FasMe: Fast and Sample-efficient Meta Estimator for Precision Matrix Learning in Small Sample Settings
Authors: Xiao Tan
Yiqin Wang
Yangyang Shen
Dian Shen
Meng Wang
Peibo Duan
Beilun Wang
Venue: NeurIPS 2024 poster (2024)
Keywords: Precision Matrix Estimation
Meta Learning
Graphical Model
Small Sample
Primary area: probabilistic_methods

Abstract: Precision matrix estimation is a ubiquitous task featuring numerous applications such as rare disease diagnosis and neural connectivity exploration. However, this task becomes challenging in small sample settings, where the number of samples is significantly less than the number of dimensions, leading to unreliable estimates. Previous approaches either fail to perform well in small sample settings or suffer from inefficient estimation processes, even when incorporating meta-learning techniques.
To this end, we propose a novel approach FasMe for Fast and Sample-efficient Meta Precision Matrix Learning, which first extracts meta-knowledge through a multi-task learning diagram. Then, meta-knowledge constraints are applied using a maximum determinant matrix completion algorithm for the novel task. As a result, we reduce the sample size requirements to $O(\log p/K)$ per meta-training task and $O(\log\vert \mathcal{G}\vert)$ for the meta-testing task. Moreover, the hereby proposed model only needs $O(p \log\epsilon^{-1})$ time and $O(p)$ memory for converging to an $\epsilon$-accurate solution. On multiple synthetic and biomedical datasets, FasMe is at least ten times faster than the four baselines while promoting prediction accuracy in small sample settings.

FasMe: Fast and Sample-efficient Meta Estimator for Precision Matrix Learning in Small Sample Settings
Xiao Tan 1.3
Yiqin Wang' Yangyang Shen'
Dian Shen'
Meng Wang* Peibo Duan' Beilun Wang!*
Schooo of Computer Science and Engineering, Southeast University, Nanjing, China CCollege of Design and Innovation, Tongji University, Shanghai, China Department of Data Science & AI, Monash University, Melboume, Australia {xtan, 213200449, seu_syy, dshen, beilun)Oseu edu. on mengwangtj@tongg edu. peibo. duan@monash. edu
Abstract
Precision matrix estimation is ubiquitous task featuring numerous applications such as rare disease diagnosis and neural connectivity exploration. However, this task becomes challenging in small sample settings, where the number of samples is significantly less than the number of dimensions, leading to unreliable estimates. Previous approaches either fail to perform well in small sample settings suffer from inefficient estimation processes, even when incorporating meta-learning techniques. To this end, we propose novel approach FasMe for Fast and Sample- efficient Meta Precision Matrix Learning, which first extracts meta- weta-knowledde through multi-task learning diagram. Then, meta-knowledger constraints applied using a maximum determinant matrix completion algorithm for the novel task. As result, meta-training task and Ollog (Gl) for the meta-testing task. Moreover, the hereby proposed model only needs (p loggo" 11) time and O(p) memory for converging an c-accurate solution. On multiple synthetic and biomedical datasets, FasMe at least ten times faster than the four baselines while promoting prediction accuracy in small sample settings.
reduce the sample size requirements to O p/ K)) per
Introduction
Precision matrix estimation aims exploit conditional dependency relationships between random variables, which plays a vital role in high-dimensional statistical learning with a wide range of applications in areas such genetics [. neuroscience 22]. and social networks. For instance, researchers are examining the gene network derived from the gene expression data of patients to investigate a rare disease. However, cases involving rare diseases, the analysis is hindered by the high dimensionality of genetic datasets, where the number of features significantly exceeds the number of samples. For example, the Cholangiocarcinoma dataset TCGA contains only 51 samples but thousands of genes. In this context, we define "small sample setting" as scenario in which the number of samples less than one-tenth of the number of dimensions. Although inferring such relationship graphs in small sample settings is challenging, it can provide valuable insights into the genetic characteristics underlying this discase and its etiology.
Precision matrix estimator inherently has the capability to handle datasets where the number of dimensions p is equal to or greater than the number of samples [3. However, this capability has its
Corresponding Author.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

---

limits, and maintaining effectiveness requires keeping the sample size at a certain level. provides the lower bound of the sample size as [dd log for well-performed graph recovery under the sub-Gaussian assumption, where d is the maximum node degree of the graph. Consequently, when the sample size is reduced to the aforementioned "small sample" levels, the sample size falls short of this theoretical lower bound with a relatively high probability due to the uncertainty of d. This indicates insufficient data support for most methods, like GLasso [5] (detailed in Background), to recover the graph structure effectively, as shown in Figure [ To address this issue, some methods use multi-task learning, integrating heterogeneous datasets to reduce the sample size bound. For example, the regularized methods [6] and [7] show that each task only needs Olog K + log p) and O(K + log p) samples, respectively, for proper estimation, and K denotes the number of tasks. Although these methods are more capable of handling small sample problems, they concurrently introduce significant computational burdens. Specifically, integrating a new task necessitates repeated joint retraining, thereby resulting in substantial computational inefficiencies.
To address these challenges, meta-learning emerges as a promis- ing paradigm, as it equips models to efficiently learn a new task using minimal data through the application of nnta-e.noweedge. Such methods are particularly attractive for precision matrix es- timation in small sample settings, as they diminish the reliance on extensive data and reduce computational demands. For example, state-of-thee the-thhe-aae art approaches from the class of Model-
Dimension 200
Sample vample beunden 2
Frobemius Norm Ecror
6225
Agnostic Meta-Learning (MAML) [8 have achieved favorable performance in many meta-learning tasks. The strength of MAML lies in its ability quickly adapt new tasks (meta- testing dataset) with fewer samples by optimizing initial param- eters using data from related tasks (meta-training dataset).
100
150
220
Sample sizee
However, the design of MAML often emphasizes sample effi- ciency and fast adaptation, typically without comprehensive and robust theoretical guarantee for its performance. While empirical improvements are sufficient for many neural network applications, they may lead to incorrect predictions when ap- plied to precision matrix estimation, where precise and reliable results are crucial. Such empirical-only strategies typically rely on extensive training data. However, in precision matrix es- timation, the available data is often significantly limited. For example, the dropout technique (9] which lacks strict theo- retical foundation, is frequently used in neural network training with millions of samples. In contrast, in precision matrix estimation, the volume of available data often falls short of the number of feature dimensions. Furthermore, MAML-based methods adapt to new tasks by performing gradient descent from meta-learned parameters. but as noted by [10], the computation and memory demands escalate dramatically as the number of features increases, presenting significant challenges.
Figure 1: Frobenius norm error (L) and Matthews Correlation Coeffi- cient (MCC, 1) V.S. sample size us- ing Glasso. We fix p = 200 and vary from 10 to 400 with step 10. The model performance drops sharply around π = 20, indicating its inadequacy for small sample.
Recent studies 12 have made initial attempts to successfully integrate meta-learning with preci- sion matrix estimation. Despite achieving sample-efficiency through meta-learning, their methods exhibit several limitations: Restrictive Data Assumptions, and Ineffective Adaption New Tasks. Firstly, they assume that the edges of a new task must be subsets of existing edges (where existing edges can be seen optimized initial parameters in MAML), thereby constraining the discovery of novel connections in real-world applications. Besides, these works state that they use the state-of-the- art algorithm BigQuic, which speeds up the gradient calculation step with a block-coordinate descent Newton method, to solve new task. The algorithm uses between (p)) and O(p²) time and O(p²) and Opp) memory per iteration. When the number of samples is extremely small, it takes a large number of iterations to converge to an accurate solution, leading to poor generalization performance.
Herein, we propose a novel approach for Fast and Sample-efficien Meta Precision Matrix Learning, FasMe. Our approach first extracts meta-knowledge as the shared pattern across auxiliary tasks through a multi-task learning diagram. We then adopt maximum determinant matrix completion algorithm with meta-knowledge constraints to achieve fast adaptation in high-dimensional settings. In detail, our contributions can be summarized follows:
2

---

A novel meta-learning based model: We propose a multi- task precision matrix estimator that extracts the -nawleee from the auxiliary tasks. We then utilize the maximum determinant matrix completion algorithm to learn a new precision matrix in the meta-testing.
Efficient adaptation: We design a recursive closed-form algorithm to speed up the adaptation in the novel task. Our model converges to e-accurate solution in O(p log e time and (p) memory. It significantly outperforms the state-of-the-arr ones.
Theoretical guarantee: We also theoretically prove that for p-dimensional multivariate sub-Gaussian random vectors and K' auxiliary tasks with meta-knowledge 0, the sam- ple complexity of our method is Ollog p/ K) per auxiliary task for meta- weta-knowledge and (log (l) for the new task. It relaxes sample size requirements with the application of meta-learning, equipping FasMe with the ability to handle small sample settings.
Experimental evaluation: On multiple synthetic datasets, FasMe outperforms the other baselines. Specifically, FasMe is at least ten times faster than the four baselines while promoting prediction accuracy. In the real-world experiment, FasMe obtains the minimum log-determinant Bregman divergence and performs better than the other baselines.
2 Background
2.1
Meta-Learning
The last years have witnessed a tremendous interest in methods for meta-learning, especially in a wide range of data-limited applications including medical image analysis [13] [1], language
modeling [ [6], and object detection [7 18 Meta-learning is machine learning technique
that allows a model to learn from prior knowledge, or meta-knowledge, to improve its performance
The process typically involves two stages: 1) Meta-training: In this stage, the model is trained on set of related tasks, also known as auxiliary tasks B. The goal of this step is to extract meta- -etwllnooeeegg from these tasks, which can be used to improve the performance of the model on new tasks. This step helps the model to learn how to quickly and effectively adapt to new tasks by identifying the shared patterns or features across the auxiliary tasks [22] 2) Meta-testing: In this stage, the model is tested on new task, also known as the target task (23]. The goal of this step is to apply the meta- veeta-nnowlegge Iearned during the meta- meta-training step to improve the model's performance to quickly and effectively adapt the model to the new task, without the need for additional training data. Overall, meta-learning allows model to better generalize to new tasks by leveraging prior knowledge, thereby making it more sample-efficies and fast.
on new tasks and speed up the learning process 22 21
the target task. Meta-know knowledge is used
2.2 Precision Matrix Estimation
Precision matrix estimation is the problem of finding the inverse of the covariance matrix of a multivariate random variable f2 - It is particularly useful in scenarios where the number of samples is equal to or smaller than the number of variables since the covariance matrix is not invertible. The graphical lasso, namely GLasso [5) is a typical penalized maximum likelihood estimator for precision matrix Ω inference under the Gaussian assumption on the dataset X E TRa The popular estimator can be written as:
Ω = arg min log (ett(())+ < Ω. > +
(1)
8-0
where À 0, -.
X) - Σー X,, and Ω denotes that Ω is positive definite and symmetric. ||0|| is the -norr of matrix f2. However, [24 proved that this estimator is sensible even for non-Gaussian X, since corresponds to minimizing an érrreuularized log-determinant Bregman divergence. To solve the problem Eq. ( we take the derivative of it Q-1 Σ = xa / Inspired by this derivative, (25) proposed the constrained é minimization method for inverse matrix estimation (CLIME) estimator. This estimator can be used to estimate the precision matrix via an l constrained optimization:
arg migmin||s1pp Q--
s.t. | A|0 S A.
CLIME can be solved column-by-colnn boooummm Compared with Glasso, CLIME has presented more favorable theoretical properties and can be solved by column-wise linear programming.
3

---

3 Method
The section elaborates on how the proposed meta-learning framework works. We begin by providing formal problem setting in Section 3.1 which covers the task and data assumptions consider. Next, we formulate the two stages our meta-learning framework Section 3.2
3.1 Problem Setting
- (x() X ...... ,(()) denotes K datasets generated by K auxiliary tasks. We use ( E R to represent the k-th dataset with ni samples and p features. (2(k) e TRxx
Notations
represents the k-th precision matrix corresponding to (k), {2(k) { ..... (2()) } is the set of the precision matrices corresponding to the auxiliary data (x))).. new denotes the precision matrix of new task Xuew Rnxp to be estimated. We use 0 to represent the meta-knowledge (i.e., the common substructure (s2())) and 0%h)) to represent the rest of (2(A) respectively.
Assumptions We consider sub-Gaussian distributions, which cover Gaussian variables, bounded random variables, and so on. This relaxed data assumption significantly improves the flexibility and robustness of our method. We assume that the auxiliary tasks share some metaaknowledge with the target task Xnew- Specifically, assume that supp(0) C supp(Snew). Here the meta-knowledge / takes the form of common substructure among the tasks. The assumption has been widely adopted [26 27 28 and proved to be feasible and applicable in the biological and genetic domains [29].
general class of distributions than most previous studies, i.e.,
The paper aims to estimate the target precision matrix Qnew of sub-Gaussian distribution with sparse structure and relatively limited samples given sufficient samples from auxiliary tasks (x()), J. The auxiliary datasets follow a family of multivariate sub-Gaussian distributions (see Definition 2 in Appendix (F). To address the problem, we propose a novel meta-learning framework that leverages the meta keeta-knowwedge learned from auxiliary tasks efficiently estimate Qnew-
3.2 A Meta-learning Framework for Small Sample Precision Matrix Estimator
This part gives more details regarding the meta- meta-learning framework we established. FilureZillussrates the pipeline of the framework in an intuitive way.
Stage I: Meta-know ledge Extraction
Task
Brain
Task X
Cancer
hidden strecture
Iaput
Stage 2: Efficient Adaptation
Task 2
knowledge
kidda tructurs 2
8.8.2.2.
Meta- tracher
student
Precision Matrix
Graph Recovery
St strectuee
20002...
Common Edge
Audividual Edge
Edge
Added Eatry
Sample
Figure 2: The pipeline of the established framework. The entire pipeline can be divided into two stages: meta- kettlednowwedge extraction (meta- treennng and efficient adaptation (meta-testing). our meta-teacher extracts the neta-knowledge, namely the shared structure, from the related auxiliary tasks with sufficient samples. In Stage 2, we obtain a prior sparsity pattern from the knawlenoweedgee and target dataset with few samples. Then our meta-studed aims to recover the edge structure by solving matrix completion problem rapidly.
Stage

---

3.2.1 Meta-teacher
We extract the common substructure across the tasks as the meta-knowledge The goal of pre-training is to extract meta-knowledge 0 from auxiliary tasks (x()).
To obtain the meta-knowledge, every precision matrix is modeled as
n 0
where 0 E Rexp is the common substructure among all graphs and 0. E RPPP represents the rest for k-th graph.
We adopt & peenalizaaion for and 03*)) since they are expected to be sparse for better interpretability. Therefore, we write Eq. (3) into ||0]] + pll00{))11 |l,, where p > 0. The value of the hyper-panee parameter depends on the properties of the graphs. This hyper-parameter controls the difference of sparsity level between and 0 Concretely, with a smaller p. the shared part gets denser and the rest gets sparser.
Then we apply the formulation to Eq. ( thus obtaining the single-task recovering method:
(0, 0 arg min
|10|] + p|
| (Ip -1p-0p - S A.
0,0,
Here (k) represents the covariance matrix corresponding to ( This single-task method, however, ignores the inner relationship between the auxiliary tasks. To this end, we sum up all the single-task estimators and devise the optimization problem in the following form:
(0, (0.10(0))= arg min K K| + 0
100
11.
s.t. |022(00 (1p - -0.4))) S A, (5)
l=
where k = 1.2,..., K.
3.2.2 Meta-student
To speed up the estimation of Ωnew, we propose our efficient estimator, which is favorable even with a few samples of the new task available. Additionally, we obtain sparsity pattern from the meta-knowledge and target dataset with few samples.
Different from most fi-penalized methods, our estimator aims to estimate the precision matrix by solving Maximum Determinant Matrix Completion (MDMC) problem. MDMC is effective technique that aims to pick the unique maximum determinant completion (when it exists) for partially observed matrix from all the possible matrices 300
Consider a maximum determinant matrix completion problem for a covariance matrix with some partially observed entries:
Σ arg max log det Σ
s.t. Eig- Mij V(i.j) E,
(6)
where M represents the partially observed matrix, and the set g contains all the observed entries. Thus, the Lagrangian dual of this problem can be derived as:
Ω arg min log det Ω (Ω, M) +p 0000
s.t. Ω S
with first-order optimality condition [M) o 22-1 M, where the operator sign(-) computes the sign of the matrix and @ denotes the Hadamard product. Herein, the set Sp refers to the p x symmetric matrices with sparsity pattern G. Strong duality indicates straightforward relation back to the primal Σ = _ Note that while Σ is (in general) dense matrix, 52 is always sparse. Instead of solving the primal problem for dense matrix, we choose to solve the dual problem for a sparse matrix, which also satisfies the optimality condition.
To solve the precision matrix Qnew, the first step is to find reliable sparsity pattern. Previous works 311 obtain the pattern by thresholding the covariance matrix. However, (31] also proved the invalidity of such a method in the case of relatively small sample size, the covariance matrix
5

---

cannot be accurately estimated. To meet the challenge, our estimator combines the support set of the thresholded covariance matrix and meta-knowledge to obtain the sparsity pattern
= supp(0) Suupp(S,, (Encw))
(8)
Here, Sey denotes the element- soft-thresholding operator with parameter 1). Note that the problem 77 has a recursive closed-fome formf solution whenever the graph is chordal, the embedding operation is conducted to satisfy the chordal property. To be specific, the chordal embedding of G is represented as & and 7 is rewritten into the following formulation,
new arg min log det S2 + (Ω, 11g (2ine))
s.t. Riy = (i,j) E GgG.
(9)
0-0
where lgg is the projection operator from @o onto S i.e. by setting [T((A,)) 0 (i. $ E.
3.3 Optimization Algorithms
The learning steps of meta-knowledge extraction (5) is solved efficiently through a formulation of multiple independent sub-problems of linear programming, which can be accelerated in a parallel form. With the assistance of meta-knowledder we solve the optimization problem (9) using Newton Conjugate Gradient method. The key idea is construct an inner conjugate gradients loop solution to the Newton subproblem of an outer Newton's 's method. The detailed description, complexity analysis, and pseudo-code (Algorithm T of the full algorithm are shown in Appendix C
Theoretical Analysis
To present the analysis more concisely, we first define Evot == diag(E E(K) (ott)
θ == diag(O..... .). O = diag(o!").... ..... 0(*), Prot == diag(?? ..... ,..., = θ + O,,
x()) We assume that the precision matrix & belongs to the uniformity Xeot := diag(X()) class of matrices,
Kp Σ
u = (g, = Ω: Q 0, |||2|| S v. 11 I Φ. max LS
I (10)
Here q, v, ái are some constants, 0 S q < 1 and Ω (wij) ) Kpx Kp. s(K)) represents the sparsity level of Ω in the uniformity class. Note that the sparsity level is related to p without an analytic form of relationships between them.
Then some important conditions and definitions are stated as the following.
Exponential Tail Condition Suppose there exists constant 7 S 3. so that logg mi
& y and
E[exp(((X: - -100(((Xx,-)))))))) s C s oo, AM S 7, Vi E (l.....P)
(11)
where C' is a constant.
Irrepresentable Condition
There exists some (0, 1] such that
(\'')((())) S a-,
(12)
where n := - ② -1
(x) represents the Kronecker matrix product) and g = {1,..., [.....,]]?[99 p22
We define KT- [II (g)
-'lle oo- Note that l'gg is a +p) x (s +p) matrix indexed by vertex pairs,
where [g). We define new
min
min [ieewsiy and wot (cJ)Esupp(
[
(k.))ssupp())..))
Definition 1. (32 Given matrix M E SP, define GM ((i, Mij & 0} its sparsity pattern. Then M is called inverse-consistent if there exists a matrix E S" such that
M+N20 0, V(i.j) E GM Ny =0 0, (M +N E
(13)
Gas
The mutrix N is called inverse-consisent complement of M and is denoted by M(s), Furthermore, M is called sign-consistent if for every (i,j) E Grs, the (i, elements of M and (M + - have opposite signs.

---

Then we define the B a:) function with respect to the sparsity pattern G and scalar a > 0
a) max M=D
|| max s.t. M E Si and | Imax a, Me, -,, is inverse-consistent.
Here i = 2, n, Illmx denotes the norm, c.g., || max43e |Ayl.
The Appendix F provides all the detailed proofs of the lemmas, theorems, and corollaries.
4.1 Main Theorems
In this section, we mainly study the theoretical properties of the meta-knowledge 0 in (5) and the precision matrix &new in (7). The first theorem specifies probability lower bound of recovering the true meta-knowledge by our estimator in (5) for multiple random multivariate sub-Gaussian distributions.
Theorem 1. Suppose that Moot u and N Ex= Tug. When the variables are sab-Gaussian, the
tail condition holds. Let X = CoPV /log( /lo((K) Ky) where the constant Co = 27 (2 + To + + c
and constant To > 0. 1f wht > 4Tm, then we have
1. |10 0" 1000 - 8C022 / (o) + +2;
2. supp(0) = supp(0*):
with a probability greater than 1p mu where 2v is tuning parameter.
m.. Here is threshold estimator with 0.g = 0, 2 Tn).
According Theorem 1 if n1 n2
7i. sample complexity O
log K +oog
K per task is
sufficient for the recovery of the meta-knowledge. In most cases, O log Hlgge) + 0 Log since
K
p >> K.
The following theorem specifies probability lower bound of recovering a correct precision matrix Qnow by our estimator in 7 for a multivariate sub-Gua Gaussian distribution.
Theorem 2. Suppose we have recovered the true sparsity pallern G of family of p-dimensional random multivariate sub-Gut Sab-Gasssian distribution. Then for new task of multivariate sub- Gaussian distribution with the precision matrix New such that supp(0) supp(( and satisfying irrepre-
sentable condition, consider the estimator Qnew with = Cav
log 191
Then we have
log | n
11Qnew D2ee 1l00 S 2223v
(14)
with probability 1 p- where > O.
This theorem shows that n E Ollog 9 is sufficient for estimating correct precision matrix of the new task with our estimation. Then, efforts should be made to prove that the max-det matrix completion method guarantees the consistency of the support set of Qnew with G.
Theorem 3. g coincides with the sparsity patlern of the optimal solution Diew if the normal- ized matrix Enew D- DD-1/TT D-1/2 where D diag([Ig (Enew)) satisfies the following conditions:
1. Snew is positive definite and sign-consistent
2. We have
7- ( Enew)) / (Enew) D)s (Il (new )m
1 (g. (.@Suewwllar
min (k.leg
(15)
We set the diagonal elements of &new to and its off-diagonal elements between -1 and 1. The projection operation 1lg(-) leads to many zero elements in Enew. resulting in Enew being positive
7

---

definite or even diagonally dominant in most cases. As shown by [32] when 9 induces an acyclic structure, condition 2 is automatically satisfied by condition 1. More generally, [33] shows that Vnew sign-consistent if (Enew + Siew> - is close to its first order Taylor expansion. This assumption holds in practice due to the fact that the magnitude of the off-diagonal elements of Enow + Site) -new
is small. Moreover, [32 shows that the left side of (50) is upper bounded by a ||_ llo Ia for some a > O. That implies that when [|Enew is small, or equivalently 7) is large, condition 3 is automatically satisfied. If the conditions in Theorem 3 are satisfied, the support set of Qnew is
consistent with G according to (7).
5 Related Work
There have been several methods proposed to estimate precision matrices, among which, three state-
of-the-art methods, QUIC [TO], Neighborhood Selection 34 Meta-IE [1]] and gRankLasso [35)
are chosen as baselines in the experiments (See reasons in Appendix E). These approaches all
consider nigh-dimensional settings which are similar
work. QUIC: This approach utilizes quadratic approximation of the negative log-likelihook function to estimate sparse inverse covariance matrices to accelerate computation speed, which reduces the computational cost from O(p²) to (p) at each iteration. Neighborhood Selection: This method is a type of regularized maximum likelihood estimation method for estimating precision matrices. Given some i.i.d. observations, it estimates the conditional independence restrictions separately for each node in the graph. The complexity of the neighborhood selection for one node with the Lasso (nnp min (n.p) p}) using LARS algorithm [36.. Meta-IE (BigQuic): [TT] first proposes a meta-lcarning- based method that is composed of two steps. First, Meta-IE takes the support union of all the precision matrices of the auxiliary tasks as prior knowledge. Then the method states that the testing step can be solved by an interior point method (37 in polynomial time Op log c!! or BigQuic [8] with better time complexity around Opp log e (See analysis in Appendix). gRankLasso: gRankLasso [35] is a method for sparse precision matrix estimation that realizes automatic parameter tuning, making it completely tuning-free. It achieves robustness and accuracy by integrating a group penalty with Lasso regularization, effectively estimating sparse structures in high-dimensional data.
6 Experiment
6.1 Synthetic Experiment
Baselines:
We compare our method with the following baselines mentioned in Section 5 1) The QUIC baseline, 2) the Neighborhood Selection (NS) baseline, 3) Meta-IE (BigQuic) baseline that is accelerated with 32 threads and 4) gRankLasso baseline.
Metric: We use the Frobenuis- norm ||Se Diew IF and Mathews correlation coefficient (MCC) as metrics to evaluate the performance. MCC is widely used in machine learning as a measure of binary classifiers, defined as MCC = (TP x TN- FP x FN) / / (TP + FPP + FN) + FPN + FN). Here the true positive (TP), true negative (TN), false positive (FP), and false negative (FN) values indicate the number of true non-zes zero entries, true-zero entries, false non-zero entries, and false zero entries, respectively. It produces high score if the classifier generates desirable estimations.
Simulated Datasets: Each graph model contains a new task and K prior learning tasks. The sample matrices follow multivariate normal distributions where the corresponding precision matrices are generated using two sparse models: (I) Random graph and (2) Tree graph. More details about the data simulation are moved into Appendix B
Time Comparison The time-complexity experiment on simulated datasets is conducted observe the changes of all the estimators in computation time with the varying p and As shown in Fig. B and Table D 2 FasMe performs better than all the baselines. Using our method, we solve a sparse inverse covaniance estimation problem containing 2500 variables, with time cost around second. By comparing all the subfigures Fig. 3 we find that all the methods except the neighborhood selection approach are insensitive to the change of n. This is because the neighborhood selection deals with the sample matrix X new R directly instead of the covariance matrix Enew E RRP*,, However, it remains computationally costly compared to FasMe when the sample size is much
8

---

150
VS. Time
P220
2002 vs. Time
p/10
(c) p VS. Time B R/5
(d) P. VS. Time
time sec
1.00
time sec
200
be Seleiie
100-
50
100
100
1500 dime (p)
2000 dimension (p)
dimension (p)
dimension (p)
Figure 3: Time cost of FasMe vs. baselines on simulated datasets with the feature dimension p varying in (500, 1000, 1500, 2000, 2500). Subfigure (a) records the time required for each method to be implemented on a series of datasets with different sample sizes n = p/20, p/10, p/5, p. respectively. Note that the missing points of the baselines mean the time cost is out of range.
smaller than the number of features. Interestingly, despite Meta-IE (BigQuie) method can speed up by parallelizing multiple threads, it is less time-efficient because of the time assumption of the I/O reads and writes. Moreover, BigQuic cannot converge within the maximum iterations (maxiter = 100), especially when only few samples are provided (n p/20, p/ p/1 Regarding gRankLasso, its tuning mechanism requires a substantial amount of time, leading to significant time costs in high- dimensional scenarios. In contrast, method converges significantly faster within 20 iterations, regardless of small sample setting.
Table I: Comparison estimation error (in terms MCC and F-norm= SS running time (seconds) on synthetic dataset. Q is generated from Random graph. time exceeds 30 minutes.
Deew PP) and means that the
p00,p
1000
100,p
200.p F-nomz7])
Methods
MCCTT)
F-oorm)))
Timel)
MCCC1
F-nommr.)
lime(.)
MCC())
F-sormt))
(imel))
MICC11
Time(
QUIC
0.013
23.02
3,028
0.085
22.46
3.676
0.039
39.87
61.062
0.065
31.65
22.020
NS
0.287
66.08
8.724
0.438
54.21
11.829
0.388
88.98
43.003
0.624
67.02
87.193
Men-1F. (BigQuic)
0.403
23.10
27.370
0.460
22.42
16.655
0.251
41.92
96. 96.850
0.310
32. 10
87.140
gRankLans
0.008
198 36
0.012
176.00
0.005
218.12
0.008
212.43
Cuis
"0.650
22
0/661
16108
T6.37
0.005
66668
26.61
0.'10
Table 2: Comparison of estimation error (in terms of MCC and F-norm= running time (seconds) on synthetic dataset. L2 is generated from Tree graph.
S new
D Dnew lI and
50,
1000
1000
100
2000
200. 2000
Methods
MCCIT)
F-aoron())
Timel))
MCC(1
F-moon())
Time(.)
MCCXT)
F-oorm)))
Time))
MCC(T)
F-nomx()) Time(()
QUIC
3.212
35:07
58.972
0,306
66.002
8.701
0.485
11.147
0..30
87.87
0.655
06.08
89.237
Mets-E (BigQuic)
0.451
28.61
26.321
0.482
20.88
17,052
0.905
30.07
00 00.780
0.340
31.52
85.176
gRankLasso
0.007
201
0.013
185.60
0.000
223.45
0.008
200.77
Guns
0.668
22.05
0.058
6.112
15.98
0.065
0.681
24.24
1.002
0.745
24.2
0.806
Accuracy Comparison We conduct several experiments on the simulated datasets to compare the prediction power of all methods with four different small sample settings < p/10. 10. Table 2 indicates that FasMe obtains lower Frobenius-ne norrm and higher MCC under all the conditions. suggests that FasMe outperforms the other baselines in high-dimensional settings. The comparison of the two tables shows that FasMe can effectively recover the special structure from the data. In addition, the MCC of Meta-IE becomes small as the feature dimension increases. This is because the predicted edges must be a subset of the support union. As p increases, the estimation of the support union is not relatively accurate, thus significantly reducing the prediction accuracy of Meta-IE. The poor performance of gRankl Lasso asso can be attributed to the inefficacy of its automatic tuning mechanism in small sample settings, which degrades the model performance. We draw one subnetwork for the experimental result of every method and compare them with the ground truth in Fig. 4 (b). They show that our method bears the highest similarity with the ground truth. The results consistent with the MCC value of every method shown in Table &&2
6.2 Real-world Experiment: Application to Gene and fMRI Data
We further evaluate our method for estimating precision matrices on two real-world datasets: ChIP- Seq dataset (ENCODE project (39) and fMRI dataset (OpenfMRI project 40), accession number: ds000002). Regarding the ChIP-Seg dataset, aim to exploit the gene network of transcription

---

factors (TFs). We randomly selected 300 gene features. 27 samples of HI-hESC (embryonic stem cells) and GM12878 (Blymphocyte) are chosen as two auxiliary tasks for meta- weta-knowledge and 20 samples of K562 (leukemia) are chosen as novel task. Regarding the fMRI dataset, we aim to exploit brain connectivity among different brain regions. We randomly selected 200 regions of interest (ROI) as features. 34 samples of PCL and DCL datasets are chosen as the auxiliary tasks for meta weta-knnoledde and 20 samples Mixed-Event dataset are chosen as a novel task. Additional experimental details including configuration choices and extended experiments in the economic domain are provided in Appendix
(n Rendaes Goople
Groedd Truihs
Geople
Bralk CounEdReN
Figure 4: Subfigure (a)(b) display graph recovery results on two synthetic datasets for different methods compared with the ground truth. Subfigure (c) shows the gene network predicted by our proposed method for 300 genes on ChIP-Seq dataset. Subfigure (d) shows the brain connectome recovered by our proposed method for 200 regions on fMRI dataset. Positive and negative correlations are represented by orange and green edges, respectively.
In Subfigure (c) of Fig. 4 we visualize the experimental results on the two real-world datasets respectively. The predicted gene network exhibits scale-free and clustering behaviors, which is consistent with its biological properties. In another experiment, the majority of brain connections are found in the left ROIs. Besides, the left brain has positive connections, while the cross-hemisphere ones are negative. this case, the subjects are asked to solve a classification problem, which mainly relies on their left brain's 's analysis function. The results align with our expectations.
7 Conclusion
In this work, we introduced FasMe, a Fast and Sample-efficient Meta Estimator designed to address the challenges of precision matrix learning in small sample settings. Under novel meta-learning- based framework, our approach first leverages multi-task precision matrix estimator to extract shared meta-knowledge from auxiliary tasks, enabling efficient adaptation to tasks with minimal data. FasMe then incorporates a maximum determinant matrix completion strategy to enhance precision matrix estimation, ensuring both computational efficiency and theoretical robustness. Our experiments on synthetic and real-world datasets demonstrate that FasMe significantly outperforms state-of-the-an baselines in terms of accuracy and speed. These results highlight the potential of FasMe address real-world challenges in high-dimg ditddmmnniionlll settings, such genetics and neuroscience, where data is scarce. Moving forward, we plan to extend our work by relaxing the sub- Gab-sanssian data assumption and further reducing the sample size requirements for training meta- meta--nooleedee. This will extend the framework other domains and improve scalability for even larger datasets.
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Numbers 61906040, 61972085, 62276063, 62272101, 6509009710), the National Key Research and Development Program of China (Grant Number 2022YFF0712400 12400,, the Natural Science Foundation of Jiangsu Province (Grant Number BK20221457, BK20230083), the Fundamental Research Funds for the Central Universities (Grant Number 2242021R41177) and SIP Support-startup funding (Grant Number MSRI8001004).
10

---

References
[1] T. TONY CAI, HONGZHE LI, WEIDONG LIU, and JICHUN XIE. Covariate-adjusted precision matrix estimation with application in genetical genomies. Biometrika, 100(1): 133 156, 2013.
[2] Azam Kheyri, Andrictte Bekker, and Mohammad Arashi. High-dimensional precision matrix estimation through gsos with application in the foreign exchange market. Mathematics, 10(22), 2022.
[3] Jianqing Fan, Yuan Liao, and Han Liu. overview of the estimation of large covariance and precision matrices. The Econometrics Journal, 19(1)C11C33, 2016.
[4] Pradeep Ravikumar, Martin J Wainwright, and John Lafferty. High-dimensional ising model selection using f f1-regularized logistic regression. The Annals of Statistics, 38(3):1287-1319, 2010.
[5] Ming Yuan and Yi Lin. Model selection and estimation in the gaussian graphical model. Biometrika, 94(1):19-35, 2007.
[6] Jean Honorio, Tommi Jaakkola, and Dimitris Samaras. On the statistical efficiency of l.p multi-task learning of gaussian graphical models. arXiv preprint arXiv:1207.4255. 2012.
[7] Jing Ma and George Michailidis. Joint structural estimation of multiple graphical models. The Journal of Machine Learning Research, 17(1):5777-5824, 2016.
[8] Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model-agnostic meta-learning for fast adap- tation of deep networks. In International conference on machine learning, pages 1126-1133 1135. PMLR, 2017.
[9] Federico Girosi, Michael Jones, and Tomaso Poggio. Regularization theory and neural networks architectures. Neural computation, 7(2):219-269, 1995.
[10] Cho-Jui Hsieh, Mátyás A Sustik, Inderjit S Dhillon, Pradeep Ravikumar, et al. Quic: quadratic approximation for sparse inverse covariance estimation. Mach. Learn. Res., 1511:2911 2014.
[11] Qian Zhang, Yilin Zheng, and Jean Honorio. Meta learning for support recovery in high- dimensional precision matrix estimation. International Conference on Machine Learning, pages 12642-12652. PMLR, 2021.
[12] Huiming Xie and Jean Honorio. Meta learning for high-dimensional ising model selection using l l-regularized logistic regression. arXiv preprint 2rXiv:22088.05533 2022.
[13] Gabriel Maicas, Andrew P Bradley, Jacinto C Nascimento, Ian Reid, and Gustavo Carneiro. Training medical image analysis systems like radiologists. In International Conference on Medical Image Computing and Computer-Assisted Intervention, pages 546-554. Springer, 2018
[14] Smart task design for meta learning medical image analysis systems: Unsupervised, weakly- supervised, and cross-domain design of meta learning tasks. In Hien Van Nguyen, Ronald Summers, and Rama Chellappa. editors, Meta Learning With Medical Imaging and Health Informatics Applications, pages 209- Academic Press, 2023.
[15] Po-Sen Huang, Chenglong Wang, Rishabh Singh, Wen-tau Yih, and Xiaodong He. Natural language structured query generation via meta-learning. arXiv preprint arXiv: arXiv:1803.00 1803.02400, 2018.
[16] Yanda Chen, Ruiqi Zhong, Sheng Zha, George Karypis, and He He. Meta-learning via language model in-context tuning, 2021.
[17] Guangting Wang, Chong Luo, Xiaoyan Sun, Zhiwei Xiong, and Wenjun Zeng. Tracking by instance detection: meta-learning approach. In Proceedings of the IEFF/CVF conference on computer vision and pattern recognition, pages 6288-6297 2020.

---

[18] Xiongwei Wu, Doyen Sahoo, and Steven C. H. Hoi. Meta- (eta-(renn)> Meta learning for few-shot object detection, 2020.
[19] Yu-Xiong Wang. Deva Ramanan, and Martial Hebert. Meta-learning detect rare objects. In 2019 IEEE/CVE International Conference Computer Vision (ICCV), pages 9924-9933,
2019.
[20] Ricardo Vilalta and Youssef Drissi. A perspective view and survey of meta-learning. Artificial intelligence review, 18:77-95, 2002.
[21] I. Hospedales, A. Antoniou, P. Micaelli, and A. Storkey. Meta-learning in neural networks: A survey. IEEE Transactions on Pattern Analysis & Machine Intelligence, 44(09):519-5119, sep 2022.
[22] Timothy Hospedales, Antreas Antoniou, Paul Micaelli, and Amos Storkey. Meta-learning in neural networks: A survey. IEEE transactions on pattern analysis and machine intelligence, 44(9):5149-5163 2021.
[23] Joaquin Vanschoren. Meta-learning survey, arXiv preprint 1rXx::8110003548 2018.
[24] Pradeep Ravikumar, Martin J Wainwright, Garvesh Raskutti, and Bin Yu. High-dimensional covariance estimation by minimizing & ei-penalized log-determinant divergence. Electronic
Journal of Statistics, 5:935-980, 2011.
[25] Tony Cai, Weidong Liu, and Xi Luo. A constrained & minimization approach sparse precision matrix estimation. Journal of the American Statistical Association, 106(494):594-607,, 2011.
[26] Wonyul Lee and Yufeng Liu. Joint estimation of multiple precision matrices with common structures. The Journal of Machine Learning Research, 16()):1035-102, 2015.
[27] Satoshi Hara and Takashi Washio. Learning a common substructure of multiple graphical gaussian models. Neural Networks, 38, 2013.
[28] Weidong Liu. Structural similarity and difference testing on multiple sparse gaussian graphical models. 2017.
[29] Data Coordinating Center Burton Robert 67 Jensen Mark 53 Kahn Ari 53 Pihl Todd Pot David 53 Wan Yunhu 53 and Tissue Source Site Levine Douglas A 68. The cancer genome atlas pan-cancer analysis project. Nature genetics, 45(
2013.
[30] Lieven Vandenberghe, Stephen Boyd, and Shao-Po Wu. Determinant maximization with linear matrix inequality constraints. SIAM journal matrix analysis and applications, 19(2):499-533 1998.
[31] Peter Bickel and Elizaveta Levina. Covariance regularization by thresholding. The Annals of statistics, 36())25577-2644 2008.
[32] Salar Fattahi and Somayeh Sojoudi. Graphical lasso and thresholding: Equivalence and closed- form solutions. Journal of machine learning research, 2019.
[33] Somayeh Sojoudi. Equivalence of graphical lasso and thresholding for sparse graphs. The Journal of Machine Learning Research, 17(1):3943-366, 2016.
[34] Nicolai Meinshausen and Peter Bühlmann. High-dimensional graphs and variable selection with the lasso. The annals of statistics, 34(3):1436-1465 1462, 2006.
[35] Chau Tran and Guo Yu. A completely tuning- free and robust approach to sparse precision matrix estimation. International Conference on Machine Learning, pages 21733-21750. PMLR, 2022.
[36] Bradley Efron, Trevor Hastie, Jain Johnstone, and Robert Tibshirani. Least angle regression. The Annals of statistics, 32(2):407-499 2004.
[37] Stephen Boyd and Lieven Vandenberghe. Convex optimization, Cambridge university press, 2004.
12

---

[38] Cho-Jui Hsieh, Mátyás A Sustik, Inderjit S Dhillon, Pradeep K Ravikumar, and Russell Poldrack. Big & quie: Sparse inverse covariance estimation for a million variables. Advances in neural information processing systems, 26, 2013.
[39] ENCODE Project Consortium al. An integrated encyclopedia of dna elements in the human genome. Nature, 489(7414):57, 2012.
[40] Russell A Poldrack, Deanna M Barch, Jason P Mitchell, Tor D Wager, Anthony D Wagner, Joseph T Devlin, Chad Cumba, Oluwasammi Koyejo, and Michael P Milham. Toward open sharing of tusk-bases finri data: the openfmri project. Frontiers in neuroinformatiiss 7:12, 2013.
[41] Riten Mitra, Peter Müller, Shoudan Liang, Lu Yue, and Yuan Ji. A bayesian graphical model for chip-seq data on histone modifications. Journal of the American Statistical Association, 108(501):69- 80, 2013.
[42] Scott M Lundberg, William B Tu, Brian Raught, Linda 2 Penn, Michael M Hoffman, and Su-In Lee. Learning the human chromatin network from all encode chip-seg data. bioRxiv, page 023911, 2015.
[43] Felicia SL Ng. David Ruau, Lorenz Wernisch, and Berthold Göttgens. A graphical model approach visualizes regulatory relationships between genome-wide transcription factor binding profiles. Briefings in bioinformatics, 19(1):162-177) 2018
[44] Hantao Shu, Jingtian Zhou, Qiuyu Lian, Han Li, Dan Zhao, Jianyang Zeng, and Jianzhu Ma. Modeling gene regulatory networks using neural network architectures. Nature Computational Science, 1(7):491-501 2021.
[45] Michael VanInsberghe, Jeroen van den Berg, Amanda Andersson-Rolf, Hans Clevers, and Alexander van Oudenaarden. Single-cell ribo-seq reveals cell cycle-dependent translational pausing. Nature, 597(7877):561-565, 2021.
[46] Yuanyuan Qu, Xiaohui Wu, Aihetaimujiang Anwaier, Jinwen Feng, Wenhao Xu, Xiaoru Pei, Yu Zhu, Yang Liu, Lin Bai, Guojian Yang, et al. Proteogenomic characterization of mit family translocation renal cell carcinoma. Nature Communications, 13(1):7494, 2022.
[47] Praneet Chaturvedi, Yaseswini Neelamraju, Waqar Arif, Auinash Kalsotra, and Sarath Chandra Janga. Uncovering rma binding proteins associated with age and gender during liver maturation. Scientific reports, 5(1):9512, 2015.
[48] Srikanth Ryali, Tianwen Chen, Kaustubh Supekar, and Vinod Menon. Estimation of functional connectivity in fmri data using stability selection-based sparse partial correlation with elastic net penalty. Neurolmage, 59(4):3852-388 2012.
[49] Xi Luo. A hierarchical graphical model for big inverse covariance estimation with an application to fmri. arXiv preprint arXiv: 1403:v4440.44600 4698, 2014.
[50] Thaddeus Robert Sulek. An application of graphical models to fMRI data using the lasso penalty. PhD thesis, University of Georgia, 2017.
[51] Jongik Chung. Brooke S Jackson, Jennifer E Mcdowell, and Cheolwoo Park. Joint estimation and regularized aggregation of brain network in fmri data. Journal of Neuroscience Methods, 364: 109374, 2021.
[52] Martin S Andersen. Joachim Dahl, and Lieven Vandenberghe. Logarithmic barriers for sparse matrix cones. Optimization Methods and Software, 28(3):396-423, 2013.
[53] Hossein Keshavarz, George Michaildiss, and Yves Atchadé. Sequential change-pont -ointt detection in high-dimensional gaussian graphical models. Journal of machine learning research, 21(2))
57, 2020.
[54] Mitchell Krock, William Kleiber, and Stephen Becker. Nonstationary modeling with sparsity for spatial data via the basis graphical lasso. Journal of Computational and Graphical Statistics, 30(2):375-38, 1021.
13

---

[55] Aaron J Molstad. Wei Sun, and Li Hsu. A covariance-enhanced approach to multi-tissue joint equl mapping with application transcriptome-wiie association studies. The annals of applied statistics, 15(2):998, 2021.
[56] Jun Ho Yoon and Seyoung Kim. Eiglasso: Scalable estimation of cartesian product of sparse inverse covariance matrices. In Conference on Uncertainty in Artificial Intelligence, pages 1248- 1257. PMLR, 2020.
[57] Seongoh Park, Xinlei Wang, and Johan Lim. Estimating high-dimensional covariance and preci- sion matrices under general missing dependence. Electronic Journal of Statistics, 15(2))4868- 4915, 2021.
[58] Tao Liu, Ziyuan Yang. Armando Marino, Gui Gao, and Jian Yang. Polsar ship detection based on neighborhood polarimetric covariance matrix. IEEE Transactions on Geoscience and Remote Sensing, 59(6):4874 4887, 2020.
[59] Zitong Wan, Rui Yang, Mengjie Huang, Weibo Liu, and Nianyin Zeng. Eeg fading data classification based on improved manifold learning with adaptive neighborhood selection. Neurocomputing, 482:186 196, 2022.
[60] Boxin Zhao, Percy S Zhai, Samuel Wang, and Mladen Kolar. High-dimensions func- tional graphical model structure learning via neighborhood selection approach. arXiv preprint D.Xiv:2/05502448: 2021.
[61] Yan Zhu, Cangzhi Jia, Fuyi Li, and Jiangning Song. Inspector: a lysine succinylation predictor based on edited nearest-neighhor undersampling and adaptive synthetic oversampling. Analytical biochemistry, 593:1 13592, 2020.
[62] Joachim Dahl, Lieven Vandenberghe, and Vwani Roychowdhury. Covariance selection for nonehordal graphs via chordal embedding. Optimization Methods & Software, 2314):501-520, 2008.
[63] Martin S Andersen, Joachim Dahl, and Lieven Vandenberghe. Implementation of nonsym- metric interior-point methods for linear optimization over sparse matrix cones. Mathematical Programming Computation, 2(3-4):167-2011 2010.
[64] Lieven Vandenberghe, Martin S Andersen, al. Chordal graphs and semidefinite optimization. Foundations and Trends@ in Optimization, 1(4):241-433 2015.
14

---

NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification:
We summarize the paper's contributions and scope in lines 85-99.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or
NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
is fine to include aspirational goals as motivation long is clear that these goals are not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification:
Due to the space limit, we include the discussion about the limitation in
Appendix C
Guidelines:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create a separate "Limitations" section their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution is low images are taken in low lighting. Or speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.
While the authors might fear that complete honesty about limitations might used by reviewers as grounds for rejection, worse outcome might be that reviewers discover limitations that aren't 't acknowledged judgment and recognize that individual actions in favor of transparency play impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
the paper. The authors should use their best
3. Theory Assumptions and Proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
15

---

Justification:
We provide the assumptions, theorems and the corresponding proofs in Ap-
pendix F
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated or referenced the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide short proof sketch provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental Result Reproducibility
Does the paper fully disclose all the information needed to reproduce the main ex- perimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
Question:
Answer: [Yes]
Justification: Yes, we provide all the experimental details, including data generation methods, hyperparameter selection, and the full optimization algorithm.
Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is dataset and/or model, the authors should describe the steps taken make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is novel architecture, describing the architecture fully might suffice, or if the contribution a specific model and empirical evaluation, it may be necessary either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of large language model), releasing of a model checkpoint, or other means that appropriate to the research performed.
While NeurlPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) the contribution is primarily a new algorithm, the paper should make clear how reproduce that algorithm. (b) If the contribution is primarily new model architecture, the paper should describe the architecture clearly and fully.
(c) the contribution is a new model (e.g., a large language model), then there should either be way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source dataset instructions for how to construct the dataset).
(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome deseribe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should possible for other researchers to have some path to reproducing or verifying the results.
5. Open access to data and code
16

---

Does the paper provide open access to the data and code, with sufficient instruc- tions faithfully reproduce the main experimental results, as described in supplemental material?
Question:
Answer: [Yes]
Justification: We have uploaded the relevant code on our GitHub (https://gitte com/ MahjongGode Saki/FasMe).
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurIPS code and data submission guidelines (https://nips... cc/
for more details.
While encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https
for more details.
The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. only subset of experiments are reproducible, they should state which ones are omitted from the script and why.
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing much information possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code permitted.
6. Experimental Setting/eetails
Question: Does the paper specify all the training and test details (c.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Answer: [Yes]
Justification: Yes, we provide comprehensive documentation on all training and testing de- tails, including data generation methods, hyperparameter selection, reasons for configuration choices, and the complete optimization algorithm.
Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented in the core of the paper to a level of detail that is necessary appreciate the results and make sense of them.
The full details can be provided either with the code, appendix, or as supplemental material.
7. Experiment Statistical Significance
Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Answer: [Yes]
Justification: We provide the error bound of our estimators in Theorem 2 and 3 of the main page.
Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results are accompanied by emoo bars, confi- dence intervals, or statistical significance tests, least for the experiments that support the main claims of the paper.
17

---

The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
The assumptions made should be given (e.g.. Normally distributed errors).
It should be clear whether the error bar is the standard deviation the standard error of the mean.
It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of errors is not verified.
For asymmetric distributions, the authors should be careful not show in tables figures symmetric error bars that would yield results that are out range (e.g. negative error rates).
error bars are reported in tables or plots, The authors should explain the text how they were calculated and reference the corresponding figures or tables in the text.
8. Experiments Compute Resources
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed reproduce the experiments?
Question:
Answer: [Yes]
Justification: We analyze computational complexity in Appendix C and compare the execu- tion time of our method with other baselines Section
Guidelines:
The answer NA means that the paper does include experiments.
The paper should indicate type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.
The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (c.g., preliminary or failed experiments that didn't 't make into the paper).
9. Code Of Ethics
Question: Does the research conducted in the paper conform, in every respect, with the NeurlPS Code of Ethics http::/e//earppppcccp/aaaaahhaaa
Answer: [NA]
Justification:
We have no such risk.
Guidelines:
The answer NA means that the authors have reviewed the NeurIPS Code of Ethics.
the authors answer No, they should explain the special circumstances that require deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (c.g.. if there special consid- eration due to laws or regulations in their jurisdiction..
10. Broader Impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [Yes]
Justification:
We discuss them in Appendix A
Guidelines:
The answer NA means that there is no societal impact of the work performed.
18

---

If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.
If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition attacks, mechanisms for monitoring misuse, mechanisms to monitor how system learns from feedback over time, improving the efficiency and accessibility of ML).
11. Safeguards
Does the paper describe safeguards that have been put in place for responsible release of data models that have high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Question:
Answer: [NA]
Justification:
We have no such risk.
Guidelines:
The answer NA means that the paper poses no such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards challenging, and many papers do not require this, but we encourage authors to take this into account and make best faith effort.
12. Licenses for existing assets
Are the creators or original owners of assets (e.g.. code, data, models), used in the paper, properly credited and the license and terms of use explicitly mentioned and properly respected?
Question:
Answer: [Yes]
Justification:
We cite the corresponding source for all the real-world dataset we use.
Guidelines:
The answer NA means that the paper does not use existing assets.
The authors should cite the original paper that produced the code package dataset.
The authors should state which version of the asset used and, possible, include URL
The name of the license (e.g., CC-BY 4.0) should be included for each asset.
For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.
19

---

If assets are released, the license, copyright information, and terms of use in the thcode has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
package should be provided. For popular datasets, paperswi
latasets.paperssit
For existing datasets that are re-packaged, both the original license and the license of the derived asset (if has changed) should be provided.
If this information not available online, the authors are encouraged to reach out to the asset's s creators.
13. New Assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [Yes]
Justification:
We have submitted our code in our GitHub.
Guidelines:
The answer NA means that the paper does not release new assets.
Researchers should communicate the details of the dataset/code/moodl as part of their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent obtained from people whose asset used.
At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
14. Crowdsoureing and Research with Human Subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well details about compensation (if any)?
Answer: [NA]
Justification:
We have no such risk.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Including this information the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then much detail as possible should be included in the main paper.
According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or equivalent approval/review based the requirements of your country institution) were obtained?
Answer: [NA]
Justification: We have no such risk.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing research with human subjects.
Depending the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
20

---

We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors adhere the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
21

---

p/10 i.i.d. samples from Gaussian distribution with mean 0 and A Impact Statements
This paper proposes a novel approach for fast and sample-efficient meta precision matrix learning, which can leverage prior knowledge from related tasks to improve the estimation of conditional dependency relationships between random variables. This approach has potential applications in various domains such genetics, neuroscience, and social networks, where sparse and high- dimensional data are common. The ethical aspects and future societal consequences of this work depend on the specific contexts and purposes of applying this approach, which beyond the scope of this paper. However, we encourage researchers and practitioners to consider the possible henefits and risks of using this approach, such as enhancing disease diagnosis or enabling authoritarian surveillance, and to adopt appropriate measures to ensure its responsible and beneficial use.
B More Experimental Details
All experiments are performed on a machine with an Intel Core i9-10910 ten-core 3.6 GHz CPU and 64 GB RAM.
B.1 Simulation Experiment
For each prior task k = 1, ..., K. we generate ni = p/2 independently and identically distributed observations from a multivariate normal distribution with mean 0 and precision matrix (20)) For the new task, we generate precision matrix Deew (2( This model assumes that the graph is composed of two parts, namely the common substructure and the rest 0( The entire precision matrix is parameterized as 2(*k = + 0(*)
Random Graph Each off-diagonal entry of is generated independently and equals 0.5 with a probability po = 1/pp-11 and 0 with a probability 1 Popo- Each off-diagonal entry of 0 is generated independently and equals 0.5 with probability 0.2kpo and with probability 0.2kkpo Then we add every diagonal element by fixed value large enough to guarantee the positive definiteness of the precision matrix.
Tree Graph We generate + networks each consisting of e unconnected subnetworks with tree structure. All the networks share one common subnetwork with p/10 dimensions. We randomly generate the dimension of the other e subnetworks that range from to p/10. Every subnetwork follows tree structure. Most subnetworks consist of only one node because we choose a relatively large value for c. Each off-diagonal entry corresponding to the edge equals 0.5. Then we add every diagonal element by fixed value large enough to guarantee the positive definiteness of the precision matrix.
B.2 Real-world Experiment
ChIP-Seq dataset ENCODE project is a widely used database for human genome research. For our experiment, we select the ChIP-Seq data for transcription factors (TFs). The ChIP-Seq dataset has been widely utilized as the benchmark dataset in other research papers in this domain 41 42 43 44 TFs are proteins that can control how genes work. The function of TFs is to regulate turn-on and off genes in order to make sure that they are expressed in the desired cells at the right time and in the right amount throughout the life of the cell and the organism. This experiment aims to exploit the gene network of transcription factors (TFs).
There are three kinds of human cells involved in the dataset: (1) H1- -i-EmmCCembboiicc stem cells: primary tissue). (2) GM12878 (B-lymphocyte: normal tissue) and (3) K562 (leukemia: cancerous tissue). The dataset provides simultaneous binding measurements of TFs to thousands of gene targets, each file contains 27 samples(TFss and each sample has 25183 features(gene targets). We randomly selected 300 features for our experiment, The exact shape of the used data in our experiment is (27 TF samples, 300 TF features) for each kind of human cell. H1-hESC and GM12878 are chosen as two auxiliary tasks for meta- knowledge and 20 samples of K562 are chosen as new task.
22

---

Note that the choice to utilize HI-hESC and GM12878 as auxiliary tasks, with K562 as the primary prediction task, was made with deliberate consideration of their biological significance. HI-hESC represents primary tissue, GM12878 represents normal tissue, and K562 represents cancer tissue. Given the heightened research focus on cancer cell gene networks, which are more likely to be the subject of predictive modeling and have rare cases, we opted for this configuration to use K562 as the test set.
Figure Gene network result of FasMe. Examples of discovered gene relationships that are consistent with other biological research: MYL6-GAA 45]. HIFIA-ATP6VIGI [46), WDR6-RBM4 [47).
IMRI dataset OpeniMRI is a task- fMRI database, we chose a dataset about the task Classifi- cation learning (accession number: ds000002). This database has been widely used in other research papers in this domain 48 49 [50 [5]] This dataset contains three different tasks, participants were instructed to perform probabilistic classification learning (PCL), deterministic classification learning (DCL), and mixed-up classification learning on Weather Prediction Task(WPT). WPT is commonly used to assess striatal or procedural learning capacities different populations. This fMRI dataset ef- fectively records neural responses to stimuli, delays, and negative and positive feedback components, and studying this dataset aims to address a fundamental question of whether and how the brain's memory systems interact.
In experiment, the 4D fMRT brain image is divided into spatial regions of size 2 x 2 x so the original scan data in spatial shape 64 x 64 x 30 is divided into 15360 regions. And the activation of each region during the task (the temporal dimension) is added up to represent the average activation situation of corresponding regions. We randomly selected 200 active regions to lower the computational burden. By training on the tasks of PCL and DCL, the common key brain regions and the interaction between the tasks are obtained, that is, a common substructure.
Note that we employed PCL (Probabilistic Classification Learning) and DCL (Deterministic Classifi- cation Learning) as auxiliary datasets due to their representation of distinct learning paradigms (See Appendix A.2). PCL captures learning under uncertainty, while DCL reflects learning from clear-cut rules. This diversity aids in generalizing the model to novel scenarios. The Mixed-Event dataset, embodying elements from both PCL and DCL, mirrors the intricate and multifaceted nature of human cognition. Consequently, the Mixed-Ever Event dataset emerged as the optimal choice for the test set, offering realistic and challenging environment for the model to demonstrate its predictive prowess in brain connectivity, reflective of the complexity encountered in real-world cognitive processes.
By using the prior knowledge of common substructure, effectively introduce the key areas of the brain on similar tasks and the interaction between these key areas, so that we can learn how the brain regions interact in other tasks efficiently.
In Table 3 we record the negative log- degeemmemnnnat Bregman divergence of our estimator for the new task and compare it with the other baselines. The experimental results show that method generalizes better than all other baselines for the real-world dataset since it achieves the minimum log-determinant Bregman divergence.
US Industry Portfolios dataset This is a well-known dataset of US industry portfolios' monthly returns from 1926 to 2023. It covers various economic sectors such as agriculture, healthcare, energy, machinery, and electronies. We take each industry as a feature (p- = 49) and use the records of different financial metries as the tasks. We use AVWR (Annual Voluntary Wage Reporting) dataset
23

---

Table 3: Negative log-determinant Bregman divergence of the estimated precision matrices of the new task the real-world dataset using FasMe and other baselines. larger value of negative dog-determinant Bregman divergence indicates better performance.
Negative log-determinant Bregman divergence
Method
IMRI Data
ChiP-Soq Data
QUIC
22001.26
-716332
Neighborhood Selection
-284.35
-316.21
Meta-IE
-291.66
-200.45
gRankLasso
-2982.78
-823,91
FasMe
-36.31
-64.35
and AEWR (Adverse Effect Wage Rate) dataset (721, n2 = 50) for meta-training, and AFS (Annual Financial Statement) dataset for meta-testing (n 10). With FasMe, about industry interactions from records of different financial metrics, thus resulting in a better estimation of the hidden relationships of industries in the new task. The predicted connections, e.g.. Steel (Steel Works) (Tanssspppoooaiioo))) and ElcEq (Electrical Equipment) (aappmentt)Haatteee SaarddCCoompueryy are consistent with reality.
extract meta-knowledge
Figure 6: Visualization of the financial network predicted by FasMe.
B.3 Time Complexity Analysis of Meta-IE (BigQuie)
BigQuic (38, an "big-data" extension of QUIC, aims estimate the precision matrix with thou- sands of variables. solves the optimization problem using a block-coordinate descent Newton method. BigQuic divides the Newton direction into several blocks and utilizes the memory cache to accelerate the updated process. The computation complexity of BigQuic mainly determined by ( B )TTTuter) where [B| is the number of boundary nodes, is the number of non-zero entries in the i-th generation estimated solution (2()) T the average number of Conjugate Gradient iterations and Touter is the number of computations within a block. Without loss of generality, is larger than p in f-th iteration. Hence, the time complexity of BigQuic is at least higher than O(p²) time complexity. Furthermore, the performance of BigQuic depends largely on the choice of cluster scheme. poor partition of variables leads to many "cache misses". The challenges of choosing cluster schema make BigQuic harder to implement.
B.4 Difference between Meta-lE and FasMe
Based the meta-learning framework, FasMe and Meta-IE differ in each component of this framework, namely meta meta-teacher, meta-knowledger and meta-student.
B.4.1 Meta-knowledge
Meta-IE: Meta-IE supposes the support union Ω as the meta-knowledge and assumes that the edges of the new task must be subset of the edges of Ω
FasMe:
Different from Meta-IE, we assume that the task shares common structure, i.e., meta-knowledge é, with the related auxiliary tasks.
24

---

As we stated in Section Introduction. it is impossible to make startling discoveries under this unrealistic assumption of Meta-IE. Our choice of meta- meta-knowwedde is more reasonable with biological research and existing works as evidence.
B.4.2 Meta-teacher
Meta-IE: Meta-IE simply pools all the samples from K auxiliary tasks to learn the meta-knowledge:
K Σ K
Ω arg min log det f2 +
02 + +112111,
which is data-inefficient.
FasMe: By taking the inherent heterogeneity among the auxiliary tasks into consideration, FasMe
proposes a multitakk-learinng-bass estimator to extract the common substructure across the tasks.
B.4.3 Meta-student
Meta-IE: Meta-IE uses BigQUIC method to solve GI Lasso problem with the knowledge constraint sup S supp(S)).
FasMe: FasMe aims to estimate the precision matrix by solving the dual form MDMC problem with Newton Conjugate Gradient algorithm. We also would like to emphasize that the meta-student a simple application standard MDMC. The formulation of standard MDMC problem is:
max X
log det
s.t. Xij = Mij. Vi.j
We improve the method from aspects: 1) able to deal with the partially observed matrices without chordality by utilizing the chordal embedding 2) lower the time complexity by applying Newton CG algorithm
Table Comparison of standard MDMC and our improved MDMC
standard MDMC
our improved MDMC
partially
Targeted matrix
partially observed matrices with chordality
observed
matrices
O(p'Oinv) where OINY the time required for matrix inversion
Time complexity
O(ploge-))
In summary, Meta-IE has presented preliminary idea for applying metalearning to precision matrix estimation with much potential for improvement. Compared to Meta-IE, FasMe exhibits better theoretical properties and superior performance.
B.5 Accuracy with Different Sparsity
Table 5: Random graph, n 100, = 1000
Sparsity
MCC score (%)
I/p 70.8
10/p
70.5
20/p
71.2
30/p
70.3
40/p
72.1
50/p
71.5
25

---

Table 6: Tree graph, 100, 1000
Sparsity
MCC score (%)
1/p
71.2
10/p
70.9
20/p
71.6
30/p
72.1
40/p
71.7
50/p
72.3
B.6 Hyperarameter Selection
K We actually have conducted synthetic experiment with varying K and a fixed data size (n 100, p 1000). We use MCC value as a metric measure the performance of estimating meta-knowledge. A higher score represents better performance. By varying K = 2,3. 6, Fig. shows that the MCC value smoothly experiences a slight decline from 0.85 to 0.60.
MCC (0-1)
0.5
2
3
K (number of tasks)
Figure 7: MCC values of our methods against varying K
P
This hyperparameter delineates the sparsity contrast between the shared and individual compo- nents of the model. As outlined in line 200, a lower (chosen as 0.8 in our experiments) enriches the shared structure's density while sparing the individual parts. Our preliminary investigations, spanning values from 0.1 to 0.9, indicated a stable predictive accuracy across the spectrum. We plan to enrich the appendix with these findings to demonstrate p' effect on model performance comprehensively.
This hyperparameter dictates the overall sparsity of both the shared and individual graph structures. with a higher value fostering sparser outcomes. Inspired by some previous researches[1, 1, 2), we adopt W./ [og[[p) where w varies across 0.05 x ili = 1,2,.... 30. The Bayesian Information Criterion (BIC) assists in pinpointing the optimal À for our model and is uniformly applied across baseline comparisons.
Rost
Regarding 77, functioning as the soft-thresholdigg parameter, 7) selection leans on empirical estimation of the graph sparsity by the user. It aims to adjust the number of nonzero entries in the model's 's sparsity pattern, correlating closely with those in the estimated precision matrix. A higher value 7j corresponds more sparsely estimated graph structure.
B.7 Tables
Table & & 9 & 10 present the detaild of Fig. They correspond to Subfigure (a)( correspondingly, showing time cost (seconds) comparison between FasMe and other baselines on simulated datasets, with dimension p changing from 500 to 2500.
26

---

Table 7: The time comparisons of FasMe and the baselines on the simulated datasets varying p and sample size n p/20.
Dimension(p)
500
1000
1500
2000
2,500
QUIC
0.33
3.03
11.38
64.06
84.50
Neighborhood Selection
3.00
8.72
18.92
43.00
76.46
Meta-IE:
4.33
27.27
89.23
96.85
112.63
FasMe
0.002
0.06
0.79
0.99
1.62
Table 8: The time comparisons of FasMe and the baselines on the simulated datasets varying p and sample size n p/10.
Dimension(p)
500
1000
15500
2000
2500
QUIC
0.31
3.68
9.74
22.93
36.77
Neighborhood Selection
2.93
10.83
58.03
87.19
170.83
Meta-IE
2.62
16.66
39.76
87.14
96.11
FasMc
0.002
0.06
0.85
0.965
1.404
Table 9: The time comparisons of FasMe and the baselines on the simulated datasets varying p and sample size n p/5.
Dimension(p)
500
1000
1500
2000
2500
QUIC
0.30
3.93
11.40
26.17
46.51
Neighborhood Selection
6.17
35.70
96.74
182.92
828.97
Meta-IE
1.65
9.90
10.29
14.08
14.10
FasMc
0.002
0.06
0.96
0.99
1.54
Table 10: The time comparisons of FasMe and the baselines the simulated datasets varying p and sample size n p.
Dimension(p)
500
1000
1500
2000
2500
QUIC
0.30
3.56
8.38
22.37
39.54
Neighhorhood Selection
13.56
84.21
216.44
402.84
N/A
Meta-IE
0.64
7.29
9.10
FasMe
0.001
0.05
0.68
0.97
1.34
27

---

C The full Algorithm of FasMe
Algorithm FasMe
1: Input: Auxiliary datasets (X(+)) New task new, hyper-parameter X, P. 7, 1/, a, the maximum iteration T, two thresholds 61 and 62, and a linear programming solver I.P(.) that
solves (8)
2: for k to K do
3: Calculate the covariance matrices ( 1 E XX
X)(X
4: Initialize 0-
Op pxp
5: Initialize (k) 0. ((k)
[A]]
EK
6: end for
7: for = to p do ,), k ()),
8:
b e, B
=1. I.....K K
9:
for k = to K do
0.! (k
10:
11: end for
12: 0, (K(K++1)((+t)
end for
14: Calculate the sparsity pattern G by 8 and its chordal embedding & by Cholesky factorization
Initialize y1
16: for = to 7 do
17: Ay = - ( -V(()))))(()))
18:
= 1
19:
if + aay) ((y) + yaAy Vi(y) then
20:
a.=a.c
21:
end if
22:
Calculate the Newton decrement 6% Av, Vi(y))
23: o < E1 or AAyel < E2 then
24: Break
25: else
26:
V/+1 1/+ aAy.
27: end if
28:
end for
29:
Output: Precision matrix Pnow-
Similar to CLIME, 5 can also be solved column by column:
(.. (,(8801) arg min
K||B1|1 + + Σ |
(.()
(16)
k=1
s.t.
||35|00] (e,
3 2 (
SA
where to = 1, K. Here 3 represents the corresponding column vector in the meta-knowledge and (0 represents the k-th column vector in the rest 0.50
To solve (T)), we can rewrite the following
3 arg min
||
(17)
[ B blloc S SA,k =A,==1,.... I...., K.
s.l.
Here, (k) (0.... (k)
0, ck
[k)) B [(3())T,.... ...., (B(K)),.... (B(K))T (K(3)))T
and b ey.
28

---

Through relaxation, we can convert (17 to the following linear programming formulation:
(K p
å = arg min
aj
nj
j=l
s.t.
0j S aj. ] = l,..., (K + 1)P,
(18)
0g S aj. j ,..., (K 1)p,
(B (k)) T b, c, c, = 1..... p 1,..., K,
(B
( To b; S c, ................................ I....... K.
Here a, are the slack variables. B represents the i-th row of ( and b, is the i-th entry of According
[25]
apply the same symmetric operators ( 0 + 0f obtained from
Algorithm|
Then combined the support set of the thresholded covariance and meta- meta-knowledge obtain the sparsity pattern:
G supp(0) U Suup (Snww)))))
(19)
To solve the optimization problem we first need to obtain chordal embedding by Cholesky factorization. We compute the unique lower triangular Cholesky factor L satisfying 1gg (Eueww L.I After ignoring perfect numerical cancellation, we have the adjacency matrix of the chordal embedding L, I, using symbolic Cholesky algorithm.
We define the cone of sparse positive semidefinite matrices P, and the cone of sparse matrices with positive semidefinite completions P. as the following:
P 3 n SP G
P. ((S 20:5 E
The primal and dual problem of Eq.
9 can be written as:
arg min (1lg (2), 52) + g(2) s.t. PT (i) = 0,
(20)
REP
arg max (S() s.t. S E -
(21)
SER'ER"
where P(.) m
represents linear map that converts a list of m variables into the corresponding matrix in V and g. are the "log-det" barrier functions on P and P.:
8P G
g( log det Q g-(S) min (S, 22) log det Ω. neP
Under the sparsity and chordal assumption, the gradient evaluations and Hessian matrix-vecto products can be efficiently evaluated in (pp) time and (pp) memory, using the numerical recipes described in [52]
To solve the dual problem 21). we rewrite into the following unconstrained optimization problem:
arg min ((y) g. (Igg(Encs)) P(y)). vERm
(22)
After obtaining the solution ÿ. we can recover the optimal estimator for the primal problem through Ω = Vg. (Ig (Enew) P(y)). The dual problem is easier to solve than the primal because the initial point y 0 tends lie close to the solution ÿ. Starting from this point, we can use Newton's method to converge rapidly.
We then solve the Newton direction Ay via the m system of equations
viIly)ay = VV(y).
(23)
Starting from the origin y/ 0, the method converges to an e-accurate search direction yı satisfying
(j1 Ay) (l)) Ay) < <Ay vi(y)
(24)
in at most / |
()) 1| log(? 3 conjugate gradient iterations.
29

---

Computational Complexity In this section, we mainly analyze the time and memory complexity of our estimator in the meta-testing step. The time and memory cost of our estimator can be divided into two parts. The first part is to threshold the covariance matrix. This step is quadratic O(p²) time and memory but embarrassingly parallelizable. The second part is to solve (7) using Newton Conjugate Gradient methods. If the prior sparsity pattern G is sparse and chordal, then the second part can be performed using our algorithm linear ((p) time and memory. This significantly outperforms QUIC [0), Neighborhood Selection method [34] and Bigquic used in [
D Limitations
As discussed earlier, the biological and genetic domains have adopted and validated the common structure assumption based some biological evidence. For other domains, if the edges sparse and the data are high-dimensional, it may be difficult to recover the common structure from the data without some prior evidence. In the worst case, the meta-feacher is no common edge among the tasks. The meta-student has to learn from scratch. The optimization problem in the meta-testing process can be seen as a graphical lasso problem (introduced in Section 2.2). The neighborhood selection baseline is one type of algorithm that solves the graphical lasso. Consequently, our proposed method remains least as good the baseline, if not stronger, in handling such challenges.
learn the knowledge that there
E Explanations about the Choices of the Baselines
We chose QUIC and Neighborhood Selection since they continue to be widely utilized in current research and serve as critical benchmarks for new methodologies within the realm of state-of-the-art works.
We take QUIC as an example. 53 utilizes the QUIC method to detect structural changes in high- dimensional Gaussian graphical models. QUIC's ability perform fast and accurate estimation underpins the methodology for identifying thange-point in the graphical model structure over time. Similarly, studies like those in recent works like [54] 55 tasks downstream.
also employ QUIC for various analytical
Additionally, while both (56] and [57 utilize QUIC baseline for comparison, [56] relies
structural assumption of Eronecker-summ-ttuuuuree inverse covariance, and [57] operates under general missing dependency assumption. These specific premises limit their applicability, rendering them less generalizable to our scenario.
Parallel to QUIC's applicability, Neighborhood Selection is similarly employed in contemporary studies, demonstrating its ongoing relevance and utility [58] 59 60 61].
As mentioned above, we observed two main categories. The first category of recent works did not propose improvements to QUIC; instead, they applied it to more specific tasks. The second category, while outperforming QUIC as a baseline and demonstrating improvements, often relied on stricter structural assumptions not compatible with our problem context. Consequently, both categories were not suitable baselines for our study.
We specifically chose Meta-IE method due to its closeness to our work. Meta-IE represents the latest advancement in meta-learning applied to sparse precision matrix estimation, making it an ideal candidate for direct comparison.
To this end, we have decided to incorporate an evaluation against gRankL 35
a recent devel- opment in graph learning, into our revised manuscript. With completely tuning-free technique, gRankl asso Lasso extensive time consumption for hyperparameter simulations and rank calculations poses a drawback. particularly underperforming with small datasets.
shows better accuracy performance than CLIME, GLasso, and TIGER. However, its
F
Proof of Theorems
Definition Let x(x) x(A) Rp be i...d. random vectors for 1 S k S K. Let X.ş be the j-th entry of x (K) for 1 S j p. We use ( 00 to represent the real covariance
30

---

matrix. precision matrix, and meta-knowin veta-knowledge, respectively. {X } follows family of random p-dimensional multivariate sub-Gaussian distributions with parameter a if
conditionally independent given )
IsiSne.IsKKK
(2)
conditioned on (D2k))) is Sab-Gaussian with parameter o for i n 1
(3)
E|X (- 0, Cov X (M) (= (E())*
((2())))) for S siS nR. 1
K.
Then we remove the assumption on the support set of thus relaxing the data assumption of M- (4) (1 = 8 + of with 8, 035)) E -RPPP. The meta-knowledde is deterministic, and 0,*)
KS K, are i..dd random matrices.
Lemma 1, Given a simple optimization problem, & and ÿ are the optimal solution.
f + 1 - arg min |r| + plul
2.y
(25)
s.t. x x
where p >> and C/ a constant. Then we have ā 120. O.
Proof. If f ÿ < 0, let x'
f + D and y' = 0. Without loss of generality, we assume that |
|0
follows that
|a'| + ply' | + أl
[ + plūl,
(26)
which means that the pair x', s/ a better solution. This contradicts the earlier stated premise that ãt, are the optimal solution. The proof is completed by contradiction.
satisfies (0). then Duot also satisfies condition Qhot > with a high probability.
Lemma If each 0%/²
10 Thus Piot satisfies the
Corollary 1. We assume that 0 is the optimal solution of Eq. 5 and 0 = (2()) 0. Based on
Lemma it follows that (1k), 0.j Then it is simple to that
|10,1lo S 11 1101 = ||ô, + + |
(27)
Theorem 1. We use θ, 4* to represent the estimated precision matrix and real precision matrix, respectively. Suppose that Siot u. If 2 v(max و.. Tiyl l)= we have that
j 10 S S2A + 26.
(28)
where v = || and Φ = 11 oo lo..
Proof. According to the condition Theorem
X
Erot Ei x
(29)
|
ll1
Then show that
|| Erot Foolloo || Eror) Ix
1| 1i Pii Etoello S A
(30)
where AB < Abo B), for any matrices A, of appropriate sizes.
Then we can rewrite (5) as:
( (oo,,
arg min (0( Hra
|| |l + (1-
(31)
s.t.
I - 11100 S A.
This is because ||Qeat 1| + (1 +)
k=1 |10 111 = Σk 110
(k) ||2 + (1 p) Σ
1 ln.
Notice that the constraint related O,. Therefore Prm is also the solution of the following optimization problem:
Piot arg min Qrolll
s.t. Pror 1ll00 S X.
(32)
Ooo
31

---

Since Rot and Ooor satisfy
32 we have
Snat 11 S ||
(33)
suggests that
[pEtot (Pror - loc I Pror - 1 11o0 + ||| SnotMiolller s 2d.
(34)
Consequently, it is straightforward to show that
D (Deot= ll 5 I|/iio (froo Piot)ll + 11(eoot 2 ) loo
(35)
I 2X 1$2pot
Sior QiotllillSton E I 4d.
Then, it follows that
|i Rorlle S || 11 |Ei (Ωtor - -oo S 41l ll 1 4vd,
(36)
where |Riorll |I
Based on the triangle inequality for norm, we have that
||6 0"1 || O, - (Diot - 0,
S l|Ω - Noot ac + 1|0, - 0
(37)
|14 S al + |1011 + ||0
In terms of O", O we assume Wot,,y 2i = 0 and Wottiy + S2j ilf
Wot. Qi
Thus we can derive that ||0* I [| Iloo. We are now turning to the proof of Eq. 37
Combining
Corollary and Theorem [
we can show that
| 0-
||$20o Diot llco + ||S 1o0 + +112 o0
||Seot - Pine 1o.0 + ||Snoo Rorlloo + 21
211 Piot 11 + 21
(38)
Sv + 20.
where || 1100 = 6. The proof is completed.
Theorem 2. Suppose thut Piot EM and N = = Th. When the variables are sub-Gaussiar, the
tail condition holds. Let X = Corv /oR(Kp), where the constant Co = 27 22 To + + 22223
N
and constant To 0. what 2Tn then have
1. 100 0r 1loc < <80nu2 /
(og(KD) + 2i;;
2. supp(0) = supp(O*):
with probability greater than 1- 4p To. Here ê is threshold estimator with 0.g 0,)[(10s)) Tx).
where Tn 2 4v is a tuning parameter.
Proof. To obtain the first conclusion of Theorem 2 we need to prove
log Kp
max (āij - ofy S Coyy N
(39)
with a probability greater than 1 - 4p To under the exponential tail condition. Without loss of generality, we assume that E(X Ba..kk-E((X... E(X Ktot,kk X kto) We then have Enot Σ X XT
O. Let Eiot N EX
tor. and Y kij tot Lett = 7//og Kp/N.
Using the inequality je* - < 52 max((s)) for any E R and let C₁ 2 + + C 123 Then by some basic calculations, we can get
N
Yug 2 1I
S expp(-C, log
< <exp(-C log Kp + No kij exp(1
(40)
exp( log Kp y C log Kd)
S exp(-(To + 2) log Kp).
32

---

Then we obtain
log Kp N
P |E Exoello 2 27
$ 2p To
(41)
Using the inequality ex S +0 for > O, we have
Ee S eC, vt S v7.
(42)
Let
C2 = 2 + Tu + + c²
(43)
and 0n Clog Kp/N) As before, we can have that
N
P
for 2 2
2 an log Kp/))/ Sp
Σ
X 2 C2V N log Kp
+ pmax
Σ X
122 N loog Kp
22p 70-1
(44)
and the inequality > 1 + 2 we see that (39 holds. The proof of the first incquality is completed.
By 41
To prove the sign-cons conn--sssittenyyyccc of Ô, we define threshold estimator Peot with w, T [ 2 Tn min [Pror.y and threshold where Tn 2 4v is a tuning parameter. We define wot (s.))
estimator with i, = θ, [[O.,] 2 T..). where 2 4vA. It is known that the resulting elements in Pror will exceed the threshold level if the corresponding element Dot is large magnitude. In contrast, the entries of Quo outside the support of oi will remain below the threshold level with high probability. As result, if what > 2Tn, we have supp((2ro) supp(S Since the nonzero entries in correspond to the intersection of the nonzero entries in (?(*), J. straightforward get that supp(@) = supp(e") if tot > 2Tn.
Lemma 3. Consider zero-mean random vector "X....,Xpp) with covariance Σn such that each is Gab--aassiinn with parameter a. We define W Ennw Dnew- Given n. i..d samples, the V
associated sample covariance E satisfies the tail bound if ['neewllo S sp:
nd² P|l > 8) s 4[g1 128(1 + 22
(45)
for all 5 E (0, 8ip(1 + 402))
Theorem 3. Suppose we have recovered the true sparsity pattern G of a family of d-dimensional random multivariate sub- Gaussian distribution. Then for new task of multivariate sub- Gaussian
distribution with the precision matrix Dee such that supp(0) C. supp(()eew) and satisfying irrepre-
sentable condition, consider the estimator Snew with
Czpy
log 9
Then we have
log |
l1Qeew 2iee Ilo0 S 2223v2 KT
(46)
m
with probability 1-p-- where Tı > O.
The proof is similar to Theorem 2 and will be omitted.
Definition 3. (32] Given matrix M E SP, define 9M (i.j): Mú * 0 as its sparsity pattern. Then M is called inverse-consistent if there exists matrix N E 3" such that
M + N 00
N = 0 V(i,j) GM
(47)
(M - S
33

---

The matrix N is called an theers-conssstent complement of M and is denoted by M Furthermore, M is called sign-consistent for every (i.j) 9s, the (i. elements of M and (M + M have opposite signs.
Then we define the B(G, a)) function defined with respect to the sparsity pattern and scalar a
a(G,a)
max M=0
[| IImax
s.t.
M E & and |M max S a
M1, =1 Vi [I. M is inverse-consistent.
Lemma 4. [32] Any arbitrary matrix with positive-defintt completion is inverse-consisent and has unique inverse-consisee complement.
Lemma 5. Ω is the optimal solution if and only if it satisfies the following conditions for every (1,5) E (1, (p,...,p??
Riy Σnj i
2,j Σnj + sign(S
ifhy #
(48)
Dig 17 S 2iy محكك +
Proof.
The proof is straightforward and omitted for brevity.
Then, consider the following optimization problem:
min 2E8
log det(St) + (2, +
Σ
71 + 2 max Eis
Σ
(49)
(3,))
(t))eg)
where
7
Ex x Dij
VEa
Ejj
We denote Ω as the optimal solution of (49) and define D as diagonal matrix with Dú Eig for every i ,p.
Lemma 6. We have = D- x D
Theorem 4. coincides with the sparsity pattern of the optimal solution D if the normal- ized matrix Enew D D- 1D11//11g/E,/// (Eueew)))) where (oog(IIggEee/) satisfies the following conditions:
1. Snow is positive definite,
2. Dnww is sign-consistent,
3. We have
g ['sowlllng max
(Il Enew)) Dacl
(50)
17
min (G)
VIIg D (Enew))) Du
Proof. We define an intermediate optimization problem as the following:
Ω arg (f2) = log det Ω + (2,2) +
Σ (L
AtylSual + max Σ
14yll (51)
(..))eg-
First, we show that Ω Q. Trivially, f2 a feasible solution for 51 and hence f(2t) < fî).. Now, we prove that Q is feasible solution. We show that 2 = for every (i.j) E go. By
contradiction, suppose 21 * for some (i,j) E go. Note that, due to the positive definiteness of
(11)-1, we obtain
(2t) (0t) ((92))))) 0.
(52)
34

---

Then, based on Lemma 6 it follows that
(Ω EEú + max Exk x sign(
(53)
Considering the fact that Σ 0, we have Exyl I max; Σsa- It implies that (227)
max; Esk-
Furthermore, due to Lemma we have (S2) ) = 94 and (5 Azz-- This leads to
(1t) x (2t), ((0)))) 5))² Ea x Ejj max (ERs) S 0,
(54)
which contradicts with 52 Therefore, S2 feasible solution. This implies that f(21) 2 f(2)
and hence. (S2) f(l2). Due the uniqueness the solution of (5I). we have Qt. Now,
note that 51 can be reformulated
min BES?
log (et + (E, + Σ
n/ + 2 max De.k
Σ (iJ)Ege
(55)
(1.))eg
Upon defining
Ω = 1V2
(56)
and following some algebra, one can verify that (51) is equivalent to
min - log (e)( + (Σ, 22 + fted
Σ
ū
+ 2 max
EEa
Σ
Kug + log de)) (57)
&
(i.))s
(1.))
Dropping the constant term in (57 gives rise to the (49) As a result, & D D holds. The proof is completed.
Next, prove that the newton subproblem can be solved in O(1) CG iteration. Let S be the set of p x real symmetric matrices. Given a sparsity pattern V. we define Sy S & as the set of x p real symmetric matrices with this sparsity pattern. We consider the following minimization problem
arg min ((y) g. g. P(y)).
(58)
yERm
Here, the problem data P. R" Sp is orthogonal basis for sub-sparsity pattern F C V that excludes the matrix Σ. In other words, the operator satisfies
P(P (2) PF(2), VQ E
(59)
(T{E)=
The penalty function g. is the convex conjugate of the "log-det" barrier function on Sp
g. (.))
min (S Ω log det !2} s2E
Assuming that V is chordal, the function g. (S), its gradient Vg. (S), and its Hessian matrix-vector
product V (Y] can all be evaluated in closed-form 62 63 52 also [64]. Furthermore,
the pattern is sparse, i.e. its number of elements in the pattern satisfies [V| =Op)), then all of these operations can performed arbitrary accuracy Opp) time and memory.
is standard to solve 58 using Newton's method. Starting from some initial point y1 E dom
uk+1 = + Ag Ayk- = (y)) VI (yx),
which the step-size ar determined by backtracking line search, selecting the first instance sequence (1, p², 03 ..... that satisfies the Armijo-Goldstein condition
1(y + aAy) S (()) + yaAy" TV(()).
35

---

in which E (0, 0.5) and p E (0, 1). The function is strongly celf-conn,,d,nt, and / inherits this property from g.. Accordingly, classical analysis shows that we require most
I (wi) 1(0) 0.05%p
+ log (1g(1/c) (1)) Newton steps
an e-optimal point satisfying (wx) (()) S 6. The bound very pessimistic, and practice, no more than 20-30 Newton steps are ever needed for convergence.
The most computationally expensive of Newton's method the solution of the Newton direction Ay via the m x m system of equations
V (yn) -V (yx)..
(60)
The main result in this section is proof that the condition number of (2((y) is independent of the problem dimension n.
Theorem S. At any y satisfying I(y) - I (31) and Vi(y) (y yn) S Omax the condition number Ki of the Hessian matrix (yy is bound
22 Amax (S2)
K1 S 4
1
(61)
Amin (92)
where:
max = (y1) - I(0) is the suboptimality of the initial point, = [vec AJ...., vec Am is the vectorized version of the problem data,
Pg = f (So) and So = Σ - (yo) are the initial primal-dual pair,
Ω (5) and S - - A()) are the solution primal-dual pair.
As consequence, each Newton direction can be computed in O (loge-2) iterations using conjugate gradients, over O (log co) total Newton steps. The overall minimization problem is solved to c-accuracy in
O (log Tloge-loe log log -1) O(1CC iterations.
The leading constant here is dependent polynomially on the problem data and the quality of the initial point, but independent the problem dimensions.
Proof To prove the first bound 6 we will instead prove
Amax (9)
20 max Amax (So)
2 +
(62)
Amin (52)
Amin (22)
which yields the desired condition number bound on V2g(y). Writing d1 = Amax (22 and A, (S2), we have from the two lemmas above:
2
Omax
Aunin (5t)
-
> 0,
20haax Amin (2 (VTi vnn)
0.
Multiplying the two upper-bounds and substituing Amin (26 1/>max (go) yields
36

---

2
2 Amax (oo)
A1
A
A,
Amin (52)
An
A
Am
A
Finally, bounding //111 yields
62
37


Reviews:
[{"review_id": "rQ0ddaNf4B", "reviewer": "Reviewer_PtUj", "summary": "The authors propose a method to estimate sparse precision matrices from few samples. Theoretical properties of the proposed method are studied, and experiments on synthetic and brain fMRI data are presented.", "questions": "* How are the meta learning tasks linked to learning the new precision matrix in the theoretical part? It would be nice if the authors can elaborate on the assumptions for the learnability of the new matrix.\n2. Why is N=0 in Eq. 13?", "limitations": "N/A", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Strengths:\n* The paper is overall well written, and fairly easy to follow and comprehend.\n* Theoretical guarantees for sub-Gaussian distributed random variables are presented, and they seem to be novel contribution. \n* The experiments on the synthetic dataset clearly demonstrate improvement over the relevant competing methods.", "weaknesses": "Weaknesses:\n* Currently quantitative results are presented for synthetic data. It would be nice to see more quantitative evaluations on benchmark datasets.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate your kind feedback and perceptive questions.\n\n### 1. Response to Weakness\n\nIn addition to the synthetic datasets, we have conducted extensive experiments on real-world datasets. Specifically, we used the ChIP-Seq dataset from the ENCODE project and the fMRI dataset from the OpenfMRI project, as described in Section 6.2 (Page 9, Line 335-345) and Appendix B.2 (Page 13-14, Line 518-565). Additionally, we have referenced Appendix B.2 in Line 345.\n\nThe datasets used in our work have been widely utilized as the benchmark datasets in other research papers in this domain. We list some papers as evidence: \n\n- ChIP-Seq dataset from the ENCODE project: \n  - Mitra et al. (2013) [1] introduced a Bayesian graphical model for inferring chromatin states from **ChIP-Seq data** on histone modifications. This approach identifies complex dependencies among various histone modifications, enabling a better understanding of their roles in gene regulation.\n  - Lundberg et al. (2015) [2] developed ChromNet, a computational method to infer the chromatin network from **ChIP-Seq datasets**, which identifies conditional dependencies among regulatory factors to discern direct from indirect interactions more effectively. This approach enables the analysis of large-scale ChIP-Seq data, revealing both known and previously unidentified interactions among regulatory factors.\n  - Ng et al. (2018) [3] developed a graphical model to visualize regulatory relationships between genome-wide transcription factor binding profiles from **ChIP-Seq datasets**, demonstrating an innovative approach to discern direct versus indirect transcription factor interactions and enhance the understanding of transcriptional regulation.\n  - Shu et al. (2021) [4]  introduced a computational framework utilizing neural networks for the inference and visualization of gene regulatory networks (GRNs) from single-cell RNA-sequencing data. Their model significantly enhances the accuracy of gene interaction analysis within and across various cell types, leveraging **ChIP-Seq datasets** to validate inferred GRNs and providing a robust method for biological data integration and interpretation.\n\n- fMRI dataset from OpenfMRI project: \n  - Ryali et al. (2012) [5] developed a method for estimating functional connectivity in fMRI datasets using stability selection-based sparse partial correlation with an elastic net penalty. This approach provides more accurate identification of functional connections, even with large **fMRI datasets**, enhancing the understanding of brain function and connectivity.\n  - Using **fMRI datasets**, Luo (2014) [6] proposes a hierarchical graphical model to effectively estimate large inverse covariance matrices, facilitating improved inference of functional brain networks and their hierarchical interactions. This model addresses the challenges of high dimensionality in brain data, offering a more accurate understanding of brain connectivity.\n  - Sulek (2017) [7] utilized graphical models with a lasso penalty to analyze functional magnetic resonance imaging **(fMRI) datasets,** focusing on brain activities during saccadic eye movement tasks. The study applied the graphical lasso method to construct sparse graphs that elucidate the connections between different brain regions, comparing these connections before and after specific tasks to understand changes in brain connectivity.\n  - Chung et al. (2021) [8] developed a robust graphical model that jointly estimates multiple precision matrices from **fMRI datasets**, facilitating a detailed analysis of brain networks by integrating regularized aggregation to account for individual variability and detect outliers, significantly enhancing the accuracy and reliability of connectivity assessments across subjects.\n\nThe above references can be seen in the global rebuttal.\n\nWe appreciate your suggestion and will ensure to highlight these real-world datasets more clearly in the revised version of our paper. Additionally, we will include the aforementioned references in the revised version as they provide significant evidence of the relevance and utilization of these benchmark datasets in the field. Thank you again for your valuable feedback.\n\n### 2. Response to Q1\n\nWe appreciate the opportunity to clarify this point. Due to the space limitation in the main text, we have provided a detailed theoretical explanation of the assumptions that elaborate on the link between the auxiliary tasks and the new task in Appendix F, specifically in Definition 2 (Page 21, Lines 748-759). Additionally, we have referenced this in the main text on lines 153-154. We hope this addresses your concerns and are grateful for your attention to detail. \n\n### 3. Response to Q2\n\nWe apologize for any confusion our notation may have caused. The $N=0$ in Eq. 13 should be read in conjunction with the subsequent $\\forall(i, j) \\in \\mathcal{G}\\_M$. To avoid any misunderstanding, we will revise this equation to $\\forall(i, j) \\in \\mathcal{G}\\_M, N_{ij}=0$. We appreciate your attention to detail and your helpful suggestion.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6MDj0we5Xu", "reviewer": "Reviewer_gJdm", "summary": "This paper introduces FasMe, a meta-learning approach for efficient precision matrix estimation in small sample settings. By leveraging meta-knowledge and maximum determinant matrix completion, FasMe reduces sample size requirements and improves computational efficiency. Experimental results show FasMe to be significantly faster and more accurate than existing methods, particularly in low-data environments.", "questions": "Is the work primarily applicable to biological scenarios, or can it be widely used in other contexts as well?", "limitations": "nan", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1) Paper investigates a key issue in precision matrix estimation and proposes a reasonable method to address the problem.\n\n2) Paper provides thorough theoretical and experimental analyses to justify the method’s ability to reduce the sample requirement and enhance learning efficiency.\n\n3) Paper has good representation.", "weaknesses": "I have few doubts about the method and experiments presented in the article.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### Response to Question\n\nThank you for your positive feedback and insightful question.\n\nYes, our work is primarily applied to biological scenarios.  For one thing, high-dimensional, small-sample settings are more common in biological research, as exemplified by our case study on Cholangiocarcinoma at the beginning of the paper. For another thing, as stated in Line 149-150 of our paper, \"The assumption has been widely adopted [28, 18, 30] and proved to be feasible and applicable in the biological and genetic domains [2].\" This highlights the strong foundation and successful application of our work in these fields. \n\nHowever, our work can also be applied to other domains, although perhaps not as widely as in the biological field. In the appendix (Line 566-575), we demonstrate the application of our method in the financial domain, presenting experimental results that show promising performance. As we mentioned above, the assumptions and dependencies leveraged by our method are more commonly found and well-studied in biological data. Nevertheless, the inherent characteristics of financial data allow for its adaptation. For example, the relationships between ships and transport or between steel and the chemical industry (shown in Figure 6 in Appendix) are common across various economic datasets, showcasing our method's versatility.\n\nWe believe this is a valuable point of discussion and have added the discussion in the main text of the revised paper to elaborate on these aspects further.\n\nThank you again for your thoughtful question.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "dvYX13qjX2", "reviewer": "Reviewer_nam2", "summary": "This paper proposes a meta-learning method for estimating the precision matrix on a new task with small data.\nThe proposed method uses common edges estimated from multiple auxiliary datasets as meta-knowledge. Then, it estimates the precision matrix on the new task, assuming its true edges contain all the estimated common edges (meta-knowledge). Some theoretical guarantees are also provided.\nExperiments with synthetic and real-world datasets show the effectiveness and efficiency of the proposed method.", "questions": "Please see the Weaknesses.", "limitations": "Yes", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper is generally well-written and easy to follow.\n- Concrete algorithm and its theoretical guarantees are presented (but, I didn't read their proof).\n- Strong performance in terms of both accuracy and efficiency in the experiments.", "weaknesses": "- As the authors stated in the Limitation section, the assumption of Eq. 8 might not be held in general machine learning tasks, although it fits well in biological domains.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### Response to Weakness\nThank you for your detailed review and for highlighting this crucial aspect of our research. As mentioned in Line 149-150: \"The assumption has been widely adopted [28, 18, 30] and has proven feasible and applicable in the biological and genetic domains.\" We have validated this assumption through numerous real-world experiments across various biological datasets, detailed in Section 6.2 and Appendix B.2.\n\nIn the Limitations Section, we acknowledge potential challenges in sparse and high-dimensional datasets from other domains, noting: \"For other domains, if the edges are sparse and the data are high-dimensional, it may be difficult to recover the common structure from the data without some prior evidence.\" However, we would like to respectfully point out that this assumption has been successfully applied in some other fields as well. For instance, references [9, 10] (See global rebuttal for references) validate this assumption in industrial sensor and stock price datasets, respectively. Additionally, we have applied this assumption to a financial dataset, as described in Lines 566-575 of the appendix. The inherent characteristics in financial data allow for the adaptation of this assumption. For example, the relationships between ships and transport or between steel and the chemical industry (shown in Figure 6 in Appendix) are common across various economic datasets, showcasing our method's versatility.\n\nMoving forward, we plan to extend our work by relaxing the sub-Gaussian data assumption and further reducing the sample size requirements for training meta-knowledge. This will enhance the applicability of our approach across a broader range of fields. We greatly appreciate your insightful feedback and constructive suggestions. We have added the above discussion to the revised paper.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
## References
[1] Mitra R, Müller P, Liang S, et al. A Bayesian graphical model for chip-seq data on histone modifications[J]. Journal of the American Statistical Association, 2013, 108(501): 69-80.

[2] Lundberg S M, Tu W B, Raught B, et al. Learning the human chromatin network from all ENCODE ChIP-seq data[J]. bioRxiv, 2015: 023911.

[3] Ng F S L, Ruau D, Wernisch L, et al. A graphical model approach visualizes regulatory relationships between genome-wide transcription factor binding profiles[J]. Briefings in bioinformatics, 2018, 19(1): 162-173.

[4] Shu H, Zhou J, Lian Q, et al. Modeling gene regulatory networks using neural network architectures[J]. Nature Computational Science, 2021, 1(7): 491-501.

[5] Ryali S, Chen T, Supekar K, et al. Estimation of functional connectivity in fMRI data using stability selection-based sparse partial correlation with elastic net penalty[J]. NeuroImage, 2012, 59(4): 3852-3861.

[6] Luo X. A hierarchical graphical model for big inverse covariance estimation with an application to fmri[J]. arXiv preprint arXiv:1403.4698, 2014.

[7] Sulek T R. An application of graphical models to fMRI data using the lasso penalty[D]. University of Georgia, 2017.

[8] Chung J, Jackson B S, Mcdowell J E, et al. Joint estimation and regularized aggregation of brain network in FMRI data[J]. Journal of Neuroscience Methods, 2021, 364: 109374.

[9] Hara S, Washio T. Learning a common substructure of multiple graphical Gaussian models[J]. Neural Networks, 2013, 38: 23-38.

[10] Banerjee S, Ghosal S. Bayesian structure learning in graphical models[J]. Journal of Multivariate Analysis, 2015, 136: 147-162.

Meta-review: This paper introduces a meta learning technique for precision matrix estimation, e.g. estimating the precision matrix of a dataset X' given a set of training datasets X1,...,Xk. Overall the reviewers appreciated the theoretical results for sub-Gaussian RVs and strong empirical efficiency improvements. There were no major concerns with the paper; I might simply recommend that the authors' make the results currently in Appendix B.2 more prominently visible.
Decision: Accept (poster)