Title: Non-Asymptotic Analysis Of Data Augmentation For Precision Matrix Estimation
Venue: NeurIPS 2025
Accepted: yes
Keywords: Data Augmentation

Precision Matrix

High Dimensional Statistics

Random Matrix Theory
Abstract: This paper addresses the problem of inverse covariance (also known as precision matrix) estimation in high-dimensional settings. Specifically, we focus on two classes of estimators: linear shrinkage estimators with a target proportional to the identity matrix, and estimators derived from data augmentation (DA). Here, DA refers to the common practice of enriching a dataset with artificial samples—typically generated via a generative model or through random transformations of the original data—prior to model fitting.
For both classes of estimators, we derive estimators and provide concentration bounds for their quadratic error. This allows for both method comparison and hyperparameter tuning, such as selecting the optimal proportion of artificial samples.
On the technical side, our analysis relies on tools from random matrix theory. We introduce a novel deterministic equivalent for generalized resolvent matrices, accommodating dependent samples with specific structure. We support our theoretical results with numerical experiments.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper considers the problem of estimating the inverse covariance matrix in a high-dimensional setting. Suppose that $X_1,\\dots,X_n$ are i.i.d. copies of some random vector $X\\in \\mathbb{R}^d$, where $n$ and $d$ are of the same order. The true covariance matrix is $\\Sigma = \\mathbb{E}XX^\\top$ and the goal is to estimate $\\Sigma^{-1}$ from the samples. Since the empirical estimator $\\hat{\\Sigma} = (1/n) \\sum_i X_i X_i^\\top$ could be close to being singular, the paper considers the resolvent $R(\\lambda) = (\\hat{\\Sigma} + \\lambda I)^{-1}$ and studies the error $E(\\lambda) = (1/d)\\Vert R(\\lambda) - \\Sigma\\Vert_F^2$. \n\nIt proposes an estimator for $E(\\lambda)$ and, under certain assumptions (such as Lipschitz concentration) on the distribution, derives a concentration inequality result for the estimator. This enables the use of the estimator as a surrogate objective for tuning $\\lambda$.\n\nIt also considers an estimator based on data augmentation. Let $g_1, \\dots, g_m$ be artificial samples generated from an auxiliary distribution depending on $X$. The augmented resolvent is $R_{aug}(\\lambda) = ((n+m)^{-1}(\\sum X_i X_i^\\top + \\sum g_i g_i^\\top)^{-1} + \\lambda I)^{-1}$ with the associated error $E_{aug}(\\lambda) = (1/d)\\Vert R_{aug}(\\lambda) - \\Sigma\\Vert_F^2$. Under assumptions on the auxiliary distribution, the paper develops an estimator $E_{aug}(\\lambda)$ and proves a concentration bound.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: These are interesting results to find a surrogate to optimal lambda in the resolvent. \n\nWeaknesses: It does not seem that authors have placed a lot of effort to optimize the bounds; the bounds are not trivial but they seem largely based on the recent progress in random matrix theory. (The dependence on some parameters look bad; see below)\n\nQUESTIONS:\n1. Proposition 1: What is B? Why is the right-hand side of the inequality independent of B? If I rescale B up by a large factor, the tail probability should be much smaller (depending on the factor)?\n2. Can authors comment on the dependence on eta in their results? Delta_X(lambda) has a dependence of 1/eta^6 and the tail has a dependence of eta^3, which do not look good for small eta.\n3. Can eta depend on n or d? For instance, can eta be at the order of 1/sqrt(n)?\n\nMinor points:\n- Line 16: reads better by changing it to “of a random mean-zero vector from i.i.d. Samples”\n- Line 79: isn’t E a measurable subset of E^{d x k}?\n- Equation (3): “tr” in “tr(R_X(lambda))” should be in upright font\n- Line 228: what is A?\n- Line 247: add a space before “we also”\n- Line 291: what is Equation (66)? It is bad for the main text to point to something in the appendix in a theorem statement.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-19"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe objective of this paper is to derive non-asymptotic guarantees for estimating the precision matrix (i.e., the inverse of the covariance matrix). The quality of the estimation is assessed using the Frobenius norm between the estimator and the true precision matrix.\n\nIn the first part of the paper, the precision matrix is estimated using a ridge-like estimator of the form $(C_X + \\lambda I)^{-1}$, and the associated risk, which depends on the regularization parameter $\\lambda$, is estimated. It is shown that this risk estimator increasingly approximates the true risk. Consequently, the parameter $\\lambda$ can be optimized—provided the estimated risk has a computable gradient—in order to improve the precision matrix estimation.\n\nThe second part of the paper investigates the effect of augmenting the data matrix $X$ with additional samples (e.g., Gaussian isotropic noise or dropout-based augmentations), and then applying the same estimation and optimization procedure to this enriched dataset.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\n* The paper provides a solid theoretical foundation, offering non-asymptotic results for the inverse of a random matrix—a notably challenging problem.\n* The use of deterministic equivalents and the companion Stieltjes transform reflects a modern approach in non-asymptotic random matrix theory.\n* Table 1 presents a broad set of data augmentation (DA) examples, and the stated assumptions are satisfied.\n* The proposed method includes a practical component, enabling gradient-based optimization of the estimated risk, which is fully computable.\n\n**Weaknesses**\n\n* The choice of the Frobenius norm as the error metric is not sufficiently justified. For instance, in some applications it might be more relevant to focus on estimation accuracy along the leading eigen-directions of $\\Sigma$.\n* The assumption $\\gamma = d/n < 1$ (required for Assumption H2) appears inconsistent with the introduction’s emphasis on the high-dimensional regime and the double descent phenomenon.\n* The assumptions in Section 2 are overly complex. Assumption H1, in particular, includes conditions both on the entire matrix $X$ and on its columns. These could be simplified by focusing only on column-level assumptions (e.g., i.i.d. Gaussian or bounded entries). Definition 1 seems unnecessary for the main narrative and could be moved to the appendix.\n* The term $\\Delta_\\lambda$, intended to represent a bound on the bias, is not clearly defined. Its formulation includes the same quantity in both the constant and the expression, making it difficult for the reader to identify the key parameters. Moreover, the homogeneity of the expression is not respected.\n* Section 3 is promising, but somewhat disappointing. While it hints at the potential to replace the regularization term $+\\lambda I$ with a data augmentation strategy that serves a similar role, this idea is not fully proved in the upper bound and the number of augmented samples is not optimized. Although the paper alludes to the implicit regularization of DA, it does not concretely leverage this concept.\n\n**Writing advices**\n\n* The manuscript may include an excess of theoretical detail (e.g., Proposition 1 and the subsequent discussion) without sufficient commentary or illustrative examples to aid understanding.\n* The choice $\\lambda_1 < \\cdots < \\lambda_d$ is unconventional (and somewhat inexacte due to multiplicity). \n\n**Typos (no response needed)**\n\n* $\\mathrm{tr}$ instead of $tr$ in Equation (3).\n*  $:=$ is not consistently used for definitions.\n* $\\bar\\Lambda$ not defined.\n\n**Summary**\n\nIn conclusion, while the paper is built on a solid mathematical foundation and uses modern tools, its overall narrative lacks clarity, which makes it difficult to follow. Nonetheless, the approach and results are compelling. I encourage the authors to clarify their exposition and would be open to revising my evaluation.\n\nAfter rebuttal period: borderline reject -> accept\n\nQUESTIONS:\nYou can responds on the weakness(soudness) bullet points to improve your rating.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 1, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper analyses common methods for estimating the inverse covariance of a matrix in high-dimensional settings. In particular it develops data-dependent estimates for the squared error of linear shrinkage and data augmentation estimators, and provides guarantees for these data-dependent estimates. It uses these empirical estimates to help tune certain hyper parameters used in covariance matrix estimation.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. The paper addresses the important problem of estimating inverse covariance matrices from a theoretical perspective.\n2. The paper focuses on the important subproblem of estimating the squared error of both kinds of inverse covariance estimators\n3. The paper establishes theoretical guarantees for these squared error estimators\n4. The paper’s brief empirical experiments help strengthen their claim about the reliability of their empirical estimators\n\nWeaknesses:\n1. Part ii of H3 severely limits the applicability of their result, and the work does not discuss how their results would change if parts of the statement were relaxed (such as the conditioning of Lambda_G(X))\n2. The Wasserstein distance part of H4 seems quite strong, it would be nice if the paper elaborated on the “weaker convergence guarantees” they mention below it.\n3. Most important: This is my biggest qualm. The theoretical results are not well thought out. Their expression uses both the operator norm of Sigma_X and lambda_1(Sigma_X), which are literally the same thing unless I am gravely mistaken. Further, the dependence of the error in squared-error estimation on d and n is not clear since the operator norm of a matrix can grow with d and n. This is further exacerbated by the missing expressions for C_1 and C_2 and no discussion about their growth with d and n. I am leaning reject primarily because the theory is not properly addressed, which is supposed to be the main strength of the paper in my eyes.\n4. While the paper is about non-asymptotic guarantees, it is important to have a sanity check where they recover some familiar or believable asymptotic guarantees under perhaps additional distributional assumptions on X.\n5. While experiments were perhaps not the main focus of this paper when it was being written, I think a paper addressing an estimation problem and submitted to an ML conference with claims about its utility in choosing hyperparameters should have more extensive experiments across multiple datasets. Unlike some theory papers, this paper’s theory is less about conceptual insight and more about estimating bounds. I think such papers need to have extensive experiments.\n\nQUESTIONS:\n1. Can you discuss the weaker convergences guarantees guarantees you promise when one relaxes H4?\n2. Can you perform an asymptotic analysis of your guarantees under some simplifying assumptions as a sanity check for your readers?\n3. Could you discuss the growth of your error in estimating the squared-error with respect to d and n?\n\nLIMITATIONS:\nMany have been discussed by the authors, I have discussed some other limitations that inform my decision to lean towards rejecting the paper above.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a theoretical and empirical study of inverse covariance (precision matrix) estimation in high-dimensional settings. It compares two classes of estimators: (1) standard linear shrinkage estimators and (2) estimators based on data augmentation (DA), including both transformative and generative DA schemes. The authors develop non-asymptotic concentration bounds for the quadratic error of these estimators, leveraging advanced tools from random matrix theory, including novel deterministic equivalents. They also provide practical estimation formulas and validate them using experiments on synthetic and real datasets (e.g., MNIST).\n\nSTRENGTHS AND WEAKNESSES:\nStrengths：\n1.\tThe paper makes a substantial contribution by offering a non-asymptotic framework for analyzing precision matrix estimation with DA, an area where prior work is sparse.\n\n2.\tIn this paper, the author gives a comprehensive and detailed prove to support the proposed claim.\n\n3.\tNumerical results support the claims, demonstrating the tightness of the theoretical bounds.\n\nWeaknesses：\n\n1.\tThe theorical contribution is substantial, but the definitions, lemma, theorem are too many resulting certain sections are notation-heavy. Is it possible to simplify some statement and make it more reader-friendly.\n2.\tThis paper is more about theory analysis and lack of practical demonstration. The experiment part need to consolidate. MNIST dataset is too simple to verify the theory is  consistent with real world scenario. This paper focus on high dimension data but MNIST data (28*28) is low dimension in reality. And also the experiment part lack comparison with prior key method.\n3.\tWhile the theory is well-documented, the experimental section lacks of code and implementation details. Hard to reproduce and understand the connection between theory and practice if lack of code.\n\nQUESTIONS:\nAny results on more complex datasets, besides the simple MNIST?\n\nLIMITATIONS:\nNo, the authors showed that the proposed method works on high-dimension settings, but the experiments were conducted on a very simple dataset (MNIST).", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-09"}