Bayesian network (BN) structure discovery algorithms typically either make
assumptions about the sparsity of the true underlying network, or are limited
by computational constraints to networks with a small number of variables.
While these sparsity assumptions can take various forms, frequently the
assumptions focus on an upper bound for the maximum in-degree of the underlying
graph $\nabla_G$. Theorem 2 in Duttweiler et. al. (2023) demonstrates that the
largest eigenvalue of the normalized inverse covariance matrix ($\Omega$) of a
linear BN is a lower bound for $\nabla_G$. Building on this result, this paper
provides the asymptotic properties of, and a debiasing procedure for, the
sample eigenvalues of $\Omega$, leading to a hypothesis test that may be used
to determine if the BN has max in-degree greater than 1. A linear BN structure
discovery workflow is suggested in which the investigator uses this hypothesis
test to aid in selecting an appropriate structure discovery algorithm. The
hypothesis test performance is evaluated through simulations and the workflow
is demonstrated on data from a human psoriasis study.

本文介绍了一种用于判断线性贝叶斯网络最大入度是否大于 1 的假设检验方法，并提出了一种基于该方法的结构发现工作流程。通过对模拟数据和人体银屑病研究数据的实验验证，证明该方法的有效性和可行性。

贝叶斯网络中稀疏假设的测试

Testing Sparsity Assumptions in Bayesian Networks

We consider the linear regression model with observation error in the design.
In this setting, we allow the number of covariates to be much larger than the
sample size. Several new estimation methods have been recently introduced for
this model. Indeed, the standard Lasso estimator or Dantzig selector turn out
to become unreliable when only noisy regressors are available, which is quite
common in practice. We show in this work that under suitable sparsity
assumptions, the procedure introduced in Rosenbaum and Tsybakov (2013) is
almost optimal in a minimax sense and, despite non-convexities, can be
efficiently computed by a single linear programming problem. Furthermore, we
provide an estimator attaining the minimax efficiency bound. This estimator is
written as a second order cone programming minimisation problem which can be
solved numerically in polynomial time.

本文研究考虑设计误差的线性回归模型，针对在实践应用中常见但引入噪声的自变量情况，提出了一种基于稀疏性假设的估计方法，并指出该方法在极小化期望风险下是近乎最优的。所提出的估计方法可以通过解线性规划问题实现高效计算，同时具有达到极小化效率界的估计量。

高维误差变量模型中的线性和锥形规划估计器

Linear and Conic Programming Estimators in High-Dimensional  Errors-in-variables Models

We consider the linear regression problem, where the number $p$ of covariates
is possibly larger than the number $n$ of observations $(x_{i},y_{i})_{i\leq i
\leq n}$, under sparsity assumptions. On the one hand, several methods have
been successfully proposed to perform this task, for example the LASSO or the
Dantzig Selector. On the other hand, consider new values $(x_{i})_{n+1\leq i
\leq m}$. If one wants to estimate the corresponding $y_{i}$'s, one should
think of a specific estimator devoted to this task, referred by Vapnik as a
"transductive" estimator. This estimator may differ from an estimator designed
to the more general task "estimate on the whole domain". In this work, we
propose a generalized version both of the LASSO and the Dantzig Selector, based
on the geometrical remarks about the LASSO in pr\'evious works. The "usual"
LASSO and Dantzig Selector, as well as new estimators interpreted as
transductive versions of the LASSO, appear as special cases. These estimators
are interesting at least from a theoretical point of view: we can give
theoretical guarantees for these estimators under hypotheses that are relaxed
versions of the hypotheses required in the papers about the "usual" LASSO.
These estimators can also be efficiently computed, with results comparable to
the ones of the LASSO.

文章提出了基于稀疏假设的广义 LASSO 和 Dantzig Selector 方法，同时解决了 transductive 任务的问题。