Motivated by recent work on atomic norms in inverse problems, we propose a new approach to line spectral estimation that provides theoretical guarantees for the mean-squared-error (MSE) performance in the presence of noise and without knowledge of the model order. We propose an abstract theory of denoising with atomic norms and specialize this theory to provide a convex optimization problem for estimating the frequencies and phases of a mixture of complex exponentials. We show that the associated convex optimization problem can be solved in polynomial time via semidefinite programming (SDP). We also show that the SDP can be approximated by an l1-regularized least-squares problem that achieves nearly the same error rate as the SDP but can scale to much larger problems. We compare both SDP and l1-based approaches with classical line spectral analysis methods and demonstrate that the SDP outperforms the l1 optimization which outperforms MUSIC, Cadzow's, and Matrix Pencil approaches in terms of MSE over a wide range of signal-to-noise ratios.
Cadzow @cite_49 proposed a heuristic that improves over MUSIC by exploiting the Toeplitz structure of the matric of moments by alternately projecting between the linear space of Toeplitz matrices and the space of rank @math matrices where @math is the desired model order. Cadzow's technique is very similar @cite_18 to a popular technique in time series literature @cite_25 @cite_48 called Singular Spectrum Analysis @cite_29 , which uses autocorrelation matrix instead of the matrix of moments for projection. Both these techniques may be viewed as instances of structured low rank approximation @cite_14 which exploit additional structure beyond low rank structure used in subspace based methods such as MUSIC and ESPRIT. Cadzow's method has been identified as a fruitful preprocessing step for linear prediction methods @cite_45 . A survey of classical linear prediction methods can be found in @cite_45 @cite_20 and an extensive list of references is given in @cite_8 .
@cite_18 @cite_14 @cite_8 @cite_48 @cite_29 @cite_45 @cite_49 @cite_25 @cite_20
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An unsupervised learning algorithm for the separation of sound sources in one-channel music signals is presented. The algorithm is based on factorizing the magnitude spectrogram of an input signal into a sum of components, each of which has a fixed magnitude spectrum and a time-varying gain. Each sound source, in turn, is modeled as a sum of one or more components. The parameters of the components are estimated by minimizing the reconstruction error between the input spectrogram and the model, while restricting the component spectrograms to be nonnegative and favoring components whose gains are slowly varying and sparse. Temporal continuity is favored by using a cost term which is the sum of squared differences between the gains in adjacent frames, and sparseness is favored by penalizing nonzero gains. The proposed iterative estimation algorithm is initialized with random values, and the gains and the spectra are then alternatively updated using multiplicative update rules until the values converge. Simulation experiments were carried out using generated mixtures of pitched musical instrument samples and drum sounds. The performance of the proposed method was compared with independent subspace analysis and basic nonnegative matrix factorization, which are based on the same linear model. According to these simulations, the proposed method enables a better separation quality than the previous algorithms. Especially, the temporal continuity criterion improved the detection of pitched musical sounds. The sparseness criterion did not produce significant improvements The problem of recovering a sparse signal x Rn from a relatively small number of its observations of the form y = Ax Rk, where A is a known matrix and k « n, has recently received a lot of attention under the rubric of compressed sensing (CS) and has applications in many areas of signal processing such as data cmpression, image processing, dimensionality reduction, etc. Recent work has established that if A is a random matrix with entries drawn independently from certain probability distributions then exact recovery of x from these observations can be guaranteed with high probability. In this paper, we show that Toeplitz-structured matrices with entries drawn independently from the same distributions are also sufficient to recover x from y with high probability, and we compare the performance of such matrices with that of fully independent and identically distributed ones. The use of Toeplitz matrices in CS applications has several potential advantages: (i) they require the generation of only O(n) independent random variables; (ii) multiplication with Toeplitz matrices can be efficiently implemented using fast Fourier transform, resulting in faster acquisition and reconstruction algorithms; and (iii) Toeplitz-structured matrices arise naturally in certain application areas such as system identification. This paper explores a seemingly counter-intuitive idea: the possibility of accelerating the solution of certain linear equations by adding even more equations to the problem. The basic insight is to trade-off problem size by problem structure. We test this idea on Toeplitz equations, in which case the expense of a larger set of equations easily leads to circulant structure. The idea leads to a very simple iterative algorithm, which works for a certain class of Toeplitz matrices, each iteration requiring only two circular convolutions. In the symmetric definite case, numerical experiments show that the method can compete with the preconditioned conjugate gradient method (PCG), which achieves O(nlogn) performance. Because the iteration does not converge for all Toeplitz matrices, we give necessary and sufficient conditions to ensure convergence (for not necessarily symmetric matrices), and suggest an efficient convergence test. In the positive definite case we determine the value of the free parameter of the circulant that leads to the fastest convergence, as well as the corresponding value for the spectral radius of the iteration matrix. Although the usefulness of the proposed iteration is limited in the case of ill-conditioned matrices, we believe that the results show that the problem size problem structure trade-off deserves further study. We present a probabilistic generative model for timing deviations in expressive music performance. The structure of the proposed model is equivalent to a switching state space model. The switch variables correspond to discrete note locations as in a musical score. The continuous hidden variables denote the tempo. We formulate two well known music recognition problems, namely tempo tracking and automatic transcription (rhythm quantization) as filtering and maximum a posteriori (MAP) state estimation tasks. Exact computation of posterior features such as the MAP state is intractable in this model class, so we introduce Monte Carlo methods for integration and optimization. We compare Markov Chain Monte Carlo (MCMC) methods (such as Gibbs sampling, simulated annealing and iterative improvement) and sequential Monte Carlo methods (particle filters). Our simulation results suggest better results with sequential methods. The methods can be applied in both online and batch scenarios such as tempo tracking and transcription and are thus potentially useful in a number of music applications such as adaptive automatic accompaniment, score typesetting and music information retrieval.