Machine-learning methods and apparatus are provided to solve blind source separation problems with an unknown number of sources and having a signal propagation model with features such as wave-like propagation, medium-dependent velocity, attenuation, diffusion, and/or advection, between sources and sensors. In exemplary embodiments, multiple trials of non-negative matrix factorization are performed for a fixed number of sources, with selection criteria applied to determine successful trials. A semi-supervised clustering procedure is applied to trial results, and the clustering results are evaluated for robustness using measures for reconstruction quality and cluster separation. The number of sources is determined by comparing these measures for different trial numbers of sources. Source locations and parameters of the signal propagation model can also be determined. Disclosed methods are applicable to a wide range of spatial problems including chemical dispersal, pressure transients, and electromagnetic signals, and also to non-spatial problems such as cancer mutation.
Please help me write a proper abstract based on the patent claims. CLAIM: 1. A method, comprising: with a computer: performing a plurality of non-negative matrix factorization (NMF) trials on mixed signals generated by one or more unidentified sources and detected by a plurality of sensors, each of the NMF trials producing a predetermined number of candidate sources referred to each of the sensors according to a signal propagation model; identifying clusters of the candidate sources and determining basis sources, by performing clustering on the candidate sources; evaluating results of the clustering by: calculating a first parameter representing reconstruction error of the basis sources, and calculating a second parameter representing separation of the identified clusters; and determining a first number of the unidentified sources for which a combination of the calculated first parameter and the calculated second parameter is optimized. 2. The method of claim 1, wherein the clustering is constrained to retain equal numbers of the candidate sources within each cluster. 3. The method of claim 1, further comprising: from first clusters of candidate sources and corresponding first basis sources obtained from clustering performed on the candidate sources produced by NMF trials that each produce the first number of candidate sources: identifying the unidentified sources from the first basis sources. 4. The method of claim 3, further comprising: responsive to identifying the unidentified sources, adjusting one or more conditions of a physical environment in which the sensors are located, wherein the adjusting comprises at least one or more of the following acts, for at least a first one of the identified sources and/or a first one of the sensors: probing local conditions at or near a location of the first identified source, counteracting the first identified source to reduce a signal strength of the first identified source, or making changes in the physical environment that will reduce a contribution from the first identified source to the mixed signal detected by the first sensor, without changing the signal strength of the first identified source. 5. The method of claim 3, wherein the sources are sources of one or more of: atmospheric pollution, water pollution, pressure transients, acoustic signals, seismic disturbances, or electromagnetic interference. 6. The method of claim 1, further comprising determining spatial locations for one or more of the unidentified sources. 7. One or more computer-readable storage media storing computer-readable instructions that, when executed by a computer, cause the computer to perform the method of claim 1. 8. The method of claim 1, wherein the propagation model is a wave-like propagation model. 9. The method of claim 8, wherein each NMF trial further produces a shift matrix comprising coefficients, each of the coefficients representing a signal shift from one of the candidate sources to one of the sensors. 10. The method of claim 9, wherein each shift matrix comprises a shift vector for each produced candidate source, and further comprising: for at least a first determined basis source corresponding to a first identified cluster of first candidate sources, producing a final shift vector dependent on the shift vectors of the first candidate sources. 11. The method of claim 8, wherein each of the NMF trials satisfies at least one of the following constraints: (i) a reconstruction quality measure is greater than or equal to a first predetermined threshold; (ii) reconstructed signals at each of the sensors comprise a contribution that is greater than or equal to a second predetermined threshold from each of the candidate sources determined from the NMF trial; or (iii) every coefficient of the produced shift matrix corresponds to a shift that is less than or equal to a third predetermined threshold. 12. The method of claim 1, wherein the propagation model incorporates diffusion and advection. 13. The method of claim 12, wherein at least one NMF trial further produces estimates of one or more transport parameters selected from the group consisting of advection velocity and dispersion coefficient. 14. The method of claim 12, wherein at least one NMF trial performs non-linear least squares minimization of a cost function which incorporates Green's functions for each unidentified source. 15. The method of claim 12, wherein the clustering and evaluating are performed in a plurality of iterations, and each iteration comprises: removing candidate sources based on reconstruction error to leave remaining candidate sources; identifying provisional clusters of the remaining candidate sources; and evaluating the second parameter representing separation of the identified provisional clusters; wherein the identified provisional clusters of a final iteration of the plurality of iterations are the identified clusters. 16. The method of claim 1, wherein the detected mixed signals are related to the unidentified sources by a mixing matrix, the NMF trials are subject to constraints for each sensor on sums of respective mixing matrix coefficients, and the NMF trials use a non-convex optimization procedure. 17. A computer-implemented system comprising: one or more computing nodes each comprising one or more processors, memory coupled thereto, and one or more network adapters, the one or more computing nodes being interconnected by one or more network connections and configured to perform a method comprising: performing a plurality of non-negative matrix factorization (NMF) trials on mixed signals generated by one or more unidentified sources and detected by a plurality of sensors, each of the NMF trials producing a predetermined number of candidate sources referred to each of the sensors according to a signal propagation model; identifying clusters of the candidate sources and determining basis sources, by performing clustering on the candidate sources; evaluating results of the clustering by: calculating a first parameter representing reconstruction error of the basis sources, and calculating a second parameter representing separation of the identified clusters; and determining a first number of the unidentified sources for which a combination of the calculated first parameter and the calculated second parameter is optimized. 18. The computer-implemented system of claim 17, wherein the method further comprises receiving data for the mixed signals, the data being produced using the plurality of sensors responsive to detection of the mixed signals. 19. The computer-implemented system of claim 17, wherein the first parameter is a Frobenius norm and the second parameter is a silhouette value. 20. A method comprising: with a computer: receiving data for mixed signals generated by one or more unidentified sources and detected by a plurality of sensors, the data being produced using the plurality of sensors responsive to detection of the mixed signals; performing a plurality of non-negative matrix factorization (NMF) trials on the mixed signals, each of the NMF trials producing a predetermined number of candidate sources referred to each of the sensors according to a signal propagation model; identifying clusters of the candidate sources and determining basis sources, by performing clustering on the candidate sources, wherein the clustering is constrained to retain equal numbers of candidate sources within each cluster; evaluating results of the clustering by: calculating a first parameter representing reconstruction error of the basis sources, and calculating a second parameter representing separation of the identified clusters; and determining a first number of the unidentified sources for which a combination of the calculated first parameter and the calculated second parameter is optimized; wherein the NMF trials that each produce the first number of candidate sources are first NMF trials, the clusters identified from first NMF trials are first clusters, and the basis sources determined from the first NMF trials are first basis sources; identifying the unidentified sources as the first basis sources; determining spatial locations and signal amplitudes for the identified sources; and responsive to identifying the unidentified sources, adjusting one or more conditions of a physical environment in which the sensors are located, wherein the adjusting comprises making changes in the physical environment that will reduce the mixed signal(s) detected by at least one of the sensors from a first identified source without changing the signal amplitude of the first identified source.