Maximizing Energy Concentration Research Articles

Guided graph spectral embedding: Application to the C. elegans connectome.

Graph spectral analysis can yield meaningful embeddings of graphs by providing insight into distributed features not directly accessible in nodal domain. Recent efforts in graph signal processing have proposed new decompositions—for example, based on wavelets and Slepians—that can be applied to filter signals defined on the graph. In this work, we take inspiration from these constructions to define a new guided spectral embedding that combines maximizing energy concentration with minimizing modified embedded distance for a given importance weighting of the nodes. We show that these optimization goals are intrinsically opposite, leading to a well-defined and stable spectral decomposition. The importance weighting allows us to put the focus on particular nodes and tune the trade-off between global and local effects. Following the derivation of our new optimization criterion, we exemplify the methodology on the C. elegans structural connectome. The results of our analyses confirm known observations on the nematode’s neural network in terms of functionality and importance of cells. Compared with Laplacian embedding, the guided approach, focused on a certain class of cells (sensory neurons, interneurons, or motoneurons), provides more biological insights, such as the distinction between somatic positions of cells, and their involvement in low- or high-order processing functions.

When Slepian Meets Fiedler: Putting a Focus on the Graph Spectrum

The study of complex systems greatly benefits from graph models and their analysis. In particular, the eigendecomposition of the graph Laplacian lets emerge properties of global organization from local interactions; e.g., the Fiedler vector has the smallest nonzero eigenvalue and plays a key role for graph clustering. Graph signal processing focuses on the analysis of signals that are attributed to the graph nodes. Again, the eigendecomposition of the graph Laplacian is important to define the graph Fourier transform and extend conventional signal-processing operations to graphs. Here, we introduce the design of Slepian graph signals by maximizing energy concentration in a predefined subgraph given a graph spectral bandlimit. We establish a novel link with classical Laplacian embedding and graph clustering, which provides a meaning to localized graph frequencies.

A multiresolution non-negative tensor factorization approach for single channel sound source separation

We propose a single channel audio source separation method to alleviate the smearing effects caused by fixed time-frequency (TF) resolution Short-Time Fourier Transform (STFT). We introduce a multiresolution representation based on Non-negative Tensor Factorization (NTF) where each layer of the tensor represents the mixture signal at a different time-frequency resolution. In order to fuse the information at different layers, the source separation is modeled as a joint optimization problem where the optimal solution is derived based on the Kullback–Leibler (KL) divergence. The resynthesis is made through an additional adaptive weighted fusion procedure which combines the sources separated at different scales by maximizing energy concentration. Numerical results over a large sound database indicate that the proposed joint optimization scheme enhances the quality of the separated sources both in terms of the conventional and the perceptual distortion measures.

Exact solutions to the acoustic source reconstruction problem

In this paper, we generalize the recently developed analytical solutions of the radiation modes problem to the determination of closed-form expressions for the singular value expansion of a number of integral operators that map the boundary velocity of a baffled planar structure onto the acoustic pressure radiated in far-field or intermediate regions. Exact solutions to this problem involve prolate spheroidal wave functions that correspond to a set of independent distributions with finite spatial support and maximal energy concentration in a given bandwidth of the transform domain. A stable solution to the inverse source reconstruction problem is obtained by decomposing the unknown boundary velocity into a number of efficiently radiating singular velocity patterns that correspond to the number of degrees of freedom of the radiated field. It is found that the degree of ill-posedness of the inverse problem is significantly reduced, when considering a hemi-circular observation arc with respect to a linear array of sensors, by a factor scaling on the small angular aperture subtended by the observation line. Estimates are derived from the spatial resolution limits that can be achieved in the source reconstruction problem from the dimension of the efficiently radiating subspace.

Analytic solutions to the acoustic source reconstruction problem

In this paper, we generalize the recently developed analytical solutions of the radiation modes problem to the determination of closed-form expressions for the singular value expansion of a number of integral operators that map the boundary velocity of a baffled planar structure onto the acoustic pressure radiated in far-field or intermediate regions. Exact solutions to this problem involve prolate spheroidal wave functions that correspond to a set of independent distributions with finite spatial support and maximal energy concentration in a given bandwidth of the transform domain. A stable solution to the inverse source reconstruction problem is obtained by decomposing the unknown boundary velocity into a number of efficiently radiating singular velocity patterns that correspond to the number of degrees of freedom of the radiated field. It is found that the degree of ill-posedness of the inverse problem is significantly reduced, when considering a hemi-circular observation arc with respect to a linear array of sensors, by a factor scaling on the small angular aperture subtended by the observation line. Estimates are derived from the spatial resolution limits that can be achieved in the source reconstruction problem from the dimension of the efficiently radiating subspace.

Analytic solutions of the radiation modes problem and the active control of sound power

This paper explores the common mathematical foundation of two different problems: the first one arises in electrical engineering for the detection and the spectral estimation of signals in noise and the second one appears in acoustics for the calculation of the acoustic radiation modes of rectangular structures. Although apparently unrelated, it is found that both applications draw on the so–called concentration problem: of determining which functions that are band–limited in one domain have maximal energy concentration within a region of the transform domain. The analytic solutions to problems of this form are seen to involve prolate spheroidal wave functions. In particular, exact expressions are given for the radiation efficiencies and shapes of the radiation modes of a baffled beam as well as their asymptotics. It is shown that a generalization of the concentration problem to the two–dimensional case provides analytic solutions that solve with a good accuracy, although approximately, the radiation problem. The properties of these special functions provide a rigorous basis of understanding some previously observed features of these applications, namely the grouping property of the radiation modes of a baffled panel and the physical limitations for the active control of sound from a panel.

Frequency spectral properties of an ULF electromagnetic signal around the 21 July 1995, M=5.7, Yong Deng (China) earthquake

The frequency temporal analysis of the electromagnetic field recorded from 30 May 1995 to 24 August 1995 at a site about a hundred kilometer North of the epicenter of the 21 July 1995 Yong Deng earthquake (China) has shown a clear systematic variation on an electric dipole. In the ultra low frequency range from 0.01 to 0.0005Hz, a three-fold decrease of the frequency of the maximal energy concentration was observed, on 30 May 1995 from 0.005 to 0.0010Hz few hours at the time of the earthquake. This decrease was nearly monotonic, with superimposed small scale step-like fluctuations, and continued after the Yong Deng earthquake for about two weeks, when the minimum of 0.001Hz was observed just before its abrupt disappearance.The ULF signal is observed on one of the four electric dipoles. The instrumentation and the data acquisition system do not generate this signal. The signal could be generated by an electrokinetic effect depending on stress field changes in the local environment of one or two probes. The results need further understanding and interpretation.