Sparse coding in learned dictionaries has been established as a successful approach for signal denoising, source separation and solving inverse problems in general. A dictionary learning method adapts an initial dictionary to a particular signal class by iteratively computing an approximate factorization of a training data matrix into a dictionary and a sparse coding matrix. The learned dictionary is characterized by two properties: the coherence of the dictionary to observations of the signal class, and the self-coherence of the dictionary atoms. A high coherence to the signal class enables the sparse coding of signal observations with a small approximation error, while a low self-coherence of the atoms guarantees atom recovery and a more rapid residual error decay rate for the sparse coding algorithm. The two goals of high signal coherence and low self-coherence are typically in conflict, therefore one seeks a trade-off between them, depending on the application. We present a dictionary learning method with an effective control over the self-coherence of the trained dictionary, enabling a trade-off between maximizing the sparsity of codings and approximating an equi-angular tight frame.

The enhancement of speech degraded by real-world interferers is a highly relevant and difficult task. Its importance arises from the multitude of practical applications, whereas the difficulty is due to the fact that interferers are often nonstationary and potentially similar to speech. The goal of monaural speech enhancement is to separate a single mixture into its underlying clean speech and interferer components. This under-determined problem is solved by incorporating prior knowledge in the form of learned speech and interferer dictionaries. The clean speech is recovered from the degraded speech by sparse coding of the mixture in a composite dictionary consisting of the concatenation of a speech and an interferer dictionary. Enhancement performance is measured using objective measures and is limited by two effects. A too sparse coding of the mixture causes the speech component to be explained with too few speech dictionary atoms, which induces an approximation error we denote source distortion. However, a too dense coding of the mixture results in source confusion, where parts of the speech component are explained by interferer dictionary atoms and vice-versa. Our method enables the control of the source distortion and source confusion trade-off, and therefore achieves superior performance compared to powerful approaches like geometric spectral subtraction and codebook-based filtering, for a number of challenging interferer classes such as speech babble and wind noise.

The enhancement of speech degraded by non-stationary interferers is a highly relevant and difficult task of many signal processing applications. We present a monaural speech enhancement method based on sparse coding of noisy speech signals in a composite dictionary, consisting of the concatenation of a speech and interferer dictionary, both being possibly over-complete. The speech dictionary is learned off-line on a training corpus, while an environment specific interferer dictionary is learned on-line during speech pauses. Our approach optimizes the trade-off between source distortion and source confusion, and thus achieves significant improvements on objective quality measures like cepstral distance, in the speaker dependent and independent case, in several real-world environments and at low signal-to-noise ratios. Our enhancement method outperforms state-of-the-art methods like multi-band spectral subtraction and approaches based on vector quantization.

We study the problem of finding the dominant eigenvector of the sample covariance matrix, under additional constraints on the vector: a cardinality constraint limits the number of non-zero elements, and non-negativity forces the elements to have equal sign. This problem is known as sparse and non-negative principal component analysis (PCA), and has many applications including dimensionality reduction and feature selection. Based on expectation-maximization for probabilistic PCA, we present an algorithm for any combination of these constraints. Its complexity is at most quadratic in the number of dimensions of the data. We demonstrate significant improvements in performance and computational efficiency compared to other constrained PCA algorithms, on large data sets from biology and computer vision. Finally, we show the usefulness of non-negative sparse PCA for unsupervised feature selection in a gene clustering task.