GMD ++ ++ −− PCA UMAP V-GAN ++ ++ ++ ++ ++ COPODCAE −− ++−− −− HiCS ++ ++ −− CLIQUE−− −− −− −− −− ELM ++ −− GM... | https://arxiv.org/abs/2504.07522v1 |
on the data’s distribution being myopic, a property we can infere from data without any prior knowledge. Furthermore, even when the data is not myopic, V-GAN is still not outperformed by its competitors. Our findings not only validate the superior performance of V-GAN for subspace selection, but also show the potential... | https://arxiv.org/abs/2504.07522v1 |
multiple views of high-dimensional data, 2024. URL https://arxiv.org/abs/2404.14451 . E. Elhamifar and R. Vidal. Sparse Subspace Clustering: Algorithm, Theory, and Applications, Feb. 2013. URL http://arxiv.org/abs/1203.1005 . arXiv:1203.1005 [cs, math, stat]. A. M. Faden. The Existence of Regular Conditional Probabilit... | https://arxiv.org/abs/2504.07522v1 |
Ionescu, and G. H. Chen. ECOD: Unsupervised Outlier Detection Using Empirical Cumulative Distribution Functions. IEEE Transactions on Knowledge and Data Engineering , 35(12):12181–12193, Dec. 2023. ISSN 1558-2191. doi: 10.1109/TKDE.2022.3159580. URL https:// ieeexplore.ieee.org/document/9737003 . Conference Name: IEEE ... | https://arxiv.org/abs/2504.07522v1 |
first introduce all of the proofs of the statemes from Section 3, and then introduce all of the additional statements and proofs. To maintain the clarity of this section, we will re-introduce all of the statements before their proofs. Lemma 1. ConsiderHa RKHS with a characteristic kernel κ; and x,UandMMDas previously d... | https://arxiv.org/abs/2504.07522v1 |
0, forA∈F(Ux)— i.e., all realizations are mutually exclusive. ii)The set of all realizations of Uis countable. iii)There exists a meassure µsuch thatµ>>PUxandµ>>PU(ω)x,∀ω∈Ω. In this case, Px=/summationtext ω∈ΩPU(U(ω))PU(ω)xandPx=/summationtext ω∈ΩPU(U(ω))PU(ω)x. Proof.Consider all U∈Θ(X)to be defined on fibers of E. By... | https://arxiv.org/abs/2504.07522v1 |
Gθthen 12: UpdateGby descending the stochastic gradient: ∇θLkl(data,noise ;θ,ϕ) 13:trained_epochsGθ+= 1 14: iftrained_epochsGθ≥iternumGθ&trained_epochsEϕ≥iternumEϕthen 15: trained_epochsEϕ= 0, trained _epochsGθ= 0 16: end if 17:end if 18:end for 19:end for 22 Algorithm 1 takes as input the dataset D, the kernel κ, the ... | https://arxiv.org/abs/2504.07522v1 |
2024). We employed the implementation provided in their official package. We chose 15neighbors as recommended by the authors. As for the dimensionality of the underlying manifold, the authors recommend using between 10and100for downstream machine learning tasks. As the dimensionality of our datasets Dvaries, we opted t... | https://arxiv.org/abs/2504.07522v1 |
- ++ - - HiCS ++ ++ CLIQUE - - - - - - - - - - - - ELM ++ ++ - GMD + ++ ++ PCA ++ ++ - UMAP - - - - - - - - - - - - V-GAN ++ ++ + + ++ kNNCAE - - ++ - - - - - + - - HiCS ++ ++ ++ CLIQUE - - - - - - - - - - - - ELM + ++ ++ - - GMD ++ ++ ++ PCA ++ ++ ++ UMAP - - - - - - - - - - - V-GAN ++ ++ ++ ECODCAE ++ - - - HiCS ++ -... | https://arxiv.org/abs/2504.07522v1 |
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