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trade-offs, and ultimately by Recht [33], who improved both previous results, proving that Ais the unique solution to (6), given the sampling size bound |Ω| ≥Cmax{µ0, µ2 1}rNlog2N, (7) for the coherence parameters µ0andµ1defined previously. If one replaces µ1with µ0√r(see (4), the RHS becomes Cµ2 0r2Nlog2N. This attain... | https://arxiv.org/abs/2501.19224v2 |
logarithmic, at the cost of an increase in the powers of r,µ0and log N. Remark 1.1. (A problem with trying many ranks ) In practice, the common situation is that we do not know the rank rexactly, but have some estimates (for instance, ris between known values rmin andrmax). It has been suggested (see, for instance, [27... | https://arxiv.org/abs/2501.19224v2 |
in practice. In many data sets, it has been noted that the leading singular values decay rapidly, meaning κis large. For instance, Figure 1.5.3 shows this phenomenon for the Yale face database, which is often used in demonstrations of the Principal Component Analysis [40] method for dimensionality reduction. From the c... | https://arxiv.org/abs/2501.19224v2 |
can reduce the case of general case to this by scaling. Algorithm 1.2 (Approximate-and-Round ( AR)). 1. Let ˜A:=p−1AΩand compute the SVD: ˜A=˜U˜Σ˜VT=Pm∧n i=1˜σi˜ui˜vT i. 2. Let ˜ sbe the last index such that ˜ σi≥N 8rµ, where µ:=Nmax{∥U∥2 ∞,∥V∥2 ∞}is known. 3. Let ˆA:=P˜s i=1˜σi˜ui˜vT i. 4. Round off every entry of ˆAt... | https://arxiv.org/abs/2501.19224v2 |
emphasizing the presence of the noise. Recovery from noisy observation is clearly a harder problem, and most papers concerning noisy recovery aim for recovery in the normalized Frobenius norm (root mean square error; RMSE), rather than exact recovery. Continuing the nuclear norm minimization approach, Candes and Plan [... | https://arxiv.org/abs/2501.19224v2 |
and noisy case (with some modification). The reason is this: even in the noiseless case, one already views the (rescalled) input matrix p−1AΩas the sum of Aand a random matrix E. Thus, adding a new noise matrix Zjust changes EtoE+Z. This changes few parameters in the analysis, but the key mathematical arguments remain ... | https://arxiv.org/abs/2501.19224v2 |
the density bound is comparable to all previous works; see Remarks 1.6 and 1.7. Setting 1.3 (Matrix completion with noise) .Consider the truth matrix A, the observed set Ω, and noise matrix Z. We assume 1.Known bound on entries: We assume ∥A∥∞≤KAfor some known parameter KA. This is the case for most real-life applicati... | https://arxiv.org/abs/2501.19224v2 |
the sum A+Zbehaves like a random matrix; see [2, 19, 32, 12, 5, 20] for many results. For instance, the leading singular vectors of A+Zlook totally random and have nothing to do with the leading singular vectors of A. This shows that there is no chance that one can recover Afrom the (even fully observed) noisy matrix A... | https://arxiv.org/abs/2501.19224v2 |
the problem from a matrix perturbation perspective, then introduce our main tool, Theorem 2.1. We will give a short proof of Theorem 1.5 using Theorem 2.1. This concludes the next section. In Section 3, we discuss Theorem 2.1 from the matrix perturbation point of view. We first discuss the classical Davis-Kahan-Wedin t... | https://arxiv.org/abs/2501.19224v2 |
Observe that E[AΩ] =pAandE[ZΩ] = 0, we have E˜A =E p−1AΩ,Z =E p−1(AΩ+ZΩ) =E p−1AΩ +E p−1ZΩ =A. LetE:=˜A−A. The above shows that Eis a random matrix with mean 0. From a matrix perturbation theory point of view, ˜Ais an unbiased perturbation of A. Establishing a bound on (A+E)s−Asin the infinity norm is one of ... | https://arxiv.org/abs/2501.19224v2 |
random matrix with mean 0, independent entries satisfying E |Eij|l =p1−lKlby Eq. (24). Recall that we have the SVD ˆA=P iˆσiˆuiˆvT i. Similarly, denote the SVD of ˜Aby ˜A=min{m,n}X i˜σi˜ui˜vT i. We have the relation ˜ ui= ˆui, ˜vi= ˆviand ˜σi=ρ−1ˆσifor each i. From the sampling density assumption, a standard applicat... | https://arxiv.org/abs/2501.19224v2 |
∥A−ρ−1A∥∞+ρ−1 ∥˜As−As∥∞+∥As−A∥∞ ≤ε 4+ 1.2ε 4+ε 4 < .9ε. The total exceptional probability is O(N−1). The proof is complete. 3 Davis-Kahan-Wedin theorem in the infinity norm Now that our matrix completion algorithm ( AR2 ) has been verified, we will focus on proving Theorem 2.1. From this point onwards, let us put a... | https://arxiv.org/abs/2501.19224v2 |
[31, 38] exploited the improbability of such interactions when Eis random and Ahas low rank, and improved the bound significantly. For instance, O’Rourke, Vu and Wang [31] proved the following: ∥˜Vs˜VT s−VsVT s∥ ≤C√s∥E∥ σs+√r∥UTEV∥∞ δs+∥E∥2 δsσs , with high probability, effectively turning the noise-to-gap on the rig... | https://arxiv.org/abs/2501.19224v2 |
also implies that rσs∥E∥ σs+2r∥UTEV∥∞ δs+2ry δsσs is a tight upper bound for ∥˜As−As∥when r=O(1). Sharpness of the results. The terms τ1andτ2play the roles of the coherence parameters in the matrix completion setting. Practically, one replaces them with upper bounds when applying Theorem 3.2, as the theorem still wor... | https://arxiv.org/abs/2501.19224v2 |
[23, 14].The term yhas been analyzed in [38] and mentioned above. We use the trivial upper bound y≤ ∥E∥2=O(ς2N), which is enough to prove the next theorem. Regarding τ1andτ2, our analysis later will give the estimates τ1=O logNr µ(U) m+log3/2N√ N+Mlog3N√ N·r µ(V) n! , and symmetrically for τ2by swapping UandV. In the s... | https://arxiv.org/abs/2501.19224v2 |
to the random case. However, the reader can be assured that the changes needed to make Theorem 3.2 imply Theorem 3.3 are trivial, and will be discussed when we prove the latter. Proof structure. First, we will assume Theorem 3.2 and use it to prove Theorem 3.3, which directly implies Theorem 2.1. The proof contains a n... | https://arxiv.org/abs/2501.19224v2 |
≥t ≤exp −t2 P k,hE[|Xkh|2] +2 3Mt! ≤exp −t2 ∥U∥−2∞∥V∥−2∞+2 3Mt! . 26 We rescale Ykh=ς∥U∥∞∥V∥∞Xkhand replace twith t/(ς∥U∥∞∥V∥∞), the above becomes P X k,hYkh ≥t ≤exp −t2 ς2+2 3M∥U∥∞∥V∥∞t! . LetN=m+nandt= 2ς(√logN+M∥U∥∞∥V∥∞logN), we have t2≥4ς2logN, t2≥2M∥U∥∞∥V∥∞tlogN, thus t2≥12 7 ς2+2 3M∥U∥∞∥V∥∞t logN. Combin... | https://arxiv.org/abs/2501.19224v2 |
moment method, with a walk-counting argument inspired by the coding scheme in [39], to bound these terms. We put the full proof in Section B.2. Let us prove Theorem 3.3 using these lemmas. Proof of Theorem 3.3. Consider the objects from Setting 3.1. We aim to apply Theorem 3.2. By Lemma 4.1, with probability 1 −O((m+n)... | https://arxiv.org/abs/2501.19224v2 |
Asymare again σ1, . . . , σ r, but each with multiplicity 2, thus the matrices Ws:= w1, w 2, . . . w s, w r+1, . . . w r+s ,(Asym)s:=sX i=1λiwiwT i+r+sX i=r+1λiwiwT i are respectively, the singular basis of the most significant 2 svectors and the best rank-2 sapprox- imation of Asym. However, we still use the subscri... | https://arxiv.org/abs/2501.19224v2 |
. . E symλi1β1Pi1β1 z(z−λi1β1)EsymI zEsymα1 λi21Pi21 z(z−λi21)Esym. . . E symλi2β2Pi2β2 z(z−λi2β2)EsymI zEsymα2 . . . E symλihβhPihβh z(z−λihβh) EsymI zαh , which can be rewritten as Cν(I)Eα0 sym"h−1Y k=1M(ik)Eαk+1 sym# M(ih)Eαh sym, (61) where we denote I:= [i1,i2, . . . ,ih],ik:= [ik1, ik2, . . . , i kβk], and ... | https://arxiv.org/abs/2501.19224v2 |
in Setting 4.3. Lemma 4.4. Consider the objects in Setting 4.3 and define the following terms R1:=∥E∥ λS∨2r∥WTEsymW∥∞ ∆S, R 2:=√ 2r∥E∥√λS∆S, R 3:=2r λS∆Smax |i−j|/∈{0,r}|wiE2 symwj|.(68) Additionally, define L0= 2andL1=λSand τ:= max α∈[⌈10 log( m+n)⌉]1 2r2rX i=1∥wT iEα symM∥ ∥E∥α, (69) 32 and analogously for τ′andM′. S... | https://arxiv.org/abs/2501.19224v2 |
The proof is complete. 4.3 Proof of the bound on generic series In this section, we prove Lemma 4.4, which is the backbone of Theorem 3.2’s proof. We summarize the objects involved in the bound below. Note that at this point we do not need to care about A. Setting 4.5. Let the following objects and properties be given:... | https://arxiv.org/abs/2501.19224v2 |
The structure is simple: Lemma 4.4implied by← − − − − − − − − Lemma 4.7implied by← − − − − − − − − Lemma 4.6. Assuming the two lemmas above, we can prove Lemma 4.4, finishing the third step in the strategy. 35 Proof of Lemma 4.4. For convenience, let k=⌊10 log( m+n)⌋. Applying Lemma 4.7, we have kX γ=1 MTT(γ) νM′ ≤rLντ... | https://arxiv.org/abs/2501.19224v2 |
prove Lemma 4.9. Proof of Lemma 4.6. Temporarily let X:=WTEsymW. Applying Lemma 4.9 to Eq. (73) gives ∥MTTν(α,β)M′∥ ≤ ∥ X∥β−h ∞∥E∥α−α0−αh+h−1X I∈[2r]β|Cν(I)| MTEα0 symwi1 wT ihβhEαh symM′ . Temporarily let Tbe the sum on the right-hand side. Applying Eq. (76) from Lemma 4.8, we have |Cν(I)| ≤Lν 1 +∆S λSβSc(I)γ+βS(I)... | https://arxiv.org/abs/2501.19224v2 |
= 2rLνττ′γ+1X β=1γ+β−2 β−1β∧(γ+2−β)X h=1β−1 h−1γ+ 2−β h Rγ−2h+2 1 R2h−2 2.(81) Consider two cases for handγ: 1.γ≥2h−1. Let R:=R1∨R2. The contribution is at most: 2rLνττ′γ+1X β=1γ+β−2 β−1β∧(γ+2−β)X h=1β−1 h−1γ+ 2−β h R1Rγ−1 ≤2rLνττ′R1Rγ−1γ+1X β=1γ+β−2 β−1γ+ 1 β ≤2rLνττ′R1Rγ−1γ+1X β=1γ+ 1 β 2γ+β−2 ≤2rLν... | https://arxiv.org/abs/2501.19224v2 |
+r3logN N1 m+1 n log6N. (85) 40 Letρ:= ˆp/p. From the sampling density assumption, a standard application of concentration bounds [23, 14] guarantees that, with probability 1 −O(N−2). 0.9≤1−1√ N≤1−logN√pmn≤ρ≤1 +logN√pmn≤1 +1√ N≤1.1. (86) Furthermore, an application of well-established bounds on random matrix norms g... | https://arxiv.org/abs/2501.19224v2 |
1 +r3logN N1 m+1 n ≤ε/4.(90) where the last inequality comes from the condition (85) if Cis large enough. After the two steps above, we obtain ∥˜As−A∥∞≤ε/2 with probablity 1 −O(N−1). Finally, we get, using Fact (86) and the triangle inequality, ∥ˆAs−A∥∞= ρ−1˜As−A ∞≤1 ρ∥˜As−A∥∞+ 1 ρ−1 ∥A∥∞≤ε/2 .9+KA .9√ N< ε. This is... | https://arxiv.org/abs/2501.19224v2 |
the second integral: C(n0;A,m(1);B,n) ≤m+n+n0−3 m−21 an0dm+n−2≤m+n+n0−3 m−21 an0−1dm+n−1. (100) Notice that the binomial coefficients in Eqs. (99) and (100) sum to the binomial coefficient in Eq. (97), we get P1(N), which proves Eq. (97) by induction. 44 Now we can prove Eq. (95). The logic is almost identical, wit... | https://arxiv.org/abs/2501.19224v2 |
(92) and (93). Let us consider two cases for C: 1.ν= 0, so n0=γ+ 1. Let a=λS(I),d=δS(I),m=βS(I),n=n′=βSc(I),m′ i=miand n′ j=njfor all i, j, then m′+n′=β≤γ+ 1 = n0, so we can apply Lemma B.2 to get |C(n0;A,m;B,n)| ≤n0+m−2 m−1(1 +d/a)n′ an0−m−ndm+n−1lY i=11 |ai|mikY j=11 |bj|nj, or equivalently, |C0(I)| ≤ 1 +∆S(I) λS(... | https://arxiv.org/abs/2501.19224v2 |
us prove the main objective of this section, Lemma 4.2, before delving into the proof of the technical lemmas. Proof of Lemma 4.2. We follow two steps: 1.Assuming M Consider the analogue of Eq. (52) for V(we wrote the proof for Vbefore the final edit, and wanted to save the energy of changing to U) and Eq. (53), and as... | https://arxiv.org/abs/2501.19224v2 |
(j(hd)0, i(hd)0, . . . , i (hd)a). We can swap the two summation in the above to get X W11,W12,W21,...,W p2∈WE"pY h=1EWh1EWh2#X l1,l2,...,lp∈[r]pY h=1ulhi(h1)aulhi(h2)a. The second sum can be recollected in the form of a product, so we can rewrite the above as X W11,W12,W21,...,W p2∈WE"pY h=1EWh1EWh2#pY h=1UT ·,i(h1)aU... | https://arxiv.org/abs/2501.19224v2 |
LetS(x, y, z, t ) be the subset of shapes having these quantities. To further shorten the notation, let M1:=M2p(2a+1)∥u∥2p ∞. Then we can rewrite the above as: M1X x,y,z,t ∈AM−2(z+t)mz−x/2−ynt−1∥u∥−x−2y ∞|S(x, y, z, t )|, (107) where Ais defined, somewhat abstractly, as the set of all tuples ( x, y, z, t ) such that S(... | https://arxiv.org/abs/2501.19224v2 |
pa)!(l+ 1)2p(2a+1)−2l (2p(2a+ 1)−2l)!l!z!(t−1)!(16p(a+ 1)−8l−2)4p(a+1)−2l−1. 51 Proof. We use the following coding scheme for each shape S∈ S(x, y, z, t ): Given such an S, we can progressively build a codeword W(S) and an associated tree T(S) accoding to the following scheme: 1. Start with VJ={1}andVI=∅,W= [] and Tbei... | https://arxiv.org/abs/2501.19224v2 |
B.6, to uniquely determine the shape S, the general idea is the following. We first generated the preliminary codeword Wfrom S, then attempt to decode it. If we encounter a plus or neutral edge, we immediately know the next vertex. If we see a minus edge that follows from a plus edge ( u, v), we know that the next vert... | https://arxiv.org/abs/2501.19224v2 |
to the proof of Lemma B.3. Temporarily let Gl:= 2p(2a+ 1)−2land Fl:=2l+1(l+ 1)Gl Gl!l!(4Gl+ 8p−2)Gl+2p−1. Note that (2 p(a+ 1))!(2 pa)!Flis precisely the upper bound on |S(x, y, z, t )|in Claim B.6. Also let M2=M1(2p(a+ 1))!(2 pa)! =M2p(2a+1)(2p(a+ 1))!(2 pa)!∥u∥2p ∞. Replacing the appropriate terms in the bound in Cla... | https://arxiv.org/abs/2501.19224v2 |
a star now consists of walks of length 2 a: S= (S1, S2, . . . , S 2p) where Sr=jr0ir0jr1ir1. . . j ra. We have, for any shape SandP∈ P(S), E[EP]≤M4pa−2|E(S)|≤M2pa−2|V(S)|+2,|vPend| ≤ ∥v∥2p ∞,and|P(S)| ≤m|VI(S)|n|VJ(S)|−1, where the power of nin the last inequality is due to 1 having been fixed in VJ(S). Therefore X S∈S... | https://arxiv.org/abs/2501.19224v2 |
Delphine F´ eral, The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluc- tuations , Ann. Probab. 37(2009), no. 1, 1–47. MR 2489158 [13] Sourav Chatterjee, Matrix estimation by universal singular value thresholding , Ann. Statist. 43(2015), no. 1, 177–21... | https://arxiv.org/abs/2501.19224v2 |
positive semidefinite linear matrix inequality , IEEE Trans. Automat. Control 42(1997), no. 2, 239–243. MR 1438452 [31] Sean O’Rourke, Van Vu, and Ke Wang, Random perturbation of low rank matrices: improving classical bounds , Linear Algebra Appl. 540(2018), 26–59. MR 3739989 [32] Sandrine P´ ech´ e, The largest eigenv... | https://arxiv.org/abs/2501.19224v2 |
1 Model non-collapse: Minimax bounds for recursive discrete distribution estimation Millen Kanabar and Michael Gastpar School of Computer and Communication Sciences, EPFL Abstract Learning discrete distributions from i.i.d. samples is a well-understood problem. However, advances in generative machine learning prompt an... | https://arxiv.org/abs/2501.19273v3 |
synthetic samples. This has also led to fears that synthetic samples might significantly contaminate real databases such as the Internet [4]. It has been observed empirically e.g. in [5], [6] that this process, without external mitigation, leads to ‘model collapse’: worsening performance with every iteration where samp... | https://arxiv.org/abs/2501.19273v3 |
when synthetic sampling outpaces real sampling, the effective sample size of batch tis proportional to ntα2 t(as opposed to simply ntαtfor the oracle-assisted setting) in most practically conceivable conditions. As a consequence, the ratio of the minimax loss to the oracle-assisted minimax loss can grow unbounded with ... | https://arxiv.org/abs/2501.19273v3 |
Model collapse in regression models and model collapse mitigation strategies are discussed in [11]–[13]; a lower bound on the expected error for regression models that do not account for self-consumption was presented in [14]. A stochastic-approximation-style treatment showing eventual degeneration for accumulating loo... | https://arxiv.org/abs/2501.19273v3 |
, Xnt t), p)] where the expectation is with respect to the joint distribution ¯Pt,(p,{ni,αi:i≥0}). It is important to note that the minimax loss is defined for the estimator at every stage twhile fixing the estimators in previous stages. This follows from the process of designing the estimators: they are selected seque... | https://arxiv.org/abs/2501.19273v3 |
upper bound1/n0 obtained by using the order-optimal estimator for just the 0thbatch. However, it is interesting to note that in both cases, there exists a constant upper bound independent of t—the sum in the RHS of (5) is exactly1/n0plus a (lower) Riemann sum of the function1/x2from n0toPt i=0niwith intervals ni. There... | https://arxiv.org/abs/2501.19273v3 |
αi’s are comparable to the gi−1(1/4k)’s, there might also be a gap between the guarantees provided by the upper and the lower bounds. In the sequel, we comment on the impact of the error term on the behavior of the bounds and the restrictions on the order-optimal estimator, show cases where differences in the aforement... | https://arxiv.org/abs/2501.19273v3 |
lower bound—exist, and are chosen at every stage. In quite a few cases, as it turns out, we find that the error term is small as compared to the rest of the denominator, and the analysis is tight if optimal or near-optimal estimators are assumed to have been used. We now show the main results in action via two examples... | https://arxiv.org/abs/2501.19273v3 |
the analysis of the lower bound (Theorem 1). The effect of the error term growing large is seen in the following Proposition. Claim 2. Consider the self-consuming distribution estimation problem with the parameters given above. For t≥1, assume that, for stages i∈[0 :t−1], there exists a sequence of estimators for which... | https://arxiv.org/abs/2501.19273v3 |
1 and 2. We now describe the more interesting regime where the upper and lower bounds on the minimax loss match in the following Proposition. It asserts that when αtvanishes with t, the bounds match if the ℓ2 2loss vanishes faster. Proposition 2. When αt↓0, the ratio between the upper and the lower bounds on the minima... | https://arxiv.org/abs/2501.19273v3 |
lower bound m:= inf t≥1¯σ2 t≥1 512 (nα2+M·16nαk2). When kis fixed, we find that since M≤√ 2trivially, m≥c/nαfor some c≥0. This is especially stark in contrast to the oracle-assisted infimum, which, with probability 1, is proportional to1/n, corresponding to the case where all samples are real. Thus, in the data replace... | https://arxiv.org/abs/2501.19273v3 |
“Beyond model collapse: Scaling up with synthesized data requires reinforcement,” in ICML 2024 Workshop on Theoretical Foundations of Foundation Models , 2024. First published at arXiv:2406.07515. [14] E. Dohmatob, Y . Feng, A. Subramonian, and J. Kempe, “Strong Model Collapse,” Oct. 2024. arXiv:2410.04840. [15] M. Mar... | https://arxiv.org/abs/2501.19273v3 |
Theorem 1 by deriving an upper bound on the KL divergences between the distributions ¯Pt,(pv)and¯Pt,(pv−2ej)for each v∈ V. May 13, 2025 DRAFT 16 Proof of Theorem 1: Using the chain rule for KL divergences, whenever j≤k/2, v[j] = +1 , D ¯Pt,(pv) ¯Pt,(pv−2ej) =n0D pv∥pv−2ej +t−1X i=0D ˜P(pv,ˆPi, αi+1)ni+1 ˜P(pv−... | https://arxiv.org/abs/2501.19273v3 |
following Lemma: Lemma 9. LetY0, Y1be unbiased estimates of a scalar θsuch that E[Y0Y1] =θ2andE[(Yj−θ)2] =σ2 jfor j= 0,1. For a0, a1∈R, E[(a0Y0+a1Y1−θ)2] =a2 0σ2 0+a2 1σ2 1+ ((a0+a1)−1)2θ2. Combining Lemmas 8 and 9, we arrive at the following intermediate result: May 13, 2025 DRAFT 18 Lemma 10. LetˆP0= ˆp0(Xn0 0)be an ... | https://arxiv.org/abs/2501.19273v3 |
distribution of Xi[s]is P Xi[s] =j Wi[s] = 0,¯Pi−1,(p) =p[j] P Xi[s] =j Wi[s] = 1,¯Pi−1,(p) =ˆPi−1[j]. Proof of Lemma 5: Denote the conditional distribution of each sample at stage igiven the auxiliary information Wni ias˜PX|W(p,ˆPi−1, α), and for each i∈[1 :n], denote the joint distribution of (Xn0 0, Wn1 1, Xn1 1, ... | https://arxiv.org/abs/2501.19273v3 |
inequality (a)is a consequence of Lemma 11. We can thus lower bound the limit lim t↑∞qPt i=0niαi Pt i=0niα2 i≥1 ϵCD + lim t↑∞Cϵ√Pt i=0niαi=1 ϵCD>0, resulting in a contradiction. We thus have tX i=0niα2 i=O vuuttX i=0niαi May 13, 2025 DRAFT 22 We bound the error term as follows: tX i=1niαigi(1/4k)(b) ≤tX i=1niαi·16... | https://arxiv.org/abs/2501.19273v3 |
ˆPemp[j] := ˆpemp(Xn1 1)[j]is E ˆPemp[j]−p[j]2 =Eh (ˆPemp[j])2i −p[j]2 =E E ˆPemp[j]2 ˆP0[j] −p[j]2 =E αp[j] + (1−α)ˆP0[j]2 1−1 n1 +1 n1 αp[j] + (1−α)ˆP0[j] −p[j]2 =(1−α)2 1−1 n1 E ˆP0[j]−p[j]2 +p[j]−p[j]2 n1 =(1−α)2 1−1 n1 η·p[j] (1−p[j]) +p[j](1−p[j]) n1. =p[j](1−p[j])1 n1+η(1−α)2 1−1 n1... | https://arxiv.org/abs/2501.19273v3 |
What is causal about causal models and representations? Frederik Hytting Jørgensen, Luigi Gresele, and Sebastian Weichwald Copenhagen Causality Lab, Department of Mathematical Sciences, and Pioneer Centre for AI, University of Copenhagen, Denmark Abstract Causal Bayesian networks are ‘causal’ models since they make pre... | https://arxiv.org/abs/2501.19335v2 |
perfectly match the joint normal distribution LO(A, B) =N0 0 ,1 1 1 2 . I am sure that there is no unobserved confounding,1but I am not sure if Acauses BorB causes A. Do you think A→BorA←Bis correct? Sofia: I am sure that Acauses B. Omar: How do you know? Sofia: Try to intervene on B. IfAcauses B, then we would e... | https://arxiv.org/abs/2501.19335v2 |
that Omar intervened on both nodes. Instead, she proposes (P2) as an analysis of (P): (P2) If you do something to change the system such that you observe B= 5 with probability 1 while not changing the marginal distribution of A, then you will observe the distribution N(0,1)⊗δ5over ( A, B). Interpreting (P) as (P2) cann... | https://arxiv.org/abs/2501.19335v2 |
as interventions, making it possible for a causal model to be falsified. Section 6 – Implications for related research: We discuss implications of our work for causal representation learning, causal discovery, and causal abstraction. We discuss connections to the philosophical literature on the logic of conditionals. S... | https://arxiv.org/abs/2501.19335v2 |
·) and the measures νj, even though the change of measure is suppressed in the notation. Sometimes, we write do(Zj=z) and take this to mean that the kernel of jis replaced with paj7→δz, where δzis the Dirac distribution with support {z}. We also sometimes use notation like do(Zj← N (0,1)) and take this to mean that the... | https://arxiv.org/abs/2501.19335v2 |
in a causal Bayesian network and a representation Z∗= (Z∗ 1, ..., Z∗ n) of a data-generating process.8The ob- servational distribution of the representation Z∗is given by the push-forward measure 8We mark variables whose distributions are derived from the data-generating process with a superscript ∗ and the correspondi... | https://arxiv.org/abs/2501.19335v2 |
our presentation (avoiding other mathematical descriptions of data-generating processes in terms of, for example, stochastic differential equations or exhaustive enumerations of the distributions for each action). The variables in causal discovery and the latent variables in causal representation learning are high-leve... | https://arxiv.org/abs/2501.19335v2 |
graph, we have, for all interventions d, that LC;d(Z) is Markov w.r.t. the DAGGof the CBN C. These considerations might compel us to consider the following interpretation. Definition 3.1. IntC. The seemingly natural interpretation. Let a data-generating process D, representation Z∗, compatible CBN C, and set of interve... | https://arxiv.org/abs/2501.19335v2 |
La(Z∗ i|PA∗ i)∼qifor all i∈J. We believe that any reasonable interpretation satisfies D0and therefore do not consider interpretations that may violate D0. In the context of hard interventions, D0is sometimes referred to as ‘effectiveness’ [Galles and Pearl, 1998, Bareinboim et al., 2022, Ibeling and Icard, 2023]. Effec... | https://arxiv.org/abs/2501.19335v2 |
Aand every set of interventions I, rendering any compatible CBNI −Intvalid for every set of interventions Iin that CBN. 11 d∈InteI(a). By condition D2, this implies that LC;d(Z) =LC;b(Z). Proposition 3.2 then gives us that d∈IntI C(a) since La(Z∗) =LC;d(Z). IntCsatisfies D0, so desiderata D1–D4together imply D0. 5 Non-... | https://arxiv.org/abs/2501.19335v2 |
the interpretation IntAby the following rule: An intervention d=do(j← qj, j∈J)∈ Iis inIntI P(a) if and only if the following four conditions hold: 1)dis a perfect intervention. 2) For all i∈J, La(Z∗ i|PA∗ i)∼qi. That is, IntPsatisfies D0(correct conditionals on intervened nodes). 3) For all i /∈J, PA∗ iis empty and La(... | https://arxiv.org/abs/2501.19335v2 |
sets the conditionals of intervened nodes correctly, 3) every node inot in J∪Cis either a source node not in eJ, orZi̸⊥ ⊥PAiinLC;d∗(Z),13and condition 4) of Definition 5.1 trivially holds. IfLC;d∗(Z) =LC;do(j←qj,j∈J∪C)(Z), then LC;do(j←qj,j∈J∪C)(Zi|PAi)∼pC ifor all i∈Csince nodes in Care not intervened upon by d∗. Ther... | https://arxiv.org/abs/2501.19335v2 |
we cannot falsify Cas an I −IntCvalid model of (TC∗,HD∗).do(TC← N (1,2))/∈IntI C(a′) since it is not the case that La′(HD∗|TC∗)∼ pC;do(TC ←N(1,2)) HD|TC. Under action a′, the conditional distribution of heart disease given total cholesterol is different than in the observational regime. Therefore, under interpretation ... | https://arxiv.org/abs/2501.19335v2 |
single-node interventions. Consider some action a. Since Z∗is emulated by CandI∗, there is a single-node intervention d∗∈ I∗such that La(Z∗) =LC;d∗(Z). Consider some intervention d=do(j←qj, j∈J)∈IntI S(a). We want to show that LC;d∗(Z) =LC;d(Z) and do this by arguing that LC;d∗(Zi|PAi)∼pC;d i for all i∈[n]. Per conditi... | https://arxiv.org/abs/2501.19335v2 |
interventions I∗in A are fine-tuned to mimic single-node interventions in C. The lack of fine-tuning between interventions has previously been suggested as a possible fundamental property of causal models [Janzing and Sch¨ olkopf, 2010, Janzing et al., 2016]. ◦ 5.3 Int K: Interventions as simple actions The larger the ... | https://arxiv.org/abs/2501.19335v2 |
no such worlds. Are we to suppose that kangaroos have no tails but that their tracks in the sand are as they actually are? Then we shall have to suppose that these tracks 19 aspects of participants’ lives such as taking smoking breaks or carrying a lighter. b) It is undesirable because we do not want to keep heart heal... | https://arxiv.org/abs/2501.19335v2 |
makeup is as it actually is? Then we shall have to suppose that genes control growth in a way quite different from the actual way (or else that there is something, unlike anything there actually is, that removes the tails). And so it goes; respects of similarity and difference trade-off. If we try too hard for exact si... | https://arxiv.org/abs/2501.19335v2 |
Cis compatible with ( S∗, R∗). Cis not an I −IntSvalid model of ( S∗, R∗). To see this, consider action asuch that Pa(P∗= 1, R∗= 1) = 1 (such an action exists since it corresponds to an intervention in I∗). Now do(S= 0)∈IntI S(a) asLa(S∗) =δ0(correct conditionals on intervened nodes) andLa(S∗)≁pC S. However, Pa(R∗= 1) ... | https://arxiv.org/abs/2501.19335v2 |
network A(with DAG G) and interventions I∗;18the observed data X= (X∗ 1, . . . , X∗ m) is given by some mixing function f:Rn→Rm, X∗=f(Z∗), where fis commonly assumed (at least) to be a diffeomorphism onto its image (e.g., von K¨ ugelgen et al. [2024], Varici et al. [2024]). In this setting, the goal is to recover Z∗and... | https://arxiv.org/abs/2501.19335v2 |
if all interventions in eI∗are single-node. In fact, as we will illustrate in Example 6.2, eg(E)−IntScan be an invalid model of h(X∗) even ifeg(a) were a single-node intervention for all a∈E. Therefore, to ensure interventional validity we need to not only make assumptions about the observed environments (often necessa... | https://arxiv.org/abs/2501.19335v2 |
6.3. Causal Representation. Let a data-generating process Dbe given. We say that a representation Z∗ofDisanI −Intcausal representation of Dif there exists a CBN Cand a set of interventions Iin Csuch that Cis anI −Intvalid model ofZ∗. ◦ The larger Iis, the more interventions the model purports to make predictions about,... | https://arxiv.org/abs/2501.19335v2 |
of the idea that causes must be manipulable [Cartwright, 2007, Glymour and Glymour, 2014, Pearl, 2018, 2019]. •We may want that for each variable there exists a single-node intervention in Ithat intervenes on it. This criterion is reminiscent of the idea of autonomy [Aldrich, 1989]. See Janzing and Mejia [2024] for a r... | https://arxiv.org/abs/2501.19335v2 |
none) of the corresponding representations to be causal representations. Since coarser equivalence classes than ∼CRLmay preserve interventional validity (depending on the chosen interpretation and assumptions), it may be possible to simplify the learning problem. 6.2 Causal discovery What do we assume when we assume th... | https://arxiv.org/abs/2501.19335v2 |
to identify the causal direction. Unfortunately, identifiability does not ensure interventional validity. For example, if the 28 representation Z∗is emulated by CandI∗with link gand the observed environments {LC;g(a)(Z∗)}a∈E,E⊆ A, correspond to single-node interventions in C, then we can identify C[Eberhardt et al., 20... | https://arxiv.org/abs/2501.19335v2 |
and observational distribution induced by MY, and let Ibe a set of interventions in Csuch that do(Y1= 0)∈ I. Cis compatible with Y∗:=τ(X1) but is not an I−IntSvalid model of Y∗. To see this, consider asuch that La(X∗ 1) =LA;do(X1=1)(X) (such an action exists since do(X1= 1)∈ I∗). Now, do(Y2= 0)∈IntI S(a) (since Pa(Y∗ 2... | https://arxiv.org/abs/2501.19335v2 |
works on abstraction [Rubenstein et al., 2017, Beckers and Halpern, 2019, Beckers et al., 2020, Rischel and Weichwald, 2021, Massidda et al., 2023, Xia and Bareinboim, 2024] there is a map ω:IL→ I Hbetween interventions in the low-level model and the high-level model. Implicitly, this suggests an interpretation Intsuch... | https://arxiv.org/abs/2501.19335v2 |
where p⇒q≡ ¬p∨q.25The material condition does not provide the correct analysis of propositions like (P); this can be seen by considering the analogous proposition (P’) If you intervene do(B= 6), then you will observe the distribution N(0,1)⊗δ5over (A, B). SinceLdo(B=6)(B) =δ6̸=δ5, (P’) should be a false proposition. Fu... | https://arxiv.org/abs/2501.19335v2 |
be false. But now we have a contradiction since (P-strict) implies (D-strict). This follows from monotonicity of strict implication: pJq|= (p∧d)Jq. Here, (P-strict) implies (D-strict) because P(A= 0, B= 5) = 1 if and only if P(A= 0) = 1 ∧P(B= 5) = 1. Therefore, if (P1) is to provide the correct analysis of (P), we must... | https://arxiv.org/abs/2501.19335v2 |
formal framework for reasoning about the interventional validity of a causal model, which depends on the chosen interpretation of actions as interventions. We discuss different interpretations by considering five desiderata, D0–D4, some of which must be violated to escape circularity. Only when the interpretation is ma... | https://arxiv.org/abs/2501.19335v2 |
Machine Learning Research , 2024. (Cited on page 27.) George Box. Use and abuse of regression. Technometrics , 8(4):625–629, 1966. (Cited on page 34.) Johann Brehmer, Pim De Haan, Phillip Lippe, and Taco S. Cohen. Weakly supervised causal representation learning. In Advances in Neural Information Processing Systems , 2... | https://arxiv.org/abs/2501.19335v2 |
Conference on Machine Learning , 2022. (Cited on page 31.) Atticus Geiger, Duligur Ibeling, Amir Zur, Maheep Chaudhary, Sonakshi Chauhan, Jing Huang, Aryaman Arora, Zhengxuan Wu, Noah Goodman, Christopher Potts, et al. Causal abstraction: A theoretical foundation for mechanistic interpretability. Preprint arXiv:2301.04... | https://arxiv.org/abs/2501.19335v2 |
(Cited on page 23.) S´ ebastien Lachapelle, Pau Rodriguez, Yash Sharma, Katie E Everett, R´ emi Le Priol, Alexandre Lacoste, and Simon Lacoste-Julien. Disentanglement via mechanism sparsity regularization: A new principle for nonlinear ICA. In Causal Learning and Reasoning , 2022. (Cited on page 23.) S´ ebastien Lachap... | https://arxiv.org/abs/2501.19335v2 |
4.) Eigil F. Rischel and Sebastian Weichwald. Compositional abstraction error and a category of causal models. In Uncertainty in Artificial Intelligence , 2021. (Cited on page 31.) Paul K. Rubenstein, Sebastian Weichwald, Stephan Bongers, Joris M. Mooij, Dominik Janzing, Moritz Grosse-Wentrup, and Bernhard Sch¨ olkopf.... | https://arxiv.org/abs/2501.19335v2 |
page 23.) Timothy Williamson. Suppose and Tell: The Semantics and Heuristics of Conditionals . Oxford University Press, 2020. (Cited on page 33.) 40 James Woodward. Making things happen: A theory of causal explanation . Oxford university press, 2005. (Cited on page 31.) James Woodward. The problem of variable choice. S... | https://arxiv.org/abs/2501.19335v2 |
emulate a representation, see Definition 2.5. do(j←qj, j∈J) An intervention on nodes J, see Definition 2.1. d,d∗Denotes interventions, d∈ I,d∗∈ I∗. pC i,pC;d i pC idenotes the i’th kernel given by CBN C.pC;d idenotes i’th kernel given by CBN Cand intervention d= do( j←qj, j∈J), that is, pC;d i=pC ifori /∈J, and pC;d i=... | https://arxiv.org/abs/2501.19335v2 |
Int eI,f: Violating only D3 We provide an interpretation that satisfies D0–D4, except D3. We do not expect that this interpretation will be useful in itself; rather, we provide it to show that D0,D1, D2, and D4do not imply D3. We leave it for future work to investigate if there exist interesting interpretations that ma... | https://arxiv.org/abs/2501.19335v2 |
that may violate D4, but satisfies D0–D3. Definition D.3. IntM. An interpretation violating only D4. Let a data-generating process D, representation Z∗, compatible CBN C, and set of interventions Iin Cbe given. We define the interpretation IntMby the following rule: An intervention do(j←qj, j∈ J)∈ Iis inIntI M(a) if an... | https://arxiv.org/abs/2501.19335v2 |
for every d∈ I∗andu1∈[4]. 3.The third condition of Definition 3.13 in Beckers and Halpern [2019] holds since ωτ is defined for all interventions in I∗and we choose IY=ωτ(I∗). τ: [4]→ {0,1}2given by x7→( 1(x= 3) + 1(x= 4), 1(x= 2) + 1(x= 4)) is clearly surjective. For each d∈ I∗, we compute ωτ(d) using Definition 3.12 o... | https://arxiv.org/abs/2501.19335v2 |
that ωτ(do(X1=x1)) =∅for all x1∈R[Beckers and Halpern, 2019, Definition 3.12]. The second condition then implies that τ(Mdo(X1=x1) X (u1, u2, u3)) = ( x1+u2, x1+u2+u3) =MY(τU(u1, u2, u3)), 48 for every x1, u1, u2, u3∈R. But this is impossible since no function τU:R3→R2exists that simultaneously satisfies all the above ... | https://arxiv.org/abs/2501.19335v2 |
Markov w.r.t. the DAG. Therefore, whether d∈IntI S(a) depends on both the graph and the specific kernels in the CBN model. In the approach by Janzing and Mejia [2024], to decide whether a causal graph is valid, one has to determine for all iwhether the conditional distributions La(Z∗ i|PA∗ i) are the same as LO(Z∗ i|PA... | https://arxiv.org/abs/2501.19335v2 |
PUATE: Efficient ATE Estimation from Treated (Positive) and Unlabeled Units Masahiro Kato∗, Fumiaki Kozai, and Ryo Inokuchi Mizuho-DL Financial Technology, Co., Ltd. May 29, 2025 Abstract The estimation of average treatment effects (ATEs), defined as the difference in expected outcomes between treatment and control gro... | https://arxiv.org/abs/2501.19345v2 |
construction of efficient estimators . Using the efficient influence function, we develop semiparametric efficient ATE estimators that are√n-consistent and whose asymptotic variance achieves the efficiency bounds. These estimators are thus optimal under the semiparametric framework. In this study, we consider two DGPs ... | https://arxiv.org/abs/2501.19345v2 |
Edenotes the expectation under P0. 2.2 Observations with two DGPs In our setting, the observations are non-standard. We can only observe part of the treatment group and the unknown group, a mixture of the treatment and control groups. This setting is a variant of PU learning. PU learning encompasses two settings: the c... | https://arxiv.org/abs/2501.19345v2 |
setting, the observation of the treatment group is a random event, where the observation indicator Oifollows a probability P(O|X). In contrast, in the case-control setting, the label observation is deterministic, and the treatment and unknown groups are drawn independently. This difference impacts the estimator design ... | https://arxiv.org/abs/2501.19345v2 |
potential outcomes (Y(1), Y(0))are independent of treatment assignment given covariates: (Y(1), Y(0)) |=(O, D )| X. Assumption 3.3 (Common support in the censoring setting) .There exists a constant c independent of nsuch that for all x∈ X,π0(d|x), g0(d|x), ζT,0(x), ζ0(x)> chold. Under these assumptions, the ATE τ0is es... | https://arxiv.org/abs/2501.19345v2 |
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