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REVIEW 2 major objections 5 minor 35 references

Efficient Truncated Statistics with Unknown Truncation

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Efficient recovery of Gaussian parameters is possible even when the truncation set is unknown.

desk verdict Real advance in unknown truncation, but a proof gap in Lemma 3 and a heuristic lower bound mean the advertised guarantees need work before I'd trust them. read the letter →

arxiv 1908.01034 v1 pith:L4M452LX submitted 2019-08-02 math.ST cs.DScs.LGstat.COstat.MLstat.TH

Vasilis Kontonis , Christos Tzamos , Manolis Zampetakis This is my paper · ORCID
classification math.STcs.DScs.LGstat.COstat.MLstat.TH MSC 62F1062H1268Q32
keywords truncatedstatisticsunknowntruncationsetGaussiansurfaceareaVCdimensionHermitepolynomialsstochasticgradientdescentparameterestimationmomentmatching
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

This paper claims that the parameters of a Gaussian can be recovered efficiently from samples that have been filtered through an unknown set, provided the family of possible truncation sets has bounded complexity. The main result is an algorithm that learns both the mean and the covariance of a spherical or diagonal Gaussian using $d^{\mathrm{poly}(1/\alpha,1/\varepsilon)\,\Gamma(\mathcal{C})^2}$ samples, where $\Gamma(\mathcal{C})$ is the Gaussian surface area of the set family, a measure of how much Gaussian mass sits near the set's boundary, and runs in polynomial time. A complementary result shows that, for any family of sets with VC-dimension $\mathrm{VC}(\mathcal{C})$, only $\tilde O(\mathrm{VC}(\mathcal{C})/\varepsilon + d^2/\varepsilon^2)$ samples suffice, and a lower bound shows the dependence on set complexity is close to necessary. A sympathetic reader would care because this resolves the unknown-truncation case of a classical problem whose known-truncation version was only solved in 2018, and it says that not knowing the truncation set is not a fundamental barrier to estimation.

What carries the argument

The load-bearing object is the weighted characteristic function $\psi(x)=1_S(x)\alpha_*^{-1}\mathcal{N}(\mu_*,\Sigma_*;x)/\mathcal{N}(0,I;x)$, which entangles the unknown set with the unknown Gaussian and whose Hermite coefficients are exactly the expectations $\mathbb{E}_{x\sim\mathcal{N}(\mu_*,\Sigma_*,S)}[H_V(x)]$ that samples reveal. The argument is carried by three identities: the tail of the Hermite expansion of $\psi$ is bounded by $\mathrm{poly}(1/\alpha)(\sqrt{\Gamma(S)}/k^{1/4}+1/k)$; the objective $M_f(u,B)=\mathbb{E}_{x\sim\mathcal{N}(\mu_*,\Sigma_*,S)}[e^{h(u,B;x)}\mathcal{N}(0,I;x)f(x)]$ is convex in $(u,B)$, with minimizer $(\Sigma_*^{-1}\mu_*,\Sigma_*^{-1})$ when $f=\psi$; and the set can be recovered by thresholding $\psi_k\mathcal{N}(0,I;x)/\mathcal{N}(\hat\mu,\hat\Sigma;x)$ at $1/2$. The algorithm's steps are therefore: estimate low-degree Hermite coefficients from truncated samples, run projected SGD on the convex objective, then threshold to recover the set.

What would settle it

Search in $\mathbb{R}^2$ for two Gaussians with $\|\mu_1-\mu_2\|_2>1$ and $\|\Sigma_1-\Sigma_2\|_F>1$, together with sets $S_1,S_2$ of mass at least $\alpha=0.1$, such that $d_{\mathrm{TV}}(\mathcal{N}(\mu_1,\Sigma_1,S_1),\mathcal{N}(\mu_2,\Sigma_2,S_2))$ is smaller than $(\alpha^3/8)\,d_{\mathrm{TV}}(\mathcal{N}(\mu_1,\Sigma_1),\mathcal{N}(\mu_2,\Sigma_2))$. An explicit pair, or a numerical optimizer that drives the ratio to zero while respecting the mass lower bound, would directly falsify Lemma 3 and with it the parameter-recovery chain.

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Extended reading notes

Core claim

The central claim, on the paper's own terms, is that unknown truncation can be decoupled from the Gaussian parameters. The algorithm never learns the truncation set first; instead it estimates a weighted characteristic function $\psi(x)=1_S(x)\alpha_*^{-1}\,\mathcal{N}(\mu_*,\Sigma_*;x)/\mathcal{N}(0,I;x)$ from truncated samples, relying on the proof that this function has low-degree Hermite concentration when $S$ has bounded Gaussian surface area. Once a low-degree approximation $\psi_k$ is in hand, the paper defines a convex objective $M_{\psi_k}(u,B)$ whose unique minimizer is $(\Sigma_*^{-1}\mu_*,\Sigma_*^{-1})$, found by projected stochastic gradient descent. The same function, thresholded against the recovered Gaussian, also identifies the set $S$ up to small Gaussian measure. The abstract's strongest claim is Informal Theorem 2: with $N=d^k$ samples for $k=\mathrm{poly}(1/\alpha,1/\varepsilon)\,\Gamma(S)^2$, the output $\hat\mu,\hat\sigma^2I$ satisfies $d_{\mathrm{TV}}(\mathcal{N}(\mu,\sigma^2 I),\mathcal{N}(\hat\mu,\hat\sigma^2 I))\le\varepsilon$.

Load-bearing premise

The load-bearing premise is Lemma 3, which asserts that truncation cannot make two genuinely different Gaussians look alike: two Gaussians at total-variation distance $\varepsilon$ remain at least $C_\alpha \varepsilon$ apart after any truncation to sets of mass at least $\alpha$, with $C_\alpha=\Omega(\alpha^3)$. If that normalization lower bound fails, closeness of truncated distributions would not certify closeness of the underlying Gaussians, and both the VC-dimension recovery theorem and the abstract's parameter-recovery guarantee would collapse.

Editorial extensions

If this is right

  • Any set family with VC-dimension $\mathrm{VC}(\mathcal C)$ admits recovery of the Gaussian mean and covariance to total-variation $\varepsilon$ from $\tilde O(\mathrm{VC}(\mathcal C)/\varepsilon + d^2/\varepsilon^2)$ samples.
  • For spherical and diagonal Gaussians, families with Gaussian surface area $\Gamma(\mathcal C)$ can be handled in polynomial time with $d^{\mathrm{poly}(1/\alpha,1/\varepsilon)\Gamma(\mathcal C)^2}$ samples; this covers intersections of halfspaces, polynomial threshold functions, and all convex sets.
  • The same samples suffice to recover the truncation set itself: the algorithm outputs a set whose symmetric difference with the true set has Gaussian mass below $\varepsilon$.
  • Moment matching is sufficient for identification: matching moments up to degree $O(\Gamma(S)/\varepsilon^4)$ forces the truncated Gaussians, and hence the untruncated parameters, to be $\varepsilon$-close.
  • The lower bound shows that some sets with small Gaussian surface area ($\Gamma(S)=O(d)$) still force $\Omega(2^{d/2})$ samples, so the polynomial dependence on $\Gamma$ cannot be replaced by a fixed polynomial in $d$ alone.

Reading between the lines

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

  • Beyond the paper, the decoupling recipe—weighted indicator, Hermite concentration, convex objective—is not obviously tied to Gaussians; the same outline could be attempted for any exponential family with a quadratic log-density, with Gaussian surface area replaced by that family's isoperimetric profile.
  • Beyond the paper, the sample bounds make concrete predictions for structured families: for intersections of $k$ halfspaces the bound is $d^{O(\log k)}$, so experiments on that family could reveal whether the true constant is closer to the bound than the convex-set case suggests.
  • Beyond the paper, the moment-matching theorem implies truncated Gaussians are identifiable from finitely many moments; checking whether $O(\Gamma^2/\varepsilon^4)$ can be replaced by a smaller degree for axis-aligned rectangles or unions of boxes would separate the statistical from the computational content of the result.
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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 / 5 minor

Summary. The paper studies estimation of the mean and covariance of a Gaussian from samples that are observed only if they fall in an unknown truncation set S. It claims two main routes: a VC-dimension-based sample-complexity bound with an ERM/tournament algorithm (Informal Theorem 1, Theorem 1), and a computationally efficient algorithm for spherical and diagonal-covariance Gaussians whose sample complexity is controlled by the Gaussian surface area of the set family (Informal Theorems 2 and 3, Theorem 2). The technical engine is a Hermite-expansion approximation of the weighted indicator function ψ(x)=1_S(x) N*(x)/(α*N0(x)), followed by a convex stochastic-gradient optimization over the Gaussian parameters and a thresholding step to recover the set. The paper also proves a moment-matching identifiability result (Theorem 8) and a lower-bound construction (Theorem 7) intended to show that the surface-area dependence is tight. The proofs are detailed and mostly self-contained, with several technical lemmas deferred to the appendices.

Significance. If the proofs are sound, this is a significant contribution to high-dimensional truncated statistics: it is the first efficient algorithm for unknown truncation sets with guarantees governed by Gaussian surface area, it covers non-ERM families such as convex sets, and it includes a clean set-recovery corollary. The Hermite-concentration analysis of the weighted indicator and the convex SGD formulation are substantial and plausible. The paper is also honest about the distinction between VC-dimension and surface-area approaches. The main technical obstruction is Lemma 3, whose proof is not currently valid; since that lemma is load-bearing for the VC-dimension result and for the diagonal-covariance extension, the full scope of the claims is not yet established. The lower bound in Section 5 is also presented as a heuristic rather than a formal minimax argument. Both issues are fixable in a revision, so the paper merits major revision rather than rejection.

major comments (2)
  1. [Section 3, Lemma 3] The proof of Lemma 3 does not establish the claimed lower bound. The displayed chain lower-bounds E[|1-exp(p)|1_S] by E[|p|/2 1_{S\Q}] using |1-e^x|≥|x|/2 for |x|≤1, but the set S is defined by |1-exp(p)|≤T and the argument only shows N0(S\Q)≥α/4; it does not show that |p|≤1 on S\Q, and for p very negative the inequality fails since |1-e^p|≤1 while |p|/2 is unbounded. This step is load-bearing: Lemma 3 is used in Theorem 1 to transfer truncated-TV closeness to parameter-TV closeness and in the reduction from arbitrary diagonal covariance to the isotropic algorithm. I recommend replacing the pointwise inequality by the globally valid bound |1-e^p|≥min(1,|p|/2) and then controlling the mass of {p<−1} inside S\Q, or proving a uniform upper bound on T. In addition, the 'relaxed optimization' objective (1/α)∫|N(0,I;x)-(α/β)N(μ,Λ;x)|1_S(x)dx is not shown to be a lower bound on dTV(D1,D2): the true truncated densities have normalizing constants α1,α2 which need not equal α, and the free parameter β only adjusts the second density. The proof must justify this reduction before the lemma can be used.
  2. [Section 5, Theorem 7] The lower-bound proof is heuristic rather than a formal minimax argument. The claim that distinguishing N+(S+) from the reference density r requires a birthday collision is not supported by a reduction, and the event |Σ(pV−p)|<2^{-d/2} controls only one error term; it does not rule out algorithms that exploit the continuous sample coordinates inside each cube HV or that test other statistics of the sample. To establish the claimed Ω(2^{d/2}) lower bound for mean estimation, the proof should be recast as a two-point or Le Cam/Fano argument over pairs of hypotheses whose means are separated by Ω(1). As written, the tightness claim in the abstract and Section 6 is not established.
minor comments (5)
  1. [Section 3, Lemma 3 proof] The symbol S is used both for the family of sets and for the intersection set S1∩S2, which makes this proof unnecessarily confusing.
  2. [Section 4.1, Theorem 3] The equality ∑_{|V|≥k} should be ∑_{|V|>k} (or the definition of Skψ should be made consistent), since Skψ contains all Hermite terms of degree at most k.
  3. [Section 4.2, proof of Theorem 2] The sentence 'Hence we can we lose probability at most δ' contains a typo, and the probability accounting around the median-selection of the SGD runs is compressed; it should be spelled out in full.
  4. [Section 6, Lemma 14] The sentence 'Dχ2(N1‖N) Dχ2(N2‖N) are finite' is missing a conjunction and is hard to parse; it should be 'Dχ2(N1‖N) and Dχ2(N2‖N) are finite'.
  5. [Introduction, Informal Theorem 3] Informal Theorem 3 does not mention the affine-invariance assumption Γ(T(S))=O(Γ(S)) that is stated in Theorem 8; without that assumption the informal statement overclaims what is proved.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's recovery guarantees are derived from explicit sample-complexity, Hermite-concentration, and anti-concentration arguments, and its self-citations are auxiliary parameter-free lemmas from prior work rather than circular premises.

full rationale

The central derivation is not circular. The unknown-truncation recovery results are obtained by first learning a weighted characteristic function through Hermite coefficient estimation, then optimizing an explicitly defined convex objective whose unique minimizer is identified by a direct calculation, and finally using standard concentration and anti-concentration tools. Lemma 3, which transfers total-variation closeness of truncated distributions to closeness of untruncated parameters, is a genuine lower-bound argument rather than a restatement of the conclusion. The paper's self-citations to [DGTZ18] are used for auxiliary facts such as empirical conditional mean/covariance estimation, projection onto the isotropic-position domain, and a truncated-Gaussian anti-concentration bound; these are parameter-free statements with stated assumptions that do not include the target unknown-truncation recovery claim, so they count as independent evidence under the review rules. No fitted parameter is later relabeled as a prediction, and no uniqueness theorem from the authors' prior work is invoked to forbid alternatives. A possible flaw in the proof of Lemma 3's inequality would be a correctness issue, not a circularity issue; the derivation chain does not assume its own conclusion.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no free parameters fitted to data; the constants in the proofs are universal or depend on alpha and epsilon as inputs. The main domain assumptions are the known mass lower bound alpha, the near-isotropic position condition, and the affine-invariance of Gaussian surface area used in the moment-matching section. The proofs rely on standard Gaussian analysis results (Hermite expansion, noise sensitivity, Carbery-Wright) and on lemmas from the authors' prior work [DGTZ18].

assumptions (5)
  • domain assumption The truncation set S has mass at least alpha under the unknown Gaussian, and alpha is known to the algorithm (equation (3)).
    All sample complexity bounds are stated in terms of 1/alpha, and the algorithm assumes this lower bound as input.
  • domain assumption The unknown Gaussian is in (O(log(1/alpha)), 1/16)-isotropic position after a whitening step (Definition 2).
    Theorem 2 is proved under this condition; the reduction for diagonal covariance uses a search over hypotheses and Lemma 18, which is cited from [DGTZ18].
  • standard math Gaussian surface area is subadditive and bounds Gaussian noise sensitivity (Lemma 19, from [KOS08]).
    Used in Lemma 6 and Lemma 15 to bound Hermite concentration of the weighted characteristic function.
  • standard math Carbery-Wright anti-concentration for polynomials over Gaussian measure (Theorem 9) holds and is applied to degree-2 polynomials.
    Used in Lemma 3 to bound the Gaussian mass of the set where a quadratic polynomial is small, and in Lemma 12 for strong convexity.
  • domain assumption The class of sets is closed under affine transformations with Gaussian surface area growing at most as O(Gamma(S)) (Theorem 8 and Lemma 15).
    The moment-matching extension for general covariance requires this affine-invariance property; it is stated explicitly but not proven for general families.

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Cite this review

Pith. "Pith review of Efficient Truncated Statistics with Unknown Truncation." pith.science (2026). https://pith.science/paper/L4M452LX

@misc{pith2026190801034,
  author       = {Pith},
  title        = {Pith review of: Efficient Truncated Statistics with Unknown Truncation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L4M452LX}},
  note         = {Machine review of arXiv:1908.01034}
}
abstract

We study the problem of estimating the parameters of a Gaussian distribution when samples are only shown if they fall in some (unknown) subset $S \subseteq \R^d$. This core problem in truncated statistics has long history going back to Galton, Lee, Pearson and Fisher. Recent work by Daskalakis et al. (FOCS'18), provides the first efficient algorithm that works for arbitrary sets in high dimension when the set is known, but leaves as an open problem the more challenging and relevant case of unknown truncation set. Our main result is a computationally and sample efficient algorithm for estimating the parameters of the Gaussian under arbitrary unknown truncation sets whose performance decays with a natural measure of complexity of the set, namely its Gaussian surface area. Notably, this algorithm works for large families of sets including intersections of halfspaces, polynomial threshold functions and general convex sets. We show that our algorithm closely captures the tradeoff between the complexity of the set and the number of samples needed to learn the parameters by exhibiting a set with small Gaussian surface area for which it is information theoretically impossible to learn the true Gaussian with few samples.

Figures

Figures reproduced from arXiv: 1908.01034 by the authors.

Figure 1
Figure 1. Illustration of the results of our algorithm for an unknown truncation set. The [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

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