Compact and discriminative visual codebooks are preferred in many visual recognition tasks. In the literature, a number of works have taken the approach of hierarchically merging visual words of an initial large-sized codebook, but implemented this approach with different merging criteria. In this work, we propose a single probabilistic framework to unify these merging criteria, by identifying two key factors: the function used to model class-conditional distribution and the method used to estimate the distribution parameters. More importantly, by adopting new distribution functions and or parameter estimation methods, our framework can readily produce a spectrum of novel merging criteria. Three of them are specifically focused in this work. In the first criterion, we adopt the multinomial distribution with Bayesian method; In the second criterion, we integrate Gaussian distribution with maximum likelihood parameter estimation. In the third criterion, which shows the best merging performance, we propose a max-margin-based parameter estimation method and apply it with multinomial distribution. Extensive experimental study is conducted to systematically analyse the performance of the above three criteria and compare them with existing ones. As demonstrated, the best criterion obtained in our framework achieves the overall best merging performance among the comparable merging criteria developed in the literature.
: In @cite_15 , the scatter-matrix-based class separability, @math , is used to measure the goodness of @math as where @math and @math are the within-class scatter matrix and the total scatter matrix, respectively. @math denotes the trace of a matrix. They are computed with training histograms @math . At each level, the words @math and @math whose mergence minimizes @math are identified and merged To facilitate the subsequent analysis, we use the minimization of @math here. Because of the identity @math , it is equivalent to @cite_15 which maximizes @math . .
@cite_15
1537002715 1998269045 2591592591 1993446440
An @math m×n matrix @math A with column supports @math Si is k-separable if the disjunctions @math źiźKSi are all distinct over all sets @math K of cardinality k. While a simple counting bound shows that @math m>klog2n k rows are required for a separable matrix to exist, in fact it is necessary for m to be about a factor of k more than this. In this paper, we consider a weaker definition of almost k-separability', which requires that the disjunctions are mostly distinct'. We show using a random construction that these matrices exist with @math m=O(klogn) rows, which is optimal for @math k=O(n1-β). Further, by calculating explicit constants, we show how almost separable matrices give new bounds on the rate of nonadaptive group testing. Many data analysis applications deal with large matrices and involve approximating the matrix using a small number of “components.” Typically, these components are linear combinations of the rows and columns of the matrix, and are thus difficult to interpret in terms of the original features of the input data. In this paper, we propose and study matrix approximations that are explicitly expressed in terms of a small number of columns and or rows of the data matrix, and thereby more amenable to interpretation in terms of the original data. Our main algorithmic results are two randomized algorithms which take as input an @math matrix @math and a rank parameter @math . In our first algorithm, @math is chosen, and we let @math , where @math is the Moore-Penrose generalized inverse of @math . In our second algorithm @math , @math , @math are chosen, and we let @math . ( @math and @math are matrices that consist of actual columns and rows, respectively, of @math , and @math is a generalized inverse of their intersection.) For each algorithm, we show that with probability at least @math , @math , where @math is the “best” rank- @math approximation provided by truncating the SVD of @math , and where @math is the Frobenius norm of the matrix @math . The number of columns of @math and rows of @math is a low-degree polynomial in @math , @math , and @math . Both the Numerical Linear Algebra community and the Theoretical Computer Science community have studied variants of these matrix decompositions over the last ten years. However, our two algorithms are the first polynomial time algorithms for such low-rank matrix approximations that come with relative-error guarantees; previously, in some cases, it was not even known whether such matrix decompositions exist. Both of our algorithms are simple and they take time of the order needed to approximately compute the top @math singular vectors of @math . The technical crux of our analysis is a novel, intuitive sampling method we introduce in this paper called “subspace sampling.” In subspace sampling, the sampling probabilities depend on the Euclidean norms of the rows of the top singular vectors. This allows us to obtain provable relative-error guarantees by deconvoluting “subspace” information and “size-of- @math ” information in the input matrix. This technique is likely to be useful for other matrix approximation and data analysis problems. This paper considers the problem of designing maximum distance separable (MDS) codes over small fields with constraints on the support of their generator matrices. For any given @math binary matrix @math , the GM-MDS conjecture, proposed by , states that if @math satisfies the so-called MDS condition, then for any field @math of size @math , there exists an @math MDS code whose generator matrix @math , with entries in @math , fits the matrix @math (i.e., @math is the support matrix of @math ). Despite all the attempts by the coding theory community, this conjecture remains still open in general. It was shown, independently by and , that the GM-MDS conjecture holds if the following conjecture, referred to as the TM-MDS conjecture, holds: if @math satisfies the MDS condition, then the determinant of a transform matrix @math , such that @math fits @math , is not identically zero, where @math is a Vandermonde matrix with distinct parameters. In this work, we first reformulate the TM-MDS conjecture in terms of the Wronskian determinant, and then present an algebraic-combinatorial approach based on polynomial-degree reduction for proving this conjecture. Our proof technique's strength is based primarily on reducing inherent combinatorics in the proof. We demonstrate the strength of our technique by proving the TM-MDS conjecture for the cases where the number of rows ( @math ) of @math is upper bounded by @math . For this class of special cases of @math where the only additional constraint is on @math , only cases with @math were previously proven theoretically, and the previously used proof techniques are not applicable to cases with @math . Suppose the complex matrix M is partitioned into a @math array of blocks; let @math . The generalized Schur complement of A in M is defined to be @math , where @math is the Moore–Penrose inverse of A. The relationship of the ranks of M, A, and @math is determined. A new proof, under certain conditions, of Sylvester’s determinantal formula is given. A quotient formula like one previously proved for the Schur complement is obtained. Finally, several known inequalities for positive semidefinite Hermitian matrices are generalized.