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Learning High-dimensional Gaussians from Censored Data

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Two censored-Gaussian models are learnable with polynomial samples

desk verdict The self-censoring result is a solid extension of DGTZ18, but the linear-thresholding half has a real gap in the Langevin gradient-sampling analysis that needs to be fixed before the runtime claim is credible. read the letter →

arxiv 2504.19446 v1 pith:YAPI73K6 submitted 2025-04-28 cs.LG cs.CCmath.STstat.MLstat.TH

Arnab Bhattacharyya , Constantinos Daskalakis , Themis Gouleakis , Yuhao Wang This is my paper · ORCID
classification cs.LGcs.CCmath.STstat.MLstat.TH MSC 62H1262F1062G07
keywords missingnotatrandomGaussiandistributionlearningcensoreddatatruncatedsamplesself-censoringlinearthresholdingprojectedstochasticgradientdescentLangevinMonteCarlo
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks when a high-dimensional Gaussian can be recovered from samples in which coordinates are missing not at random, with the missingness rule known. It answers for two mechanisms. Under self-censoring, where coordinate $i$ is seen iff $y_i$ satisfies a known predicate, if every pair of coordinates is jointly observed with probability at least $\alpha$, both mean and covariance can be estimated to small total-variation error using $\tilde{O}(d^2(\lambda_{\max}/\lambda_{\min})^2/(\alpha\varepsilon^2))$ samples. Under linear-thresholding missingness, where coordinate $i$ is seen iff $v_i^\top y \le b_i$, the mean can be estimated in polynomial time with known covariance, provided any small subset of coordinates is observed with probability at least $\alpha$ and a $\gamma$-anchoring set of always-observed coordinates exists. If these results hold, MNAR corruption is not inherently an obstacle to efficient Gaussian estimation.

What carries the argument

For self-censoring, the machinery is coordinate-pair decomposition plus the truncated-Gaussian SGD used in dimensions 1 and 2, with Assumption 1.1 ensuring enough complete pairs. For linear thresholding, the machinery is the negative log-likelihood $\ell(\mu)$ whose gradient requires sampling from the conditional Gaussian $N(\mu,\Sigma)$ restricted to $\{y: S(y)=A,\, y_A=x\}$; because linear thresholds define convex sets, conditioning a Gaussian on a convex set can only reduce variance, which yields strong convexity of $\ell$. The gradient samples are obtained by projected Langevin Monte Carlo rather than rejection sampling, since rejection costs would grow exponentially when $\mu$ drifts from $\mu^*$; the anchoring assumption bounds the conditional pattern probabilities and keeps the sampler polynomial. A projection step keeps iterates in a Mahalanobis ball so the stochastic gradients have bounded centralized second moment.

What would settle it

Instantiate the linear-thresholding model with $d=2$, $\Sigma=I$, coordinate 2 always observed, coordinate 1 observed when $y_1 \le t$, and run Algorithm 3 as $t$ sweeps from 0 toward infinity. The theorem predicts the sample bound grows polynomially in $1/\alpha$ and $1/\gamma$; if the observed degradation in reaching $\|\hat\mu - \mu^*\|_\Sigma \le \varepsilon$ is exponential in $1/\gamma$, the assumptions as stated are not sufficient.

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Extended reading notes

Core claim

The central claim is that estimation under MNAR Gaussian data is computationally tractable exactly when the missingness rule has a decomposable or anchored structure. For self-censoring, the paper shows the $d$-dimensional problem reduces to one- and two-dimensional truncated-Gaussian subproblems, one per coordinate or coordinate pair; each subproblem is solved with the known truncated-sample SGD, and the pieces are assembled into a consistent covariance matrix. For linear-thresholding missingness with known $\Sigma$, the paper defines the population negative log-likelihood as a function of $\mu$ and proves it is $\lambda$-strongly convex with $\lambda = \alpha\beta/\lambda_{\max}(\Sigma)$, so projected SGD converges. The key algorithmic step is a projected Langevin Monte Carlo sampler that draws approximately from the conditional Gaussian over unseen coordinates given the observed pattern, with bias and second-moment bounds that make the PSGD convergence theorem applicable. The result is Theorem 1.5: with $\mathrm{poly}(d, 1/\alpha, 1/\beta, 1/\gamma, \lambda_{\max}/\lambda_{\min}, 1/\varepsilon, \log(1/\delta))$ samples, the estimator meets $\|\hat\mu - \mu^*\|_\Sigma \le \varepsilon$.

Load-bearing premise

The load-bearing premise is Assumption 1.4: there must be an anchoring set of coordinates that is observed in every sample and whose values make every missingness pattern occur with probability either 0 or at least $\gamma$; without such a set, the gradient sampler's cost becomes exponential and the strong-convexity estimate depending on pattern probabilities collapses.

Editorial extensions

If this is right

  • Under self-censoring with every pair observed with probability at least $\alpha$, both mean and covariance are recoverable to TV error $O(\varepsilon)$ with $\tilde{O}(d^2(\lambda_{\max}/\lambda_{\min})^2/(\alpha\varepsilon^2))$ samples, so heavy coordinate-wise missingness is not a barrier.
  • Under linear-thresholding missingness with known covariance, the mean can be found in polynomial time whenever small subsets are observed with probability at least $\alpha$ and a $\gamma$-anchor set exists, giving a finite-sample MNAR guarantee without an all-or-nothing observation model.
  • The self-censoring result needs only pairwise observation, not a constant fraction of fully observed samples, so it strictly extends the earlier truncated-Gaussian setting.
  • The $\Omega(1/\sqrt{\lambda_{\min}})$ lower bound shows some eigenvalue dependence is unavoidable for self-censoring, even when uncensored distributions are far apart in TV distance.
  • Because the likelihood is strongly convex only for linear-thresholding missingness, the same optimization route will not directly work for general self-censoring rules.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural next step, left open by the paper, is unknown $\Sigma$ under linear-thresholding missingness; the Hessian then involves a fourth moment, but the anchoring condition may still make a second-order method or a moment-based correction work.
  • The anchoring assumption is reminiscent of anchor words in topic modeling; if anchor coordinates are domain-known, the algorithm applies directly, and the dependence on $1/\gamma$ makes it worth measuring $\gamma$ in practice before deployment.
  • The coordinate-pair reduction for self-censoring suggests a general principle: when a missingness model factorizes over small subsets, estimation can be assembled from small truncated subproblems, potentially extending to other exponential families with pairwise-sufficient statistics.
  • For practical deployment, the $\mathrm{poly}(1/\gamma)$ and $\mathrm{poly}(1/\alpha)$ factors mean the algorithm's usefulness depends on how large these probabilities are; identifying real missingness mechanisms where $\gamma$ is not tiny would be a valuable empirical study.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies high-dimensional Gaussian estimation from missing-not-at-random (MNAR) data under two missingness models. In the self-censoring model, coordinate i is observed exactly when the underlying value lies in a known set S_i; the authors reduce estimation to one- and two-dimensional truncated-Gaussian subroutines from DGTZ18 and prove poly(d,1/epsilon) sample complexity under a pairwise-observation assumption. In the linear-thresholding model, coordinate i is observed when v_i^T y <= b_i, the covariance is known, and the goal is mean estimation; the paper proposes a projected stochastic gradient descent on the negative log-likelihood, with gradients sampled by a projected Langevin algorithm, and claims poly(d,1/alpha,1/beta,1/gamma,lambda_max/lambda_min,1/epsilon,log(1/delta)) samples and time under Assumptions 1.3 and 1.4.

Significance. If the linear-thresholding result were fully established, it would be a substantial advance: it would give the first polynomial-time algorithm for high-dimensional MNAR mean estimation under explicit, non-asymptotic conditions, going beyond the all-or-nothing censoring of DGTZ18. The self-censoring part is a clean and mostly convincing reduction that leverages published external subroutines, so it is not circular. The paper is commendably explicit about its assumptions and gives concrete pseudocode. However, the proof of the central runtime claim for the Langevin gradient sampler has a serious gap, so the main advertised theorem (Theorem 1.5) is not established as written.

major comments (2)
  1. [C.3, Lemma C.2 and Theorem 4.6] The proof that Algorithm 5 mixes in polynomial time is not sound. In the notation of Algorithm 5, K={z in R^{d-|A|} : S(x∘z)=A} is an affine slice of the missingness pattern; hence the probability Pr_{w∼N(0,Sigma)}[w in K] used in the proof of Lemma C.2 is zero, and if K is instead interpreted in the conditional space N(mu_cond,Sigma_cond), Assumption 1.4 gives no lower bound on its probability, since it conditions on the anchor set C rather than on the full observed vector y_A=x. For example, take d=3, C={1}, S(y)={1,2} iff y_3>0 (and {1} otherwise), and Corr(y_2,y_3)=-0.9. Then Assumption 1.4 holds with gamma=1/2, but for an observed sample with A={1,2} and x_2=t, the measure of K under the conditional Gaussian is Phi(-0.9t/sqrt(0.19)), which decays as exp(-Omega(t^2)). The proof does not show that such t cannot occur with non-negligible probability, nor does it adapt the Langevin radius to the actual measure of K. Consequently the polynomial bound on the number of Langevin steps in Theorem 4.6 is not established, and the running time claimed in Theorem 1.5 has no term that could absorb this exponential degradation.
  2. [4.3.1, Lemma 4.5] The strong-convexity proof is incomplete as printed. The text contains the garbled string 'enecccfckgvkkjlfnlclcggguvjldtdurlrlltnecnid' and skips the block-matrix manipulations needed to pass from Eq. (14) to the final inequality v^T nabla^2 ell(mu) v >= alpha v_H^T Sigma_HH^{-1} v_H. The displayed formulas suggest the identity may be correct, but a referee cannot verify the lemma without the missing algebra, and strong convexity is load-bearing for the SGD convergence argument. The proof needs to be rewritten with the omitted steps supplied.
minor comments (4)
  1. [Appendix C.3] Lemma C.2 is stated twice with the same label; the second occurrence should be renumbered or removed to avoid confusion.
  2. [Algorithm 5] Line 4 uses B_Sigma(W^{-1}mu_cond,R), but B_Sigma is a ball in R^d while the argument W^{-1}mu_cond lies in R^{d-|A|}; the intended metric for the projected Langevin dynamics should be clarified.
  3. [Algorithm 2] Line 7 says to remove rows containing at least one '*' but a missing entry is not previously defined; the intended definition of a missing-value indicator should be stated.
  4. [Lemma 3.4] The lower-bound proof concludes that o(max{1/epsilon, sqrt(lambda)/epsilon}) samples are insufficient, but the lemma statement is phrased as o(1/sqrt(lambda_min)) for epsilon = Omega(sqrt(lambda_min)); the relationship between these parameters should be spelled out for the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central reductions rely on published external subroutines, and no fitted parameter or prior conclusion is relabeled as a prediction.

full rationale

The paper's derivation chain is a reduction, not a restatement of its assumptions. For self-censoring, Assumption 1.1 is used only to guarantee that enough samples contain each pair of coordinates; the per-pair estimates are delegated to the published DGTZ18 truncated-Gaussian algorithm (Theorem 3.1), whose stated assumptions do not include the self-censoring model. For linear thresholding, Assumptions 1.3 and 1.4 are explicit hypotheses of Theorem 1.5, not consequences of the algorithm, and the proof uses the generic SGD convergence lemma (Lemma 6 of CDIZ22) and the projected-Langevin guarantee of BEL18; both are external, parameter-free lemmas with stated assumptions that do not contain the target mean-estimation theorem. The authors' earlier DGTZ18 and CDIZ22 works are cited heavily, but in each cited instance they supply an independent published result rather than a conclusion that is being derived in this paper. No fitted parameter is renamed as a prediction, and no uniqueness theorem from the authors' prior work is used to force the model choice. There is a separate correctness concern that is not circularity: in the proof of Lemma C.1 the paper states 'Setting δ = γ/2, we get that: Pr[v∈BΣ(0,R1)∩K]≥γ/2', which implicitly requires a lower bound on Pr[v∈K]; Assumption 1.4 bounds pattern probabilities only after conditioning on the anchor set C, not on the full observed set A, so this step is unsupported. That gap affects the validity of the Langevin runtime guarantee but does not make the claim reduce to its own inputs by definition. Therefore no structural circularity is present, and the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper does not fit any constant to data; alpha, beta, gamma are lower-bound assumptions, not fitted values. The algorithmic parameters (eta, R, M) are prescribed in terms of d, lambda_max, lambda_min, S, gamma, epsilon. The main burden falls on the modeling assumptions (pairwise visibility, small-subset visibility, anchoring) and on imported results from prior work by the same research group (DGTZ18, CDIZ22), which are published and not re-derived here.

assumptions (6)
  • domain assumption Assumption 1.1: for every pair i,j, Pr[y_i in S_i, y_j in S_j] >= alpha.
    Invoked in Section 3 to guarantee enough samples for 1D and 2D truncated subproblems; if it fails, the decomposition into pairwise problems has no sample support.
  • domain assumption Assumption 1.3: any subset A of size at most beta d is observed with probability at least alpha.
    Used in Lemma 4.5 for strong convexity and in the initialization (Algorithm 2) to ensure every block of coordinates is present with non-negligible probability.
  • domain assumption Assumption 1.4: existence of a gamma-anchoring set C as in Definition 4.1.
    Needed for the Langevin gradient estimator: conditioned on y_C the pattern probabilities are bounded below, which controls the cost of sampling from K and the bias bound in Corollary 4.7.
  • standard math DGTZ18 Theorem 3.1 (truncated Gaussian estimation) can be applied to 1D and 2D subproblems.
    Imported as a black box in Section 3; the paper does not re-prove it, so the central bounds for self-censoring inherit the correctness of this published subroutine.
  • standard math KP77 Corollary 2.1: conditioning a Gaussian on a convex set does not increase its variance in the Loewner order.
    Used in Lemma 4.2 and Lemma 4.5 to lower-bound the Hessian of the negative log-likelihood; this is a classical result cited and not proved.
  • standard math BEL18 Theorem 1: convergence of projected Langevin Monte Carlo to the truncated Gaussian.
    Basis for Theorem 4.6, which states that Algorithm 5's output is within TV epsilon of the conditional Gaussian; assumed to hold with the stated polynomial parameters.

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Pith. "Pith review of Learning High-dimensional Gaussians from Censored Data." pith.science (2026). https://pith.science/paper/YAPI73K6

@misc{pith2026250419446,
  author       = {Pith},
  title        = {Pith review of: Learning High-dimensional Gaussians from Censored Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YAPI73K6}},
  note         = {Machine review of arXiv:2504.19446}
}
abstract

We provide efficient algorithms for the problem of distribution learning from high-dimensional Gaussian data where in each sample, some of the variable values are missing. We suppose that the variables are missing not at random (MNAR). The missingness model, denoted by $S(y)$, is the function that maps any point $y$ in $R^d$ to the subsets of its coordinates that are seen. In this work, we assume that it is known. We study the following two settings: (i) Self-censoring: An observation $x$ is generated by first sampling the true value $y$ from a $d$-dimensional Gaussian $N(\mu*, \Sigma*)$ with unknown $\mu*$ and $\Sigma*$. For each coordinate $i$, there exists a set $S_i$ subseteq $R^d$ such that $x_i = y_i$ if and only if $y_i$ in $S_i$. Otherwise, $x_i$ is missing and takes a generic value (e.g., "?"). We design an algorithm that learns $N(\mu*, \Sigma*)$ up to total variation (TV) distance epsilon, using $poly(d, 1/\epsilon)$ samples, assuming only that each pair of coordinates is observed with sufficiently high probability. (ii) Linear thresholding: An observation $x$ is generated by first sampling $y$ from a $d$-dimensional Gaussian $N(\mu*, \Sigma)$ with unknown $\mu*$ and known $\Sigma$, and then applying the missingness model $S$ where $S(y) = {i in [d] : v_i^T y <= b_i}$ for some $v_1, ..., v_d$ in $R^d$ and $b_1, ..., b_d$ in $R$. We design an efficient mean estimation algorithm, assuming that none of the possible missingness patterns is very rare conditioned on the values of the observed coordinates and that any small subset of coordinates is observed with sufficiently high probability.

Figures

Figures reproduced from arXiv: 2504.19446 by the authors.

Figure 1
Figure 1. In this example, the self-censoring missingness mechanism is as follows: Each coordinate [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗
Figure 2
Figure 2. An illustration of convex sets in Section 4.3.2. Proof of Theorem 4.6. With R2 as in Lemma C.2, set R = R2 + S, so that L = BΣ(µ, R) ∩ K. Since R2 > R1, Lemma C.1 implies that L contains a ball of radius r. On the other hand, by Lemma C.2, Pr[w ∈ L/ ] ≤ ε 4 Pr[w ∈ K] for w ∼ N (µ, Σ), which implies that the truncation of N (µ, Σ) to K and to L are at most ε/2 far from each other in TV distance. We can now use the ma… view at source ↗

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Pith tools

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