Pau Bofill

Underdetermined blind source separation using sparse representations

The scope of this work is the separation of N sources from M linear mixtures when the underlying system is underdetermined, that is, when Mi N. If the input distribution is sparse the mixing matrix can be estimated either by external optimization or by clustering and, given the mixing matrix, a minimal l1 norm representation of the sources can be obtained by solving a low-dimensional linear programming problem for each of the data points. Yet, when the signals per se do not satisfy this assumption, sparsity can still be achieved by realizing the separation in a sparser transformed domain. The approach is illustrated here for M = 2. In this case we estimate both the number of sources and the mixing matrix by the maxima of a potential function along the circle of unit length, and we obtain the minimal l1 norm representation of each data point by a linear combination of the pair of basis vectors that enclose it. Several experiments with music and speech signals show that their time-domain representation is not sparse enough. Yet, excellent results were obtained using their short-time Fourier transform, including the separation of up to six sources from two mixtures. ? 2001 Elsevier Science B.V. All rights reserved.

First stereo audio source separation evaluation campaign: data, algorithms and results

This article provides an overview of the first stereo audio source separation evaluation campaign, organized by the authors. Fifteen underdetermined stereo source separation algorithms have been applied to various audio data, including instantaneous, convolutive and real mixtures of speech or music sources. The data and the algorithms are presented and the estimated source signals are compared to reference signals using several objective performance criteria.

The 2008 Signal Separation Evaluation Campaign: A Community-Based Approach to Large-Scale Evaluation

This paper introduces the first community-based Signal Separation Evaluation Campaign (SiSEC 2008), coordinated by the authors. This initiative aims to evaluate source separation systems following specifications agreed between the entrants. Four speech and music datasets were contributed, including synthetic mixtures as well as microphone recordings and professional mixtures. The source separation problem was split into four tasks, each evaluated via different objective performance criteria. We provide an overview of these datasets, tasks and criteria, summarize the results achieved by the submitted systems and discuss organization strategies for future campaigns.

Identifying Single Source Data for Mixing Matrix Estimation in Instantaneous Blind Source Separation

This paper presents a simple yet effective way of improving the estimate of the mixing matrix, in instantaneous blind source separation, by using only reliable data. n nThe paper describes how the idea of detecting single source data is implemented by selecting only the data which remain for two consecutive frames in the same spatial signature. Such data, which are most likely to belong to a single source, are then used to accurately identify the spatial directions of the sources and, hence, the mixing matrix. n nThe paper also presents a refined histogram procedure which improves on the potential function method to estimate the mixing matrix, in the two dimensional case (two sensors). n nThe approach was experimentally evaluated and submitted to the first Stereo Audio Source Separation Evaluation Campaign (SASSEC), with good results in matrix estimation both for development and test data.

Comparison of simulated annealing and mean field annealing as applied to the generation of block designs

This paper describes an experimental comparison between a discrete stochastic optimization procedure (Simulated Annealing, SA) and a continuous deterministic one (Mean Field Annealing), as applied to the generation of Balanced Incomplete Block Designs (BIBDs). A neural cost function for BIBD generation is proposed with connections of arity four, and its continuous counterpart is derived, as required by the mean field formulation. Both strategies are optimized with regard to the critical temperature, and the expected cost to the first solution is used as a performance measure for the comparison. The results show that SA performs slightly better, but the most important observation is that the pattern of difficulty across the 25 problem instances tried is very similar for both strategies, implying that the main factor to success is the energy landscape, rather than the exploration procedure used.

Underdetermined convoluted source reconstruction using LP and SOCP, and a neural approximator of the optimizer

The middle-term goal of this research project is to be able to recover several sound sources from a binaural life recording, by previously measuring the acoustic response of the room. As a previous step, this paper focuses on the reconstruction of n sources from mconvolutive mixtures when m < n (underdetermined case), assuming the mixing matrix is known. n nThe reconstruction is done in the frequency domain by assuming that the source components are Laplacian in their real and imaginary parts. By posterior likelihood optimization, this leads to norm 1 minimization subject to the mixing equations, which is an instance of linear programming (LP). Alternatively, the assumption of Laplacianity imposed on the magnitudes leads to second order cone programming (SOCP). n nPerformance experiments are run from synthetic mixtures based on realistic simulations of each source-microphone impulse response. Two sets of sources are used as benchmarks: four speech utterances and six short violin melodies. Results show S/N reconstruction ratios around 10dB. If any, SOCP performs slightly better. n nSOCP is probably too slow for real-time processing. In the last part of this paper we train a neural network to predict the response of the optimizer. Preliminary results show that the approach is feasible but yet inmature.

MBMUDs: a combinatorial extension of BIBDs showing good optimality behaviour

Abstract The construction of a balanced incomplete block design (BIBD) is formulated in terms of combinatorial optimization by defining a cost function that reaches its lower bound on all and only those configurations corresponding to a BIBD. This cost function is a linear combination of distribution measures for each of the properties of a block design (number of plots, uniformity of rows, uniformity of columns, and balance). The approach generalizes naturally to a super-class BIBDs, which we call maximally balanced maximally uniform designs (MBMUDs), that allow two consecutive values for their design parameters [ r , r +1; k , k +1; λ , λ +1]. In terms of combinatorial balance, MBMUDs are the closest possible approximation to BIBDs for all experimental settings where no set of admissible parameters exists. Thus, other design classes previously proposed with the same approximation aim—such as RDGs, SRDGs and NBIBDs of type I—can be viewed as particular cases of MBMUDs. Interestingly, experimental results show that the proposed combinatorial cost function has a monotonic relation with A- and D-statistical optimality in the space of designs with uniform rows and columns, while its computational cost is much lower.

Neural cost functions and search strategies for the generation of block designs: an experimental evaluation

A constraint satisfaction problem, namely the generation of Balanced Incomplete Block Designs (v, b, r, kappa, lambda)-BIBDs, is cast in terms of function optimization. A family of cost functions that both suit the problem and admit a neural implementation is defined. An experimental comparison spanning this repertoire of cost functions and three neural relaxation strategies (Down-Hill search, Simulated Annealing and a new Parallel Mean Search procedure), as applied to all BIBDs of up to 1000 entries, has been undertaken. The experiments were performed on a Connection Machine CM-200 and their analysis required a careful study of performance measures. The simplest cost function stood out as the best one for the three strategies. Parallel Mean Search, with several processors searching cooperatively in parallel, could solve a larger number of problems than the same number of processors working independently, but Simulated Annealing yielded overall the best results. Other conclusions, as detailed in the paper, could be drawn from the comparison, BIBDs remaining a challenging problem for neural optimization algorithms.

Learning by Back-Propagation: Computing in a Systolic Way

In this paper we present a systolic algorithm for back-propagation, a supervised, iteratived, gradient-descent, connectionist learning rule. The algorithm works on feedforward networks where connections can skip layers and fully exploits spatial and training parallelism, which are inherent to back-propagation. Spatial parallelism arises during the propagation of activity—forward—and error—backward—for a particular input-output. On the other hand, when this computation is carried out simultaneously for all input-output pairs, training parallelism is obtained. In the spatial dimension, a single systolic ring carries out sequentially the three main steps of the learning rule—forward, backward and weight increments update. Furthermore, the same pattern of matrix delivery is used in both the forward and the backward passes. In this manner, the algorithm preserves the similarity of the forward and backward passes in the original model. The resulting systolic algorithm is dual with respect to the pattern of matrix delivery—either columns or rows.