In this paper, we propose a method for image-set classification based on convex cone models, focusing on the effectiveness of convolutional neural network (CNN) features as inputs. CNN features have non-negative values when using the rectified linear unit as an activation function. This naturally leads us to model a set of CNN features by a convex cone and measure the geometric similarity of convex cones for classification. To establish this framework, we sequentially define multiple angles between two convex cones by repeating the alternating least squares method and then define the geometric similarity between the cones using the obtained angles. Moreover, to enhance our method, we introduce a discriminant space, maximizing the between-class variance (gaps) and minimizes the within-class variance of the projected convex cones onto the discriminant space, similar to a Fisher discriminant analysis. Finally, classification is based on the similarity between projected convex cones. The effectiveness of the proposed method was demonstrated experimentally using a private, multi-view hand shape dataset and two public databases.
Given a set of feature vectors @math , the basis vectors @math of a convex cone representing a distribution of @math can be obtained by non-negative matrix factorization (NMF) @cite_5 @cite_13 . Let @math and @math . NMF generates the basis vectors @math by solving the following optimization problem: where @math denotes the Frobenius norm. We use the alternating non-negativity-constrained least squares-based method @cite_13 to solve this problem.
@cite_5 @cite_13
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Non-negative matrix factorization (NMF), i.e. V ap WH where both V, W and H are non-negative has become a widely used blind source separation technique due to its part based representation. The NMF decomposition is not in general unique and a part based representation not guaranteed. However, imposing sparseness both improves the uniqueness of the decomposition and favors part based representation. Sparseness in the form of attaining as many zero elements in the solution as possible is appealing from a conceptional point of view and corresponds to minimizing reconstruction error with an L0 norm constraint. In general, solving for a given L0 norm is an NP hard problem thus convex relaxation to regularization by the L1 norm is often considered, i.e., minimizing (1 2||V - WH||F 2 + lambda||H||1).An open problem is to control the degree of sparsity lambda imposed. We here demonstrate that a full regularization path for the L1 norm regularized least squares NMF for fixed W can be calculated at the cost of an ordinary least squares solution based on a modification of the least angle regression and selection (LARS) algorithm forming a non-negativity constrained LARS (NLARS). With the full regularization path, the L1 regularization strength lambda that best approximates a given L0 can be directly accessed and in effect used to control the sparsity of H. The MATLAB code for the NLARS algorithm is available for download. We interpret non-negative matrix factorization geometrically, as the problem of finding a simplicial cone which contains a cloud of data points and which is contained in the positive orthant. We show that under certain conditions, basically requiring that some of the data are spread across the faces of the positive orthant, there is a unique such simplicial cone. We give examples of synthetic image articulation databases which obey these conditions; these require separated support and factorial sampling. For such databases there is a generative model in terms of 'parts' and NMF correctly identifies the 'parts'. We show that our theoretical results are predictive of the performance of published NMF code, by running the published algorithms on one of our synthetic image articulation databases. Nonnegative matrix factorization (NMF) determines a lower rank approximation of a matrix @math where an integer @math is given and nonnegativity is imposed on all components of the factors @math and @math . NMF has attracted much attention for over a decade and has been successfully applied to numerous data analysis problems. In applications where the components of the data are necessarily nonnegative, such as chemical concentrations in experimental results or pixels in digital images, NMF provides a more relevant interpretation of the results since it gives nonsubtractive combinations of nonnegative basis vectors. In this paper, we introduce an algorithm for NMF based on alternating nonnegativity constrained least squares (NMF ANLS) and the active set-based fast algorithm for nonnegativity constrained least squares with multiple right-hand side vectors, and we discuss its convergence properties and a rigorous convergence criterion based on the Karush-Kuhn-Tucker (KKT) conditions. In addition, we also describe algorithms for sparse NMFs and regularized NMF. We show how we impose a sparsity constraint on one of the factors by @math -norm minimization and discuss its convergence properties. Our algorithms are compared to other commonly used NMF algorithms in the literature on several test data sets in terms of their convergence behavior. We present several new variations on the theme of nonnegative matrix factorization (NMF). Considering factorizations of the form X = FGT, we focus on algorithms in which G is restricted to containing nonnegative entries, but allowing the data matrix X to have mixed signs, thus extending the applicable range of NMF methods. We also consider algorithms in which the basis vectors of F are constrained to be convex combinations of the data points. This is used for a kernel extension of NMF. We provide algorithms for computing these new factorizations and we provide supporting theoretical analysis. We also analyze the relationships between our algorithms and clustering algorithms, and consider the implications for sparseness of solutions. Finally, we present experimental results that explore the properties of these new methods.