scholarly journals AMP Chain Graphs: Minimal Separators and Structure Learning Algorithms

2020 ◽  
Vol 69 ◽  
pp. 419-470
Author(s):  
Mohammad Ali Javidian ◽  
Marco Valtorta ◽  
Pooyan Jamshidi
Keyword(s):  
Structure Learning ◽  
Ground Truth ◽  
High Dimensional ◽  
Adjacent Pair ◽  
Underlying Graph ◽  
Polynomial Time Algorithms ◽  
Disjoint Sets ◽  
Minimal Separator ◽  
Chain Graphs ◽  
Special Case

This paper deals with chain graphs (CGs) under the Andersson–Madigan–Perlman (AMP) interpretation. We address the problem of finding a minimal separator in an AMP CG, namely, finding a set Z of nodes that separates a given non-adjacent pair of nodes such that no proper subset of Z separates that pair. We analyze several versions of this problem and offer polynomial time algorithms for each. These include finding a minimal separator from a restricted set of nodes, finding a minimal separator for two given disjoint sets, and testing whether a given separator is minimal. To address the problem of learning the structure of AMP CGs from data, we show that the PC-like algorithm is order dependent, in the sense that the output can depend on the order in which the variables are given. We propose several modifications of the PC-like algorithm that remove part or all of this order-dependence. We also extend the decomposition-based approach for learning Bayesian networks (BNs) to learn AMP CGs, which include BNs as a special case, under the faithfulness assumption. We prove the correctness of our extension using the minimal separator results. Using standard benchmarks and synthetically generated models and data in our experiments demonstrate the competitive performance of our decomposition-based method, called LCD-AMP, in comparison with the (modified versions of) PC-like algorithm. The LCD-AMP algorithm usually outperforms the PC-like algorithm, and our modifications of the PC-like algorithm learn structures that are more similar to the underlying ground truth graphs than the original PC-like algorithm, especially in high-dimensional settings. In particular, we empirically show that the results of both algorithms are more accurate and stabler when the sample size is reasonably large and the underlying graph is sparse

scholarly journals An Order-Independent Algorithm for Learning Chain Graphs

2021 ◽  
Vol 34 (1) ◽  
Author(s):  
Mohammad Ali Javidian ◽  
Marco Valtorta ◽  
Pooyan Jamshidi
Keyword(s):  
Directed Acyclic Graphs ◽  
High Dimensional ◽  
P Values ◽  
Order Dependence ◽  
Error Measures ◽  
Acyclic Graphs ◽  
Simulation Results ◽  
Improved Performance ◽  
Low Dimensional ◽  
Chain Graphs

LWF chain graphs combine directed acyclic graphs and undirected graphs. We propose a PC-like algorithm, called PC4LWF, that finds the structure of chain graphs under the faithfulness assumption to resolve the problem of scalability of the proposed algorithm by Studeny (1997). We prove that PC4LWF is order dependent, in the sense that the output can depend on the order in which the variables are given. This order dependence can be very pronounced in high-dimensional settings. We propose two modifications of the PC4LWF algorithm that remove part or all of this order dependence. Simulation results with different sample sizes, network sizes, and p-values demonstrate the competitive performance of the PC4LWF algorithms in comparison with the LCD algorithm proposed by Ma et al. (2008) in low-dimensional settings and improved performance (with regard to error measures) in high-dimensional settings.

scholarly journals Stability in the high-dimensional cohomology of congruence subgroups

2020 ◽  
Vol 156 (4) ◽  
pp. 822-861
Author(s):  
Jeremy Miller ◽  
Rohit Nagpal ◽  
Peter Patzt
Keyword(s):  
Congruence Subgroup ◽  
Stability Result ◽  
High Dimensional ◽  
Vanishing Theorem ◽  
Congruence Subgroups ◽  
Codimension One ◽  
Representation Stability ◽  
Steinberg Module ◽  
Finiteness Properties ◽  
Special Case

We prove a representation stability result for the codimension-one cohomology of the level-three congruence subgroup of $\mathbf{SL}_{n}(\mathbb{Z})$. This is a special case of a question of Church, Farb, and Putman which we make more precise. Our methods involve proving finiteness properties of the Steinberg module for the group $\mathbf{SL}_{n}(K)$ for $K$ a field. This also lets us give a new proof of Ash, Putman, and Sam’s homological vanishing theorem for the Steinberg module. We also prove an integral refinement of Church and Putman’s homological vanishing theorem for the Steinberg module for the group $\mathbf{SL}_{n}(\mathbb{Z})$.

High-dimensional quantum key distribution using polarization-phase encoding: security analysis

2020 ◽  
Vol 18 (06) ◽  
pp. 2050031
Author(s):  
Ali Mehri-Toonabi ◽  
Mahdi Davoudi Darareh ◽  
Shahrooz Janbaz
Keyword(s):  
Quantum Key Distribution ◽  
Degrees Of Freedom ◽  
Single Photon ◽  
Transmission Rate ◽  
Security Analysis ◽  
Key Distribution ◽  
High Dimensional ◽  
Effective Transmission ◽  
Special Case ◽  
Polarization Phase

In this work, we introduce a high-dimensional polarization-phase (PoP)-based quantum key distribution protocol, briefly named PoP[Formula: see text], where [Formula: see text] is the dimension of a hybrid quantum state including polarization and phase degrees of freedom of the same photon, and [Formula: see text] is the number of mutually unbiased bases. We present a detailed description of the PoP[Formula: see text] protocol as a special case, and evaluate its security against various individual and coherent eavesdropping strategies, and in each case, we compare it with the BB84 and the two-dimensional (TD)-PoP protocols. In all the strategies, the error threshold and the effective transmission rate of the PoP[Formula: see text] protocol are far greater than the other two protocols. Unlike most high-dimensional protocols, the simplicity of producing and detecting the qudits and the use of conventional components (such as traditional single-photon sources and quantum channels) are among the features of the PoP[Formula: see text] protocol.

scholarly journals Kernelized movement primitives

2019 ◽  
Vol 38 (7) ◽  
pp. 833-852 ◽  
Author(s):  
Yanlong Huang ◽  
Leonel Rozo ◽  
João Silvério ◽  
Darwin G Caldwell
Keyword(s):  
Motor Skills ◽  
Parametric Representation ◽  
High Dimensional ◽  
Coordinate Systems ◽  
Motion Patterns ◽  
External Perturbations ◽  
Human Skills ◽  
The Many ◽  
Special Case ◽  
Additional Constraints

Imitation learning has been studied widely as a convenient way to transfer human skills to robots. This learning approach is aimed at extracting relevant motion patterns from human demonstrations and subsequently applying these patterns to different situations. Despite the many advancements that have been achieved, solutions for coping with unpredicted situations (e.g., obstacles and external perturbations) and high-dimensional inputs are still largely absent. In this paper, we propose a novel kernelized movement primitive (KMP), which allows the robot to adapt the learned motor skills and fulfill a variety of additional constraints arising over the course of a task. Specifically, KMP is capable of learning trajectories associated with high-dimensional inputs owing to the kernel treatment, which in turn renders a model with fewer open parameters in contrast to methods that rely on basis functions. Moreover, we extend our approach by exploiting local trajectory representations in different coordinate systems that describe the task at hand, endowing KMP with reliable extrapolation capabilities in broader domains. We apply KMP to the learning of time-driven trajectories as a special case, where a compact parametric representation describing a trajectory and its first-order derivative is utilized. In order to verify the effectiveness of our method, several examples of trajectory modulations and extrapolations associated with time inputs, as well as trajectory adaptations with high-dimensional inputs are provided.

scholarly journals Causal Datasheet for Datasets: An Evaluation Guide for Real-World Data Analysis and Data Collection Design Using Bayesian Networks

2021 ◽  
Vol 4 ◽  
Author(s):  
Bradley Butcher ◽  
Vincent S. Huang ◽  
Christopher Robinson ◽  
Jeremy Reffin ◽  
Sema K. Sgaier ◽  
...  
Keyword(s):  
Global Health ◽  
Bayesian Networks ◽  
Sample Size ◽  
Observational Data ◽  
Real World ◽  
Structure Learning ◽  
Ground Truth ◽  
Research Process ◽  
Real World Data ◽  
Real World Datasets

Developing data-driven solutions that address real-world problems requires understanding of these problems’ causes and how their interaction affects the outcome–often with only observational data. Causal Bayesian Networks (BN) have been proposed as a powerful method for discovering and representing the causal relationships from observational data as a Directed Acyclic Graph (DAG). BNs could be especially useful for research in global health in Lower and Middle Income Countries, where there is an increasing abundance of observational data that could be harnessed for policy making, program evaluation, and intervention design. However, BNs have not been widely adopted by global health professionals, and in real-world applications, confidence in the results of BNs generally remains inadequate. This is partially due to the inability to validate against some ground truth, as the true DAG is not available. This is especially problematic if a learned DAG conflicts with pre-existing domain doctrine. Here we conceptualize and demonstrate an idea of a “Causal Datasheet” that could approximate and document BN performance expectations for a given dataset, aiming to provide confidence and sample size requirements to practitioners. To generate results for such a Causal Datasheet, a tool was developed which can generate synthetic Bayesian networks and their associated synthetic datasets to mimic real-world datasets. The results given by well-known structure learning algorithms and a novel implementation of the OrderMCMC method using the Quotient Normalized Maximum Likelihood score were recorded. These results were used to populate the Causal Datasheet, and recommendations could be made dependent on whether expected performance met user-defined thresholds. We present our experience in the creation of Causal Datasheets to aid analysis decisions at different stages of the research process. First, one was deployed to help determine the appropriate sample size of a planned study of sexual and reproductive health in Madhya Pradesh, India. Second, a datasheet was created to estimate the performance of an existing maternal health survey we conducted in Uttar Pradesh, India. Third, we validated generated performance estimates and investigated current limitations on the well-known ALARM dataset. Our experience demonstrates the utility of the Causal Datasheet, which can help global health practitioners gain more confidence when applying BNs.

Weight-Equitable Subdivision of Red and Blue Points in the Plane

2018 ◽  
Vol 28 (01) ◽  
pp. 39-56 ◽  
Author(s):  
Jude Buot ◽  
Mikio Kano
Keyword(s):  
General Position ◽  
Sufficient Conditions ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Total Weight ◽  
Blue Point ◽  
Disjoint Sets ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Special Case

Let [Formula: see text] and [Formula: see text] be two disjoint sets of red points and blue points, respectively, in the plane in general position. Assign a weight [Formula: see text] to each red point and a weight [Formula: see text] to each blue point, where [Formula: see text] and [Formula: see text] are positive integers. Define the weight of a region in the plane as the sum of the weights of red and blue points in it. We give necessary and sufficient conditions for the existence of a line that bisects the weight of the plane whenever the total weight [Formula: see text] is [Formula: see text], for some integer [Formula: see text]. Moreover, we look closely into the special case where [Formula: see text] and [Formula: see text] since this case is important to generate a weight-equitable subdivision of the plane. Among other results, we show that for any configuration of [Formula: see text] with total weight [Formula: see text], for some integer [Formula: see text] and odd integer [Formula: see text], the plane can be subdivided into [Formula: see text] convex regions of weight [Formula: see text] if and only if [Formula: see text]. Using the proofs of the main result, we also give a polynomial time algorithm in finding a weight-equitable subdivision in the plane.

scholarly journals High-dimensional consistency in score-based and hybrid structure learning

2018 ◽  
Vol 46 (6A) ◽  
pp. 3151-3183 ◽  
Author(s):  
Preetam Nandy ◽  
Alain Hauser ◽  
Marloes H. Maathuis
Keyword(s):  
Structure Learning ◽  
Hybrid Structure ◽  
High Dimensional

scholarly journals On causal discovery with an equal-variance assumption

Biometrika ◽  
2019 ◽  
Vol 106 (4) ◽  
pp. 973-980 ◽  
Author(s):  
Wenyu Chen ◽  
Mathias Drton ◽  
Y Samuel Wang
Keyword(s):  
Structural Equation ◽  
Structure Learning ◽  
State Of The Art ◽  
Causal Structure ◽  
Equation Model ◽  
Causal Discovery ◽  
High Dimensional ◽  
Prior Work ◽  
Equal Variance ◽  
Error Terms

Summary Prior work has shown that causal structure can be uniquely identified from observational data when these follow a structural equation model whose error terms have equal variance. We show that this fact is implied by an ordering among conditional variances. We demonstrate that ordering estimates of these variances yields a simple yet state-of-the-art method for causal structure learning that is readily extendable to high-dimensional problems.

Multiple-cause discovery combined with structure learning for high-dimensional discrete data and application to stock prediction

2015 ◽  
Vol 20 (11) ◽  
pp. 4575-4588 ◽  
Author(s):  
Weiqi Chen ◽  
Zhifeng Hao ◽  
Ruichu Cai ◽  
Xiangzhou Zhang ◽  
Yong Hu ◽  
...  
Keyword(s):  
Structure Learning ◽  
Discrete Data ◽  
High Dimensional ◽  
Stock Prediction

scholarly journals Structure Learning for Hierarchical Regulatory Networks

2021 ◽  
Author(s):  
Anthony Federico ◽  
Joseph Kern ◽  
Xaralabos Varelas ◽  
Stefano Monti
Keyword(s):  
Breast Cancer ◽  
Biological Networks ◽  
Regulatory Networks ◽  
Structure Learning ◽  
High Dimensional Data ◽  
High Dimensional ◽  
Structural Constraints ◽  
Cancer Data ◽  
Multiple Networks ◽  
Graph Properties

Network analysis offers a powerful technique to model the relationships between genes within biological regulatory networks. Inference of biological network structures is often performed on high-dimensional data, yet is hindered by the limited sample size of high throughput "omics" data typically available. To overcome this challenge, we exploit known organizing principles of biological networks that are sparse, modular, and likely share a large portion of their underlying architecture. We present SHINE - Structure Learning for Hierarchical Networks - a framework for defining data-driven structural constraints and incorporating a shared learning paradigm for efficiently learning multiple networks from high-dimensional data. We show through simulations SHINE improves performance when relatively few samples are available and multiple networks are desired, by reducing the complexity of the graphical search space and by taking advantage of shared structural information. We evaluated SHINE on TCGA Pan-Cancer data and found learned tumor-specific networks exhibit expected graph properties of real biological networks, recapture previously validated interactions, and recapitulate findings in literature. Application of SHINE to the analysis of subtype-specific breast cancer networks identified key genes and biological processes for tumor maintenance and survival as well as potential therapeutic targets for modulating known breast cancer disease genes.

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