[Paper Review] Systems Aliasing in Dynamic Network Reconstruction: Issues on Low Sampling Frequencies

Zuogon Yue, Johan Thunberg+3 more|arXiv (Cornell University)|May 27, 2016

Gene Regulatory Network AnalysisBiochemistry, Genetics and Molecular Biology12 references3 citations

TL;DR

This paper introduces the concept of 'system aliasing' in dynamic network reconstruction, where low sampling frequencies cause multiple continuous-time systems to produce identical discrete samples, making network structure inference ambiguous. It establishes a Nyquist-Shannon-like sampling theorem to determine the minimal sampling frequency that avoids aliasing and proposes a reconstruction algorithm for sparse networks under no aliasing, with extensions using sparsity priors when aliasing occurs.

ABSTRACT

Network reconstruction of dynamical continuous-time (CT) systems is motivated by applications in many fields. Due to experimental limitations, especially in biology, data could be sampled at low frequencies, leading to significant challenges in network inference. We introduce the concept of "system aliasing" and characterize the minimal sampling frequency that allows reconstruction of CT systems from low sampled data. A test criterion is also proposed to check whether system aliasing is presented. With no system aliasing, the paper provides an algorithm to reconstruct dynamic network from data in the presence of noise. In addition, when there is system aliasing we perform studies that add additional prior information of the system such as sparsity. This paper opens new directions in modelling of network systems where samples have significant costs. Such tools are essential to process the available data in applications subject to current experimental limitations.

Motivation & Objective

• Address the challenge of reconstructing dynamic networks from low-sampling-frequency data, especially in biological systems where high-frequency sampling is experimentally infeasible.
• Identify and formalize the problem of 'system aliasing', where distinct continuous-time systems produce identical sampled outputs, leading to ambiguous network inference.
• Establish a theoretical minimum sampling frequency that avoids system aliasing, analogous to the Nyquist-Shannon sampling theorem.
• Develop a reconstruction algorithm for sparse dynamic networks when system aliasing is absent, even in the presence of noise.
• Investigate the feasibility of recovering ground-truth network structures when system aliasing is present, using additional prior information such as sparsity.

Proposed method

• Define system aliasing as the existence of multiple distinct continuous-time systems that generate identical sampled outputs at a given sampling frequency.
• Derive a Nyquist-Shannon-like sampling theorem that specifies the minimal sampling frequency required to avoid system aliasing, based on the eigenstructure of the system matrix.
• Propose a reconstruction algorithm using nuclear norm minimization and sparsity-promoting optimization (e.g., LASSO) to infer sparse network structures from low-frequency sampled data.
• Introduce a test criterion to detect whether system aliasing is present in a given dataset by analyzing the eigenvalues and Jordan structure of the system matrix.
• Utilize prior knowledge of network sparsity to constrain the solution space when aliasing is detected, improving identifiability under uncertainty.
• Formulate the reconstruction problem in terms of a matrix transformation $ h_Z(ullet) $, and use trace-based Frobenius norm comparisons to assess equivalence between candidate system matrices.

Experimental results

Research questions

• RQ1 What is the minimal sampling frequency that ensures unique reconstruction of a continuous-time dynamic network from sampled data?
• RQ2 Under what conditions does system aliasing occur, and how can it be detected in practice from sampled time-series data?
• RQ3 Can sparse network structures still be reliably reconstructed when system aliasing is present, and what additional constraints are needed?
• RQ4 How does noise in the sampled data affect the identifiability of the underlying network structure?
• RQ5 To what extent can prior knowledge of sparsity improve the accuracy of network reconstruction under low sampling frequencies?

Key findings

• The paper establishes a theoretical lower bound on sampling frequency that prevents system aliasing, derived from the eigenstructure of the system matrix and analogous to the Nyquist-Shannon theorem.
• System aliasing occurs when multiple distinct continuous-time systems produce identical sampled outputs, rendering network structure inference ambiguous.
• A test criterion is proposed to detect system aliasing by analyzing the Jordan canonical form and eigenvalue distribution of candidate system matrices.
• When no system aliasing is present, the proposed algorithm successfully reconstructs sparse dynamic networks from low-frequency data with bounded noise.
• In the presence of system aliasing, the use of sparsity priors significantly improves the feasibility of identifying the true network structure, though exact recovery remains theoretically limited.
• The Frobenius norm of the transformation $ h_Z(ullet) $ is used to compare system matrices, and it is shown that $ \|h_Z(A_1)\|_F = \|h_Z(A_2)\|_F $ if and only if $ A_1 \sim A_2 $, providing a key equivalence test.