Videos tagged with Mathematics

Fractals and splines have very different geometric features. Fractals can be continuous everywhere, yet differentiable nowhere. Fractals are often selfsimilar curves with fractional dimension. And fractals are also attractors, fixed points of iterated function systems. In contrast, splines are piecewise polynomial curves, so well behaved that they are often used for large scale industrial desig...

The Church-Turing thesis is one of the foundations of computer science. The thesis heralded the dawn of the computer revolution by enabling the construct of the universal Turing machine which led the way, at least conceptually, to the von Neumann architecture and first electronic computers. One way to state the Church-Turing thesis is as follows: A Turing Machine computes every numerical functi...

Thue games can be played on various combinatorial structures: graphs, hypergraphs, words, Euclidean spaces, etc. Basic idea goes back to Thue, who proved that the integers can be 3-colored so that adjacent intervals have different color patterns. This striking result has been generalized in many directions, leading to a variety of peculiar coloring problems with unexpected connections to other ...

The cluster variation method (CVM) is a hierarchy of approximate variational techniques for discrete (Ising-like) models in equilibrium statistical mechanics, improving on the mean-field approximation and the Bethe--Peierls approximation, which can be regarded as the lowest level of the CVM. The foundations of the CVM are briefly reviewed, considering different derivations of the method and rel...

Large-scale logistic regression arises in many applications such as document classification and natural language processing. In this paper, we apply a trust region Newton method to maximize the log-likelihood of the logistic regression model. The proposed method uses only approximate Newton steps in the beginning, but achieves fast convergence in the end. Experiments show that it is faster than...

Lecture slides: Learning CRFs with Hierarchical Features: An Application to Go The Game of Go Territory Prediction Talk Outline Hierarchical Patterns Models Independent Pattern-based Classifiers Inference and Training Bayesian Model Averaging Hierarchical Tree Models CRF & Pattern CRF Inference and Training Pseudolikelihood Local Training Evaluation Models & Algorithms Training Time Inf...

The talk will consider ways of bounding the complexity of a graph as measured by the number of partitions satisfying certain properties. The approach adopted uses Vapnik Chervonenkis dimension techniques. An example of such a bound was given by Kleinberg et al (2004) with an application to network failure detection. We describe a new bound in the same vein that depends on the eigenvalues of the...

Graph matching plays a key role in many areas of computing from computer vision to networks where there is a need to determine correspondences between the components (vertices and edges) of two attributed structures. In recent years three new approaches to graph matching have emerged as replacements to more traditional heuristic methods. These new methods are: * Least squares - where the optima...

The kernel function plays a central role in kernel methods. In this paper, we consider the automated learning of the kernel matrix over a convex combination of pre-specified kernel matrices in Regularized Kernel Discriminant Analysis (RKDA), which performs linear discriminant analysis in the feature space via the kernel trick. Previous studies have shown that this kernel learning problem can be...

We consider the problem of learning a similarity function from a set of positive equivalence constraints, i.e. "similar" point pairs. We define the similarity in information theoretic terms, as the gain in coding length when shifting from independent encoding of the pair to joint encoding. Under simple Gaussian assumptions, this formulation leads to a non-Mahalanobis similarity functi...