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74
Just Relax: Convex Programming Methods for Identifying Sparse Signals in Noise
, 2006
"... This paper studies a difficult and fundamental problem that arises throughout electrical engineering, applied mathematics, and statistics. Suppose that one forms a short linear combination of elementary signals drawn from a large, fixed collection. Given an observation of the linear combination that ..."
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Cited by 416 (2 self)
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This paper studies a difficult and fundamental problem that arises throughout electrical engineering, applied mathematics, and statistics. Suppose that one forms a short linear combination of elementary signals drawn from a large, fixed collection. Given an observation of the linear combination that has been contaminated with additive noise, the goal is to identify which elementary signals participated and to approximate their coefficients. Although many algorithms have been proposed, there is little theory which guarantees that these algorithms can accurately and efficiently solve the problem. This paper studies a method called convex relaxation, which attempts to recover the ideal sparse signal by solving a convex program. This approach is powerful because the optimization can be completed in polynomial time with standard scientific software. The paper provides general conditions which ensure that convex relaxation succeeds. As evidence of the broad impact of these results, the paper describes how convex relaxation can be used for several concrete signal recovery problems. It also describes applications to channel coding, linear regression, and numerical analysis.
Just relax: Convex programming methods for subset selection and sparse approximation
, 2004
"... Subset selection and sparse approximation problems request a good approximation of an input signal using a linear combination of elementary signals, yet they stipulate that the approximation may only involve a few of the elementary signals. This class of problems arises throughout electrical enginee ..."
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Cited by 94 (5 self)
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Subset selection and sparse approximation problems request a good approximation of an input signal using a linear combination of elementary signals, yet they stipulate that the approximation may only involve a few of the elementary signals. This class of problems arises throughout electrical engineering, applied mathematics and statistics, but small theoretical progress has been made over the last fifty years. Subset selection and sparse approximation both admit natural convex relaxations, but the literature contains few results on the behavior of these relaxations for general input signals. This report demonstrates that the solution of the convex program frequently coincides with the solution of the original approximation problem. The proofs depend essentially on geometric properties of the ensemble of elementary signals. The results are powerful because sparse approximation problems are combinatorial, while convex programs can be solved in polynomial time with standard software. Comparable new results for a greedy algorithm, Orthogonal Matching Pursuit, are also stated. This report should have a major practical impact because the theory applies immediately to many realworld signal processing problems.
On the exponential convergence of matching pursuit in quasicoherent dictionaries
 IEEE TRANS. INFORMATION TH
, 2006
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MPTK: Matching pursuit made tractable
 in Proc. Int. Conf. on Acoustic Speech and Signal Processing
, 2006
"... Matching Pursuit (MP) aims at finding sparse decompositions of signals over redundant bases of elementary waveforms. Traditionally, MP has been considered too slow an algorithm to be applied to reallife problems with highdimensional signals. Indeed, in terms of floating points operations, its typi ..."
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Cited by 51 (6 self)
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Matching Pursuit (MP) aims at finding sparse decompositions of signals over redundant bases of elementary waveforms. Traditionally, MP has been considered too slow an algorithm to be applied to reallife problems with highdimensional signals. Indeed, in terms of floating points operations, its typical numerical implementations have a complexity of ¢¤£¦¥¨§� © and are associated with impractical runtimes. In this paper, we propose a new architecture which exploits the structure shared by many redundant MP dictionaries, and thus decreases its complexity to ¢¤£¦¥�������¥¨ ©. This architecture is implemented in a new software toolkit, called MPTK (the Matching Pursuit Toolkit), which is able to reach, e.g., ������� � real time for a typical MP analysis scenario applied to a 1 hour long audio track. This substantial acceleration makes it possible, from now on, to explore and apply MP in the framework of reallife, highdimensional data processing problems. 1.
Bayesian Harmonic Models for Musical Signal Analysis
 in Bayesian Statistics 7
, 2002
"... This paper is concerned with the Bayesian analysis of musical signals. The ultimate aim is to use Bayesian hierarchical structures in order to infer quantities at the highest level, including such quantities as musical pitch, dynamics, timbre, instrument identity, etc. Analysis of real musical si ..."
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Cited by 45 (8 self)
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This paper is concerned with the Bayesian analysis of musical signals. The ultimate aim is to use Bayesian hierarchical structures in order to infer quantities at the highest level, including such quantities as musical pitch, dynamics, timbre, instrument identity, etc. Analysis of real musical signals is complicated by many things, including the presence of transient sounds, noises and the complex structure of musical pitches in the frequency domain. The problem is truly Bayesian in that there is a wealth of (often subjective) prior knwowledge about how musical signals are constructed, which can be exploited in order to achieve more accurate inference about the musical structure. Here we propose developments to an earlier Bayesian model which describes each component `note' at a given time in terms of a fundamental frequency, partials (`harmonics'), and amplitude. This basic model is modified for greater realism to include nonwhite residuals, timevarying amplitudes and partials `detuned' from the natural linear relationship. The unknown parameters of the new model are simulated using a variable dimension MCMC algorithm, leading to a highly sophisticated analysis tool. We discuss how the models and algorithms can be applied for feature extraction, polyphonic music transcription, source separation and restoration of musical sources
Environmental sound recognition with timefrequency audio
"... Abstract—The paper considers the task of recognizing environmental sounds for the understanding of a scene or context surrounding an audio sensor. A variety of features have been proposed for audio recognition, including the popular Melfrequency cepstral coefficients (MFCCs) which describe the au ..."
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Cited by 42 (0 self)
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Abstract—The paper considers the task of recognizing environmental sounds for the understanding of a scene or context surrounding an audio sensor. A variety of features have been proposed for audio recognition, including the popular Melfrequency cepstral coefficients (MFCCs) which describe the audio spectral shape. Environmental sounds, such as chirpings of insects and sounds of rain which are typically noiselike with a broad flat spectrum, may include strong temporal domain signatures. However, only few temporaldomain features have been developed to characterize such diverse audio signals previously. Here, we perform an empirical feature analysis for audio environment characterization and propose to use the matching pursuit (MP) algorithm to obtain effective time–frequency features. The MPbased method utilizes a dictionary of atoms for feature selection, resulting in a flexible, intuitive and physically interpretable set of features. The MPbased feature is adopted to supplement the MFCC features to yield higher recognition accuracy for environmental sounds. Extensive experiments are conducted to demonstrate the effectiveness of these joint features for unstructured environmental sound classification, including listening tests to study human recognition capabilities. Our recognition system has shown to produce comparable performance as human listeners. Index Terms—Audio classification, auditory scene recognition, data representation, feature extraction, feature selection, matching pursuit, Melfrequency cepstral coefficient (MFCC). I.
Instrumentspecific harmonic atoms for midlevel music representation
 IEEE Trans. on Audio, Speech and Lang. Proc
, 2008
"... Abstract—Several studies have pointed out the need for accurate midlevel representations of music signals for information retrieval and signal processing purposes. In this paper, we propose a new midlevel representation based on the decomposition of a signal into a small number of sound atoms or m ..."
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Cited by 36 (6 self)
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Abstract—Several studies have pointed out the need for accurate midlevel representations of music signals for information retrieval and signal processing purposes. In this paper, we propose a new midlevel representation based on the decomposition of a signal into a small number of sound atoms or molecules bearing explicit musical instrument labels. Each atom is a sum of windowed harmonic sinusoidal partials whose relative amplitudes are specific to one instrument, and each molecule consists of several atoms from the same instrument spanning successive time windows. We design efficient algorithms to extract the most prominent atoms or molecules and investigate several applications of this representation, including polyphonic instrument recognition and music visualization. Index Terms—Midlevel representation, music information retrieval, music visualization, sparse decomposition. I.