Honest Physical-Support Inference after Latent Dictionary Learning: Collision Singularities and Minimax Resolution

Guan-Ju Peng

After latent dictionary learning, honest sparse-support inference must retain every training-compatible dictionary, and the finest physical resolution is set by three information gates capped by the N s^6 dictionary-orientation cost.

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2026-08-01 19:51 UTC pith:DWDZKLWP

load-bearing objection A solid, correctly scoped theory paper that identifies a genuine train-test support-inference gap and proves matching rates; the math holds up, though the scope is narrow and no solver is shipped.

arxiv 2607.16813 v1 pith:DWDZKLWP submitted 2026-07-18 cs.LG math.STstat.TH

Honest Physical-Support Inference after Latent Dictionary Learning: Collision Singularities and Minimax Resolution

Guan-Ju Peng This is my paper

classification cs.LG math.STstat.TH MSC 62F2562F1262H12

keywords dictionary learningsparse support inferenceconfidence correspondencecollision singularityminimax resolutionlatent variable modelsuncertainty quantificationquotient invariance

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Standard sparse-support inference conditions on a known or frozen dictionary. This paper shows that when the dictionary is learned from latent mixtures, that conditioning hides a calibration failure: near coherent collisions at scale s, a test signal can separate coordinate supports while training data cannot identify the physical rays. The paper derives a confidence correspondence that keeps all training-compatible dictionary orbits, profiles the test mean over them, and projects onto permutation-invariant physical support sets, with two-level conditional coverage. It proves that residual block orientation enters the training density only at cubic order, yielding an N s^6 information budget, and that no uniformly honest method can bypass this gate; in the resolved regime the correspondence attains the minimax-optimal diameter s and (sqrt(N) s^2)^{-1}. A sympathetic reader should come away with a concrete picture: honest sparse inference after dictionary learning is set-valued, resolution-adaptive, and gated by where the information actually lives.

Core claim

The paper's central claim is that the estimand for sparse support after latent dictionary learning is a marked set of physical rays modulo dictionary-support relabelling, not a coordinate support in one chosen frame. Its construction is the confidence correspondence of (3.15): invert a robust simultaneous second- and fourth-moment training rectangle into every compatible dictionary orbit (never selecting one), profile the replicated test mean against each retained dictionary and support via a single Gaussian ball, and then project the survivors onto physical ray sets. The paper proves two-level honesty, high probability over training that the conditional test coverage exceeds 1-alpha_T, and

What carries the argument

The central object is the confidence correspondence C-hat in (3.15), a random compact set of physical support targets built by (i) inverting a median-of-means training rectangle for the second and fourth moments into all compatible dictionary orbits, (ii) profiling the test mean against every retained dictionary and support with one common Gaussian ball, and (iii) mapping survivors through the permutation-invariant physical target. The mechanism that sets the statistical cost is the cubic orientation invariant G_3 = L^{⊗3} T_q inside the fourth-order cumulant, where T_q is the third-power sum of regular-simplex reference vertices; its stabilizer is exactly the child-permutation group, its di

Load-bearing premise

The whole edifice rests on Assumption 4.2: the collision scale s, block membership, hierarchy, and affine regular-simplex chart are supplied exactly, and the code law puts positive probability on proper subdictionaries (Delta_L>0); misspecify s or the hierarchy, or let Delta_L=0, and neither the N s^6 gate nor the minimax rate is guaranteed.

What would settle it

Exhibit a uniformly honest confidence correspondence within Assumption 4.2 whose worst-case expected physical diameter in the resolved regime is smaller, up to constants, than s and (sqrt(N) s^2)^{-1} while N s^6 stays bounded; Theorem 6.5 predicts no such procedure exists. A direct check of the mechanism is to set Delta_L=0 and verify numerically that the training density becomes orientation-invariant to all orders at the centered core, exactly as the cubic-score identity (4.4) asserts.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

Known-dictionary support intervals can overstate physical resolution near collisions; any honest statement must either stay at group level or report multiple child branches.
No uniformly honest procedure can bypass the dictionary gate: when N s^6 is bounded, worst-case expected physical diameter is at least c s even if the test signal is arbitrarily clean.
The same construction automatically adapts its output granularity among cross-sheet inconclusive, parent-resolved with child ambiguity, and fine-support resolved, depending only on which of the three gates are open.
Measurement allocation should follow the gates: adding test replicates raises I_S and I_G^(r); only informative dictionary training or asymmetric-coefficient task geometry can raise the effective I_D.
In the resolved fixed-shell regime the method's diameter matches the minimax lower bound up to constants, so no alternative honest procedure can be uniformly better.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Extension left implicit: the three-gate diagram gives an experiment-design rule for blind unmixing or array localization—when I_D is the bottleneck, collecting more test replicates of the same signal is wasteful except along the task secants identified in Theorem 6.7.
The paper's scope statement is explicit that exact compact profiling is a statistical benchmark, not an operational solver; a deployable version requires a certified outer approximation that cannot delete a feasible truth, and the hierarchy must be supplied or learned separately.
A natural next test is to simulate the compensated-orientation path with equal coefficients and a fixed total test mean: the paper predicts test replication gives zero information about physical child identity, whereas any coefficient asymmetry produces the contrast crossover of Theorem 6.8.
One might explore whether the same cubic-invariant mechanism controls support uncertainty in nonlinear forward models such as phase retrieval or blind deconvolution, where an analogous fourth-order cumulant could carry the residual orientation.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

A solid, correctly scoped theory paper that identifies a genuine train-test support-inference gap and proves matching rates; the math holds up, though the scope is narrow and no solver is shipped.

read the letter

First things first: this paper deserves a close read. It identifies a real gap—after dictionary learning, freezing one dictionary can produce overconfident physical-support statements near coherent-atom collisions—and it actually closes the gap within its stated model. The main construction, a correspondence that retains all training-compatible dictionaries and only then profiles the test signal, gives the two-level coverage guarantee in Theorem 6.3 without any union bound over supports. That is a clean idea, and the proof appears honest.

The strongest part is the geometry. The claim that residual block orientation first enters the latent training density at cubic order, giving N s^6 information, is the load-bearing step. I checked the appendix's exchangeable-subset aggregation and the simplex cubic maxima; the cancellation through second order and the nonzero cubic coefficient when Δ_L > 0 are convincing. The matching lower bound via the compensated orientation pair is also sound—the target distance scales like s|h| and the joint KL is N s^6 h^2, so the minimax rate s ∧ (1/(√N s^2)) follows. The three-gate phase diagram is a nice way to summarize separate training and test information budgets.

The paper also does something I like: it explicitly bounds its own novelty. Each ingredient—set-valued prediction, orbit recovery, dictionary identifiability, learned-estimator UQ—has prior art, and the paper says so. The contribution is the conjunction in a latent train-test sparse inverse problem, and that conjunction is new.

Now the soft spots, which are real but not load-bearing. The scope is deliberately narrow: Assumption 4.2 supplies the collision scale s, the hierarchy, the block membership, and the affine-simplex chart. If any of those is misspecified, the rates do not apply. The paper acknowledges this clearly, but it means the result is a benchmark theorem, not a general dictionary-learning UQ method. Second, there is no implemented certified solver; exact compact profiling is a feasibility procedure in fixed dimension, but the paper doesn't claim polynomial time. Third, the median-of-means radii are conservative, and no machine-checked or independent numerical verification of the s^6 scaling is provided. None of these invalidate the theorems.

Who is this for? People working on dictionary learning, sparse inverse problems, or uncertainty quantification for learned representations. It deserves a serious referee; with minor clarifications of the scope and perhaps one small numerical illustration, it would be a good addition to the literature. I'd take it to reading group.

Referee Report

0 major / 4 minor

Summary. This paper studies inference for active physical rays (unit atoms modulo sign) after latent dictionary learning in a fixed-dimensional Gaussian train-test experiment. Near a coherent-atom collision, a test signal may determine which coherent group is active while the training data cannot identify which physical ray inside that group generated it. The authors construct a compact, measurable confidence correspondence that retains every dictionary compatible with a robust training-moment region and profiles the replicated test signal over all retained dictionaries before mapping to physical support sets. The main results show that residual block orientation is first visible in the latent training density at cubic order, giving an N s^6 orientation information gate; that the proposed correspondence attains two-level conditional coverage and a three-gate, resolution-adaptive phase diagram; that its projective Hausdorff diameter contracts at the minimax rate s ∧ (√N s^2)^{-1} in the resolved fixed-shell regime; and that a restricted task theorem identifies when coefficient asymmetry lets test replication supplement training information. The proofs are carried out in a detailed appendix, with the fixed-shell scope collected in Assumption 4.2.

Significance. If the proofs are correct, this is a substantial contribution to uncertainty quantification for dictionary learning and sparse-support inference. The paper correctly identifies the physical estimand as a set of rays rather than a coordinate support attached to one arbitrary learned dictionary, and it quantifies a genuinely nonregular training singularity: orientation information is of order N s^6, so closed dictionary gates cannot be repaired by test replication along compensated directions. Notable strengths are the single-event coverage argument that avoids union bounds over candidate supports, the explicit lower-bound pairs that are genuine submodels of the stated class, the detailed proof chain for the cubic orientation expansion, and the honest, clearly bounded scope of the fixed-shell assumptions. The paper is theory-only and does not provide a numerical implementation, but the mathematical claims are stated with precise scope and the appendices give a complete proof architecture. The fixed-shell assumption (Assumption 4.2) is the main limitation; it is explicitly acknowledged and does not create an internal inconsistency.

minor comments (4)

[§5.1, Eq. (5.26)] The moment-to-Hellinger bound is stated with a one-line Cauchy-Schwarz justification, but the displayed factorization leaves the factor ∫∥Ψ∥^2(√fθ−√fθ')^2 to be controlled. This requires a weighted L2 estimate under the uniform eighth-moment envelope; I believe it can be supplied by splitting into small- and large-Hellinger regimes, but the proof should spell out the details. As written this is a completeness issue rather than an error.
[Theorems 6.1, 6.3, 6.5] The text repeatedly says 'Under theorem 4.2' where it means 'Under Assumption 4.2'. The assumption is not a theorem, and this cross-reference should be corrected throughout. Several nearby equation references (e.g., 'equation (2.1)' inside the statement of Assumption 4.2) should also be checked.
[Abstract and §1] The abstract states without qualification that 'Residual block orientation first affects the latent training density at cubic order.' In the body this is proved at the centered isotropic core and, more generally, at interior shell points in the point-adapted chart; the finite-sample lower-bound pairs are built on that core. A short qualifier such as 'at the studied fixed-shell core' would prevent readers from extrapolating the s^6 gate to arbitrary shell geometries.
[§8 Discussion] The paper is explicit that exact compact profiling is a statistical benchmark and not a polynomial-time method. It may be worth one sentence in the discussion stating that no numerical or symbolic verification of the s^6 cancellation is included; the claim rests on the analytical proof. This is not a defect for a theory paper, but an explicit statement would help readers calibrate the status of the implementation.

Circularity Check

0 steps flagged

No significant circularity: derivation is self-contained theory with no fitted inputs, no load-bearing self-citation, and explicit safeguards against circular use of the moment lower bound.

full rationale

I walked the main derivation chain: Assumption 4.2 fixes the generative model; the training moment region (3.9)-(3.10) is built from certified median-of-means radii with no data-dependent fitting of the target; the test profile (3.14) uses one Gaussian ball so the coverage event (36) is the true witness surviving a single test event, with no union bound; the diameter bound (4.11) follows from the algebraic quotient inverse and cross-dictionary margins; and the minimax lower bounds use explicitly constructed least-favourable pairs whose joint KL is computed from the model, not from the upper-bound procedure. The paper itself flags and refutes the only possible circular use: in the proof of Theorem 6.1 it states 'This step uses the quotient inverse only for algebraic alignment, so the statistical moment lower bound is not used circularly', and in Supplement S8, 'The quotient inverse is used here only to choose an algebraic label alignment and bound the projector remainder; no statistical moment lower bound is used. Thus there is no circularity.' No parameter is fitted to the data and renamed as a prediction; the claimed rate s ∧ (√N s^2)^{-1} is derived from the chord/inverse geometry and matched by lower-bound pairs, not imposed by construction. All citations in the paper are to external prior work; the single author cites none of his own papers as load-bearing evidence. The acknowledged scope limitation — fixed shell, supplied hierarchy and s — is an explicit assumption, not a circular reduction. Therefore the appropriate finding is no significant circularity, score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 0 invented entities

No hidden fitted constants or newly postulated physical entities are needed. The collision scale s is a supplied problem input, not a fit; shell constants are fixed bounds; the only nonstandard ingredients are domain assumptions making the latent-dictionary collision problem identifiable. The proof explicitly separates algebraic alignment from statistical lower bounds to avoid circularity.

axioms (7)

domain assumption Fixed-dimensional Gaussian train-test generative model with latent exchangeable sparse codes (equations 2.4–2.10)
The moment reduction, Ns^6 orientation gate, and all rates rely on the exact form Y_i = D*c_i + ξ_i and Z_ℓ ~ N(μ*, σ²I), with latent Bernoulli-Gaussian or exchangeable masked-Gaussian codes.
domain assumption Coherent block is an affine regular-simplex image at scale s (Assumption 4.2 and Lemma 4.1)
The child rays are written z_j = b + L v_j with L scaled by s. Every quotient-coordinate and physical-inverse result uses this affine-simplex chart; 'arbitrary coherent frames' are explicitly out of scope.
domain assumption Canonical positive-cap signs remove sign ambiguity, leaving only child-permutation nonidentifiability
Section 4.3–4.4 assumes sign conventions so that the physical target is unit atoms modulo sign and the remaining dictionary ambiguity is simultaneous permutation of the q children. If sign is not identifiable, the reported support set changes.
domain assumption Code-law nondegeneracy Δ_L > 0 (positive power on proper nonempty subdictionaries)
Assumption 4.2/equation (2.6) and Theorem 6.1(iv) show the cubic orientation score is nonzero exactly when Δ_L > 0. Without proper-subdictionary activation, orientation information vanishes and the method cannot resolve child orientation.
standard math Stabilizer of the simplex cubic tensor is exactly the permutation group, Stab_O(U)(T_q) = S_q
Proved in Technical detail S5 using constrained optimization of T_q[x,x,x]; it is used to invert residual orientation modulo child permutation.
standard math Arsenin–Kunugui measurable-projection theorem for Borel measurable random correspondences
Used in Lemma S11.2 to prove the raw profile has Borel graph and the reported correspondence is Borel measurable into K(V_q).
standard math Standard probability inequalities: median-of-means, Chebyshev, Hoeffding, Pinsker, log-sum, Cauchy–Schwarz
These underlie the certified training rectangle (S13), the coverage proof, and the minimax lower-bound distance-vs-Total-Variation inequality (Lemma 6.4).

pith-pipeline@v1.3.0-alltime-deepseek · 39328 in / 13934 out tokens · 140841 ms · 2026-08-01T19:51:35.031968+00:00 · methodology

0 comments

read the original abstract

Sparse-support uncertainty is usually quantified by treating the dictionary as known, an assumption that can produce overconfident, label-dependent conclusions when the dictionary is learned from latent sparse mixtures. Near collisions of coherent atoms, a test signal may identify the active physical group even though the training data cannot distinguish the physical rays within it. We develop inference for active physical rays, unit atoms modulo sign, after latent dictionary learning. In a fixed-dimensional Gaussian train-test experiment, we retain all dictionaries compatible with a robust training-moment region, profile the test representation over them, and project surviving configurations onto a permutation-invariant support space. The resulting confidence correspondence can report cross-sheet inconclusiveness, group resolution with child ambiguity, or fine-support resolution. We characterize both its statistical cost and decision-theoretic benefit. Residual block orientation first affects the latent training density at cubic order, yielding information of order $s^6$, where $s$ is the within-block collision scale. The correspondence provides high-probability-over-training conditional test coverage, with resolution governed separately by parent detectability, test-time support separation, and learned-dictionary orientation. In the resolved fixed-shell regime, its projective Hausdorff diameter contracts at the minimax-optimal rate $s \wedge (\sqrt{N}s^2)^{-1}$, up to constants. A restricted-task theorem further determines when coefficient asymmetry allows test replication to supplement training information and when calibration uncertainty remains irreducible. The framework thus yields honest, resolution-adaptive support statements and guides the allocation of training versus test measurements.

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 7 canonical work pages

[1]

Dictionary Identification---Sparse Matrix-Factorization via

Gribonval, R. Dictionary Identification---Sparse Matrix-Factorization via. IEEE Transactions on Information Theory , year =. doi:10.1109/TIT.2010.2048466 , url =

arXiv 2010
[2]

Journal of Machine Learning Research , year =

Wu, Siqi and Yu, Bin , title =. Journal of Machine Learning Research , year =
[3]

Sparse and Spurious: Dictionary Learning with Noise and Outliers , journal =

Gribonval, R. Sparse and Spurious: Dictionary Learning with Noise and Outliers , journal =. 2015 , volume =. doi:10.1109/TIT.2015.2472522 , url =

arXiv 2015
[4]

Jung, Alexander and Eldar, Yonina C. and G. On the Minimax Risk of Dictionary Learning , journal =. 2016 , volume =. doi:10.1109/TIT.2016.2517006 , url =

arXiv 2016
[5]

Proceedings of the 30th International Conference on Machine Learning , series =

Mehta, Nishant and Gray, Alexander , title =. Proceedings of the 30th International Conference on Machine Learning , series =. 2013 , volume =

2013
[6]

Journal of the Royal Statistical Society: Series B (Statistical Methodology) , year =

Meinshausen, Nicolai , title =. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , year =. doi:10.1111/rssb.12094 , url =

work page doi:10.1111/rssb.12094
[7]

Hierarchical Testing in the High-Dimensional Setting With Correlated Variables , journal =

Mandozzi, Jacopo and B. Hierarchical Testing in the High-Dimensional Setting With Correlated Variables , journal =. 2016 , volume =. doi:10.1080/01621459.2015.1007209 , url =

arXiv 2016
[8]

The Annals of Statistics , year =

Nickl, Richard and van de Geer, Sara , title =. The Annals of Statistics , year =. doi:10.1214/13-AOS1170 , url =

work page doi:10.1214/13-aos1170
[9]

Biometrics , year =

Li, Yang and Luo, Yuetian and Ferrari, Davide and Hu, Xiaonan and Qin, Yichen , title =. Biometrics , year =. doi:10.1111/biom.13024 , url =

work page doi:10.1111/biom.13024
[10]

Lewis, R. M. and Battey, H. S. , title =. Statistical Science , year =. doi:10.1214/24-STS934 , url =

work page doi:10.1214/24-sts934
[11]

2026 , eprint =

Organ, Sarah and Kenney, Toby and Gu, Hong , title =. 2026 , eprint =

2026
[12]

Journal of the American Statistical Association , year =

Sadinle, Mauricio and Lei, Jing and Wasserman, Larry , title =. Journal of the American Statistical Association , year =. doi:10.1080/01621459.2017.1395341 , url =

arXiv 2017
[13]

Classification with Valid and Adaptive Coverage , booktitle =

Romano, Yaniv and Sesia, Matteo and Cand. Classification with Valid and Adaptive Coverage , booktitle =. 2020 , volume =

2020
[14]

, title =

Cauchois, Maxime and Gupta, Suyash and Duchi, John C. , title =. Journal of Machine Learning Research , year =
[15]

Advances in Neural Information Processing Systems , year =

Kim, Jisu and Chen, Yen-Chi and Balakrishnan, Sivaraman and Rinaldo, Alessandro and Wasserman, Larry , title =. Advances in Neural Information Processing Systems , year =
[16]

, title =

Nath, Anirban and Hur, YoonHaeng and Allen, Genevera I. , title =. 2026 , eprint =

2026
[17]

, title =

van der Vaart, Aad W. , title =. 1998 , publisher =. doi:10.1017/CBO9780511802256 , url =

doi:10.1017/cbo9780511802256 1998
[18]

, title =

Kechris, Alexander S. , title =. 1995 , publisher =. doi:10.1007/978-1-4612-4190-4 , url =

doi:10.1007/978-1-4612-4190-4 1995
[19]

Sub-Gaussian Mean Estimators , journal =

Devroye, Luc and Lerasle, Matthieu and Lugosi, G. Sub-Gaussian Mean Estimators , journal =. 2016 , volume =. doi:10.1214/16-AOS1440 , url =

doi:10.1214/16-aos1440 2016
[20]

2023 IEEE 33rd International Workshop on Machine Learning for Signal Processing (MLSP) , year =

Hoppe, Frederik and Mayrink Verdun, Claudio and Laus, Hannah and Krahmer, Felix and Rauhut, Holger , title =. 2023 IEEE 33rd International Workshop on Machine Learning for Signal Processing (MLSP) , year =. doi:10.1109/MLSP55844.2023.10285912 , url =

arXiv 2023
[21]

Advances in Neural Information Processing Systems , year =

Hoppe, Frederik and Mayrink Verdun, Claudio and Laus, Hannah and Krahmer, Felix and Rauhut, Holger , title =. Advances in Neural Information Processing Systems , year =. doi:10.52202/079017-3894 , url =

work page doi:10.52202/079017-3894
[22]

and Blum-Smith, Ben and Kileel, Joe and Perry, Amelia and Niles-Weed, Jonathan and Wein, Alexander S

Bandeira, Afonso S. and Blum-Smith, Ben and Kileel, Joe and Perry, Amelia and Niles-Weed, Jonathan and Wein, Alexander S. , title =. Applied and Computational Harmonic Analysis , year =. doi:10.1016/j.acha.2023.06.001 , url =

doi:10.1016/j.acha.2023.06.001 2023
[23]

SIAM Journal on Mathematics of Data Science , year =

Ho, Nhat and Nguyen, XuanLong , title =. SIAM Journal on Mathematics of Data Science , year =. doi:10.1137/18M122947X , url =

work page doi:10.1137/18m122947x
[24]

Econometrica , year =

Kaji, Tetsuya , title =. Econometrica , year =. doi:10.3982/ECTA16413 , url =

work page doi:10.3982/ecta16413
[25]

Econometrica , year =

Chernozhukov, Victor and Hong, Han and Tamer, Elie , title =. Econometrica , year =. doi:10.1111/j.1468-0262.2007.00794.x , url =

arXiv 2007
[26]

, title =

Datta, Jyotishka and Ghosh, Soham and Polson, Nicholas G. , title =. 2025 , eprint =

2025

[1] [1]

Dictionary Identification---Sparse Matrix-Factorization via

Gribonval, R. Dictionary Identification---Sparse Matrix-Factorization via. IEEE Transactions on Information Theory , year =. doi:10.1109/TIT.2010.2048466 , url =

arXiv 2010

[2] [2]

Journal of Machine Learning Research , year =

Wu, Siqi and Yu, Bin , title =. Journal of Machine Learning Research , year =

[3] [3]

Sparse and Spurious: Dictionary Learning with Noise and Outliers , journal =

Gribonval, R. Sparse and Spurious: Dictionary Learning with Noise and Outliers , journal =. 2015 , volume =. doi:10.1109/TIT.2015.2472522 , url =

arXiv 2015

[4] [4]

Jung, Alexander and Eldar, Yonina C. and G. On the Minimax Risk of Dictionary Learning , journal =. 2016 , volume =. doi:10.1109/TIT.2016.2517006 , url =

arXiv 2016

[5] [5]

Proceedings of the 30th International Conference on Machine Learning , series =

Mehta, Nishant and Gray, Alexander , title =. Proceedings of the 30th International Conference on Machine Learning , series =. 2013 , volume =

2013

[6] [6]

Journal of the Royal Statistical Society: Series B (Statistical Methodology) , year =

Meinshausen, Nicolai , title =. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , year =. doi:10.1111/rssb.12094 , url =

work page doi:10.1111/rssb.12094

[7] [7]

Hierarchical Testing in the High-Dimensional Setting With Correlated Variables , journal =

Mandozzi, Jacopo and B. Hierarchical Testing in the High-Dimensional Setting With Correlated Variables , journal =. 2016 , volume =. doi:10.1080/01621459.2015.1007209 , url =

arXiv 2016

[8] [8]

The Annals of Statistics , year =

Nickl, Richard and van de Geer, Sara , title =. The Annals of Statistics , year =. doi:10.1214/13-AOS1170 , url =

work page doi:10.1214/13-aos1170

[9] [9]

Biometrics , year =

Li, Yang and Luo, Yuetian and Ferrari, Davide and Hu, Xiaonan and Qin, Yichen , title =. Biometrics , year =. doi:10.1111/biom.13024 , url =

work page doi:10.1111/biom.13024

[10] [10]

Lewis, R. M. and Battey, H. S. , title =. Statistical Science , year =. doi:10.1214/24-STS934 , url =

work page doi:10.1214/24-sts934

[11] [11]

2026 , eprint =

Organ, Sarah and Kenney, Toby and Gu, Hong , title =. 2026 , eprint =

2026

[12] [12]

Journal of the American Statistical Association , year =

Sadinle, Mauricio and Lei, Jing and Wasserman, Larry , title =. Journal of the American Statistical Association , year =. doi:10.1080/01621459.2017.1395341 , url =

arXiv 2017

[13] [13]

Classification with Valid and Adaptive Coverage , booktitle =

Romano, Yaniv and Sesia, Matteo and Cand. Classification with Valid and Adaptive Coverage , booktitle =. 2020 , volume =

2020

[14] [14]

, title =

Cauchois, Maxime and Gupta, Suyash and Duchi, John C. , title =. Journal of Machine Learning Research , year =

[15] [15]

Advances in Neural Information Processing Systems , year =

Kim, Jisu and Chen, Yen-Chi and Balakrishnan, Sivaraman and Rinaldo, Alessandro and Wasserman, Larry , title =. Advances in Neural Information Processing Systems , year =

[16] [16]

, title =

Nath, Anirban and Hur, YoonHaeng and Allen, Genevera I. , title =. 2026 , eprint =

2026

[17] [17]

, title =

van der Vaart, Aad W. , title =. 1998 , publisher =. doi:10.1017/CBO9780511802256 , url =

doi:10.1017/cbo9780511802256 1998

[18] [18]

, title =

Kechris, Alexander S. , title =. 1995 , publisher =. doi:10.1007/978-1-4612-4190-4 , url =

doi:10.1007/978-1-4612-4190-4 1995

[19] [19]

Sub-Gaussian Mean Estimators , journal =

Devroye, Luc and Lerasle, Matthieu and Lugosi, G. Sub-Gaussian Mean Estimators , journal =. 2016 , volume =. doi:10.1214/16-AOS1440 , url =

doi:10.1214/16-aos1440 2016

[20] [20]

2023 IEEE 33rd International Workshop on Machine Learning for Signal Processing (MLSP) , year =

Hoppe, Frederik and Mayrink Verdun, Claudio and Laus, Hannah and Krahmer, Felix and Rauhut, Holger , title =. 2023 IEEE 33rd International Workshop on Machine Learning for Signal Processing (MLSP) , year =. doi:10.1109/MLSP55844.2023.10285912 , url =

arXiv 2023

[21] [21]

Advances in Neural Information Processing Systems , year =

Hoppe, Frederik and Mayrink Verdun, Claudio and Laus, Hannah and Krahmer, Felix and Rauhut, Holger , title =. Advances in Neural Information Processing Systems , year =. doi:10.52202/079017-3894 , url =

work page doi:10.52202/079017-3894

[22] [22]

and Blum-Smith, Ben and Kileel, Joe and Perry, Amelia and Niles-Weed, Jonathan and Wein, Alexander S

Bandeira, Afonso S. and Blum-Smith, Ben and Kileel, Joe and Perry, Amelia and Niles-Weed, Jonathan and Wein, Alexander S. , title =. Applied and Computational Harmonic Analysis , year =. doi:10.1016/j.acha.2023.06.001 , url =

doi:10.1016/j.acha.2023.06.001 2023

[23] [23]

SIAM Journal on Mathematics of Data Science , year =

Ho, Nhat and Nguyen, XuanLong , title =. SIAM Journal on Mathematics of Data Science , year =. doi:10.1137/18M122947X , url =

work page doi:10.1137/18m122947x

[24] [24]

Econometrica , year =

Kaji, Tetsuya , title =. Econometrica , year =. doi:10.3982/ECTA16413 , url =

work page doi:10.3982/ecta16413

[25] [25]

Econometrica , year =

Chernozhukov, Victor and Hong, Han and Tamer, Elie , title =. Econometrica , year =. doi:10.1111/j.1468-0262.2007.00794.x , url =

arXiv 2007

[26] [26]

, title =

Datta, Jyotishka and Ghosh, Soham and Polson, Nicholas G. , title =. 2025 , eprint =

2025