Learning dictionaries suitable for sparse coding instead of using engineered bases has proven effective in a variety of image processing tasks. This paper studies the optimization of dictionaries on image data where the representation is enforced to be explicitly sparse with respect to a smooth, normalized sparseness measure. This involves the computation of Euclidean projections onto level sets of the sparseness measure. While previous algorithms for this optimization problem had at least quasi-linear time complexity, here the first algorithm with linear time complexity and constant space complexity is proposed. The key for this is the mathematically rigorous derivation of a characterization of the projection's result based on a soft-shrinkage function. This theory is applied in an original algorithm called Easy Dictionary Learning (EZDL), which learns dictionaries with a simple and fast-to-compute Hebbian-like learning rule. The new algorithm is efficient, expressive and particularly simple to implement. It is demonstrated that despite its simplicity, the proposed learning algorithm is able to generate a rich variety of dictionaries, in particular a topographic organization of atoms or separable atoms. Further, the dictionaries are as expressive as those of benchmark learning algorithms in terms of the reproduction quality on entire images, and result in an equivalent denoising performance. EZDL learns approximately 30 % faster than the already very efficient Online Dictionary Learning algorithm, and is therefore eligible for rapid data set analysis and problems with vast quantities of learning samples.
The $k$-dimensional coding schemes refer to a collection of methods that attempt to represent data using a set of representative $k$-dimensional vectors, and include non-negative matrix factorization, dictionary learning, sparse coding, $k$-means clustering and vector quantization as special cases. Previous generalization bounds for the reconstruction error of the $k$-dimensional coding schemes are mainly dimensionality independent. A major advantage of these bounds is that they can be used to analyze the generalization error when data is mapped into an infinite- or high-dimensional feature space. However, many applications use finite-dimensional data features. Can we obtain dimensionality-dependent generalization bounds for $k$-dimensional coding schemes that are tighter than dimensionality-independent bounds when data is in a finite-dimensional feature space? The answer is positive. In this paper, we address this problem and derive a dimensionality-dependent generalization bound for $k$-dimensional coding schemes by bounding the covering number of the loss function class induced by the reconstruction error. The bound is of order $\mathcal{O}\left(\left(mk\ln(mkn)/n\right)^{\lambda_n}\right)$, where $m$ is the dimension of features, $k$ is the number of the columns in the linear implementation of coding schemes, $n$ is the size of sample, $\lambda_n>0.5$ when $n$ is finite and $\lambda_n=0.5$ when $n$ is infinite. We show that our bound can be tighter than previous results, because it avoids inducing the worst-case upper bound on $k$ of the loss function and converges faster. The proposed generalization bound is also applied to some specific coding schemes to demonstrate that the dimensionality-dependent bound is an indispensable complement to these dimensionality-independent generalization bounds.
In this note, we focus on smooth nonconvex optimization problems that obey: (1) all local minimizers are also global; and (2) around any saddle point or local maximizer, the objective has a negative directional curvature. Concrete applications such as dictionary learning, generalized phase retrieval, and orthogonal tensor decomposition are known to induce such structures. We describe a second-order trust-region algorithm that provably converges to a global minimizer efficiently, without special initializations. Finally we highlight alternatives, and open problems in this direction.
In this paper, a novel semi-supervised dictionary learning and sparse representation (SS-DLSR) is proposed. The proposed method benefits from the supervisory information by learning the dictionary in a space where the dependency between the data and class labels is maximized. This maximization is performed using Hilbert-Schmidt independence criterion (HSIC). On the other hand, the global distribution of the underlying manifolds were learned from the unlabeled data by minimizing the distances between the unlabeled data and the corresponding nearest labeled data in the space of the dictionary learned. The proposed SS-DLSR algorithm has closed-form solutions for both the dictionary and sparse coefficients, and therefore does not have to learn the two iteratively and alternately as is common in the literature of the DLSR. This makes the solution for the proposed algorithm very fast. The experiments confirm the improvement in classification performance on benchmark datasets by including the information from both labeled and unlabeled data, particularly when there are many unlabeled data.
We consider discriminative dictionary learning in a distributed online setting, where a network of agents aims to learn a common set of dictionary elements of a feature space and model parameters while sequentially receiving observations. We formulate this problem as a distributed stochastic program with a non-convex objective and present a block variant of the Arrow-Hurwicz saddle point algorithm to solve it. Using Lagrange multipliers to penalize the discrepancy between them, only neighboring nodes exchange model information. We show that decisions made with this saddle point algorithm asymptotically achieve a first-order stationarity condition on average.
We propose a robust approach for performing automatic species-level recognition of fossil pollen grains in microscopy images that exploits both global shape and local texture characteristics in a patch-based matching methodology. We introduce a novel criteria for selecting meaningful and discriminative exemplar patches. We optimize this function during training using a greedy submodular function optimization framework that gives a near-optimal solution with bounded approximation error. We use these selected exemplars as a dictionary basis and propose a spatially-aware sparse coding method to match testing images for identification while maintaining global shape correspondence. To accelerate the coding process for fast matching, we introduce a relaxed form that uses spatially-aware soft-thresholding during coding. Finally, we carry out an experimental study that demonstrates the effectiveness and efficiency of our exemplar selection and classification mechanisms, achieving $86.13\%$ accuracy on a difficult fine-grained species classification task distinguishing three types of fossil spruce pollen.
The refugee crisis is perhaps the single most challenging problem for Europe today. Hundreds of thousands of people have already traveled across dangerous sea passages from Turkish shores to Greek islands, resulting in thousands of dead and missing, despite the best rescue efforts from both sides. One of the main reasons is the total lack of any early warning-alerting system, which could provide some preparation time for the prompt and effective deployment of resources at the hot zones. This work is such an attempt for a systemic analysis of the refugee influx in Greece, aiming at (a) the statistical and signal-level characterization of the smuggling networks and (b) the formulation and preliminary assessment of such models for predictive purposes, i.e., as the basis of such an early warning-alerting protocol. To our knowledge, this is the first-ever attempt to design such a system, since this refugee crisis itself and its geographical properties are unique (intense event handling, little or no warning). The analysis employs a wide range of statistical, signal-based and matrix factorization (decomposition) techniques, including linear & linear-cosine regression, spectral analysis, ARMA, SVD, Probabilistic PCA, ICA, K-SVD for Dictionary Learning, as well as fractal dimension analysis. It is established that the behavioral patterns of the smuggling networks closely match (as expected) the regular burst and pause periods of store-and-forward networks in digital communications. There are also major periodic trends in the range of 6.2-6.5 days and strong correlations in lags of four or more days, with distinct preference in the Sunday-Monday 48-hour time frame. These results show that such models can be used successfully for short-term forecasting of the influx intensity, producing an invaluable operational asset for planners, decision-makers and first-responders.
Sparse representations has shown to be a very powerful model for real world signals, and has enabled the development of applications with notable performance. Combined with the ability to learn a dictionary from signal examples, sparsity-inspired algorithms are often achieving state-of-the-art results in a wide variety of tasks. Yet, these methods have traditionally been restricted to small dimensions mainly due to the computational constraints that the dictionary learning problem entails. In the context of image processing, this implies handling small image patches. In this work we show how to efficiently handle bigger dimensions and go beyond the small patches in sparsity-based signal and image processing methods. We build our approach based on a new cropped wavelet decomposition, which enables a multi-scale analysis with virtually no border effects. We then employ this as the base dictionary within a double sparsity model to enable the training of adaptive dictionaries. To cope with the increase of training data, while at the same time improving the training performance, we present an Online Sparse Dictionary Learning (OSDL) algorithm to train this model effectively, enabling it to handle millions of examples. This work shows that dictionary learning can be up-scaled to tackle a new level of signal dimensions, obtaining large adaptable atoms that we call trainlets.
Dictionary learning is the problem of estimating the collection of atomic elements that provide a sparse representation of measured/collected signals or data. This paper finds fundamental limits on the sample complexity of estimating dictionaries for tensor data by proving a lower bound on the minimax risk. This lower bound depends on the dimensions of the tensor and parameters of the generative model. The focus of this paper is on second-order tensor data, with the underlying dictionaries constructed by taking the Kronecker product of two smaller dictionaries and the observed data generated by sparse linear combinations of dictionary atoms observed through white Gaussian noise. In this regard, the paper provides a general lower bound on the minimax risk and also adapts the proof techniques for equivalent results using sparse and Gaussian coefficient models. The reported results suggest that the sample complexity of dictionary learning for tensor data can be significantly lower than that for unstructured data.
Despite the fact that different objects possess distinct class-specific features, they also usually share common patterns. Inspired by this observation, we propose a novel method to explicitly and simultaneously learn a set of common patterns as well as class-specific features for classification. Our dictionary learning framework is hence characterized by both a shared dictionary and particular (class-specific) dictionaries. For the shared dictionary, we enforce a low-rank constraint, i.e. claim that its spanning subspace should have low dimension and the coefficients corresponding to this dictionary should be similar. For the particular dictionaries, we impose on them the well-known constraints stated in the Fisher discrimination dictionary learning (FDDL). Further, we propose a new fast and accurate algorithm to solve the sparse coding problems in the learning step, accelerating its convergence. The said algorithm could also be applied to FDDL and its extensions. Experimental results on widely used image databases establish the advantages of our method over state-of-the-art dictionary learning methods.
We target the problem of sparse 3D reconstruction of dynamic objects observed by multiple unsynchronized video cameras with unknown temporal overlap. To this end, we develop a framework to recover the unknown structure without sequencing information across video sequences. Our proposed compressed sensing framework poses the estimation of 3D structure as the problem of dictionary learning, where the dictionary is defined as an aggregation of the temporally varying 3D structures. Given the smooth motion of dynamic objects, we observe any element in the dictionary can be well approximated by a sparse linear combination of other elements in the same dictionary (i. e. self-expression). Moreover, the sparse coefficients describing a locally linear 3D structural interpolation reveal the local sequencing information. Our formulation optimizes a biconvex cost function that leverages a compressed sensing formulation and enforces both structural dependency coherence across video streams, as well as motion smoothness across estimates from common video sources. We further analyze the reconstructability of our approach under different capture scenarios, and its comparison and relation to existing methods. Experimental results on large amounts of synthetic data as well as real imagery demonstrate the effectiveness of our approach.
In support of art investigation, we propose a new source sepa- ration method that unmixes a single X-ray scan acquired from double-sided paintings. Unlike prior source separation meth- ods, which are based on statistical or structural incoherence of the sources, we use visual images taken from the front- and back-side of the panel to drive the separation process. The coupling of the two imaging modalities is achieved via a new multi-scale dictionary learning method. Experimental results demonstrate that our method succeeds in the discrimination of the sources, while state-of-the-art methods fail to do so.
In this paper, we invest the domain transfer learning problem with multi-instance data. We assume we already have a well-trained multi-instance dictionary and its corresponding classifier from the source domain, which can be used to represent and classify the bags. But it cannot be directly used to the target domain. Thus we propose to adapt them to the target domain by adding an adaptive term to the source domain classifier. The adaptive function is a linear function based a domain transfer multi-instance dictionary. Given a target domain bag, we first map it to a bag-level feature space using the domain transfer dictionary, and then apply a the linear adaptive function to its bag-level feature vector. To learn the domain-transfer dictionary and the adaptive function parameter, we simultaneously minimize the average classification error of the target domain classifier over the target domain training set, and the complexities of both the adaptive function parameter and the domain transfer dictionary. The minimization problem is solved by an iterative algorithm which update the dictionary and the function parameter alternately. Experiments over several benchmark data sets show the advantage of the proposed method over existing state-of-the-art domain transfer multi-instance learning methods.
Dictionary learning is a cutting-edge area in imaging processing, that has recently led to state-of-the-art results in many signal processing tasks. The idea is to conduct a linear decomposition of a signal using a few atoms of a learned and usually over-completed dictionary instead of a pre-defined basis. Determining a proper size of the to-be-learned dictionary is crucial for both precision and efficiency of the process, while most of the existing dictionary learning algorithms choose the size quite arbitrarily. In this paper, a novel regularization method called the Grouped Smoothly Clipped Absolute Deviation (GSCAD) is employed for learning the dictionary. The proposed method can simultaneously learn a sparse dictionary and select the appropriate dictionary size. Efficient algorithm is designed based on the alternative direction method of multipliers (ADMM) which decomposes the joint non-convex problem with the non-convex penalty into two convex optimization problems. Several examples are presented for image denoising and the experimental results are compared with other state-of-the-art approaches.
Sparse matrix factorization is a popular tool to obtain interpretable data decompositions, which are also effective to perform data completion or denoising. Its applicability to large datasets has been addressed with online and randomized methods, that reduce the complexity in one of the matrix dimension, but not in both of them. In this paper, we tackle very large matrices in both dimensions. We propose a new factoriza-tion method that scales gracefully to terabyte-scale datasets, that could not be processed by previous algorithms in a reasonable amount of time. We demonstrate the efficiency of our approach on massive functional Magnetic Resonance Imaging (fMRI) data, and on matrix completion problems for recommender systems, where we obtain significant speed-ups compared to state-of-the art coordinate descent methods.
The ability to grasp ordinary and potentially never-seen objects is an important feature in both domestic and industrial robotics. For a system to accomplish this, it must autonomously identify grasping locations by using information from various sensors, such as Microsoft Kinect 3D camera. Despite numerous progress, significant work still remains to be done in this field. To this effect, we propose a dictionary learning and sparse representation (DLSR) framework for representing RGBD images from 3D sensors in the context of determining such good grasping locations. In contrast to previously proposed approaches that relied on sophisticated regularization or very large datasets, the derived perception system has a fast training phase and can work with small datasets. It is also theoretically founded for dealing with masked-out entries, which are common with 3D sensors. We contribute by presenting a comparative study of several DLSR approach combinations for recognizing and detecting grasp candidates on the standard Cornell dataset. Importantly, experimental results show a performance improvement of 1.69% in detection and 3.16% in recognition over current state-of-the-art convolutional neural network (CNN). Even though nowadays most popular vision-based approach is CNN, this suggests that DLSR is also a viable alternative with interesting advantages that CNN has not.
The problem of probabilistic forecasting and online simulation of real-time electricity market with stochastic generation and demand is considered. By exploiting the parametric structure of the direct current optimal power flow, a new technique based on online dictionary learning (ODL) is proposed. The ODL approach incorporates real-time measurements and historical traces to produce forecasts of joint and marginal probability distributions of future locational marginal prices, power flows, and dispatch levels, conditional on the system state at the time of forecasting. Compared with standard Monte Carlo simulation techniques, the ODL approach offers several orders of magnitude improvement in computation time, making it feasible for online forecasting of market operations. Numerical simulations on large and moderate size power systems illustrate its performance and complexity features and its potential as a tool for system operators.
We study the theoretical properties of learning a dictionary from $N$ signals $\mathbf x_i\in \mathbb R^K$ for $i=1,...,N$ via $l_1$-minimization. We assume that $\mathbf x_i$'s are $i.i.d.$ random linear combinations of the $K$ columns from a complete (i.e., square and invertible) reference dictionary $\mathbf D_0 \in \mathbb R^{K\times K}$. Here, the random linear coefficients are generated from either the $s$-sparse Gaussian model or the Bernoulli-Gaussian model. First, for the population case, we establish a sufficient and almost necessary condition for the reference dictionary $\mathbf D_0$ to be locally identifiable, i.e., a local minimum of the expected $l_1$-norm objective function. Our condition covers both sparse and dense cases of the random linear coefficients and significantly improves the sufficient condition by Gribonval and Schnass (2010). In addition, we show that for a complete $\mu$-coherent reference dictionary, i.e., a dictionary with absolute pairwise column inner-product at most $\mu\in[0,1)$, local identifiability holds even when the random linear coefficient vector has up to $O(\mu^{-2})$ nonzeros on average. Moreover, our local identifiability results also translate to the finite sample case with high probability provided that the number of signals $N$ scales as $O(K\log K)$.
In support of art investigation, we propose a new source separation method that unmixes a single X-ray scan acquired from double-sided paintings. In this problem, the X-ray signals to be separated have similar morphological characteristics, which brings previous source separation methods to their limits. Our solution is to use photographs taken from the front and back-side of the panel to drive the separation process. The crux of our approach relies on the coupling of the two imaging modalities (photographs and X-rays) using a novel coupled dictionary learning framework able to capture both common and disparate features across the modalities using parsimonious representations; the common component models features shared by the multi-modal images, whereas the innovation component captures modality-specific information. As such, our model enables the formulation of appropriately regularized convex optimization procedures that lead to the accurate separation of the X-rays. Our dictionary learning framework can be tailored both to a single- and a multi-scale framework, with the latter leading to a significant performance improvement. Moreover, to improve further on the visual quality of the separated images, we propose to train coupled dictionaries that ignore certain parts of the painting corresponding to craquelure. Experimentation on synthetic and real data - taken from digital acquisition of the Ghent Altarpiece (1432) - confirms the superiority of our method against the state-of-the-art morphological component analysis technique that uses either fixed or trained dictionaries to perform image separation.
Is it possible to find the sparsest vector (direction) in a generic subspace $\mathcal{S} \subseteq \mathbb{R}^p$ with $\mathrm{dim}(\mathcal{S})= n < p$? This problem can be considered a homogeneous variant of the sparse recovery problem, and finds connections to sparse dictionary learning, sparse PCA, and many other problems in signal processing and machine learning. In this paper, we focus on a **planted sparse model** for the subspace: the target sparse vector is embedded in an otherwise random subspace. Simple convex heuristics for this planted recovery problem provably break down when the fraction of nonzero entries in the target sparse vector substantially exceeds $O(1/\sqrt{n})$. In contrast, we exhibit a relatively simple nonconvex approach based on alternating directions, which provably succeeds even when the fraction of nonzero entries is $\Omega(1)$. To the best of our knowledge, this is the first practical algorithm to achieve linear scaling under the planted sparse model. Empirically, our proposed algorithm also succeeds in more challenging data models, e.g., sparse dictionary learning.