Joachim Dahl

Algorithms and software for total variation image reconstruction via first-order methods

This paper describes new algorithms and related software for total variation (TV) image reconstruction, more specifically: denoising, inpainting, and deblurring. The algorithms are based on one of Nesterov’s first-order methods, tailored to the image processing applications in such a way that, except for the mandatory regularization parameter, the user needs not specify any parameters in the algorithms. The software is written in C with interface to Matlab (version 7.5 or later), and we demonstrate its performance and use with examples.

Covariance selection for nonchordal graphs via chordal embedding

We describe algorithms for maximum likelihood estimation of Gaussian graphical models with conditional independence constraints. This problem is also known as covariance selection, and it can be expressed as an unconstrained convex optimization problem with a closed-form solution if the underlying graph is chordal. The focus of the paper is on iterative algorithms for covariance selection with nonchordal graphs. We first derive efficient methods for evaluating the gradient and Hessian of the log-likelihood function when the underlying graph is chordal. The algorithms are formulated as simple recursions on a clique tree associated with the graph. We also show that the gradient and Hessian mappings are easily inverted when the underlying graph is chordal. We then exploit these results to obtain efficient implementations of Newtons method and the conjugate gradient method for large nonchordal graphs, by embedding the graph in a chordal graph.

Proportional fairness in multi-carrier system: upper bound and approximation algorithms

The solution for the optimal multi-carrier (MC) proportional fair scheduling (PFS) is prohibitively complex to obtain. In this letter we obtain an upper bound on the achievable proportional fairness (PF) performance in the case of finite PFS window size. Next, three approximation algorithms are proposed. The first one is computationally complex, but achieves near-optimal PF. The two others achieve a good tradeoff between throughput and PF with low complexity

Implementation of nonsymmetric interior-point methods for linear optimization over sparse matrix cones

We describe an implementation of nonsymmetric interior-point methods for linear cone programs defined by two types of matrix cones: the cone of positive semidefinite matrices with a given chordal sparsity pattern and its dual cone, the cone of chordal sparse matrices that have a positive semidefinite completion. The implementation takes advantage of fast recursive algorithms for evaluating the function values and derivatives of the logarithmic barrier functions for these cones. We present experimental results of two implementations, one of which is based on an augmented system approach, and a comparison with publicly available interior-point solvers for semidefinite programming.

Proportional fairness in multi-carrier system with multi-slot frames: upper bound and user multiplexing algorithms

Optimal Proportional Fair Scheduling (PFS) in a multi-carrier system is a prohibitively complex combinatorial problem. In this paper we consider practical time frames with multiple time slots, where this optimal allocation becomes even more complex. Therefore, we derive bounds for the optimal proportional fair allocation, by means of convex optimization, and propose approximation algorithms where several users can be time-multiplexed on a same subchannel. With a much lower complexity than the optimal allocation, these algorithms achieve an excellent tradeoff between throughput and proportional fairness, even with the increased signaling overhead.

Joint estimation of short-term and long-term predictors in speech coders

In low bit-rate coders, the near-sample and far-sample redundancies of the speech signal are usually removed by a cascade of a short-term and a long-term linear predictor. These two predictors are usually found in a sequential and therefore suboptimal approach. In this paper we propose an analysis model that jointly finds the two predictors by adding a regularization term in the minimization process to impose sparsity constraints on a high order predictor. The result is a linear predictor that can be easily factorized into the short-term and long-term predictors. This estimation method is then incorporated into an Algebraic Code Excited Linear Prediction scheme and shows to have a better performance than traditional cascade methods and other joint optimization methods, offering lower distortion and higher perceptual speech quality.

Logarithmic barriers for sparse matrix cones

Algorithms are presented for evaluating gradients and Hessians of logarithmic barrier functions for two types of convex cones: the cone of positive semidefinite matrices with a given sparsity pattern and its dual cone, the cone of sparse matrices with the same pattern that have a positive semidefinite completion. Efficient large-scale algorithms for evaluating these barriers and their derivatives are important in interior-point methods for nonsymmetric conic formulations of sparse semidefinite programs. The algorithms are based on the multifrontal method for sparse Cholesky factorization.

Approximate maximum-likelihood estimation using semidefinite programming

We consider semidefinite relaxations of a quadratic optimization problem with polynomial constraints. This is an extension of quadratic problems with Boolean variables. Such combinatorial problems cannot, in general, be solved in polynomial time. Semidefinite relaxation has been proposed as a promising technique to give provable good bounds on certain Boolean quadratic problems in polynomial time. We formulate the extensions from Boolean variables to quarternary variables using (i) a polynomial relaxation or (ii) standard semidefinite relaxations of a linear transformation of Boolean variables. We compare the two different relaxation approaches analytically. The relaxations can all be expressed as semidefinite programs, which can be solved efficiently using e.g. interior point methods. Applications of our results include maximum likelihood estimation in communication systems, which we explore in simulations in order to compare the quality of the different relaxations with optimal solutions.

Robust implementation of the MUSIC algorithm

The problem of estimating frequencies of sinusoids in noise has been studied intensively by the signal processing community during the last decades. Traditionally high resolution subspace-based techniques suffer from high computational complexity, and generally sensitive to the colored noise. We present here a frequency-domain based subspace parameter estimation algorithm termed frequency-selective MUltiple SIgnal Classification (F-MUSIC) that is based on the signal and noise subspace orthogonality property. The method is computationally efficient in providing estimates in the selected subband compared to the classic MUSIC. The performance of F-MUSIC is evaluated and compared to both MUSIC and Cramér-Rao lower bound (CRLB). In a low signal to noise ratio (SNR) with colored noise scenarios, F-MUSIC outperforms MUSIC.

Multiple-Description l1-Compression

Multiple descriptions (MDs) is a method to obtain reliable signal transmissions on erasure channels. An MD encoder forms several descriptions of the signal and each description is independently transmitted across an erasure channel. The reconstruction quality then depends on the set of received descriptions. In this paper, we consider the design of redundant descriptions in an MD setup using l1-minimization with Euclidean distortion constraints. In this way we are able to obtain sparse descriptions using convex optimization. The proposed method allows for an arbitrary number of descriptions and supports both symmetric and asymmetric distortion design. We show that MDs with partial overlapping information corresponds to enforcing coupled constraints in the proposed convex optimization problem. To handle the coupled constraints, we apply dual decompositions which makes first-order methods applicable and thereby admit solutions for large-scale problems, e.g., coding entire images or image sequences. We show by examples that the proposed framework generates non-trivial sparse descriptions and non-trivial refinements. We finally show that the sparse descriptions can be quantized and encoded using off-the-shell encoders such as the set partitioning in hierarchical trees (SPIHT) encoder, however, the proposed method shows a rate-distortion loss compared to state-of-the-art image MD encoders.