Matricial Wasserstein and Unsupervised Tracking

The context of this work is spectral analysis of multivariable times-series as this may arise in processing signals originating in antenna and sensor arrays. The salient feature of these time signals is that they contain information about moving scatterers/targets which may not be known a priori. That is, neither the number nor the physical properties of scatterers may be known in advance, a fact which necessitates that analysis needs to be model free. Thus, what is important is to attain reliable and high resolution spectral estimates based on short-time observations due to the expected motion of objects within the scattering field. Traditional spectral analysis methods such as spectrograms and maximum entropy, Capon, etc. techniques, are often severely constrained by the non-stationary nature of timeseries, which necessitates very short observation records. Thus, our goal has been to develop natural regularization techniques that allow smooth interpolation of spectrograms in time, thereby improving resolution and robustness. Since power spectra are matrix-valued measure, we sought to develop geometric tools that are based on weak∗ continuous metrics, such as Wasserstein metrics, only for matrix-valued functions. The present work is largely based on [1] where such a theory was laid out. Introduction Traditional techniques in sensor arrays, detection, and estimation rely on periodogram-based methods, maximumentropy techniques, or beamforming. While these are ubiquitous, they are severely limited when dealing with nonstationary time-series. Assimilation of data from disparate sources and dealing with systemic biases are quite challenging on their own, and they are even more so, when dealing with non-stationary time-series; the non-stationary nature of the signal content necessitates that spectral analysis is based on short observation records. Our approach has been to seek natural ways to quantify distance between spectra, and to develop geometric tools based on that. More specifically, power spectra and probability distributions can be viewed as points on a suitable manifold. Then, slowly time-varying power spectra can be viewed as flows (geodesics) on this manifold, and interpolation/extrapolation of power spectral estimates as well as L. Ning is with Brigham and Women’s Hospital, Harvard Medical School, Psychiatry Neuroimaging Laboratory 1249 Boylston St. Boston, MA 02421; email: [email protected] T.T. Georgiou is with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, Minnesota MN 55455, USA; email: [email protected] R. Sandhu and A. Tannenbaum are with the Departments of Computer Science and Applied Mathematics/Statistics, Stony Brook University, Stony Brook, NY 11794. email: [email protected] The research was partially supported by the AFOSR under Grants FA9550-12-1-0319 and FA9550-15-1-0045. uncertainty quantification can all be carried out within the associated metric topology. A desirable feature of metrics that would make them suitable for applications is weak∗ continuity; this is the property that small changes in a distribution affects measurements in a continuous manner. Interesting, L2, L1, Kullback-Leibler divergence and several many other popular notions of distance fail in this regard. To this end we sought to generalize the notion of Wasserstein distance, which is natural in this respect for scalar distributions. Our interest in matrix valued measures stems from the fact that those represent power spectra of vector-valued time-series. In turn, vector-valued time-series may represent measurements of different modalities across a distributed array of sensors that reflect frequency/color, polarization, spatial characteristics, and other attributes that are thought to characterize target properties. Thus, we are interested in a “transport-based geometry” for such matrix-valued distributions as well as a “transport theory” that is flexible with regard to the preservation of mass/power across time. Advances on this front make it possible to tackle in a natural way smoothing and interpolation between inconsistent data sets or time-varying characteristics of a series as well as the computation of optical flow for object tracking in series of frames. Besides the relevance of geodesics as a tool for modeling, tracking, morphing, data assimilation/association, etc., for power spectra and images alike, we expect to advance concepts of resolution and the quantification of uncertainty in rigorous terms. Background Our topic, to define and explore Wasserstein-like metrics and the corresponding geometry for matrix-valued densities or measures, is an attempt to build on the classical subject of optimal mass transport (OMT). This subject has in recent years witnessed a fast developing phase with many applications in physics, probability theory, economics, etc. Standard references are [2], [3], [4] and some of the most significant recent developments that in particular have inspired our work are traced to [5], [6], [7]. Example We wish to exemplify the spirit of our approach by a simple example, which would be quite challenging (unless, one uses hindsight and tailors a parametric method to the task). The task is to track slowly time-varying sinusoidal signals in noise that represent the echo from a pair of targets, whose position changes with regard to an antenna array at the same time. We wish to emphasize that we seek nonparametric techniques, and although our example with be treated parametrically (since a model is known), that in general would not be the case. Hence, our interest in nonparametric techniques. Fig. 1: Sensors and sources – correlated time-series Consider two sources of sound (equivalently, scatterers in an antenna array’s field of view) as shown in Figure 1 moving relative to each other in opposite directions and, exchanging positions in the process, relative to the pair of stationary microphones. The emitted sounds are recorded by each microphone in considerable amount of ambient noise. Their respective frequencies and intensities vary due to Doppler shift and due to the change in their relative proximity to the two sensors. Computer simulated signals are also shown in Figure 1. The task is to distinguish the relative position of the two sources using frequency analysis of the recorded time-series. The time-series is vector-valued (having two entries = number of sensors/channels). Thus, the power spectrum is matrix-valued (2 × 2 in this case). Short time maximum entropy reconstruction of the power spectrum is shown in Figure 2. Our convention is to show in the (1,1)-subplot the spectrogram of the first sensor, the (2,2)-subplot that of the second sensor, in the (1,2)-subplot the absolute value of the cross spectrum and in the (2,1)-subplot its phase. Next, in Figure 3 we display, following the same convention, the regularized spectrogram where we used optimal mass transport geometry and interpolated the time-distributions in Figure 2 by an OMT geodesic. At each frequency and time, either plot represents the (color-coded) intensity of a Hermitian matrix, namely, the value of the power spectral matrix density. (The (1,1) and (2,2) entries are real, while the (1,2) and (2,1) are complex conjugate of each other and their real part and phase are displayed accordingly as indicated above.) Singular value decomposition reveals the directionality of the incoming energy. Thus, by identifying at each point in time the maxima of the power, we determine the corresponding singular vectors which reflect the relative portion of power in each sensor for the corresponding power source, thereby, pointing to its relative position (after suitable calibration). Figures 4 and 5 show the paths that the corresponding singular vectors traced over time, based on the maximum entropy spectrogram of Figure 2 and that of the OMT geodesic reconstruction in Figure 3, respectively. What is especially revealing is the dramatic improvement in resolution and consistency afforded by the use of OMT geodesics. To some degree, this is to be expected since the OMT geometry induces a natural (weakly continuous) metric where the approximation takes place and, automatically, the optimal path determined. time fr eq u en cy