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arXiv:2503.17538v1 [stat.ML] 21 Mar 2025A Statistical Theory of Contrastive Learning via Approxima te Sufficient Statistics Licong Lin∗Song Mei∗† Abstract Contrastive learning—a modern approach to extract useful r epresentations from unlabeled data by training models to distinguish similar samples from dissim ilar ones—h... | https://arxiv.org/abs/2503.17538v1 |
Department of EECS, UC Berkele y. Email: songmei@berkeley.edu . 1 equivalent forms of the definition, we establish that minimizing the con trastive loss (e.g., the InfoNCE loss [OLV18]) is essentially finding approximate sufficient statistics that are adap table to downstream tasks. (2) We focus on data augmentation-based ... | https://arxiv.org/abs/2503.17538v1 |
in Theorem 1is independent of the batch sizeKand thus allows for large or full-batch learning. The most related wor k to ours is [ OLCM25 ], which introduced the concept of approximate sufficiency to assess the quality of representations. They also demonstrated that the learned representation from CLIP [ RKH`21] can be e... | https://arxiv.org/abs/2503.17538v1 |
by minimizing RfpS˝Tqover a sufficiently rich spaceS. Consequently, VFS provides a loss minimization framework for findin gTwith low sufficiency by minimizing the f-contrastive loss RfpSqoverSin some space Sand extracting TfromS. Moreover, an extension of approximate sufficiency to similarity scores Sis introduced in Appendix... | https://arxiv.org/abs/2503.17538v1 |
either discrete or has a continuous density w.r.t. some base measu re onXb3. We abuse the notation Pp¨qto refer to either discrete distributions or the density of continuo us distributions, with the intended meaning clear from the context. Als o, we occasionally omit the subscript kl when referring to KL-sufficiency. Sim... | https://arxiv.org/abs/2503.17538v1 |
concentration propert ies of functions with bounded differences. Interestingly, it depends only on the total sample size n“n1Kratherthan the batch size Korthe number of batchesn1. This allows our results to account for large or full-batch training, a s used in SimCLR [ CKNH20 ] and CLIP [ RKH`21]. WhennÑ 8, the generali... | https://arxiv.org/abs/2503.17538v1 |
replaced by the minimum error rǫG:“infhEx„PX,g„PGrphpgpxqq ´h‹pxqq2s ďǫG. We refer to the proof for more details. 6 Adaptation to downstream classification task. We next turn to classification tasks. Suppose in the downstreamweare givensamples px,yqfrom somejoint distribution PonXˆrKs, wherex„PXis the input andyP rKsis t... | https://arxiv.org/abs/2503.17538v1 |
logp1{δq `B2 τż2psBS`Bτq 0b logNpu,} ¨ }2,8,Fqduı , approximation error :“inf fPFRχ2pSfq ´Rχ2pS‹q for some absolute constant cą0. The proof of Theorem 4is provided in Appendix B.5. Note that we do not assume the boundedness of S‹as in Theorem 1. 4.3.2 Implications of low f-Sufficiency Similar to the KL case in Section 4.... | https://arxiv.org/abs/2503.17538v1 |
Opd{mq, Theorem 6achieves a smaller excess risk of order rOpp{mqwhenp!dandfpgpxqqis a “good” representation of x, in the sense that Suff fpfqandǫGare sufficiently small. A similar bound can be established for the risk Rlinprhpηqwith high probability under additional sub-Gaussian assumptions on the representation fpzq “Wgp... | https://arxiv.org/abs/2503.17538v1 |
we provide theoretical guarantees for contrastive learnin g and its downstream performance in a classi- fication setting. Let Y“ t1,2,...,M urepresent a set of classes. A sample xis generated by first selecting a classyPYfrom some distribution PY, and then drawing x“ pxc1,xc2q P rSs ˆ rSsconditioned on y, with the joint ... | https://arxiv.org/abs/2503.17538v1 |
exists some absolute constants c,c1ą0such that, given the encoder pfand suppose Suffχ2pSpfaugq ďc1σ2 E‹ S2M, with probability at least 1´δ1 RclspshpΓq:“Ex,y,grDKLpPpy|xq||hpΓppfpgpxqqqqs ďc´ ” ǫcls G`SexppBq σ2 E‹¨Suffχ2pSpfaugqı loooooooooooooooooooomoooooooooooooooooooon approximation error`B?m”a logp1{δ1q `Mpa logBΓ`?... | https://arxiv.org/abs/2503.17538v1 |
ence on artificial intelligence and statistics, JMLR Workshop and Con ference Proceedings, 2010, pp. 297–304. [GKKW06] L´ aszl´ o Gy¨ orfi, Michael Kohler, Adam Krzyzak, and Har ro Walk, A distribution-free theory of nonparametric regression , Springer Science & Business Media, 2006. [GSA`20] Jean-Bastien Grill, Florian ... | https://arxiv.org/abs/2503.17538v1 |
and multimodal generative ai , arXiv preprint arXiv:2501.04641 (2025). [OLV18] Aaron van den Oord, Yazhe Li, and Oriol Vinyals, Representation learning with contrastive predictive coding , arXiv preprint arXiv:1807.03748 (2018). [POVDO`19] Ben Poole, Sherjil Ozair, Aaron Van Den Oord, Alex Alemi, and Geo rge Tucker, On... | https://arxiv.org/abs/2503.17538v1 |
III 14, Springer, 2016, pp. 649–66 6. [ZMKB23] Xiaohua Zhai, Basil Mustafa, Alexander Kolesnikov, and Lu cas Beyer, Sigmoid loss for lan- guage image pre-training , Proceedings of the IEEE/CVF International Conference on Com - puter Vision, 2023, pp. 11975–11986. [ZSS`21] Roland S Zimmermann, Yash Sharma, Steffen Schnei... | https://arxiv.org/abs/2503.17538v1 |
. . . . . . . 17 A.3 Sufficiency of similarity scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 B Proofs in Section 4 20 B.1 Proof of Eq. (6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 B.2 Proof of Theorem 1 . . . . . . . . . . . . . . . . .... | https://arxiv.org/abs/2503.17538v1 |
. . . . . . . . . . 32 C.4.2 Proof of Eq. (15) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 C.5 An auxiliary lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 15 A Properties of approximate sufficient statistics In this section, we discuss some ... | https://arxiv.org/abs/2503.17538v1 |
convex conjuga tes that∇f˚pf1pxqq “x. Lemma 3 (A general bound on DTVpPpy|xq||Ppy|Tpxqqqbased on sufficiency.) .Forfin Definition 1that is twice continuously differentiable, and for any statistic T, we have EPpxqrDTVpPpy|xq||Ppy|Tpxqqqs ďc2¨b Suffcb,fpTq, (17) wherec2:“´ 2inf px,yqPsupp px,yqf2´ Ppy|xq Ppyq¯¯´1{2 , andsupp ... | https://arxiv.org/abs/2503.17538v1 |
definition of the pVFS q, we have Suffvf,fpSq “RfpSq ´RfpS‹q “EPpx,yqrS‹´sSpx,yqs `EPpxqPpyqrf˚psSpx,yqq ´f˚pS‹px,yqqs piq“EPpx,yqrS‹´sSpx,yqs `EPpxqPpyq” f´Ppx,yq PpxqPpyq¯ ´Ppx,yq PpxqPpyqS‹px,yqı “ ´EPpx,yqrsSpx,yqs `EPpxqPpyqrf˚psSpx,yqqs `EPpxqPpyq” f´Ppx,yq PpxqPpyq¯ı piiq“ ´EPpx,yqrsSpx,yqs `EPpxqPpyq” f´Ppx,yq Pp... | https://arxiv.org/abs/2503.17538v1 |
( 19).Recall the definition of pRsimclr,Kin Eq. (3) and adopt the shorthand pRKforpRsimclr,K. LetBf:“a BτplogBS`Bτq, B:“cpB6 S`1qBfBτfor some absolute constant cą0. It can be verified by Assumption 2thatFmust satisfies }f}2,8ďBffor allfPFfor Assumption 1to hold. Define the zero-mean random process Xf:“pRKpSfq ´ErpRKpSfqs, ... | https://arxiv.org/abs/2503.17538v1 |
proof of Eq. ( 20a), for any given index pi´1qK`j, we have |U4psz1,..., szpi´1qK`j,..., sznq ´U4psz1,..., rzpi´1qK`j,..., sznq| “ˇˇˇ1 2nKÿ k“1” log´Ukpszq rUkpszq¯ `log´Vkpszq rVkpszq¯ ´log´Ukprzq rUkprzq¯ ´log´Vkprzq rVkprzq¯ıˇˇˇ ďB2 S 2nKÿ k“1«ˇˇˇUkpszq rUkpszq´Ukprzq rUkprzqˇˇˇ`ˇˇˇVkpszq rVkpszq´Vkprzq rVkprzqˇˇˇff ,... | https://arxiv.org/abs/2503.17538v1 |
Eq. ( 7a) and (7b) hold for general (expected) losses ℓpx,yqthat satisfy (1) ℓpx,yqis nonnegative; (2) ℓpx,yqis symmetric in px,yqand convex in x´y; and (3) ℓpx,zq ďcpℓpx,yq `ℓpy,zqqfor some absolute constant cą0 and allx,y,z PR. This includes the absolute loss, Huber loss, losses induced by norms, etc. B.4 Proof of Th... | https://arxiv.org/abs/2503.17538v1 |
constant cą0 P´ˇˇsup fPF|Xf| ´Ersup fPF|Xf|sˇˇět¯ ď2exp´ ´cnt2 sB4 S¯ ,for alltě0. (30a) Ersup fPF|Xf|s ďEr|Xf0|s `Ersup f,rfPF|Xf´Xrf|s ďcsB2 S?n`32B?n¨ż2Bf 0b logNpu,} ¨ }2,8,Fqdu.(30b) Combining the two bounds and noting RKpSpfq ´inf fPFRKpSfq ď2sup fPF|pRKpSfq ´RKpSfq| “2sup fPF|pRKpSfq ´ErpRKpSfqs| “2sup fPFXf, yi... | https://arxiv.org/abs/2503.17538v1 |
as from the proof of Theorem 1, we only need |τpxfpzp1qq,zp2qyq| ďlogBS, which follows from the boundedness of F). 28 Approximation error. The approximation error inf fPFRsimclr,KpSfq ´Rsimclr,KpS‹q “0 sinceS‹`c1 is realized by f‹and the link function τpxq “κxfor some normalizing constant c1andRsimclr,KpS‹q “ Rsimclr,K... | https://arxiv.org/abs/2503.17538v1 |
}x´z}2is small). Thus, compared with the OLS estimator which has a risk of or derOpd{mq, the two- step estimator achieves a small risk of order Opp{mqwhen the errorfrom SimCLR training is of higher order. Proof of Theorem 9.First, we have from Corollary 1that, with probability at least 1 ´δ, the learned encoder satisfie... | https://arxiv.org/abs/2503.17538v1 |
v, it can be verified that (when choosing the absolute constant in Bsufficiently large) T5ď2sσ2{m2. Putting the bounds on T3,T5 (and hence T4) together, we conclude that ErRlinprholsqs ´ sσ2ěcsσ2d m´d´1for some absolute constant cą0. Proof of claim (35)and(36).Claim (35) follows directly from properties of the inverse Wis... | https://arxiv.org/abs/2503.17538v1 |
follows from Bayes’ formula and the fact that xc1K Kxc2|y. Forzp1q“zp2q“z, using Eq. ( 39b) and properties of conditional distribution, we have ÿ z1PrSsPcpzp1q“z,zp2q“z1q Pcpzp1q“zqPcpzp2q“z1q“1 Pcpzp2q“zq“1 Ppzp2q“zq“ÿ z1PrSsPpzp1q“z,zp2q“z1q Ppzp1q“zqPpzp2q“z1q. Combining this with Eq.( 41) for allzp2q‰zp1qand noting... | https://arxiv.org/abs/2503.17538v1 |
}2,Γb¯¸ du ďcżcB{?m 0d M2¨log´ 1`4BΓsB?mu¯ duďcBMlog1{2pBΓsBq?mďcBM plog1{2BΓ`? Bq?m, 35 whereΓw:“ tΓwPRMˆM:| | |Γw| | |opďBΓuandΓb:“ tΓbPRM:}Γb}2ďBΓu, and the last line uses the covering number bound of unit balls. Putting pieces together yields t he desired bound. C.5 An auxiliary lemma Lemma 6 (Upper bound on the te... | https://arxiv.org/abs/2503.17538v1 |
on rSs, by the definition of pfaugand claim ( 37) in the proof of Eq. ( 14) | | |pIS´P1SqppEJpE`pwISq ´ pIS´P1SqpE‹JE‹`S¨IS{2q| | |2 fro “ | | |pIS´P1SqpSm pfaug´S‹mq| | |2 fro “S2¨T1, where T1:“Ezp1q,zp2q„PzˆPzrppSpfaug´S‹qpzp1q,zp2qq ´Ezp2q„PzrpSpfaug´S‹qpzp1q,zp2qqsq2s. Finally, by a second-order Taylor expansion of ... | https://arxiv.org/abs/2503.17538v1 |
A new tail bound for the sum of bounded independent random variables Jackson Loper Department of Statistics, University of Michigan and Jeffrey Regier Department of Statistics, University of Michigan March 25, 2025 Abstract We construct a new tail bound for the sum of independent random variables for situations in whic... | https://arxiv.org/abs/2503.17594v1 |
each i, and (iii) the expected sum over the vector is at most µ. Hoeffding (1963, Theorem 2) provides a tail bound for the variable S=Pm i=1Xiwhen Xis drawn according to Mb,µ, for fixed bandµ. To construct this bound, Hoeffding (1963) first considers a tail bound for a particular p∈ M b,µ. Let φ(p, s) = inf t≥0 mX i=1l... | https://arxiv.org/abs/2503.17594v1 |
as φ∗ b,µ(s) = min t,λ≥0g(t, λ;s). Moreover, the mapping t7→min λg(t, λ;s)is convex and the mapping λ7→g(t, λ)is convex for each t≥0. 4 0.00.51.0Bounds onP(S≥s) Tight Chernoff, µ=0.80 Hoeffding Theorem 2, µ=0.80 Tight Chernoff, µ=0.90 Hoeffding Theorem 2, µ=0.90 Tight Chernoff, µ=0.95 Hoeffding Theorem 2, µ=0.95 0.800 0.825 ... | https://arxiv.org/abs/2503.17594v1 |
formality is necessary to clarify that Mb,µis, in fact, convex. Theorem (Restatement of Theorem 3.1) .Fixb= (b1, . . . b n),µ∈[0,Pn i=1bi], and s∈ [0,Pn i=1bi]. Let T={τ∈Qm i=1[0, bi] :Pm i=1τi=µ}. Then φ∗ b,µ(s)from Equation (4)can be expressed as φ∗ b,µ(s) = min t≥0 max τ∈TmX i=1log (1 + τi(exp( bit)−1)/bi)−ts! . Pro... | https://arxiv.org/abs/2503.17594v1 |
so strong duality holds and max τ∈Q i[0,bi]P iτi=µX ilog (1 + ξ(bi, t)τi)−ts= min λg(t, λ) as desired. We now demonstrate that gis convex. First, observe that Lis affine in λand convex int. Thus gis a pointwise maximum of a family of convex functions: it is convex. Finally, we demonstrate that t7→min λg(t, λ) is convex... | https://arxiv.org/abs/2503.17594v1 |
Poisson-Process Topic Model for Integrating Knowledge from Pre-trained Language Models Morgane Austern, Yuanchuan Guo, Zheng Tracy Ke, and Tianle Liu March 25, 2025 Abstract Topic modeling is traditionally applied to word counts without accounting for the context in which words appear. Recent advancements in large lang... | https://arxiv.org/abs/2503.17809v1 |
pre-trained large language models (LLMs). LLMs are typi- cally large, transformer-based neural networks pre-trained on extensive text corpora using self-supervised learning techniques, such as masked token prediction or next-token prediction. Many pre-trained LLMs (e.g., BERT [6], LLaMA [29]) are open-source, providing... | https://arxiv.org/abs/2503.17809v1 |
sha1_base64="7p1/TBCTwv/kR5jZFbSV9uSYykg=">AAAC1nicbVHNbtQwEPaGv7IUuoUjF0NVqQdYJT0UuFWtEHAoWsRuu2J3FTnObGvVsSN70nYVhRtw5Ql4Aq7wLjwKJ7CTHsiWkSx//r4Zz1+SS2ExDH91gmvXb9y8tXK7e2f17r213vr9Q6sLw2HEtdRmnDALUigYoUAJ49wAyxIJR8npvtePzsBYodUQFznMMnasxFxwho6Ke/RDXKqoelLf2+6e8lSjbd5vY1XFvY2wH9ZGr4LoEmzsPvrz27x6+WIQr3e+TVPNiwwUcsmsn... | https://arxiv.org/abs/2503.17809v1 |
Heuristically, the signals of low-frequency anchor words are significantly enhanced by semantically similar high-frequency words, under the assumption of smoothness forAk(·).Second, the estimated model captures the context in which words appear. For example, the word bond can appear in multiple contexts—such as governm... | https://arxiv.org/abs/2503.17809v1 |
to [2], and encodes word relationships, addressing limitations of non-contextualized embeddings as used in [7] and its 6 variants. Additionally, most existing methods require specialized estimation procedures, such as designing new neural network architectures or adapting EM algorithms with variational inference. In co... | https://arxiv.org/abs/2503.17809v1 |
more precisely, tokens.3A self-attention block maps each xjto three vectors: the query qj=Wqxj∈Rd1, the key kj=Wkxj∈Rd1, and the value vj=Wvxj∈Rd2, where Wq,Wk, and Wvare learnable parameter matrices. The output for token jis computed as zj=PN m=1αj,mvm, where αj,m= Softmax q′ jkm/√d1 represents the attention weights... | https://arxiv.org/abs/2503.17809v1 |
than a simplistic embedding vector in Rd, as in BERTopic [8], or samples from multinomial distributions, as in traditional topic models [4, 19]. This framework combines the strengths of both perspectives, enabling a richer representation for topic modeling. 4Positional encodings are typically added either directly to t... | https://arxiv.org/abs/2503.17809v1 |
apply traditional topic modeling to the hyperword count matrix Xnet. Most existing meth- ods can be grouped into two main categories: Bayesian approaches and anchor-word-based approaches. Bayesian approaches, including the well-known LDA algorithm [4] and its extensions, typically assume a Dirichlet prior on w1, w2, . ... | https://arxiv.org/abs/2503.17809v1 |
z∈RdKh(z−z0)bAnaive k(z)dz, (2.4) which means that bAk(·)is the convolution (more precisely, cross-correlation) of bAnaive k(·)with the rescaled kernel function Kh(·). This motivates the term kernel smoothing for this estimation step (see Figure 1). Notably, (2.4) immediately implies thatR z0∈RdbAk(z0)dz0= 1. If the ke... | https://arxiv.org/abs/2503.17809v1 |
the embeddings to d-dimensional space. Topic Modeling : 1.Net-rounding : Perform k-means clustering on {zij}1≤i≤n,1≤j≤Ni. Denote the cluster centers as bx1, . . . ,bxM, and define the hyperwords R1, . . . ,RMas regions in the corresponding V oronoi di- agram. Construct the hyperword count matrix Xnetas in (2.2). 2.Trad... | https://arxiv.org/abs/2503.17809v1 |
(2.7) By assumption, the weight vectors wiare non-negative and sum to one. However, due to randomness in the model assumptions and potential mis-specification when applied to real data, the estimators bwiare not guaranteed to satisfy these constraints. To address this, we mention two approaches to enforce valid weights... | https://arxiv.org/abs/2503.17809v1 |
h(z) =KX j=1Aj(z),ΣW=n−1WW′,and ΣA(k, ℓ) =Z z∈ZAk(z)Aℓ(z) h(z)dz. (3.2) Here, h(z)characterizes the variation of word frequency over the embedding space, ΣWis the topic-topic covariance matrix [1, 19], and ΣAis analogous to the topic-topic overlapping matrix [19] in the traditional topic model. By the definitions, ΣWan... | https://arxiv.org/abs/2503.17809v1 |
max z0∈Zh|eAk(z0)− A k(z0)| ≤C0 hβ+ min ϵβ∧1, h−(d+1)ϵ1+β∧1 . In classical kernel density estimation (KDE), the bias depends solely on the bandwidth h. In contrast, in our framework, the bias is influenced by both hand the side length ϵof the net. This distinction arises because, instead of applying kernel smoothin... | https://arxiv.org/abs/2503.17809v1 |
evaluate the optimality of the rate established in Theorem 3.1, we provide an information-theoretic lower bound. When K= 1,Ωi(·) =A1(·)for all 1≤i≤n. It then follows from the equivalent form of PPTM in (2.1) that all embeddings are sampled from A1(·). Hence, the problem reduces to the classical KDE, and the optimal rat... | https://arxiv.org/abs/2503.17809v1 |
not change. Hence, we can choose a properly large ϵso that the first term is dominated by the second term: Corollary 3.2. Under the conditions of Lemma 3.2, if we choose ϵsuch that n−1log(n)≤ϵd≤log−3(n), then with probability 1−o(n−2),n−1Pn i=1∥bwi−wi∥1≤CN−1/2p log(n). We observe that the choices of ϵfor estimating top... | https://arxiv.org/abs/2503.17809v1 |
a normalizing constant f1, . . . , f pfor each word, and define ak,j=fjebk,jto enforce the conditionPp j=1ak,j=Pp j=1fjebk,j= 1for1≤k≤K. Such solutions exist for p≥Kand are obtained by solving the constrained optimization: minimizing PK k=1(Pp j=1fjebk,j−1)2subject to fj≥0for1≤j≤pusing the projected gradient descent al... | https://arxiv.org/abs/2503.17809v1 |
as hgrows, with the optimal bandwidth roughly between 0.1and0.5. A smaller his generally less harmful than an overly large h. On the other hand, when h≤0.5is fixed, the loss first decreases and then increases as Mgrows from 100to 2500 , with the optimal performance achieved at M= 400 or800, aligning with the bias-varia... | https://arxiv.org/abs/2503.17809v1 |
of the singular values from this matrix. Subsequently, the Topic- SCORE algorithm [19] is applied to the hyperword frequency matrix with K= 7. Finally, the topic functions bA1(·), . . . ,bA7(·)are computed using a Gaussian kernel, with the bandwidth hselected based on the maximum entropy criterion. Further details on t... | https://arxiv.org/abs/2503.17809v1 |
defined as follows: Let bv1,bv2, . . . ,bv7denote the vertices of a standard heptagon in R2. For a fixed word (e.g., bond ), each embedding zof this word is mapped to a point ξ(z) :=P7 k=1bBk(z)bvkwithin the standard heptagon, where bB1(·), . . . ,bB7(·)represent the re- normalized topic densities. The resulting point ... | https://arxiv.org/abs/2503.17809v1 |
list of representative words per topic. The top 20 representative words for each topic are presented in Table 1. Our findings reveal that the topics generated by these methods are less interpretable compared to those produced by TRACE. These methods exhibit greater mutual overlap across topics and less internal clarity... | https://arxiv.org/abs/2503.17809v1 |
sakharov, kash- mir, amal, sinhalese, kasparov4two, united, new, american, year, national, three, po- lice, dukakis, state, end, march, university, help, gor- bachev, force, war, good, say, british4chairs, exams, courses, dances, medals, performances, universities, lights, clubs, photos, genes, railroads, maps, classes... | https://arxiv.org/abs/2503.17809v1 |
orthogonalExperimental Design test examination check alternative null wilcoxon hypothesis option powerful rank precedence possibility homogeneity homoscedastic wald candidate selection variable select lasso auc bic aic akaike choice decide choose model oracle penalty tune predictor variable vector tensor identical rand... | https://arxiv.org/abs/2503.17809v1 |
low productivity” Survival Analysis (0.80) “for renewal processes with infinite mean” Probability Theory (1.0) “the processing time required to solve ... ” Computational Statistics (0.38) words related to genetics (e.g., genome ,dna,chromosome ) are now associated with Variable/Model Selec- tion, while those pertaining... | https://arxiv.org/abs/2503.17809v1 |
model with a vocabulary size of M. This can be viewed as a form of regularization, and our simulation results indeed show that the performance of our method improves when Mis chosen wisely and that as Mvaries we observe a bias-variance trade-off. It is worth noting that this reduction in vocabulary size is only feasibl... | https://arxiv.org/abs/2503.17809v1 |
[6] Devlin, J., M.-W. Chang, K. Lee, and K. Toutanova (2019, June). BERT: Pre-training of deep bidi- rectional transformers for language understanding. In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Sh... | https://arxiv.org/abs/2503.17809v1 |
the 2019 Conference on Empirical Methods in Natural Language Process- ing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP) , pp. 3982–3992. [27] Sparck Jones, K. (1972). A statistical interpretation of term specificity and its application in retrieval. Journal of documentation 28... | https://arxiv.org/abs/2503.17809v1 |
Two-Sample Tests for Optimal Lifts, Manifold Stability and Reverse Labeling Reflection Shape Do Tran Van∗1, Susovan Pal†2, Benjamin Eltzner‡1, and Stephan F. Huckemann§1 1Institute for Mathematical Stochastics, University of G¨ ottingen, Germany 2Mathematics and Data Science Group, Vrije Universiteit Brussel, Belgium M... | https://arxiv.org/abs/2503.17879v1 |
limit theorems on manifolds (ranging from Bhattacharya and Patrangenaru (2005) to Eltzner and Huckemann (2019) giving a rather general version including smeariness) can be applied. From a.e. continuity of optimal lifts in the basepoint we derive a strong law of large numbers for optimal lifts (Theorem 5.1). This is req... | https://arxiv.org/abs/2503.17879v1 |
g.p′) for all p, p′∈Mandg∈G. 7. Then the quotient Q:=M/G ={[p] :p∈M}with orbit [p] :={g.p:g∈M}is Hausdorff when Qis equipped with the canonical quotient topology, i.e. the unique topology making the canonical projection π:M→Q, p7→[p] continuous and open. Further, for every p∈M, [p]⊂Mis a closed embedded submanifold and... | https://arxiv.org/abs/2503.17879v1 |
3 Manifold Stability and Optimal Lifts The following main result of this section states that every Fr´ echet mean lies on the manifold part, if the latter is assumed with positive probability. This strengthens Huckemann (2012, Corollary 1) who assumed that there were only countably many point masses on the singular par... | https://arxiv.org/abs/2503.17879v1 |
=Z Qexp−1 µ◦ℓ′ µ(q)dPπ◦X(q).(3.1) Now, in order to see (iv), it suffices to show that Z Bexp−1 µ◦ℓµ(q)dPπ◦X(q) =Z Bexp−1 µ◦ℓ′ µ(q)dPπ◦X(q) (3.2) for all Borel B⊆Qand to this end, fix such a Band define ℓ′′ µ(q) =ℓµ(q) if q∈B ℓ′ µ(q) if q∈Q\B, which is yet another optimal lift through µ. Then, due to (3.1), Z Q\Bexp−1 ... | https://arxiv.org/abs/2503.17879v1 |
is v∈TqM, varying smoothly in q∈Q∗, such that q′=π(p′) = expQ∗ qvand hence q= expQ∗ q′(−θq,q′(v)) where θq,q′:TqQ∗→Tq′Q∗is the parallel transport which is smooth. Similarly, for p∈L′ p′(see first assertion), p′= exppv, where we have identified HpM∼=TqQ∗, i.e. ℓp′(q) =p= expp′(−θq,q′(v)) which varies smoothly in q∈U⊆Q∗.... | https://arxiv.org/abs/2503.17879v1 |
which is CSin the incomplete Mwhich is the plane minus a slit along the positive first axis. For otherwise, if fwas differentiable at p∈CS, there would be S∋s1̸=s2∈Swith d(si, p) =f(p) for i= 1,2. In particular, since Mis complete, there would be two different unit speed geodesics γi:t7→expp(tvi) with TpM∋v1̸=v2∈TpM,∥v... | https://arxiv.org/abs/2503.17879v1 |
there is g′∈Gsuch that the first component of Φ−1 q(p′) (Φ qmaps from the tangent bundle into the manifold, cf. (4.1) is ( g′)−1.p, i.e. g′.p′is in optimal position to p. However, also gg′.p′̸=g′.p′ (since p′∈M∗) is in optimal position to p, yielding U∩Fp=∅. Since vol(U)>0, thus p̸∈F. M∗⊆F: Let p∈M∗with π(p) =: q∈Q∗. I... | https://arxiv.org/abs/2503.17879v1 |
Hotelling T2-distribution with ( d, k) degrees of freedom, see, e.g. Mardia et al. (2024), if covn,m[X, Y] is of rank d. Here we have utilized the expected value E[f(X)] =R Rdf(x)dPX(x) forPX-integrable functions. 11 Further in case of arbitrary X, Y which are not necessarily normally distributed but feature second mom... | https://arxiv.org/abs/2503.17879v1 |
H0ifTpooledI > Td,m+n−2,1−α. Alternatively we can lift each sample separately. Test 7.4 (Individual Lifting) .With the notation from Test 7.2 let µW n∈π−1(νW n)∩LµW,Z n,m, µZ n∈π−1(νZ n)∩LµW,Z n,m, i.e. both are a.s. uniquely (due to Theorems 3.1 and 5.7) in optimal position to µW,Z n,mand let ℓµWnbe an optimal lift th... | https://arxiv.org/abs/2503.17879v1 |
data. One of their test is based on quantiles of a χ2-distribution and the other is a bootstrap alternative (see also Bhattacharya and Lin (2017)). 8 Reverse Labeling Reflection Shape Spaces Classical shape spaces and reflection shape spaces (Dryden and Mardia 2016) model, modulo certain group actions, landmark configu... | https://arxiv.org/abs/2503.17879v1 |
the left by eiϕ. In consequence, Σk 2carries a canonical Riemannian manifold structure of the complex projective space CPk−2of complex dimension k−2 (real dimension 2 k−4). Reflection then corresponds to joint complex conjugation: R.z=R.(z1, . . . , z k) := (¯ z1, . . . , ¯zk) = ¯z and hence RΣk 2also has a singular st... | https://arxiv.org/abs/2503.17879v1 |
− w2+w1√ 3 2 = 2( |w1|2− |w2|2)−4√ 3Re(w1¯w2). Hence, under reverse labeling, x1 x2 x3 = 2Re(w1¯w2) 2Im(w1¯w2) |w1|2− |w2|2 16 is mapped to 1 2Re w2√ 3−w1 w2+w1√ 3 1 2Im w2√ 3−w1 w2+w1√ 3 1 4 w2√ 3−w1 2− w2+w1√ 3 2 =−1 2 x1+√ 3x3 2x2 √ 3x1−x3 , i.e. accounting for a rotation by ... | https://arxiv.org/abs/2503.17879v1 |
2. Top row for equal sample sizes (rows 4 – 7 in Table 2) and bottom row for different sample sizes (rows 8 – 10 in Table 2). In contrast to Table 2 each test has been repeated 500times. The horizontal axis records the reverse relabeling reflection shape distance between the corresponding two population Fr´ echet means... | https://arxiv.org/abs/2503.17879v1 |
“pooled lifting” ( T) and “pooled lifting intrinsically” ( T0) do not. Notably, after Bonferroni correction for four tests, only the test “individual asymmetric lifting” ( T1) detects the 19 difference. For convenience, Table 3 also records the p-value of test TJfrom Preston and Wood (2010). T T0 T1 T2 TJ p-value 0.08 ... | https://arxiv.org/abs/2503.17879v1 |
mean shape: Manifold stability, locus and the two sample test. Annals of the Institute of Statistical Mathematics 64 (6), 1227–1259. Huckemann, S. and T. Hotz (2009). Principal components geodesics for planar shape. Journal of Multivariate Analysis (100), 699–714. Huckemann, S., T. Hotz, and A. Munk (2010a). Intrinsic ... | https://arxiv.org/abs/2503.17879v1 |
Non-Bayesian Learning in Misspecified Models* Sebastian Bervoets†& Mathieu Faure‡& Ludovic Renou§ Latest version: April 8, 2025 First version: June 26, 2024 Abstract Deviations from Bayesian updating are traditionally categorized as biases, errors, or fallacies, thus implying their inherent “sub-optimality.” We offer a... | https://arxiv.org/abs/2503.18024v2 |
1For comprehensive reviews, see Benjamin (2019) and Ortoleva (2022). 2 the closest to the true data generating process (in the sense of the Kullback-Leibler divergence). In other words, the theorem states that it is as if the agent considers all possible mixtures of the candidate processes (thus, “convexifying” his mod... | https://arxiv.org/abs/2503.18024v2 |
weight γbecomes arbitrarily small, the occupation measure concentrates on the mixture process the closest to the true data generating process – Theorem 3. We illustrate our main result with a simple example, inspired by the work of Spiegler (2016) on causal models. Suppose that each observation is a vector of variables... | https://arxiv.org/abs/2503.18024v2 |
the sets XandΘare finite and that the support of p∗and each pθisX.5The latter assumption guarantees that the agent is never surprised, that is, the agent cannot observe a realization x, believed impossible under all data-generating processes. We think of xas financial returns in investment problems, losses in insurance... | https://arxiv.org/abs/2503.18024v2 |
prediction accuracy. While a fre- quentist approach would guarantee correct learning in this setting (a direct application of the law of large numbers), its applicability is limited in many other settings. For instance, in all settings where the agent observes a single realization, the frequentist approach has no bite.... | https://arxiv.org/abs/2503.18024v2 |
to the associated (de- terministic) ODE: ˙q(t) =H(q(t)). When applicable, this method reduces the analysis of the Robbins-Monro algorithm (ER) to the study of the associated ODE. We prove two central results about the ODE. (All proofs are in the Appendix.) The first result – Lemma 1 – states that the function q7→V(q)is... | https://arxiv.org/abs/2503.18024v2 |
logp(x). It follows immediately that: Corollary 1 Ifp∗∈coP, then the agent’s predictive process converges to the true data- generating process p∗. 9 Thus, if p∗∈coP, the learning outcome is the same as the one of a Bayesian learner with the richer set of (non-parametric) models coP, assuming that the agent’s prior assi... | https://arxiv.org/abs/2503.18024v2 |
10 agent’s belief comes closer, and sometimes strictly so, to the true state under updating (ER) (in the sense of the Kullback-Leibler divergence) if, and only if, the agent does not learn the true state under Bayesian updating. The updating rule (ER) may also serve as a “ misspecification test .” Indeed, if the proces... | https://arxiv.org/abs/2503.18024v2 |
{θ∈Θ : ˆqθ= 0, q∗ θ>0}. From the construction of the components Ck,Θ3is non-empty. (IfΘ3was empty, then Ck=Ck∗.) From the strict concavity of V,∂ ∂τV(τq∗+ (1−τ)ˆq)>0. Therefore, 0<X θ∈Θ(q∗ θ−ˆqθ)fθ(ˆq) =X θ∈Θq∗ θfθ(ˆq)−1 =X θ∈Θ2∪Θ3q∗ θfθ(ˆq)−1 =X θ∈Θ2q∗ θ+X θ∈Θ3q∗ θfθ(ˆq)−1, where the first and third equalities follows... | https://arxiv.org/abs/2503.18024v2 |
start with two immediate observations. First, since ||Un+1||is bounded by 2, the updating process is sub-Gaussian. From Proposition 4.4. of Bena ¨ım (1999), Theorem 1 remains true with the weaker condition:P ne−c/γn<+∞for some positive constant c. E.g., the series (1/√n)n∈Nsatisfies this condition (sinceP ne−√n<+∞), bu... | https://arxiv.org/abs/2503.18024v2 |
measure is parame- terized by γ. Also, recall that qmis random, as it depends on the realized observations. We first prove that the sequences of occupation measures converge to invariant distri- butions π∈∆(∆(Θ)) of the Markov chains. This result is, however, of limited value as there are multiple invariant measures. F... | https://arxiv.org/abs/2503.18024v2 |
most of its mass on Ck∗, but not all of it. OBSERVATION -DEPENDENT WEIGHTS .Biases such as the self-confirmation bias re- quire the weights of the updating rule (ER) to depend on the realized signal xnat period nand, possibly, the belief at period n−1(see, e.g., Rabin and Schrag (1999)). Now, if each γn(x, q)is an arbi... | https://arxiv.org/abs/2503.18024v2 |
predictive process a Bayesian agent would converge to. If, however, the prior dependance is weaker, say α(a) = 2 /3andα(b) = 1 /2, then the predictive process converges to (8/11,3/11), which is strictly closer to the true data-generating process than pA. So far, we have assumed that the weights are independent of the c... | https://arxiv.org/abs/2503.18024v2 |
1n+1X i=1logpθ∗(xi)− min i=1,...,n+1γi1 n+ 1n+1X i=1,...,n+1logpθ(xi)i . The strong law of large numbers implies that limn→+∞1 n+1Pn+1 i=1logpˆθ(xi) =V(pˆθ)almost surely for all ˆθ, hence the term into bracket converges to a strictly positive number since V(pθ∗)> V(pθ). Therefore, log qn+1(θ∗) qn+1(θ) nconverges ... | https://arxiv.org/abs/2503.18024v2 |
problem as an active learning problem, they also explain how we can reframe it as a learning problem with non-Bayesian updating. For instance, the model accommodates underreaction for specifications of QandbQsuch that xn>bxn>0andxn<bxn<0. The 22 model can also accommodate base-rate neglect and the confirmatory bias. If... | https://arxiv.org/abs/2503.18024v2 |
n)maximizes the agent’s expected payoff given the observations (x1, . . . , x n) and the model Pi. (The authors assume Bayesian updating, but as already explained, some choices of Piare equivalent to non-Bayesian updating.) The model P1outper- forms the model P2if there exists n∗such that for all n≥n∗,EP∗[f1 F(x1, . . ... | https://arxiv.org/abs/2503.18024v2 |
implies that the only invariant sets for the flow are connected components of E, the set of zeroes of H. By Proposition 6.4 in Bena ¨ım (1999), it also implies that L(qn), the (random) limit set of (qn)n, is included in a closed connected subset of E.15We now characterize E. Lemma 2 (i) The set Eis a finite union of di... | https://arxiv.org/abs/2503.18024v2 |
Therefore, q∈SbΘfor some bΘ∈ Ck. Consequently, the collection {Ck}is composed of closed, connected, pairwise disjoint sets, and E=K[ k=1Ck. (7) We now argue that Ckis convex, for k= 1, ..., K . Define W: ∆(X)→Ras follows: p∈∆(X)7→W(p) :=X xp∗(x) logp(x). Since p∗has full support on X,Wis strictly concave on ∆(X).17LetL... | https://arxiv.org/abs/2503.18024v2 |
>0such thatP θ∈Θ3µθfθ(ˆq)>1 + 3 ε, for all ˆq∈Ck. By continuity of fθ, there exists an open neighborhood UofCk(in∆(Θ) ) such that inf ˆq∈UX θ∈Θ3µθfθ(ˆq)>1 + 2 ε. Note that, if qθ>0, then 0≤Bθ(q,x) qθ=pθ(x)P θ′qθ′pθ′(x)≤pθ(x) minθ′∈Θpθ′(x)<+∞since pθ′(x)>0for all(θ′, x).18LetC(θ) := max x∈Xpθ(x) minθ′∈Θpθ′(x). Ifqn,θ>0,... | https://arxiv.org/abs/2503.18024v2 |
n+γB(qγ n, xn+1). (10) It will be convenient to rewrite the recursive formula (10) component-wise as follows: forθ∈Θ, qγ n+1,θ qγ n,θ=Fγ θ(qγ n, xn+1),where Fγ θ(q, x) := (1 −γ) +γpθ(x)P θ′pθ′(x)qθ′. (11) LetPγbe the transition kernel associated to the Markov chain (qγ(n))n. Given h∈ C(S), the set of continuous maps fr... | https://arxiv.org/abs/2503.18024v2 |
Lemma 6 Letπγbe a weak* limit point of (Πγ n)n. Then πγ∈Inv(Pγ), and X θλ∗ θrγ θ(πγ)≤0,for all λ∗∈Ck∗. Proof of Lemma 6 Letπγbe a weak* limit point of the sequence (Πγ n)n. By definition, there exists an increasing sequence nksuch that, for any h∈ C(S), Z Sh(q)πγ(dq) = lim kZ Sh(q)Πγ nk(dq) = lim k1 nknk−1X m=0h(qγ m) ... | https://arxiv.org/abs/2503.18024v2 |
R∗(γ, q) =R∗(0, q) +γ∂R∗ ∂γ(0, q) +o(γ),for all q∈S. LetO∗be an open subset of Ssuch that [ k̸=k∗Ck⊆O, C k∗⊆O∗, where O:=S\Cl(O∗). (Recall that the components (Ck)k=1,...,k∗are compact, connected and disjoint.) Fixε >0. Choose O∗small enough so that supq∈Cl(O∗)∂R∗ ∂γ(0, q) ≤ε 2. (Recall that the derivative of R∗is zero... | https://arxiv.org/abs/2503.18024v2 |
130, 2594–2642. DIACONIS , P. AND D. F REEDMAN (1986): “On the consistency of Bayes estimates,” The Annals of Statistics , 1–26. ——— (1999): “Iterated random functions,” SIAM review , 41, 45–76. EDWARDS , W. (1968): “Conservatism in human information processing,” Formal repre- sentation of human judgment . EPSTEIN , L.... | https://arxiv.org/abs/2503.18024v2 |
models and stochastic approximations,” The Annals of Probability , 18, 698–712. PHILLIPS , L. D. AND W. E DWARDS (1966): “Conservatism in a simple probability infer- ence task.” Journal of Experimental Psychology , 72, 346. 38 RABIN , M. AND J. L. S CHRAG (1999): “First impressions matter: A model of confirma- tory bia... | https://arxiv.org/abs/2503.18024v2 |
arXiv:2503.18400v1 [math.ST] 24 Mar 2025Asymptotically uniformly most powerful tests for diffusion processes with nonsynchronous observations Teppei Ogihara∗and Futo Ueno∗ ∗Graduate School of Information Science and Technology, Uni versity of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113–8656, Japan March 25, 2025 Abstract. ... | https://arxiv.org/abs/2503.18400v1 |
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