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Bogdan Dumitrescu - Dictionary Learning Algorithms and Applications

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Bogdan Dumitrescu Dictionary Learning Algorithms and Applications

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This book covers all the relevant dictionary learning algorithms, presenting them in full detail and showing their distinct characteristics while also revealing the similarities. It gives implementation tricks that are often ignored but that are crucial for a successful program. Besides MOD, K-SVD, and other standard algorithms, it provides the significant dictionary learning problem variations, such as regularization, incoherence enforcing, finding an economical size, or learning adapted to specific problems like classification. Several types of dictionary structures are treated, including shift invariant; orthogonal blocks or factored dictionaries; and separable dictionaries for multidimensional signals. Nonlinear extensions such as kernel dictionary learning can also be found in the book. The discussion of all these dictionary types and algorithms is enriched with a thorough numerical comparison on several classic problems, thus showing the strengths and weaknesses of each algorithm. A few selected applications, related to classification, denoising and compression, complete the view on the capabilities of the presented dictionary learning algorithms. The book is accompanied by code for all algorithms and for reproducing most tables and figures.

  • Presents all relevant dictionary learning algorithms - for the standard problem and its main variations - in detail and ready for implementation;
  • Covers all dictionary structures that are meaningful in applications;
  • Examines the numerical properties of the algorithms and shows how to choose the appropriate dictionary learning algorithm.

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Bogdan Dumitrescu - Dictionary Learning Algorithms and Applications
Dictionary Learning Algorithms and Applications
Bogdan Dumitrescu
Dictionary Learning Algorithms and Applications

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Springer International Publishing AG, part of Springer Nature 2018
Bogdan Dumitrescu and Paul Irofti Dictionary Learning Algorithms and Applications
1. Sparse Representations
Bogdan Dumitrescu 1 and Paul Irofti 2
(1)
Department of Automatic Control and Systems Engineering, Faculty of Automatic Control and Computers, University Politehnica of Bucharest, Bucharest, Romania
(2)
Department of Computer Science, Faculty of Mathematics and Computer Science, University of Bucharest, Bucharest, Romania
Abstract
Sparse representations using overcomplete dictionaries have found many signal processing applications. We present the main ways of formulating sparse approximation problems and discuss their advantages over the classical orthogonal transforms. The foremost difficulty is the computation of sparse representations, since it amounts to find the sparsest among the infinite number of solutions of an underdetermined linear system, a problem that has a combinatorial character. The most successful classes of algorithms are based on greedy approaches and convex relaxation. We describe in detail a representative algorithm from each class, namely Orthogonal Matching Pursuit and FISTA. In some circumstances, the algorithms are guaranteed to find the sparsest solution and we present sets of conditions that ensure their success. In preparation for stating the dictionary learning problem, we debate the advantages and drawbacks of learned dictionaries with respect to fixed ones. Since learning is based on training signals from the application at hand, adapted dictionaries have the potential of more faithful sparse representations, an advantage that overwhelms the necessity of (mainly off-line) extra computation.
1.1 The Sparse Model
The main character in this book is a matrix Dictionary Learning Algorithms and Applications - image 1 called dictionary. Its columns are named atoms; we denote d j the j th column of D ; we assume that all atoms are normalized
Dictionary Learning Algorithms and Applications - image 2
(1.1)
where Picture 3 is the Euclidean norm, which is the default norm hereafter. In most cases the dictionary is overcomplete, which means m < n .
The typical use of a dictionary is for sparse representation. A vector called here signal has a sparse representation if it can be written as a - photo 4 , called here signal, has a sparse representation if it can be written as a linear combination of few atoms, i.e.,
Dictionary Learning Algorithms and Applications - image 5
(1.2)
where most of the coefficients x j are zero. We name Dictionary Learning Algorithms and Applications - image 6 the support of the signal. For example, in Fig. ) is also named sparsity level ; we will typically denote it s and note that Dictionary Learning Algorithms and Applications - image 7 . The notion of sparsity is not exactly defined, but it assumes that x 0 n and, more often than not, x 0 m .
Fig 11 Sparse representation of a signal The used atoms are red and the - photo 8
Fig. 1.1
Sparse representation of a signal. The used atoms are red and the nonzero coefficients are blue. The unused atoms are pink
Example 1.1
Let us see first what kind of sparsity is usually not interesting. Figure shows a signal y that is sparse in the canonical basis, which means that many of its elements are equal to zero. In this case, the dictionary is trivially the unit matrix, D = I , and the representation coincides with the signal, y = x . Such sparse signals may appear in practice, but are mostly devoid of content (exactly because they are sparse). A sound with these values would be mostly silence; an image would be nearly black, with a few gray dots. Fig 12 The uninteresting case a signal that is sparse in the canonical base - photo 9
Fig 12 The uninteresting case a signal that is sparse in the canonical base - photo 10
Fig. 1.2
The uninteresting case: a signal that is sparse in the canonical base
Example 1.2
Sinusoidal signals are not sparse in the time domain, but their Discrete Fourier Transform (DFT) is sparse. Sinusoids are easily recognizable in sounds, since they are pure tones, and they are important ingredients in music and in speech signals. A basis of sinusoids is easy to make; since we aim at real transforms and thus avoid the DFT, the Discrete Cosine Transform (DCT) is an immediate candidate. However, for other signals than sinusoids, the representation in the DCT base may not be even approximately sparse. To increase the family of sparsely representable signals, we can add more atoms to the DCT, obtaining an overcomplete dictionary. An example is the overcomplete DCT , that can be built by setting first i 1 m j 1 n the average of each column is subtracted from that - photo 11 , i =1: m , j =1: n ; the average of each column is subtracted from that column, excepting the first, which has equal elements; finally, the atoms are normalized. (Note that this construction can be different, depending on what type of DCT transform is taken as prototype; here it is the type I DCT, slightly modified.) Figure shows the atoms of this overcomplete DCT with m =8, n =16; although they are discrete signals, the atoms are drawn with continuous line, to better visualize the shape of the corresponding signals. The first and third columns of subplots contain the atoms of the standard DCT (since n =2 m , they are obtained when j 1 is even).
Fig 13 The atoms of an 816 overcomplete DCT transform An example of sparse - photo 12
Fig. 1.3
The atoms of an 816 overcomplete DCT transform
An example of sparse signal in the overcomplete DCT basis is y =0.5 d 10.2 d 6, since it can be represented with only two atoms of the dictionary. However, if we represent the same signal in the DCT base (the odd-numbered atoms in Fig. ), then all the coefficients of the representation are nonzero. Picture 13
Remark 1.3
Orthogonal transforms have found many uses in signal processing. They have many advantages, amongst which the easiness of representation. If Picture 14 is such a transform, then the transform of a signal y is x = Ay . The inverse transform is simply A T , which in our case can be viewed as the dictionary, since y = A T x . So, finding the representation consists of a matrix multiplication. Moreover, for many transforms, like the DFT or DCT, there are fast algorithms that reduce the complexity to Dictionary Learning Algorithms and Applications - image 15
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